Doctoral Thesis - UNIGRAZphysik.uni-graz.at/~wap/images/thesis-stickler.pdfDoctoral Thesis for...

Doctoral Thesis

for obtaining the academic degree of

Doctor rerum naturalium

written by

BENJAMIN A. STICKLER

THEORY AND MODELING OFSPIN-TRANSPORT ON THE MICROSCOPIC

AND THE MESOSCOPIC SCALE

supervised by

WALTER PÖTZ Karl-Franzens-University Graz

Institute of Physics - Theoretical Physics

Karl-Franzens-University Graz

i

STATUTORY DECLARATION

I declare that I have authored this thesis independently, that I have notused other than the declared sources / resources, and that I have explicitlymarked all material which has been quoted either literally or by contentfrom the used sources.

Date Benjamin A. Stickler

iii

ABSTRACT

It is the aim of this thesis to contribute to the description of spin dynam-ics in solid state systems. In the first part of this work we present a fullquantum treatment of spin-coherent transport in halfmetal / semiconduc-tor CrAs / GaAs heterostructures. The theoretical approach is based on theab-initio determination of the electronic structures of the materials involvedand on the calculation of the band offset. These ingredients are in the sec-ond step cast into an effective nearest-neighbor tight-binding Hamiltonian.Finally, in the third step, we investigate by means of the non-equilibriumGREEN’s function technique the current which flows through such a het-erostructure if a finite bias is applied. With the help of this strategy it ispossible to identify CrAs / GaAs heterostructures as probable candidatesfor all-semiconductor room-temperature spin-filtering devices, which oper-ate without externally applied magnetic fields. In the second part of thisthesis we derive a linear semiclassical spinorial BOLTZMANN equation. Formany (mesoscopic) device geometries a full quantum treatment of transportdynamics may not be necessary and may not be feasible with state-of-the-arttechniques. The derivation is based on the quantum mechanical descriptionof a composite quantum system by means of VON NEUMANN’s equation.The BORN-MARKOV limit allows us to derive a LINDBLAD master equationfor the reduced system plus non-Markovian corrections. Finally, we per-form a WIGNER transformation and take the semiclassical limit in order toobtain a spinorial BOLTZMANN equation, suitable for the description of spintransport on the mesoscopic scale. It has to be emphasized that the spinorialBOLTZMANN equation constitutes the missing link between a full quantumtreatment and heuristically introduced mesoscopic models for spin trans-port in solid state systems.

v

ZUSAMMENFASSUNG

Ziel dieser Dissertation ist es, zur theoretischen Beschreibung von Spin-Transport in Festkörpersystemen beizutragen. Der erste Teil der Arbeit be-schäftigt sich mit einer vollständig quantenmechanischen Modellierung vonspin-kohärentem Transport durch magnetische Heterostrukturen. Mit Hilfedieser Modellierung werden CrAs / GaAs Heterostrukturen auf ihre Ein-setzbarkeit als effiziente Raumtemperatur-Spinfilter-Bauteile untersucht.Der theoretische Zugang besteht aus drei Schritten: Im ersten werden dieelektronischen Strukturen der Materialien sowie der band offset zwischendiesen bestimmt. Im zweiten Schritt wird diese Information auf ein ef-fektives nearest-neighbor tight-binding Modell abgebildet. Auf Basis diesesModells wird dann, im dritten Schritt, über Nichtgleichgewichts - GREEN’s- Funktionen der durch die Heterostruktur fließende Strom berechnet. Un-sere theoretischen Ergebnisse weisen darauf hin, dass sich CrAs / GaAsHeterostrukturen tatsächlich zur Realisierung effizienter Spinfilter-Bautei-le eignen. Im zweiten Teil dieser Dissertation diskutieren wir einen meso-skopischen Zugang zur Beschreibung von Spin-Transport. MesoskopischeModelle sind im Besonderen dann von Interesse, wenn eine vollständigquantenmechanische, also mikroskopische Beschreibung, zu kompliziert,beziehungsweise zu aufwendig ist. Wir leiten aus der VON NEUMANN Glei-chung für ein zusammengesetztes Quantensystem eine Gleichung für diesemiklassische effektive Dynamik eines einzelnen Elektrons in der Umge-bung magnetischer Streuzentren her, also eine Spin - BOLTZMANN - Glei-chung. Diese Gleichung kann dann, gemeinsam mit entsprechenden Quan-tenkorrekturen, verwendet werden, um weitere mesoskopische Spin - Trans-portmodelle herzuleiten. Es wird besonders hervorgehoben, dass die Spin- BOLTZMANN - Gleichung eine bisher fehlende systematische Verbindungzwischen einer mikroskopisch quantenmechanischen und einer heuristischmesoskopischen Beschreibung darstellt.

CONTENTS

1 Preface 11.1 Introductory Overview . . . . . . . . . . . . . . . . . . . . . . . 11.2 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . 6

I Quantum Transport in Spintronic Devices 7

2 Preliminary Comments 9

3 Ab-initio Electronic Structures 113.1 Basic Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.2 Density Functional Theory . . . . . . . . . . . . . . . . . . . . . 163.3 Muffin Tin Orbitals . . . . . . . . . . . . . . . . . . . . . . . . . 203.4 Linear Muffin Tin Orbitals . . . . . . . . . . . . . . . . . . . . . 24

4 Empirical Tight Binding Hamiltonian 314.1 Some Introductory Remarks . . . . . . . . . . . . . . . . . . . . 314.2 The TB - LCAO Approach . . . . . . . . . . . . . . . . . . . . . 334.3 The ZB nn ETB Hamiltonian . . . . . . . . . . . . . . . . . . . . 394.4 Including Spin-Orbit Coupling . . . . . . . . . . . . . . . . . . 464.5 Nearest Neighbor ETB Fits . . . . . . . . . . . . . . . . . . . . . 49

5 Multiband Quantum Transport 535.1 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 535.2 The Transport Hamiltonian . . . . . . . . . . . . . . . . . . . . 535.3 The Current Density . . . . . . . . . . . . . . . . . . . . . . . . 635.4 Calculation of the System’s GREEN’s Function . . . . . . . . . 69

vii

viii

5.5 The Surface GREEN’s Functions . . . . . . . . . . . . . . . . . . 72

6 Some Remarks 75

7 Spin Injection and Filtering in Halfmetal / Semiconductor (CrAs /GaAs) Heterostructures 777.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 777.2 Theoretical Approach and Results . . . . . . . . . . . . . . . . 78

8 Theoretical Investigation of Spin -Filtering in CrAs / GaAs Heterostructures 838.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 848.2 LMTO Electronic Structure Calculations and the fcc CrAs /

fcc GaAs [1,0,0] Band Offset . . . . . . . . . . . . . . . . . . . . 868.3 Effective sp3d5s∗ Empirical Tight-Binding Model . . . . . . . . 928.4 Steady-State Transport . . . . . . . . . . . . . . . . . . . . . . . 958.5 Summary, Discussion, Conclusions, and

Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1048.A Empirical Tight-Binding Parameters . . . . . . . . . . . . . . . 107

9 Charge Transport Through Interfaces: a Tight-Binding Toy Modeland its Implications 1099.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1099.2 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1119.3 The Dilemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

9.3.1 An Artificial Heterostructure . . . . . . . . . . . . . . . 1169.3.2 A Genuine Heterostructure . . . . . . . . . . . . . . . . 117

9.4 The Interface Matching Problem - A Possible Solution . . . . . 1209.4.1 Tight-Binding Interface Matching for Single-Atomic

Chain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1219.4.2 The Genuine Heterostructure Revisited . . . . . . . . . 125

9.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

II Derivation of a Linear Spinorial Boltzmann Equation131

10 Introductory Remarks 133

11 Non-Markovian Quantum Dynamics from Environmental Relax-ation 13711.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13711.2 Physical Model and Scaling . . . . . . . . . . . . . . . . . . . . 141

ix

11.3 Derivation of Master Equations . . . . . . . . . . . . . . . . . . 14411.3.1 HILBERT Expansion of the State Operator . . . . . . . . 14411.3.2 LINDBLAD Master Equation . . . . . . . . . . . . . . . . 150

11.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15111.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15311.A Analysis of the Operator Q . . . . . . . . . . . . . . . . . . . . . 15511.B Existence and Uniqueness . . . . . . . . . . . . . . . . . . . . . 15911.C Second-Order Contribution . . . . . . . . . . . . . . . . . . . . 161

12 Derivation of a Linear Collision Operator for the Spinorial WignerEquation and its Semiclassical Limit 16512.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16612.2 Notations and Modeling . . . . . . . . . . . . . . . . . . . . . . 16912.3 Integral Kernel of the Dissipator . . . . . . . . . . . . . . . . . 17312.4 WIGNER Representation of the LINDBLAD

Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17612.5 The Quantum Collision Operator . . . . . . . . . . . . . . . . . 17912.6 Semiclassical Limit: Spinorial BOLTZMANN Equation . . . . . 186

12.6.1 Short-Range Interactions (` = λ0) . . . . . . . . . . . . . 19012.6.2 Long-Range Interactions (` = λs) . . . . . . . . . . . . . 191

12.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19412.A Derivation of Eq. (12.24) . . . . . . . . . . . . . . . . . . . . . . 19512.B Derivation of the MOYAL Bracket . . . . . . . . . . . . . . . . . 19712.C The FOURIER Transform . . . . . . . . . . . . . . . . . . . . . . 19912.D Rewriting the Dissipator . . . . . . . . . . . . . . . . . . . . . . 20012.E The Kernel of Q(2)

~ . . . . . . . . . . . . . . . . . . . . . . . . . . 20112.F Derivation of the MOYAL Product . . . . . . . . . . . . . . . . . 20212.G Derivation of Eqs. (12.80) . . . . . . . . . . . . . . . . . . . . . . 20312.H Derivation of Eqs. (12.84) . . . . . . . . . . . . . . . . . . . . . . 205

13 Concluding Summary 20713.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20713.2 Future Prospects . . . . . . . . . . . . . . . . . . . . . . . . . . . 209

Chapter 1

PREFACE

1.1 Introductory Overview

Over the last years spintronics, i.e. spin-electronics or spin-transport-electronics,developed into a huge field of active research, which is still growing rapidly.It is mainly driven by two motivations [1], i.e. the quest for a profound un-derstanding of spin physics in semiconductors and other materials and bythe expectation that such insight might allow the harvesting of the spin de-grees of freedom in information technology. In particular, it is an attractiveidea to exploit both the charge and the spin degrees of freedom of carriersin order to transfer information. Thus far, the only technologically realizeddevices in mass production are based on the giant magentoresistance andon the tunneling magnetoresistance effect [2, 3], which are both related tometallic compounds. On the other hand, semiconductor spintronics, althoughvery interesting for technical reasons, still has not found its way into practi-cal applications.

Within this thesis we shall concentrate on the theory and modeling ofspin transport in semiconductor-based devices. This thesis is composed oftwo main parts, each can be attributed to one of the two major topics ofthe super-ordinated FWF project P221290-N16. This project’s scope is, onthe one hand, the modeling of spin-selective transport in solid state de-vices and, on the other hand, the description of spin-transfer-torque dy-namics. The first part of this thesis extends previous work of C. ERTLER onthe dynamics of spin selective tunneling in GaMnAs based quantum-wellheterostructures. The second part provides the theoretical framework for

1

2 1.1. INTRODUCTORY OVERVIEW

several semiclassical spin transport models such as the spin drift-diffusionmodels, which have been employed by M. WENIN to study the dynamicsof spin-transfer-torques. Let us briefly overview these two parts in moredetail.

Part I - Quantum Transport in Spintronic Devices. This part containsa numerical study of spin coherent-transport in halfmetal / semiconduc-tor CrAs / GaAs heterostructures. It is one of the major goals of modernresearch to design all-semiconductor spintronic devices which efficientlyoperate at room temperature without an externally applied magnetic field[2, 3]. It is our aim to contribute to this search by demonstrating that on thebasis of CrAs / GaAs heterostructures all-semiconductor spin-filters andspin injectors should indeed be realizable. In particular, we show that CrAs/ GaAs heterostructures are probable candidates for efficient spin-filters orspin injectors which fulfill the above listed requirements, i.e. they operateat room temperature without an externally applied magnetic field. Suchdevices could be used to efficiently inject or detect spin polarized currentsinto or from conventional semiconductors, such as GaAs. Furthermore, thetheoretical approach developed to demonstrate this stunning property canbe easily applied to further promising zinc-blende material combinations.It is a straight-forward but tedious exercise to extend the formalism to notzinc-blende type materials.

The first part of this thesis contains eight chapters. The first gives a briefintroduction into the current status of experimental as well as ab-initio stud-ies of CrAs and GaAs / CrAs heterostructures. The subsequent three chap-ters, Chap. 3 - Chap. 5, are devoted to a detailed account of the theoreticalapproach employed. More precisely, within Chap. 3 we present the methodof LMTOs (linear muffin tin orbitals) [4, 5] which is utilized to determine thebulk properties of the materials constituting the heterostructure. Moreover,we discuss in great detail a method to determine the band offset betweenthese materials [6, 7]. Then, within Chap. 4 we define a zinc-blende nearest-neighbor empirical tight-binding (ETB) Hamiltonian [8, 9, 10]. The ab-initioelectronic structures are mapped onto this effective model by minimizingthe function K(ξ):

K(ξ) =∑nk

an(k)[εLSDAn (k)− εn(k, ξ)

]2, (1.1)

where n labels the bands, k is the k-vector, an(k) are normalized weights andεLSDAn (k) and εn(k, ξ) are the ab-initio and ETB band structures, respectively.

Chapter 1: PREFACE 3

Here, ξ denotes the vector of ETB parameters. Finally, the transport Hamil-tonian of the entire heterostructure is formulated in Chap. 5. It is obtainedby performing a partial FOURIER transformation of the basis states of thebulk Hamiltonians [11], while the different materials are linked via the vir-tual crystal approximation (VCA). The VCA states that the onsite element ofthe atom situated at the interface has to be averaged [12, 11, 13]. It is empha-sized, in particular, that this approximation does not respect the parametersymmetries stemming from the bulk ETB Hamiltonians. Hence, a slightlymodified fitting routine is proposed which is then applied to the particularcase of halfmetal / semiconductor CrAs / GaAs heterostructures in Chaps.7 and 8. In addition, it is demonstrated that CrAs / GaAs heterostruc-tures might operate as efficient room-temperature all-semiconductor spin-filtering devices. Chapter 7 has been accepted for publication in the AIPConference Proceeding to the International Conference on the Physics of Semicon-ductors (ICPS) - Zurich, 2012 [14], and Chap. 8 is still in the draft-stadiumand has been uploaded to the arXiv [15]. In conclusion, we discuss the ETBmodeling of charge transport across interfaces in Chap. 9, which has beenpublished in Journal of Computational Electronics, 2013 [16]. This chapter con-tains a detailed discussion of the caveats of the VCA as well as a possibleremedy to the thus arising dilemma.

Part II - Derivation of a Linear Spinorial BOLTZMANN Equation. Wederive the semiclassical spinorial linear BOLTZMANN equation from the quan-tum dynamics of a composite system. This derivation constitutes a novelapproach to the derivation of mesoscopic spin transport equations. It is per-formed in four steps, which have been published in two papers [17] and[18], see Chaps. 11 and 12. The first was published in Physical Review A(PRA) [17] while the second has been accepted for publication in PRA. Letus briefly sketch the main results.

We investigate a composite quantum system AB, characterized by thestate operator ρ. The evolution equation of ρ is supposed to be of the form

∂tρ = −i [HAB, ρ]− i

α[HI , ρ] +

1

α2Q(ρ). (1.2)

Here, HAB = HA ⊗ 1B + 1A ⊗HB are the Hamiltonians of the isolated sub-systems A and B, HI describes the interaction between A and B and α ∈ Ris some (small) parameter.1 Furthermore, Q(ρ) is a superoperator acting on

1For convenience, we set ~ = 1.

4 1.1. INTRODUCTORY OVERVIEW

the state operator ρ according to

Q(ρ) = trB (ρ)⊗ χB − ρ, (1.3)

where χB is some predefined state of subsystem B. Hence, Eq. (1.2) de-scribes the dynamics of the composite system AB under the assumptionthat the interaction between A and B is very strong, i.e. α 1, while thereis a further influence, which drives ρ into the BORN approximation ρA⊗χB,where ρA := trB (ρ). This further influence is modeled with the superopera-tor Q(ρ) and is assumed to be the fastest contribution to the dynamics witha frequency proportional to α−2. It is demonstrated in Chap. 11 that, by in-voking an expansion of the total state operator ρ in powers of α, a hierarchyof master equations for the reduced state operator ρA follows. In zeroth or-der, the well-established LINDBLAD master equation [19, 20] is retrieved.Higher order corrections include non-Markovian effects and account forcorrelations as well as entanglement between subsystem A and B. Hence,we found a very convenient way of incorporating non-Markovian correc-tions in a perturbative manner to the reduced dynamics of open quantumsystems. The main advantage of this approach in comparison to the pro-jection operator technique [19, 21, 22] or the time-convolutionless projectionoperator method [19] is, on the one hand, that it is guided by a clear physi-cal picture and, on the other hand, that it may be much simpler to solve forcertain cases [17].

In the next step, Chap. 12, [18], we redefine subsystemA to be composedof a single spin-1/2 particle while subsystem B is assumed to contain Nspin-1/2 particles. We then perform a WIGNER transformation of the LIND-BLAD master equation and introduce the semiclassical parameter ε (scaled ~),which is suited to perform the semiclassical limit ε → 0. Hence, we regardthe LINDBLAD equation in a situation in which quantum effects cease to beobservable. Scaling assumptions to some characteristic functions involvedallow us to perform a semiclassical analysis. It is demonstrated in Chap.12 that numerous linear mesoscopic spin transport models can be obtainedsystematically by such an approach. As intriguing examples we derive theBLOCH equations as well as the linear semiclassical spinorial BOLTZMANNequation for a distribution matrix F ,

∂tF − h1, Fx,η + i[~Ω · ~σ, F ] = Q(F ). (1.4)

Here, h1 denotes the (classical) HAMILTON function multiplied by the 2× 2

unity matrix, ·, · is the POISSON bracket, ~Ω is some external exchange fieldand Q(·) is the spinorial collision operator. For suitable interaction poten-tials, this collision operator describes momentum relaxation as well as spin

Chapter 1: PREFACE 5

decoherence. It has been demonstrated by POSSANNER and NEGULESCUthat the spin drift-diffusion equations, see for instance Ref. [3], can be ob-tained from the spinorial BOLTZMANN equation (1.4) [23]. Further spinortransport models may be obtained in analogous fashion from the BOLTZ-MANN equation by following the strategies outlined in [24].

6 1.2. ACKNOWLEDGMENTS

1.2 Acknowledgments

I am especially indebted to my supervisor WALTER PÖTZ. First of all, hegave me the opportunity to begin with my PhD (solely) equipped with aChemistry degree and before I finished my Physics studies. In particular,he allowed me to take my last Physics exams during the first year of myPhD. Furthermore, he permanently supported me in my PhD, he gave methe freedom to develop my own ideas, to follow my additional projects andhe even assisted me in these. I highly profited from his interest in and lexicalknowledge of many different fields in modern theoretical physics. It is noexaggeration to emphasize that this thesis as well as my additional workswould have never been possible without his support.

I am also extremely grateful to EWALD SCHACHINGER for supportingand supervising my additional projects. All these would not have been re-alizable without his help. I am particularly thankful to E. SCHACHINGERfor proof-reading the publications in the second part of this thesis, for for-mulating great parts of the papers on LÉVY flights and on space fractionalquantum mechanics, which we published together. Last but not least, he tookthe necessary but tedious task of correcting our book into his hands, thusgiving me the possibility to simultaneously work on different projects. Hisinput enriched this thesis, as well as my further projects in a way whichshould not be undervalued.

In addition, I want to thank STEFAN POSSANNER for our very successfulcollaboration. In particular, the entire second part of this thesis can be at-tributed to numerous fruitful discussions between us. The fact that he wasworking on a similar topic and that we are good friends, allowed for a verysuccessful cooperation to develop. I am certain that these are not the last pa-pers we have been working on together and I am curiously looking forwardto these projects.

I was also fortunate to get the opportunity to cooperate with CHRISTIANERTLER and LIVIU CHIONCEL, who contributed to the first part of this the-sis. In particular, I am very thankful to C. ERTLER for helping me with thetransport code and to L. CHIONCEL for assisting me with the ab-initio cal-culations. I am also very grateful to ULRICH HOHENESTER for his financialsupport during the last months of my PhD.

Finally, I want to thank my friends and colleagues RENÉ HAMMER andTHOMAS OBERMÜLLER for stimulating discussions and relaxing conversa-tions. I am also extremely grateful to my family and friends for support-ing me. This work was financially supported by the Austrian Science Fund(FWF): P221290-N16.

Part I

QUANTUM TRANSPORT IN SPINTRONICDEVICES

7

Chapter 2

PRELIMINARY COMMENTS

It is an experimental observation that bulk CrAs crystallizes in an orthorhom-bic crystal structure and that it shows anti-ferromagnetic behavior [25]. Moreinteresting, however, when grown epitaxially on top of GaAs it seems to bepossible to stabilize a few layers in the zinc-blende (ZB) crystal structure, inwhich CrAs shows ferromagnetic behavior with a CURIE temperature wellabove room temperature [26, 27, 28, 29]. Since the discovery of ZB CrAsnumerous theoretical as well as experimental work has been published onthis topic.1 For instance, it has been predicted with the help of ab-initiotechniques [27, 30] that ZB CrAs is a halfmetallic ferromagnet (HFM), i.e. amaterial which shows a 100 % spin polarization at the FERMI energy, witha CURIE temperature up to 1000 K [27, 30, 31, 32]. These observations in-dicate that ZB CrAs may be an interesting candidate for room temperaturespintronic applications. Moreover, the electronic structure of GaAs / CrAsheterostructures, as well as the transport properties through GaAs / CrAs /GaAs trilayers have been studied by BENGONE et al. [33]. Although most ofthese theoretical results had been based on the local spin density approxima-tion, see Chap. 3, it has been demonstrated by CHIONCEL et al. [34] that theinclusion of dynamic correlation effects induces only minor changes to theelectronic structure. All these findings highly support the proposition thatCrAs may be suitable candidate for versatile spintronic applications.

On the other hand, there is also an experimental and a theoretical re-port [25, 35] which claim that stable thin films of ZB CrAs cannot be grown.

1For a review on the early history of ZB CrAs research see the article by MAVROPOULOSand GALANAKIS [27].

9

10 Chapter 2: PRELIMINARY COMMENTS

However, the accuracy of these theoretical results might be too low to ex-clude stable ZB CrAs thin films since the minimization of total energy ofbulk GaAs resulted in a lattice constant of aGaAs = 5.74 Å [25], which isroughly 1.6 % larger then the experimentally observed value of aGaAs = 5.65Å. Hence, this is an indication that the theoretical approach might not besophisticated enough for predicting the stability of CrAs thin films on GaAssamples. In summary, however, it has to be emphasized that the final an-swer to the question whether or not CrAs thin films are stable can only begiven by additional experiments, although several theoretical works seemto support the previous reports of half-metallic ZB CrAs. It is our aim tostimulate further experimental studies by demonstrating that CrAs / GaAsheterostructures should indeed function as very efficient room temperaturespin filtering devices provided that stable CrAs / GaAs heterostructures canbe grown.

For this purpose, we have developed a model to calculate the non-equili-brium steady state current flowing through a ZB semiconductor heterostruc-ture within a full quantum mechanical approach. It is based on determiningthe ab-initio electronic structures, mapping these onto an effective Hamilto-nian and, finally, performing the transport calculation within the frameworkof non-equilibrium GREEN’s function techniques. The details of the theoreti-cal approach are summarized in Chaps. 3 - 5, while the results are presentedand discussed in the publications in Chaps. 7 - 9. It is important to remarkthat the scheme presented in Chaps. 3 - 5 is not primarily restricted to CrAs/ GaAs heterostructures. It is rather straight-forward to extend all steps tonot-ZB crystal structures.

Chapter 3

AB-INITIO ELECTRONIC STRUCTURES

3.1 Basic Concepts

It is our aim to describe non-equilibrium charge transport in solid state sys-tems on a quantum mechanical level. In particular, we imagine a junctioncomposed of an arbitrary material combination sandwiched between twosemi-infinite leads. For such a system we seek the electric current throughthe device under given applied bias. Furthermore, we want a descriptionwhich captures the influence of exchange effects in the material. Hence, thespin degree of freedom of the carriers has to be taken into account.

We give the total Hamiltonian of such a system. However, in a first stepwe shall refrain from explicitly noting the spin degrees of freedom for thesake of a simplified notation. The corresponding indices are introducedwhenever clarity of notation demands. Hence, in absence of an externalmagnetic field, the total Hamiltonian H of the system may be written as[36, 37, 38]

H =∑i

[− ~2

2me

∇2i + Vext(ri)

]+∑i

∑I

Ven(ri, RI) +1

2

∑iji6=j

Vee(ri, rj)

+∑I

[− ~2

2MI

∇2I + Vext(RI)

]+

1

2

∑IJI 6=J

Vnn(RI , RJ), (3.1)

where i, j ∈ N and I, J ∈ N label electrons and nuclei, respectively, ri, RI ∈R3 are the position space coordinates of electron i and nucleus I , ∇i/I de-notes the gradient with respect to the coordinates of particle i/I , ~ is the

11

12 3.1. BASIC CONCEPTS

reduced PLANCK constant, me is the electron’s mass and MI the mass of thenucleus I . Furthermore, Vext is an externally applied potential (which weassume to be independent of time t), Ven, Vee and Vnn are the potentials ofelectron-nucleus, electron-electron and nucleus-nucleus interaction, respec-tively. The interpretation of the different terms on the right-hand side (rhs)of Eq. (3.1) is straight-forward [36, 37]: the first term describes the free-flightof the electrons in an externally applied field, the second term accounts forthe interaction between electrons and nuclei, the third term accounts forelectron-electron interaction, the fourth term describes the free motion ofnuclei and, finally, the fifth term accounts for the interaction between nu-clei. In order to explicitly introduce the spin into our notation we have toreplace, for instance, the coordinate ri in Eq. (3.1) by the object (αi, ri), whereαi is a discrete index accounting for the spin of electron i. The HamiltonianEq. (3.1) states a complex many-body problem for which, at least in princi-ple, we have to solve the time dependent SCHRÖDINGER equation. We shallnow formulate two crucial assumptions in order to simplify the Hamilto-nian (3.1).

In a first step we apply the celebrated BORN - OPPENHEIMER or adia-batic approximation [36]. It is based on the observation that the mass of anindividual nucleus vastly exceeds the mass of a single electron, i.e. me

MI

1, ∀I ∈ N. Under certain restrictions it is, therefore, legitimate to neglect thefree motion of the nuclei and regard the total wavefunction as a product ofelectron- and nuclei-wavefunction.1 The remaining contributions to the to-tal Hamiltonian (3.1) are: (i) the free-flight of the electrons together with theinteraction of the electrons with the nuclei, which provides a one-electroncontribution, (ii) the interaction between electrons, which is a two-electroncontribution and, finally, (iii) the interaction between nuclei, which yieldsa constant energy shift and will, therefore, be neglected in what follows.We define the potential acting on the electrons which arises from the (static)nuclei as

V (ri) :=∑I

Ven(ri, RI). (3.2)

This is commonly referred to as the lattice potential. Hence, with the help ofthe adiabatic approximation (3.2) the effective Hamiltonian H reads

H =∑i

[− ~2

2me

∇2i + Vext(ri) + V (ri)

]+

1

2

∑iji 6=j

Vee(ri, rj). (3.3)

1For a detailed discussion of the caveats of the adiabatic approximation we refer thereader to, for instance, MADELUNG’s book [36].

Chapter 3: AB-INITIO ELECTRONIC STRUCTURES 13

The Hamiltonian (3.3) still poses a very complicated many-particle prob-lem due to the electron-electron interaction Vee. However, in a second ap-proximation we assume that

1

2

∑iji6=j

Vee(ri, rj) ≡∑i

Vint(ri), (3.4)

which is referred to as the one-electron approximation.2 Please note that theeffective interaction potential Vint(ri) does not need to be independent of themany-body wavefunction |Ψ〉, however, it only depends on the position ri ofone particular particle. Inserting approximation (3.4) into the Hamiltonian(3.3) yields a many-particle Hamiltonian H which is a sum of effective one-electron Hamiltonians, i.e.

H =∑i

[− ~2

2me

∇2i + Vext(ri) + V (ri) + Vint(ri)

]. (3.5)

Hence, we have to solve the time-dependent SCHRÖDINGER equation

i~∂tψ(r, t) = Hψ(r, t), (3.6)

for a single-particle wavefunction ψ(r, t) = 〈r|ψ〉, r ∈ R3, where we omittedthe particle index for the sake of a more transparent notation. Moreover, weintroduced the one-particle Hamiltonian H given by

H = − ~2

2me

∇2 + Vext(r) + V (r) + Vint(r). (3.7)

Equivalently, we may solve the eigenvalue problem

Hψm(r) = Emψm(r), (3.8)

i.e. the stationary SCHRÖDINGER equation where ψm(r) = 〈r|ψm〉 denotesthe eigenstate to energy Em and m ∈ N. Since we assumed the exter-nal potential Vext(r) to be time-independent it follows that the eigenstatesψm(r) are also independent of time t. The solution of the time-dependentSCHRÖDINGER equation (3.6) can then be constructed from ψm(r), see, forinstance, Ref. [39].

We emphasize that the total Hamiltonian (3.1) as well as the effectiveHamiltonians (3.5) and (3.7) are not restricted to the particular case of peri-odic solids. In fact, the assumption of periodic boundary conditions is onlynecessary for BLOCH’s theorem to be valid [36].

2Clearly, this approximation is rather severe and has far-reaching consequences. Again,for a detailed discussion we refer the interested reader to, for instance, MADELUNG [36].

14 3.1. BASIC CONCEPTS

At this point it seems advisory to pause for a moment and to recapitulate:We reduced the general many-body problem described by the Hamiltonian(3.1) with the help of certain assumptions to an effective one-body problemwith Hamiltonian (3.7). However, if we want to solve the correspondingeigenvalue problem (3.8) numerically, we are still confronted with two dif-ficulties: (i) The lattice potential V (r) is infinitely extended and the systemunder investigation, i.e. a multi-component device, does not obey trans-lational invariance.3 This makes it impossible to solve the problem withstate-of-the-art techniques without posing further simplifications. (ii) Theeffective electron-electron interaction Vint(r) and the particular form of theatomic potentials Ven(r, Ri) remained completely unspecified.

Let us briefly elaborate on point (i): Obviously, further simplificationsare required. Suppose we investigate a composite system consisting of twodifferent materials, say material A and material B. Then, in a first step,we follow the straight-forward approach of neglecting interfacial effects.Hence, we solve the electronic structure problem for A and B separatelyand determine the band alignment in an additional step. This simplifica-tion comes with the advantage that we may apply BLOCH’s theorem for theisolated materials A and B, which allows us to solve Eq. (3.8) for each ma-terial locally within the unit-cell. The relative alignment of the bands is thendetermined with the help of a supercell4 calculation which includes the in-terface [6, 7]. In particular, the band offset between two materials A and Bis given by

∆ = ∆A −∆B + ∆AB, (3.9)

where ∆A/B is the energy difference between a reference band, such as thevalence band maximum, and a core level, such as the As-s level in GaAs, inmaterial A and B, respectively. Moreover, ∆AB is the difference between thetwo core levels in the composite material, i.e. as obtained from the supercellcalculation. A schematic illustration of these quantities is depicted in Fig.3.1.

3This is also due to the externally applied potential Vext(r) which explicitly depends onposition r.

4A supercell is a unit cell containing several layers of material A and B as well as theinterface between the materials. This cell is then periodically extended. Choosing the pe-riod sufficiently large allows to study the interface as if it were in the genuine bulk case. Ofcourse, a supercell approach comes with huge computational effort.


Figure 3.1: Schematic illustration of the quantities involved in the definition(3.9) of the band offset ∆ between materials A and B.

Knowing the electronic structure of the different (isolated) materials aswell as the band offsets between them, we may construct an effective Hamil-tonian H ′ which replaces the Hamiltonian H , Eq. (3.7). The gain of such anapproach is that we try to define the effective Hamiltonian H ′ in such a waythat we can solve the corresponding transport problem with state-of-the-artnumerical methods. The construction of this effective Hamiltonian is by nomeans a trivial task and will be extensively discussed in Chap. 4. Clearly,if the materials A and B have very similar properties, the simplification ofneglecting interfacial effects will not be too severe. But, if this is not thecase, we have to improve the effective Hamiltonian H ′. The first idea mightbe to construct an effective description H ′I of the interface from a supercellcalculation. However, it turns out that in most realistic cases the requiredsupercells are too large to extract any useful information. We come back tothis point within Secs. 3.4 and 4.5 as well as within Chap. 9. We brieflymention at this point, that there is a completely different approach to thisproblem, which was extensively discussed by BRANDBYGE et al. [40].

We first solve the eigenvalue problem (3.8) for the two separate mate-rials A and B in order to obtain the electronic structures. We can thusturn our attention to point (ii) of the above discussion. In summary, theelectron-electron interaction is treated within the local spin density approxi-mation (LSDA) of density functional theory (DFT) which will be discussed inSec. 3.2. The atomic potentials are approximated by muffin tin potentials.These potentials also introduce a proper basis for the solution of the station-ary SCHRÖDINGER equation, i.e. the muffin tin orbitals (MTOs), see Sec. 3.3.Linearization of the resulting equations yields the method of linear muffintin orbitals ( LMTO) which will be described in Sec. 3.4.

16 3.2. DENSITY FUNCTIONAL THEORY

3.2 Density Functional Theory

The following presentation of spin dependent density functional theory (DFT)is substantially based on the articles collected in [41], as well as on thebook by SKRIVER [4]. We want to solve the eigenvalue problem (3.8) fora many-electron Hamiltonian H of the form (3.5) where the lattice poten-tial V (r) describes a solid state bulk material. It is periodic in space, andwe set the external potential Vext(r) ≡ 0. Let us assume that we knowthe ground state wavefunction |ψ0〉 to the ground state energy E0 of theeffective many-electron Hamiltonian H given by Eq. (3.3). In this case wecould calculate the particle density in the ground state n0(r) = n(r, |ψ0〉) ≡∑

m pm|φm(r)|2, where |φm〉 are single-particle orbitals reproducing the chargedensity n0(r), pm are some weights which still have to be determined andr ∈ R3. Since the ground state energy depends on the ground state wave-function we can write E0 as a functional of n0, i.e. E0 = E[n0]. In particu-lar, the HOHENBERG-KOHN theorem states that if the ground state is non-degenerate, the ground state energy E0 can be written as a unique functionalof the ground state charge density n0.

If we regard a system containing N electrons, the stationarity conditionof the ground state is of the form [41]

δ

E[n(r)]− µ

(∫drn(r)−N

)= 0, (3.10)

where δ denotes the variation with respect to the electron density n(r) andµ is a LAGRANGE multiplier.5 In the spirit of the discussion in Sec. 3.1we express the energy-density functional E[n(r)] as a sum of several one-particle contributions. In particular, we write

E[n(r)] = T [n(r)] + Vlat[n(r)] + Vel[n(r)], (3.12)

where T [n(r)] is the kinetic energy functional, Vlat[n(r)] is the functional per-taining to the interaction with the lattice potential and Vel[n(r)] accounts forelectron-electron interactions. The kinetic energy functional T [n(r)] can bewritten as

T [n(r)] = − ~2

2me

∑m

pm

∫drφ∗m(r)∇2φm(r), (3.13)

5In particular, for an arbitrary functional F [n(r)] we define the variation δF via [41]

δF = F [n(r) + δn(r)]− F [n(r)] =

∫dr

(δF [n(r)]

δn(r)

)δn(r), (3.11)

where (δF [n(r)]/δn(r)) is the functional derivative of F with respect to n.


and the functional accounting for interaction with the lattice can be ex-pressed as

Vlat[n(r)] =

∫drV (r)n(r), (3.14)

where V (r) is the lattice potential (3.2). We write the electron-electron inter-action functional Vel[n(r)] in terms of the one-particle interaction potential(3.7)

Vel[n(r)] =

∫drVint(r)n(r)

=e2

2

∫drdr′

n(r′)n(r)

|r′ − r|+ Vxc[n(r)], (3.15)

where e is the elementary charge and Vxc[n(r)] = Vx[n(r)] + Vc[n(r)] is re-ferred to as the exchange-correlation functional. It has to be kept in mindthat the single-particle orbitals |φm〉 reproduce the densityn(r) =

∑m pm|φm(r)|2. Furthermore, we assume that these orbitals are nor-

malized, i.e.∫

dr|φm(r)|2 = 1. The variation with respect to n(r) can nowbe replaced by a variation with respect to φ∗m since the orbitals uniquely de-termine the electron density [41].6 The resulting variational problem can bewritten as

δφ∗m

E[n(r)]−

∑m

εm

(∫dr|φm(r)|2 − 1

)= 0, (3.16)

which yields the well-known KOHN-SHAM equations7

Hφm(r) = εmφm(r), (3.17)

where the effective one-particle Hamiltonian H reads

H = − ~2

2me

∇2 + Veff(r). (3.18)

The effective one-particle potential Veff(r) can be expressed as

Veff(r) := V (r) +e2

2

∫dr′

n(r′)

|r − r′|+δVxc[n(r)]

δn(r). (3.19)

6Please note that the LAGRANGE multiplier µ is related to the KOHN - SHAM energiesεm via the temperature dependence of the weights pm [41]. For instance, for T = 0 we havepm = Θ(µ − εm) where Θ(·) denotes HEAVISIDE’s Θ and µ has to be determined in such away that the total number of particles is N .

7In a similar fashion, the one-particle orbitals |φm〉 are referred to as KOHN - SHAMorbitals.

18 3.2. DENSITY FUNCTIONAL THEORY

Please note that the Hamiltonian (3.18) is of the same structure as the Hamil-tonian (3.7) obtained in the one-electron approximation (3.4). However, theexchange correlation functional has remained unspecified and one has torely on approximations in order to treat this term. One of the most promi-nent approximations is referred to as the local density approximation (LDA).It is of the particular form

V LDAxc [n(r)] =

∫drεLDAxc [n(r)]n(r), (3.20)

where εLDAxc [n(r)] = εLDAx [n(r)] + εLDAc [n(r)] is the exchange-correlation en-ergy density. The exchange-energy density is derived from the theory of auniform electron gas [41], for which it is known to be of the form

εLDAx [n(r)] = −3

4

(3

π

) 13

n(r)13 . (3.21)

Analytic expressions for the correlation functional are only known in thehigh- and low density limit [42]. However, intermediate values for the cor-relation energy density can be obtained numerically [41]. If spin degrees offreedom are taken into account the LDA is referred to as local spin density ap-proximation (LSDA) and the following modifications are incorporated: Firstof all we note that the total electron density n(r) at position r ∈ R3 can bewritten as

n(r) = n↑(r) + n↓(r), (3.22)

where ↑, ↓ label the two opposite spin orientations. The exchange energydensity (3.21) is then replaced by

εLSDAx [n↑(r), n↓(r)] =1

2

εLDAx [2n↑(r)] + εLDAx [2n↓(r)]

. (3.23)

In order to describe the spin dependence of the correlation functional it iscommon to define the relative spin polarization ζ(r) as

ζ(r) :=n↑(r)− n↓(r)n↑(r) + n↓(r)

. (3.24)

The two limiting cases, i.e. ζ(r) = 0, i.e. no spin polarization (’paramagneticcase’), and ζ(r) = 1, i.e. complete spin polarization (’ferromagnetic case’), areused to interpolate the correlation energy density for intermediate values.8

Finally, the self consistency loop of DFT takes on the form:

8For a detailed discussion of the correlation energy and related topics we refer the inter-ested reader to the literature [41] and, in particular, to Ref. [42].


(i) Choose some initial density n(0)(r).

(ii) Calculate the effective potential Veff(r).

(iii) Solve the KOHN-SHAM equations by expanding the single-particlewavefunctions |φm〉 in some basis.

(iv) Calculate the density according to n(t)(r) =∑

m |φm(r)|2Θ(µ − εm),where t is the iteration index, Θ(·) is HEAVISIDE’s Θ and we set thetemperature T = 0.

(v) If the solution is not converged, i.e. if |n(t)(r) − n(t−1)(r)| > η, withη > 0 the required accuracy, we return to point (ii), otherwise the cal-culation is terminated and desired output, such as the band structure,is generated.

In the context of the above iteration scheme there are numerous crucialpoints, however, a detailed discussion of these concepts is certainly far be-yond the scope of this section. For instance, the choice of the basis in itera-tion step (iii), as well as the modeling of the atomic potential Vlat(r), are ofprofound significance for the solution of the KOHN - SHAM equations (3.17).In particular, these two questions are intertwined. For a detailed discussionof these and further points we refer the interested reader to the literature,for instance Refs. [4, 36, 38, 41]. However, let us briefly point out the mainstrategies:

Generally speaking, there are two main strategies, i.e. the fixed basisset methods and the partial wave methods. In the first case one uses a set offixed basis functions, like in the case of linear combination of atomic or-bitals (LCAO) which we will discuss in Chap. 4, plane wave methods, orGAUSS - or SLATER - type orbitals [43]. The advantage of a such a procedurelies in its mathematical and numerical simplicity. However, the requirementfor large basis sets needed to achieve reasonable accuracy can be regardedas their major disadvantage. On the contrary, the partial wave methodsintroduce an energy-dependent basis, which is closely connected to the par-ticular modeling of the atomic potential as a muffin tin (MT) sphere, as it willbe introduced in Sec. 3.3. Well known methods arising from such a strategyare, for instance, augmented plane wave methods, the KORRINGA - KOHN -ROSTOKER (KKR) method, or cellular methods. The mathematical difficul-ties arising in such a treatment are the main disadvantage of these methods.Anyhow, they allow the use of a minimal basis set. Moreover, the partialwaves are independent of the atomic species and they yield good solutionsfor closely packed systems.

20 3.3. MUFFIN TIN ORBITALS

The LMTO method, which is also referred to as linearized KKR, tries tocombine the advantages of both. It uses a minimal basis set and, due tothe linearization scheme, the equations can be regarded as mathematicallysimple. This method will be the topic of Sec. 3.4. It has to be noted at thispoint that there are also other linear methods in band theory, such as themethod of linear augmented plane-waves, which we shall not discuss here[44].

3.3 Muffin Tin Orbitals

In what follows we introduce the concept of muffin-tin potentials VMT(r),which replace the exact electron-nucleus interaction potential Ven(r, R), andthe concept of the thus defined muffin tin orbitals (MTOs). This introductionis based on the book by SKRIVER [4], the original papers by ANDERSEN etal., e.g. [45, 46, 47, 48], and on lecture notes from the university of Frankfurt[49] and from the Technical University Graz [44].

It is a reasonable assumption that the bare potential Ven(r, R) induced bya single ion situated at position R ∈ R3, is spherically symmetric aroundthat nucleus, i.e. Ven(r, R) ≡ Ven(|r − R|). Furthermore, in many cases wemay regard the potential as almost constant in the interstitial region, i.e. inthe region between the nuclei. This leads to the natural definition of themuffin tin potential VMT(r) as

VMT(r −R) =

Ven(r −R)− V0 |r −R| ≤ s,

0 |r −R| > s,(3.25)

where s is the radius of the muffin tin sphere and V0 is the constant potentialin the interstitial region. Hence, the effective one particle Hamiltonian (3.18)can be written as

H = − ~2

2me

∇2 + Vint(r) +∑I

VMT(r −RI) + V0. (3.26)

Note that the sum in Eq. (3.26) covers the whole crystal and the MT spheresare also allowed to overlap, with the important consequence that the poten-tial inside the spheres does no longer need to be spherically symmetric.

However, in order to illustrate the main idea of MTOs we shall regardthe case of non-overlapping MT spheres where we restrict ourselves fur-ther to one MT potential per unit cell situated at R = 0. We follow thestandard procedure, well-known from basic quantum mechanics [39], andpartition the domain into distinct regions. We then solve the Schrödinger


equation locally, while the complete solution is obtained from matching thewavefunction, as well as its first derivative at the boundaries. Hence, in theinterstitial region (IR) we have to solve the equation[

− ~2

2me

∇2 + Vint(r) + V0 − E]φ(IR)m (r, E) = 0, (3.27)

where we denote φ(IR)m (r) ≡ φ

(IR)m (r, E) in order to emphasize the explicit

dependence of the wavefunction on energy E. Furthermore, we demandthat E−V0 ≥ 0 for physical reasons.9 In the particular case of a one-electronproblem, i.e. Vint(r) ≡ 0, the solutions φm(r, E) are plane waves with wave

vector k ∈ R3 and |k| =√

2me

~2 (E − V0).Within a single, not superimposed, MT sphere we have to solve[

− ~2

2me

∇2 + Vint(r) + VMT(r)− E]φ(MT )m (r, E) = 0. (3.28)

In the one-electron case we can, due to the spherical symmetry of the po-tential VMT(r), express the solution in terms of partial waves, which are wellknown objects from scattering theory [50].

Hence, since we want to match the wavefunction at the sphere bound-aries, it is beneficial to expand the plane wave solution of Eq. (3.27) interms of spherical harmonics YL(r), where L = (`,m) is a double index andr = (θ, φ) denotes the angular part of r ∈ R3, and a spherical function forthe radial part R`(|r|). We note that the solution of the radial part of theSCHRÖDINGER equation in the interstitial region yields two linear indepen-dent solutions [4]. These are spherical BESSEL functions j`(|r|) and sphericalNEUMANN functions n`(|r|), where the first is referred to as the regular so-lution and the second is called the irregular solution. For |r| → 0 they showthe asymptotic behavior [4]

j`(|r|) ∝ |r|`, (3.29)

andn`(|r|) ∝ |r|−(`+1). (3.30)

We define

JL(r) = j`(|r|)YL(r) =1

2(2`+ 1)

(|r|λ

)`YL(r), (3.31)

9Setting the kinetic energy in the interstitial region equal to zero, i.e. E − V0 = 0 isknown as the atomic sphere approximation (ASA).

22 3.3. MUFFIN TIN ORBITALS

and the envelope function

NL(r) = n`(|r|)YL(r) =

(λ

|r|

)`+1

YL(r), (3.32)

where we included the quantity λ of dimension [length] for the sake of di-mensionless functions. Within the MT sphere, i.e. |r| ≤ s, we write thewavefunction φ(MT )

m (r, E) as

φ(MT )m (r, E) =

∑L

a(m)L (E)ΨL(r, E), (3.33)

whereΨL(r, E) = YL(r)Φ`(|r|, E), (3.34)

and Φ`(|r|, E) is the radial part of the wavefunction within the MT sphere.Hence, the solution of our problem consists of Eq. (3.34) within the MTsphere and a linear combination of Eqs. (3.31) and (3.32) in the interstitialregion. It is well known, that the conservation of particles requires thatthe wavefunction is continuous and differentiable across the boundary [39].Matching the functions at the boundary and including a normalization fac-tor results in the definition of an MTO as [4]

χL(r, E) = YL(r)

Φ`(|r|, E) + P`(E)j`(|r|) |r| ≤ s,

n`(|r|) |r| > s.(3.35)

The total wavefunction φm(r, E) is then expressed as

φm(r, E) =∑L

a(m)L (E)χL(r, E). (3.36)

We note that ΨL(r, E) solves the SCHRÖDINGER equation (3.28) while NL(r)solves Eq. (3.27). The additional term P`(E)j`(|r|) ensures the continuity ofthe wavefunction as well as of its first derivative, however, does not solve theSCHRÖDINGER equation (3.28). In particular, we define the potential functionP`(E) by

P`(E) =D`(E) + `+ 1

D`(E)− `, (3.37)

where D`(E) is the logarithmic derivative function,

D`(E) =s

Φ`(s, E)

∂

∂rΦ`(|r|, E)

∣∣∣|r|=s

. (3.38)


Hence, the MTO (3.35) is not yet a solution of the SCHRÖDINGER equation(3.8) with Hamiltonian (3.26).

In the case of many-electrons as well as multiple, possibly overlappingMT potentials with radii sI within one unit cell, we simply expand the totalwavefunction φm(r, E) in terms of MTOs centered at different positions RI ,i.e.

φm(r, E) =∑IL

a(m)IL (E)χIL(r, E). (3.39)

It is the final task to find a condition for the coefficients a(m)IL (E), which en-

sures that the total wavefunction φm(r, E) is a genuine solution of the sta-tionary SCHRÖDINGER equation (3.8) with Hamiltonian (3.26). Therefore,we expand the spherical functions nI`(r) = n`(|r − RI |) inside the spherescentered at RJ 6= RI in the form

nI`(r) = −∑L′

SIL,JL′jJL′(r) for |r −RJ | < sJ , RI 6= RJ . (3.40)

The corresponding MTO takes on the form

χIL(r, E) =

ΨIL(r, E) + PI`(E)JIL(r) |r −RI | ≤ sI ,

−∑

L′ SIL,JL′JJL′(r) |r −RJ | < sJ RI 6= RJ ,

NIL(r) |r −RI | ∈ interstitial region,(3.41)

where we applied the notation JIL(r) = JL(|r − RI |), et cetera. The expan-sion coefficients SIL,JL′ are referred to as the structure constants and can bedetermined from [4]

SIL,JL′ =∑L′′

(−1)`+1 8π(2`′′ − 1)!!cL′′LL′

(2`− 1)!!(2`′ − 1)!!NL′′(RJ −RI), (3.42)

where the coefficients cLL′L′′ are given by

cLL′L′′ =

∫drYL(r)Y ∗L′(r)YL′′(r). (3.43)

It has to be emphasized at this point that the structure constants SIL,JL′ aresolely a function of the crystal structure. In contrast, the potential functionPI`(E) is a local function, i.e. depends on the local muffin tin sphere.

Again, we emphasize that the function ΨIL(r, E) is a solution of SCHRÖ-DINGER’s equation while the whole MTO is not because of the appearanceof the term PI`(E)JIL(r). Now, the tail cancellation theorem states that thisterm has to be canceled by the tails of the expansion (3.40) (at RI 6= RJ ).

24 3.4. LINEAR MUFFIN TIN ORBITALS

Then the MTOs χIL(x,E) is a solution of the SCHRÖDINGER equation, as canbe seen immediately from inspection of Eq. (3.41). This condition leads tothe so called KKR-ASA, i.e. KORRINGA - KOHN - ROSTOKER- atomic sphereapproximation, equations,∑

IL

a(m)IL (E) [2(2`+ 1)PI`(E)δIL,JL′ − SIL,JL′ ] = 0. (3.44)

It can be reformulated as

det [2(2`+ 1)PI`(E)δIL,JL′ − SIL,JL′ ] = 0, (3.45)

where det(·) denotes the determinant operation.Equations (3.45) are commonly very complicated, non-linear equations

in energy E, the solution of which can be rather cumbersome. However,there is a possible remedy to this drawback, which we shall discuss next.

3.4 Linear Muffin Tin Orbitals

Since the MTO χIL(r, E) still depends on energyE, the problem of Eq. (3.45)lies in its complex energy dependence. Therefore, one expands the partialwaves ΨIL(r, E) in terms of energy up to first order,

ΨIL(r, E) ≈ ΨIL(r) + (E − E0)ΨIL(r), (3.46)

where we introduced the obvious notation ΨIL(r) = ΨIL(r, E0) and

ΨIL(r) =∂

∂EΨIL(r, E)

∣∣∣E=E0

. (3.47)

Here, E0 is a fixed energy, which should be in the range of the eigenenergiesof the problem to be solved. The corresponding linear MTOs (LMTOs) arenow energy independent for E = E0, i.e. χIL(E0) = 0. This allows us toreplace the secular equations (3.45) by the linear equation

det (M − EO) = 0, (3.48)

where M and O are energy-independent matrices.Further details of the LMTO method can be found in, for instance, Ref.

[4] but are certainly far beyond the scope of this section. However, let usbriefly summarize the benefits and caveats of the LMTO method in particu-lar and of LSDA in general. First of all, the LMTO code as provided by AN-DERSEN and JEPSEN [5] does not feature any relativistic corrections, hence


spin-orbit interaction is not included. Furthermore, the fact that the LMTOcode employed within this work has been implemented in the ASA10 [5]makes the method particularly applicable for closely packed systems. Themajor source of systematic errors constitutes the LSDA, see Sec. 3.2. Asa rule of thumb we may state that the LSDA works well for very homo-geneous systems. In particular, if the electrons are strongly correlated, theLDA fails due to the approximations made in its introduction [41]. In such acase one has to go beyond LDA and employ more sophisticated exchange-correlation functionals [41] or even go beyond DFT and use methods, suchas dynamical mean field theory (DMFT) [51] or the variational cluster approach[52, 53]. Furthermore, a well known caveat of DFT in general is that, inmany cases the bandgap of conventional semiconductors is clearly under-estimated. This is of particular interest for our treatment of charge transportin CrAS / GaAs heterostructures.

In Fig. 3.2 we show the electronic structure of GaAs (lattice constanta = 5.65 Å) as obtained with the LMTO-ASA code [5] together with the’correct’ electronic structure as obtained by JANCU et al. [8], see Chap. 4. Ob-viously, the bandgap is profoundly underestimated when compared to theexperimentally observed value of 1.43 eV at room temperature (or 1.52 eV atlow temperatures) [8]. Moreover, since spin-orbit coupling is not taken intoaccount, the heavy- and the light hole bands, as well as the split-off band,do not emerge. While state-of-the-art GW methods [54] seem to improveon the band gap [55], the overall features of the electronic structure aroundthe conduction band minimum are reproduced rather well by LSDA. In Fig,3.3 we show a scissored version of the ab-initio electronic structure of GaAstogether with the ’exact’ bands. Hence, if the absolute value of the bandgapcan be regarded as of minor importance the method of LMTO is sufficient.It has been demonstrated by STICKLER et al. [14, 15], see Chaps. 7 and 8,that the absolute value of the bandgap indeed does not affect the questionwhether or not CrAs / GaAs heterostructures act as efficient spin filters.This is due to the particular electronic structure and band lineup of the ma-terials involved.

In Fig. 3.4 we show the band structures of minority-spin and majority-spin bulk zinc-blende CrAs as obtained with the LMTO-ASA code. Ob-viously, the majority spin bands cross the FERMI energy while the minor-ity spin bands do not. Hence, bulk zinc-blende CrAs is predicted to be ahalfmetallic ferromagnet. We shall discuss the band structures in more detailin Chaps. 7 and 8. However, as emphasized in Ref. [15] and Chap. 8, the fi-nal answer to the question of accuracy of the overall features of the ab-initio

10Atomic sphere approximation: see footnote 9 in Sec. 3.3.


L G X W L K G−13

−10

−7

−4

−1

2

5

E / e

V

k

Figure 3.2: Ab-initio electronic structure of GaAs as obtained with theLMTO-ASA code [5] (black dashed line) in comparison with the electronicstructure from JANCU et al. (red solid line) [8]. The FERMI energy is indi-cated by a solid blue line.

L G X W L K G−13

−10

−7

−4

−1

2

5

E / e

V

k

Figure 3.3: Ab-initio electronic structure of GaAs as obtained with theLMTO-ASA code [5] scissored (black dashed line) in comparison with theelectronic structure from JANCU et al. (red solid line) [8]. The FERMI energyis indicated by a solid blue line.


L G X W L K G−5

−4

−3

−2

−1

0

1

2

3

4

5

E / e

V

k

Figure 3.4: Ab-initio electronic structure of bulk zinc-blende CrAs as ob-tained with the LMTO-ASA code [5] for minority-spin (red solid line) andmajority-spin (black dashed line). The FERMI energy is indicated by a solidblue line.

predictions has to be answered by experiments.Moreover, we show the band structure of a supercell containing six lay-

ers of CrAs and six layers of GaAs, i.e. (CrAs)6/(GaAs)6 in Fig. 3.5. Dueto the increased number of atoms in the unit cell the number of bands alsoincreases dramatically.11

A final, but very important ab-initio study of bulk CrAs regards the sta-bility of the unit cell under a longitudinal distortion, i.e. normal to the inter-face. Comparison of the absolute energies reveals, that when the in-planelattice constant ap is fixed at ap = aGaAs, then the energetic minimum ofCrAs is reached for a longitudinal lattice constant al ≈ 0.98aGaAs, whereaGaAs is the lattice constant of GaAs. This is also illustrated in Fig. 3.6. Thein-plane lattice constant is kept constant at ap = aGaAs because breaking ofthe in-plane periodicity is assumed to be unlikely. Hence, the unit cell is notperfectly cubic but tetragonal.

In Fig. 3.7 and 3.8 we compare the electronic structures of the originalzinc-blende unit cell with the band structure of the distorted tetragonal unitcell of bulk CrAs for minority- and majority-spin, respectively. It is an es-

11This makes it rather difficult to extract ’useful’ information other than the band offsetfrom such calculations.


G X M GZ R A−2

−1

0

1

2

3

E / e

V

k

Figure 3.5: Ab-initio electronic structure of a (CrAs)6/(GaAs)6 supercell asobtained with the LMTO-ASA code [5] for minority-spin (red solid line) andmajority-spin (black dashed line). The FERMI energy is indicated by a solidblue line.

0.95 1 1.05 1.1−0.45

−0.445

−0.44

−0.435

−0.43

al / ap

E / a

rb. units

Figure 3.6: Absolute energy of CrAs as a function of the longitudinal latticeconstant al. The energetic minimum is reached for al = 0.98ap where ap isthe in-plane lattice constant. Hence, the unit cell is not cubic but tetragonal.


Z G X W L−5

−4

−3

−2

−1

0

1

2

3

4

5

E / e

V

k

Figure 3.7: Ab-initio electronic structure of minority spin CrAs as obtainedwith the LMTO-ASA code [5] for the fcc (red solid line) and the distortedunit cell (black dashed line). The FERMI energy is indicated by a solid blueline.

Z G X W L−5

−4

−3

−2

−1

0

1

2

3

4

5

E / e

V

k

Figure 3.8: Ab-initio electronic structure of majority spin CrAs as obtainedwith the LMTO-ASA code [5] for the fcc (red solid line) and the distortedunit cell (black dashed line). The FERMI energy is indicated by a solid blueline.


sential result that the electronic structure remains almost unchanged. Thishas two important consequences: First of all, this signifies that the elec-tronic properties and, therefore, the high degree of spin polarization of theelectronic structure is stable with respect to modest deviations from the fcclattice geometry.12 Moreover, this means that we can model the transportsafely by assuming an fcc geometry of the CrAs layers. The introduced sys-tematic error is negligible. This is a very important observation since, aswe shall see in Chaps. 4 and 5, it simplifies the mapping onto the effec-tive Hamiltonian and the transport calculations. Therefore, from now on weshall assume an fcc crystal structure for bulk CrAs with aCrAs = aGaAs.

In conclusion, we return to our original motivation and summarize whathas been gained so far. It is our goal to describe spin polarized transport inmagnetic heterostructures. With the methods and the mathematical appa-ratus introduced in this chapter, we are able to tackle the first step. We candetermine the electronic structures of all materials involved, as well as theband offset between them. In the next step we condense this informationinto a tractable model Hamiltonian which allows the calculation of the non-linear current response of CrAs / GaAs heterostructures. For this purposewe chose a tight-binding Hamiltonian.

12Please note that a distortion of 2 % in longitudinal direction is not a small deviation[38].

Chapter 4

EMPIRICAL TIGHT BINDING HAMILTONIAN

4.1 Some Introductory Remarks

We want to map the band structures obtained with the methods describedin the previous chapter onto an empirical nearest-neighbor tight-bindingHamiltonian. Some introductory remarks are required.

We chose a tight-binding (TB) Hamiltonian for the transport calcula-tion because it is suited ideally for a non-equilibrium GREEN’s function ap-proach, see Chapter 5. Generally speaking, if one chooses a fixed basis setto expand the wavefunctions φm(r) of Eq. (3.17), see the discussion in Sec.3.2, then there are, from the physical point of view, two phenomenologi-cally distinct limiting cases. In the first case we assume that the electronsare nearly free and the wavefunctions are, therefore, well approximated byplane waves. The second extreme is the assumption of tightly bound elec-trons, i.e. the crystal orbitals or molecular orbitals are assumed to be of ashape very similar to that of atomic orbitals. In this case it might be favor-able to write the total wavefunctions φm(r) as a linear combination of atomicorbitals (LCAO), which is the original tight-binding approach [38].

It is important to remark that both approaches, the plane wave or theLCAO approach, are in principle exact because both sets of basis functionsspan the entire HILBERT space, i.e. both sets are complete. However, ex-pressing, for instance, a localized orbital with plane waves might requiremore basis functions than expressing the same orbital in a localized basis.Hence, the LCAO approach is particularly useful if we regard, for instance,valence electrons which are tightly bound to nuclei. The great advantage

31

32 4.1. SOME INTRODUCTORY REMARKS

of the TB approach lies in its mathematical simplicity. In particular, if theorbitals are truly localized, the overlap between two orbitals which are notsituated at neighboring atoms is very small. In this case the nearest-neighborapproximation is justified and the Hamiltonian is characterized by a few TBparameters. However, for realistic materials this is hardly ever the case.

When designing an effective TB Hamiltonian from an ab-initio bandstructure, the situation is more involved. Basically, we could solve theSCHRÖDINGER equation (3.17) with the help of the method of LMTOs as de-scribed in Chapter 3 and then simply calculate the TB parameters ab-initio.But it turns out that, since the TB assumptions are in general not fulfilled, anearest-neighbor Hamiltonian is not sufficient to capture the important fea-tures of the electronic structure. It is therefore not trivially possible to obtainthe effective Hamiltonian from the ab-initio electronic structure code. It hasto be emphasized that there are numerous approaches to obtain maximallylocalized basis states from ab-initio codes, for instance Refs. [56, 57].

An alternative method is the empirical tight-binding (ETB) method [10, 9,8]. Its main idea is to regard the TB parameters as genuine fitting parame-ters. These ETB parameters are determined in such a fashion that the part ofthe electronic structure, relevant for the transport properties, is representedwith sufficient accuracy. A benefit of ETB is its conceptual simplicity. Aslong as the electronic structure is the only relevant property and as long asit is sufficiently well reproduced, the approach is useful.

However, the first problems arise as soon as a junction of different mate-rials is considered. Since the ETB Hamiltonian is obtained via a fitting pro-cedure of its matrix elements, the underlying basis functions are unknown.Hence, it is impossible to establish a connection between two separately ob-tained Hamiltonians in order to join them to a heterojunction. Additionalinformation is needed. We shall come back to this problem within Sec. 5.2and within Chap. 9.

This Chapter is organized as follows: In Sec. 4.2 we introduce the basicideas of the TB - LCAO approach, in Sec. 4.3 we concretize the ideas for theparticular case of a zinc-blende nearest-neighbor sp3d5s∗ ETB Hamiltonian,in Sec. 4.4 we discuss how the spin-orbit interaction can be introduced and,finally, in Sec. 4.5 we explain the fitting procedure used to obtain the ETBparameters. This introduction is mainly based on Refs. [10, 38, 58, 59].

Chapter 4: EFFECTIVE TIGHT BINDING HAMILTONIAN 33

4.2 The TB - LCAO Approach

Let ψJn(r −RIJ), r ∈ R3 and n ∈ N be an atomic orbital defined via

HJψJn(r −RIJ) = εJnψJn(r −RIJ), (4.1)

where HJ is the atomic Hamiltonian given by

HJ = T + VJ(r −RIJ) + Vint(r), (4.2)

centered at position RIJ ∈ R3, where Vint(r) is the effective electron-electroninteraction, see Sec. 3.1. Here, I ∈ N labels the unit cells while J ∈ N labelsthe atoms within one particular unit cell and n labels the orbitals of atomJ .1 Moreover, the position of the Jth atom is written as RIJ = RI + τJ , withRI the position of the Ith primitive cell and τJ the vector pointing from theorigin of the cell to the Jth atom within this primitive cell. Furthermore, theεJn denote the corresponding energy eigenvalues, see Eq. (4.1).

The Hamiltonian of the total crystal reads

H = T +∑IJ

VJ(r −RIJ) + Vint(r), (4.3)

which is equivalent to the Hamiltonian described by Eq. (3.5) for Vext(r) ≡ 0.In the next step we write the total wavefunctions φm(r), see Eq. (3.17), aslinear combination of atomic orbitals. In particular, we apply BLOCH’s theo-rem, which introduces a further index on the wavefunction φm(r)→ φmk(r)which is referred to as the k-vector [36], k ∈ R3. We represent the wavefunc-tion φmk(r) as a linear combination of BLOCH sums ΦJnk(r):

φmk(r) =∑Jn

c(m)Jn (k)ΦJnk(r), (4.4)

where the BLOCH sum (or WANNIER function) ΦJnk(r) localized at atom Jis given by

ΦJnk(r) =1√N

∑I

exp (iRIJ · k)ψJn(r −RIJ). (4.5)

Here N denotes the number of unit cells in the crystal. The correspondingstationary SCHRÖDINGER equation is

Hφmk(r) = Em(k)φmk(r), (4.6)1We shall not discuss the case that the number of orbitals n is an explicit function of the

type of the atom J , i.e. n 6= nJ , for reasons of simplicity. The more general treatment isstraight-forward.

34 4.2. THE TB - LCAO APPROACH

where Em(k) is referred to as the band structure and the Hamiltonian H isgiven by Eq. (4.3). We multiply the left-hand-side (lhs) of Eq. (4.6) withΦ∗Jnk(r) from the left, insert Eqs. (4.4) and integrate over r and obtain∫

drΦ∗Jnk(r)Hφmk(r) =∑J ′n′

c(m)J ′n′(k)

∫drΦ∗Jnk(r)HΦJ ′n′k(r)

=:∑J ′n′

HJnJ ′n′(k)c(m)J ′n′(k). (4.7)

In the very last step we implicitly defined the matrix elements HJnJ ′n′(k) ofthe Hamiltonian matrix H(k), which we can rewrite with the help of Eq. (4.5)as

HJnJ ′n′(k) =

∫drΦ∗Jnk(r)HΦJ ′n′k(r)

=1

N

∑II′

exp [i (RI′J ′ −RIJ) · k]

×∫

drψ∗Jn(r −RIJ)HψJ ′n′(r −RI′J ′). (4.8)

The Hamiltonian matrix H(k) is independent of I , i.e. it only depends onthe indices J and n of the atoms within one unit cell.

In analogy, we multiply the rhs of Eq. (4.6) from the left with the BLOCHsum Φ∗Jnk(r), integrate over positions r and obtain

Em(k)

∫drΦ∗Jnk(r)φmk(r) = Em(k)

∑J ′n′

c(m)J ′n′(k)

∫drΦ∗Jnk(r)ΦJ ′n′k(r)

=: Em(k)∑J ′n′

c(m)J ′n′(k)SJnJ ′n′(k). (4.9)

Again, in the very last step we implicitly defined the matrix elementsSJnJ ′n′(k) of the overlap matrix S(k). We express these matrix elements interms of the atomic orbitals by inserting Eq. (4.5):

SJnJ ′n′(k) =

∫drΦ∗Jnk(r)ΦJ ′n′k(r)

=1

N

∑II′

exp [i (RI′J ′ −RIJ) · k]

×∫

drψ∗Jn(r −RIJ)ψJ ′n′(r −RI′J ′)

=∑I

exp [i(RIJ ′ − τJ) · k]

×∫

drψ∗Jn(r − τJ)ψJ ′n′(r −RIJ ′). (4.10)


In the last step we performed the variable transformation r → r + RI andrenamed RI −RI′ → RI in order to get rid of the prefactor 1/N .

Hence, we rewrite SCHRÖDINGER’s equation (4.6) with the help of thedefinitions (4.8) and (4.10) as∑

J ′n′

HJnJ ′n′(k)c(m)J ′n′(k) = Em(k)

∑J ′n′

SJnJ ′n′(k)c(m)J ′n′(k). (4.11)

We assume that the atomic wavefunctions ψJn(r) are orthonormal and thatthe overlap between two wavefunctions localized at different atomic sites Jand J ′ is negligible, i.e.∫

drψ∗Jn(r − τJ)ψJ ′n′(r −RIJ ′) = δnn′δJJ ′δI0, (4.12)

where R0 = (0, 0, 0)T denotes the center of the crystal. Please note that thisassumption is unrealistic for realistic materials, however, the more generaltreatment is straight-forward and leads to a generalized eigenvalue problem[43]. Hence, we obtain for the overlap matrix elements SJnJ ′n′(k) defined inEq. (4.10) that

SJnJ ′n′(k) = δnn′δJJ ′ , (4.13)

i.e. the identity. In a similar fashion, we investigate the matrix elements ofthe Hamiltonian matrix H(k) given by Eq. (4.8) by inserting Eq. (4.3). Weobtain

HJnJ ′n′(k) =1

N

∑II′

exp [i(RI′J ′ −RIJ) · k]

∫drψ∗Jn(r −RIJ)

×HψJ ′n′(r −RI′J ′)

=1

N

∑II′


∫drψ∗Jn(r −RIJ)

×

[T +

∑I′′J ′′

VJ ′′(r −RI′′J ′′) + Vint(r)

]×ψJ ′n′(r −RI′J ′)

=1

N

∑II′


∫drψ∗Jn(r +RI′J ′ −RIJ)

×

[T +

∑I′′J ′′

VJ ′′(r +RI′J ′ −RI′′J ′′) + Vint(r +RI′J ′)

]×ψJ ′n′(r), (4.14)

where we substituted r → r +RI′J ′ in the last step.


A closer inspection of Eq. (4.14) reveals that the sum over unit cells Icontains three different contributions. We rewrite Eq. (4.14) in order todemonstrate this:

HJnJ ′n′(k) = A1(J, J ′) +∑J ′′

A2(J, J ′′, J ′), (4.15)

where we define

A1(J, J ′) :=1

N

∑II′



× [T + Vint(r +RI′J ′)]ψJ ′n′(r), (4.16)

as well as

A2(J, J ′′, J ′) :=1

N

∑II′I′′



×VJ ′′(r +RI′J ′ −RI′′J ′′)ψJ ′n′(r). (4.17)

Again, A1(J, J ′) consists of two different contributions: For I = I ′ and J =J ′ both wavefunctions are centered at the same atom. This contribution isreferred to as the one-center contribution. In all other cases, i.e. either I 6= I ′

or J 6= J ′ or both, the wavefunctions are centered at different nuclei, whichis referred to as a two-center contribution. Contributions of a similar typealso appear in A2(J, J ′′, J ′), see Eq. (4.17). However, in addition we obtaina three-center contribution if I 6= I ′ 6= I ′′ or J 6= J ′ 6= J ′′ or both, i.e. acontribution where both wavefunctions as well as the atomic potential areall centered at different nuclei. Neglecting this contribution is the celebratedthree-center approximation. It has been pointed out by SLATER and KOSTER[10] that there is no general reason why these contributions should indeedbe negligible. However, for the bare purpose of a fitting routine such anapproach might be justified.

Let us concretize this step: The three-center approximation states thatthe contributions to A2(J, J ′′, J ′) as given by Eq. (4.17) are only non-zero if,(a) RI′J ′ −RI′′J ′′ = 0 or, (b) RI′J ′ −RIJ = RI′J ′ −RI′′J ′′ . Thus, we have eitherI ′′ = I ′ and J ′′ = J ′ or I ′′ = I and J ′′ = J . All other contributions are setequal to zero. Applying the three-center approximation to A2(J, J ′′, J ′) Eq.


(4.17) gives for J 6= J ′

A2(J, J ′′, J ′) =1

N

∑II′



× [VJ ′(r)δJ ′′J ′ + VJ(r +RI′J ′ −RIJ)δJ ′′J ]ψJ ′n′(r)

=∑I


∫drψ∗Jn(r +RIJ ′ − τJ)

× [VJ ′(r)δJ ′′J ′ + VJ(r +RIJ ′ − τJ)δJ ′′J ]ψJ ′n′(r), (4.18)

where we substituted RI′ − RI → RI in the last step in order to get rid ofthe prefactor 1/N . Hence, also A2(J, J ′′, J ′) contains only one-center (RIJ =RI′J ′) and two-center (RIJ 6= RI′J ′) contributions. Let us denote by B1(J, J ′)the one-center contributions and by B2(J, J ′) the two-center contributions.Then,

HJnJ ′n′(k) = B1(J, J ′) + B2(J, J ′), (4.19)

with

B1(J, J ′) =1

N

∑I

∫drψ∗Jn(r) [T + Vint(r +RIJ) + VJ(r)]ψJn′(r)δJJ ′

=1

N

∑I

∫drψ∗Jn(r −RIJ) [T + Vint(r) + VJ(r −RIJ)]

×ψJn′(r −RIJ)δJJ ′

=: εJnδnn′δJJ ′ , (4.20)

where we substituted r → r − RIJ and employed Eq. (4.1). Hence, theenergy eigenvalues εJn of the atomic Hamiltonian (4.2) are referred to asonsite energies.

In analogy, we obtain the two-center approximations to be of the form

B2(J, J ′) = 2∑I


∫drψ∗Jn(r +RIJ ′ − τJ)

×VJ(r +RIJ ′ − τJ)ψJ ′n′(r), (4.21)

where RIJ ′ − τJ 6= 0, i.e. I 6= 0 if J = J ′. Let λJJ ′` ∈ R3 with ` ∈ N denote theset of vectors connecting all atoms of type J and J ′ within the whole crystal.We may then write the two-center contributions B2(J, J ′) as

B2(J, J ′) = 2∑`

exp(iλJJ

′

` · k)∫

drψ∗Jn(r + λJJ′

` )

×VJ(r + λJJ′

` )ψJ ′n′(r). (4.22)


The different types of overlap integrals which appear in Eq. (4.22) are com-monly referred to as hopping parameters.

Based on the three-center approximation is the nearest-neighbor approxi-mation. It introduces the assumption that all two-center approximations areequal to zero if the two centers are not nearest-neighbors.2 Let λJJ ′` ∈ R3 bethe vectors pointing from nucleus J to those nuclei J ′ which are its nearest-neighbors. Here, ` = 1, . . . ,M and |λ`| = λ ∈ R for all `.3 Then we canrewrite Eq. (4.22) as

B2(J, J ′) = 2M∑`=1

exp(iλJJ

′

` · k)∫

drψ∗Jn(r + λJJ′

` )

×VJ(r + λJJ′

` )ψJ ′n′(r). (4.23)

We abbreviate∫drψ∗Jn(r + λJJ

′

` )VJ(r + λJJ′

` )ψJ ′n′(r) =: αJnJ′n′

` tJJ′

nn′ , (4.24)

where αJnJ ′n′` ∈ R determines the sign as well as the relative strength ofthe overlap integral and tJJ

′

nn′ is the hopping parameter. With the help of thisdefinition, the matrix elements of the Hamiltonian matrix H(k) (4.14) takeon the convenient form

HJnJ ′n′(k) = εJnδJJ ′δnn′ + gJJ′

nn′ (k)tJJ′

nn′ , (4.25)

which defines the structure factor gJJ ′nn′ (k) as

gJJ′

nn′ (k) =M∑`=1

αJnJ′n′

` exp(iλJJ

′

` · k). (4.26)

It is important to remark that the structure factors gJJ ′nn′ (k) are completelydetermined by the crystal structure and the type of the orbitals involved butare independent of the actual material. In other words, they represent thesymmetry of the Hamiltonian and contain the only k-dependence of the TBmodel.4 On the other hand, the parameters which determine the final form

2Obviously, this approximation is rather severe and, hence, the main source of sys-tematic errors in this approach. However, as we shall observe in Chapter 5, the nearest-neighbor approximation substantially simplifies the numerics of the transport calculations.

3If |λ`| 6= λ for some `, then λ` would not point to a nearest-neighbor.4In this sense, the structure factors gJJ

′

nn′ (k) are comparable to the structure constantsSIL,JL′ of LMTO, see Eq. (3.42).


of the band structure are the onsite energies εJn and the hopping parame-ters tJJ ′nn′. Hence, we defined an effective Hamiltonian matrix (4.25), whichonly depends on a finite number of TB parameters εJn and tJJ ′nn′. Let|Jnk〉 denote the basis-kets5 spanning the Hilbert space together with c†Jn(k)and cJn(k) the corresponding creation and annihilation operators. Then thecorresponding Hamiltonian can be written as

H =∑k

∑Jn

εJn |Jnk〉〈Jnk|+∑k

∑〈JJ ′〉

∑nn′

gJJ′

nn′ (k)tJJ′

nn′ |Jnk〉〈J ′n′k|

≡∑k

H(k), (4.27)

or in in second quantized form as

H =∑k

∑Jn

εJnc†Jn(k)cJn(k) +

∑k

∑〈JJ ′〉

∑nn′

gJJ′

nn′ (k)tJJ′

nn′c†Jn(k)cJ ′n′(k)

≡∑k

H(k), (4.28)

where 〈JJ ′〉 indicates that the sum is restricted to J and J ′ nearest-neighbors.The simplifications invoked in the derivation of Eq. (4.27) make this

Hamiltonian useless for the ab-initio description of realistic materials. Itis the idea of empirical tight-binding (ETB) [10] to regard the TB parametersεJn and tJJ ′nn′ as pure fitting parameters. These have to be determined insuch a fashion that the electronic structure, which has been obtained withthe help the LMTO method in Chap. 3, is reproduced with sufficient ac-curacy. Then, the simple, parametric form of Eq. (4.27) allows a tractableformulation of the non-equilibrium transport problem. In what follows weconcretize the above Hamiltonian matrix to a particular example, which werequire in order to describe spin polarized transport in CrAs / GaAs het-erostructures. This example is the zinc-blende nearest-neighbor ETB Hamil-tonian, for brevity referred to as ZB nn ETB Hamiltonian.

4.3 The ZB nn ETB Hamiltonian

Within this section we investigate the zinc-blende nearest-neighbor ETBHamiltonian, the ZB nn ETB Hamiltonian which is of particular interest

5If no approximations would have been invoked, the position space representation ofthe basis-kets |Jnk〉 would be the WANNIER functions, i.e. 〈r|Jnk〉 = ΦJnk(r). However,due to the numerous approximations, the basis functions are unknown.

40 4.3. THE ZB NN ETB HAMILTONIAN

for our purpose to investigate transport in CrAs / GaAs heterostructuresin Chaps. 7 and 8. The ZB unit cell contains two different atoms, say A andB, and is of the form of two superimposed fcc lattices. It is depicted in Fig.4.1.

Figure 4.1: Graphical illustration of the zinc-blende crystal structure. Pictureadapted from WIKIPEDIA [60].

Thus, the index J in Eq. (4.25) takes on the values J ∈ 1, 2. Moreover,it is straight-forward to establish that to every atom there are four nearest-neighbors and the corresponding connecting vectors λJJ ′` are of the form

λ11` = λ22

` = 0 for ` = 1, . . . , 4, (4.29)

andλ12` = λ21

` ≡ λ` for ` = 1, . . . , 4. (4.30)

In particular, the vectors λ` ∈ R3 can be expressed as, see Fig. 4.1,

λ1 =a

4(1, 1, 1)T , (4.31a)

λ2 =a

4(1,−1,−1)T , (4.31b)

λ3 =a

4(−1, 1,−1)T , (4.31c)

and

λ4 =a

4(−1,−1, 1)T , (4.31d)

with a ∈ R the lattice constant. For instance, for GaAs we have a = 5.65Å [8]. Since we consider only nearest-neighbor hopping, we can neglectthe index JJ ′ on the hopping parameters (4.24). We therefore define theparameters

Vnn′ := t12nn′ = t21

n′n, (4.32)


the hopping parameters of the ZB crystal.6 Hence, the value of the hoppingparameter only depends on the type of the orbitals n and n′ involved in theoverlap integral (4.24). The particular relative orientation of these orbitalswithin the crystal enters through the structure factors (4.26). It turns out tobe convenient to define the k-dependent ZB structure factors g`(k) ∈ C for` = 1, . . . , 4 as

g1(k) =1

4[exp (iλ1 · k) + exp (iλ2 · k)

+ exp (iλ3 · k) + exp (iλ4 · k)] , (4.33a)

g2(k) =1

4[exp (iλ1 · k) + exp (iλ2 · k)

− exp (iλ3 · k)− exp (iλ4 · k)] , (4.33b)

g3(k) =1

4[exp (iλ1 · k)− exp (iλ2 · k)

+ exp (iλ3 · k)− exp (iλ4 · k)] , (4.33c)

and

g4(k) =1

4[exp (iλ1 · k)− exp (iλ2 · k)

− exp (iλ3 · k) + exp (iλ4 · k)] . (4.33d)

In many cases it is beneficial to express the hopping parameters Vnn′ interms of the overlap parameters, such as for instance (spσ), since not all hop-ping parameters are mutually independent by symmetry. An exhaustivediscussion of the relationship between these parameters can be found inRef. [38] or in the original paper by SLATER and KOSTER [10]. For the ZBconfiguration, we summarize the dependency in Tab. 4.1. In ETB, these hop-ping parameters together with the onsite energies serve as fitting-parametersfor the band structure.

Let us give an illustrating example: If we only consider s and p orbitals,i.e. we use an sp3 basis on each atom which gives eight basis functions perunit cell, we may express the total Hamiltonian matrix as

H(k) =

(Hss Hsp

H†sp Hpp

), (4.34)

6Please note that we reversed the order of n and n′ in the second equality of Eq. (4.32).This means that it is of no influence whether we regard the hopping from, for instance, theA− s orbital to the B − p orbital or from B − p to A− s, where A and B are the atoms ands, p, d, etc., are the orbitals. However, it is of importance whether we regard the A − s toB − p or the A− p to B − s hopping.


Table 4.1: Summary of the hopping parameters expressed in terms of theoverlap parameters for s − s, s − p, p − p, s − d, p − d and d − d integrals.Notation according to SLATER and KOSTER [10].

Vmjm′j′

s− s Vss 4(ssσ)s− p Vsp

4√3(spσ)

pp Vxx43

[(ppσ) + 2(ppπ)]Vxy

43

[(ppσ)− (ppπ)]s− d Vsxy

4√3(sdσ)

Vsx2−y2 0Vs3z2−r2 0

p− d Vxxy43

[(pdσ) + 1√

3(pdπ)

]Vxyz

43

[(pdσ)− 2√

3(pdπ)

]Vxzx VxxyVxx2−y2

4√3(pdπ)

Vyx2−y2 −Vxx2−y2Vzx2−y2 0Vx3z2−r2 − 1√

3Vxx2−y2

Vy3z2−r2 − 1√3Vxx2−y2

Vz3z2−r22√3Vxx2−y2

d− d Vxyxy43

[(ddσ) + 2

3(ddπ) + 4

3(ddδ)

]Vxyyz

43

[(ddσ)− 1

3(ddπ)− 2

3(ddδ)

]Vxyzx VxyyzVxyx2−y2 0Vyzx2−y2

43

[−(ddπ) + (ddδ)]Vzxx2−y2 −Vyzx2−y2Vxy3z2−r2

2√3Vyzx2−y2

Vyz3z2−r2 − 1√3Vyzx2−y2

Vzx3z2−r2 − 1√3Vyzx2−y2

Vx2−y2x2−y243

[2(ddπ) + (ddδ)]Vx2−y23z2−r2 0V3z2−r23z2−r2 Vx2−y2x2−y2


where we suppress the k-dependence of the sub-matrices for the sake of amore compact notation. The particular form of these sub-matrices can befound in the literature, for instance in Ref. [9], to be

Hss =

(ε1s g1Vssg∗1Vss ε2s

), (4.35a)

for the s− s integrals,

Hsp =

(0 0 0 g2Vsp g3Vsp g4Vsp

−g∗2Vps −g∗3Vps −g∗4Vps 0 0 0

)(4.35b)

for the s− p integrals,

Hpp =

ε1p 0 0 g1Vxx g4Vxy g3Vxy0 ε1p 0 g4Vxy g1Vxx g2Vxy0 0 ε1p g3Vxy g2Vxy g1Vxx

g∗1Vxx g∗4Vxy g∗3Vxy ε2p 0 0g∗4Vxy g∗1Vxx g∗2Vxy 0 ε2p 0g∗3Vxy g∗2Vxy g∗1Vxx 0 0 ε2p

. (4.35c)

for the p−p integrals. Again, we suppress the k-dependence of the structurefactors g`(k) for the sake of a more compact notation and, in the spirit of theabove notation, εJn are onsite energies. Please note that we account for thedegeneracy of the onsite energies of the px, py and pz orbitals by definingonly one p onsite energy εJp per atom.

Of particular interest for our purpose is the Hamiltonian in an sp3d5s∗

basis, which has been used by JANCU et al. [8] to successfully reproducethe band structure of several ZB semiconductors.7 The artificial s∗ orbitalshave been introduced for the first time by VOGL et al. [9] in order to capturethe indirect band gap in Si and Ge. We extend the Hamiltonian (4.34) to ansp3d5s∗ basis (ten orbitals per atom) and write the total Hamiltonian in theform

H(k) =

Hss Hsp Hsd Hs∗s

H†sp Hpp Hpd Hs∗p

H†sd H†pd Hdd Hs∗d

H†s∗s H†s∗p H†s∗d Hs∗s∗

. (4.36)

The sub-matricesHss,Hpp andHsp have already been provided in Eqs. (4.35).In analogue to Hss and Hsp we write Hs∗s∗ as

Hs∗s∗ =

(ε1s∗ g1Vs∗s∗

g∗1Vs∗s∗ ε2s∗

), (4.37a)

7These values have also been employed to generate the exact GaAs band structure forFigs. 3.2 and 3.3.


and Hs∗p as

Hs∗p =

0 −g2Vs∗p0 −g3Vs∗p0 −g4Vs∗p

g∗2Vps∗ 0g∗3Vps∗ 0g∗4Vps∗ 0

. (4.37b)

Moreover, the coupling between s∗ and s is of the form

Hss∗ =

(0 g1Vs1s∗2

g∗1Vs2s∗1 0

). (4.37c)

Introducing d-orbitals into the Hamiltonian matrix profoundly increasesthe degree of complexity. We give the final form of the sub-matrices of Eq.(4.36) which can be obtained by taking the symmetry of crystal field splittinginto account [8, 61]. In particular, due to this symmetry we only have twoindependent d onsite energies per atom. The corresponding sub-matricesare

Hsd =

(0 0 0 0 0 g4Vsxy g2Vsxy g3Vsxy 0 0

g∗4Vxys g∗2Vxys g∗3Vxys 0 0 0 0 0 0 0

),

(4.38a)for s− d integrals and

Hpd(k) =

(0 H

(1)pd

H(2)pd 0

), (4.38b)

with

H(1)pd =

g3Vxxy g1Vxyz g4Vxxy g2Vxx2−y2 g2Vx3z2−r2

g2Vxxy g4Vxxy g1Vxyz g3Vyx2−y2 g3Vy3z2−r2

g1Vxyz g3Vxxy g2Vxxy g4Vzx2−y2 g4Vz3z2−r2

. (4.38c)

and

H(2)pd =

−g∗3Vxyx −g∗1Vyzx −g∗4Vxyx −g∗2Vx2−y2x −g∗2V3z2−r2x−g∗2Vxyx −g∗4Vxyx −g∗1Vyzx −g∗3Vx2−y2y −g∗3V3z2−r2y−g∗1Vyzx −g∗3Vxyx −g∗2Vxyx −g∗4Vx2−y2z −g∗4V3z2−r2z

.

(4.38d)for p− d integrals. In a similar fashion, the s∗ − d integrals enter the matrix


Hs∗d in the following form

Hs∗d =

0 g4Vs∗xy0 g2Vs∗xy0 g3Vs∗xy0 00 0

g∗4Vxys∗ 0g∗2Vxys∗ 0g∗3Vxys∗ 0

0 00 0

. (4.38e)

Please note that the overlap of the s∗ orbitals with the d-orbitals of eg sym-metry is zero, as it is the case for Hsd. Finally, we give the d − d overlapHamiltonian Hdd. For a clearer notation we separate it into different contri-butions, i.e.

Hdd =

(H

(1)dd H

(2)dd

H(2)†dd H

(3)dd

), (4.38f)

where

H(1)dd =

ε1d1 0 0 0 00 ε1d1 0 0 00 0 ε1d1 0 00 0 0 ε1d2 00 0 0 0 ε1d2

(4.38g)

and

H(3)dd =

ε2d1 0 0 0 00 ε2d1 0 0 00 0 ε2d1 0 00 0 0 ε2d2 00 0 0 0 ε2d2

, (4.38h)

are the onsite matrices and εJd1 are the t2g-symmetric orbitals while εJd2 arethe eg-symmetric orbitals. Moreover, the d− d overlap matrix is given by

H(2)dd =

g1Vxyxy g3Vxyyz g2Vxyzx g4Vxyx2−y2 g4Vxy3z2−r2

g3Vxyzx g1Vxyxy g4Vxyyz g2Vyzx2−y2 g2Vyz3z2−r2g2Vxyzx g4Vxyyz g1Vxyxy g3Vzxx2−y2 g3Vxz3z2−r2g4Vxyx2−y2 g2Vyzx2−y2 g3Vzxx2−y2 g1Vx2−y2x2−y2 g1Vx2−y23z2−r2

g4Vxy3z2−r2 g2Vyz3z2−r2 g3Vzx3z2−r2 g1Vx2−y23z2−r2 g1V3z2−r23z2−r2

.

(4.38i)

46 4.4. INCLUDING SPIN-ORBIT COUPLING

Some concluding remarks are required: The total sp3d5s∗ ZB nn ETBHamiltonian (4.36) is a 20 × 20 matrix for every k ∈ R3. The inclusion ofspin-orbit interactions, see Sec. 4.4, doubles the size. Note that this Hamil-tonian has to be diagonalized for each value k in order to obtain its spectrumEm(k), Eq. (4.11). The above expressions for the sub-matrices describe theHamiltonian’s dependence on the hopping parameters Vnn′ , however, in agenuine fitting routine the overlap parameters of Tab. 4.1 have to enter sincethe hopping parameters are not mutually independent. For an sp3d5s∗ basiswithout spin-orbit interaction this gives in total 31 independent parametersfor a ZB nn ETB Hamiltonian per spin. A detailed description of the fittingroutine will be given in Sec. 4.5.

4.4 Including Spin-Orbit Coupling

It is our aim to explicitly introduce the spin degree of freedom into the ETBformalism in order to be able to account for spin-orbit (SO) interaction inthe transport calculations. Thus, we replace the basis-kets |Jnk〉 by basis-kets |Jnkα〉 = |Jnk〉 ⊗ |α〉 where α ∈ ↑, ↓ labels the electron’s spin degreeof freedom. Hence, the number of basis functions doubles. The completeHamiltonian matrix H(k) can then be written as

H(k) =

(H↑(k) V↑↓(k)

V †↑↓(k) H↓(k)

), (4.39)

where H↑(k) and H↓(k) denote the two HAMILTON matrices for spin ↑ and↓ which are of the form (4.36), respectively, and V↑↓(k) denotes the cou-pling between the two different spin orientations. Since we regard a single-electron Hamiltonian, this coupling stems, for instance, from the interac-tion of the electron’s spin with an externally applied magnetic field or withsome background magnetization. However, we shall focus on the particu-lar case that the matrix elements of V↑↓(k) result from the interaction of theelectron’s spin with its own motion, i.e. SO interaction. We follow the strat-egy outlined by CHADI [62, 63] and include SO interaction by regarding theETB matrix elements of the atomic SO Hamiltonian. The total SO couplingHamiltonian of the crystal can then be written in the form

HSO =~

4m2ec

2[∂rV (r)× ~p] · ~σ, (4.40)

where c is the speed of light, ∂rV (r) is the gradient of the lattice potential,~p is the momentum vector operator and ~σ is the vector of PAULI matrices.


If we restrict ourselves to contributions on the same atom and assume aspherically symmetric atomic potential, we can write [39, 13, 64]

HatomSO =

1

2m2ec

2

1

|r|∂Vatom(|r|)

∂|r|~L ~S, (4.41)

where Vatom(|r|) is the (rotationally symmetric) atomic potential, ~L is theangular momentum vector operator, ~S = (~/2)~σ is the spin vector operatorand denotes the tensorial scalar product:

~L ~S ≡ Lx ⊗ Sx + Ly ⊗ Sy + Lz ⊗ Sz. (4.42)

Hence, the ETB matrix elements of HSO are of the form

〈Jnkα|HSO|J ′n′kβ〉 (4.43)

=

⟨Jnk

∣∣∣∣ κ|r| ∂Vatom(|r|)∂|r|

~L

∣∣∣∣J ′n′k⟩ · 〈α |~σ |β〉, (4.44)

where κ denotes the constant prefactor

κ =~

4m2ec

2. (4.45)

In the following we use the notation employed by DI CARLO [13]. If onlyonsite (J ′ = J) contributions between p orbitals are considered, the onlynon-zero elements are

〈J, px, ↑ |HSO |J, py, ↑〉 = −iλJ (4.46a)〈J, px, ↓ |HSO |J, py, ↓〉 = iλJ (4.46b)〈J, pz, ↑ |HSO |J, px, ↓〉 = −λJ (4.46c)〈J, pz, ↓ |HSO |J, px, ↑〉 = λJ (4.46d)〈J, py, ↑ |HSO |J, pz, ↓〉 = −iλJ (4.46e)〈J, py, ↓ |HSO |J, pz, ↑〉 = −iλJ , (4.46f)

and their complex conjugates. This result is easily demonstrated by rewrit-ing

~L ~S =1

2

(~J 2 − ~L2 − ~S2

), (4.47)

where ~J is the vector operator of total angular momentum, ~J = ~L + ~S.Hence, the only non-zero matrix elements are those for which total angularmomentum is conserved. In particular, let j = l + s, with s = 1/2 and

48 4.4. INCLUDING SPIN-ORBIT COUPLING

l = 0, 1, 2 for s, p and d orbitals. Then, splitting off the radial parts of thewavefunctions and suppressing the nucleus index J , leaves us with matrixelements of the form

⟨jmj

∣∣∣ ~L ~S∣∣∣j′mj′

⟩=

~2

2[j(j + 1)− l(l + 1)− s(s+ 1)] δjj′δmjmj′

, (4.48)

wheremj = ml+ms ∈ −j,−j+1, . . . , j−1, jwithml ∈ −l,−l+1, . . . , l−1, l and ms ∈

−1

2, 1

2

[39, 64]. Furthermore, we remember that |pz〉 =

|l = 1,ml = 0〉while [39]

|px〉 =1√2

(|l = 1,ml = 1〉+ |l = 1,ml = −1〉) , (4.49)

and

|py〉 =i√2

(|l = 1,ml = −1〉+ |l = 1,ml = 1〉) . (4.50)

Finally, we neglect the elements which only contribute to the onsite energiesin order to arrive at the relations (4.46).

A similar argument can be employed in order to obtain the matrix el-ements involving d orbitals (l = 2). However, this discussion is straight-forward but beyond the scope of this section because we do not need thesematrix elements in order to account for the peculiarities of the electronicstructures we want to describe. In the particular case of a ZB crystal struc-ture, see Sec. 4.3, the parameters λJ in Eqs. (4.46) are defined for J = 1, 2,and are referred to as the splitting parameters of the two different atoms inthe unit cell [8, 62]. Note that the above Hamiltonian in the sp3d5s∗ basis isof dimension 40 × 40. In Tab. 4.2 we give the ETB parameters as obtainedby JANCU et al. [8]. The resulting band structure with and without SO in-teraction is illustrated in Fig. 4.2. It is interesting to observe that with theinclusion of SO coupling the heavy- and the light-hole band, as well as thesplit-off band, emerge.8

8See also Figs. 3.2 and 3.3 in Sec. 3.4.


Table 4.2: Tight binding parameters for bulk GaAs [8], Fig. 4.2. All valuesare given in eV.

Onsite Overlapε1s −5.9818 (ddσ) −1.1409ε2s −0.4028 (ddπ) 2.2030ε1p 3.5820 (ddδ) −1.9770ε2p 6.3853 (ss∗σ) −1.5648ε1d1 13.1023 (s∗sσ) −1.9927ε1d2 13.1023 (s∗pσ) 2.1835ε2d1 13.1023 (ps∗σ) 2.2086ε2d2 13.1023 (s∗dσ) −0.6906ε1s∗ 19.4220 (ds∗σ) −0.6486ε2s∗ 19.4220 (s∗s∗σ) −3.6761

Overlap SO coupling(ssσ) −1.6187 λ1 0.1824(spσ) 2.4912 λ2 0.0408(psσ) 2.9382(ppσ) 4.4094(ppπ) −1.4572(sdσ) −2.7333(dsσ) −2.4095(pdσ) −1.7811(pdπ) 1.7821(dpσ) −1.8002(dpπ) 2.0709

4.5 Nearest Neighbor ETB Fits

It is the aim to determine the overlap parameters listed in Tabs. 4.1, as wellas the onsite energies, in such a way that they reproduce the required elec-tronic structure. This means that the electronic structure arising from theHamiltonian (4.36) or (4.39) mimics the ab-initio band structure sufficientlywell. As already emphasized in the previous chapter, the band structurewhich we want to reproduce was, in a first step, obtained with the helpof the LMTO method, see Sec. 3.4. Due to the symmetry of the electronicstructure, it suffices to reproduce it along its characteristic path through theBRILLOUIN zone. This characteristic path is determined by the characteris-tic points within the BRILLOUIN zone. For ZB materials these characteristic

50 4.5. NEAREST NEIGHBOR ETB FITS

L G X W L K G−13

−10

−7

−4

−1

2

5

E / e

V

k

Figure 4.2: Band structure as obtained for GaAs with an sp3d5s∗ ZB nn ETBHamiltonian with the parameters given in Tab. 4.2 [8]: the black dashed linewhere spin-orbit coupling is taken into account; the red solid line where itis neglected.

points are listed in Tab. 4.3 [8, 58]. In this case the path starts at the L-pointand terminates at the Γ-point, see Fig. 4.2.

Table 4.3: Characteristic points in the BRILLOUIN zone for the fcc crystal.k

L 2πa

(12, 1

2, 1

2)T

Γ 2πa

(0, 0, 0)T

X 2πa

(0, 1, 0)T

W 2πa

(12, 1, 0)T

K 2πa

(34, 3

4, 0)T

Let us briefly discuss which relations the ETB parameters have to obeyin order to be regarded as reasonable. First of all, they have to reproduce theband structure within prescribed accuracy. If the solid is composed of one


material only, for instance GaAs, and no further conditions are imposed9,this already suffices and we use a very simple fitting routine. Let ξ denotethe vector of ETB parameters and let Em(k; ξ) denote the band structureobtained from the Hamiltonian matrix H(k), either Eq. (4.36) or Eq. (4.39),while ELSDA

m (k) denotes the ab-initio electronic structure. Then we obtainthe ETB parameters ξ by minimizing the highly non-linear cost functionK(ξ) which is of the form

K(ξ) =∑mk

am(k)[ELSDAm (k)− Em(k; ξ)

]2, (4.51)

where am(k) are weights which determine the relevance of the energy eigen-valueELSDA

m (k).10 ETB fits with different strategies have been performed, forinstance, in Refs. [8, 9, 67, 68].

It is very important to emphasize that the ETB parameters obtained fromminimizing K(ξ), Eq. (4.51), are not necessarily unique. Moreover, in mostcases it can even be demonstrated that the band structure does not uniquelydefine the ETB parameters due to internal symmetries of the ETB Hamilto-nian matrix H(k) which leave the band structure invariant.11 A more de-tailed discussion of this symmetry argument can be found in Ref. [16] andChapter 9. Nevertheless, let us briefly discuss the situation of two or moredifferent materials sandwiched together. Then the internal symmetry of theETB Hamiltonian matrix poses additional constraints on the ETB parame-ters, i.e. the ETB parameter sets of the two materials are no longer indepen-dent. This can easily be incorporated into Eq. (4.51) by augmenting K(ξ)with a similar cost function for the second band structure, taking into ac-count that the parameter sets are not mutually independent. We shall comeback to this point in the course of Chapter 5, when discussing the virtualcrystal approximation.

Moreover, it is clear that if the unit cell is too large, i.e., if there are toomany bands in the relevant energy region, then the Hamiltonian matrix isvery large and minimization of K(ξ) will not be feasible. One faces thisproblem when information about a genuine interface from a supercell cal-culation is to be extracted, see Fig. 3.5. To be more precise, in order to have

9For instance, it has been required by VOGL et al. [9], by JANCU et al. [8] and by HARRI-SON [38] that the ETB parameters also imply certain scaling rules.

10The cost function K(ξ) defines a greatly non-trivial minimization problem in the formof a least-squares fit, see for instance Ref. [65], which may, in principle, be treated by nu-merous methods. We chose a genetic algorithm, for instance Refs. [65, 66], as provided byMATLAB.

11It is intuitively clear that the reversed statement is true, i.e. the ETB parameters definethe electronic structure uniquely.

52 4.5. NEAREST NEIGHBOR ETB FITS

a bulk-like interface within the supercell it is necessary to increase the su-percell’s size until some predefined threshold has been reached.12 However,this usually means that several layers of both materials have to be included.This makes the supercell rather large as compared to one single genuine unitcell. This also increases the number of possible parameters which makesthe fitting procedure impossible from a numerical point of view, even if alltechnical questions, such as the non-uniqueness of parameters, the questionof ETB parameter symmetries in a supercell or the validity of the nearest-neighbor approximation, can be answered satisfactorily.

12The band offset as obtained with Eq. (3.9) for different sizes of supercells can, forinstance, be used as a convergence criterion. Convergence then means that |∆n−∆n+1| ≤ ηwhere η > 0 is the required accuracy and n labels the different sizes of the supercell inincreasing order.

Chapter 5

MULTIBAND QUANTUM TRANSPORT

5.1 Outline

Let us briefly summarize what has been achieved in Chaps. 3 and 4. InChap. 3 we discussed how to obtain the electronic structures of the separatedmaterials of which the heterostructure is composed, as well as a methodto compute the band offset between these materials. Then, in Chap. 4 wediscussed the design of an effective nn ETB Hamiltonian from the respectiveelectronic structures. Within this chapter we introduce the mathematicalformalism of quantum transport through a heterostructure. In a first step,Sec. 5.2, we formulate the transport Hamiltonian Htr and discuss how it canbe obtained from the band offset, Eq. (3.9), together with the Hamiltonianmatrix, Eq. (4.39), of the different materials. Furthermore, we concretize thisprocedure for the ZB nn sp3d5s∗ ETB Hamiltonian introduced in Sec. 4.3. Ina second step we determine the current density at a finite applied voltage Va

in Sec. 5.3. The numerical details of the method are explicated in Secs. 5.4and 5.5.

5.2 The Transport Hamiltonian

In Sec. 4.2 we defined the Hamiltonian matrix H(k) as a function of k, Eq.(4.36), where we also denoted the respective basis-kets by |Jnk〉. In thissection we specify the transport Hamiltonian Htr which describes the het-erostructure in form of a quasi one-dimensional chain in the [100] direction.

53

54 5.2. THE TRANSPORT HAMILTONIAN

The heterostructure is composed of different materials. Each is character-ized by a bulk Hamiltonian matrix Hn(k), n = 1, . . . , K, K ∈ N. As a start-ing point, we derive a quasi one-dimensional chain, which consists of onlyone material with the bulk Hamiltonian matrix H(k). This can be achievedby performing a partial FOURIER transform [11] of the basis-kets.1 We intro-duce the new basis-kets

∣∣Jnjk‖⟩ as∣∣Jnjk‖⟩ =1

LBZ

∫dkx exp

(−ikxj

a

4

)|Jnk〉 , (5.1)

where j labels the atomic layers2 in the one-dimensional chain in x-direction,Fig. 5.3, k‖ = (ky, kz)

T denotes the parallel k-vector and LBZ = 8πa

is thelength of the BRILLOUIN zone. Please note that this particular form of thepartial FOURIER transform (5.1) applies to ZB systems.

Inserting the inverse of Eq. (5.1),

|Jnk〉 =∑j

exp(ikxj

a

4

) ∣∣Jnjk‖⟩ , (5.2)

into the Hamiltonian (4.27) gives

H =∑k

∑〈JJ ′〉

∑nn′

HJnJ ′n′(k) |Jnk〉〈J ′n′k|

=∑jj′

∑〈JJ ′〉

∑nn′

∑k

HJnJ ′n′(k) exp[ikx (j − j′) a

4

]×∣∣Jnjk‖⟩ ⟨J ′n′j′k‖∣∣ . (5.3)

We replace the sum over k by∑k

=∑k‖

∑kx

≈ 1

LBZ

∑k‖

∫dkx, (5.4)

and obtain

H =∑jj′

∑〈JJ ′〉

∑nn′

∑k‖

Hjj′

JnJ ′n′(k‖), (5.5)

1The particular partial FOUIER transform (5.1) is sometimes also referred to as partialWANNIER transform, due to the WANNIER functions |Jnk〉.

2It is important to realize that j labels the atomic layers rather than the unit cells. Hence,the sum over J , i.e. over the nearest-neighbors in all three spatial dimensions, and the sumover j are not independent. In particular, J ∈ 1, 2 while j ∈ Z. For instance, for ZBsystems we could label the cations by J = 1 and center them at even j. Hence, if j = 2i+ 1,

with i ∈ Z, it follows immediately that J != 2.

Chapter 5: MULTIBAND QUANTUM TRANSPORT 55

where we defined the partially FOURIER transformed Hamiltonian matrixelements Hjj′

JnJ ′n′(k‖) as

Hjj′

JnJ ′n′(k‖) :=1

LBZ

∫dkxHJnJ ′n′(k) exp

[ikx (j − j′) a

4

]. (5.6)

We note that1

LBZ

∫dkx exp

[ikx (j − j′) a

4

]= δjj′ . (5.7)

Since the only k-dependence of the matrix elements of the Hamiltonian ma-trix H(k) occurs in the structure factor g`(k), we have to evaluate integralsof the form

I` :=1

LBZ

∫dkx exp

[ikx(j − j′)

a

4

]g`(k). (5.8)

We introduce the vectors d1, d2 ∈ R2

d1 =a

2(1, 1)T and d2 =

a

2(1,−1)T , (5.9)

as well as the reduced structure factors ci(k‖) and si(k‖) for i = 1, 2

ci(k‖) :=1

2cos(k‖ · di

), (5.10a)

si(k‖) :=1

2sin(k‖ · di

). (5.10b)

Then, we insert definitions (4.33) into (5.8) and obtain with the help of Eq.(5.7) together with the above definitions (5.9) and (5.10),

I1 = c1(k‖)δj+1,j′ + c2(k‖)δj−1,j′ , (5.11a)I2 = c1(k‖)δj+1,j′ − c2(k‖)δj−1,j′ , (5.11b)I3 = s1(k‖)δj+1,j′ + s2(k‖)δj−1,j′ , (5.11c)I4 = s1(k‖)δj+1,j′ − s2(k‖)δj−1,j′ . (5.11d)

Thus, with the help of Eq. (4.25) the matrix elements of the partially FOURIERtransformed Hamiltonian matrix Hjj′(k‖) take on the form

Hjj′

JnJ ′n′(k‖) = εJnδJJ ′δnn′δjj′ +[gJJ ′nn′ (k)

]jj′

(k‖)tJJ ′

nn′ , (5.12)

where we have to insert for [gJJ ′nn′ (k)]jj′

(k‖) the respective partial FOURIER

transform of g`(k) (5.11). Denoting by Hjj′(k‖)∣∣jk‖⟩ ⟨j′k‖∣∣ the operator

Hjj′(k‖)∣∣jk‖⟩ ⟨j′k‖∣∣ =

∑〈JJ ′〉

∑nn′

Hjj′

JnJ ′n′(k‖)∣∣Jnjk‖⟩ ⟨J ′n′j′k‖∣∣ , (5.13)


allows us to rewrite the Hamiltonian (5.3) in the well known form of anearest-neighbor ETB chain as

H =∑k‖

∑j

[Hjj

∣∣jk‖⟩ ⟨jk‖∣∣+Hjj+1(k‖)

∣∣jk‖⟩ ⟨j + 1k‖∣∣+Hjj−1(k‖)

∣∣jk‖⟩ ⟨j − 1k‖∣∣] , (5.14)

where the onsite matrix Hjj obeys3

Hjj∣∣jk‖⟩ ⟨jk‖∣∣ =

∑J

∑n

εJn∣∣Jnjk‖⟩ ⟨Jnjk‖∣∣ , (5.15)

and the coupling matrices Hjj+1(k‖) and Hjj−1(k‖) contain the couplings tothe left- and to the right-hand atomic planes, respectively. These couplingmatrices result from the KRONECKER-δs in relations (5.11). For ZB materials,it is convenient to rewrite Eq. (5.14) as a Hamiltonian matrix H1D(k‖) ofinfinite dimension, i.e. as a matrix in the indices J , n and j. Clearly, for adiatomic material, J ∈ 1, 2, it is of the form

H1D(k‖) =

. . . ... 0. . . E1 U †(k‖)

U(k‖) E2 V (k‖)V †(k‖) E1 U †(k‖)

U(k‖) E2 V (k‖)V †(k‖) E1 . . .

0... . . .

, (5.16)

where the onsite matrices EJ are diagonal matrices containing the onsiteenergies εJn, i.e.

EJ = diag(εJn), (5.17)

and U and V are the coupling matrices which couple the different atomiclayers. By convention, we call U the matrix which couples from j − 1 to jand V the matrix which couples from j to j + 1.4 A graphical illustration ofthe ETB chain is given in Fig. 5.1.

3Please note that Hjj is independent of k‖ because j labels atomic planes rather thanunit cells.

4We remark that the matrices EJ , U and V are essentially the matrices Hjj , Hjj+1 andHjj−1 of Eq. (5.14). However, we adopt the notation from the literature [11].


Figure 5.1: Graphical illustration of the elements of the one-dimensionalETB chain with Hamiltonian matrix H1D(k‖), Eq. (5.16).

For a ZB nn sp3d5s∗ basis, see Sec. 4.3, we obtain with the help of Eqs.(5.11), (4.35), (4.37), and (4.38) the following form for U ∈ C10×10 and V ∈C10×10, if SO coupling is neglected: We separate U into two sub-matrices,

U = (U1U2) , (5.18a)

which are of the form

U1 =

=

c2Vss −c2Vsp is2Vsp −is2Vsp −is2Vsxyc2Vps c2Vxx −is2Vxy is2Vxy is2Vxxy−is2Vps −is2Vxy c2Vxx −c2Vxy −c2Vxxyis2Vps is2Vxy −c2Vxy c2Vxx c2Vxyz−is2Vxys −is2Vxyx c2Vxyx −c2Vyzx c2Vxyxyis2Vxys −c2Vyzx is2Vxyx −is2Vxyx is2Vxyzx−c2Vxys is2Vxyx −c2Vyzx c2Vxyx −c2Vxyzx

0 c2Vx2−y2x −is2Vx2−y2y is2Vx2−y2z −is2Vxyx2−y20 −c2V3z2−r2x −is2V3z2−r2y is2V3z2−r2z −is2Vxy3z2−r2

c2Vs∗s −c2Vs∗p is2Vs∗p −is2Vs∗p −is2Vs∗xy

,

(5.18b)


and

U2 =

=

−c2Vsxy is2Vsxy 0 0 c2Vss∗c2Vxyz −is2Vxxy −c2Vxx2−y2 −c2Vx3z2−r2 c2Vps∗−is2Vxxy c2Vxyz is2Vyx2−y2 is2Vy3z2−r2 −is2Vps∗is2Vxxy c2Vxxy −is2Vzx2−y2 −is2Vz3z2−r2 is2Vps∗is2Vxyyz −c2Vxyzx −is2Vxyx2−y2 −is2Vxy3z2−r2 −is2Vxys∗c2Vxyxy −is2Vxyyz −c2Vyzx2−y2 −c2Vyz3z2−r2 −c2Vxys∗−is2Vxyyz c2Vxyxy is2Vzxx2−y2 is2Vxz3z2−r2 is2Vxys∗−c2Vyzx2−y2 is2Vzxx2−y2 c2Vx2−y2x2−y2 c2Vx2−y23z2−r2 0−c2Vyz3z2−r2 is2Vzx3z2−r2 c2Vx2−y23z2−r2 c2V3z2−r23z2−r2 0is2Vs∗xy −c2Vs∗xy 0 0 c2Vs∗s∗

,

(5.18c)

where we suppress the argument k‖ for a more compact notation. In anal-ogy, we divide V according to

V = (V1V2) , (5.18d)

with

V1 =

=

c1Vss c1Vsp is1Vsp is1Vsp is1Vsxy−c1Vps c1Vxx is1Vxy is1Vxy is1Vxxy−is1Vps is1Vxy c1Vxx c1Vxy c1Vxxy−is1Vps Vxy c1Vxy c1Vxx c1Vxyzis1Vxys −is1Vxyx −c1Vxyx −c1Vyzx c1Vxyxyis1Vxys −c1Vyzx −is1Vxyx −is1Vxyx is1Vxyzxc1Vxys −is1Vxyx −c1Vyzx −c1Vxyx c1Vxyzx

0 −c1Vx2−y2x −is1Vx2−y2y −is1Vx2−y2z is1Vxyx2−y20 c1V3z2−r2x −is1V3z2−r2y −is1V3z2−r2z is1Vxy3z2−r2

c1Vs∗s c1Vs∗p is1Vs∗p is1Vs∗p is1Vs∗xy

,

(5.18e)


and

V2 =

=

is1Vsxy c1Vsxy 0 0 c1Vss∗c1Vxyz is1Vxxy c1Vxx2−y2 c1Vx3z2−r2 c1Vps∗is1Vxxy c1Vxyz is1Vyx2−y2 is1Vy3z2−r2 is1Vps∗is1Vxxy c1Vxxy is1Vzx2−y2 is1Vz3z2−r2 is1Vps∗is1Vxyyz c1Vxyzx is1Vxyx2−y2 is1Vxy3z2−r2 is1Vxys∗c1Vxyxy is1Vxyyz c1Vyzx2−y2 c1Vyz3z2−r2 c1Vxys∗is1Vxyyz c1Vxyxy is1Vzxx2−y2 is1Vxz3z2−r2 is1Vxys∗c1Vyzx2−y2 is1Vzxx2−y2 c1Vx2−y2x2−y2 c1Vx−y23z2−r2 0c1Vyz3z2−r2 is1Vzx3z2−r2 c1Vx2−y23z2−r2 c1V3z2−r23z2−r2 0is1Vs∗xy c1Vs∗xy 0 0 c1Vs∗s∗

.

(5.18f)

Therefore, by performing a partial FOURIER transform of the basis-kets|Jnk〉 we obtain the effective one-dimensional ETB chain H1D(k‖) (5.16) inx-direction for a single material, which is described by the bulk Hamilto-nian matrix H(k). It is important to observe that the simple form of thisHamiltonian is a result of the nearest-neighbor approximation invoked inSec. 4.2. However, in order to obtain the transport Hamiltonian we haveto clarify two main points: First of all, H1D(k‖) describes an infinite chainconsisting only of one particular material, i.e. it does not describe a het-erostructure. Thus, the question is how we can connect different ETB chains.And, secondly, the Hamiltonian matrix H1D(k‖) does not yet take care of anexternally applied scalar field Vext(r) or contact regions.

The inclusion of an externally applied potential Vext(x) isstraight-forward.5 It follows from the arguments presented in this chapter,that it can be incorporated by adding the potential to the diagonal elementsof the Hamiltonian (5.14), i.e. we substitute EJ → EJ + Vext(xj) where xjdenotes the x-coordinate of the jth atomic plane. In the matrix (5.16) thiscan be indicated by explicitly denoting the onsite matrices by EJ ≡ EJ(xj).Furthermore, the influence of an externally applied magnetic field can bemodeled by a strategy outlined by GRAF and VOGL [69].

On the other hand, extending the Hamiltonian H1D(k‖) (5.16) to the caseof a genuine heterostructure is not trivial and cannot be achieved withoutmaking further assumptions. In what follows we shall restrict our discus-sion to the particular case of a heterostructure consisting of an arbitrary

5For reasons of simplicity, we assume that Vext(r) ≡ Vext(x) depends solely on thex-coordinate. Otherwise, the periodicity in y- and z-direction would be violated, whichwould profoundly complicate the case.


number of ZB materials sandwiched together, where it is assumed that allmaterials share one atomic constituent.6 For pedagogical reasons we restrictthe discussion within this section to the specific case of two semi-infinite ZBETB chains, where the first consists ofAB and the second of CB. A straight-forward route to set up the total transport Hamiltonian matrixHtr(k‖) of thissystem is to apply the virtual crystal approximation (VCA) for the (shared)atom situated at the interface. Again, we note that such an approach is onlyfeasible because of the nearest-neighbor approximation invoked in Sec. 4.2.Hence, the transport Hamiltonian matrix Htr(k‖) is of the form [11], see Fig.5.2,

Htr(k‖) =

. . . ... 0

. . . EABA U †AB

UAB EABB VABV †AB EAB

A U †ABUAB EI

B VCBV †CB ECB

C U †CBVCB EB UCB

V †CB ECBB . . .

0... . . .

. (5.19)

Here, EXBX , EXB

B , UXB and VXB for X = A,C are obtained from the bulkproperties of the material XB by including the band offset ∆, as describedin the previous chapters. EI

B denotes the interfacial onsite energy of theshared atom B.7 The VCA states that the interfacial onsite energy is wellapproximated by [11, 12, 13, 38, 70, 71, 72]

EIB =

1

2

(EABB + ECB

B

). (5.20)

It is interesting to note that also the method described by BRANDBYGE et al.[40] utilizes an ad-hoc approximation of a form comparable to (5.20). Weshall come back to this point in the course of Chapter 9.

6More general cases will not be discussed here since the discussion is rather complicatedand not interesting for our purpose. However, the interested reader is referred to Refs.[16, 38, 58].

7We refrain from explicitly denoting the dependence on k‖ as well as the externallyapplied potential Va(xj) for the sake of a simplified notation.


Figure 5.2: Graphical illustration of the elements of the transport Hamilto-nian Htr(k‖) (5.19).

Defining the onsite Hamiltonian matrices

HAB(k‖) =

(EABA U †AB(k‖)

UAB(k‖) EABB

), (5.21)

and

HCB(k‖) =

(ECBC U †CB(k‖)

UCB(k‖) ECBB

), (5.22)

allows us to rewrite the transport Hamiltonian matrix (5.19) in the simplerform

Htr(k‖) =

. . . ... 0. . . HAB WAB

W †AB HI WCB

W †CB HCB

0... . . .

, (5.23)

where we also define the interface Hamiltonian HI . In the VCA it can bewritten as

HI =

(EABA U †AB

UAB12

(EABB + ECB

B

) ) . (5.24)

The coupling matrices WXB(k‖) in Eq. (5.23) are given by

WXB(k‖) =

(0 0

VXB(k‖) 0

). (5.25)


Let us briefly discuss the benefits and caveats of the VCA. The VCA as-sumes that the coupling elements towards the interface and away from theinterface are completely determined by the bulk couplings UAB, VAB andUCB, VCB, however, the onsite element of the shared atom B which is situ-ated at the interface is given by the arithmetic mean between the bulk val-ues of the two separate bulk materials, Eq. (5.20) and Eq. (5.24). It can beclearly regarded as an advantage that the transport ETB Hamiltonian in theVCA is rather simple. Improving the approximation by, for instance, takinginto accounts hoppings which cross the interface and, hence, also connectnext-nearest-neighbors is rather difficult due to the requirement of large su-percell calculations as outlined in Sec. 4.5. On the other hand, it is by nomeans clear whether or not the VCA is justified at all. In particular, if theETB parameters, and therefore also the onsite energies of the two materials,are bare fitting parameters, it is obvious that the VCA violates an importantsymmetry. This symmetry states that the electronic structure, and, thus, thetransport properties of the bulk materials, are independent of the actual ori-entation of the unit cell, i.e. we may simply exchange atom A and atomB, for instance, and the bulk electronic structure stays unaffected. This, inthe last instance, implies that we cannot know which onsite energies per-tain to which atomic species. Clearly, this symmetry is broken by Eq. (5.20)because this equation requires the knowledge of which atom has which on-site energies. A more detailed discussion of such symmetry arguments ispostponed to Chapter 9. However, we briefly mention two possible waysout of this dilemma: (i) We may try to formulate an interface matching con-dition which preserves the symmetries of the ETB parameters of the bulkHamiltonians. In some cases this is possible, as it has been demonstrated bySTICKLER and PÖTZ in Ref. [16] and Chap. 9. (ii) We may explicitly try toincorporate this symmetry into our fitting procedure, thus, in the definitionof the cost function K(ξ), Eq. (4.51). This approach has been employed bySTICKLER et al. in Refs. [14, 15] and Chaps. 7 and 8. In the specific case of aGaAs/CrAs interface, we perform the fit in such a fashion that all As onsiteenergies in bulk GaAs and bulk CrAs are equal, i.e.

EGaAsAs

!= ECrAs

As . (5.26)

Anyhow, whether or not the VCA gives reasonable results is hard tojudge. Intuitively, we could argue that the VCA works best if the basis sets|Jnk〉AB and |Jnk〉CB are very similar. However, since the ETB parametersare obtained within the framework of an ETB model, the basis functionsare unknown. Posing the additional constraint on the fitting procedure thatall hopping parameters should have similar orders of magnitude can be re-garded as an attempt to make the basis sets similar. Nevertheless, it has to


be admitted that the modeling of the interface clearly is the weak point ofthe whole approach, as it will be pointed out in Chap. 9.8

5.3 The Current Density

Having set up the transport Hamiltonian for a quasi one-dimensional ETBchain, we aim at calculating the current density J(Va) through the device asa function of applied voltage Va. We separate the system into three differentparts, illustrated in Fig. 5.3: the left lead L, the system S, and the right leadR.9 The leads are assumed to spatially extend to infinity, i.e. are semi-infinite,while the dimension of the system is regarded to be finite. Within eachof the three parts we regard a set of discrete grid-points which display thepositions of the unit cells. The grid-points in the left lead L will be labeledby iL = 1, . . . ,∞ (starting from the right), in the system by i = 1, . . . , N ,and in the right semi- infinite lead R by iR = 1, . . . ,∞ (both starting fromthe left). The applied voltage is defined to be zero in the left lead L, finite−Va in the right lead R, and, for simplicity, is approximated to drop linearlyfrom L to R across S.

Figure 5.3: Schematic illustration of the one-dimensional system under in-vestigation.

Hence, we write

Htr(k‖) = HL(k‖) +WLS(k‖) +HS(k‖) +HR(k‖) +WRS(k‖), (5.27)

8Moreover, interfacial lattice reconstruction is not captured by such a fitting procedure.The band lineup, however, is independent from the interface matching procedure, at leastunder zero or weak electric bias since it is determined from an ab-initio calculation.

9For a more detailed and self-contained discussion of the concepts introduced withinthis and the following sections we refer the reader to the book by S. DATTA [73], as well asto the publications cited within this chapter.

64 5.3. THE CURRENT DENSITY

where we introduce the system’s Hamiltonian matrix HS(k‖) together withthe Hamiltonian matrices of the leads,HL/R(k‖). Further,WLS(k‖) andWRS(k‖)account for the coupling between L and S and R and S, respectively. Thedirect coupling between L and R is zero because of the nearest-neighborapproximation, Sec. 4.2. In particular, we have

WLS(k‖) =

(0 VL(k‖)

V †L(k‖) 0

)and WRS(k‖) =

(0 VR(k‖)

V †R(k‖) 0

).

(5.28)In what follows we refrain from explicitly denoting k‖.

Furthermore, in the particular case that the leads are made of the samematerial, characterized by an onsite Hamiltonian10 H0 with coupling V0, theHamiltonians of the left and the right semi-infinite lead are given by

HL = HL0 , (5.29)

andHR = HR

0 − Va1, (5.30)

where Va ∈ R and

HL0 =

. . . .... . . H0 V0

V †0 H0

, (5.31)

and

HR0 =

H0 V0

V †0 H0 . . .... . . .

. (5.32)

The assumption that both leads are made of the same material also simpli-fies the coupling matrices VL and VR which couple the leads to the system.Due to the nearest-neighbor approximation we have

VL =

...00V0

, (5.33)

andVR =

(V0 0 0 · · ·

), (5.34)

10For the definition of an onsite Hamiltonian, see Eq. (5.21).


if we assume that our system is symmetric, in the sense that on both sidesthe same atomic constituent is positioned at the interface.

The retarded GREEN’s function of the one-dimensional ETB chain withHamiltonian (5.27) is given by

GR = [(E + iη)1−Htr]−1 , (5.35)

where Htr is the Hamiltonian of the whole system, i.e. of L, S, and R, seeEq. (5.27), where we also drop the argument E of the GREEN’s function.Further, η > 0 is an infinitesimal real number and 1 is the unity operator.The advanced GREEN’s function GA is connected to the retarded GREEN’sfunction GR by

GA =(GR)†. (5.36)

With the help of Eq. (5.27) we rewrite Eq. (5.35) as GRL GR

LS GRLR

GRSL GR

S GRSR

GRRL GR

RS GRR

=

(E + iη)1−HL VL 0

V †L (E + iη)1−HS VR0 V †R (E + iη)1−HR

−1

, (5.37)

which is equivalent to the following set of nine equations [73],

[(E + iη)1−HL]GRL + VLG

RSL = 1, (5.38a)

[(E + iη)1−HL]GRLS + VLG

RS = 0, (5.38b)

[(E + iη)1−HL]GRLR + VLG

RSR = 0, (5.38c)

V †LGRL + [(E + iη)1−HS]GR

SL + VRGRRL = 0, (5.38d)

V †LGRLS + [(E + iη)1−HS]GR

S + VRGRRS = 1, (5.38e)

V †LGRLR + [(E + iη)1−HS]GR

SR + VRGRR = 0, (5.38f)

V †RGRSL + [(E + iη)1−HR]GR

RL = 0, (5.38g)

V †RGRS + [(E + iη)1−HR]GR

RS = 0, (5.38h)

V †RGRSR + [(E + iη)1−HR]GR

R = 1. (5.38i)

We solve Eq. (5.38b) for GRLS and (5.38h) for GR

RS and obtain

GRLS = −gRLVLGR

S , (5.39a)

andGRRS = −gRRV

†RG

RS , (5.39b)


where we define the retarded GREEN’s function of the isolated semi-infiniteleads, gRM as

gRM = [(E + iη)1−HM ]−1 , (5.40)

for M = L,R. Substituting (5.39b) and (5.39a) into (5.38e) and solving forGRS yields

GRS =

[E1−HS − ΣR

L − ΣRR

]−1, (5.41)

where with ΣRL and ΣR

R we introduce the retarded self energies of the semi-infinite leads. They are related to the leads’ GREEN’s functions by

ΣRL = V †Lg

RLVL and ΣR

R = VRgRRV†R. (5.42)

Moreover, we may define the advanced self energies via

ΣAM =

(ΣRM

)†, (5.43)

for M = L,R. Moreover, in order to derive Eq. (5.41) we carry out the limitη → 0+. Let ΣR denote the self energy of the system, where

ΣR = ΣRR + ΣR

L , (5.44)

then we obtain for GRS the compact expression

GRS =

[E1−HS − ΣR

]−1. (5.45)

Note thatHS is of finite dimension, see Eq. (5.27). The influence of the semi-infinite leads is taken into account in an exact fashion via the self energiesΣRL and ΣR

R. Moreover, for reasons of numerical convenience it is more ad-vantageous to calculate the system’s GREEN’s function GR

S iteratively ratherthen by inversion, Eq. (5.45). The numerical procedure will be discussed inSec. 5.4.

It is convenient to define the Γ-matrices ΓM , also referred to as couplingfunctions, of the semi-infinite leads as

ΓM = i(ΣRM − ΣA

M

), (5.46)

for M = L,R. We proceed by discussing how the current density J(Va) isobtained for a given transport Hamiltonian matrix Htr(k‖) (5.27). We em-ploy the LANDAUER - BÜTTIKER formula [73]

J(Va) = − qh

∫dET (E)[fL(E)− fR(E)], (5.47)


where q is the elementary charge, h is PLANCK’s constant, and T (E) is thetransmission from lead L to lead R as a function of the energy E.11 Further-more, fL(E) and fR(E) are given by

fL(E) = fFD(E) and fR(E) = fFD(E + Va). (5.48)

Here, fFD(E) denotes the FERMI-DIRAC distribution function

fFD(E) =

[exp

(E − EFkBT

)+ 1

]−1

, (5.49)

with BOLTZMANN’s constant kB, the temperature T , and the FERMI energyEF . The transmission function T (E) is determined within a non-equilibriumGREEN’s function approach by12

T (E) = tr(ΓRG

RΓLGA). (5.50)

It has to be emphasized at this point, that we assume that the leads are inlocal equilibrium in order to obtain Eq. (5.47) [73].

Let us subject Eq. (5.50) to a closer inspection. First, we rewrite (5.50) bynoting the explicit energy-dependence

T (E) = tr[ΓR(E)GR(E)ΓL(E)GA(E)

]. (5.51)

In a second step we specify the trace operation in Eq. (5.51): for a generaloperator A the trace reads13

tr (A) =∑Jn

∑k‖

∑j

⟨Jnjk‖

∣∣A ∣∣Jnjk‖⟩, (5.52)

where we use the basis-kets∣∣Jnjk‖⟩ defined in Eq. (5.1). Note that j =

1, . . . , N labels the atomic planes within our system S, Fig. 5.3, n labelsthe orbitals per atom, and J labels the atoms within the unit cell. In thefollowing we will denote spatial matrix elements by

〈i |A |j〉 = Aij, (5.53)11Please note that, strictly speaking, Eq. (5.47) expresses the total current and can be

written as the sum of majority- and minority-spin current. However, since we regard spin-coherent transport, Eq. (5.47) applies to the current for majority- and minority-spin inde-pendently. In what follows we shall, therefore, neglect the trace over the spin degree offreedom for the sake of a simplified notation.

12Alternatively, one may determine the transmission function, for instance, within atransfer-matrix approach [13], with an effective mass model, Chap. 9, or some othermethod.

13By convention, we neglect the trace over the spin degree of freedom, see footnote 11 inthis section.


where Aij is still a matrix in, for instance, the orbital degrees of freedom.Since the Γ-matrices ΓM are determined by the leads’ self energy, Eq.

(5.46), and since the leads self energies are defined by Eq. (5.42) it followsimmediately, that the Γ-matrices are of the form

ΓR =

0 · · · 0... . . .

0 γR

, (5.54)

and

ΓL =

γL 0 · · · 00 0... . . .0 0

, (5.55)

where γL/R are matrices. This is a further consequence of the nearest-neighborapproximation in Sec. 4.2. Hence, noting only the spatial part of the traceoperation in Eq. (5.51) we obtain∑

j

⟨j∣∣ΓRGRΓLG

A∣∣j⟩ = γRG

RN1γLG

A1N , (5.56)

or, by taking into account the additional degrees of freedom,

tr(ΓRG

RΓLGA)

=∑Jn

∑k‖

⟨Jnk‖

∣∣γRGRN1γLG

A1N

∣∣Jnk‖⟩. (5.57)

Introducing the explicit dependence of all functions on energy E and paral-lel momentum k‖ yields

T (E) =:∑k‖

T (E, k‖)

=∑k‖

∑Jn

〈Jn| γR(E, k‖)GRN1(E, k‖)

×γL(E, k‖)GA1N(E, k‖) |Jn〉 . (5.58)

Hence, Eq. (5.47) takes on the form

J(Va) = − qh

∑k‖

∫dET (E, k‖)[fL(E)− fR(E)]. (5.59)


One final remark is required. Because of the nearest-neighbor approx-imation, the expressions for the self energies of the semi-infinite leads canbe substantially simplified. This can be observed from inserting Eqs. (5.33)and (5.34) into Eqs. (5.42). We demonstrate it for the right semi-infinite leadR: We write gRR in the form

gRR =

GRR G01 · · ·G10 G11 · · ·

...... . . .

, (5.60)

where we call GRR the right surface GREEN’s function. Then, with the help ofEqs. (5.34) and (5.42), we have

ΣRR = V0GRRV

†0 . (5.61)

Thus we only need to know the surface GREEN’s function in order to deter-mine ΣR

R. A similar result is easily obtained for the left semi-infinite lead Land, finally, relations (5.42) are replaced by

ΣRL = V †0 GRLV0 and ΣR

R = V0GRRV†

0 , (5.62)

where GRL denotes the retarded surface GREEN’s function of the left semi-infinite lead. The surface GREEN’s functions GRL and GRR are determined nu-merically by applying an algorithm suggested by SANCHO et al. [74]. Thismethod will be the topic of Sec. 5.5.

5.4 Calculation of the System’s GREEN’s Function

Let us briefly present a very efficient iterative method to calculate the sys-tem’s GREEN’s functionGR

S . In the following we assume that the system un-der investigation consists of N grid-points, see Fig. 5.3. Due to the nearest-neighbor approximation the corresponding Hamiltonian takes on the form

HS =

H1 V1 0

V †1 H2 V2

V †2. . . . . .. . . VN−1

0 V †N−1 HN

, (5.63)

where the onsite Hamiltonians Hj are still Hermitian matrices. We splitthe system S into subsystems (S1, S2, . . . , SN) which are coupled through

70 5.4. CALCULATION OF THE SYSTEM’S GREEN’S FUNCTION

hopping matrices Vi, i = 1, . . . , N − 1, which are of the form given in Sec.5.2. Each isolated system Sk is completely determined by its HamiltonianHk. Furthermore, we denote by Uk the union of the first k isolated systems,where we include interactions, i.e.

Uk =k⋃i=1

Si, (5.64)

and by GRk the corresponding retarded GREEN’s function of Uk. Hence Uk isdescribed by Hamiltonian consisting of the first k lines and rows of the com-plete system’s Hamiltonian, (5.63). We note that UN = S and, correspond-ingly, GRN = GR

S . Let gRk be the retarded GREEN’s function of the isolatedgrid-point Sk, i.e.

gRk =[(E + iη)1−Hk − ΣR

Lδk1 − ΣRRδkN

]−1, (5.65)

where ΣRL and ΣR

R are the self energies accounting for the semi-infinite leadsas introduced in the previous section, Sec. 5.3. We connect the isolatedgrid-points iteratively by employing DYSON’s equation. Let us write theequation for arbitrary k, i.e.

GRk = ΓRk + ΓRk V(k)GRk , (5.66)

where V (k) denotes the matrix coupling Sk to Uk−1 and ΓRk denotes

ΓRk = gRk ⊗ 1Uk−1+ GRk−1 ⊗ 1Sk

. (5.67)

Here we use the identity matrices 1Uk−1and 1Sk

which are of the same di-mension as Uk−1 and Sk, respectively. In the following we will sacrifice theexplicit notation of these identity operators for the sake of a more compactformulation. We consider matrix elements of Eq. (5.66) in the LCAO basis|n〉, n = 1, . . . , k. Note that here the basis function |n〉 label lattice points andnot orbitals. Hence, the basis-ket |n〉 accounts for all atoms and their orbitalssituated in the cell around the grid-point with index n. In this case we canwrite the coupling matrix V (k) as

V (k) = |k − 1〉Vk−1 〈k|+ |k〉V †k−1 〈k − 1| . (5.68)

Inserting Eq. (5.68) into Eq. (5.66) we obtain for a particular matrix element⟨n∣∣GRk ∣∣m⟩⟨

n∣∣GRk ∣∣m⟩ =

⟨n∣∣ΓRk ∣∣m⟩+

∑γδ

⟨n∣∣ΓRk ∣∣γ⟩〈γ |V |δ〉⟨δ ∣∣GRk ∣∣m⟩

=⟨n∣∣ΓRk ∣∣m⟩+

⟨n∣∣ΓRk ∣∣k − 1

⟩Vk⟨k∣∣GRk ∣∣m⟩

+⟨n∣∣ΓRk ∣∣k⟩V †k ⟨k − 1

∣∣GRk ∣∣m⟩. (5.69)


We identify four different scenarios, (i) n = m = k, (ii) n 6= k, m 6= k, (iii)n = k, m 6= k, and (iv) n 6= k, m = k. We give the resulting equations. Forcase (i), n = m = k we obtain⟨

k∣∣GRk ∣∣k⟩ =

⟨k∣∣gRk ∣∣k⟩+

⟨k∣∣gRk ∣∣k⟩V †k ⟨k − 1

∣∣GRk ∣∣k⟩, (5.70a)

for case (ii), n 6= k, m 6= k,⟨n∣∣GRk ∣∣m⟩ =

⟨n∣∣GRk−1

∣∣m⟩+⟨n∣∣GRk−1

∣∣k − 1⟩Vk⟨k∣∣GRk ∣∣m⟩. (5.70b)

Case (iii), n = k, m 6= k, and case (iv), n 6= k, m = k, yield⟨k∣∣GRk ∣∣m⟩ =

⟨k∣∣gRk ∣∣k⟩V †k ⟨k − 1

∣∣GRk ∣∣m⟩, (5.70c)

and ⟨n∣∣GRk ∣∣k⟩ =

⟨n∣∣GRk−1

∣∣k − 1⟩Vk⟨k∣∣GRk ∣∣k⟩, (5.70d)

respectively.In a first step, we insert Eq. (5.70c) into Eq. (5.70b) and obtain⟨

n∣∣GRk ∣∣m⟩ =

⟨n∣∣GRk−1

∣∣m⟩+⟨n∣∣GRk−1

∣∣k − 1⟩Vk⟨k∣∣gRk ∣∣k⟩

×V †k⟨k − 1

∣∣GRk ∣∣m⟩. (5.71)

In a second step we consider in Eq. (5.71) the particular case of n = k − 1and solve the equation for

⟨k − 1

∣∣GRk ∣∣m⟩, m = 1, . . . , k − 1. This yields⟨k − 1

∣∣GRk ∣∣m⟩ = (1−Mk)−1 ⟨k − 1

∣∣GRk−1

∣∣m⟩, (5.72)

where weMk is

Mk =⟨k − 1

∣∣GRk−1

∣∣k − 1⟩Vk⟨k∣∣gRk ∣∣k⟩V †k . (5.73)

Inserting (5.72) into (5.71) results for n,m ∈ 1, . . . , k − 1 in⟨n∣∣GRk ∣∣m⟩ =

⟨n∣∣GRk−1

∣∣m⟩+⟨n∣∣GRk−1

∣∣k − 1⟩Vk⟨k∣∣gRk ∣∣k⟩V †k (1−Mk)

−1

×⟨k − 1

∣∣GRk−1

∣∣m⟩. (5.74)

We now consider the particular case of n = k − 1 in Eq. (5.70d), insert theresult into Eq. (5.70a) and solve the resulting expression for

⟨k∣∣GRk ∣∣k⟩ to

obtain ⟨k∣∣GRk ∣∣k⟩ = (1−Nk)−1 ⟨k ∣∣gRk ∣∣k⟩. (5.75)

Here, we denote

Nk =⟨k∣∣gRk ∣∣k⟩V †k ⟨k − 1

∣∣GRk−1

∣∣k − 1⟩Vk. (5.76)

72 5.5. THE SURFACE GREEN’S FUNCTIONS

The remaining matrix elements of GRk can be obtained by inserting Eq. (5.72)into Eq. (5.70c) and by inserting Eq. (5.75) into Eq. (5.70d). In summary, theresulting expressions for the matrix elements of GRk read⟨n∣∣GRk ∣∣m⟩ =

⟨n∣∣GRk−1

∣∣m⟩+⟨n∣∣GRk−1

∣∣k − 1⟩Vk⟨k∣∣gRk ∣∣k⟩V †k (1−Mk)

−1

×⟨k − 1

∣∣GRk−1

∣∣m⟩, (5.77a)

for n,m ∈ 1, . . . , k − 1 and whereMk is given by Eq. (5.73),⟨k∣∣GRk ∣∣k⟩ = (1−Nk)−1 ⟨k ∣∣gRk ∣∣k⟩, (5.77b)

where Nk was introduced in Eq. (5.76),⟨k∣∣GRk ∣∣m⟩ =

⟨k∣∣gRk ∣∣k⟩V †k (1−Mk)

−1 ⟨k − 1∣∣GRk−1

∣∣m⟩, (5.77c)

for m ∈ (1, . . . , k − 1) and⟨n∣∣GRk ∣∣k⟩ =

⟨n∣∣GRk−1

∣∣k − 1⟩Vk (1−Nk)−1 ⟨k ∣∣gRk ∣∣k⟩, (5.77d)

for n ∈ (1, . . . , k − 1). This iterative scheme is initialized with gR1 , whichalready includes the coupling to the semi-infinite left lead L through the selfenergy ΣR

L , and is terminated by attaching gRN , which contains the couplingto the semi-infinite right lead R via ΣR

R, see Eq. (5.65).

5.5 The Surface GREEN’s Functions

The following discussion is based on the work of SANCHO et al. [74]. Let|n〉, n ≥ 0, denote the basis-ket corresponding to grid-point n in lead R.Furthermore, we introduce the notation

Gnn′ =⟨n∣∣gRR ∣∣n′⟩ (5.78)

where gRR denotes the GREEN’s function of lead R, see Eq. (5.62). From Eq.(5.32)

Hnn′ =⟨n∣∣HR

0

∣∣n′⟩ =

H0 n′ = n,

V0 n′ = n+ 1,

V †0 n′ = n− 1,

0 else,

(5.79)

where we set Va = 0 for simplicity. The more general case is immediatelyobtained by replacing H0 → H0 − Va1, see Eq. (5.32). We obtain from[

(E + iη)1−HR0

]gRR = 1, (5.80)


the relations

[(E + iη)1−H0]G00 = 1 + V0G10, (5.81a)[(E + iη)1−H0]G10 = V †0 G00 + V0G20 (5.81b)

......

[(E + iη)1−H0]Gn0 = V †0 G(n−1)0 + V0G(n+1)0. (5.81c)

Solving (5.81c) for Gn0 yields

Gn0 = [(E + iη)1−H0]−1(V †0 G(n−1)0 + V0G(n+1)0

). (5.82)

Inserting the same expressions for G(n+1)0 and G(n−1)0 as functions of Gn0

and G(n+2)0 or Gn0 and G(n−2)0 (note that n ≥ 2), respectively, into Eq. (5.82)yields

AGn0 = V0 [(E + iη)1−H0]−1 V0G(n+2)0

+V †0 [(E + iη)1−H0]−1 V †0 G(n−2)0. (5.83)

where we use

A = (E + iη)1−H0 − V0 [(E + iη)1−H0]−1 V †0 − V†

0 [(E + iη)1−H0]−1 V0.(5.84)

We defineα1 = V0 [(E + iη)1−H0]−1 V0, (5.85a)

β1 = V †0 [(E + iη)1−H0]−1 V †0 , (5.85b)

εs1 = H0 + V0 [(E + iη)1−H0]−1 V †0 , (5.85c)

and

ε1 = H0 + V0 [(E + iη)1−H0]−1 V †0 + V †0 [(E + iη)1−H0]−1 V0. (5.85d)

Hence,[(E + iη)1− εs1]G00 = 1 + α1G20, (5.86a)

[(E + iη)1− ε1]Gn0 = β1G(n−2)0 + α1G(n+2)0, (5.86b)

[(E + iη)1− ε1]Gnn = 1 + β1G(n−2)n + α1G(n+2)n, (5.86c)

where we employ that

[(E + iη)1−H0]Gnn = 1 + V †0 G(n−1)n + V0G(n+1)n

= 1 + β1G(n−2)n + (ε1 −H0)Gnn

+α1G(n+2)n. (5.87)

74 5.5. THE SURFACE GREEN’S FUNCTIONS

These equations couple second nearest-neighbors of the original chain as ifthey were nearest-neighbors with effective hopping matrices α1 and β1. Wecan repeat this argument and obtain an effective interaction between i-thnearest-neighbors. The corresponding equations take on the form

αi = αi−1 [(E + iη)1− εi−1]−1 αi−1, (5.88a)βi = βi−1 [(E + iη)1− εi−1]−1 βi−1, (5.88b)εi = εi−1 + αi−1 [(E + iη)1− εi−1]−1 βi−1

+βi−1 [(E + iη)1− εi−1]−1 αi−1, (5.88c)

andεsi = εsi−1 + αi−1 [(E + iη)1− εi−1]−1 βi−1. (5.88d)

The strategy is clear: we increase the number of layers coupled to the in-terface iteratively until αi and βi are sufficiently small, i.e. |εsi − εsi−1|M < δ,where δ > 0 is the required accuracy and | · |M is some matrix norm. The un-perturbed GREEN’s function of the surface between lead R and the systemS, GRR , consequently reads

GRR ≡ G00 ≈ [E1− εsi ]−1 , (5.89)

where η → 0+ has been carried out. We obtain the surface GREEN’s functionfor lead, L, in a similar fashion by interchanging αi and βi, i.e.

GRL ≈ [E1− εsi ]−1 , (5.90)

where εsi is iterated as

εsi = εsi−1 + βi−1 [(E + iη)1− εi−1]−1 αi−1. (5.91)

The iteration is initialized by setting ε0 = εs0 = εs0 = H0, α0 = V0 and β0 = V †0 .

Chapter 6

SOME REMARKS

We apply the methods presented in Chaps. 3 - 5 to the particular scenarioof CrAs / GaAs heterostructures. In particular, we investigate spin-filteringin these heterostructures in Refs. [14] and [15] and Chaps. 7 and 8. Fur-thermore, in Ref. [16] and Chap. 9 we discuss a possible solution to theinconsistency, which arises from the treatment of the interface within theVCA. Let us briefly summarize these three studies.

The investigation of spin-filtering in CrAs / GaAs heterostructures isbased on the ab-initio electronic structure of CrAs as obtained with the helpof the LMTO method in Chap. 3. It is demonstrated that the electronic struc-ture of CrAs is stable with respect to moderate longitudinal distortions ofthe bulk CrAs unit cell, see Sec. 3.4, and that it is, therefore, a legitimateapproach to assume that CrAs is cubic with aCrAs = aGaAs. Based on thisobservation, the band structures of CrAs / GaAs supercells are determinedin order to calculate the band offset between CrAs and GaAs. The bandoffset is predicted to be in the range of 0.5 eV for majority spin and 0.6 eVfor minority spin with a generous error estimate of ±0.2 eV. Nevertheless,whether or not CrAs / GaAs heterostructures operate as spin-filters is notaffected by corrections of the band offset within 0.2. eV due to the particularband alignment. Moreover, it is an important result that half-metalicity ofthe heterostructure is not required for efficient spin-filtering to persist be-cause of the particular band lineup. On the basis of these results we predictthat CrAs / GaAs heterostructures might act as efficient spin-filtering de-vices. In order to quantify this statement we calculate the non-equilibriumcurrent through a device consisting of several atomic layers of CrAs embed-

75

76

ded in a GaAs matrix. The ETB parameters are obtained via a fit to the ab-initio electronic structures, where all As onsite energies have to have equalvalues in order to take care of the interface identification problem withinthe VCA, see Sec. 4.5. The voltage is assumed to drop linearly from the leftto the right lead. The non-linear current response is then calculated withina non-equilibrium GREEN’s function approach, see Sec. 5.3. We determinethe current-voltage (I-V) characteristics for majority and minority spin fordifferent layer thicknesses of CrAs and different temperatures. Our theo-retical results clearly predict that room-temperature spin-filtering should beachievable with CrAs / GaAs heterostructures with a current spin polariza-tion of up to 97 %.

Within Ref. [16] and Chap. 9 we point out the inconsistency of the VCAwithin the ETB theory by utilizing a simple one-dimensional two-orbital TBtoy-model. It is emphasized that the electronic structure is invariant undercertain parameter symmetries of the ETB parameters. This symmetries areclearly violated by the standard application of the VCA. We discuss a pos-sible solution to this inconsistency by presenting two interfacial matchingconditions which are familiar from continuous space quantum mechanics.The matching conditions are motivated by the requirement of current con-tinuity, however, cannot be derived on a rigorous level. In the limit k → 0an effective mass model becomes appropriate to calculate the transmissionthrough the interface and the matching conditions are exact. It is stressedthat these matching conditions provide a possible solution only if one in-channel and one out-channel are present. For more in- and/or out-channelsfurther assumptions, which are discussed in some detail, have to be in-voked.

Chapter 7

SPIN INJECTION AND FILTERING IN HALF-METAL / SEMICONDUCTOR (CRAS / GAAS)HETEROSTRUCTURES

B.A. Stickler, C. Ertler, L. Chioncel, E. Arrigoni and W. Pötz

published as a proceeding to the 31st International Conference on thePhysics of Semiconductors (ICPS), Zurich, 29th July to 3rd August, 2012.

Abstract. Theoretical investigations of spin-dependent transport in GaAs/ CrAs / GaAs halfmetal-semiconductor heterostructures indicate that thissystem is a candidate for an efficient room temperature spin injector andfilter. The spin dependent electronic structure of zinc-blende CrAs and theband offset between GaAs and CrAs are determined by ab-initio calcula-tions within the method of linear muffin tin orbitals (LMTO). This bandstructure is mapped onto an effective sp3d5s∗ nearest neighbor tight-binding(TB) Hamiltonian and the steady-state transport characteristic is calculatedwithin a non-equilibrium GREEN’s function approach. Even at room tem-perature we find current spin polarizations up to 97%.

7.1 Introduction

Theoretical investigations of bulk CrAs by means of density functional the-ory (DFT) have revealed that, as a bulk material, CrAs crystallizes in an

77

78 Chapter 7: PROCEEDINGS ICPS - ZURICH, 2012.

orthorhombic structure and shows antiferromagnetic behavior [25]. Inter-estingly, when grown epitaxially on GaAs, CrAs can crystallize in the zinc-blende (ZB) structure showing half-metallic ferromagnetic behavior, i.e., 100% spin polarization at the FERMI energy, with a CURIE temperature wellabove room temperature [26]. These findings suggest that CrAs is a po-tential candidate for the realization of room-temperature spin injection intosemiconductors. Here we study the current-voltage (I-V) characteristics ofa CrAs sheet consisting of 10 atomic layers sandwiched between two GaAsleads. Our theoretical results, founded upon the assumption of clean het-erointerfaces, support the hypothesis that GaAs/CrAs heterostructures actas efficient room temperature spin filters.

7.2 Theoretical Approach and Results

The theoretical approach consists of three major steps: (i) the calculation ofthe ab-initio electronic structure of the materials involved and the determi-nation of the band offset between them, (ii) the mapping of the electronicstructure onto an effective sp3d5s∗ nearest-neighbor TB model, and (iii) thecalculation of the I-V characteristic within a non-equilibrium GREEN’s func-tion approach.

(i) Electronic Structure and Band Offset

The electronic structure of bulk ZB GaAs, bulk ZB CrAs, and a (GaAs)6 /(CrAs)6 supercell was calculated with the LMTO-ASA code as developedby JEPSEN and ANDERSEN [5]. Based on these calculations we determinethe [1,0,0] band offset ∆ between bulk GaAs and bulk CrAs by using thealmost flat, deep lying As-s band as the reference state [6]. With respect tothe center of mass of this band we obtain ∆1 = 0.601 and ∆2 = 0.545 eV forthe majority and minority spin.

(ii) Effective sp3d5s∗ Tight-Binding Model

The energy bands of GaAs and CrAs which, according to the ab-inito bandstructure calculation, contribute to transport are fitted within a nearest-neigh-bor sp3d5s∗ TB Hamiltonian via minimization of the cost functional

K(ξ) =

√∑nk

an(k)|εLSDAn (k)− εTBn (k, ξ)|2, (7.1)

Chapter 7: PROCEEDINGS ICPS - ZURICH, 2012. 79

L G X W L K G−1

0

1

2

3

4

E / e

V

k

Figure 7.1: (Color online) Majority spin CrAs ab-initio band structure (redsolid line), CrAs TB-fit (green dashed line), and GaAs TB fit (blue dash-dotted line). The Fermi energy EF = 0.01 eV above the conduction bandminimum of GaAs is indicated by the horizontal solid line.

using a genetic algorithm. Here, ξ are the TB-parameters, n denotes theband index, k is the k-vector, an(k) are normalized weights and εLSDAn (k)and εTBn (k, ξ) denote the ab-initio and the TB band structure, respectively.Our results for the ab-initio electronic structure of majority and minorityspin CrAs, the respective fits, and the fitted band structure of GaAs are plot-ted in Figs. 7.1 and 7.2.

(iii) Steady State Transport

An effective spin-dependent single-particle Hamiltonian of the GaAs / CrAs/ GaAs structure is set up by performing a partial WANNIER transform ofthe basis states from the wave-vector k to (x, k‖) [11], where x denotes thegrowth direction of the crystal and k‖ is the in-plane momentum.

The transmission function under bias is calculated from the system’sretarded (R) and advanced (A) GREEN’s functions GR/A and the couplingfunctions ΓL/R functions of the left (L) and right (R) lead, respectively,

T (E, k‖) = Tr[ΓRG

RΓLGA], (7.2)

whereE is the total energy and Tr [·] denotes the trace operation. The steady

80 Chapter 7: PROCEEDINGS ICPS - ZURICH, 2012.

L G X W L K G−1

0

1

2

3

4

E / e

V

k

Figure 7.2: (Color online) Minority spin CrAs ab-initio band structure (redsolid line), CrAs TB fit (green dashed line) and GaAs TB fit (blue dash-dotted line). The Fermi energy is indicated by the horizontal solid line.

state current J is then calculated by the "LANDAUER - BÜTTIKER" formula

J =e

h

∑k‖

∫dE T (E, k‖) [fL(E)− fR(E)] . (7.3)

Here, e is the elementary charge, h is PLANCK’s constant, and fL/R de-note the FERMI - DIRAC distributions of the left and right lead, respectively,which provide the sole temperature dependence in our model. The I-V char-acteristics of the GaAs / CrAs / GaAs heterostructure for several temper-atures is shown in Fig. 7.3 , whereby a linear voltage drop from the left tothe right lead is assumed. Even at room temperature the current of one spinspecies remains strongly suppressed compared to the other’s, which leadsto the desired spin filtering and spin injection into the right GaAs lead.

A more detailed investigation of spin filtering in CrAs/GaAs heterostruc-tures and its dependence on the modeling of the interfaces will be given inthe near future.

Chapter 7: PROCEEDINGS ICPS - ZURICH, 2012. 81

0 0.05 0.1 0.15 0.20

0.5

1

1.5

2

2.5x 10

9

V / V

j /

A m

−2

spin 1: 4.2 K

spin 1: 77 K

spin 1: 300 K

spin 2: 4.2 K

spin 2: 77 K

spin 2: 300 K

Figure 7.3: (Color online) Spin-resolved current-voltage characteristics fordifferent temperatures (T = 4.2 K, T = 77 K, and T = 300 K).

Ackowledgments. This work has been supported by the Austrian ScienceFund (FWF) P21289-N16.

Chapter 8

THEORETICAL INVESTIGATION OF SPIN -FILTERING IN CRAS / GAAS HETEROSTRUC-TURES

B.A. Stickler, C. Ertler, L. Chioncel and W. Pötz

draft version,uploaded to the ARXIV: 1301.2933 [condens-mat.mes-hall], 2013.

Abstract. The electronic structure of bulk fcc GaAs, fcc and tetragonal CrAs,and CrAs/GaAs supercells, computed within LMTO local spin-density func-tional theory, is used to extract the band alignment (band offset) for the[1,0,0] GaAs/CrAs interface in dependence of the spin orientation. Withthe lateral lattice constant fixed to the experimental bulk GaAs value, a lo-cal energy minimum is found for a tetragonal CrAs unit cell with a slightly(≈ 2 %) reduced longitudinal ([1,0,0]) lattice constant. Due to the identifiedspin-dependent band alignment, half-metallicity of CrAs no longer is a keyrequirement for spin-filtering. Encouraged by these findings, we study thespin-dependent tunneling current in [1,0,0] GaAs/CrAs/GaAs heterostruc-tures within the non-equilibrium GREEN’s function approach for an effec-tive tight-binding Hamiltonian derived from the LMTO electronic structure.Results indicate that these heterostructures are probable candidates for effi-cient room-temperature all-semiconductor spin- filtering devices.

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84 Chapter 8: ARXIV: 1301.2933 [CONDENS-MAT.MES-HALL], 2013.

8.1 Introduction

The identification and design of high-efficiency all-semiconductorspin-filtering devices which operate at room temperature and zero exter-nal magnetic field are of profound interest for spintronic applications [2, 3].In spite of some success, for example regarding improved spin injectionfrom ferromagnets into Si, progress has been moderate. Diluted magneticsemiconductors, such as GaMnAs, which have been grown most success-fully with good interfacial quality onto nonmagnetic fcc semiconductors,generally, have critical temperatures far below room temperature and arehampered by disorder [75, 76, 77]. Growing high-quality GaMnAs layerswithin heterostructures proves to be more difficult than in the bulk. Signa-tures of weak quantum confinement effects associated with GaMnAs quan-tum wells in GaAlAs / GaMnAs single- and double barrier heterostructureshave been reported in the literature [78]. Using one Ga0.96Mn0.04As contactlayer on an asymmetric GaAlAs / GaMnAs heterostructure magnetization-dependent negative-differential-resistance has been observed [79]. A sim-ilar TMR experiment has been performed recently, in which resonant tun-neling in non-magnetic AlGaAs / GaAs / AlGaAs was used to control thehole current [80].

It is an experimental fact that bulk MAs or bulk MSb compounds, withM denoting a transition metal, such as V, Cr, and Mn, in the bulk, do nothave their ground state in the fcc phase. Even as thin layers on an fcc sub-strate, such as GaAs, they appear to be difficult to grow in a lattice-matchedform [81, 82, 25, 27]. Important exceptions are the reports of the experimen-tal realization of fcc MnAs quantum dots on GaAs and the hetero-epitaxialgrowth of thin layers of CrAs on GaAs substrates [83, 26, 84, 28, 28, 27].Recent experiments have suggested that CrAs can be grown epitaxially inthe fcc structure on top of GaAs and displays ferromagnetic behavior wellabove room temperature [26, 84, 85]. Ab-initio studies of the system haverevealed that fcc CrAs is a half-metallic ferromagnet and have led to theprediction that the CURIE temperature of fcc CrAs may be as high as 1000K [30, 31, 27, 32]. The electronic structure of GaAs / CrAs heterostructuresand transport properties through GaAs/CrAs/GaAs tri-layers have beenstudied by BENGONE et al. [33].

Recently the effect of electronic correlations upon the half-metallicity ofstacked short period (CrAs)`/(GaAs)` (` ≤ 3) superlattices along [001] hasbeen investigated. Results indicate that the minority spin half-metallic gapis suppressed by local correlations at finite temperatures and continuouslyshrinks on increasing the heterostructure period. As a consequence, at the

Chapter 8: ARXIV: 1301.2933 [CONDENS-MAT.MES-HALL], 2013. 85

FERMI level, the polarization is significantly reduced, while dynamic cor-relations produce only a small deviation in magnetization [34]. Essentiallythe same effect was found for defect-free digital magnetic heterostructuresδ-doped with Cr and Mn [86]. In addition both studies indicate that the mi-nority spin highest valence states below the FERMI level are localized morein the GaAs layers while the lowest conduction band states have a many-body origin derived from CrAs. Therefore independent of the presence ofelectronic correlations in these heterostructures holes and electrons may re-main separated among different layers which may be detected in photo-absorption measurements.

Another density-functional theory (DFT) based calculation, as well as,an experimental report have lead to the claim that the fcc-structure of thinfilm CrAs is energetically unstable [35, 25]. It was argued that the exper-imentally observed ferromagnetic behavior reported in Refs. [26, 84, 85]may be caused by magnetic defects near the heterointerface and that nohalf-metallicity may be present at all. An additional problem to our knowl-edge, not studied in detail as of yet, may be the uncontrollable diffusionof Cr into GaAs thereby forming deep traps [87]. It has to be noted thatthe final answer to the structural properties and stability of thin films ofCrAs on top of GaAs, as well as CrAs/GaAs heterostructures, can only begiven by or in conjunction with further experiments. It is indeed the aimof this article to stimulate these experimental studies by demonstrating thatepitaxial CrAs/GaAs heterostructures should function as very efficient all-semiconductor room-temperature spin-filtering devices.

Exploring spin-filtering in GaAs - CrAs - GaAs heterostructures, firstwe investigate the stability of the CrAs unit cell under longitudinal distor-tion. It is a central result of this work that a tetragonal, local-minimum bulkphase of CrAs can be found which is very near to being fcc lattice matchedto bulk GaAs. This implies that one may perform a transport calculationsunder the assumption of perfect fcc lattice matching without introducing alarge systematic error. The band alignment is determined with the help oftwo different approaches, which both yield rather similar results. This pro-vides a reasonable error estimate for the method employed, and leads us tothe conclusion that half-metallicity is not a necessary ingredient for efficientspin-filtering. Rather, the interfacial properties and the spin-selective bandalignment between the CrAs and GaAs layers appear to be essential. Wemap the electronic structure of the bulk materials onto an effective 20-orbitalsp3d5s∗ nearest-neighbor empirical tight-binding (ETB) model, which is par-ticularly suited for non-equilibrium transport calculations [13, 11, 73, 72] .In a final step we calculate the spin-selective current-voltage (I-V) character-


istics within the non-equilibrium GREEN’s function formalism for differentlayer thicknesses and temperatures.

This article is structured as follows. In Sec. 8.2 we summarize the the-oretical approach and present the results for the electronic structure withinthe LMTO method. In Sec. 8.3 we describe the mapping procedure onto thetight-binding model. The transport model and results for the spin-currentresponse are discussed in Sec. 8.4. Summary and conclusions are given inSec. 8.5.

8.2 LMTO Electronic Structure Calculations andthe fcc CrAs / fcc GaAs [1,0,0] Band Offset

The electronic structures of bulk zinc-blend (ZB) GaAs, bulk ZB and tetrag-onal CrAs, as well as that of several CrAs / GaAs supercells, have beendetermined employing the LMTO-ASA code, as developed by JEPSEN andANDERSEN [5, 4]. This code has been used previously to explore the elec-tronic structure of bulk ZB CrAs and thin-layer fcc GaAs / CrAs superlat-tices. In particular the half-metallic behavior as a function of superlatticeperiod and lattice constants has been investigated [88, 34]. Details of thisapproach, as well as its benefits and caveats, can be found in the literature[4].

The electronic structure model is based on LSDA omitting spin-orbit in-teraction and corrections for strong correlations as provided, for example,by the dynamic-mean-field-theory (DMFT) [51] or the variational cluster ap-proach (VCA) [53, 52]. However, it has been shown by CHIONCEL et al.[88, 34] that the inclusion of correlations does not affect the magnetizationand only leads to minor corrections to the band structure. It is well knownthat LSDA or its gradient-corrected approximation (GGA) produce bandgaps that are typically at least 30% smaller than the experimental values foralmost all semiconductors and insulators. In our case, too, the experimen-tally verified GaAs band gap of 1.52 eV at low temperatures is strongly un-derestimated by the present spin-DFT method, predicting a value of about0.35 eV. However, the overall features in the vicinity of the band gap are re-produced reasonably well. Since we do not study transport across the mainenergy gap, such as in ZENER tunneling, this shortcoming is without anyfurther disadvantage. Therefore, in what follows we consider two distinctcases: (i) we leave the electronic structure unchanged, i.e. we use the GaAsband structure with the underestimated band gap, and (ii) we scissor theGaAs electronic structure to the experimentally observed value of 1.52 eV.


As we shall see, this does not affect the statement that CrAs / GaAs het-erostructures function as spin-filters. Furthermore, we consider n-dopedGaAs such that the spin-orbit effects, at least in GaAs, can be neglected. Inthe case of half-metals one may speculate that the problems of LSDA arenot as serious, since the dielectric response of half-metals is of metallic type.Previous results have shown that the detailed nature of states around EFin bulk CrAs is changed when different lattice constants or models for elec-tronic correlations are considered, while the overall features of the electronicstructure are preserved [86]. Clearly the final answer as to the accuracy ofthe electronic structure model can only be given by comparison to experi-ment.

Since the GaAs band structure is well known and, except for the bandgap, is fairly well reproduced within the LMTO-ASA code, here we confineourselves to results obtained for CrAs. In a first approach we assume thatthe fcc CrAs lattice constant is equal to the GaAs lattice constant aGaAs = 5.65Å. This choice is motivated by the hope that thin layers of CrAs can begrown onto a GaAs substrate in lattice-matched fashion. The electronicstructure for majority and minority carriers is shown, respectively, in Figs.8.4 and 8.5 (red solid lines). For the minority carriers a gap of about 1.8eV is predicted. A more detailed analysis of the electronic structure showsthe origin of the valence band edge in the As-p - Cr-d hybrid orbitals, whileconduction band edge states are dominated by the Cr-d orbitals, see CHION-CEL et al. [88, 34]. Further, we observe from the ab-initio calculations thatthe lattice constant chosen for bulk fcc CrAs determines whether CrAs ishalf-metallic or not, see also [88, 34]. Since the bulk band structures of GaAsand CrAs combined in a heterostructure are shifted relative to each otherby the band offset, as discussed below, we are led to the conclusion thathalf-metallicity of the CrAs layer is not compulsory for the realization of aspin-filter. A necessary ingredient is a highly spin-dependent CrAs bandstructure.

Moreover, we investigate the total energy of the bulk CrAs unit cell asa function of the longitudinal, i.e. the out-of-the-plane, lattice constant a⊥while leaving the in-plane lattice constant a|| fixed thus making the unit celltetragonal. It is reasonable to assume that, for thin lattice-matched layersof CrAs on top of GaAs, the in-plane lattice constant a|| = aGaAs takes onthe value of bulk GaAs, i.e. that the in-plane symmetry is preserved. It isfound, that the minimal energy is achieved for 98 % of the GaAs lattice con-stant, as depicted in Fig. 8.1, i.e. a⊥ = 0.98a||. Hence, the actual unit cell ofCrAs on the GaAs [1,0,0] surface is not perfectly cubic but tetragonal. Themere existence of such a (local) minimum close to the GaAs lattice constant


0.95 1 1.05 1.1−0.45

−0.445

−0.44

−0.435

−0.43

a⊥ / a||

E / a

rb. units

Figure 8.1: (Color online) Total energy of the bulk CrAs unit cell as a func-tion of lateral lattice constant a⊥/a||.

is non-trivial and an important result and motivation for the spin transportanalysis to follow. Moreover, a comparison of the bands of fcc CrAs with thebands of tetragonal CrAs with a⊥ = 0.98a|| reveals that the electronic struc-ture is almost unaffected by this distortion, see Figs. 8.2 and 8.3. Thereforethe systematic error introduced by assuming that the CrAs unit cell is per-fectly cubic with a|| = a⊥ ≡ aCrAs = aGaAs is negligible when compared toother uncertainties. We shall from now on assume that bulk CrAs (on [1,0,0]GaAs) has the fcc crystal structure.

In the spirit of core level spectroscopy, the band offset at the [1,0,0] GaAs/ CrAs interface was determined using the energy of a low-lying referencestate that is present in both bulk materials, as well as in the CrAs / GaAssupercell. We chose two different reference states and compared the valuesobtained with each of them in order to have a reasonable error estimate forthe method. The band offset ∆ is calculated according to [6]

∆ = ∆G −∆C + ∆GG −∆CC . (8.1)

In the first evaluation, we use the low-lying As-4s bands for our refer-ence states. We compute the center of energy of these bands for bulk fccCrAs ∆C , bulk fcc GaAs ∆G, as well as for the As atoms embedded in theGa and Cr environment in a (GaAs)6(CrAs)6 supercell ∆GG and ∆CC . Allenergies are taken relative to the respective FERMI energy. The center of en-ergy of one particular band is obtained by calculating the fat-band weights


Z G X W L−5

−4

−3

−2

−1

0

1

2

3

4

5

E / e

V

k

Figure 8.2: (Color online) Ab-initio electronic structure of majority spinCrAs for the fcc (green solid line) and the tetragonal unit cell (black dashedline). The FERMI energy is indicated by a solid blue line.

Z G X W L−5

−4

−3

−2

−1

0

1

2

3

4

5

E / e

V

k

Figure 8.3: (Color online) Ab-initio electronic structure of minority spinCrAs for the fcc (green solid line) and the tetragonal unit cell (black dashedline). The FERMI energy is indicated by a solid blue line.


L G X W L K G−1

0

1

2

3

E / e

V

k

Figure 8.4: (Color online) Majority spin CrAs ab-initio band structure (redsolid line), scissored, and un-scissored GaAs band structure (blue-dashed,respectively, green-dash-dotted line). The FERMI energy EF = 0.01 eVabove the conduction band minimum of GaAs is indicated by the horizontallines (solid: scissored GaAs, dashed: un-scissored GaAs).

with the help of the ab-initio LMTO-ASA code and using them for a weightfactor when performing the average over the first BRILLOUIN zone. Thesecond way of calculating the band offset is based on including the As-3dorbitals into the self-consistent loop of the ab-initio calculation. The refer-ence state is then taken as the center of band as obtained from the LMTO-ASA code [5]. In both cases we obtain a band offset ∆ in the range of∆1 = 0.6 ± 0.2 eV, for majority spin, and a band offset of ∆2 = 0.5 ± 0.2eV, for minority spin bands. Although this error may seem considerable, inour calculations, it is of little significance as to the performance of CrAs /GaAs heterostructures as a spin-filter. In the calculations below we chose∆1 = 0.60 eV and ∆2 = 0.55 eV, respectively, for majority and minority spin,as obtained within the method based on the As-4s reference states.

In Figs. 8.4 and 8.5 we show, on top of the electronic structure of majority-and minority-spin CrAs, the electronic structure of GaAs for case (i) (un-scissored gap) and case (ii) (scissored gap) taking into account the band off-set. It can now clearly be seen that carriers injected from n-GaAs near theΓ-point do not reach the main energy gap region of bulk CrAs. Even in caseof p-doping of the GaAs contact layers, this region can only be reached un-


L G X W L K G−1

0

1

2

3

E / e

V

k

Figure 8.5: (Color online) Minority spin CrAs ab-initio band structure (redsolid line) and scissored as well as un-scissored GaAs band structure (blue-dashed, respectively, green-dash-dotted line). The FERMI energy EF = 0.01eV above the conduction band minimum of GaAs is indicated by the hori-zontal lines (solid: scissored GaAs, dashed: un-scissored GaAs).


der an applied bias exceeding approximately 0.3 V, as can be observed froman inspection of Fig. 8.5. Moreover, scissoring of the GaAs band gap doesnot significantly change the spectral overlap of the lowest GaAs conductionband with the CrAs bands near the Γ point. The situation is rather differentfor minority carriers, however. Here the overlap of the lowest GaAs conduc-tion band occurs, in both cases, with fairly flat CrAs bands of low mobility.Hence, the actual value of the band offset is not too important which al-lows the claim that the presence of half-metallicity in the CrAs layer is notmandatory for spin-filtering in CrAs / GaAs heterostructures. In view ofthese results and the broad-band form of the electronic structure of CrAsin the relevant energy window, in particular, it appears that spin-filteringis rather stable with respect to moderate modifications to the value of theband offsets arising, for example, from interfacial defects or local lattice dis-tortions.

8.3 Effective sp3d5s∗ Empirical Tight-Binding Model

The results obtained within the LMTO electronic structure investigation en-courage one to go one step further and to investigate spin-selectivity inthe electric current for CrAs/GaAs heterostructures under bias. Since theLMTO calculations are based on thermal equilibrium we first map the rel-evant segments of the electronic structure onto an empirical tight-binding(ETB) model on which we then base the non-equilibrium transport study.Specifically, each of the ab-initio electronic structure εLSDAn of GaAs andof majority- and minority-spin CrAs are mapped onto an effective sp3d5s∗

nearest-neighbor ETB model [10, 8, 67]. We chose an ETB model because itis particularly well-suited for non-equilibrium steady state transport calcu-lations with the help of GREEN’s functions [13, 11, 72, 73].

In principle, this step can be avoided if the LSDA wave functions wereused to express the transmission function of the heterostructure, as for ex-ample proposed within the SIESTA DFT approach [40]. The trade-off of anapproach which is based on the L(S)DA one-particle wave functions, how-ever, is that its validity ad-hoc is questionable, since the wave functions usedin the KOHN-SHAM variational principle do not allow a direct physical in-terpretation. Their connection to the S-matrix of the (many-body) system isnot obvious. The use of ground state wave-functions definitely limits oneto the linear-response regime, since the transmission function would be ob-tained for zero external bias only. In general, L(S)DA bulk band structurecalculations do not produce the correct energy gap. A simple scissoringstrategy cannot be performed for a heterostructure, thus, any deficiencies


in the ab-intio electronic structure are carried over inevitably into the trans-port calculation. These difficulties have convinced us to follow a mappingapproach from the ab-initio band structure calculation to an empirical tight-binding model and to take care of the interface problem in an additionalstep.

A mapping of the electronic structure onto an ETB model does not, in thefirst place, introduce any further systematic error, as long as the electronicstructure in the relevant energy region is well approximated. Moreover, thisapproach comes with further benefits, such as computational effectiveness.It does not only allow one to account for known deficiencies of the elec-tronic structure within SDFT, such as a scissoring of the GaAs main bandgap to the experimentally verified value, but also to control the particularmodeling of the interface (e.g. by the inclusion of defects) and to treat agenuine non-equilibrium situation self-consistently. Bias-dependent mean-field corrections or disorder effects can be added in self-consistent fashion,as utilized by some of us recently for similar systems [89, 90].

The formal mapping process for a given bulk material is executed us-ing a genetic algorithm, as implemented in MATLAB, to minimize the costfunctional

K(ξ) =

√∑nk

an(k) [εLSDAn (k)− εETBn (k, ξ)]2 . (8.2)

Here, an(k) are normalized weights where n is the band index and k thewave vector, ξ denotes the set of 31 independent ETB parameters in thesp3d5s∗ basis [10] and εETBn (k, ξ) is the ETB band structure as a function of kand ξ. The weights an(k) are used to restrict and/or focus the fit to the partof the band structure which contributes to charge transport. This ensuresthat the energy bands are well represented by the ETB fit and no "spuriousbands" appear inside a chosen energy window.

Subsequently, the Hamiltonian of a given heterostructure can be put to-gether layer by layer in a straight-forward fashion. This convenient layer-by-layer construction can be carried over to the construction of the non-equilibrium GREEN’s function components and has been used in the cal-culations below. A further advantage is that we know the k-dependenceof the bulk ETB Hamiltonian matrices analytically via the structure factors[10]. Hence, the influence of small deviations in the ETB binding parameterscan be investigated systematically.

One problem, however, arises when the interface between GaAs andCrAs has to be modeled in this modular approach [16].1 Let us briefly dis-

1Note that the SIESTA DFT approach, too, employs an ad-hoc approximation at the


cuss the problem of how to model the interface in an adequate way, consis-tent with the available information, which are the ETB bulk Hamiltoniansand the band offsets. The standard approach is to invoke the virtual crystalapproximation (VCA) [11, 12, 13, 38, 70, 71, 72]. It has been demonstratedthat this approximation is, in general, inconsistent since it violates symme-tries underlying the bulk ETB Hamiltonians [16]. In particular, the ETB pa-rameters are not uniquely defined by the electronic structure alone, how-ever, the VCA requires a unique identification of the ETB parameters [16].There are two possible remedies to this apparent inconsistency. First, wecan formulate matching conditions which are a discrete form of the matchingconditions in continuous space quantum mechanics. This approach worksfor pairs of band-to-band transitions and has been discussed extensively bySTICKLER and PÖTZ [16]. Here, in this more complex situation, we shallfollow an alternative approach made possible by the presence of As as acommon anion: we can ensure that the ETB parameters can be uniquely at-tributed to one atomic species by posing a further constraint onto the fittingprocedure Eq. (8.2): We require the As onsite energies (under zero bias andband offset ) to be constant throughout the device. This is in accordancewith ETB theory [10]. Hence, the mapping onto a ETB model is executed intwo steps: In a first step we fit the majority and minority spin band struc-ture of CrAs independently without any further restrictions. In a secondstep we fit the GaAs band structure under the constraint that all As onsiteenergies have to have the same value as in CrAs. Let us briefly discuss theimplications of such a fitting procedure: First of all, we obtain two differentsets of ETB parameters for GaAs since we fit the two CrAs band structuresindependently and then restrict the GaAs parameters depending on spinorientation. Had we included the spin-orbit interaction into our model, twodistinct sets of ETB parameters for GaAs would come more natural, how-ever, in any case the values obtained must be considered best fits undergiven constraints. Moreover, we emphasize that the particular form of theETB parameters has no influence on the transport as long as the electronicstructure including the band offset is reproduced reasonably well [16].

All in all we perform the down-folding process for four different combi-nations: (A) majority-spin CrAs and scissored GaAs, (B) majority-spin CrAsand un-scissored GaAs, (C) minority-spin CrAs and scissored GaAs, and(D) minority-spin CrAs and un-scissored GaAs. The ETB parameters whichwere identified as optimal for each of the four cases are listed in the Ap-pendix. The fits for GaAs together with the ab-initio band structure and thefits of majority- and minority-spin CrAs are given in Figs. 8.6, 8.7, 8.8 and

interface which may be regarded as rather questionable [40].


L G X W L K G−1

0

1

2

3

4

E / e

V

k

Figure 8.6: (Color online) Scenario (A): majority spin CrAs ab-initio bandstructure (red solid line), CrAs ETB-fit (blue dashed line), scissored GaAsband structure (green solid line) and GaAs ETB fit (black dashed line). TheFERMI energy EF = 0.01 eV above the conduction band minimum of GaAsis indicated by the horizontal solid line.

8.9. In view of the fact that the computed ab-initio electronic structure willat best capture the overall features of the actual electronic band structurethe ETB fits achieved are highly satisfactory.

In principle, the spin-orbit interaction can be included in the ETB modelfollowing the work of CHADI [62]. However, since the LMTO-ASA codeitself currently does not feature spin-orbit interactions, its implementationat the ETB level would require the introduction of further (and somewhatarbitrary) parameters into our model, in particular for CrAs. Furthermore,we shall focus on transport with n-doped GaAs buffer layers so that a de-tailed account of the spin-orbit interaction in the GaAs electronic structurewill not really be important here.

8.4 Steady-State Transport

The I-V-characteristics of the heterostructure is calculated within a non-equilibrium GREEN’s function approach which has been adapted from ourrecent study of GaMnAs-based heterostructures, to which we refer for fur-ther details and references [89, 90, 13, 11]. The Hamiltonian of the GaAs -


L G X W L K G−1

0

1

2

3

4

E / e

V

k

Figure 8.7: (Color online) Scenario (B): majority spin CrAs ab-initio bandstructure (red solid line), CrAs ETB-fit (blue dashed line), un-scissored GaAsband structure (green solid line) and GaAs ETB fit (black dashed line). TheFERMI energy EF = 0.01 eV above the conduction band minimum of GaAsis indicated by the horizontal solid line.


L G X W L K G−1

0

1

2

3

4

E / e

V

k

Figure 8.8: (Color online) Scenario (C): minority spin CrAs ab-initio bandstructure (red solid line), CrAs ETB-fit (blue dashed line), scissored GaAsband structure (green solid line) and GaAs ETB fit (black dashed line). TheFERMI energy EF = 0.01 eV above the conduction band minimum of GaAsis indicated by the horizontal solid line.


L G X W L K G−1

0

1

2

3

4

E / e

V

k

Figure 8.9: (Color online) Scenario (D): minority spin CrAs ab-initio bandstructure (red solid line), CrAs ETB-fit (blue dashed line), un-scissored GaAsband structure (green solid line) and GaAs ETB fit (black dashed line). TheFERMI energy EF = 0.01 eV above the conduction band minimum of GaAsis indicated by the horizontal solid line.


CrAs - GaAs heterostructure is obtained by performing a partial WANNIERtransformation from the wave vector k to (x, k‖), where x denotes the [1,0,0]growth direction of the crystal and k‖ is the in-plane (parallel) k-vector [11],∣∣nbjk‖⟩ =

1

LBZ

∫dkx exp

(−ikxj

a⊥4

)|nbk〉 , (8.3)

where LBZ = 8πa⊥

with j labeling the layer. The resulting one-particle Hamil-tonian is of the single-particle form

H(k‖) =∑i,σ

ε(i)σ,σ(k‖)c

†i,σ(k‖)ci,σ(k‖)

+∑i,σσ′

t(i)σσ′(k‖)c

†i+1,σ(k‖)ci,σ′(k‖) + h.c., (8.4)

with c†i,σ(k‖) [ci,σ(k‖)] denoting the creation (annihilation) operator for sitei and orbital σ. ε

(i)σσ′(k‖) and t

(i)σσ′(k‖), respectively, are onsite and hopping

matrix elements. The semi-infinite GaAs “leads" are taken into account byevaluating the associated self-energies and feeding them into the system’sDYSON equation [73]. The surface GREEN’s functions are obtained with thehelp of an algorithm suggested by SANCHO et al. [74]. For each carrier type(majority and minority), the transmission function T (E, k‖) for total energyE and in-plane momentum k‖ is calculated via

T (E, k‖) = tr(ΓRG

RΓLGA). (8.5)

Here, GR/A are the system’s retarded (R) and advanced (A) GREEN’s func-tions, ΓL/R are the coupling functions to the left (L) and right (R) GaAs leadsand tr· is the trace operation. We then compute the steady-state currentj(Va) assuming local thermal equilibrium among the electrons injected froma particular contact using the standard expression from stationary scatteringtheory[73]

j(Va) =2e

h

∑k‖

∫dET (E, k‖) [fL(E)− fR(E)] , (8.6)

with e, h, and fL/R denoting, respectively, the elementary charge, PLANCK’sconstant, and the FERMI-DIRAC distribution function for the left and rightelectric contact. The applied voltage Va enters Eq. (8.6) in two places: thetransmission function T (E, k‖) and the difference in the quasi-FERMI lev-els between left and right contact. In order to cut computational cost, weassume a linear voltage drop from the left to the right lead across the simu-lated structure. This implies a somewhat artificial relationship between the


electric field across the structure and the applied bias [91, 92]. In principle,both an effective single-particle potential, an effective exchange splitting,and a self-consistent treatment of charge injection can be implemented intothe present model [90]. However, while providing a significant reductionin computation time, omission of self-consistency may not significantly re-duce the quality of our results. Note that the FERMI-DIRAC distributionsof the GaAs contacts provide the sole temperature dependence in the cur-rent model since a temperature dependence of the electronic structure is notconsidered here. For a discussion of the latter, we refer to recent work [34].

In what follows we present results for the I-V characteristics of GaAs /(CrAs)` / GaAs heterostructures for ` = 4, 6, 8, 10. While thin layers of fccCrAs may be easier to realize experimentally, thicker layers thereof are de-scribed more realistically within our approach. The free carrier density inthe n-doped GaAs regions is about 4.5 × 1017 cm−3 at T = 300 K (7.9 × 1016

cm−3 at T = 0 K), with the quasi-FERMI level held constant at 10 meV abovethe conduction band edge. The applied bias was varied between zero and0.2 V. Results for scissored and un-scissored GaAs, respectively, and ` = 10are shown in Figs. 8.10 and 8.11 (mind the semi-logarithmic plot). The over-all features of the I-V characteristics agree for both cases: the majority cur-rent density clearly dominates over the minority current density and this, inmost bias regions, by several orders of magnitude. However, spin-filteringis more pronounced for the scissored GaAs model. While the majority cur-rent is rather insensitive to scissoring, the minority current density is not(we believe that the small oscillations for the 77 K minority case near 1.5 Vin Fig. 8.10 are of numerical origin). The reason is found by inspection ofFigs. 8.4 and 8.5. It shows that, for un-scissored GaAs and low applied bias,there is a resonance between the GaAs conduction band minimum (dashedline) and CrAs-associated bands near the Γ-point. Near the Γ-point thesebands are rather flat and so the group velocity is almost zero. Under mod-erate bias, however, these bands are moved further into resonance (to re-gions with higher group velocity) with the conduction band of GaAs at theemitter side, and the minority current rises steeply with applied bias.

The current spin polarization P (Va) as a function of applied voltage Va isdefined as

P (Va) =

∣∣∣∣jmaj(Va)− jmin(Va)

jmaj(Va) + jmin(Va)

∣∣∣∣ . (8.7)

Here, jmaj/min refers to the majority and minority spin current density, re-spectively. In Figs. 8.12 and 8.13, respectively, we display the computedcurrent spin polarization for scissored and un-scissored GaAs and the threedifferent temperatures discussed above.


0 0.05 0.1 0.15 0.2

−2

0

2

4

6

8

10

Va (V)

log

10 j

(A m

−2)

4.2 K

77 K

300 K

Figure 8.10: (Color online) Spin-resolved current-voltage characteristics forGaAs/(CrAs)10/GaAs and case (A) and (C) (scissored GaAs) for differenttemperatures (T = 4.2 K, T = 77 K, and T = 300 K). The majority spin cur-rent clearly dominates the minority spin current for all voltages and tem-peratures.


0 0.05 0.1 0.15 0.20

2

4

6

8

10

Va (V)

log

10 j (

A m

−2)

4.2 K

77 K

300 K

Figure 8.11: (Color online) Spin-resolved current-voltage characteristics forGaAs/(CrAs)10/GaAs and case (B) and (D) (un-scissored GaAs) for differ-ent temperatures (T = 4.2 K, T = 77 K, and T = 300 K). The majority spincurrent clearly dominates the minority spin current for all voltages and tem-peratures.

0 0.05 0.1 0.15 0.20.97

0.975

0.98

0.985

0.99

0.995

1

1.005

1.01

Va (V)

P

4.2 K

77 K

300 K

Figure 8.12: (Color online) Current spin polarization P (Va) for scissoredGaAs for different temperatures (T = 4.2 K, T = 77 K, and T = 300 K).


0 0.05 0.1 0.15 0.20.4

0.5

0.6

0.7

0.8

0.9

1

Va (V)

P

4.2 K

77 K

300 K

Figure 8.13: (Color online) Current spin polarization P (Va) for un-scissoredGaAs for different temperatures (T = 4.2 K, T = 77 K, and T = 300 K).

In Fig. 8.14 we show the I-V characteristics as obtained for different lay-ers thicknesses, i.e. (GaAs)m / (CrAs)` / (GaAs)m, where ` = 4, 6, 8, 10 andm = 5. The results shown in Fig. 8.14 stem from simulations in which thenumber of layers of GaAs to the left and the right of CrAs was set to m = 5and kept constant, i.e. the electric field across the CrAs layer at a given volt-age increases with decreasing layer thickness. This trend follows the actualphysical trend within the device and that of a self-consistent model. More-over, we note that the actual form of the computed I-V characteristic is onlyslightly changed if our simulations are performed under equal-electric-fieldconditions, i.e. m = 10 − `

2because the form of the bands involved in the

transport of majority carriers (determining their transmission coefficient) isvery robust under a slight change of the energy offset, see Figs. 8.6 and 8.7.

From Fig. 8.14 we clearly observe non-Ohmic behavior which is dueto the rather complicated electronic structure involved in the transmissionprobability. We observe, for instance, that the absolute value of the currenttransmitted through a structure consisting of six layers CrAs is higher forall voltages than when transmitted through four layers. These I-V charac-teristics indicate that spin-filtering should also be realizable with very thinstructures of CrAs, see Fig. 8.15, which might be easier to fabricate. Forn=4, the minority current shows nonlinearities which we attribute to reso-nant transport mediated by states which, in the bulk, give rise to the bands


0 0.05 0.1 0.15 0.20

2

4

6

8

10

Va (V)

log

10 j

(A m

−2)

l = 4

l = 6

l = 8

l = 10

Figure 8.14: (Color online) I-V characteristics for different layer thicknesses` = 4, 6, 8, 10.

discussed above. Nevertheless, it has to be kept in mind that the systematicerror of our approach is larger for very thin structures because (i) the mod-eling of the CrAs layers is based on the Hamiltonian of bulk ZB CrAs and(ii) effects from the interface will become more important for thin layers.Nevertheless, spin-filtering should be observable.

8.5 Summary, Discussion, Conclusions, andOutlook

We have performed a model study of transport in CrAs / GaAs heterostruc-tures which is based on the assumption that sufficiently thin layers of CrAscan be grown in between a GaAs substrate in lattice matched fashion. Thebulk electronic structure of fcc GaAs, fcc and tetragonal CrAs, as well aslattice matched single [1,0,0] GaAs / CrAs heterointerfaces were calculatedwithin an LSDA LMTO model and used to determine the band offsets be-tween the two materials for minority and majority carriers. As a remarkableresult, we find a (local) total energy minimum for a hexagonal bulk CrAsunit cell, when the transverse lattice constant is held fixed at the bulk GaAsvalue aGaAs = 5.65 and the longitudinal [1,0,0] lattice constant a⊥ is varied.The minimum is found for a⊥ ≈ 0.98aGaAs. Although this local equilibrium


0 0.05 0.1 0.15 0.20.8

0.85

0.9

0.95

1

Va (V)

P

l = 4

l = 6

l = 8

l = 10

Figure 8.15: (Color online) Current spin polarization for different layerthicknesses ` = 4, 6, 8, 10.

unit cell of CrAs is tetragonal rather than cubic, the systematic errors intro-duced by assuming a perfectly lattice matched fcc crystal structure at aGaAs

is found to be negligible. For a lattice constant of aCrAs = 5.65 we find thatfcc CrAs is a half-metal, with zero gap for one spin orientation (majority car-riers) and a gap of 1.8 eV at theX point for the other (minority carriers). Thecomputed band offset for a lattice matched [1,0,0] hetero-interface betweenthe two materials is found to be about 0.5 − 0.6 ± 0.2 eV. The latter impliesan alignment of the gap region of minority CrAs with the central region ofthe uppermost valence bands of GaAs. For spin-filtering, therefore, it is notimportant whether the sheet of CrAs is half-metallic or not.

The ab-initio spin-dependent electronic band structures are mapped ontoa ETB model which is used to construct the effective Hamiltonian of the n-GaAs/CrAs/n-GaAs heterostructures consisting, respectively, of` = 4, 6, 8, 10 mono-layers of CrAs. This down-folding was constrainedby the requirement that As ETB onsite parameters for a given spin orien-tation be constant throughout the system, thereby, eliminating the need forthe introduction of ad-hoc ETB parameters at the heterointerface. The cur-rent response for majority and minority carriers is obtained within a non-equilibrium GREEN’s function approach. We consider carrier injection fromn-doped GaAs and our calculations show efficient spin-filtering over a wideparameter range, in particular, regarding the precise band alignment be-


tween the GaAs conduction band edge with the CrAs bands, temperature,and layer thickness. Spin-polarization of up to 99 percent, as well as room-temperature spin-filtering, is predicted within this model.

A number of potential improvements to the present theoretical approach,such as a more realistic account of correlations, the inclusion of the spin-orbit interaction, a self-consistent treatment of transport, a more detailedinclusion of the interface, scattering, etc., can readily be listed and be ad-dressed in future studies. However, we are convinced that the main state-ment of this publication, i.e. that lattice matched CrAs/GaAs heterostruc-tures are strong candidates for room-temperature spin-filters, is not affectedby these details. A potential source for major corrections could be a strongbias dependence of the exchange interaction, i.e. a bias anomaly, in CrAs,similar to the one predicted and reported for heterostructures containingGaMnAs [78, 77].

At this point, however, we believe that experimental assistance is essen-tial to make further progress. It is our hope that with the present results wecan stimulate a renewed interest in the fabrication of fcc hetrerostructurescontaining layers of transition metal compounds, such as CrAs, MnAs, orVAs, and conventional fcc semiconductors. For MnAs, apart from Mn δ-doped GaAs structures and strained fcc MnAs quantum dots on GaAs, thegrowth in the fcc or tetragonal phase apparently has not been successful.Evidence for fcc (or tetragonal) CrAs layers on GaAs substrates still seemsto be controversial. It is hoped that these promising theoretical results re-garding the CrAs electronic structure, the favorable band alignment withGaAs, and the predicted high spin-polarization in charge transport encour-age the materials growth and experimental physics community in a con-tinued study of semiconductor heterostructures containing transition metalcompounds.

Acknowledgments: The authors thank R. HAMMER, M. AICHHORN, andE. ARRIGONI for fruitful discussions. This work was supported financiallyby FWF project P221290-N16.


8.A Empirical Tight-Binding Parameters

Table 8.1: tight-binding parameters for bulk CrAs majority and minorityspin and the respective GaAs parameters. The anion onsite energies areindicated by the number 1 while the cation is labeled by the number 2. Forfurther notations see [8].

CrAs maj GaAs maj (A) GaAs maj (B) CrAs min GaAs min (C) GaAs min (D)Es1 2.6574 2.6574 2.6574 4.5703 4.5703 4.5703Es2 4.3837 4.1719 -11.8119 1.0108 5.5583 -16.4426Ep1 1.2291 1.2291 1.2291 2.0184 2.0184 2.0184Ep2 13.8494 26.2033 11.8069 -6.5536 5.4491 27.2734Ed11 8.2985 8.2985 8.2985 9.9412 9.9412 9.9412Ed12 7.4409 7.4409 7.4409 6.3685 6.3685 6.3685Ed21 -1.3823 15.6055 14.1494 4.5335 4.5941 9.9138Ed22 -1.9278 14.1638 10.6150 0.8616 5.5512 14.2845Es∗1 18.5203 18.5203 18.5203 -0.6061 -0.6061 -0.6061Es∗2 1.8607 10.4258 9.3580 9.0261 -2.4055 16.4261(ssσ) -1.1307 15.6358 8.1308 1.3311 0.2533 -0.2897

(s1p2σ) 0.9972 -2.2562 0.5449 2.9666 0.2245 -1.8973(s2p1σ) -0.1885 7.2869 5.2848 -1.4504 0.7586 7.7539(ppσ) 3.2695 -4.9820 3.2030 -3.5148 -2.3875 -0.9973(ppπ) -1.9484 2.7602 -2.0813 -0.1283 1.3346 2.9221

(s1d2σ) -3.3586 -4.3212 3.9507 0.4038 -0.6059 -1.3790(s2d1σ) 3.1659 -3.3578 -1.4149 2.4298 0.6413 -4.6244(p1d2σ) -1.1220 -0.2759 0.1416 0.5525 2.7231 3.8984(p1d2π) 0.8269 2.5960 2.5355 0.5372 0.3951 2.0218(p2d1σ) 0.1535 2.1035 0.4901 3.7059 -0.2031 0.7540(p2d1π) -1.9975 -3.6283 -2.8348 3.0276 0.9990 1.6321(ddσ) -2.1444 -1.4358 0.7386 -2.1594 0.1633 -0.3069(ddπ) -0.5411 0.1829 -0.7089 0.3193 -0.1910 -1.7396(ddδ) 0.3357 -0.2430 -1.2726 0.6818 0.4668 2.9257

(s1s∗2σ) -0.9709 2.3385 7.1707 0.0642 -0.0772 1.9910(s2s∗1σ) 3.3994 -5.0239 0.4364 3.0872 -0.2349 12.4053(s∗1p2σ) -2.7234 -4.8008 1.7586 -2.5451 0.5351 16.9906(s∗2p1σ) 4.5299 5.7515 0.6181 0.6516 4.6082 4.5138(s∗1d2σ) 2.8069 -1.7050 3.4076 1.0414 -3.8518 12.5601(s∗2d1σ) 0.2943 -2.6707 -1.6950 -2.1579 2.2004 -3.5301(s∗s∗σ) 3.8433 -3.8555 0.3662 1.4586 0.7470 -5.2658

Chapter 9

CHARGE TRANSPORT THROUGH INTERFACES:A TIGHT-BINDING TOY MODEL AND ITS IM-PLICATIONS

B.A. Stickler and W. Pötz

published in the Journal of Computational Electronics, 2013, DOI:10.1007/s10825− 013− 0466− 7

Abstract. With the help of a tight-binding (TB) electronic-structure toy modelwe investigate the matching of parameters across hetero-interfaces. Wedemonstrate that the virtual crystal approximation, commonly employedfor this purpose, may not respect underlying symmetries of the electronicstructure. As an alternative approach we propose a method which is mo-tivated by the matching of wave functions in continuous-space quantummechanics. We show that this method obeys the required symmetries andcan be applied in simple band to band transitions. Extension to multipleinterfaces and to more sophisticated TB models is discussed.

9.1 Introduction

The modeling of quantum transport in nano-structured semiconductor de-vices is one of the major challenges in applied solid-state physics [70, 71, 72,

109

110 Chapter 9: J. COMP. ELEC., 2013.

11, 13, 93, 90, 89, 77, 73]. In most cases a reliable quantum mechanical treat-ment of charge transport based on ab-initio electronic structure calculationsseems to be impossible with state-of-the-art methods. For many materialclasses even "ab-initio" methods require empirical input in order to reliablycapture the electronic structure in equilibrium (T = 0). Furthermore, theygenerally cannot capture the non-linear-response regime [33, 94].

As a resort, an empirical framework for the modeling of electronic de-vices under a finite bias has commonly been used. The calculation is sub-divided into three steps: (i) the electronic properties of (the components of)the device are identified by means of computational or experimental efforts,(ii) the findings of step (i) are implemented into a model Hamiltonian whichin step (iii) is used in the transport calculations, usually, with the inclusionof (self-consistent) corrections to account for the specific non-equilibriumsituation. Typically, step (i) yields the (bulk) electronic structure of individ-ual components of the device, as well as the band offset, for example, froma DFT super-cell calculation. In step (ii) this information is combined intoa model Hamiltonian for the entire device. Here one faces the problem ofmatching two or more regions with limited information regarding the in-terface. In the simplest case, one may know the electronic structure of twobulk semiconductors and their relative band alignment and be faced withthe task to design an effective Hamiltonian for the study of charge transportacross the hetero-interface.

In the case of single-band electronic transport in the parabolic regime, aneffective-mass model (EMM) may be appropriate. There is a plethora of ana-lytic, as well as numerical, methods available to solve the associated bound-ary value problem. In particular, if a linear voltage drop is assumed, GUND-LACH’s method may be employed to solve the problem analytically [95].However, the eigenvalues of the problem are substantially influenced bythe boundary and matching conditions imposed upon the SCHRÖDINGERequation. We note that the EMM may be viewed as a special one-band limitof the envelope-function approach. The matching strategy for the latter alsofaces problems when the overlaps of the BLOCH functions for the two bulkmaterials to be joined are not known [96].

For more complex band structure situations, empirical tight-binding (TB)models provide a popular modeling tool [13, 11, 73, 70, 89, 90, 71]. Inthis case the TB parameters have to be regarded as pure fitting parame-ters, [9, 8, 67, 97], to be distinguished from the case where they stem fromab-initio calculations [4, 47, 56]. Within this model, transport calculationsusually are performed within the framework of non-equilibrium GREEN’sfunctions (NEGFs) or the transfer matrix method [73, 13].

Chapter 9: J. COMP. ELEC., 2013. 111

Besides its numerous benefits, such as conceptual simplicity and compu-tational effectiveness, the empirical TB approach also comes with caveats.Recently it has been argued by TAN et al. [56] that the reliability of trans-port calculations based on empirical TB parameters is rather questionableand, therefore, a direct mapping procedure based on ab-initio methods ispreferable. Here, we shall address an even more serious disadvantage ofcommonly employed techniques in cases where the only information avail-able consists of the bulk electronic structure and the band-offsets betweenthe materials involved [6, 7]. A common approach to the modeling of theinterface for binary compounds is the virtual crystal approximation (VCA)[38, 12, 11, 70, 71, 72, 13]. It is based on a simple (linear) interpolation ofTB hopping and/or on-site parameters at the interface for establishing amatching between materials.

Within this work and utilizing a toy model we demonstrate that the com-monly used VCA for constructing the interface TB elements may not respectall symmetries of the underlying TB model and we propose an alternativeapproach which preserves these symmetries. It is important to remark thatthe method proposed by BRANDBYGE et al. [40] also employs an ad hocapproximation which is of the form of the VCA. This paper is structuredas follows: In Sec. 9.2 we define a rather simple model Hamiltonian andbriefly discuss the NEGF method, as well as the EMM, as required for ourpurpose. In Sec. 9.3 we demonstrate that the VCA is likely to lead to in-consistencies by investigating an artificial and a genuine interface. Finally,in Sec. 9.4 we propose an alternative formulation of the interface matchingproblem and demonstrate that it respects the symmetries supplied by theinput information. Conclusions are drawn in Sec. 9.5.

9.2 The Model

We define a bulk toy model in order to formulate the matching problemand to demonstrate the inconsistencies which may arise. It is an infiniteone-dimensional two component tight-binding chain with nearest-neighborhopping only. Each element of the chain contains one orbital and the grid-spacing is given by a > 0. The Hamiltonian may be written as

H =∑lσ

εσ |l, σ〉〈l, σ|+ t12

∑l

(|l, 2〉〈l + 1, 1|+ |l, 1〉〈l, 2|

+h.c.) , (9.1)

where l ∈ Z labels the unit cells, σ ∈ 1, 2 labels the atoms within the unitcells and |l, σ〉 are the basis-kets localized at lattice point (l, σ). Furthermore,


εσ ∈ R are the onsite energies of atom σ and t12 is the hopping element.Please note that in writing Eq. (9.1) we assume that t12 ∈ R for reasons ofsimplicity. Let us define an onsite matrix ε and a hopping matrix t via

ε =

(ε1 t12

t12 ε2

)(9.2)

and

t =

(0 0t12 0

). (9.3)

The eigen-energies of the Hamiltonian (9.1) can be expressed as

E1,2(k) = B ±√B2 − A(k), (9.4)

where we introduce the wavenumber k ∈[−πa, πa

]and define

A(k) = det [h(k)] , (9.5)

together with

B =1

2tr [h(k)] . (9.6)

Moreover, we define the Hamiltonian matrix h(k) as the representation ofthe Hamiltonian H in reciprocal space,

h(k) = ε+ t exp(ika) + t† exp(−ika). (9.7)

Please note that we refer to the operator (9.1) as the Hamiltonian while wedenote the matrix h(k) as the Hamiltonian matrix. The band structure de-fined by Eq. (9.4) is completely determined by the set of TB parametersξ = (ε1, ε2, t12) ∈ R3. In order to emphasize this dependence we denoteEn(k) ≡ En(k, ξ), where n = 1, 2.

We shall now consider two different bulk materials, each characterizedby a Hamiltonian of the form (9.1) with TB parameters ξL and ξR, respec-tively. We bring these two materials into contact and investigate the result-ing heterostructure assuming that the band-offset is known and no furtherinterface effects are taken into account. In what follows we present twocommon approaches to calculate the transmission function T (E) throughthe interface for some particular energy E. Please note that we restrict ourdiscussion to the case of zero bias for reasons of simplicity. The generalarguments also apply to heterostructures under bias.

As a first technique we shall discuss the NEGF approach with the VCA.We introduce two Hamiltonians of the form Eq. (9.1), however, each defined


only on one half axis, i.e. l ∈ Z− and l ∈ Z+. Furthermore, we denote HL

and HR the Hamiltonians defined on the semi-infinite domains of the leftmaterial L (l = −1,−2, . . .) and the right material R (l = 1, 2, . . .). Diago-nal elements of the TB Hamiltonians are shifted to comply with the bandalignment. These two Hamiltonians are then connected with the help of aninterface Hamiltonian HI such that the total HamiltonianH of the system isof the form

H = HL +HI +HR. (9.8)

It is obvious that the particular choice of HI significantly influences thetransport physics of the system, yet, in most applications its exact struc-ture is unknown. In the particular case of two diatomic materials AB andCB which share the atomic constituent B, as for instance in the case of aGaAs / AlAs heterostructure [11], a common approach is the virtual crystalapproximation, for instance [38, 12], in which HI contains one element fromthe left chain, associated with A, while the onsite element of the commonatom B is averaged at the interface, i.e.

εI2 = xεL2 + (1− x)εR2 (9.9)

where x ∈ [0, 1]. Here, εL/R2 denote the onsite energy of atom B according tothe TB parameter sets ξL/R. The coupling to the left semi-infinite Hamilto-nian is then described according to tL12, while the coupling to the right semi-infinite Hamiltonian is described by the hopping between atoms B and C,i.e. tR12.

The transmission function T (E) across the interface associated with Hamil-tonian (9.8) may, for instance, be calculated within the NEGF formalism,dividing the TB chain into three segments: the left semi-infinite lead con-sisting of the unit cells l = −2,−3, . . ., the interface region referred to assystem S given by the unit cells l = −1, 0, 1, and the right semi-infinite leadconsisting of the cells l = 2, 3, . . .. T (E) is given by [73]

T (E) = tr(ΓRG

RSΓLG

AS

), (9.10)

where tr(·) denotes the operator trace, ΓR/L are the coupling functions to theright (R) and the left (L) semi-infinite leads, respectively, and G

R/AS denote

the retarded and advanced system’s GREEN’s function. The latter are givenby [73]

GRS =

[(E + iη)I −HS − ΣR

L − ΣRR

]−1, (9.11)

and GAS =

(GRS

)† and HS is the system Hamiltonian. In Eq. (9.11) η > 0 is asmall parameter, I is the identity, and ΣR

L/R is the retarded self-energy of the


leads. The retarded self-energy of the leads is calculated from the surfaceGREEN’s functions of the left and the right lead GR

L/R, respectively, via [73]

ΣRL = t†GR

Lt, and ΣRR = tGR

Rt† (9.12)

where we take into account that t12 ∈ R. The surface GREEN’s functions, ingeneral, can be determined recursively with the help of layer doubling, assuggested by SANCHO et al. [74]. The coupling functions appearing in Eq.(9.10) are given by

ΓL/R = i(ΣRL/R − ΣA

L/R

). (9.13)

An entirely different approach is to calculate the transmission with thehelp of an EMM. We calculate the effective mass of the band of interest, sayn, at k = 0 via

m(ξ) = ~2

[d2En(k, ξ)

dk2

]−1

k=0

. (9.14)

In what follows, we employ the notation

m =

mL x ≤ 0,

mR x > 0,(9.15)

where we replace the discrete index l ∈ Z by the continuous variable x ∈ R.Again, we partition the domain into two different regimes: (I) with m = mL

for x ≤ 0 and (II) with m = mR for x > 0 and solve the correspondingeigenvalue problem analytically. In particular, in region (I) we have

ψ′′(x) +2mL

~2Eψ(x) = 0, (9.16)

with the solution

ψ(x) = exp(ikLx) +R exp(−ikLx). (9.17)

Here, R is the reflection amplitude and the wavenumber kL is given by

kL =

√2mL

~2E. (9.18)

In similar fashion we obtain for regime (II) the solution

ψ(x) = T exp(ikLx), (9.19)


when no right-incident waves are considered. Here T is the transmissionamplitude and the wavenumber kR reads

kR =

√2mR

~2(E + ∆), (9.20)

with ∆ ∈ R the bandoffset. If we consider a linear voltage drop betweentwo leads, one may solve the corresponding SCHRÖDINGER equation ana-lytically with the help of AIRY functions [95]. The numerical constants Rand T are determined by the continuity conditions of the wave function andthe current density. These conditions read

1 +R = T , (9.21a)kLmL

(1−R) =kRmR

T . (9.21b)

The transmission function T (E) is then calculated via

T (E) =kRmL

kLmR

|T |2. (9.22)

A major benefit of the EMM is clearly its conceptual simplicity, but, inmost cases the parabolic approximation may not be justified and the methodof choice is the determination of T (E) within a NEGF approach as discussedabove. However, by writing Eq. (9.8) for the total Hamiltonian H we em-ploy information which we actually do not have. Even worse, it turns outthat the electronic structure is invariant under certain transformations ofthe TB parameters and that these symmetries are destroyed by the particu-lar choice of HI in the VCA (9.9). Let us exemplify this dilemma within thenext section in more detail.

9.3 The Dilemma

Let us define the operators A and B acting on vectors ξ ∈ R3 via

A : (ε1, ε2, t12) 7→ (ε1, ε2,−t12), (9.23a)

andB : (ε1, ε2, t12) 7→ (ε2, ε1, t12). (9.23b)

Then we note from Eq. (9.4) - (9.7) that the electronic structureEn(k, ξ) obeysthe invariances

En (k,Aξ) = En (k, ξ) , (9.24a)


andEn (k,Bξ) = En (k, ξ) , (9.24b)

for all ξ ∈ R3. Thus, the electronic structure described by the HamiltonianEq. (9.1) is invariant under a change of the sign of the hopping parame-ter connecting the two atomic species, see Eq. (9.24a). Furthermore, theelectronic structure is also invariant under a relabeling of the atoms withinthe unit cell, as expressed in Eq. (9.24b). This has the consequence thatTB parameters are not uniquely determined by the band structure while theTB parameters uniquely determine the bandstructure. Moreover, given aset of TB parameters we can immediately construct three more sets of pa-rameters which yield completely identical bands by simply employing theinvariances (9.24a) and (9.24b).

In this light, the VCA for TB models may become problematic becauseone needs to unambiguously identify the associated atomic species, whichaccording to Eq. (9.24b) is not possible in many cases. In particular, ε2(BξL) =ε1(ξL). On the other hand, the EMM respects the symmetries Eqs. (9.24a)and (9.24b), since the effective mass is solely based on the electronic struc-ture, see Eq. (9.14).

In summary, if the only information available is the electronic structureEn(k) then the set of TB parameters defining the underlying Hamiltonian(9.1) is not unique. This is not problematic as long as one deduces fromthe Hamiltonian (9.1) observables which too are invariant under a changein the TB parameters according to A and B. On the other hand, if the ob-servables are not invariant under these transformations one introduces aninconsistency because one utilizes information which is not contained in theelectronic structure, as for instance, the sign of t12.

In what follows we investigate this problem in terms of the toy model inorder to quantify the error.

9.3.1 An Artificial Heterostructure

For a first example we consider a homogeneous system with a bulk elec-tronic structure described by a Hamiltonian of the form (9.8), i.e. we modela diatomic material comparable to, for instance, GaAs. The TB parame-ters ξ = (2,−1, 1) uniquely determine the electronic structure displayed inFig. 9.1. Now an artificial interface between two semi-infinite linear chainsis constructed by employing the symmetry operators A and B to the rightchain. For the left semi-infinite chain we keep ξL = ξ while, for the rightsemi-infinite chain, ξR is one of the parameter sets: ξa = ξL, ξb = AξL,ξc = BξL and ξd = ABξL. Since the transmission obtained with an EMM


−2 0 2

−2

0

2

4

k/a

E / e

V

Figure 9.1: (Color online) Electronic structure of the toy model Eq. (9.1) withTB parameters ξ = (2,−1, 1).

is invariant under these substitutions we shall restrict the following discus-sion to the NEGF formalism with the VCA. The electronic structure, as wellas the resulting transmission function for the lower band with x = 0.5, areillustrated in Fig. 9.2.

The transmission function resulting from the correct treatment of the ma-terial (ξR = ξa, red solid line) agrees with the result obtained with the helpof ξR = ξb and with that of an EMM. However, the transmission originatingfrom an exchange of the parameters 1↔ 2 according to B strongly deviatesfrom the original curve. The reason obviously stems from modeling an in-terface of the form · · ·ABABBABA · · · instead of · · ·ABABABAB · · · . Thisexample clearly illustrates a major problem in the TB treatment of interfaceswith the help of the VCA. Moreover, we do not know which value of x inthe VCA (9.9) should give the correct result. We illustrated this dependencyfor the case ξR = ξc in Fig. 9.3 for three different values of x.

9.3.2 A Genuine Heterostructure

For the second example we investigate two truly different materials char-acterized by ξL and ξR and which have one atom in common, i.e. we arelooking at an interface of the form · · ·ABABCBCB · · · . According to theabove discussion it is not possible to assign TB parameters to a particularatom within the unit cell as expressed in Eq. (9.24b). This point is par-ticularly crucial if the TB parameters are genuine fitting parameters stem-ming from the electronic structure solely, as in the case of empirical TB ap-


−2 −1.5 −10

0.2

0.4

0.6

0.8

1

E / eV

T

ξa

ξc

ξd

Figure 9.2: (Color online) The transmission function through the lower bandresulting from an artificial interface as obtained by applying the symmetryoperators A and B to ξ. Please note that we did not plot the transmission ofξb because it is equivalent to the correct case ξa.

−2 −1.5 −10

0.2

0.4

0.6

0.8

1

E / eV

T

ξa

x = 0.25

x = 0.375

x = 0.5

Figure 9.3: (Color online) The transmission function for the lower band re-sulting from an artificial interface with ξc for different values of x.


−2 0 2

−2

0

2

4

k/a

E / e

V

Figure 9.4: (Color online) Electronic structure according to the TB param-eters ξL = (2,−1, 1) (red solid lines) and ξR = (2.6398,−0.0602, 1.5) (bluedashed lines).

proaches [9, 8, 67]. Then it is not possible to unambiguously identify certainparameters with one atomic species in general.

The two electronic structures depicted in Fig. 9.4, respectively, corre-spond to the TB parameters ξL = (2,−1, 1), for AB, andξ = (2.6398,−0.0602, 1.5), for material BC. Again, ξR is equivalent to oneof the TB parameters ξa = ξ, ξb = Aξ, ξc = Bξ and ξd = ABξ. The resultingtransmission functions for the lower band are depicted in Fig. 9.5.

In summary, for a simple model system we have identified the reasonswhy the VCA does not yield satisfactory results. The problem is that (i)an arbitrary mixing parameter x is introduced at the interface, and (ii) thesymmetry of the electronic structure under the operators A and B acting onξ, Eqs. (9.24a) and (9.24b), is not respected within the VCA. It is clear thatambiguities grow dramatically with increasing complexity of the TB model.A determination of the transmission function within the EMM respects thissymmetry, however, the EMM itself often is too crude for an electronic struc-ture model. In the EMM the preservation of symmetry in the transmissionfunction is a result of the matching condition at the interface, see Sec. 9.2. Inwhat follows we propose an approach for TB models as a possible methodfor avoiding ambiguities in the interface problem.


−2 −1.5 −10

0.2

0.4

0.6

0.8

1

E / eV

T

ξa

ξc

eff. mass

Figure 9.5: (Color online) Transmission T (E) for the composite materialwith the NEGF technique with parameters ξa (red solid), ξc (magenta dashdotted) and for the EMM (black solid line). The curves for ξb and ξd havebeen omitted since they are the same as for ξa and ξc, respectively.

9.4 The Interface Matching Problem - A PossibleSolution

Within this section we propose a discrete matching approach for TB modelswhich is adopted from the matching procedure within the EMM. The mainprocedure consists in solving the bulk problem for the two different ma-terials separately and matching the wave functions, as well as the currentbetween unit cells, at the interface. Such an approach has two main advan-tages: (i) it does not introduce arbitrary parameters at the interface and (ii)it respects the symmetries of the electronic structure, e.g. under variationof ξ according to A and B, Eqs. (9.24a) and (9.24b), for the case of the TBHamiltonian (9.1). However, this approach still cannot account for interfa-cial effects as long as no additional information is made available. It has,therefore, to be regarded as a best solution based on the information given.We shall first illustrate the procedure for the matching of two single-atomiclinear TB chains in subsection 9.4.1 since it allows an entirely analytic ex-position of the method. Subsequently, the method is generalized and thenapplied to the toy model in subsection 9.4.2.

The assumption of an abrupt interface between two semi-infinite crystalsinevitably represents an approximation which leads to a loss of informationregarding microscopic details at the interface. In an empirical TB model


parameters should become position dependent near the interface. How-ever, the mapping of ab-initio band structure calculations onto empiricalTB models numerically is rather intractable (due to supercell size) and onemay resort to an approach based on stationary scattering theory. Consid-ering a quasi-1D system, such as a heterostructure, a stationary solution forgiven energyE with in-asymptote

∣∣n, k||, ki⟩, with band index n and parallelk-vector k|| , may be written as

∣∣E, k‖, ki⟩ =

∣∣n, k||, ki⟩+

∑(out)m,j rn,i;m,j(E, k||)

∣∣m, k||, kj⟩ for z < 0,∑(out)m′,j′ tn,i;m′,j′(E, k||)

∣∣m′, k||, kj′⟩ for z > 0,

(9.25)when the interface is positioned at "z = 0". The sum is over the out-channelsfor which E = Em(k||, kj) = Em′(k||, kj′). In- and out-channels, respec-tively, are identified as having group velocity z-components towards theinterface and away from the interface. In general, degeneracy may implymore than two out-channels m, kj for each in-channel. Unitarity of the Smatrix and possibly other symmetries, such as time-reversal invariance, re-duce the number of independent elements rn,i;m,j(E, k||) and tn,i;m′,j′(E, k||)but are not sufficient to uniquely specify them if the interface potential atz ≈ 0 is unknown.

Let NL and NR denote the number of available in-channels on the left-and the right-hand side of the interface at a given energy E. Under time-reversal symmetry, we also have NL and NR out-channels on each side.Then, unitarity of the S matrix leads to (NL+NR)2 conditions for 2(NL+NR)2

unknowns. For NL = NR = 1, i.e., one out-channel on each side, one is leftwith two unknowns. For this case, and this represents our suggested alter-native for the ad-hoc VCA, the condition of continuity in charge and currentdensity across the interface determine the transmission and reflection at thehetero-interface, in analogy to the effective-mass case and a finite potentialstep. For higher degeneracy, additional information is necessary to deter-mine all of the S matrix elements.

9.4.1 Tight-Binding Interface Matching for Single-AtomicChain

Two semi-infinite single-atomic linear TB chains are to be connected at thegrid point l = 0. The connection between these two chains is not establishedby introducing an ad-hoc hopping parameter t across the interface, as forinstance suggested by HARRISON [38], but rather by matching the left andright solutions at the interface. For this purpose, the left- and right-semi-


infinite TB Hamiltonian both are extended to site l = 0. Specifically wepropose the matching conditions

|ψL(0)|2 = |ψR(0)|2, (9.26a)

andJL−1/2 = JR+1/2. (9.26b)

Here, ψL/R are the wave functions at site l = 0 expressed respectively, interms of the left (L) and right (R) TB orbitals. The second condition (9.26b)guarantees stationarity of the solution, i.e. the current flowing between thegrid points l = −1 and l = 0 in the left chain equals the current flowingbetween l = 0 and l = 1 in the right chain. While the matching conditions(9.21) can be derived from the SCHRÖDINGER equation, these two condi-tions (9.26) may be viewed as a (stationary version of the) particle conti-nuity equation in terms of TB orbitals. Condition (9.26a) ensures that theprobability for finding a particle at the interface is the same when approach-ing the interface from left or right: at the interface this probability can beexpressed either in terms of the L or R basis functions. In the parabolicregime above conditions (9.26) become exact (i.e., agree with the ones forthe SCHRÖDINGER equation) and coincide with Eqs. (9.21).

The isolated Hamiltonian of the L andR single-particle TB chain, respec-tively, is denoted by HL and HR with onsite energies εL, εR and hoppingelements tL, tR, respectively. The corresponding energy bands are

EL/R(k) = εL/R + 2tL/R cos(aL/Rk). (9.27)

For both bulk Hamiltonians, we write the eigenfunctions as linear combina-tion of localized states

∣∣lL/R⟩∣∣ΨL/R

⟩=∑l

aL/Rl

∣∣lL/R⟩ , (9.28)

where the general form of the coefficients aL/Rl for given k is well known tobe of the form

aL/Rl (k) = AL/R exp(ikla) +BL/R exp(−ikla). (9.29)

We assume for simplicity that the lattice constants coincide aL = aR ≡ a. Inanalogy to Eq. (9.17) we choose

AL = 1, BL = R, AR = T, and BR = 0. (9.30)


For given energy E, condition (9.26a) reads

|aL0 [kL(E)]|2 = |aR0 [kR(E)]|2, (9.31)

where we define the wave numbers kL/R(E) = |E−1L/R(E)| with the help of

the inverse of Eq. (9.27). With Eq. (9.30) we rewrite this condition as

|1 +R(E)|2 = |T (E)|2. (9.32)

We calculate the current from lattice point −1 to lattice point 0 via

J−1/2 =i

~[(aL0 )∗tLa

L−1 − (aL−1)∗tLa

L0

]= −2

~tL(|R|2 − 1) sin(kLa), (9.33)

and from 0 to 1 asJ+1/2 =

2

~|T |2tR sin(kRa). (9.34)

Furthermore, it is convenient to define the velocities

vL/R(k) =1

~d

dkEL/R(k) = −2a

~tL/R sin(ka), (9.35)

and rewrite condition (9.26b) as

|R(E)|2 +vR[kR(E)]

vL[kL(E)]|T (E)|2 = 1. (9.36)

Hence, we solve the coupled equations (9.32) and (9.36). Under the assump-tion that R, T ∈ R these equations are easily solved to give

T (E) =2vL(kL)

vL(kL) + vR(kR), (9.37a)

andR(E) =

vL(kL)− vR(kR)

vL(kL) + vR(kR). (9.37b)

Clearly, the quantities of physical interest are

vL(kL)

vR(kR)|T (E)|2 =

4vR(kR)vL(kL)

[vL(kL) + vR(kR)]2, (9.38a)

and

|R(E)|2 =

(vL(kL)− vR(kR)

vL(kL) + vR(kR)

)2

. (9.38b)


0 1 2 3 40

0.2

0.4

0.6

0.8

1

E / eV

T

R

Figure 9.6: (Color online) Reflection and transmission coefficient throughthe interface between two semi-infinite single atom TB chains as obtainedwith the matching method (solid lines) and with the EMM (dashed lines).

The transmission and reflection as a function of energyE can be obtainedby inverting Eq. (9.27), however, it has to be kept in mind that, due to ourchoice of Eq. (9.30), one has to take the branch vL(kL), vR(kR) ≥ 0. In Fig.9.6 we illustrate the transmission and reflection coefficients Eq. (9.38) forεL = 2, εR = 1, tL = −1, tR = −0.5 in comparison with the result obtainedwith an EMM. The corresponding electronic structures Eq. (9.27) togetherwith the parabolic approximations are illustrated in Fig. 9.7.

From Fig. 9.6 we observe that the reflection and the transmission coeffi-cients from the left to the right TB chain are well approximated by the EMMonly for energies near to the band minimum. This is in accordance withthe effective mass approximation, see Fig. 9.7. Moreover, we note that thereflection correctly approaches 1 as the energy E approaches 2 eV, see Fig.9.6. This is due to the reduced band width of the electronic structure of theright semi-infinite chain, see Fig. 9.7. The transmission is maximal for lowenergies.

In what follows we shall formulate the matching method for more gen-eral TB Hamiltonians and then apply it to the diatomic TB chain model dis-cussed in subsection 9.3.2.


−2 0 20

1

2

3

4

5

k / a

E / e

V

EL(k)

ER

(k)

Figure 9.7: (Color online) Electronic structures of the two isolated semi-infinite chains (solid lines) and parabolic approximations (dashed lines).

9.4.2 The Genuine Heterostructure Revisited

Let us begin with a brief review of some important properties of generalnearest neighbor TB models. The Hamiltonian H is of the form

H =∑lσ

εσ |lσ〉〈lσ|

+∑ll′σσ′

tσσ′

ll′ |lσ〉〈l′σ′| (9.39)

where l and l′ label the unit cell and σ = 1, . . . , N is an additional index.This additional index may include, for instance, the atoms in the unit cells,as well as their orbitals and the parallel momentum k‖, if the Hamiltonian(9.39) is derived from a three-dimensional TB model by partial WANNIERtransformation [11]. The hopping elements tσσ′ll′ couple only up to neigh-boring unit cells. This is a valid assumption for any finite-range TB modelsince the unit cell can be chosen as large as necessary [98]. We expand thewavefunction as

|ψ〉 =∑lσ

clσ |lσ〉 ≡∑l

cl · |l〉 , (9.40)

where in the very last step we introduce theN -component vectors cl = clσand a "vector ket" |l〉 = |lσ〉. BLOCH’s theorem allows one to extract thespace dependence of the vector cl at a given energy E in form of a phase


factor

cl(E) = c[k(E)] exp [ik(E)la]

+c[−k(E)] exp [−ik(E)la] , (9.41)

where k(E) is the inverse of the (doubly degenerate) electronic structureE = En(k) for a given energy E and c(k) is the eigenvector of the Hamilto-nian matrix h(k) with eigenvalue E. The Hamiltonian matrix h(k) reads

h(k) = ε+ exp (ika) t+ exp (−ika) t†, (9.42)

where we define an onsite matrix ε with matrix elements εσδσσ′ + tσσ′

ll anda coupling matrix t with elements tσσ′ll+1, respectively. Please note that theHamiltonian matrix Eq. (9.7) is a special case of this general form. Further-more, the current Jl+1/2 between unit cells l and l + 1 can be written as

Jl+1/2 =i

~

(c†l tcl+1 − c†l+1t

†cl

). (9.43)

Hence, the general matching conditions for given energy E read

|cL[kL(E)] + cL[−kL(E)]R|2 = |TcR[kR(E)]|2 , (9.44a)

and

JL−1/2 = JR+1/2, (9.44b)

where we useJL−1/2 =

i

~

[(cL−1

)†tLc

L0 −

(cL0)†t†Lc

L−1

], (9.44c)

with

cLl = cL[kL(E)] exp [ikL(E)laL]

+RcL[kL(E)] exp [−ikL(E)laL] , (9.44d)

andJR+1/2 =

i

~

[(cR0)†tRc

R1 −

(cR1)†t†Rc

R0

], (9.44e)

withcRl = TcR[kR(E)] exp [ikR(E)laR] . (9.44f)

Here, cL/R[kL/R(E)] are the eigenvectors to the Hamiltonian matrices hL/R(k)with eigenenergies E, which are of the form Eq. (9.7), however, correspond-ing to the Hamiltonians HL and HR, respectively. Moreover, with kL/R(E)


for a given energy E we define the inverse of the dispersion relations ELn (k)

and ERn (k), respectively. Finally, aL/R denote the lattice constants of the two

materials. Note that these matching conditions respect time-reversal sym-metry.

We solve the above equations (9.44) for the problem discussed in Sec.9.3.2. The resulting transmission and reflection functions are plotted in Fig.9.8, in comparison to the result obtained with the help of Eqs. (9.38). Itis very important to notice that the result depicted in Fig. 9.8 is indepen-dent of the actual choice of the parameter sets ξL and ξR, i.e. the transmis-sion function is independent under the action of the symmetry operators Aand B. This follows from the matching conditions (9.38), since all quantitiesentering these equations are solely determined by the bulk HamiltoniansHL/R. In Fig. 9.9 we illustrate the transmission through the diatomic TB het-erostructure discussed in Sec. 9.3.1 in comparison with the results obtainedwith the help of the virtual crystal approximation, see Sec. 9.3.1.

The transmission and reflection function versus energy computed in thisfashion may subsequently be mapped back onto a TB model. This can beachieved by constructing a Hamiltonian H of the form

H = HL + V (E) +HR, (9.45)

where one introduces an energy dependent coupling Hamiltonian V (E) be-tween the two chains. Following the procedure outlined in Sec. 9.2 to obtainthe transmission T (E) with the help of a GREEN’s function approach, the el-ements of the hopping matrix V (E) is then determined in such a way thatthe transmission obtained by solving Eqs. (9.38) is reproduced. Hence, V (E)will depend on the particular choice of ξL and ξR, however, is determined insuch a way that it reproduces a parameter-independent result, such as thetransmission T (E). Moreover, as long as the voltage drop at the interface issmall, V (E) may provide a good approximation for the heterostructure un-der bias. An extension to multiple interfaces of this type is straight-forward.

This method works as long as to every in-channel BLOCH state thereis only one out-channel state, respectively, to the left and to the right. Inother words, as long as one deals with the problem of matching a doublydegenerate band in one material with another one in the other. The pro-posed matching conditions (9.44) alone are not sufficient to tackle the case ofmulti-channel scattering where, for given in-channel E, n, k|| (energy, bandindex, and k-parallel) the degenerate out-channel states lie in different en-ergy bands n, or in the same band n when a degeneracy of greater than twois present. A unique solution cannot be obtained without further assump-tions. This is most easily observed by inspection of a situation with one


−2 −1.5 −1 −0.5 00

0.2

0.4

0.6

0.8

1

E / eV

T

R

Ts

Rs

Figure 9.8: (Color online) Transmission and reflection coefficient throughthe interface between two semi-infinite diatomic TB chains as obtained withthe matching method (solid lines) and as obtained when employing rela-tions (9.38) (dashed lines).

−2 −1.5 −1 −0.5 00

0.2

0.4

0.6

0.8

1

E / eV

T

T

Tfit

TVCA1

TVCA2

Figure 9.9: (Color online) Transmission coefficient throught the interfacebetween two semi-infinite diatomic TB chains as obtained with the NEGFformalism where the hopping has been fitted to the result of the matchingmethod, (solid red and dashed blue lines). Furthermore, we present thesolutions obtained with the VCA in Sec. 9.3.2.


in-channel on the left hand side i and two out-channels on the right handside. In this case the wave function to the right of the interface is given bya linear combination of the first and the second channel states with weightsti,1(E) and ti,2(E) in Eq. (9.25). It is clear, that the matching conditions Eqs.(9.44) may be solved only if the ratio ti,1(E)/ti,2(E) is known. An estimatefor this ratio might be obtained by "blanking off" one out-channel at a timeand determining the two transmission functions at a time. However, in thisprocedure, phase information is lost. It should also be mentioned that inmany multi-channel problems, there may be dominant out-channels (thosewhich couple dominantly with a given in-channel). This may be used toreduce the problem in an approximation to a tractable 2 by 2 form of oneout-channel on each side of the interface. In addition, of course, empiricalinput from experiment may be helpful in the fitting procedure.

9.5 Conclusions

We have studied the electron transmission through a hetero-interface asmodeled by the linking of two diatomic single-orbital TB chains. The twoseparate TB chains are characterized solely by their bulk properties whileno particular information about the interface, other than the relative bandalignment, is available. For this example we demonstrate that the com-monly employed VCA does not respect the underlying symmetries of theelectronic structure in relation to the TB model used in the fitting proce-dure. Commonly, however, the latter is the key input information available.Hence, the VCA introduces an arbitrary error which is hard to estimate. Asa remedy to this ambiguity we suggest a matching method of the wave func-tion and current density which is motivated by the continuity relations ofcontinuous space quantum mechanics. In particular, the obtained transmis-sion functions respect the symmetries of the band structures associated withthe TB model and can therefore be regarded as a best result under availableinformation. The transmission function determined with the help of theproposed matching method can subsequently be used to construction of ahopping Hamiltonian which reproduces this transmission function. Thus,multiple interfaces can be treated in this fashion.

We stress that the proposed matching method does not give a physicallycomplete description of the interface. In fact, this is not possible since therequired information regarding the coupling matrix at the interface whichenters the total HamiltonianH is considered to be unknown. Hence, the ap-proach discussed here is no substitute for a full (microscopic) study of thehetero-interface. But, since such a study is almost always based on large su-


percell calculations, one is confronted with the problem of fitting numerousbands which, in most cases, practically is not feasible.

The matching method is applicable to the case of one out-channel oneach side (per in-channel). If more in- or out-channels are present, a simplematching technique cannot yield the channel resolved transmission sincethe problem is underdetermined (i.e. the Hamiltonian at the interface isunknown). However, if the ratio of the transmission amplitudes betweenall available out-channels is known, the proposed method can still be em-ployed in order to obtain the total transmission function.

Acknowledgments: This work was supported financially by FWF projectP221290-N16.

Part II

DERIVATION OF A LINEAR SPINORIALBOLTZMANN EQUATION

131

Chapter 10

INTRODUCTORY REMARKS

The simulation of electronic transport properties of solid-state devices isone of the greatest challenges in modern condensed-matter theory. Sincein most cases a full quantum treatment of the system is not feasible andmaybe not reasonable, it is a common and legitimate approach to employeffective models. Hence, the derivation of such effective models, togetherwith possible corrections to the simplified dynamics, plays a major role inactive research. One of the most intriguing examples for such an approachis the description of diffusion processes within the framework of stochastics[65, 99]. A further example is the application of the semiclassical (scalar)BOLTZMANN equation to describe charge transport in a condensed-matterenvironment [24, 100]. Here we present a novel derivation of the semiclassi-cal BOLTZMANN equation which respects the spin degree of freedom of thecarriers. Let us briefly review some basic notions of the dynamics of quan-tum systems in order to position the following works, Chaps. 11 and 12, inan appropriate context.

We regard a closed quantum system, characterized by the state operatorρ ≡ ρ(t),1 which evolves according to the (time-independent) HamiltonianH . Hence, VON NEUMANN’s equation is of the form [39, 64]

∂tρ = −i [H, ρ] , (10.1)

where we set ~ = 1 for convenience. We rewrite the above equation with the

1In what follows we prefer a density operator formalism over a wavefunction formula-tion since the latter does not allow for mixed states, see for instance Ref. [39].

133

134 Chapter 10: INTRODUCTORY REMARKS

help of the superoperator2 L, which is commonly referred to as the quantumLIOUVILLE operator [19], as

∂tρ = Lρ. (10.2)

Given the initial condition ρ(0) = ρ0, we write the formal solution of Eq.(10.2) in the form

ρ(t) = exp (Lt) ρ0 ≡ V (t)ρ0. (10.3)

It is immediately obvious, that V (t)V (t′) = V (t + t′), i.e. V (t) obeys theCHAPMANN - KOLMOGOROV equation [19, 99]. Furthermore, it is wellknown [19, 39, 64] that the time evolution (10.3) is completely reversible.

Now, let us suppose that this closed quantum system can be separatedinto two subsystems A and B. Hence, the total Hamiltonian is of the form

H = HA ⊗ 1B + 1A ⊗HB +HI , (10.4)

where HA and HB are the Hamiltonians of the isolated subsystems, 1A/Bare the respective unities and HI accounts for the interaction between sub-system A and subsystem B. Furthermore, let trB (·) denote the partial traceover the degrees of freedom of subsystem B. Then, the reduced state op-erator ρA = trB (ρ) obeys, according to Eqs. (10.2) and (10.4), the evolutionequation

∂tρA = trB (Lρ)

= −i[HA, ρA]− itrB ([HI , ρ]) . (10.5)

If the dynamics of subsystem A are, again, assumed to be Markovian, wecan write

trB (Lρ) = LρA, (10.6)

where L is the generator of a quantum dynamical semigroup [19]. It hasbeen demonstrated by LINDBLAD that the most general form of L (with afew assumptions) is [20]

LρA = −i[HA +H0, ρA] +∑k

γk

(AkρAA

†k −

1

2ρAA

†kAk −

1

2A†kAkρA

), (10.7)

where H0 is some Hermitian operator, Ak are referred to as the LINDBLADoperators and γk ≥ 0. Equations of the form (10.7) are commonly referredto as LINDBLAD equations or equations of LINDBLAD form. The applica-tions of LINDBLAD equations in quantum optics and solid state physics are

2A superoperator is an operator acting on operators. In the present case, the superopera-tor L is given by L· = −i [H, ·].

Chapter 10: INTRODUCTORY REMARKS 135

numerous, see for instance Ref. [19]. Within Chap. 11 we will present a sys-tematic derivation of LINDBLAD equations in the particular case that thecoupling between subsystem A and subsystem B is assumed to be verystrong. Furthermore, a common approximation employed in the deriva-tions of LINDBLAD equations (10.7) is the BORN approximation [19]. TheBORN approximation states that the total state operator ρ of the compositesystem AB may be written as ρ = ρA ⊗ χB, where χB is some equilibriumstate of subsystem B. By augmenting the VON NEUMANN equation (10.2)by a relaxation operator Q(ρ), which drives the total state ρ into the BORNapproximation ρA⊗χB within a finite time interval τ , it is possible to obtainnon-Markovian corrections to (10.7) in a systematic way, [17] and Chap. 11.

It is, then, a natural approach to derive semiclassical evolution equa-tions from the LINDBLAD equation (10.7). However, an alternative approachis to perform the semiclassical limit in Eq. (10.2) in a first step, and per-form the MARKOV approximation afterwards. In particular, we could per-form a WIGNER transformation (see Chap. 12) of Eq. (10.2) to obtain themany body WIGNER equation [101]. Drawing the semiclassical limit scaled-~ → 0 gives the classical many-body LIOUVILLE equation [24, 101]. Fromthis equation we may derive a BOLTZMANN equation with the help of theBBGKY (BOGOLIUBOV - BORN - GREEN - KIRKWOOD - YVON) hierarchy[24, 102]. The scalar BOLTZMANN equation is of the form

∂tf − h, fx,η = C[f ], (10.8)

where f ≥ 0 is a classical distribution function, ·, ·x,η denotes the POIS-SON bracket with respect to position x ∈ Rd and momentum η ∈ Rd, h(x, η)is a classical HAMILTON function, and C[f ] is the collision integral or collisionoperator. The collision integral C[f ] accounts for the change of the distribu-tion f(x, η, t) according to collisions between different particles. The impor-tance of the BOLTZMANN equation (10.8) stems from its capability to handlenon-equilibrium phenomena. We note BOLTZMANN’s H-theorem as an in-triguing example. Moreover, it is possible to derive a plethora of classicaltransport models, such as the diffusion equations or hydrodynamic models,from BOLTZMANN’s equation [24, 102].

Quantum derivations of the BOLTZMANN equation have been achievedwithin the framework of KADANOFF - BAYM GREEN’s functions for quasi-particles [103], from the SCHRÖDINGER equation with Gaussian random po-tentials [100, 104, 105], as well as with a quantum version of the BBGKYhierarchy [106]. Within Chap. 12 we shall employ an alternative strategy.We base our derivation on the LINDBLAD master equation as obtained inChap. 11, in which the spin degree of freedom of subsystem A is still in-corporated. We then derive a WIGNER transport equation equipped with a

136 Chapter 10: INTRODUCTORY REMARKS

proper collision operator stemming from the LINDBLAD dissipator. Draw-ing the semiclassical limit allows one to derive several semiclassical spino-rial transport models [18]. As examples we derive the BLOCH equations, aswell as the linear spinorial BOLTZMANN equation.

Chapter 11

NON-MARKOVIAN QUANTUM DYNAMICSFROM ENVIRONMENTAL RELAXATION

S. Possanner and B.A. Stickler

published in Physical Review A, PRA 85, 062115 (2012).

Abstract. We consider the dynamics of composite quantum systems inthe particular case that the state operator relaxes towards the BORN approx-imation. For this we augment the VON NEUMANN equation by a relaxationoperator imposing a finite relaxation time τr. Under the premise that therelaxation is the dominant process we obtain a hierarchy of non-Markovianmaster equations. The latter arises from an expansion of the total state oper-ator in powers of the relaxation time τr. In the BORN-MARKOV limit τr → 0the LINDBLAD master equation is recovered. Higher order contributions en-able a systematic treatment of correlations and non-Markovian dynamics ina recursive manner.

11.1 Introduction

The notion of quantum dissipation and decoherence arising from system-environment coupling is becoming increasingly important in many branchesof physics such as quantum computation [107], quantum optics [108], orsemiconductor spintronics [109]. The progress in atomic and molecular

137

138 Chapter 11: PRA 85, 062115 (2012).

interferometry made over the last decade [110, 111, 112] enables the test-ing of these important concepts of the theory of open quantum systems[19, 113, 114]. The latter is the most prominent tool for tackling such funda-mental problems as the collapse of the wave function during measurement[115, 116] or the transition between the micro- and the macroscopic worldin general [117].

The peculiar nature of quantum states (coherent, delocalized, correlated,entangled) makes the treatment of non-equilibrium processes considerablymore complicated than in the classical case. The usual approach is to startfrom a closed quantum system consisting of interacting degrees of freedomA and B. The state operator ρ of the composite system AB undergoes uni-tary (Hamiltonian) time evolution,

∂tρ = −i[H, ρ] , (11.1)

where H denotes the system’s Hamiltonian, the square brackets [·, ·] standfor the commutator and we set the reduced PLANCK constant ~ = 1. In thecomposite state spaceH = HA ⊗HB, the most general form of the Hamilto-nian H reads

H = HA ⊗ 1B + 1A ⊗HB +HI , (11.2)

where the operator subscript A(B) indicates an operator acting inHA (HB),1A (1B) denotes the identity and the operator HI accounts for the interac-tions between A and B. Taking the partial trace, trB (·), over the subsystemB in the VON NEUMANN equation (11.1) yields the exact equation of motionfor the “relevant” degrees of freedom A, i.e.

∂tρA = −i[HA, ρA]− trB (i[HI , ρ]) , (11.3)

where we introduced the reduced state operator ρA via

ρA := trB (ρ) . (11.4)

In general, the reduced equation of motion (11.3) is an integro-differentialequation, featuring memory effects in B that cause the second term on theright-hand-side to be non-local in time. It describes the subsystem A asan open quantum system that exchanges energy with the environment B.In the special case of Markovian time evolution, memory effects becomenegligible and equation (11.3) takes on the form

∂tρA = LρA . (11.5)

Chapter 11: PRA 85, 062115 (2012). 139

Here, the operatorL is the infinitesimal generator of a dynamical semigroup[118, 119, 120]. In its most general form L is given by [20]

LρA = −i[HeffA , ρA]

+K−1∑k=0

Γk

(L†kρALk −

1

2L†kLkρA −

1

2ρAL

†kLk

).

(11.6)

where HeffA is an effective Hamiltonian, Γk ≥ 0 are transition rates (chan-

nels) and Lk is an operator basis in the K-dimensional space 1 of hermitianoperators in HA. Equation (11.5) is commonly referred to as a master equa-tion of LINDBLAD form, or LINDBLAD master equation. The second term onthe right hand side of the generator (11.6) may account for quantum deco-herence as well as dissipation in A due to interactions with its environmentB. Master equations of the LINDBLAD form (11.5) are frequently encoun-tered in various fields of quantum physics, in particular in the context ofquantum Brownian motion or quantum optics [121, 122, 123, 124, 125, 126,127, 128, 129].

The LINDBLAD master equation is obtained by performing Markovianapproximations to the exact dynamics (11.3). This usually means that a typ-ical parameter α of the composite system such as the correlation time, massratio or timescale ratio, tends towards zero or infinity [19, 113, 114]. The dy-namics (11.5) are, therefore, only exact in the respective limiting case, whichmight not necessarily be a good approximation of the physical system con-sidered. It is, thus, desirable to study the corrections to the Markovian case(11.5) which arise when the limiting parameter mentioned above is small,but not zero (or large, but still finite). One expects to obtain non-Markoviancorrections which account for correlations between system A and environ-ment B. The enhanced model will be more difficult to treat, but it shouldstill be much less involved than a full treatment of the composite systemAB.

Over the last decade, considerable effort has been put into the deriva-tion of non-Markovian corrections to the LINDBLAD master equation (11.5)[130, 131, 132, 133, 134, 135, 136, 137]. Two well-established approachesproved to be particularly fruitful, i.e. the projection operator technique andthe time-convolutionless projection operator method. The projection op-erator technique results in the NAKAJIMA-ZWANZIG equation [21, 22, 138]which is an exact equation for open quantum systems and its solution iscomparably difficult to the solution of Eq. (11.1). A series expansion of

1To the knowledge of the authors, rigorous proofs for the existence of (11.6) in the caseK =∞ and for unbounded Hamiltonians H are still lacking.

140 Chapter 11: PRA 85, 062115 (2012).

the NAKAJIMA-ZWANZIG integral kernel yields non-Markovian evolutionequations which are non-local in time [19]. This drawback is remedied byelimination of the non-locality in time with the help of a back propagator asdeveloped by SHIBATA et al. [139, 140]. The resulting equation is referredto as the time-convolutionless master equation and it provides the meansfor the derivation of time-local, non-Markovian contributions to Eq. (11.5)in ascending orders of the coupling strength between degrees of freedom Aand B.

It might be interesting to note that the projection techniques describedabove have been motivated by a technical point of view. The aim is to elimi-nate from the VON NEUMANN equation the irrelevant degrees of freedomBwithout employing any further assumptions on the dynamics of the physi-cal system. Subsystem B is usually described by an arbitrary reference stateχB, which is why a physical interpretation of the results obtained appearsto be difficult. Nevertheless, these methods are exact, but difficult to treat inthe general case.

In this work we present an alternative approach towards non-Markoviancontributions to Eq. (11.5). This approach is based on a particular physicalpicture and is closely related to the diffusion limit of the linear BOLTZMANNequation in classical kinetic theory [24, 141, 23]. In our approach, the non-Markovianity arises from the relaxation of parts of the environment towardsan equilibrium state χB on a finite timescale τr. By explicitly accounting forthis relaxation process by means of a relaxation operator Q in Eq. (11.1), weuse a HILBERT expansion technique to derive a hierarchy of master equa-tions for subsystem A. In the limit τr → 0, we retrieve the LINDBLAD masterequation (11.5). It has to be emphasized that by introducing the operator Qwe depart from the exact description of the system’s dynamics. However,this approach as well as the resulting equations of motion follow a clearphysical picture and, therefore, allow for an easy interpretation.

The paper is organized as follows. In section 11.2 we specify the physicalpicture of our approach. Moreover, we introduce the relaxation operator Qand a scaled version of the resulting equation of motion for the state oper-ator ρ of the composite system AB. In section 11.3 we employ a HILBERTexpansion of ρ and derive a hierarchy of master equations for the reducedstate operator ρA. Section 11.4 contains a discussion of the results obtained.The paper is summarized in section 11.5 and a short outlook for possiblefuture work is presented. A mathematical analysis of the relaxation oper-ator Q as well as the proof of existence and uniqueness of solutions of theequation of motion for ρ can be found in the appendices 11.A and 11.B, re-spectively. In App. C we explicitly compute the second order contribution

Chapter 11: PRA 85, 062115 (2012). 141

to the hierarchy of master equations obtained.

11.2 Physical Model and Scaling

A common approximation to the state operator ρ of a composite system ABin which subsystem A obeys Markovian dynamics is

ρ(t) = ρA(t)⊗ χB , (11.7)

where ρA(t) is the solution of Eq. (11.5) and χB is some reference state in theenvironment B. This approximation is known as the BORN approximation.It clearly depends on the physical system whether or not the state (11.7)represents a good approximation to the exact solution of Eq. (11.1). For sys-tems where this is not the case, it might be desirable to have corrections tothe BORN approximation that can be expanded in orders of a typical param-eter α which is zero in the Markovian limit. In order to achieve this, let usregard the BORN approximation as a sort of equilibrium state of the compos-ite system AB and let τr denote the corresponding relaxation time. We shallexplicitly account for the relaxation of ρ towards the BORN approximationby rewriting the equation of motion (11.1) as

∂tρ = −i[H, ρ] +1

τrQ(ρ) . (11.8)

Here we introduced the relaxation operator Q as

Q(ρ) := trB (ρ)⊗ χB − ρ , (11.9)

where trB (χB) = 1 and we remark that

trB (Q(ρ)) = 0 ∀ ρ . (11.10)

In what follows the limit τr → 0 in Eq. (11.8) will be denoted as the BORN-MARKOV limit. Hence, taking in Eq. (11.8) the partial traces over degrees offreedom A and B, respectively, yields

∂tρA = −i[HA, ρA]− trB (i[HI , ρ]) , (11.11)

∂tρB = −i[HB, ρB]− trA (i[HI , ρ]) +χB − ρB

τr, (11.12)

where trA (ρ) = ρB is the reduced state operator of the environment B. Al-though Eqs. (11.3) and (11.11) might seem to be identical on a first glance,the total state operator ρ will be different in these two equations, because

142 Chapter 11: PRA 85, 062115 (2012).

Figure 11.1: (Color online) Schematic representation of an observed “Sys-tem” A that interacts with an “Environment” that consists of two parts Band C, respectively. Subsystem B interacts with A on a time scale τI . More-over, on a timescale τr, B may exchange energy (information) with sub-system C, which is assumed to be completely isolated from the observedsystem A.

of the introduction of the relaxation operator Q in Eq. (11.8). It depends onthe particular situation whether the Hamiltonian HB contains interactionswithin the environmentB or whether these interactions have been absorbedinto the relaxation term Eq. (11.12).

We point out that Eq. (11.8) does not conserve the total energy of thecomposite system AB, which is, consequently, not a closed quantum sys-tem. Eq. (11.8) rather resembles a configuration in which subsystem B iscoupled to a third subsystem C, which can be regarded as isolated from A.This situation is sketched in Figure 11.1. Hence, the environment of A is acomposite system BC. In this case Eq. (11.8) results from tracing out thedegrees of freedom C from the total equation of motion for the compositesystem ABC.

The remaining effect of subsystem C is that it relaxes the state operatorρB to a particular equilibrium state χB on a timescale τr. In the case that C isa reservoir, i.e. features an infinite number of degrees of freedom, χB couldbe the minimizer of a certain entropy functional in B. For instance, systemA could contain the conduction band electrons in a semiconductor, whereassystem B describes the lattice phonons coupled to an external heat bath C.On the other hand, one could imagine that a probe C prepares the stateχB with a mean frequency 1/τr. Such a scenario could be realized by twointeracting spins, where one of the two spins is constantly monitored andprepared to be in state χB. Another possible scenario could be a compositequantum system, where subsystem A interacts solely with a part of the totalenvironment due to short range iteractions.

If the state χB is a pure state the corresponding state of the compositesystem AB must be uncorrelated [142], i.e. of the form (11.7). In writing Eq.

Chapter 11: PRA 85, 062115 (2012). 143

(11.8), we presuppose that even for a mixed state χB, the coupling of C to Bleads to decorrelations in AB. Thus, correlations between A and B due tothe interaction HI are gradually destroyed on a timescale τr by the couplingof B to C.

The aim of the following sections is to find approximate solutions to Eq.(11.8) in cases where the time scale τr is small compared to all other relevanttimescales of the system. For technical reasons which will become clear inthe next section, let us introduce the mean-field operator Hmf

A acting inHA,

HmfA := trB (HIχB) . (11.13)

We define furthermore,

HI := HI −HmfA ⊗ 1B ,

HA := HA +HmfA ,

HAB := HA ⊗ 1B + 1A ⊗HB ,

(11.14)

and rewrite Eq. (11.8) with the help of the definitions (11.14):

∂tρ = −i[HAB, ρ]− i[HI , ρ] +1

τrQ(ρ) . (11.15)

As a next step we present a scaled version of Eq. (11.15) which is appro-priately suited for the BORN-MARKOV limit. For this suppose one can de-fine a timescale τAB induced by HAB as well as a timescale τI induced byHI . The former timescale is a characteristic for the evolution of the isolated,mean-field-corrected subsystems A and B, respectively, whereas the latteris a characteristic for the mean-field-corrected interaction between A and B.The introduction of a typical parameter α 1 via

τIτAB

= O(α) ,τrτAB

= O(α2) , (11.16)

and of the timescale τAB to describe the dynamics,

t′ =t

τAB, (11.17)

yields

∂t′ρ = −i[HAB, ρ]− i

α[HI , ρ] +

1

α2Q(ρ) . (11.18)

Equation (11.18) corresponds to the equation of motion (11.15) for the com-posite system AB in the BORN-MARKOV scaling. We remark that sinceα 1, Eq. (11.18) implies strong interactions between system A and en-vironment B while the relaxation towards the BORN approximation is thedominant process.

144 Chapter 11: PRA 85, 062115 (2012).

11.3 Derivation of Master Equations

11.3.1 HILBERT Expansion of the State Operator

It is the aim of this section to search for an approximate solution ρ(α)(t′) ofEq. (11.18) with initial condition ρ(α)(0) = ρ

(α)i . For small values of α this

approximate solution is supposed to be close to the exact solution ρ. In whatfollows we write t instead of t′ for the scaled time (11.17). Thus, we considerthe following initial value problem,

∂tρ(α) = −i[HAB, ρ

(α)]− i

α[HI , ρ

(α)] +1

α2Q(ρ(α)) ,

ρ(α)(0) = ρ(α)i . (11.19)

The first question of interest is whether or not the initial value problem(11.19) has a unique solution. The proof of existence and uniqueness of asolution ρ(α)(t) on a finite time interval [0, T ] to the initial value problem(11.19) is given in App. 11.B.

Let us proceed with the approximate solution of Eq. (11.19). We shallemploy a series expansion of the solution in powers of α, thus assumingρ(α) to be analytic in α within a certain radius around α = 0. By insertingthe HILBERT expansion into Eq. (11.19),

ρ(α) =∞∑n=0

αnρn , (11.20)

subsequently multiplying by α2 and sorting the terms in orders of α, oneobtains the following system of equations

Q(ρ0) = 0 , (11.21a)

Q(ρ1) = i[HI , ρ0] , (11.21b)

Q(ρ2) = ∂tρ0 + i[HAB, ρ0] + i[HI , ρ1] , (11.21c)

Q(ρ3) = ∂tρ1 + i[HAB, ρ1] + i[HI , ρ2] , (11.21d)

Q(ρn) = ∂tρn−2 + i[HAB, ρn−2] + i[HI , ρn−1] , (11.21e)

for n ≥ 4 .

We remark that even though Eqs. (11.21c) and (11.21d) are of the generalform (11.21e), they have been written explicitly for the purpose of a betterunderstanding of the concepts elaborated in this section.

Chapter 11: PRA 85, 062115 (2012). 145

Regarding Eqs. (11.21), the question immediately arises whether or notthe system is well-posed, i.e. whether or not the right-hand-sides of Eqs.(11.21) lie in the image of the operator Q, such that a solution ρ(α) of theform (11.20) can be obtained, at least in principle. It is therefore necessaryto investigate the operator Q, defined in Eq. (11.9), in more detail. We notein passing that Q is very similar to one of the projection operators used inthe projection operator techniques mentioned in the introduction [21, 22,139, 140]. However, strictly speaking it is not a projection operator sinceQ2 = −Q.

For the subsequent analysis, let us introduce the following notations:

• H : space of hermitian operators inH.

• HA,B: space of hermitian operators inHA,B.

Moreover, let D(Q) ⊂ H stand for the domain of Q, thus the operator Q isa mapping

Q : D(Q)→H . (11.22)

We assume that D(Q) is a linear space (a detailed analysis of the operator Qcan be found in App. 11.A). Here, we briefly repeat the main results of App.11.A needed in what follows:

(i) Let KerQ denote the kernel of Q. One has

D(Q) = KerQ⊕ (KerQ)⊥ , (11.23)

where (KerQ)⊥ denotes the space orthogonal to the kernel ofQ. Henceany X ∈ D(Q) can be decomposed into

X = XKer +X⊥ , (11.24)

where XKer ∈ KerQ and X⊥ ∈ (KerQ)⊥.

(ii) For XKer ∈ KerQ one has

XKer = XA ⊗ χB , XA ∈HA. (11.25)

(iii) For X⊥ ∈ (KerQ)⊥ one has

trB(X⊥)

= 0 . (11.26)

(iv) Let ImQ denote the image of Q. One has

ImQ = (KerQ)⊥ . (11.27)

146 Chapter 11: PRA 85, 062115 (2012).

(v) The equation Q(X) = Y is well-posed (and thus has a solution) if Y ∈(KerQ)⊥. Moreover, it has a unique solution in (KerQ)⊥ denotedX⊥.It follows immediately from Eq. (11.26) that this solution is given by

X⊥ = −Y . (11.28)

We begin now with the investigation of well-posedness of Eqs. (11.21).We use Eqs. (11.24) and (11.25) to decompose each term ρn of the HILBERTexpansion,

ρn = ρ(n)A ⊗ χB + ρ⊥n . (11.29)

Moreover, we note the important property

trB

(i[HI , ρ

(n)A ⊗ χB]

)= 0 ∀ ρ(n)

A ∈HA , (11.30)

which is a consequence of the introduction of the mean-field operator, c.f.Eq. (11.14). Let us take the trace over the degrees of freedom B in Eqs.(11.21) and let us, furthermore, use the property (11.30) to obtain

0 = trB

(i[HI , ρ

⊥0 ]), (11.31a)

0 = ∂tρ(0)A + i[HA, ρ

(0)A ] + trB

(i[HI , ρ

⊥1 ]), (11.31b)

0 = ∂tρ(1)A + i[HA, ρ

(1)A ] + trB

(i[HI , ρ

⊥2 ]), (11.31c)

0 = ∂tρ(n−2)A + i[HA, ρ

(n−2)A ] + trB

(i[HI , ρ

⊥n−1]

),

for n ≥ 4 , (11.31d)

where we omitted the result 0 = 0 obtained from Eq. (11.21a). From prop-erty (v) of the relaxation operator Q it is clear that the system (11.21) iswell-posed if and only if Eqs. (11.31) are fulfilled. In what follows we shallpresent an inductive proof that this can be indeed achieved. Furthermore,we shall prove that a system consisting of the firstN ∈ N equations (11.31) isclosed and that its solution can be computed recursively from Eqs. (11.21).

We know a priori that Eq. (11.21a) is well-posed and that its solution isobtained as

ρ0 = ρ(0)A ⊗ χB , ρ⊥0 = 0 . (11.32)

Assuming Eqs. (11.21b) to (11.21e) are also well-posed, we can employ Eq.

Chapter 11: PRA 85, 062115 (2012). 147

(11.28) to determine their unique solutions ρ⊥n ∈ (KerQ)⊥ as

ρ⊥1 = −i[HI , ρ0] , (11.33a)

ρ⊥2 = −∂tρ0 − i[HAB, ρ0]− i[HI , ρ1] , (11.33b)

ρ⊥3 = −∂tρ1 − i[HAB, ρ1]− i[HI , ρ2] , (11.33c)

ρ⊥n = −∂tρn−2 − i[HAB, ρn−2]− i[HI , ρn−1] , (11.33d)

for n ≥ 4 .

From Eq. (11.32) we deduce that Eq. (11.31a) is fulfilled trivially and, thus,Eq. (11.21b) is well-posed. This enables us to insert the result (11.33a) intoEq. (11.31b) to obtain

∂tρ(0)A = −i[HA, ρ

(0)A ]− trB

([HI , [HI , ρ

(0)A ⊗ χB]]

). (11.34)

Equation (11.34) is a master equation of LINDBLAD form, as will be elab-orated later in more detail in subsection 11.3.2. The second term on theright-hand-side of Eq. (11.34) is the dissipative part; thus, let us define the“dissipator” D : D(D) ⊂HA →HA,

D(XA) := −trB

([HI , [HI , XA ⊗ χB]]

). (11.35)

Since XA ⊗ χB = XKer ∈ KerQ, the operator D can also be viewed asa mapping from KerQ to HA. Using the short notation (11.35), equation(11.34) reads

∂tρ(0)A = −i[HA, ρ

(0)A ] +D(ρ

(0)A ) . (11.36)

For now we suppose the LINDBLAD master equation (11.36) to have a uniquesolution. This assumption is sufficient for completing the inductive proof ofwell-posedness of Eqs. (11.21), as will become transparent in the remainderof this subsection.

A first consequence of well-posedness of the LINDBLAD equation (11.36)is that Eq. (11.21c) is also well-posed and, thus, that its unique solutionρ⊥2 ∈ (KerQ)⊥ given in Eq. (11.33b) is valid. Inserting this into Eq. (11.31c)results in

∂tρ(1)A =− i[HA, ρ

(1)A ] (11.37)

+ trB

(i[HI , ∂tρ0 + i[HAB, ρ0] + i[HI , ρ1]]

).

148 Chapter 11: PRA 85, 062115 (2012).

This equation can be simplified by use of Eq. (11.32), the property (11.30),and the result (11.33a) which yields

∂tρ(1)A = −i[HA, ρ

(1)A ] +D(ρ

(1)A ) + S1 , (11.38)

withS1 =− trB

(i[HI , ρ

(0)A ⊗ [HB, χB]

)+ trB

(i[HI , [HI , [HI , ρ

(0)A ⊗ χB]]]

).

(11.39)

We remark that the first two terms on the right-hand-side of Eq. (11.38) formexactly the LINDBLAD generator from equation (11.36). The additional termS1 does not depend on ρ(1)

A and, thus, can be viewed as a well-defined, localsource term. Hence, Eq. (11.38) has a unique solution. We deduce that Eq.(11.21d) is well-posed and its unique solution ρ⊥3 ∈ (KerQ)⊥ given in Eq.(11.33c) is valid. One can already see the evolving pattern that will resultin the well-posedness of the entire system (11.21). In order to complete theproof we shall proceed by induction. Therefore, suppose that Eqs. (11.21e)are well-posed up to order n−1. The solution to the (n−1)-th order equationis then written as

ρn−1 = ρ(n−1)A ⊗ χB + ρ⊥n−1 . (11.40)

Due to Eq. (11.33d), ρ⊥n−1 is given by

ρ⊥n−1 = −∂tρn−3 − i[HAB, ρn−3]− i[HI , ρn−2] . (11.41)

The aim is now to specify under which condition the n-th order Eq. (11.21e)is also well-posed. From Eq. (11.31d) one deduces that this condition reads

∂tρ(n−2)A = −i[HA, ρ

(n−2)A ]− trB

(i[HI , ρ

⊥n−1]

). (11.42)

Inserting Eq. (11.41) into Eq. (11.42) yields

∂tρ(n−2)A =− i[HA, ρ

(n−2)A ] + trB

(i[HI , ∂tρn−3]

)− trB

([HI , [HAB, ρn−3]]

)(11.43)

− trB

([HI , [HI , ρn−2]]

).

Again we employ the decompositions

ρn−2 = ρ(n−2)A ⊗ χB + ρ⊥n−2 , (11.44)

ρn−3 = ρ(n−3)A ⊗ χB + ρ⊥n−3 , (11.45)

Chapter 11: PRA 85, 062115 (2012). 149

and profit from the fact that ρ⊥n−2 ∈ (KerQ)⊥ is uniquely defined by Eq.(11.28), with the result

ρ⊥n−2 = −∂tρn−4 − i[HAB, ρn−4]− i[HI , ρn−3] . (11.46)

We note that we were able to obtain Eqs. (11.41), (11.44) and (11.45) becausewe supposed Eq. (11.21e) to be well-posed up to order n− 1. Moreover,

∂tρn−3 = ∂tρ(n−3)A ⊗ χB + ∂tρ

⊥n−3 , (11.47)

and thus property (11.30) yields

trB

(i[HI , ∂tρn−3]

)= trB

(i[HI , ∂tρ

⊥n−3]

). (11.48)

The decompositions (11.44) and (11.45) are applied to Eq. (11.43) and oneobtains, also using Eq. (11.48),

∂tρ(n−2)A =− i[HA, ρ

(n−2)A ] + trB

(i[HI , ∂tρ

⊥n−3]

)− trB

([HI , [HAB, ρ

(n−3)A ⊗ χB + ρ⊥n−3]]

)− trB

([HI , [HI , ρ

⊥n−2]]

)(11.49)

− trB

([HI , [HI , ρ

(n−2)A ⊗ χB]]

).

In the last term on the right-hand-side one can introduce the definition(11.35) of the dissipator D in order to obtain, finally

∂tρ(n−2)A = −i[HA, ρ

(n−2)A ] +D(ρ

(n−2)A ) + Sn−2 , (11.50)

where Sn−2, n ≥ 4, is given by

Sn−2 =trB

(i[HI , ∂tρ

⊥n−3]

)−trB

([HI , ρ

(n−3)A ⊗ [HB, χB]]

)−trB

([HI , [HAB, ρ

⊥n−3]]

)+trB

([HI , [HI , ∂tρn−4]]

)+trB

(i[HI , [HI , [HAB, ρn−4]]]

)+trB

(i[HI , [HI , [HI , ρn−3]]]

).

(11.51)

150 Chapter 11: PRA 85, 062115 (2012).

Here ρ⊥n−2 was expressed with the help of Eq. (11.46). Equation (11.50) iscalled the (n − 2)-th order master equation for n ≥ 4. It arises solely fromthe requirement that Eq. (11.21e) of order n is well-posed. Besides the LIND-BLAD generator, which has already been found in the zeroth and first ordermaster equations (11.36) and (11.38), respectively, Eq. (11.50) comprises theadditional source term Sn−2. This term depends solely on operators ρk ob-tained from Eqs. (11.21e) of order k < n − 2. Therefore, under the premisethat the LINDBLAD master equation (11.36) yields sufficiently well-behavedsolutions, Eqs. (11.50) are solvable up to arbitrary order n, which provesthe well-posedness of Eqs. (11.21). It follows, moreover, from the particu-lar form of the source term Sn−2 that the first n− 2 equations (11.50) form aclosed system of equations, in which solutions can be computed recursively.For n = 4, the source term (11.51) is evaluated in App. 11.C.

11.3.2 LINDBLAD Master Equation

We shall briefly elaborate on the LINDBLAD master equation (11.36). Thisequation will also be called zeroth order master equation. Recalling thatHI = HI −Hmf

A ⊗ 1B, a straightforward calculation results in the followingform of the dissipator (11.35),

D(ρ(0)A ) = trB

(2HIρ0HI −H2

I ρ0 − ρ0H2I

)−2Hmf

A ρ(0)A Hmf

A +(HmfA

)2

ρ(0)A + ρ

(0)A

(HmfA

)2

.(11.52)

We note that the interaction Hamiltonian HI can be written in the form [19]

HI =∑i

Ai ⊗Bi , (11.53)

where Ai ∈HA and Bi ∈HB. Therefore, the mean-field operator reads

HmfA =

∑i

AitrB (BiχB) . (11.54)

By inserting relations (11.53) and (11.54) into Eq. (11.52), one obtains

D(ρ(0)A ) =

∑ij

Γij

(2Ajρ

(0)A Ai − AiAjρ(0)

A − ρ(0)A AiAj

), (11.55)

where the coefficients Γij are defined as

Γij = 〈BiBj〉χB− 〈Bi〉χB

〈Bj〉χB. (11.56)

Chapter 11: PRA 85, 062115 (2012). 151

Here we made use of the standard definition of correlation functions

〈BiBj〉χB= trB (BiBjχB) , (11.57)

and expectation values,

〈Bi〉χB= trB (BiχB) . (11.58)

In fact, the coefficients (11.56) stand for the covariance of Bi and Bj in thestate χB. In summary, the zeroth order master equation for the reducedsystem A can be written as

∂tρ(0)A =− i

[HA, ρ

(0)A

]− i∑i

〈Bi〉χB

[Ai, ρ

(0)A

](11.59)

+∑ij

Γij

(2Ajρ

(0)A Ai − AiAjρ(0)

A − ρ(0)A AiAj

).

This equation can be transformed into the LINDBLAD form (11.5) by expand-ing the operators Ai, Aj in an appropriate basis Lk ∈ HA. The second termon the right-hand-side of Eq. (11.59) represents an energy shift induced bythe mean-field approximation of the interaction between system A and en-vironment B. We emphasize that this energy shift has to occur in the zerothorder equation of the reduced system, because otherwise the HILBERT ex-pansion (11.20) would result in an ill-posed equation (11.21b). For the samereason the coefficients (11.56) stand for the covariance of Bi and Bj , ratherthan their correlation.

11.4 Discussion

Let us briefly summarize what has been accomplished so far. The goal ofthe present work was to find approximate solutions ρ(α) to Eq. (11.19) inthe case that the parameter α is small but not zero. For this, we invoked aHILBERT expansion of the form

ρ(α) =∞∑n=0

αn(ρ

(n)A ⊗ χB + ρ⊥n

), (11.60)

where ρ(n)A ⊗ χB ∈ KerQ and ρ⊥n ∈ (KerQ)⊥. We note that in this represen-

tation, the reduced state operator reads

trB(ρ(α))

:= ρ(α)A =

∞∑n=0

αnρ(n)A . (11.61)

152 Chapter 11: PRA 85, 062115 (2012).

After inserting the ansatz (11.60) into Eq. (11.19) we required equality of theleft- and the right-hand-side of the equation in each power αn. The furtherrequirement of well-posedness of the resulting equations (11.21) gave riseto the following hierarchy of master equations for the ρ(n)

A :

∂tρ(0)A = Lρ(0)

A , (11.62a)

∂tρ(1)A = Lρ(1)

A + S1 , (11.62b)

∂tρ(n−2)A = Lρ(n−2)

A + Sn−2 for n ≥ 4 . (11.62c)

Here, L is the generator of a dynamical semigroup of the LINDBLAD form(11.6), specified in Eq. (11.59). The source terms S1 to Sn−2 are given by Eqs.(11.39) and (11.51), respectively. Under the initial conditions

ρ(0)A (0) = ρ

(0)A,i ∈HA , (11.63a)

ρ(1)A (0) = ρ

(1)A,i ∈HA , (11.63b)

ρ(n−2)A (0) = ρ

(n−2)A,i ∈HA for n ≥ 4 , (11.63c)

the formal solution of Eqs. (11.62) can be obtained via DUHAMEL’s formula

ρ(0)A (t) = eLtρ

(0)A,i , (11.64a)

ρ(1)A (t) = eLtρ

(1)A,i +

∫ t

0

ds eL(t−s)S1(s) , (11.64b)

ρ(n−2)A (t) = eLtρ

(n−2)A,i +

∫ t

0

ds eL(t−s)Sn−2(s)

for n ≥ 4 , (11.64c)

where eLt with t ≥ 0 denotes the dynamical semigroup generated by L.Under the assumption that the power series (11.61) converges for all t ≥ 0(which is reasonable for small α), one can perform the sum in the results(11.64) in order to obtain

ρ(α)A (t) = eLtρ

(α)A,i +

∫ t

0

ds eL(t−s)S(α)(s) . (11.65)

Here we defined the initial values ρ(α)A,i and the operator S(α) as

ρ(α)A,i :=

∞∑n=0

αnρ(n)A,i , (11.66)

S(α) :=∞∑n=1

αnSn , (11.67)

Chapter 11: PRA 85, 062115 (2012). 153

where we made use of S0 = 0. The integral on the right-hand-side of Eq.(11.65) makes the non-Markovianity of the time evolution of the reducedstate (11.61) transparent, since S(α) depends on ρ(α)

A in a rather complicatedway. In the BORN-MARKOV limit α → 0, the term S(α) vanishes and onerecovers the Markovian dynamics for the reduced system induced by theLINDBLAD generator L.

In writing the formal solution (11.65), we note that the total state opera-tor ρ(α) of the composite system has been determined entirely. This followsfrom the fact that the terms ρ⊥n in Eq. (11.60) are uniquely defined by Eqs.(11.33). Therefore, if the power series (11.60) is convergent for all t ≥ 0(which is reasonable for small α), it represents the unique solution of theinitial value problem (11.19). Let us focus briefly on the “orthogonal” termsρ⊥n . With the definition

ρ⊥ :=∞∑n=1

αnρ⊥n , (11.68)

where we used that ρ⊥0 = 0, we point out that the contribution (11.68) to thesolution (11.60) is traceless,

tr(ρ⊥)

= 0 . (11.69)Therefore, it solely describes correlations between system A and environ-ment B. Moreover, it is obvious that the power series (11.68) vanishes in theBORN-MARKOV limit α→ 0, thereby confirming the absence of correlationsin the Markovian time evolution.

At first glance it might seem that nothing has been gained because theevaluation of Eqs. (11.61) and (11.68) requires the calculation of an infi-nite number of terms. However, provided that these power series converge,their benefits can be found in the fact that one can successively approach theexact solution ρ(α) of Eq. (11.19) until a desired accuracy has been reached.For instance, truncating the series (11.60) after two terms appears to be avalid approximation in the case that α 1. In general, once the LINDBLAD

master equation (11.59) has been solved for ρ(0)A , the source terms Sn and

thus the higher order corrections ρ(n)A and ρ⊥n can be computed recursively

in order to achieve the desired accuracy. In this way, correlations betweensystem A and environment B can be incorporated rather easily in the re-duced dynamics for A.

11.5 Conclusion

In this work we employed a HILBERT expansion in order to obtain approx-imate solutions of a VON NEUMANN equation which was augmented by a

154 Chapter 11: PRA 85, 062115 (2012).

relaxation operator Q. This operator relaxes the state operator of a compos-ite quantum systemAB towards the BORN approximation on a timescale τr.This approach resulted in the hierarchy of master equations (11.62) for thereduced state operator ρA. In zeroth order, which accounts for the exact dy-namics in the BORN-MARKOV limit τr → 0, a master equation of LINDBLADform was recovered. The transition rates derived are exactly the covariancefunctions between different components of the environmental part of theinteraction Hamiltonian HI in the reference state χB. Moreover, the dis-cussed approach allows to systematically incorporate correlations and non-Markovian effects in the reduced system dynamics. These effects can becalculated recursively from the solution to the LINDBLAD master equation,as was achieved in Eqs. (11.64). Such an approach might be advantageousin physical systems for which the BORN approximation is nearly justifiedor for which a full treatment on the basis of projection techniques is far toocomplex.

We point out that the non-Markovian quantum dynamics derived herefollow from a transparent physical picture, namely the relaxation of the to-tal state operator towards the BORN approximation. This appears to be areasonable scenario, for instance, if the environment contains degrees offreedom that are completely isolated from the observed system, c.f. Fig.11.1. Nevertheless, the results obtained are merely valid under three as-sumptions:

• the coupling of different degrees of freedom (B and C) of the envi-ronment diminish correlations between observed system and environ-ment,

• the BORN relaxation is by far the fastest process in the composite sys-tem,

• strong system-environment interaction (singular coupling scaling).

The derived model could be enhanced rather easily by replacing the relax-ation operator Q, introduced in Eq. (11.9), by a more sophisticated dissipa-tive term for subsystemB; for instance by a LINDBLAD dissipator. However,this would result in minor changes only, because the hierarchy (11.31) is ageneral result which does not depend on the particular form of the relax-ation operator Q. This hierarchy is a mere consequence of the complete iso-lation of the observed systemA from the environmental reservoir (or probe)C, which manifests itself in the relation trB (Q(ρ)) = 0.

Chapter 11: PRA 85, 062115 (2012). 155

Ackowledgments. The authors are very grateful to E. SCHACHINGER forcarefully reading the manuscript. We also thank W. PÖTZ and C. NEG-ULESCU for precious input and fruitful discussions. The first author ac-knowledges the support from the Austrian Science Fund, Vienna, underthe contract number P21326-N16; the second author was supported by theAustrian Science Fund (FWF): P221290-N16.

Note added in proof. We are thankful to A. ARNOLD for pointing out thatthe proof in Appendix 11.B of existence and uniqueness of a solution to theinitial value problem (11.19) is achieved by Theorem 1.1 of Chapter 3 in[143].

Appendix.

11.A Analysis of the Operator Q

Defining for X, Y ∈H the scalar product

(X, Y ) := tr(χ−1/2Xχ−1/2Y

), (11.70)

where χ ∈H satisfies

χ = χA ⊗ χB , tr (χ) = 1 , (χv, v)H > 0 ∀v ∈ H \ 0 , (11.71)

the space H becomes a HILBERT space 2 with the associated norm

||X|| =√

(X,X) ∀X ∈H . (11.72)

In (11.71), (·, ·)H denotes the scalar product inH. Note that

(X, Y ) = (Y,X) ,

(X,X) ≥ 0 ,(11.73)

∀X, Y ∈H . The positive-definiteness of (11.70) follows from

(X,X) = tr(χ−1/4Xχ−1/4χ−1/4Xχ−1/4

)= tr

(Y 2)≥ 0 ,

(11.74)

2In what follows we denote by H the space of Hermitian operators in H for which||X|| <∞.

156 Chapter 11: PRA 85, 062115 (2012).

where Y = χ−1/4Xχ−1/4 ∈H . Moreover, (11.70) is linear in both arguments.We shall further define a scalar product in HA,

(XA, YA)A := trA

(χ−1/2A XAχ

−1/2A YA

), (11.75)

and denote the corresponding norm by

||XA||A =√

(XA, XA)A , (11.76)

∀XA ∈HA3. For the operator Q defined in Eq. (11.9),

Q(X) = trB (X)⊗ χB −X , X ∈ D(Q) ⊂H , (11.77)

one has the following properties:

(i) Q is linear, bounded, self-adjoint and non-positive.

(ii) For X ∈ KerQ (the kernel of Q) we have

X = XA ⊗ χB , XA ∈HA. (11.78)

Moreover, we have

(KerQ)⊥ = X ∈ D(Q) | trB (X) = 0 . (11.79)

(iii) Let P : H → KerQ denote the orthogonal projection operator ontoKerQ. Then there exists d > 0 such that

− (Q(X), X) ≥ d||X − P(X)||2 (11.80)

∀X ∈ D(Q).

(iv) The image of Q, denoted by ImQ, is closed and we have

ImQ = (KerQ)⊥ . (11.81)

Further, the equation Q(X) = Y has a solution in D(Q) if and only ifY ∈ ImQ. The solution is moreover unique in (KerQ)⊥.

In the following we denote,

XA := trB (X) . (11.82)

3We denote by HA the space of Hermitian operators inHA for which ||X||A <∞.

Chapter 11: PRA 85, 062115 (2012). 157

andQA(X) = trB (Q(X)) = 0 , ∀X ∈ D(Q) . (11.83)

Moreover, in what follows we shall frequently make use of the identity

(XA ⊗ χB, Y ) = (XA, YA)A . (11.84)

We shall now prove the above statements.(i) The linearity of Q is obvious. Let us show that Q is a well-defined,

bounded operator. For this one needs a constant c > 0 such that

||Q(X)||2 ≤ c||X||2 , ∀X ∈ D(Q) . (11.85)

Using the identities (11.83) and (11.84) one obtains

||Q(X)||2 = (Q(X), Q(X))

= (XA ⊗ χB, Q(X))− (X,Q(X))

= (XA, QA(X))A − (X,Q(X))

= −(X,XA ⊗ χB) + (X,X)

= −||XA||2A + ||X||2

≤ ||X||2 ,

(11.86)

which proves (11.85). Therefore Q is bounded (and thus also continous).The self-adjointness follows from

(Q(X), Y ) = (XA ⊗ χB, Y )− (X, Y )

= (XA, YA)A − (X, Y )

= (YA, XA)A − (Y,X)

= (Q(Y ), X)

= (X,Q(Y )) .

(11.87)

In order to prove the non-positivity of Q we estimate the term ||XA||2A usingEq. (11.84) together with the CAUCHY-SCHWARZ inequality,

||XA||2A = (X,XA ⊗ χB)

≤ ||X|| ||XA ⊗ χ||= ||X|| ||XA||A≤ ||X||2 .

(11.88)

It follows the non-positivity of Q,

(Q(X), X) = ||XA||2A − ||X||2 ≤ ||X||2 − ||X||2 = 0 . (11.89)

158 Chapter 11: PRA 85, 062115 (2012).

We remark that Eq. (11.88) implies

X ∈H ⇒ trB (X) ∈HA . (11.90)

(ii) For X ∈ KerQ we have (Q(X), X) = 0 which can be written as

0 = ||XA||2A − ||X||2

= (X,XA ⊗ χB)− (X,X) .(11.91)

The solutions of Eq. (11.91) are given by X = XA ⊗ χB, XA ∈ HA arbitrary.Conversely,

Q(XA ⊗ χB) = XA ⊗ χB −XA ⊗ χB = 0 . (11.92)

Moreover, for Y ∈ (KerQ)⊥, we have (X, Y ) = 0 for X ∈ KerQ and thus

(XA ⊗ χB, Y ) = (XA, YA) = 0 ∀XA ∈HA . (11.93)

Since Eq. (11.93) must hold for arbitrary XA ∈HA we conclude

YA = trB (Y ) = 0 ∀Y ∈ (KerQ)⊥ . (11.94)

(iii) We now prove the coercitivity relation (11.80). For X ∈ KerQ thisrelation is fulfilled trivially becauseQ(X) = 0 andP(X) = X . Now supposeX ∈ (KerQ)⊥. Then, according to Eq. (11.94),

− (Q(X), X) = −||XA||2A + ||X||2 = ||X||2 , (11.95)

which completes the coercitivity proof.(iv) First we show that the image of Q is closed. Let Xn be a sequence in

D(Q) and let Jn be a sequence in ImQ such that Q(Xn) = Jn. Moreover, letJn → J as n→∞. We have to prove that J ∈ ImQ, i.e. that there existsX ∈D(Q) such that Q(X) = J . To any sequence Xn ∈ D(Q) one can constructa corresponding sequence Yn ∈ (KerQ)⊥ by setting Yn = Xn − P(Xn) andone has Q(Xn) = Q(Yn) = Jn. The coercitivity relation then yields

− (Q(Yn − Ym), Yn − Ym) ≥ d||Yn − Ym||2 ∀n,m ∈ N . (11.96)

In addition, from the CAUCHY-SCHWARZ inequality we have

||Q(Yn)−Q(Ym)|| ||Yn − Ym|| ≥ −(Q(Yn − Ym), Yn − Ym) , (11.97)

and consequently

1

d||Q(Yn)−Q(Ym)|| = 1

d||Jn − Jm|| ≥ ||Yn − Ym|| . (11.98)

Chapter 11: PRA 85, 062115 (2012). 159

Since Jn is a CAUCHY sequence in ImQ we obtain that Yn is a CAUCHYsequence in D(Q). By assumption D(Q) is complete and therefore Yn →Y ∈ D(Q). We already proved that Q is continuous, i.e. Q(Yn) → Q(Y ).One obtains Q(Y ) = J with Y ∈ D(Q). Thus, the image of Q is closed andwe have ImQ = (KerQ)⊥.

We finally prove that the equation Q(X) = Y has a unique solution X ∈(KerQ)⊥. It is obvious that there exists a solution if Y ∈ ImQ. Let X besuch a solution, then X − P(X) ∈ (KerQ)⊥ is also a solution. Assume thatthere are two solutions X1, X2 ∈ (KerQ)⊥ such that Q(X1) = Q(X2) = Y .Then

Q(X1)−Q(X2) = Q(X1 −X2) = 0 . (11.99)

It follows that X1 −X2 ∈ KerQ ∩ (KerQ)⊥ = 0 and therefore X1 = X2.

11.B Existence and Uniqueness

In this section we demonstrate the existence and uniqueness of a solutionto the initial value problem (11.19) on the basis of a fixed point argument.Let 0 < T < ∞ and α = 1 for convenience (the proof holds for α > 0). Thefollowing proposition holds:

• Let Q denote the operator defined in Eq. (11.9). Furthermore, letH ∈ H such that the Liouville operator L(·) = −i[H, ·] in H is theinfinitesimal generator of a bounded one-parameter semigroup eLt inH , i.e. one has C > 0 such that

||eLtρ||2H ≤ C||ρ||2H ∀ρ ∈H , t ∈ [0, T ] , (11.100)

where || · ||H denotes the norm (11.72). Then, the intial value problem∂tρ− L(ρ) = Q(ρ)

ρ(0) = ρi ∈H(11.101)

admits a unique solution ρ ∈ L2([0, T ],H ).

In order to prove this claim we denote the norm and the correspondingscalar product in L2([0, T ],H ) by

||ρ||L2 =

∫ T

0

dt ||ρ(t)||H , (11.102)

and

(ρ, σ)L2 =

∫ T

0

dt (ρ(t), σ(t))H , (11.103)

160 Chapter 11: PRA 85, 062115 (2012).

respectively. Let us now define the following fixed point map,

F : L2([0, T ],H )→ L2([0, T ],H ) , σ 7→ ρ , (11.104)

where ρ is the solution of∂tρ− L(ρ) + ρ = trB (σ)⊗ χBρ(0) = ρi ∈H .

(11.105)

In the following, we denote Sσ(t) = trB (σ(t)) ⊗ χB. The formal solution of(11.105) reads

F(σ)(t) = ρ(t) = eT tρ(0) +

∫ t

0

ds eT (t−s)Sσ(s) , (11.106)

where T := L−1. The proof is performed in two steps. First we demonstratethat the mapping (11.104)-(11.106) is well-defined and we prove then thatthe mapping is a contraction, possessing thus a unique fixed point, solutionof equation (11.101). From Eq. (11.100) it follows that

||eT tρ||2H = e−2t||eLtρ||2H ≤ C||ρ||2H ∀t ∈ [0, T ] . (11.107)

Therefore, Eq. (11.106) yields

||ρ(t)||H ≤√C||ρ(0)||H +

√C

∫ t

0

ds ||Sσ(s)||H ,

≤√C||ρ(0)||H +

√C

∫ T

0

ds ||Sσ(s)||H ,

≤√C||ρ(0)||H +

√C

∫ T

0

ds ||σ(s)||H ,

(11.108)

where the last inequality follows from Eq. (11.88),

||Sσ(s)||H = ||σA(s)||A ≤ ||σ(s)||H . (11.109)

Integration of (11.108) over t results in

||ρ||L2 ≤ T√C||ρ(0)||H + T

√C||σ||L2 , (11.110)

which proves that the fixed point map (11.104) is well-defined.In order to show that (11.106) is a contraction we introduce the following

norm in L2([0, T ],H ),

||ρ||2δ :=

∫ T

0

dt e−δt||ρ(t)||2H . (11.111)

Chapter 11: PRA 85, 062115 (2012). 161

For F to be contractive it is required that

||F(σ1)−F(σ2)||2δ ≤ k||σ1 − σ2||2δ , k < 1 . (11.112)

One obtains

||F(σ1)−F(σ2)||2δ =

=

∫ T

0

dt e−δt∫ t

0

ds ||eT (t−s)[Sσ1(s)− Sσ2(s)]||2H

≤ C

∫ T

0

dt e−δt∫ t

0

ds ||Sσ1(s)− Sσ2(s)||2H

= C

∫ T

0

ds

∫ T

s

dt e−δt||Sσ1(s)− Sσ2(s)||2H

≤ C

δ

∫ T

0

ds e−δs||Sσ1(s)− Sσ2(s)||2H .

(11.113)

Due to the inequality (11.88) one has

||Sσ1(s)− Sσ2(s)||2H = ||trB (σ1(s)− σ2(s))⊗ χB||2H= ||trB (σ1(s)− σ2(s)) ||2A≤ ||σ1(s)− σ2(s)||2H ,

(11.114)

and this results in

||F(σ1)−F(σ2)||2δ ≤C

δ||σ1 − σ2||2δ . (11.115)

Here, δ can be chosen in such a way that relation (11.112) is fulfilled. ThusF is a contraction, its unique fixed point is the solution of Eq. (11.101).

11.C Second-Order Contribution

We compute the source term (11.51) for n = 4. For this, we need the solu-tions of Eqs. (11.21a) and (11.21b), namely ρ0 and ρ⊥1 , given by (11.32) and(11.33a), respectively. It follows that

∂tρ0 = ∂tρ(0)A ⊗ χB (11.116)

= −i[HA, ρ(0)A ]⊗ χB +D(ρ

(0)A )⊗ χB ,

and

∂tρ⊥1 = −i[HI , ∂tρ0] (11.117)

= −[HI , [HA, ρ(0)A ]⊗ χB]− i[HI ,D(ρ

(0)A )⊗ χB] .

162 Chapter 11: PRA 85, 062115 (2012).

We now compute all the terms appearing on the right-hand side of Eq.(11.51). Whenever possible, we make use of definition (11.35) of the op-erator D in order to simplify the notation.

1. With the help of Eq. (11.117) the first term on the right-hand-side ofEq. (11.51)

trB

(i[HI , ∂tρ

⊥1 ])

= −trB

([HI , [HI , i[HA, ρ

(0)A ]⊗ χB]]

)+ trB

([HI , [HI ,D(ρ

(0)A )⊗ χB]]

)= D(i[HA, ρ

(0)A ])−D2(ρ

(0)A ) ,

(11.118)

where D2(·) = D(D(·)).

2. The second term results in

− trB

([HI , [HAB, ρ

⊥1 ]])

= trB

(i[HI , [HAB, [HI , ρ0]]]

). (11.119)

3. In the third term we set ρ(n−3)A → ρ

(1)A .

4. Equation (11.116) helps in evaluating term four to

trB

([HI , [HI , ∂tρ0]]

)=

= −trB


(0)A ]⊗ χB]]

)+ trB

([HI , [HI ,D(ρ

(0)A )⊗ χB]]

)= D(i[HA, ρ

(0)A ])−D2(ρ

(0)A ) .

(11.120)

5. We get for term five

trB

(i[HI , [HI , [HAB, ρ0]]]

)=

= trB


(0)A ]⊗ χB]]

)+ trB

(i[HI , [HI , ρ

(0)A ⊗ [HB, χB]]]

)= −D(i[HA, ρ

(0)A ])

+ trB

(i[HI , [HI , ρ

(0)A ⊗ [HB, χB]]]

).

(11.121)

Chapter 11: PRA 85, 062115 (2012). 163

6. Finally, term six becomes

trB

(i[HI , [HI , [HI , ρ1]]]

)=

= trB

([HI , [HI , [HI , [HI , ρ

(0)A ⊗ χB]]]]

)+ trB

(i[HI , [HI , [HI , ρ

(1)A ⊗ χB]]]

).

(11.122)

Adding the results of Eqs. (11.118)-(11.122) gives the desired result for S2:

S2 =− 2D2(ρ(0)A ) +D(i[HA, ρ

(0)A ])

+ trB

(i[HI , [HAB, [HI , ρ

(0)A ⊗ χB]]]

)− trB

([HI , ρ

(1)A ⊗ [HB, χB]

)+ trB

(i[HI , [HI , ρ

(0)A ⊗ [HB, χB]]]

)+ trB

([HI , [HI , [HI , [HI , ρ

(0)A ⊗ χB]]]]

)+ trB

(i[HI , [HI , [HI , ρ

(1)A ⊗ χB]]]

).

(11.123)

Chapter 12

DERIVATION OF A LINEAR COLLISION OP-ERATOR FOR THE SPINORIAL WIGNER EQUA-TION AND ITS SEMICLASSICAL LIMIT

B.A. Stickler and S. Possanner

Physical Review A in print, ARXIV: 1304.4772 [quant-ph]

Abstract. We systematically derive a linear quantum collision operator forthe spinorial WIGNER transport equation from the dynamics of a compositequantum system. For suitable two particle interaction potentials, the par-ticular matrix form of the collision operator describes spin decoherence oreven spin depolarization as well as relaxation towards a certain momentumdistribution in the long time limit. It is demonstrated that in the semiclas-sical limit the spinorial WIGNER equation gives rise to several semiclassi-cal spin-transport models. As an example, we derive the BLOCH equationsas well as the spinorial BOLTZMANN equation, which in turn gives rise tospin drift-diffusion models which are increasingly used to describe spin-polarized transport in spintronic devices. The presented derivation allowsto systematically incorporate BORN-MARKOV as well as quantum correc-tions into these models.

165

166 Chapter 12: ARXIV: 1304.4772 [quant-ph]

12.1 Introduction

The modeling of transport phenomena in electronic devices is one of the ma-jor challenges in modern solid state physics. While in most physical applica-tions a full quantum mechanical treatment by means of the SCHRÖDINGERequation or the VON NEUMANN equation is far too complex, it is a ben-eficial and legitimate approach to employ effective models. Well-knownand prosperous examples are the drift-diffusion equations to treat systemsin local thermal equilibrium and the BOLTZMANN equation (BE) to capturenon-equilibrium phenomena.

It is a major challenge to clarify the simplifactions and approximationsposed in a microscopic theory which lead to such an effective model. Westate spintronics as an example [144, 145] where spin-drift-diffusion equa-tions proved to be a powerful tool for describing spin-polarized transport[146] and spin-transfer torques [147, 148] in magnetic mulitlayers. It hasbeen demonstrated that these equations can be derived from a spinorial BE[149, 150, 23]. Hence, the missing link in a systematic, qualitative under-standing is the derivation of a spinorial BE starting from a full quantummechanical treatment. This is the main goal of the present study.

Let us briefly discuss some well-established results in order to positionthe present work in an appropriate context: The generic form of the scalarBE is

∂tf − h, fx,η = C(f) , (12.1)

where f(x, η, t), f ≥ 0, is a probability distribution on the 2d-dimensionalphase space Rd

x × Rdη, h(x, η) stands for the energy of a non-interacting par-

ticle and h, fx,η = ∇xh · ∇ηf − ∇ηh · ∇xf denotes the POISSON bracketwith respect to the position coordinate x and the momentum coordinate η.The collision operator C on the right-hand-side (rhs) of (12.1) models shortrange interactions between particles or with obstacles, e.g. impurity cen-ters or phonons in case of electronic transport in semiconductor devices.C is usually an integral operator and, moreover, non-linear in case that itdescribes interactions between identical particles or accounts for quantumstatistics. Eq. (12.1) is referred to as the semiclassical BE since microscopicproperties like the electronic bandstructure and quantum scattering ratescan be described in terms of h(x, η) and C(f), respectively.

The incorporation of further quantum phenomena like coherence andentanglement [107, 109, 110] creates a need for either quantum correctionsto the BE or quantum versions thereof, called quantum BOLTZMANN equa-tions (QBEs) [151, 103, 152]. Moreover, the recent emergence of spintronics[153] raised the question of how to describe scattering of spin-coherent elec-

Chapter 12: ARXIV: 1304.4772 [quant-ph] 167

tron states in magnetic multilayers or domain walls by means of a kineticequation [154, 149, 155, 156, 157]. In the spin-coherent regime, the BE (12.1)is replaced by

∂tF − h1, Fx,η + i[Ω, F ] = Q(F ) , (12.2)

where F (x, η, t), Ω(x, η, t) are hermitian 2× 2 matrices defined on the phasespace and [Ω, F ] = ΩF − FΩ denotes the commutator. F is the distributionmatrix, the eigenvalues of which give the scalar distribution functions of thetwo spin species. The term Ω is an exchange field that mixes the two spindistributions. Equation (12.2) is referred to as the spinorial or matrix BOLTZ-MANN equation (SBE) [23, 150]. POSSANNER and NEGULESCU [23] studiedlinear collision operators Q which feature spin-dependent scattering rates,for example

Q(F )(η) =

∫dη′(S ′1/2F (η′)S ′1/2 − 1

2SF (η)− 1

2F (η)S

), (12.3)

Here, S = S(η, η′) is a strictly positive, hermitian 2 × 2 matrix, whoseeigenvalues denote the scattering rates from η to η′ for the two spin species[158, 159] and S ′ = S(η′, η). The left-hand-side (lhs) of Eq. (12.2) has beenderived on a rigorous basis by HAJJ [150]. A derivation of (12.3), which isable to relate the scattering matrices S to a microscopic Hamiltonian will beaccomplished in the course of this work.

The derivation of QBEs or the SBE starts at the microscopic level bydefining a suitable model Hamiltonian. Then, the natural framework topass from the quantum to the kinetic level is the WIGNER-WEYL formalismof quantum mechanics [101, 160]. There exists a plethora of results regard-ing this passage for the scalar (spin-less) case, some of them we shall brieflymention here (for further information the reader is urged to view the ref-erences in the articles cited below). We remark that for the case that theeigenvalues of S are identical, performing the trace in Eq. (12.2) leads thescalar BE (12.1). QBEs have been obtained in the framework of general-ized KADANOFF-BAYM non-equilibrium GREEN’s functions [151, 103, 161]and by a monitoring technique [152]. On the rigorous level, the linear BEhas been obtained from the single particle SCHRÖDINGER equation with aGaussian random potential in the weak-coupling [105, 104] and in the low-density limit [100], respectively. The non-linear BE was derived by startingfrom the many-body SCHRÖDINGER equation with weak pair interactionpotential and by studying the quantum version of the BBGKY-hierarchy[106].

In this work we apply a different strategy for passing to the kinetic level.Our starting point for the semiclassical analysis will be a master equation


von Neumann

~→0

reduced dynamics

Born-Markov limit// Lindblad

~→0

semiclassical limit

classical Liouville BBGKYmolecular chaos

// Boltzmann

Figure 12.1: Schematic illustration of the strategies to pass from the quan-tum level to a semiclassical BE. The path employed in the present work isindicated by solid arrows while an alternative approach is indicated by dot-ted arrows.

of the LINDBLAD form [20] describing a single quantum particle in contactwith its environment [19, 114]. Semiclassical limits of LINDBLAD type mas-ter equations have already been considered [162]. In particular, we shallstart from the hierarchy of master equations derived by POSSANNER andSTICKLER [17]. Master equations of LINDBLAD form describe the quantumevolution in terms of a semigroup law (quantum dynamical semigroups)[120], just as the BE does for the classical evolution on the kinetic level.Therefore, by starting the semiclassical analysis from the LINDBLAD equa-tion instead of the VON NEUMANN equation, the passage from the quantumto the kinetic level has been decomposed into two stages as sketched in Fig.12.1:

(1) In the quantum regime, one performs a Markovian limit that leadsto dynamics described in terms of a quantum dynamical semigroup(BORN-MARKOV limit [17]),

(2) in the Markovian regime, one performs the semiclassical limit (scaled~→ 0) in order to obtain the BE.

Corrections to the BE arise at each of the two stages. One obtains non-Markovian corrections at the first stage and quantum corrections in ascend-ing powers of ~ (scaled) at the second stage. This paper deals solely withthe second stage, while the first has been accomplished by POSSANNER andSTICKLER [17].

This paper is organized as follows: in Sec. 12.2 we shall agree on somenotations and specify the physical system under investigation. In Sec. 12.3we explicitly evaluate the integral kernel of the dissipator, in Sec. 12.4 we


introduce the WIGNER transform of the state operator and transform thewhole master equation into the WIGNER representation. Here we derivea WIGNER equation equipped with a linear collision operator which fea-tures momentum relaxation as well as spin decoherence. The details of thederivation are explicated in App. 12.C - App. 12.E. In Sec. 12.5 we discussthe quantum collision operator and, finally, in Sec. 12.6 we introduce thesemiclassical scaling and define the different semiclassical scenarios whichwe will regard in this work. Moreover, we draw the semiclassical limit forthese scenarios and, thus, derive the spinorial BE with a collision operatorof the form (12.3) as well as the BLOCH equations. Conclusions are drawnin Sec. 12.7.

12.2 Notations and Modeling

We consider the evolution of a single quantum particle with two spin de-grees of freedom (spin 1/2-particle), henceforth called the ’system’. Thisparticle interacts with the ’environment’ B which in turn is composed of Nidentical spin 1/2-particles.

The dynamics of the system’s particle are governed by a master equationof the LINDBLAD form [17],

∂tρ = − i~

[Hmf0 , ρ] +D(ρ). (12.4)

Here, ρ is the density matrix defined on a one-particle Hilbert space H, ~denotes the Planck constant, Hmf

0 and D stand for the system Hamiltonianand the dissipator, respectively. The HAMILTON operator Hmf

0 acting in Hreads Hmf

0 = H0 + Hmf , where H0 is a one-particle Hamiltonian and onedefines the mean-field operator

Hmf = trB

(HI 1⊗ χB

). (12.5)

Here, HI is an operator acting in the composite HILBERT space H ⊗ HB,which describes the interaction between the system’s particle and the en-vironment (or bath), 1 acting on H is the unity operator in H and χB is apredefined equilibrium density matrix on HB. The operation trB (·) standsfor taking the trace over the degrees of freedom of the bath. Moreover, theaction of the dissipator D is defined by

D(·) := − τ0

~2trB

([Hmf

I , [HmfI , · ⊗ χB]]

), (12.6)


where τ0 denotes the characteristic timescale1 of the system’s dynamics Hmf0

and HmfI = HI − Hmf ⊗ 1B. Please note that in writing (12.6) we assume

that the Hamiltonian HmfI is associated with the same characteristic energy

ε0 as Hmf0 , i.e. we rescaled Hmf

I → HmfI ε0/εI where εI denotes the character-

istic mean-field corrected interaction energy [17]. Equations (12.4)-(12.6) arevalid for very fast relaxation of bath states towards χB and τI/τ0 1. It hasto be emphasized that due to the assumptions incorporated in the deriva-tion of Eq. (12.4) we restrict our discussion to a case in which the system’sparticle is distinguishable from the particles constituting the environment.Furthermore, we note that under certain premises, Eq. (12.4) may accountfor the dynamics of the system’s particle towards a unique equilibrium state[163].

Let us briefly comment on the physical picture employed: We assumethat Eq. (12.4) provides a proper description of the quantum dynamics ofthe system’s particle in contact with its environment. It is the aim of thiswork to draw the semiclassical limit of Eq. (12.4), i.e. to regard the dynamicsof the system’s particle in a regime in which quantum effects cease to be ob-servable. This goal is achieved in three steps: in a first step we shall rewriteEq. (12.4) as an equation for the integral kernel ρ(x, x′, t) = 〈x | ρ(t) |x′〉 inposition space, in a second step we shall derive the WIGNER representationof Eq. (12.4) and, finally, in a third step we shall draw the semiclassical limitof Eq. (12.4).

However, we need to clarify some notations first. In what follows weshall denote the position and momentum coordinates of the system’s parti-cle by x ∈ Rd

x and η ∈ Rdη, respectively, and the position coordinate of the

n-th particle in the environment B by zn ∈ Rdzn . Moreover, we introduce the

short-hand notation Z = (z1, z2, . . . , zN) ∈ RNdZ . For multiple integrals we

use the abbreviations∫dZ =

∫dz1

∫dz2 . . .

∫dzN , (12.7a)∫

dZn =

∫dz1 . . .

∫dzn−1

∫dzn+1 . . .

∫dzN , (12.7b)∫

dZnm =

∫dz1 . . .

∫dzn−1

∫dzn+1 . . .

∫dzm−1

∫dzm+1 . . .

∫dzN .

(12.7c)

The spin degrees of freedom of the system will be labeled by roman lowercase letters, such as i, j ∈ (1, 2), while the spin degrees of freedom of the

1Denoting by τ0 the characteristic timescale of the system corresponding to the Hamil-tonian Hmf

0 , the characteristic energy is defined via ε0τ0 = ~.


n-th particle in B are labeled by Greek letters, i.e. αn, βn ∈ (1, 2). In whatfollows we shall use the multi-index notations

α = (α1, α2, . . . , αN) , (12.8a)αn = (α1, . . . , αn−1, αn+1, . . . , αN) , (12.8b)αnm = (α1, . . . , αn−1, αn+1, . . . , αm−1, αm+1, . . . αN) , (12.8c)α, βn = (α1, . . . , αn−1, βn, αn+1, . . . , αN) . (12.8d)

In particular, for the multi-indices (12.8a)-(12.8d) we employ the abbrevia-tion

∑α ≡

∑α1α2···αN

and write the matrix elements of a 2N × 2N matrix Γ

as Γα,β ≡ Γα1α2···αN ,β1β2···βN .In what follows the set of hermitian 2× 2 matrices is termed H2(C) and

we shall use the notation ~σ = (σ1, σ2, σ3), where

σ1 =

(0 11 0

), σ2 =

(0 −ii 0

), σ3 =

(1 00 −1

), (12.9)

are the three PAULI matrices. Hence, any matrix G ∈ H2(C) can be writtenin the PAULI basis (1, ~σ) with coefficients g0 ∈ R and ~g = (g1, g2, g3) ∈ R3,respectively,

G = g01 + ~g · ~σ , (12.10a)

g0 =1

2tr (G) , ~g =

1

2tr (~σG) , (12.10b)

where 1 is the 2× 2 unity matrix. The eigenvalues of G can be expressed asg0 ± |~g|, hence we call |~g| the spin polarization of G. Moreover, we refer to~g as the spin (spinorial) part or directional spin polarization and to ~g/|~g| asthe direction of spin polarization of G.2 Moreover, we remark the followingproperty: let A,B ∈H2(C), then with the help of the decomposition (12.10)and the properties of the PAULI matrices we have

i[A,B] = i[~a · ~σ,~b · ~σ] = 2(~b× ~a) · ~σ, (12.11)

where ~a and~b denote the spinorial part of A and B, respectively.With the help of these notations, let us further concretize the physics

of the system under investigation. In the following we denote operatorswith a hat, such as A, and their integral kernels (matrix elements in the po-sition space basis) without the hat, i.e. A(x, y) =

⟨x∣∣∣ A ∣∣∣y⟩. The integral

2We note that in the case of more than two spin degrees of freedom one may express allquantities in a form analogous to Eq. (12.10) by employing the generalized GELL-MANNmatrices [164].


kernel of H0 is assumed to be of the general form⟨x∣∣∣H0

∣∣∣y⟩ = H0(x, y) ∈H2(C). Similarly, for the bath reference state χB we write 〈Z | χB |Z ′〉 =

χB(Z,Z ′) ∈⊗N

n=1 H2(C), i.e. it is a hermitian 2N × 2N matrix at everypoint (Z,Z ′) ∈ RNd

Z × RNdZ . The interaction Hamiltonian accounts for spin-

dependent two particle interactions and is assumed to be diagonal in posi-tion space,

⟨x, Z

∣∣∣HI

∣∣∣x′, Z ′⟩ = HI(x, Z, x′, Z ′) = HI(x, Z)δ(x−x′)δ(Z−Z ′) ∈⊗N+1

n=1 H2(C), where

HI(x, Z) =N∑n=1

(n−1⊗k=1

1k

)⊗V(x− zn)⊗

(N⊗

k=n+1

1k

). (12.12)

Here, 1k denotes the 2× 2 identity matrix referring to the k-th bath particleand V(r) ∈H2⊗H2 stands for the spin-dependent pair interaction potentialthat depends on the distance r > 0 between the system’s particle and onebath particle as well as on their spins. With the help of the multi-indexnotation, we can write the matrix elements of Eq. (12.12) as

HαβI (x, Z) =

N∑n=1

δα1β1 · · · δαn−1βn−1Vαnβn(x−zn)δαn+1βn+1 · · · δαNβN , (12.13)

where δij denotes KRONECKER’s δ and we note that Vαnβn(r) are 2× 2 matri-ces in the system’s spin degrees of freedom which obey V †αnβn

(r) = Vβnαn(r)due to the hermiticity of V(r). The particular form of the pair interactionV(r) is given by the system of interest and, therefore, depends on the typeof particles or quasiparticles constituting the system and the environment.Independent of its actual form we may express V(r) with the help of Eqs.(12.10) as

V(r) = V0(r)⊗ 1 + ~V (r) ~σ, (12.14)

where ~V (r) ~σ =∑3

i=1 Vi(r)⊗ σi and we defined the hermitian matrices

V0(r) =1

2[V11(r) + V22(r)] , (12.15a)

V1(r) =1

2

[V12(r) + V †12(r)

], (12.15b)

V2(r) =i

2

[V †12(r)− V12(r)

], (12.15c)

V3(r) =1

2[V11(r)− V22(r)] . (12.15d)

Here, we already employed that V21(r) = V †12(r) and, hence, Vi(r) ∈ H2(C).We shall frequently employ the representation (12.14) in what follows.


Hence, we regard the dynamics of a single quantum particle in contactwith its environment. It is assumed that the dynamics are described in aproper fashion by the LINDBLAD Eq. (12.4) where the matrix element ofthe free particle Hammiltonian H0 is of the most general form

⟨x∣∣∣H0

∣∣∣y⟩ =

H0(x, y) and the interaction HI between the system’s particle and the envi-ronment, Eq. (12.12), is composed of two particle interactions V(r) which isa function of the distance r between the interacting particles.

12.3 Integral Kernel of the Dissipator

It is the aim of this section to rewrite Eq. (12.4) as an equation for the integralkernel ρ(x, y, t) = 〈x | ρ(t) |y〉 ∈ H2(C) in order to prepare the transforma-tion of Eq. (12.4) into the WIGNER representation in Sec. 12.4. Starting fromEq. (12.4), a straight-forward calculation gives

∂tρ(x, y, t) +i

~L(Hmf

0

)(x, y, t) = Q(ρ)(x, y, t), (12.16)

where we defined the integral kernel of the commutator

L(A)(x, y, t) =⟨x∣∣∣ [A, ρ]

∣∣∣y⟩ =

∫dx′ [A(x, x′)ρ(x′, y, t)− ρ(x, x′, t)A(x′, y)] ,

(12.17)for some general operator A(x, x′) =

⟨x∣∣∣ A ∣∣∣x′⟩ ∈ H2(C) and the integral

kernel of the dissipator

Q(ρ)(x, y, t) = 〈x |D(ρ) |y〉. (12.18)

We note that Eq. (12.16) is entirely equivalent to Eq. (12.4) and will serve asbasis to derive the WIGNER equation in Sec. 12.4. The particular form of thedissipatorQ as well as of the matrix element of the mean-field HamiltonianHmf (x) in Eq. (12.16) will be determined in what follows. For this sake, weinsert into Eq. (12.6) the definition Hmf

I = HI − Hmf , see Sec. 12.2, in orderto obtain

D(ρ) =τ0

~2

[trB

(2HI ρ⊗ χBHI − HIHI ρ⊗ χB − ρ⊗ χBHIHI

)−2Hmf ρHmf + HmfHmf ρ+ ρHmfHmf

]. (12.19)

To further simplify expression (12.19), we note that inserting the interactionHamiltonian (12.12) gives contributions in which we can take the partial


trace over all but one or two bath particles. It is therefore advantegeous todefine the partial traces of χB over all but one or two bath particles, respec-tively, as

χ(1)βnαn

(zn) :=∑αn

∫dZnχ

α,βnαB (Z,Z), (12.20a)

χ(2)βnγmαnβm

(zn, zm) :=∑βnm

∫dZnmχ

β,γmβ,αnB (Z,Z) , (12.20b)

where the indices n,m still refer to specific particles. Since we assume thatthe bath particles are indistinguishable (Sec. 12.2), we may omit these in-dices. It follows immediately that the corresponding matrices χ(1)(z) ∈H2(C) and χ(2)(z, z′) ∈H2(C)⊗H2(C) are normalized, i.e.∑

α

∫dzχ(1)

αα(z) = 1,∑αβ

∫dz′dz′χ

(2)αβαβ(z, z′) = 1. (12.20c)

Moreover, due to the indistinguishability it turns out to be beneficial to de-fine the one- and two-particle spin-density matrices N(1)(z) ∈ H2(C) andN(2)(z, z′) ∈H2(C)⊗H2(C) with matrix elements

n(1)αβ(z) := Nχ

(1)αβ(z), (12.21a)

n(2)αα′ββ′(z, z

′) := N2χ(2)αα′ββ′(z, z

′), (12.21b)

respectively. Let us briefly comment on this definitions: Employing the no-tation (12.10), we call

N(1)(z) = n0(z)1 + ~n(z) · ~σ, (12.22)

the spinorial density of bath particles. Here the scalar part n0(z) is the den-sity of bath particles at z and the spin part ~n(z) is the directional spin polar-ization of the bath at z. The spin part ~n(z) is proportional to the magnetiza-tion ~m(z) of the environment [165], thus, the spin-density matrix N(1)(z) atposition z contains the complete spin resolved information about the prob-ability of finding a bath particle at position z. In a similar fashion we regardthe matrix N(2)(z, z′) as the spin-resolved two-particle density matrix of theenvironment.

With the help of the above definitions a straight forward calculation al-lows to express the mean-field interaction Hamiltonian Hmf (x),

Hmf (x) =

∫dzVαβ(x− z)n

(1)βα(z) , (12.23)


as well as the integral kernel of the dissipator (see App. 12.A)

Q(ρ)(x, y, t) =2τ0

~2

∫dzdz′κββ′αα′(z, z

′)[Vαβ(x− z)ρ(x, y, t)Vα′β′(y − z′)

− 1

2Vαβ(x− z)Vα′β′(x− z′)ρ(x, y, t)

− 1

2ρ(x, y, t)Vαβ(y − z)Vα′β′(y − z′)

],

for the system under investigation. Here and in what follows we shall em-ploy EINSTEIN’s sum convention. Moreover, καα′ββ′(z, z′) are the matrix el-ements of the modified density-density covariance matrix K(z, z′) definedvia

K(z, z′) = C(z, z′) + D(z, z′), (12.24)

where D(z, z′) ∈ H2(C) ⊗ H2(C) is given via dαα′ββ′(z, z′) := n

(1)αβ(z)

δαα′δββ′δ(z− z′) and we introduced the spinorial density-density covariancematrix (or environmental covariance matrix)

C(z, z′) = N(2)(z, z′)− N(1)(z)⊗ N(1)(z′). (12.25)

Hence, we determined all components of the evolution equation of theintegral kernel ρ(x, y, t), Eq. (12.16). The particular form of the integral ker-nel of the commutator as well as of the dissipator were determined for theinteraction potential (12.12) to be of the forms (12.17) and (12.24). The actionof the dissipator Q on ρ is, thus, determined by the matrix elements of themodified spin-resolved density-density covariance matrix K(z, z′) (12.24)and the interaction potential V(r). Furthermore, the explicit form of themean-field Hamiltonian Hmf (x) was obtained, Eq. (12.23). The derivationof the WIGNER equation in Sec. 12.4 will essentially be based on these equa-tions.

However, before proceeding to the next section let us briefly discuss theintegral kernel of the mean-field interaction,Hmf (x), Eq. (12.23), in more de-tail. The mean-field interaction Hmf (x) can be regarded as the partial tracetr1(·) over the bath’s particle spin degrees of freedom of the convolution ofthe potential V(r), Eq. (12.12), with the spinorial density N(1)(z),

Hmf (x) =

∫dztr1

[V(x− z)N(1)(z)

]=

∫dzVi(x− z)ni(z), (12.26)

where the sum runs from i = 0, . . . , 3 and the matrices Vi(r) ∈ H2(C) havebeen defined in Eq. (12.15).


The mean field interaction (12.26) is easily concretized for two particu-larly interesting physical situations. As a first example we regard the casethat the interaction is spin independent, i.e.

V(r) = v(r)1⊗ 1, (12.27)

where v(r) ∈ R and we obtain the familiar expression

Hmf (x) =

∫dzv(x− z)n0(z)1, (12.28)

which is known from spin independent mean-field theory[59]. Here themean-field interaction is substantially determined by the total density ofbath particles n0(z) at position z and, therefore, insensible for any spin-polarization of the environment. On the other hand, in the case that

V(r) = v(r)σ3 ⊗ σ3, (12.29)

i.e. a distance-dependent 3-polarized spin-spin interaction, we have

Hmf (x) =

∫dzv(x− z)n3(z)σ3, (12.30)

which is a mean-field interaction that depends on the 3-spin polarizationof the environment n3(z) at position z. Since the spin polarization of thebath may be connected to a magnetization m3(z) [165] via m3(z) ∝ n3(z),we interpret Eq. (12.30) as the interaction of the system’s spin with a posi-tion dependent effective magnetic field in 3-direction. Since we may writethe mean-field interaction as Hmf (x) = hmf3 (x)σ3 we regard hmf3 (x) as the 3-component of the effective magnetic field at position x. Here, the3-component of the effective magnetic field hmf3 (x) is given by a convolu-tion between v(x− z) and the magnetization m(z) ∝ n3(z).

We shall come back to these two particular examples in the course of thefollowing sections.

12.4 WIGNER Representation of the LINDBLAD

Equation

It is the aim of this section to apply the WIGNER transformW to Eq. (12.16).The ensuing transport equation for W =W(ρ) is a spinorial WIGNER equa-tion equipped with a quantum collision operator stemming from the dissi-pator Q defined in Eq. (12.24). This result will serve as a starting point forthe semiclassical analysis performed in Sec. 12.6.


In what follows we shall refrain from an explicit notation of the timeargument t. Let us define the WIGNER transform of ρ(x, y) ∈ H2(C) and itsinverse. The element-wise WIGNER transform reads

wij(x, η) :=W(ρij) =1

(2π~)d

∫dyρij

(x+

y

2, x− y

2

)exp

(− i~y · η

),

(12.31)and the corresponding inverse transform is given by

ρij(x, y) =W−1(wij) =

∫dηwij

(x+ y

2, η

)exp

[i

~(x− y) · η

]. (12.32)

By convention, we shall understand all WIGNER transforms in an element-wise fashion as defined in Eq. (12.31) and (12.32), i.e. we denote W =W(ρ). We remark that the WIGNER function W must not be interpreted as aphase space distribution function since its eigenvalues also take on negativevalues. The size of negative regions of the WIGNER function display thewavefunction’s ability to interfere and may be used as a measure for thenon-classicality of the system [166].

We now apply the WIGNER transform (12.31) to Eq. (12.16) for the inte-gral kernel ρ. For this sake, let a(x, η) = (2π~)dW [A(x, y)] ∈ H2(C) denotethe phase space symbol of an operator A acting in the system’s HILBERTspace H. Then we write for any phase space symbol a the MOYAL bracket[101] as (see App. 12.B)

L~ (a) (x, η, t) := W [L(A)] (x, η, t)

=1

(2π~)2d

∫dzdz′

∫dξdξ′

[a

(z′ +

1

4z, ξ +

1

4ξ′)

W

(z′ − 1

4z, ξ − 1

4ξ′, t

)−W

(z′ +

1

4z, ξ +

1

4ξ′, t

)a

(z′ − 1

4z, ξ − 1

4ξ′)]

exp

[i

~ξ′ · (x− z′)

]× exp

[− i~z · (η − ξ)

], (12.33)

which is the WIGNER transform of the integral kernel of the commutator,L(A) defined in Eq. (12.17). In the general case, we remember thatHmf

0 (x, y) = H0(x, y) + Hmf (x, y), where the mean-field interactionHmf (x, y) = Hmf (x)δ(x − y) was concretized in Eq. (12.23). Let h0(x, η) =

(2π~)dW(H0) denote the symbol of H0 in phase space and let hmf0 = h0 +

Hmf (x) be the phase space symbol of Hmf0 . Then the WIGNER-transformed


Eq. (12.16) may be written in a compact form as

∂tW +i

~L~(h

mf0 ) =

τ0

~2Q~(W ). (12.34)

where the WIGNER representation of the dissipator (12.24) reads

Q~(W ) := W [Q(ρ)]

=2

(2π~)d

∫dx′dzdz′

∫dη′κββ′αα′(z, z

′)

×[Vαβ

(x+

1

2x′ − z

)W ′Vα′β′

(x− 1

2x′ − z′

)−1

2Vαβ

(x+

1

2x′ − z

)Vα′β′

(x+

1

2x′ − z′

)W ′

−1

2W ′Vαβ

(x− 1

2x′ − z

)Vα′β′

(x− 1

2x′ − z′

)]× exp

[− i~x′ · (η − η′)

], (12.35)

with W ′ = W (x, η′, t).Eq. (12.34) is the WIGNER equation for a single spin-1/2 particle in inter-

action with its environment and is entirely equivalent to Eq. (12.16) of op-erator symbols in position space, from which it was derived. The lhs of Eq.(12.34) represents the free flight of this particle and is therefore analogousto the equation obtained from the VON NEUMANN equation in the WIGNER

picture, except for the mean-field correction. For instance, if Hmf0 (x, y) gives

the symbol hmf0 (x, η) = [η2/(2m) + u(x)]1 + ~Ω(x, η) · ~σ + Hmf (x), with theparticle’s mass m, the scalar external potential u(x) and the exchange field~Ω(x, η) · ~σ, one obtains the free flight term of the VLASOV equation

i

~L~

(hmf0

)=

η

m· ∇xW +

i

~L~ (u1) +

i

~L~

(~Ω · ~σ +Hmf

). (12.36)

Here, L~ (u1) is of the well known form (see App. 12.B)

L~(u1) =1

(2π~)d

∫dx′∫

dη′[u

(x+

x′

2

)− u

(x− x′

2

)]W ′

× exp

[− i~x′ · (η − η′)

]. (12.37)

The mean-field correction in Eq. (12.36) is a first contribution stemmingfrom the interaction between the system’s particle and the environment and


its particular action depends on the form of the two particle interactionV(r). As illustrated in Eqs. (12.28) and (12.30) the mean-field interactionmay give rise to a scalar as well as to a spinorial contribution to the one-particle Hamiltonian H0(x, y). In particular, if V(r) is spin-independent, i.e.of the form Eq. (12.27), the mean field contribution to Eq. (12.36) is of theform Eq. (12.37) and Hmf (x) behaves like an external scalar potential u(x).On the other hand, if V(r) models a spin-dependent interaction of the formEq. (12.29) a brief calculation demonstrates that the mean field contributionto Eq. (12.36) takes on the form

L~

(hmf3 σ3

)=

∫dη′hmf3 (η′)

[σ3W

(x, η − η′

2, t

)−W

(x, η +

η′

2, t

)σ3

]× exp

(i

~x · η′

). (12.38)

Here, hmf3 (η′) denotes the FOURIER transform of the 3-component of the vec-tor ~hmf (z), see App. 12.C. This term may be interpreted in the sense that itis accounting for the mixing of momentum components of different spinspecies due to the effective magnetic field hmf3 (z) induced by the spin de-grees of freedom of the environment (Sec. 12.3).

The rhs of Eq. (12.34) is a quantum mechanical collision integral (or col-lision operator) for the WIGNER equation. We note that Eq. (12.34) togetherwith Eqs. (12.33) and (12.35) will serve as a starting point for the semiclas-sical analysis carried out in Sec. 12.6, however, let us comment on somegeneral properties of the quantum collision operator first.

12.5 The Quantum Collision Operator

We discuss some general features of the collision operator Q~(W ) acting onthe spinorial WIGNER function W according to Eq. (12.35) in order to un-derstand its action in more detail. This will prove to be crucial in the semi-classical analysis of Eqs. (12.34), (12.33) and (12.35) in Sec. 12.6. We insertinverse FOURIER transforms (see App. 12.C) of the interaction potential,

Vαβ(x) =

∫dξVαβ(ξ) exp

(i

~x · ξ

), (12.39a)

and of the modified spin resolved density-density covariance,

κββ′αα′(z, z′) =

∫dξdξ′κββ′αα′(ξ, ξ

′) exp

[i

~(z · ξ + z′ · ξ′)

], (12.39b)


into Eq. (12.35) and subsequently integrate over x′, z and z′ in order to ob-tain

Q~(W )(x, η, t) = 2(2π~)2d

∫dξdξ′κββ′αα′(ξ, ξ

′)

×[Vαβ (ξ)W

(x, η − ξ − ξ′

2, t

)Vα′β′ (ξ

′)

−1

2Vαβ (ξ) Vα′β′ (ξ

′)W

(x, η − ξ + ξ′

2, t

)−1

2W

(x, η +

ξ + ξ′

2, t

)Vαβ (ξ) Vα′β′ (ξ

′)

]× exp

[i

~x · (ξ + ξ′)

]. (12.40)

We denote this as the representation of the collision operator (12.35) as a mo-mentum space integral. Whether or not this representation is more conve-nient than Eq. (12.35) depends on the system under investigation. However,we shall now discuss the special case that K(z, z′) = K(z − z′) and it willturn out that in this case representation (12.40) is the more convenient one.We stress that it has to be checked carefully whether or not the simplificationK(z, z′) = K(z− z′) is valid for the system under investigation. For the sakeof a simplified discussion, we shall assume it to be a valid approximation inwhat follows.

From definition (12.24) we remember that K(z, z′) = C(z, z′) + D(z, z′).For translational invariant systems the covariance C(z, z′) = C(z − z′). Ifwe further restrict our discussion to the case of a constant particle distribu-tion, i.e. N(1)(z) = const, we also have D(z, z′) = D(z − z′). It is, therefore,a sufficient condition to assume a space homogeneous environment whosespinorial density is constant in space in order to justify the above simplifi-cation. Hence, we regard the case that K(ξ, ξ′) = K

(ξ+ξ′

2

)δ(ξ+ξ′), see App.

12.C, and, therefore, Eq. (12.40) simplifies to

Q~(W )(x, η, t) = 2(2π~)2d

∫dη′κββ′αα′(η − η′)

×[Vαβ (η − η′)W ′Vα′β′ (η − η′)

−1

2Vαβ (η − η′) Vα′β′ (η − η′)W

−1

2WVαβ (η − η′) Vα′β′ (η − η′)

], (12.41)


with W = W (x, η, t) and W ′ = W (x, η′, t). Please note that we employedthat Vαβ(ξ) = Vαβ(−ξ) according to Eq. (12.12). In what follows we shallhighlight some general features of the quantum collision operator (12.41).

It is demonstrated in App. 12.D that we may rewrite Eq. (12.41) in theeven more convenient form

Q~(W )(x, η, t) = 2(2π~)2d

∫dη′ρi(η − η′)

× [Si (η − η′)W ′Si (η − η′)

−1

2Si (η − η′)Si (η − η′)W

−1

2WSi (η − η′)Si (η − η′)

], (12.42)

i.e. as a sum of collision operators of the form of the BOLTZMANN equation,Eq. (12.3). Hence, in the particular case that the modified covariance fulfillsK(z, z′) = K(z − z′), the quantum collision operator (12.35) turns out to beof the form proposed for the semiclassical SBE, Eq. (12.3). However, sinceW can take negative eigenvalues it is not a collision integral in the classicalsense as already emphasized at the beginning of this section. Here, ρi(η′) ∈R, Si(η′) ∈ H2(C) and i = 0, . . . , 3. As demonstrated in App. 12.D thehermitian matrices Si are linear combinations of the matrices Vi, where theweights are determined by the matrix K(η′). The scalar functions ρi(η′) ∈ Rare the eigenvalues of K(η′).

Let us briefly clarify the terminology used in the subsequent analysis:According to Eq. (12.10) we may express W as W = w1 + ~w · ~σ, wherewe denote by w = 1/2 tr (W ) the scalar part and by ~w = 1/2 tr (W~σ) thespin part of W . Furthermore, we call N =

∫dηW the spinorial particle den-

sity (or distribution) andM =∫

dxW the spinorial momentum distribution,which may also be decomposed according to Eq. (12.10). Again, the corre-sponding terms are referred to as the scalar and the spin part of N and M ,respectively.

The collision operator (12.42) may be decomposed according to

Q~(·) = Q(1)~ (·) +Q(2)

~ (·), (12.43)

where

Q(1)~ (W ) = 2(2π~)2d

∫dη′ρi(η − η′)Si (η − η′) [W ′ −W ]

×Si (η − η′) . (12.44)


and

Q(2)~ (W )(x, η, t) = (2π~)2d

∫dη′ρi(η

′) [[Si (η′) ,W ] , Si (η

′)] . (12.45)

We note that Q(1)~ is of the general master equation form ’gain term − loss

term’. It is easily observed that we have∫dηQ(1)

~ (W ) = 0, (12.46)

for arbitrary W , i.e. the spinorial particle density N is conserved by Q(1)~ .

To be more specific, the particle distribution N(x, t) is not affected by theaction of Q(1)

~ while the scalar as well as the spin part of the momentumdistribution M(η, t) are changed.

In a similar fashion we obtain for Q(2)~ that

tr[Q(2)

~ (W )]

= 0, (12.47)

for arbitrary W , i.e. the scalar part w of the Wigner function W is conservedby Q(2)

~ . Thus Q(2)~ acts solely on the spin part ~w. Interestingly, the conser-

vation of w signifies that both, the scalar particle distribution n(x, t) as wellas the scalar momentum distribution m(η, t) remain unaffacted. The oper-ator Q(2)

~ may therefore be identified as accounting solely for local spin-flipprocesses.

Combining Eq. (12.47) and Eq. (12.46) gives the property∫dηtr [Q~(W )] = 0, (12.48)

for arbitrary W . Thus the scalar part of the spinorial particle density, n(x, t)is conserved byQ~ as in the well-known scalar case [24]. However, the spinpart and in particular the local spin polarization |~n(x, t)| is changed, i.e. Q~

accounts for spin-flip processes due to its part Q(2)~ .

We now investigate in more detail the operatorQ(2)~ . Due to its particular

form (12.45) we may regard the arguments x, η and t of W as parametersand assume that the domain of Q(2)

~ is H2(C) in the following analysis. Atfirst, we are interested in its kernel denoted by Ker(Q(2)

~ ), because, as willbecome apparant in what follows, we may under certain premises reasonfrom the structure of Ker(Q(2)

~ ) on the stationary spin distribution of W for


some given V(r). The Ker(Q(2)~ ) is defined as the set of all matrices A ∈

H2(C) for which Q(2)~ (A) = 0. As demonstrated in App. 12.E we have

Q(2)~ (A) = 0⇔ [Si(η

′), A] = 0, ∀i and ∀η′, (12.49)

if detailed balance is requred for A. SinceQ(2)~ is linear and self-adjoint with

respect to the HILBERT-SCHMITDT scalar product tr (AB), A,B ∈H2(C), itsdomain D(Q(2)

~ ) may be decomposed into

D(Q(2)~ ) = H2(C) = Ker(Q(2)

~ )⊕Ker(Q(2)~ )⊥, (12.50)

where

Ker(Q(2)~ )⊥ :=

A ∈H2(C)|tr (AB) = 0∀B ∈ Ker(Q(2)

~ ), (12.51)

is the space orthogonal to Ker(Q(2)~ ).

If we knew the projection of some particular state W on Ker(Q(2)~ ), we

could assume that Q(2)~ can be approximated by a relaxation time ansatz

Q(2)

~ (W ) :=W −Wτ2

, (12.52)

where W denotes the projection of W on the kernel of Q(2)~ for some given

V(r) and τ2 is the mean relaxation time. The action of the operator Q(2)

~as defined in Eq. (12.52) on some function W is that it relaxes W into W ,where tr (W ) = tr

(W)

since tr(Q(2)

~

)= 0. We stress that, again, whether or

not Eq. (12.52) is a valid approximation has to be checked carefully for theproblem of interest.

It is the aim of the following paragraphs to determine the equilibriumdistribution W ∈ Ker

(Q(2)

~

)for a given two particle interaction poten-

tial V(r). First of all, we deduce from Eq. (12.49) and (12.11) that forA ∈ Ker(Q(2)

~ ) we have

[Si(η′), A] = 2i [~si(η

′)× ~a] · ~σ != 0. (12.53)

Since A does not depend on η′, this is for ~si(η′) 6= 0 and ~a 6= 0 only pos-sible if the direction of the spin part of Si(η′) does not depenend on η′, i.e.~si(η

′)/|~si(η′)| = const. On the other hand, if it does depend on η′, i.e. if~si(η

′)/|~si(η′)| 6= const, the only possible solution to Eq. (12.53) is ~a = 0. In


particular, it follows that in this case the spin part of W is zero. Hence, theunique projection W 7→ W reads

W =1

2tr (W )1, (12.54)

i.e. the operator Q(2)

~ destroys spin polarization on a finite time scale τ2

whenever the interaction potential’s direction of the spin polarization~si(η

′)/|~si(η′)| 6= const. In a similar fashion, we note that Eq. (12.54) is thesole solution to Eq. (12.49) if the directions of the spin parts of Si are notidentical for different i. We highlight that this mechanism of spin depolar-ization is not based on spin orbit interaction [144, 145] but on direct spin-spin interaction between different particles. Again, we emphasize: When-ever the direction of the spin part of the interaction Si is a function of η′ ornot equivalent for different i, the action of Q(2)

~ unavoidably leads to spindepolarization on a finite time scale, as described in Eq. (12.54).

We now concentrate on the special case that the direction of the spin partof Si(η′) is constant in η′ and identical for all i. If ~si(η′) = 0 then Q(2)

~ (·) = 0and the operatorQ~ does not account for spin-decoherence at all. However,if ~si(η′) 6= 0 then ~si(η′)/|~si(η′)| = const is identical for all i, i.e. we may write

~si(η′) = γi(η

′)~λ, (12.55)

where γi(η′) is some scalar function and |~λ| = 1. Hence,

Si(η′) = si(η

′)1 + γi(η′)~λ · ~σ, |~λ| = 1. (12.56)

Let Σ be the matrix which diagonalizes ~λ · ~σ and, according to Eq. (12.49),also W ∈ Ker(Q(2)

~ ). We then obtain that the unique equilibrium spin con-figuration W reads

W =1

2tr (W )1 +

1

2tr(σ3Σ†WΣ

)Σσ3Σ†. (12.57)

In order to prove the result Eq. (12.57) we note that we may decomposean arbitrary matrix W according to W = D(W ) + O(W ), where D(W ) de-notes the matrix whose off-diagonal elements are zero and O(W ) denotesthe matrix whose diagonal elements are zero. Now, according to Eq. (12.49)we have

W ∈ Ker(Q(2)~ )⇔ Σ†WΣ = D(Σ†WΣ), (12.58)

is solely diagonal. In a similar fashion

W ′ ∈ Ker(Q(2)~ )⊥ ⇔ Σ†W ′Σ = O(Σ†W ′Σ), (12.59)


is solely off-diagonal.3 Hence, with the help of Eq. (12.50), since Σ†WΣ =D(Σ†WΣ) + O(Σ†WΣ), we note that

Σ†WΣ = D(Σ†WΣ), (12.61)

defines the unique projection onto Ker(Q(2)~ ). Finally, inserting into (12.61)

the expression

D(Σ†WΣ) =1

2tr (W )1 +

1

2tr(σ3Σ†WΣ

)σ3, (12.62)

and solving for W proves statement (12.57).Hence, it was possible to identify the equilibrium spin configuration W

for a given two particle interaction potential V(r), provided that Q~ leadsto a unique equilibrium distribution in the kernel of Q(2)

~ . This analysis ofthe quantum collision operator will show to be a crucial ingredient in thefollowing section in order to derive the BLOCH as well as the BOLTZMANNequation. Furthermore, in the above discussion we identified the necessarycriteria which allow for the treatment of spin depolarization via a relaxationtime operator of the form Eq. (12.52) together with W given by Eq. (12.54).Such an operator was employed by POSSANNER and NEGULESCU [23] toderive spin drift-diffusion equations from a semiclassical BE.

However, before we proceed to the semiclassical analysis of the WIGNERequation (12.34) let us briefly discuss the two particular examples alreadyillustrated in Secs. 12.3 and 12.4.

First, we regard an interaction potential of the form (12.27), i.e. a spin-independent interaction. In this case Q(2)

~ vanishes for all W , i.e. spin po-larization is not destroyed. Hence, the quantum collision operator (12.44)takes on the form

Q~(W ) =

∫dη′ω1(η − η′)|v(η − η′)|2 [W (x, η′, t)−W (x, η, t)] , (12.63)

where we definedω1(η′) = 2

∑i

ρi(η′). (12.64)

3This statement is is easily proved: suppose D(Σ†W ′Σ) 6= 0. We may then define amatrix A = ΣD(Σ†W ′Σ)Σ† ∈ Ker(Q(2)

~ ), which is a contradiction to the assumption W ′ ∈Ker(Q(2)

~ )⊥, which proofs ’⇐’. On the other hand, let W ′ ∈ Ker(Q(2)~ )⊥ and B ∈ Ker(Q(2)

~ ).Then we have

0 = tr (W ′B) = tr(Σ†W ′ΣΣ†BΣ

)= tr

(Σ†BΣD(Σ†BΣ)

)⇒ Σ†W ′Σ = O(Σ†W ′Σ),

(12.60)which demonstrates ’⇒’ and the proof is completed.


We note that the collision integral Eq. (12.63) independently acts on bothspin directions and for each spin species it is of the classical scalar form,however, it has to be kept in mind that W is not a proper distribution func-tion. The weight ω1(η′) is given by the matrix trace of the corrected density-density covariance K(η′) (note that the trace is basis independent).

In a similar fashion we obtain for a spin-spin interaction with a potentialof the form (12.29) the collision integral

Q~(W ) =

∫dη′ω2(η − η′)[v(η − η′)]2 [σ3W (x, η′, t)σ3 −W (x, η, t)] , (12.65)

whereω2(η′) = 2(−1)α+α′καα′αα′(η

′). (12.66)

In particular, according to Eq. (12.57) the projection of W on the kernel ofQ(2)

~ is of the form

W =1

2tr (W )1 +

1

2tr (σ3W )σ3, (12.67)

i.e. all particles are spin-polarized in 3-direction in the equilibrium state.

12.6 Semiclassical Limit: Spinorial BOLTZMANN

Equation

We shall now study the LINDBLAD Equation (12.4) [or, equivalently, Eq.(12.16) or the WIGNER equation (12.34)] in a regime in which quantum ef-fects other then spin coherence cease to be observable. The passage from thequantum (or microscopic) world to the classical (or macroscopic) world willbe described in terms of a small parameter ε, referred to as the semiclassicalparameter or scaled PLANCK constant. This parameter tends towards zeroas quantum effects become more and more negligible.

Before discussing the mathematical subtleties of this transition we shallconnect the semiclassical limit to the following physical picture: suppose λ0

is a characteristic length on which quantum effects dominate the physics ofthe system and let V0 = λd0 be the volume of a thus defined quantum box B0.Moreover, suppose Eq. (12.34) provides a proper description of the systemin the quantum box B0. In the semiclassical picture we are interested insolving Eq. (12.34) in a domain Bs with volume Vs = V0/ε

d for ε 1 withsuitable intial and boundary conditions. Hence, ε−d is a measure of howmany quantum boxes with volume V0 fit into the regarded domain Bs. All


wavefunctions are normalized within the domain Bs. The limit ε → 0 canbe understood as zooming out of the microscopic world B0, such that wave-characteristics of particles are on a very small length scale compared to themacroscopic domain Bs, i.e. V0/Vs = εd → 0.

On the other hand, we define a microscopic time scale τ0 on which quan-tum effects are dominant and which defines the corresponding quantum timedomain T0 = [0, τ0]. Then, in the semiclassical limit, we regard a timescaleτs = τ0/ε characterizing the macroscopic time domain Ts. With the helpof λ0 and τ0 we may define the characteristic scales of the microscopic worldB0 × T0 by the relations

ε0τ0

~=λ0π0

~= 1, (12.68)

where ε0 and π0 stand for the energy scale and the momentum scale associ-ated with the microscopic world B0 × T0, i.e. with Eq. (12.34). It is the aimof the following scaling considerations to express Eq. (12.34) in variables ofthe macroscopic world Bs × Ts.

This is performed in three stages: (i) all functions f(x, η, t) which appearin Eq. (12.34) are rewritten in such a form that we explicitly emphasizetheir characteristic wavelength and amplitudes. In particular, we rewritef(x, η, t) = αcf

(xλc, ηπc, tτc

), where αc denotes the characteristic amplitude

and λc, πc and τc denote characteristic length-, momentum- and time scales.Thus, the function f(x, η, t) is of order one with gradients of order one.

In a second step, we introduce dimensionless variables

t′ =t

τs, x′ =

x

λs, η′ =

η

πsE ′ =

E

εs, (12.69)

where we introduced the semiclassical scales of the macroscopic world Bs×Ts. According to the above considerations the semiclassical scales are con-nected to the quantum scales via

ε =λ0

λs=τ0

τs. (12.70)

It follows from the definition (12.68) of quantum scales that the above defi-nition implies that

ε =~εsτs

=~

λsπs, (12.71)

with εs = ε0 and πs = π0, i.e. the energy and momentum scales remainunaffacted while, by expressing Eq. (12.34) in the dimensionless variables(12.69), we regard a long-time, large-scale limit.


In the third and final step, we define the characteristic lengthscale as wellas the characteristic amplitude of all quantities which appear in Eq. (12.34)by posing suitable scaling assumptions. These assumptions determine theproperties of the system under investigation. To be specific we assume inthe following semiclassical analysis:

h(ε)0 (x, η) = ε0h

(x

λs,η

πs

)1 + εε0~Ω

(x

λs,η

πs

)· ~σ, (12.72a)

V(ε)(r) = aε0V(r`

), (12.72b)

N(1)(z) =b

λdsN(1)

(z

λs

), (12.72c)

where we omitted the overlines on the rescaled functions, however, indexedall quantities by (ε) which implicitly depend on ε. Further, a, b and ` are scal-ing parameters which will assume different definite values in the followingsubsections. Depending on the particular choice of these scaling parameterswe shall derive different macroscopic transport models.

Let us briefly interpret the scaling assumptions (12.72). The free particleHamiltonian h0 varies on the macroscopic scale λs. We note that the weakscaling of the spin part ~Ω ·~σ of the free particle’s Hamiltonian h0 is necessaryin order to preserve spin coherence in the limit ε→ 0, as it has been demon-strated by HAJJ [150]. The particle distribution N(1)(z) varies on the macro-scopic scale λs. The scale on which the interaction potential V(r) varies isdenoted by `. For instance, for ` = λ0, Eqs. (12.72) describe short-range in-teractions between the system’s particle and an environment whose densityN(1)(z) varies on the macroscopic scale λs. On the other hand, for ` = λs Eq.(12.72b) accounts for long-range interactions. The parameters a and b arethe characteristic amplitudes of V(r) and N(1)(z), respectively. For instance,the combination (a = 1, b = ε) is referred to as the ’low-density scaling’[104, 100] while the scenario (a = ε, b = 1) is denoted the ’weak-couplingscaling’ [104, 105].

In what follows we shall denote the rescaled WIGNER distribution ma-trix W by W (ε) in order to emphasize its dependence on the semiclassicalparameter ε. A short calculation based on the steps illustrated above showsthat the rescaled WIGNER equation (12.34) is obtained by replacing τ0/~2 infront of the collision integral as well as 1/~ in front of the commutator by1/ε. This follows from changing to the variables Eq. (12.69) and employingdefinition (12.71). In a similar fashion, all ~-s appearing in Eqs. (12.33) and(12.35) are replaced by ε. We obtain the rescaled version of Eq. (12.34) as

∂tW(ε) +

i

εLε(hmf,(ε)0

)=

1

εQε(W (ε)

), (12.73)


whereLε andQε denote the rescaled operators (12.33) and (12.35) and hmf,(ε)0 =

h(ε)0 + H

(ε)mf . For ε = 1, the domain on which Eq. (12.73) is defined is equiv-

alent to the microscopic world B0 × T0 while for ε → 0 we approach themacroscopic world Bs × Ts.

We assume that a solution W (ε) of Eq. (12.73), for suitable intial andboundary conditions, can be written asW (ε) = F+εW1+. . ., whereF (x, η, t) ∈H2(C) is a positive definite, normalized, slowly varying function of x, η andt, called the spinorial distribution function. This assumption is necessary inorder to obtain a well posed semiclassical transport equation because thenfor ε→ 0, W (ε) → F is a proper distribution matrix in the classical sense.

Before we refer to the specific limits, we shall briefly regard the system’sparticle free Hamiltonian h

(ε)0 since this part is independent of the scaling

parameters (a, b, `). It is demonstrated in App. 12.F that one obtains

i

εLε(h(ε)

0 ) =i

ε(h

(ε)0 ,W (ε))

−1

2

(h

(ε)0 ,W (ε)

x,η−W (ε), h

(ε)0

x,η

)+O(ε), (12.74)

which is commonly referred to as MOYAL’s bracket [101]. From the scal-ing assumption (12.72a) we deduce that in zeroth order the spin part Ω ofh

(ε)0 appears in the commutator while the scalar part h1 enters POISSON’s

bracket. Hence, provided the semiclassical limit of the mean-field term Lεand of the quantum collision operator Qε exist, we may write Eq. (12.73) inthe macroscopic world Bs × Ts, i.e. for ε→ 0 as

∂tF − h1, Fx,η + i[~Ω · ~σ, F ] + L(F ) = Q(F ), (12.75)

where L(F ) denotes the semiclassical limit of the mean-field term and Q(F )the semiclassical limit of the collision integral, see Eq. (12.73). Please notethat the quantity F in Eq. (12.75) is a proper distribution matrix, i.e. Eq.(12.75) represents a macroscopic transport equation. It is the aim of the fol-lowing subsections to specify the particular form of L(·) and Q(·) for differ-ent scaling scenarios.

The requirement that L(F ) must not diverge, i.e. that Eq. (12.75) exists,poses constraints on the different combinations of free scaling parameters(a, b, `). In particular, inserting Eqs. (12.72) together with the definition Eq.(12.69) into the rescaled mean-field interaction (12.23) gives

H(ε)mf (x) = ab

∫dzn

(1)βα (z)Vαβ

[(x− z)

λs`

]. (12.76)


Let us briefly discuss the two different scenarios for ` for arbitrary a, b: (i) If` = λs the above integral is of order ab. (ii) If ` = λ0 the interaction H

(ε)mf (x)

is strongly varying in x and N(1)(z) enters only as constant, i.e. N(1)(0). Weremark that the contribution of a strongly varying mean-field interactionH

(ε)mf (x) to Eq. (12.75) vanishes in the semiclassical limit ε→ 0 for most cases.

We shall come back to this point in Subsec. 12.6.1. Due to the prefactor of1/ε in front of Lε

(H

(ε)mf

)in Eq. (12.73) we require that ab = ε. Particularly

interesting are the two specific cases (a = 1, b = ε) and (a = ε, b = 1), i.e. thelow-density and the weak-coupling scaling.

12.6.1 Short-Range Interactions (` = λ0)

Here, we study the semiclassical limit of Eq. (12.34) under the scaling as-sumption Eq. (12.72) for (a = 1, b = ε, ` = λ0), i.e. a short-range lowdensity scaling and (a = ε, b = 1, ` = λ0), i.e. a short-range weak couplingscaling. It is important to realize that we restrict our discussion to poten-tials which are integrable in Rd, i.e. Vαβ(r) ∈ L1(Rd), such as the YUKAWApotential (screened COULOMB potential). Please note that this excludes thebare COULOMB interaction. However, since in this case V (r/ε) = εV (r), thestrongly varying COULOMB interaction is equivalent to the slowly varyingCOULOMB interaction in the weak coupling scaling and will therefore bediscussed in Subsec. 12.6.2.

First, we note from Eqs. (12.76) that the mean-field contribution vanishesin both scalinges, i.e. L(F ) = 0, since

H(ε)mf (x) = ε

∫dz

∫dξn

(1)βα(εz)Vαβ(ξ) exp(−iz · ξ) exp

(i

εx · ξ

)→ 0, (12.77)

due to the RIEMANN-LEBESGUE lemma and Vαβ(r) ∈ L1(Rd). Here, theFOURIER transform Vαβ(ξ) of Vαβ(x/ε) is independent of ε, see App. 12.C.

In order to evaluate the semiclassical limit of the collision integral, wemake the following hypothesis on the environmental covariance C definedin Eq. (12.25),

C(ε)(z, z′) = γC(ε)(z − z′)

= γΓs (z − z′) +γ

ε2dΓ0

(z − z′

ε

), (12.78)

i.e. we regard a space-homogeneous environment whose covariance has amicroscopic and a macroscopic part, Γ0 and Γs, which are assumed to bebehave as γ when approaching the macroscopic regime.


In the low density scaling, the resulting collision integral is for γ = ε ofthe form

Q(F ) = Q0(F ), (12.79)

where we have (App. 12.G)

Q0(F ) =

∫dη′ρ

(0)i (η − η′) [Si (η − η′)F ′Si (η − η′)

−1

2Si (η − η′)Si (η − η′)F

−1

2FSi (η − η′)Si (η − η′)

], (12.80)

with F = F (x, η, t) and F ′ = F (x, η′, t). Here, the matrices Si(ξ) are, again,hermitian linear combinations of the matrices Vi which are obtained by di-agonalizing the matrix Γ0, see App. 12.D. The scalars ρ(0)

i ∈ R are the eigen-values of Γ0. In the low density scaling, Q0 vanishes for γ = ε2 or weakerwhile it diverges for γ = 1 or stronger. The same collision integral (12.80) isobtained in the short-range weak coupling scaling for γ = 1/ε.

We note that the operator (12.80) is a linear collision integral of BOLTZMANN-form, i.e. in the case of a strongly varying density-density covariance C(ε)

we obtain the semiclassical BOLTZMANN equation. The complete discus-sion of the collision operator can be adapted from Sec. 12.5, however, inthe present case it was not necessary to assume that the bath density N(1) =const. A spinorial BOLTZMANN equation of such a form has already beenpostulated by POSSANNER and NEGULESCU [23], however, it has not beenemployed as a basis for the derivation of spin drift-diffusion equations. Thecollision operator used to derive drift-diffusion equations can be obtainedfrom (12.80) by rewriting the collision operator as sum of an operator ac-counting for momentum relaxation Q1 and a second operator accountingfor spin decoherence Q2, see Sec. 12.5. The latter is then replaced by a re-laxation time approximation of the form Eq. (12.52) where the projection ofF onto the kernel of Q2 has to be assumed to be of the form F = 1

2tr (F )1.

Within this work it was possible to identify the necessary criteria allowingfor such a treatment, see Sec. 12.5.

12.6.2 Long-Range Interactions (` = λs)

Let us regard an interaction potential varying on the macroscopic scale, i.e.` = λs in Eq. (12.72). Again, we shall regard the low density (a = 1, b = ε,` = λs) as well as the weak coupling (a = ε, b = 1, ` = λs) case. Please notethat the strongly varying COULOMB potential is equivalent to the slowly


varying COULOMB potential in the weak coupling scaling. From Eq. (12.76)we deduce for both scalings

L(F ) = i[Hmf , F ], (12.81)

where Hmf is the macroscopic mean-field interaction of the form (12.23), i.e.independent of ε.

In the case of long range interactions we employ the following ansatz forthe environmental covariance C:

C(ε)(z, z′) = γC(ε)(z, z′)

= γΓs(z, z′) +

γ

ε2dΓ0

(z

ε,z′

ε

), (12.82)

i.e. in contrast to Eq. (12.78) we do not restrict to the space homogeneouscase.

In the low density scaling, the collision operator Q(F ) takes on the form(App. 12.H)

Q(F ) = QLD(F ) = Qs(F ) +Qn(F ), (12.83)

where for γ = ε

Qs(F ) =2

(2π)d

∫dzdz′γ

(s)ββ′αα′(z, z

′)

×[Vαβ (x− z)FVα′β′ (x− z′)

−1

2Vαβ (x− z)Vα′β′ (x− z′)F

−1

2FVαβ (x− z)Vα′β′ (x− z′)

], (12.84a)

stems from the slowly varying part of the environmental covariance and

Qn(F ) =2

(2π)d

∫dzn

(1)βα(z)

×[Vαβ (x− z)FVαβ (x− z)

−1

2Vαβ (x− z)Vα′β′ (x− z)F

−1

2FVαβ (x− z)Vα′β′ (x− z)

], (12.84b)

stems from the matrix D(ε). In the long-range weak coupling limit Q(F )reads for γ = 1/ε (App. 12.H)

Q(F ) = QWC(F ) = Qs(F ). (12.85)


Please note that we refrained from writing Eqs. (12.84a) and (12.84b) as mo-mentum space integrals and in the basis described in App. 12.D for the sakeof a more transparent notation. Of course, we may obtain such a representa-tion by replacing the interaction potential by its (macroscopic) FOURIER rep-resentation and carrying out the steps in App. 12.D. (In particular, for Eq.(12.84a) we note that in the case that the environmental covariance obeysΓs(z, z

′) = Γs(z − z′) the representation as a momentum space integral willturn out to be more convenient. Moreover, the form of Qs(F ) will in thiscase be very similar to Eq. (12.80), i.e. one obtains a BOLTZMANN collisionintegral.)

Let us briefly discuss the ensuing transport equation in the long-rangeweak coupling limit for γ = 1/ε. According to the above discussion we have

∂tF − h1, Fx,η + i[~Ω · ~σ +Hmf , F ] = Qs(F ), (12.86)

The transport equation of the spin part ~f of F is of the form

∂t ~f +(∇xh · ∇η

~f −∇ηh · ∇x~f)

+ 2~f ×(~Ω + ~hmf

)= −

∫dzdz′γ

(s)i (z, z′)~si(x, z, z

′)× ~si(x, z, z′)× ~f, (12.87)

where we employed Eqs. (12.10) and (12.11) and rewrote Qs with the helpof the steps outlined in App. 12.D. In particular, the vectors ~si(x, z, z′) arelinear combinations of the spin parts ~vi(x − z) of Vi(x − z), see Eq. (12.15),however, the weights may be functions of z and z′. The quantities γ(s)

i (z, z′)are the eigenvalues of Γs(z, z

′). Equation (12.87) describes the precession ofthe directional spin polarization ~f under the influence of an external field ~Ωand an interaction with the environment. This interaction induces an addi-tional field ~hmf and gives rise to the dissipator Qs, which relaxes the vector~f into a predefined direction, see Sec. 12.5.

Let us consider a spin located at a certain lattice point (’system’) whichinteracts with other spins located at different lattices sites (’environment’).Since the system’s spin cannot move on the lattice the scalar part of thefree particle Hamiltonian vanishes, h = 0, i.e. it has no kinetic part and noscalar field is externally applied. We recognize that in this particular case Eq.(12.87) gives rise to the macroscopic BLOCH equations of magnetism. To bemore specific, we replace in Eq. (12.87) the dissipatorQs by a relaxation timeansatz as discussed in Sec. 12.5 and interpret the spin part ~f = 1/2tr (F~σ)as the magnetization of the system.

We emphasize that it was, therefore, possible to derive the BOLTZMANNas well as the BLOCH equations from the spinorial WIGNER equation de-rived within this work. In a similar fashion one may impose further scaling


assumptions and study the resulting classical transport models as well astheir quantum corrections.

12.7 Summary

We derived a linear quantum collision operator for the spinorial WIGNERequation. Furthermore, it was demonstrated that the WIGNER equationgives rise to several linear semiclassical spin-transport models. We derivedthe BLOCH equations as well as the linear BOLTZMANN equation as an ex-ample. Let us briefly summarize the main aspects of the derivation.

We investigated the dissipative dynamics of a spin-1/2 quantum particle,referred to as the system, in contact with its environment, which is in ther-mal equilibrium. It has been shown by POSSANNER and STICKLER [17] thatin the limit of vanishing system-environment correlations these dynamicsare properly described by the LINDBLAD equation (12.4). The latter servedas a basis of the current study. The WIGNER representation of the LIND-BLAD equation (12.4) is a spinorial WIGNER equation (12.34) equipped witha quantum collision operator (12.40). It is then demonstrated that the lattercan be cast into the form of a BOLTZMANN collision integral, Eq. (12.41),provided that the spinorial density of the environment is constant and thatthe spinorial density-density covariance of the environment is space homo-geneous. The 2×2 hermitian scattering matrices Si are uniquely determinedby the spin-dependent two-particle interaction potential V(r) and by themodified spinorial density-density covariance K(z, z′) of the environment,defined in Eq. (12.24). The eigenvalues of the scattering matrices Si are thescattering rates for the two spin species.

Moreover, the quantum collision operator (12.41) is composed of twoqualitatively rather different parts. The first part changes solely the spino-rial momentum density of the system while the second part accounts forlocal spin flip processes. Hence, the second part modifies the local spin po-larization of the system in a fashion uniquely determined by the scatteringmatrices Si and the eigenvalues of the modified spinorial density-densitycovariance K(z, z′). Furthermore, it is possible to identify clear criteria un-der which the interaction between the system’s particle and the environ-ment leads to spin decoherence or even spin depolarization in the long timelimit.

Finally, we performed a semiclassical analysis of the spinorial WIGNERequation (12.34), i.e. we regarded the dynamics of the system’s particlein a regime in which quantum effects other than spin-coherence cease tobe observable. We restricted our discussion to the well established low-


density and weak-coupling limits. In principle, several semiclassical evolu-tion equations for a positive definite, hermitian distribution matrix F can beobtained. As two particularly interesting examples we note the derivationof the BLOCH equations for long range interactions and the spinorial BOLTZ-MANN equation for short range interactions and a spatially homogeneousenvironmental density-density covariance.4 This form of the BOLTZMANNequation has already been used for deriving spin- coherent drift-diffusionequations in magnetic multilayers [23].

In summary, we remark that within this work it was possible to system-atically establish the link between a full quantum-mechanical treatment of acomposite quantum system by means of the VON NEUMANN equation andmacroscopic linear spin-transport models featuring dissipation such as thespin drift-diffusion models. This makes the derived equations particularlyinteresting for applications involving graphene [167] (pseudo-spin formal-ism) or magnetically doped semiconductors [144, 145].

Ackowledgments. The authors are very grateful to E. SCHACHINGER forcarefully reading the manuscript. B.A.S. was supported by the AustrianScience Fund (FWF): P221290-N16.

Appendix.

12.A Derivation of Eq. (12.24)

We evaluate the matrix elements of the dissipator (12.19) by investigatingthe terms containing trB

(HIHI 1⊗ χB

). In what follows we employ EIN-

STEIN’s summation convention in order ot simplify the notation. We obtain

trB

(HIHI 1⊗ χB

)(x, y) =

∫dzdz′

[N(N − 1)χ

(2)ββ′αα′(z, z

′)

+Nχ(1)βα(z)δαα′δββ′δ(z − z′)

]×Vαβ(x− z)Vα′β′(y − z′)δ(x− y) (12.88)

4It is interesting to note that the covariance in the collision integral is a direct conse-quence of the requirement of well-posedness of the hierarchy of master equations on thequantum scale [17].


where χ(2)ββ′αα′(z, z

′) ∈ C stems from Eq. (12.20b) for identical particles (in-dices n,m omitted). In a similar fashion one obtains the following relations

trB

(HI ρ⊗ χBHI

)(x, y) = N(N − 1)

∫dzdz′χ


′)

×Vαβ(x− z)ρ(x, y)Vα′β′(y − z′)

+N

∫dzχ

(1)βα(z)Vαβ(x− z)ρ(x, y)

×Vαβ(y − z), (12.89a)

trB

(H2I ρ⊗ χB

)(x, y) = N(N − 1)

∫dzdz′χ


′)

×Vαβ(x− z)Vα′β′(x− z′)ρ(x, y)

+N

∫dzχ

(1)βα(z)Vαβ(x− z)

×Vαβ(x− z)ρ(x, y), (12.89b)

trB

(ρ⊗ χBH2

I

)(x, y) = N(N − 1)

∫dzdz′χ


′)

×ρ(x, y)Vαβ(y − z)Vα′β′(y − z′)

+N

∫dzχ

(1)βα(z)ρ(x, y)

×Vαβ(y − z)Vαβ(y − z), (12.89c)

(Hmf ρHmf

)(x, y) = N2

∫dzdz′χ

(1)βα(z)χ

(1)β′α′(z

′)Vαβ(x− z)ρ(x, y)Vα′β′(y− z′),(12.89d)(

H2mf ρ

)(x, y) = N2

∫dzdz′χ

(1)βα(z)χ

(1)β′α′(z

′)Vαβ(x− z)Vα′β′(x− z′)ρ(x, y),

(12.89e)and(

ρH2mf

)(x, y) = N2

∫dzdz′χ

(1)βα(z)χ

(1)β′α′(z

′)ρ(x, y)Vαβ(y − z)Vα′β′(y − z′).(12.89f)

Using thatN(N−1) ≈ N2, applying definitions (12.21) together with (12.24)and (12.25) and inserting the relations (12.89) into Eq. (12.19) yields the finalresult Eq. (12.24).


12.B Derivation of the MOYAL Bracket

Within this appendix we derive the phase space symbol of the commutatorEq. (12.33), i.e. the MOYAL bracket. We calculate the Wigner transform ofthe element

⟨x∣∣∣ Aρ ∣∣∣y⟩ =

(Aρ)

(x, y) for some operator A as

W[(Aρ)

(x, y)]

=1

(2π~)d

∫dx′dzA

(x+

1

2x′, z

)ρ

(z, x− 1

2x′)

exp(−ix′ · η)

=1

(2π~)2d

∫dx′dz

∫dξdξ′a

(x+ z

2+

1

4x′, ξ

)×W

(z + x

2− 1

4x′, ξ′

)exp

[i

~ξ ·(x+

1

2x′ − z

)]× exp

[i

~ξ′ ·(z − x+

1

2x′)]

exp

(− i~x′ · η

), (12.90)

where we used Eq. (12.32) and⟨x∣∣∣ A ∣∣∣y⟩ = A(x, y) = (2π~)−dW−1[a(x, η)].

We rearrange the exponential terms as

i

~x · (ξ − ξ′) +

i

~z · (ξ′ − ξ)− i

~x′ ·[η − 1

2(ξ + ξ′)

]which suggests the substitution ξ = ξ− ξ′ and ξ = 1

2(ξ+ ξ′). Hence, we have

W[(Aρ)

(x, y)]

=1

(2π~)2d

∫dx′dz

∫dξdξa

(x+ z

2+

1

4x′, ξ +

1

2ξ

)×W

(x+ z

2− 1

4x′, ξ − 1

2ξ

)× exp

[i

~(x− z) · ξ +

i

~x′ · (ξ − η)

]. (12.91)

This expression may be rewritten in a more convenient form by replacingz′ = x+z

2and also ξ = 2ξ. Finally, we obtain the first term of Eq. (12.33) as

W[(Aρ)

(x, y)]

=1

(2π~)2d

∫dx′dz′

∫dξdξa

(z′ +

1

4x′, ξ +

1

4ξ

)×W

(z′ − 1

4x′, ξ − 1

4ξ

)× exp

[i

~(x− z′) · ξ +

i

~x′ · (ξ − η)

]. (12.92)


A similar calculation for the term⟨x∣∣∣ ρA ∣∣∣y⟩ =

(ρA)

(x, y) gives

W[(ρA)

(x, y)]

=1

(2π~)2d

∫dx′dz′

∫dξdξW

(z′ +

1

4x′, ξ +

1

4ξ

)×a(z′ − 1

4x′, ξ − 1

4ξ

)× exp

[i

~(x− z′) · ξ +

i

~x′ · (ξ − η)

]. (12.93)

Combining this relation with Eq. (12.92) gives Eq. (12.33) as

W [L(A)(x, y)] (x, η) =1

(2π~)2d

∫dzdz′

∫dξdξ′

[a

(z′ +

1

4z, ξ +

1

4ξ′)

W

(z′ − 1

4z, ξ − 1

4ξ′, t

)−W

(z′ +

1

4z, ξ +

1

4ξ′, t

)a

(z′ − 1

4z, ξ − 1

4ξ′)]

exp

[i

~ξ′ · (x− z′)

]× exp

[− i~z · (η − ξ)

]. (12.94)

We shall now investigate the particular case⟨x∣∣∣ B ∣∣∣y⟩ = B(x, y) = B(x)

δ(x − y) where it follows from Eq. (12.31) that b(x, η) ≡ B(x). We retur toEq. (12.90) in order to obtain

W[(Bρ)

(x, y)]

=1

(2π~)d

∫dx′B

(x+

1

2x′)ρ

(x+

1

2x′, x− 1

2x′)

× exp(−ix′ · η)

=1

(2π~)d

∫dx′∫

dξB

(x+

1

2x′)W (x, ξ)

× exp

[i

~(ξ − η) · x′

], (12.95)

and, therefore, for B(x) = b(x)1 the well known result Eq. (12.37)

W [L (b1) (x, y)] =1

(2π~)d

∫dx′∫

dξ′[b

(x+

1

2x′)− b(x− 1

2x′)]

W ′

× exp

[− i~x′ · (η − ξ′)

], (12.96)

follows, where we defined W ′ = W (x, ξ′, t).


12.C The FOURIER Transform

We define the FOURIER transform of an operator, which is diagonal in posi-tion space, i.e. V (x, y) = V (x)δ(x− y) as

V (η) =1

(2π~)d

∫dxV (x) exp

(− i~x · η

), (12.97)

and, therefore, its inverse as

V (x) =

∫dηV (η) exp

(i

~x · η

). (12.98)

In a similar fashion we define the FOURIER transform of a functionK(x, y)which stems from a two particle operator i.e. K(x, y, x′, y′) = K(x, y)δ(x −x′)δ(y − y′) as

K(ξ, ξ′) =1

(2π~)2d

∫dxdyK(x, y) exp

[− i~

(x · ξ + y · ξ′)]. (12.99)

In particular, if K(x, y) = K(x− y) we obtain

K(ξ, ξ′) =1

(2π~)d

∫dxK(x) exp

(− i~x · ξ

)δ(ξ + ξ′)

= K

(ξ − ξ′

2

)δ(ξ + ξ′). (12.100)

Moreover, we will frequently employ the identities

δ(x) =1

(2π)d

∫dη exp(iη · x), (12.101)

andδ(η) =

1

(2π)d

∫dx exp(−iη · x), (12.102)

where δ(·) denotes DIRAC’s delta distribution.The semiclassical FOURIER transform is obtained by replacing all ap-

pearing ~-s in Eq. (12.97) by ε. Hence, we have

V (ε)(η) =1

(2πε)d

∫dxV (ε)(x) exp

(− iεx · η

), (12.103)

and, therefore, also

V (ε)(x) =

∫dηV (ε)(η) exp

(i

εx · η

), (12.104)


for a single particle operator which is diagonal in position space. Here, theindex (ε) signifies that V may still be a function of ε. In the particular casethat V (ε)(x) = V (x/ε) we obtain the important result that

V (ε)(η) =1

(2πε)d

∫dxV

(xε

)exp

(− iεx · η

)= V (η), (12.105)

is independent of ε. In a similar fashion we obtain that

K(ε)(ξ, ξ′) =1

(2πε)2d

∫dxdyK(ε) (x, y) exp

[− iε

(x · ξ + y · ξ′)], (12.106)

is independent of ε for K(ε)(x, y) strongly varying. In particular,

K(ε)(ξ, ξ′) =1

(2πε)2d

∫dxdyK

(xε,y

ε

)exp

[− iε

(x · ξ + y · ξ′)]

= K(ξ, ξ′).

(12.107)

12.D Rewriting the Dissipator

We shall derive the representation (12.42) of the dissipator. In what followswe shall drop the explicit notation of the momentum argument ξ as well asthe tildes in order to simplify the expressions. We note that we can expressthe matrix K with the help of the PAULI base (12.10) as

K = K0 ⊗ 1 + ~K ~σ, (12.108)

where ~K ~σ = Ki ⊗ σi where i runs from 1 to 3 and we defined the compo-nents Ki ∈ H2(C) for i = 0, . . . , 3 according to Eq. (12.10). We express theremaining hermitian matrices Ki in a similar fashion in order to obtain

Ki = ki1 + ~ki · ~σ = ki01 + kijσj, (12.109)

and, hence,

K = k001⊗ 1 + k0jσj ⊗ 1 + ki01⊗ σi + kijσj ⊗ σi. (12.110)

It follows from the definition (12.10) that kij ∈ R. Furthermore, from the in-distinguishability of bath particles we note that καα′ββ′ = κα′αβ′β and, there-fore,

kij = kji i, j = 0, . . . , 3. (12.111)

In a similar fashion we decompose the interaction potential V ∈ H2(C) ⊗H2(C) with respect to the second particle with the help of Eq. (12.10) as Eq.


(12.15) where Vi ∈ H2(C) are given by Eqs. (12.15). With the help of thesedefinitions we rewrite a typical sum which appears in Eq. (12.41) as

κββ′αα′VαβVα′β′ = kijViVj, (12.112)

where the sum goes over all i, j = 0, . . . , 3. We note that we can understandthe above sum Eq. (12.112) as the scalar product between a vector ~V =(V0, V1, V2, V3)T ∈ R4 ⊗H2(C) and the rotated vector KV , where the matrixK = kij ∈ R4×4 is symmetric due to Eq. (12.111), i.e. KT = K. Since allelements ofK are real and sinceK is symmetric, it follows that theKmay bediagonalized by an orthogonal matrix U ∈ R4×4, where UT = U−1. Hence,denoting by (·, ·) the scalar product in R4, we have

kijViVj = (V ,KV) = (V ,UTRUV) = (UV ,RUV) = (S,RS), (12.113)

where we defined S = UV ∈ R4 ⊗H2(C). Denoting by Si the componentsof the vector S, and by ρi the diagonal elements ofRwhere Si ∈H2(C) andρi ∈ R, we obtain

κββ′αα′VαβVα′β′ = ρiSiSi. (12.114)

We remark that this convenient form of the sum Eq. (12.112) is a resultof the indistinguishability of bath particles, which assures that K is a real,symmetric matrix.

12.E The Kernel of Q(2)~

We shall briefly demonstrate that Q(2)~ (A) = 0 is equivalent to [Si(ξ), A] = 0

for all ξ and all i if detailed balance is required for A, where A, Si(ξ) ∈H2(C). Detailed balance means that each term contributing to Q(2)

~ , Eq.(12.45), vanishes individually. Hence, it suffices to investigate the operator

Q(2)~ (A) =

1

2

∫dξ[[Λ(ξ), A],Λ(ξ)], (12.115)

where Λ(ξ) ∈ H2(C). We assume that A 6= 1 however Q(2)~ (A) = 0. Then,

clearly

tr[Q(2)

~ (A)A]

=

∫dξtr

[Λ(ξ)AΛ(ξ)A− Λ(ξ)Λ(ξ)A2

]= 0. (12.116)

We rewrite the first term of this equation for all ξ as

tr [Λ(ξ)AΛ(ξ)A] = λii(ξ)λkk(ξ)|aik(ξ)|2, (12.117)


where the λii are the eigenvalues of Λ(ξ) and the aik(ξ) are matrix elementsof A represented in the eigenbasis of Λ(ξ). In a similar fashion, we obtainfor all ξ

tr[Λ(ξ)Λ(ξ)A2

]= λ2

ii(ξ)|aik(ξ)|2. (12.118)

Since Λ(ξ) ∈H2(C),

tr[Λ(ξ)AΛ(ξ)A− Λ(ξ)Λ(ξ)A2

]= − [λ11(ξ)− λ22(ξ)]2 |a12(ξ)|2 ≤ 0, (12.119)

and, therefore,

Q(2)~ (A) ≤ 0, Q(2)

~ (A) = 0⇒ [Λ(ξ), A] = 0 ∀ξ. (12.120)

The statement [Λ(ξ), A] = 0⇒ Q(2)~ (A) = 0 is trivial and, therefore,

Q(2)~ (A) = 0⇔ [Si(ξ), A] = 0 ∀ξ and ∀i, (12.121)

if detailed balance is required for A.

12.F Derivation of the MOYAL Product

In this Appendix we briefly present the derivation of Eq. (12.74), i.e. theMOYAL product, where the MOYAL bracket has been specified in App. 12.B.We start with Eq. (12.92). We rewrite this equation in dimensionless vari-ables, see Eq. (12.69), in order to obtain

Wε

[(Aρ)

(x, y)]

=1

(2πε)2d

∫dx′dz′

∫dξdξa(ε)

(z′ +

1

4x′, ξ +

1

4ξ

)×W (ε)

(z′ − 1

4x′, ξ − 1

4ξ

)× exp

[i

ε(x− z′) · ξ +

i

εx′ · (ξ − η)

]=

1

(2π)2d

∫dx′dz′

∫dξdξa(ε)

(z′ +

ε

4x′, ξ +

ε

4ξ)

×W (ε)(z′ − ε

4x′, ξ − ε

4ξ)

× exp[i(x− z′) · ξ + ix′ · (ξ − η)

]. (12.122)

The expression (12.122) can now be expanded in a TAYLOR series in termsof ε around ε = 0 under the assumption that a(ε) and W (ε) are both slowly


varying functions with amplitudes of order one. Then, we obtain in zerothorder

O0 = a(0)W (0), (12.123)

where Eqs. (12.101) and (12.102) have been used in order to eliminate ofthe integrals in Eq. (12.122) for ε = 0. A similar calculation in first orderdemonstrates that

O1 =i

2∇xa

(0) · ∇ηW(0) − i

2∇ηa

(0) · ∇xW(0). (12.124)

Higher order contributions can be obtained in a similar fashion but will notbe discussed here.

The same calculation can be carried out for the other term of the com-mutator and one obtains

Wε

[(ρA)

(x, y)]

= W (0)a(0) +iε

2

[∇ηW

(0) · ∇xa(0)

−∇xW(0) · ∇ηa

(0)]

+O(ε2). (12.125)

Combining Eqs. (12.123), (12.124) and (12.125) finally gives the desired re-sult, Eq. (12.74),

Wε [L (A) (x, y)] =[a(0),W (0)

]+iε

2

[a(0),W (0)

x,η−W (0), a(0)

x,η

]+O(ε2), (12.126)

where ·, ·x,η denotes POISSON’s bracket.

12.G Derivation of Eqs. (12.80)

We rescale Eq. (12.40) in order to obtain

1

εQε(W (ε))(x, η, t) =

2(2πε)2d

ε

∫dξdξ′κ

(ε)ββ′αα′(ξ, ξ

′)

×[Vβα (ξ)W (ε)

(x, η − ξ − ξ′

2, t

)Vβ′α′ (ξ

′)

−1

2Vβα (ξ) Vβ′α′ (ξ

′)W (ε)

(x, η − ξ + ξ′

2, t

)−1

2W (ε)

(x, η +

ξ + ξ′

2, t

)Vβα (ξ)Vβ′α′ (ξ

′)

]× exp

[i

εx · (ξ + ξ′)

]. (12.127)


According to Eq. (12.105), where we used that Vβα(ξ) is independent ofε since V(ε)(r) is strongly varying in position space. Furthermore, from thedefinition of K, Eq. (12.24), we obtain that

K(ε)(z, z′) = γC(ε)(z − z′) + bD(z, z′), (12.128)

where we already inserted assumption (12.78) and employed that due toEq. (12.72c) D(ε)(z, z′) = bD(z, z′). Hence, with the help of Eq. (12.107) weobtain for the matrix elements of Eq. (12.128)

κ(ε)(ξ, ξ′) = γc(ε)αα′ββ′(ξ)δ(ξ + ξ′) + εd

(ε)αα′ββ′ (ξ, ξ

′) . (12.129)

We shall now discuss these two contributions to the integral in Eq. (12.127)separately. We note that the FOURIER transform of the strongly varying partof C(ε), see Eq. (12.78), is independent of ε, i.e. Γ

(ε)0 = Γ0. Hence, if γ = ε the

strongly varying part results in a collision integral of the form (12.80), forγ = ε2 or even weaker, this contribution vanishes and for γ = 1 or stronger,the collision integral diverges. In a similar fashion, for the slowly varyingpart of C(ε) we have

Γ(ε)s (ξ) =

1

(2πε)d

∫dzΓs(z) exp

(− iεξ · z

)≡ 1

εdΓs

(ξ

ε

), (12.130)

hence, the resulting collision integral vanishes as ε approaches zero.We shall now study the contribution to (12.127) arising from D. Apply-

ing the FOURIER transform (12.107) to D [see Eq. (12.21)] gives the matrixelements

d(ε)αα′ββ′(ξ, ξ

′) =δαα′δββ′

(2πε)2d

∫dznαβ(z) exp

[− iεz · (ξ + ξ′)

]=

δαα′δββ′

(2πε)2dnαβ

(ξ + ξ′

ε

). (12.131)

Inserting this expression into Eq. (12.127) yields that this contribution tendsto zero after the substitution ξ → εξ and ξ′ → εξ′. The final form of Eq.(12.80) is obtained by performing the steps outlined in App. 12.D.


12.H Derivation of Eqs. (12.84)

In order to derive this result we rewrite Eq. (12.35) in rescaled variables as

Qε(W (ε)) :=2

(2πε)d

∫dx′dzdz′

∫dη′κ

(ε)ββ′αα′(z, z

′)

×[Vαβ

(x+

1

2x′ − z

)W (ε)′Vα′β′

(x− 1

2x′ − z′

)−1

2Vαβ

(x+

1

2x′ − z

)Vα′β′

(x+

1

2x′ − z′

)W (ε)′

−1

2W (ε)′Vαβ

(x− 1

2x′ − z

)Vα′β′

(x− 1

2x′ − z′

)]× exp

[− iεx′ · (η − η′)

], (12.132)

where W (ε)′ = W (ε)(x, η′, t). We note that Vαβ(r) is independent of ε since Vis slowly varying in position space.

We now study the different scenarios arising from Eq. (12.82). For theslowly varying part of C(ε) we substitute x′ → εx′, draw the limit and inte-grate with respect to x′ and η′ in order to obtain Eq. (12.84a) for γ = ε in thelow density limit and for γ = 1/ε in the weak coupling limit.

For the strongly varying part C(ε) we also substitute z → εz and z′ → εz′.Hence, z and z′ only appear in the matrix elements of Γ0 as ε→ 0. However,according to Eq. (12.25) this integral vanishes at all scales, thus, no contri-bution arises from the strongly varying part Γ

(ε)0 in the semiclassical limit.

Finally, in the low density limit the contribution of D(ε)(z, z′) = εD(z, z′) iseasily seen to be of the form (12.84b). We remark this term vanishes in theweak-coupling limit.

Chapter 13

CONCLUDING SUMMARY

13.1 Summary

Let us summarize this thesis. Its aim is to contribute to the theoretical de-scription of spin transport in solid state systems. The rapid development ofspintronics requires highly sophisticated theoretical models which are ca-pable of describing spin dynamics on very different length- and time-scales.Naturally, a full quantum treatment of such a system is in most situationsfar too complex and, moreover, not feasible with state-of-the-art numericaltechniques.1 Hence, one has to rely on suitable and controlled simplifica-tions in order to achieve a realistic account of spin related transport phe-nomena in a solid state environment.

This thesis is composed of two main parts: Within the first part wepresent a full quantum mechanical treatment of spin-coherent transport inhalfmetal / semiconductor heterostructures. It is then applied to the partic-ular scenario of zinc-blende CrAs / GaAs spin-filtering devices. Within thesecond part we derive in a systematic fashion a linear semiclassical spinorialBOLTZMANN equation from the dynamics of a composite quantum system.Let us briefly summarize these two parts in more detail.

It is the aim of the first part to describe spin polarized transport throughhalfmetal / semiconductor heterostructures. The theoretical approach isbased on three major steps: The first determines the bulk electronic struc-tures of the materials involved with the help of the method of linear muffin

1In fact, in many situations a semiclassical approach is sufficient to capture the mainphysical properties of the system.

207

208 Chapter 13: CONCLUDING SUMMARY

tin orbitals in the local spin density approximation. In particular, we com-pute the band structures of bulk zinc-blende GaAs and bulk zinc-blendeCrAs, as well as of a supercell, which contains a lattice-matched heteroin-terface between the two materials. The latter serves to calculate the bandoffset between GaAs and CrAs. In a second step, these ingredients are usedto design an effective Hamiltonian. This is achieved by mapping the elec-tronic structures onto a nearest neighbor sp3d5s∗ empirical tight-binding(ETB) model. Additional constraints on the fitting procedure ensure thatthe symmetry properties of the bulk materials are respected. In a thirdstep, a partial FOURIER transformation allows us to obtain a quasi one-dimensional transport Hamiltonian. Finally, the current density at a finiteapplied voltage is obtained within a non equilibrium GREEN’s function ap-proach. The results indicate that CrAs / GaAs heterostructures act as ef-ficient room-temperature spin-filtering devices, making this material com-bination a particularly attractive candidate for spintronic applications. Inaddition, we discuss some subtleties of ETB modeling of charge transportacross heterointerfaces with the help of a simple toy model. It is shown thatimportant symmetries stemming from the bulk ETB Hamiltonians may beviolated under naive matching conditions. The proposed remedy to this in-consistency are two matching conditions at the interface, which are latticeversions of the continuity conditions in continuous space quantum mechan-ics.

Within the second part we follow an alternative path towards a properdescription of spin transport in solid state devices. In particular, we presenta microscopic derivation of the linear semiclassical spinorial BOLTZMANNequation, which, in turn, may be regarded as the basis of a plethora of semi-classical spin transport models. The investigation is based on the treatmentof a composite quantum system AB with VON NEUMANN’s equation whichhas been augmented with a relaxation superoperator Q(ρ) acting on thestate operator ρ of the composite system. Here, subsystemA is a single spin-1/2 particle which interacts with magnetic scatterers (subsystemB). It is theaction of this superoperator to relax the composite state ρ into the BORN ap-proximation ρA⊗χB within some finite time τ , where χB is some predefinedequilibrium state and ρA = trB (ρ) is the reduced state operator. If it is as-sumed that this relaxation is by far the fastest process in the dynamics and,further, if strong interactions between subsystem A and B are considered,one can derive a hierarchy of master equations for the reduced state op-erator ρA = trB (ρ), whereby the zeroth order contribution is a LINDBLADmaster equation. Higher order corrections include non-Markovian effects,as well as correlations and entanglement between subsystem A and B. In

Chapter 13: CONCLUDING SUMMARY 209

a second step, we perform the semiclassical limit of this LINDBLAD mas-ter equation. This is achieved by transforming the whole equation into theWIGNER representation and posing appropriate scaling assumptions ontothe functions characterizing the dynamics. It is demonstrated that, undersuitable conditions, the resulting equation is a linear spinorial BOLTZMANNequation equipped with a collision operator which describes momentum re-laxation as well as spin decoherence. Hence, this approach allows the treat-ment of spin polarized transport on a semiclassical level, where quantumcorrections as well as non-Markovian effects can be incorporated perturba-tively in a straight-forward manner.

13.2 Future Prospects

We give a brief overview of possible extensions and future perspectives ofthe work presented in this thesis. It is definitively a possible future taskto apply the transport code and the related theoretical approach developedin the first part to further interesting materials. Potential candidates are,for instance, VAs / GaAs or MnAs / GaAs heterostructures, but also non-magnetic heterostructures are tractable and offer many exciting open prob-lems. In the course of such studies it might be necessary to extend thepresent formulation of the transport code to non-ZB type materials or toincorporate spin-orbit interactions, thus leaving the realm of spin-coherenttransport.2 Moreover, the incorporation of self-consistent corrections, i.e.coupling the equations of motion to the POISSON equation, might be verychallenging from a numerical point of view. Because of the dependence ofthe transport Hamiltonian on k‖ one has to find a very efficient method tocompute the charge density which is inserted into POISSON’s equation. Anadditional improvement of the current approach is to develop an advancedmodel of the interface between the two different materials. Such interfacialmodels would have to be based on high-end ab-initio studies. The results ofsuch studies can then be compared in a systematic manner with the resultsobtained with the current approach, with a matching condition as proposedin Chap. 9, as well as, with an alternative model reported by BRANDBYGEet al. [40].

All these possible extensions concern the theoretical approach in generaland not spin-filtering in CrAs / GaAs heterostructures in particular. Ofcourse, one can include spin-orbit coupling, self-consistent corrections oradvanced interface models into the CrAs / GaAs calculations. However,

2For instance, a study of spin-flip processes at interfaces in GaMnAs has been carriedout in our group by P. SENEKOWITSCH.

210 Chapter 13: CONCLUDING SUMMARY

it is my personal opinion that these extensions do not affect the validity ofthe statement that CrAs / GaAs heterostructures are probable candidatesfor efficient spin-filters. I expressed and explained this point in Chap. 8 inmore detail. On the other hand, additional experimental investigations ofCrAs / GaAs heterostructures are highly desirable. Clearly, the quantitiesof interest for these future studies are the precise crystal structure of thinlayers of CrAs on GaAs and the electronic properties of CrAs thin-films.

There are numerous possible routes to continue the work presented inthe second part of this thesis. For instance, it is still not entirely clear whatthe role of the non-Markovian corrections to the LINDBLAD equation is, seeChap. 11. It would be an interesting task to solve the LINDBLAD equa-tion as well as the first order correction for a simple model system, suchas a spin chain or a two-level system, in interaction with an environment.As soon as the role of these corrections is understood in more detail onemight investigate their influence on the semiclassical evolution. This is astraight-forward exercise which consists of repeating the steps presented inChap. 12 for the first order correction in Chap. 11. On the other hand, itis also a very interesting question how further semiclassical spin transportmodels can be obtained from the spinorial BOLTZMANN equation. In anal-ogy to the derivation of the hydrodynamic models [24] it should be possi-ble to derive spin-hydrodynamic models which might lead to new physics.Again, the role of non-Markovian effects offers numerous interesting stud-ies. Moreover, the influence of quantum corrections to the BOLTZMANNequation has not yet been investigated which is a challenging task also fromthe numerical point of view. It opens the door for an alternative approachto the physics described within the first part of this thesis by starting witha semiclassical transport equation and successively incorporating quantumcorrections. One could also imagine more complicated device geometries,such as a spin-transistor configuration which means that the BOLTZMANNequation has to be solved on a two-dimensional domain.

An entirely different extension of the work presented in the second partis to investigate the case of identical (bosonic or fermionic) particles. It isemphasized in Chap. 12 that due to the particular choice of Q(ρ) in Chap.11, the resulting BOLTZMANN equation is linear. In order to obtain non-linear transport models one has to take the indistinguishability of particlesinto account. In such a way it should, at least in principle, be possible toderive numerous non-linear transport models, such as the BLOCH - BOLTZ-MANN - PEIERLS equations or the LANDAU - LIFSHITZ equation.

BIBLIOGRAPHY

[1] C. Ertler, A. Matos-Abiague, M. Gmitra, M. Turek, and J. Fabian, Jour-nal of Physics: Conference Series 129, 012021 (2008).

[2] I. Zutic, J. Fabian, and S. Das Sarma, Rev. Mod. Phys. 76, 323 (2004).

[3] J. Fabian, A. Matos-Abiague, C. Ertler, P. Stano, and I. Zutic, ActaPhysica Slovaca. Reviews and Tutorials 57, 565 (2007).

[4] H. Skriver, The LMTO method: Muffin Tin Orbitals and Electronic Struc-ture (Springer, Berlin, 1984).

[5] O. Jepsen and O. Andersen, The Stuttgart LMTO-TB program (MaxPlanck Institut für Festkörperforschung, Stuttgart, 2000), link:http://www.fkf.mpg.de/andersen/LMTODOC/LMTODOC.html.

[6] S.-H. Wei and A. Zunger, Applied Physics Letters 72, 2011 (1998).

[7] Y.-H. Li et al., Applied Physics Letters 94, 212109 (2009).

[8] J.-M. Jancu, R. Scholz, F. Beltram, and F. Bassani, Phys. Rev. B 57, 6493(1998).

[9] P. Vogl, H. P. Hjalmarson, and J. D. Dow, Journal of Physics and Chem-istry of Solids 44, 365 (1983).

[10] J. C. Slater and G. F. Koster, Phys. Rev. 94, 1498 (1954).

[11] J. A. Støvneng and P. Lipavský, Phys. Rev. B 49, 16494 (1994).

[12] M. D. Ventra, Eletrical Transport in Nanoscale Systemambridges (Cam-bridge University Press, Cambridge, 2008).

211

212 BIBLIOGRAPHY

[13] A. D. Carlo, Semiconductor Science and Technology 18, R1 (2003).

[14] B. A. Stickler, C. Ertler, E. Arrigoni, L. Chioncel, and W. Pötz, AIPConference Proceedings - ICPS 2012, Zurich (2012).

[15] B. A. Stickler, C. Ertler, L. Chioncel, and W. Pötz, draft version, arXiv:1301.2933 [cond-mat.mes-hall] .

[16] B. A. Stickler and W. Pötz, Journal of Computational Electronics , DOI:10.1007/s10825 (2013).

[17] S. Possanner and B. A. Stickler, Phys. Rev. A 85, 062115 (2012).

[18] B. A. Stickler and S. Possanner, arXiv: 1304.4772 [quant-ph] .

[19] H. P. Breuer and F. Petruccione, The theory of open quantum systems(Oxford University Press Oxford, 2002).

[20] G. Lindblad, Communications in Mathematical Physics 48, 119 (1976).

[21] S. Nakajima, Progress of Theoretical Physics 20, 948 (1958).

[22] R. Zwanzig, The Journal of Chemical Physics 33, 1338 (1960).

[23] S. Possanner and C. Negulescu, Kinetic and Related Models 4, 1159(2011).

[24] C. Cercignani, The Boltzmann equation and its applications (Springer,1988).

[25] S. J. Hashemifar, P. Kratzer, and M. Scheffler, Phys. Rev. B 82, 214417(2010).

[26] H. Akinaga, T. Manago, and M. Shirai, Japanese Journal of AppliedPhysics 39, L1118 (2000).

[27] P. Mavropoulos and I. Galanakis, Journal of Physics: Condensed Mat-ter 19, 315221 (2007).

[28] M. Mizuguchi et al., Journal of Magnetism and Magnetic Materials239, 269 (2002).

[29] M. Mizuguchi et al., Surface Review and Letters 09, 331 (2002).

[30] J. Kübler, Phys. Rev. B 67, 220403 (2003).

BIBLIOGRAPHY 213

[31] I. Galanakis and E. Sasioglu, Journal of Applied Physics 109, 113912(2011).

[32] I. Galanakis and P. Mavropoulos, Phys. Rev. B 67, 104417 (2003).

[33] O. Bengone et al., Phys. Rev. B 70, 035302 (2004).

[34] L. Chioncel et al., Phys. Rev. B 83, 035307 (2011).

[35] V. H. Etgens et al., Phys. Rev. Lett. 92, 167205 (2004).

[36] O. Madelung, Introduction to Solid-State Theory (Springer - Berlin,1996).

[37] H. Ibach and H. Lüth, Solid-State Physics: An Introduction to Principlesof Materials Science (Springer - Heidelberg, 2002).

[38] W. Harrison, Elementary Electronic Structure (World Scientific - Singa-pore, 1999).

[39] A. Messiah, Quantum mechanics (North-Holland Publishing Com-pany, 1996).

[40] M. Brandbyge, J.-L. Mozos, P. Ordejón, J. Taylor, and K. Stokbro, Phys.Rev. B 65, 165401 (2002).

[41] C. Fiolhais, F. Noguiera, and M. Marques, A Primer in Density Func-tional Theory (Springer, 2003).

[42] J. P. Perdew and Y. Wang, Phys. Rev. B 45, 13244 (1992).

[43] A. Szabo and N. Ostlund, Modern Quantum Chemistry: Introductionto Advanced Electronic Structure Theory (Dover Books on Chemistry,1996).

[44] H. Sormann and E. Schachinger, Theoretische Festkörperphysik (LectureNotes - TU Graz, 2011).

[45] O. K. Andersen, Phys. Rev. B 12, 3060 (1975).

[46] O. K. Andersen and O. Jepsen, Phys. Rev. Lett. 53, 2571 (1984).

[47] O. K. Andersen et al., eprint arXiv:cond-mat/9804166 (1998),arXiv:cond-mat/9804166.

[48] E. Zurek, O. Jepsen, and O. K. Andersen, ChemPhysChem 6, 1934(2005).

214 BIBLIOGRAPHY

[49] NN, The LMTO method (Universität Frankfurt, XXXX), link: itp.uni-frankfurt.de/∼valenti/CHAP5_two.pdf.

[50] J. Taylor, Scattering Theory: The Quantum Theory of nonrelativistic Colli-sions (Dover Publications, 2006).

[51] A. Georges, G. Kotliar, W. Krauth, and M. J. Rozenberg, Rev. Mod.Phys. 68, 13 (1996).

[52] M. Potthoff, M. Aichhorn, and C. Dahnken, Phys. Rev. Lett. 91, 206402(2003).

[53] C. Gros and R. Valentí, Phys. Rev. B 48, 418 (1993).

[54] L. Hedin, Phys. Rev. 139, A796 (1965).

[55] M. Shishkin and G. Kresse, Phys. Rev. B 75, 235102 (2007).

[56] Y. Tan et al., Journal of Computational Electronics 12, 56 (2013).

[57] A. Urban, M. Reese, M. Mrovec, C. Elsässer, and B. Meyer, Phys. Rev.B 84, 155119 (2011).

[58] P. Yu and M. Cardona, Fundametnals of Semicinductors (Springer -Berlin, 2005).

[59] N. Ashcroft and N. Mermin, Festkörperphysik (Oldenburg Verlag -München, 2005).

[60] WIKIPEDIA, http://en.wikipedia.org/wiki/Cubic _ crystal _ system.

[61] S. Y. Ren, X. Chen, and J. D. Dow, J. Phys. Chem. Solids 49 (1997).

[62] D. J. Chadi, Phys. Rev. B 16, 790 (1977).

[63] S. Schulz, S. Schumacher, and G. Czycholl, European Physical JournalB 64, 51 (2008).

[64] J. Sakurai, Modern Quantum Mechanics (Addison Wesley Longman -Reading, Massachusetts, 1994).

[65] B. A. Stickler and E. Schachinger, Basic Concepts in ComputationalPhysics (in print. Springer - Heidelberg, 2013).

[66] A. Hartmann and H. Rieger, Optimization Algorithms in Physics (Wiley-VCH - Berlin, 2002).

BIBLIOGRAPHY 215

[67] F. Starrost, S. Bornholdt, C. Solterbeck, and W. Schattke, Phys. Rev. B53, 12549 (1996).

[68] S. Sapra, N. Shanthi, and D. D. Sarma, Phys. Rev. B 66, 205202 (2002).

[69] M. Graf and P. Vogl, Phys. Rev. B 51, 4940 (1995).

[70] R. Lake, G. Klimeck, R. C. Bowen, and D. Jovanovic, Journal of Ap-plied Physics 81, 7845 (1997).

[71] C. Strahberger and P. Vogl, Phys. Rev. B 62, 7289 (2000).

[72] T. B. Boykin, J. P. A. van der Wagt, and J. S. Harris, Phys. Rev. B 43,4777 (1991).

[73] S. Datta, Electronic Transport in Mesoscopic Systems (Cambridge Studiesin Semiconductor Physics and Microelectronic Engineering, 1999).

[74] M. P. L. Sancho, J. M. L. Sancho, J. M. L. Sancho, and J. Rubio, Journalof Physics F: Metal Physics 15, 851 (1985).

[75] K. Burch, D. Awschalom, and D. Basov, Journal of Magnetism andMagnetic Materials 320, 3207 (2008).

[76] K. Sato et al., Rev. Mod. Phys. 82, 1633 (2010).

[77] C. Ertler and W. Pötz, Phys. Rev. B 86, 155427 (2012).

[78] S. Ohya, K. Takata, and M. Tanaka, Nature Physics 7 (2011).

[79] E. Likovich et al., Phys. Rev. B 80, 201307 (2009).

[80] I. Muneta, S. Ohya, and M. Tanaka, Applied Physics Letters 100,162409 (2012).

[81] S. Sanvito and N. A. Hill, Phys. Rev. B 62, 15553 (2000).

[82] Y.-J. Zhao and A. Zunger, Phys. Rev. B 71, 132403 (2005).

[83] K. Ono et al., Journal of Applied Physics 91, 8088 (2002).

[84] M. Mizuguchi et al., Journal of Applied Physics 91, 7917 (2002).

[85] J. F. Bi et al., Applied Physics Letters 88, 142509 (2006).

[86] F. Beiuseanu et al., Phys. Rev. B 83, 125107 (2011).

216 BIBLIOGRAPHY

[87] G. Strasser, private communications.

[88] L. Chioncel, M. I. Katsnelson, G. A. de Wijs, R. A. de Groot, and A. I.Lichtenstein, Phys. Rev. B 71, 085111 (2005).

[89] C. Ertler, W. Potz, and J. Fabian, Applied Physics Letters 97, 042104(2010).

[90] C. Ertler and W. Pötz, Phys. Rev. B 84, 165309 (2011).

[91] W. Pötz, Journal of Applied Physics 71, 2297 (1992).

[92] W. Pötz, Journal of Applied Physics 66, 2458 (1989).

[93] A. Di Carlo, P. Vogl, and W. Pötz, Phys. Rev. B 50, 8358 (1994).

[94] P. Mavropoulos, N. Papanikolaou, and P. H. Dederichs, Phys. Rev. B69, 125104 (2004).

[95] K. Gundlach, Solid-State Electronics 9, 949 (1966).

[96] W. Pötz and D. K. Ferry, Superlattices and Microstr. 3 (1987).

[97] T. B. Boykin, G. Klimeck, and F. Oyafuso, Phys. Rev. B 69, 115201(2004).

[98] J. N. Schulman and Y.-C. Chang, Phys. Rev. B 27, 2346 (1983).

[99] N. van Kampen, Stochastic processes in physics and chemistry (Elsevier -Amsterdam, 2007).

[100] D. Eng and L. Erdös, Reviews in Mathematical Physics 17, 669 (2005).

[101] C. Zachos, D. Fairlie, and T. Curtright, Quantum mechanics in phasespace: an overview with selected papers (World Scientific, 2005).

[102] R. Liboff, Kinetic Theory - Classical, Quantum and Relativistic Descrip-tions (Springer - New York, 2003).

[103] V. Spicka and P. Lipavský, Phys. Rev. B 52, 14615 (1995).

[104] H. Spohn, Journal of Statistical Physics 17, 385 (1977).

[105] L. Erdös and H.-T. Yau, Communications on Pure and Applied Math-ematics 53, 667 (2000).

BIBLIOGRAPHY 217

[106] D. Benedetto, F. Castella, R. Esposito, and M. Pulvirenti, Journal ofStatistical Physics 116, 381 (2004).

[107] M. Nielsen and I. Chuang, Quantum Computation and Quantum Infor-mation (Cambridge University Press, Cambridge, 2000).

[108] D. Walls and G. Milburn, Quantum optics (Springer verlag, 2008).

[109] D. Awschalom, D. Loss, and N. Samarth, Semiconductor Spintron-ics and Quantum ComputationSeries on Nanoscience and Technology(Springer-Verlag Berlin, 2002).

[110] K. Hornberger et al., Physical review letters 90, 160401 (2003).

[111] L. Hackermüller, K. Hornberger, B. Brezger, A. Zeilinger, andM. Arndt, Nature 427, 711 (2004).

[112] A. Cronin, J. Schmiedmayer, and D. Pritchard, Reviews of ModernPhysics 81, 1051 (2009).

[113] U. Weiss, Quantum dissipative systems (World Scientific Pub Co Inc,2008).

[114] A. Rivas and S. Huelga, Open Quantum Systems: An Introduction(Springer Verlag, 2011).

[115] M. Arndt, K. Hornberger, and A. Zeilinger, Physics world 18, 35(2005).

[116] R. Alicki, Physical Review A 65, 034104 (2002).

[117] E. Joos and H. Zeh, Zeitschrift für Physik B Condensed Matter 59, 223(1985).

[118] A. Kossakowski, Reports on Mathematical Physics 3, 247 (1972).

[119] E. Davies, Communications in mathematical Physics 39, 91 (1974).

[120] V. Gorini, A. Frigerio, M. Verri, A. Kossakowski, and E. Sudarshan,Reports on Mathematical Physics 13, 149 (1978).

[121] L. Diosi, EPL (Europhysics Letters) 30, 63 (1995).

[122] C. Gardiner and P. Zoller, Physical Review A 55, 2902 (1997).

[123] P. Dodd and J. Halliwell, Physical Review D 67, 105018 (2003).

218 BIBLIOGRAPHY

[124] B. Vacchini, Physical Review Letters 84, 1374 (2000).

[125] B. Vacchini, Journal of Mathematical Physics 42, 4291 (2001).

[126] K. Hornberger, Physical review letters 97, 60601 (2006).

[127] K. Hornberger, EPL (Europhysics Letters) 77, 50007 (2007).

[128] B. Vacchini and K. Hornberger, Physics Reports 478, 71 (2009).

[129] M. Knap, E. Arrigoni, W. von der Linden, and J. Cole, Physical ReviewA 83, 023821 (2011).

[130] A. Royer, Physical review letters 77, 3272 (1996).

[131] S. Barnett and S. Stenholm, Physical Review A 64, 33808 (2001).

[132] A. Budini, Physical Review A 69, 042107 (2004).

[133] Y. Yan and R. Xu, Annu. Rev. Phys. Chem. 56, 187 (2005).

[134] H. P. Breuer, J. Gemmer, and M. Michel, Physical Review E 73, 016139(2006).

[135] H. P. Breuer, Physical Review A 75, 022103 (2007).

[136] K. Liu and H. Goan, Physical Review A 76, 022312 (2007).

[137] D. Chruscinski and A. Kossakowski, Physical review letters 104,70406 (2010).

[138] I. Prigogine and R. Landshoff, Physics Today 16, 76 (1963).

[139] F. Shibata, Y. Takahashi, and N. Hashitsume, Journal of StatisticalPhysics 17, 171 (1977).

[140] F. Shibata and T. Arimitsu, J. Phys. Soc. Jap 49, 891 (1980).

[141] N. Abdallah and P. Degond, Journal of Mathematical Physics 37, 3306(1996).

[142] L. Ballentine, Quantum Mechanics - A Modern Development (World Sci-entific Publishing, 1998).

[143] A. Pazy, Semigroups of linear operators and applications to partial differen-tial equations (Springer, 1983).

BIBLIOGRAPHY 219

[144] I. Zutic, J. Fabian, and D. S. S., Rev. Mod. Phys. 76, 323 (2004).

[145] J. Fabian, A. Matos-Abiague, E. C., P. Stano, and I. Zutic, Acta PhysicaSlovaca 57, 565 (2007).

[146] S. Zhang, P. Levy, and A. Fert, Phys. Rev. Lett. 88, 236601 (2002).

[147] C. Garcia-Cervera and X.-P. Wang, J. Comp. Phys. 224, 699 (2007).

[148] S. Possanner and N. Ben Abdallah, Proceeding of the 2010 Interna-tional Conference on Simulation of Semiconductor Processes and De-vices (SISPAD) (2010).

[149] E. Simanek, Phys. Rev. B 63, 224412 (2001).

[150] R. El Hajj, Etude mathématique et numï¿12rique de modèles de transport:

application à la spintronique, PhD thesis, Institut de Mathématiques deToulouse (IMT), Université Paul Sabatier, 2008.

[151] L. Reggiani, P. Lugli, and A. P. Jauho, Phys. Rev. B 36, 6602 (1987).

[152] K. Hornberger and B. Vacchini, Phys. Rev. A 77, 022112 (2008).

[153] A. Fert, Rev. Mod. Phys. 80, 1517 (2008).

[154] F. Piéchon and A. Thiaville, Phys. Rev. B 75, 174414 (2007).

[155] J. Xiao, A. Zangwill, and M. Stiles, Eur. Phys. J. B 59, 415 (2007).

[156] D. Culcer et al., Phys. Rev. Lett. 93, 046602 (2004).

[157] J. Zhang, P. Levy, S. Zhang, and V. Antropov, Phys. Rev. Lett. 93 (2004).

[158] C. Vouille et al., Phys. Rev. B 60, 6710 (1999).

[159] M. Viret et al., Phys. Rev. B 53, 8464 (1996).

[160] P. Gérard, P. A. Markowich, N. J. Mauser, and F. Poupaud, Commu-nications on Pure and Applied Mathematics 50, 323 (1997).

[161] H. Haug and A.-P. Jauho, Quantum kinetics in transport and optics ofsemiconductors (Springer - Berlin, 1996).

[162] A. M. O. de Almeida, P. de M Rios, and O. Brodier, Journal of PhysicsA: Mathematical and Theoretical 42, 065306 (2009).

[163] M. Esposito and S. Mukamel, Phys. Rev. E 73, 046129 (2006).

220 BIBLIOGRAPHY

[164] R. A. Bertlmann and P. Krammer, Journal of Physics A: Mathematicaland Theoretical 41, 235303 (2008).

[165] R. White, Quantum Theory of Magnetism: magnetic properties of materials(Springer, 2007).

[166] A. Kenfack and K. Zyczkowski, Journal of Optics B: Quantum andSemiclassical Optics 6, 396 (2004).

[167] O. Morandi and F. Schürrer, Journal of Physics A: Mathematical andTheoretical 44, 265301 (2011).

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