The Wigner Monte Carlo Method for Nanoelectronic Devices (Querlioz/The Wigner Monte Carlo Method for...

Chapter 1

Theoretical Framework of Quantum Transport in Semiconductors and Devices

This chapter aims to introduce the basic concepts of quantum transport in nanodevices and the possible formalisms that may be used for their modeling. The fundamentals of device quantum mechanics are briefly summarized in section 1.1, including the concepts of quasi-electrons and envelope function. The common semi-classical approach to electron transport in semiconductor devices is then presented, together with the possible quantum corrections that it may include (section 1.2). This leads naturally to a discussion on electron delocalization/localization. Three formalisms of quantum transport are then described in section 1.3: the density matrix, the Wigner function and the non-equilibrium Green’s functions. The modeling of contacts, which is a crucial problem of device physics, is addressed in section 1.4.1. Finally, we show how the effect of scattering by phonons and impurities may be integrated in quantum transport models (section 1.4.2). This leads to a general understanding of the quantum phenomena and formalism, on which the next chapters will be built.

In this chapter, we have tried to keep quantum theory accessible to readers with a general knowledge of it, as typically taught in applied physics or electrical engineering curricula. The second quantization formalism is avoided as much as possible, and we have tried to present advanced topics with an intuitive language. We hope this chapter will provide the basic physics needed for a graduate student or a researcher to approach the quantum mechanics of semiconductor electron devices. Further details and advanced material can be found using the bibliography.

2 Wigner Monte Carlo Method for Nanodevices

1.1. The fundamentals: a brief introduction to phonons, quasi-electrons and envelope functions

Here we briefly review the main concepts of solid-state physics used in this book, to keep in mind what they represent and the meaning of approximations that will be made to model quantum transport. This section is not intended to re-examine the details of solid-state physics that may be found in many textbooks.

1.1.1. The basic concepts: band structure and phonon dispersion

1.1.1.1. The crystal Hamiltonian

The quantum mechanical study of a semiconductor device starts with its general Schrödinger equation. For a system of N electrons and N ′ nuclei described by a general wave function ψ this equation is written as

( ) ( )1 1 '1 1 '

, , , , ,H , , , , , N N

N N it

ψψ

∂=

∂r r R R

r r R R… …

… … [1.1]

where we have introduced a crystal Hamiltonian H that is given by

( ) ( ) ( )2 2

, , ,

ˆ ˆ1 1ˆ ˆ ˆ ˆˆ ˆ ˆH2 2 2 2

l in n l m e e i j e n i l

l l m i i j i lV V V

M m− − −= + − + + − + −∑ ∑ ∑ ∑ ∑p pR R r r r R

[1.2]

In these two equations, electrons and nuclei are labeled by indices i, j and l, m, respectively. The electrons are characterized by their mass m and position operator

ir while M and ˆlR stand for the nuclei mass and position operator, and p is the

momentum operator with appropriate subscript to label electrons and nuclei.

( )ˆ ˆn n l mV − −R R is the interacting potential between the l-th and m-th nuclei,

( )ˆê n i lV − −r R is the interacting potential between the i-th electron and the l-th

nucleus and ( ) 20ˆ ˆ ˆ ˆ4e e i j i jV e πε− − = −r r r r is the Coulomb interacting potential

between the i-th and j-th electrons.

Unfortunately, solving the Schrödinger equation [1.1] with the Hamiltonian [1.2] is a many-body problem of tremendous complexity. Solid-state physics has, however, established some reasonable approximations and techniques which make it possible to extract the realistic behavior of electrons in typical materials and devices. With the aim of keeping in mind the physical ideas behind these approximations, we

Quantum Transport in Semiconductors and Devices 3

recall here the main steps of the derivation leading to the widely-accepted transport equations.

The first step consists of separating the electrons in core electrons of low energy which are strongly bound to the nuclei and valence electrons of high energy which are weakly bound to the nuclei. The latter are delocalized over the crystal and are likely to participate in the conduction. A nucleus and its core electrons thus form an entity called an ion or ion core, and the electron dynamics in the crystal only involves the valence electrons. The second approximation is the adiabatic or Born-Oppenheimer approximation which clearly distinguishes the dynamics of ions from that of valence electrons. The ions being much heavier than electrons, they move much slower than electrons do, and the electron quantum states can be regarded as instantaneously reacting to the changes of ion position. One can consider that the ions only see a charged cloud of electrons and not the details of their dynamics. The total wave function can then be separated as

( ) ( ) ( ), ,ion eψ ψ ψ=r R R r R [1.3]

where ( )ionψ R is the wave function for the ions and ( ),eψ r R is the wave function for electrons. The Schrödinger equation [1.1] splits into a purely ionic equation and a purely electronic equation with ionic and electronic Hamiltonians, written as

( )

( )

2

,2 2

, ,0

ˆ 1ˆ ˆ ˆH2 2

ˆ 1ˆ ˆHˆ ˆ2 2 4

lion i i l m

l l m

ie e i i l

i i j i li j

VM

eV

m πε

−

−

⎧= + −⎪

⎪⎪⎨⎪ = + + −⎪ −⎪⎩

∑ ∑

∑ ∑ ∑

pR R

pr R

r r

[1.4]

1.1.1.2. Phonons

It is usually assumed that the displacements of ions around their equilibrium position are small and the interaction between them is commonly described by harmonic potentials. The problem of ion dynamics then becomes a problem of coupled harmonic oscillators. Each ion excitation induces a collective response from the neighboring ions. The eigen-modes of angular frequency ωq of the harmonic

oscillators form a basis of the lattice vibration response, which is often shown as a textbook problem [COH 06]. The excitations of these modes can be described using the common quantum approach of the harmonic oscillator. In this harmonic approximation the quanta of vibration, the phonons, are uncoupled, which makes their description simple. In the second quantization language of phonon creation and


annihilation, operators †aq and aq , the normal-mode representation of the ionic

Hion Hamiltonian becomes [KIT 87]

† 1ˆ ˆ ˆH2ph a aω ⎛ ⎞= +⎜ ⎟

⎝ ⎠∑ q q qq

[1.5]

and the displacement operator of the l-th ion can be obtained from the usual quantum treatment of harmonic oscillators [KIT 96] as

( )1 2

†,ˆ ˆ Û

2j ji i

j ja e a eM Nω

⋅ − ⋅⎛ ⎞⎜ ⎟= +⎜ ⎟⎝ ⎠

∑ q R q Rq q q

qqε [1.6]

where ,j qε is the unit vector along the direction of the ion vibration and N is the

number of ions per unit cell of the crystal.

Neutron diffraction measurements provide comprehensive information on the modes and the phonon dispersion in the reciprocal space [WAU 63], [KUL 94]. For instance, Figure 1.1 gives the phonon dispersion in silicon for some crystallographic directions, obtained both experimentally [KUL 94] and by modeling using the adiabatic bond-charge model [WEB 77], [VAL 08b]. One can identify the acoustic (LA, TA) and optical (LO, TO) phonon branches.

02468

10121416

Freq

uenc

y ν

(TH

z)

Γ X LΓΔ Σ Λ

TA

LA

LO

TO

Figure 1.1. Phonon dispersion in silicon. Symbols: experiments from [KUL 94]. Lines: calculation using the adiabatic bond charge model [WEB 77]. Figure from [VAL 08a]


1.1.1.3. Quasi-electrons and envelope functions

Now we consider electron dynamics. The fundamental idea is first to assume the ions to be fixed at their equilibrium position 0

lR . The impact of their motion, i.e. of the phonons, on the electrons will be treated later as a perturbation. The electron Hamiltonian thus simplifies to

( )2 2

0

, ,0

ˆ 1ˆ ˆH2 2 ˆ ˆ4

ie e i i l

i i j i li j

eV

m πε−= + + −

−∑ ∑ ∑

pr R

r r [1.7]

where e iV − is the interaction potential between ions and valence electrons. This many-body Hamiltonian is still extremely difficult to treat, even numerically. Its behavior is, however, well understood in many situations. At zero temperature the electron gas is in the fundamental state but at increasing temperatures the excitations of the electron gas manifest. What do such excitations look like? The many-body theory has made it possible to answer this question. These excitations essentially behave as single electrons surrounded by a cloud of charges corresponding to the repulsion of the rest of the electronic cloud by the excited electron. To highlight this similarity, they are called “quasi-electrons” [ASH 76].

An excited electron generates a vacancy in the normally-occupied states of low energy. The dynamics of this vacancy are similar to that of a single particle of positive charge +e and influenced by the cloud of non-excited electrons. Such an excitation is called a quasi-hole.

Determining the basis for these excitations is a non-trivial problem. The common approach consists of considering that, since the quasi-particles have a long lifetime, they are solutions of a conventional one-body equation. The problem thus turns out to be a case of looking for the stationary states of the single-particle Schrödinger equation

( ) ( ) ( )2ˆ

ˆ2 c e eV E

mΨ Ψ

⎡ ⎤+ =⎢ ⎥

⎢ ⎥⎣ ⎦

pr r r [1.8]

where the potential CV includes the interaction of the quasi-electron with both the ions and the cloud of non-excited electrons. This potential, called crystal potential, is periodical with the same period as a perfect crystal. The states of quasi-electrons thus satisfy the well-known Bloch Theorem [ASH 76] and take the form

( ) ( )ie uΨ ⋅= k rk kr r [1.9]


where ( )uk r is a periodic function which contains the “atomic” details of the wave

function, while the plane wave ie ⋅k r is an envelope function. The parameter k is then called the pseudo-momentum. The solution of the Schrödinger equation [1.8] for each k vector to extract the band structure ( )E k is still a difficult problem of crucial importance in modern microelectronic devices, in particular to include the non-trivial effects of mechanical strain [RID 06]. Different numerical techniques have been developed, based on the empirical pseudo-potential [CHE 76], the linear combination of atomic orbitals (or tight-binding approximation) [SLA 54], [NIQ 00], [JAN 07], the k.p expansion of the Hamiltonian [CAR 66], [RIC 04], [RID 06] or even first-principle ab initio calculation [NIE 83], [NIE 85a], [NIE 85b], [BLA 94]. Figure 1.2 shows a typical example of the bulk-silicon band structure with Δ valleys for the quasi-electrons and heavy-hole (hh), light-hole (lh) and split-off (so) bands for the quasi-holes.

L Γ X K,U Γ[111] [100] [110]L Γ X K,U Γ[111] [100] [110]

4

2

0

-2

-4

Ener

gy(e

V)

4

2

0

-2

-4

Ener

gy(e

V)

so

hh,lh

Δ

so

hhlh

Figure 1.2. Band structure of unstrained bulk silicon along axes of strong symmetry, obtained from 30-band k.p calculation [RID 06]. Figure from [HUE 08]

1.1.1.4. Dynamics of envelope functions

Once the band structure is known, the quasi-electron states can be built by superposition of Bloch states [1.9], characterized by a function ( )g k , i.e.

( ) ( ) ( )id e u gψ ⋅= ∫ k rkr k r k [1.10]


A common approximation then consists of forgetting the atomic details of the wave function and replacing the “actual” wave function [1.10] by an envelope function defined by [KIT 87]

( ) ( )id e gψ ⋅′ = ∫ k rr k k [1.11]

where the function ( )g k is the same as in [1.10]. For the envelope function the pseudo-momentum k plays the same role as the momentum operator for free

particles. One can thus write ˆˆ i= ∇k . The Hamiltonian finally reduces to the band structure ( )E k and the Schrödinger equation is written as

( ) ( ) ( )Ê it

ψψ

′∂′ =

∂r

k r [1.12]

A most common and useful approximation consists of fitting the relation ( )E k

near the valley bottoms with a parabolic function 2 2 *2mk where *m is called the effective mass. Under this approximation, the Schrödinger equation of quasi-electrons is simply written

( ) ( )2 2

*

ˆ

2i

tm

ψψ

⎡ ⎤ ′∂′ =⎢ ⎥

∂⎢ ⎥⎣ ⎦

rk r [1.13]

The dynamics of the envelope function is then similar to that of a free electron of mass *m , with the pseudo-momentum operator in place of the momentum operator.

In device physics, all phenomena likely to modify the coherent transport described by equations [1.12] or [1.13] are usually treated as perturbations of these equations. In the transport problems addressed in this book, these phenomena are essentially electron-phonon scattering, electron-electron scattering, electron-ionized impurity scattering, surface roughness scattering, and the presence of hetero-junctions and of Ohmic contacts. But one thing that is essential to highlight is that this perturbative approach is appropriate only if these phenomena do not play any role at the atomistic level, i.e. in other words, if they do not distort the band structure of the material. In modern nanodevices, this may be questionable, for instance, in carbon nanostructures like carbon nanotubes and graphene sheets [ROC 05] and may also be in some other devices of the future.


We can now consider the treatment of two examples of important scattering mechanisms in electron devices: quasi-electron/phonon scattering and quasi-electron/quasi-electron scattering. Both are essential for the understanding of devices, and both raise pretty complex questions.

1.1.2. Quasi-electron/phonon scattering

We now have to consider the full system of electrons and phonons to evaluate the impact of phonons on electron behavior. We first assume a longitudinal non-polar phonon of wave vector q and angular frequency ωq . The phonon-induced

strain of the elementary cell can be expressed through the divergence of the ion displacement (as seen in [1.6])

( )1 2

†ˆ ˆ ˆ ˆ ˆS .2

i ii q a e a eM Nω

⋅ − ⋅⎛ ⎞⎜ ⎟= ∇ = −⎜ ⎟⎝ ⎠

∑ q r q rq q

q qU [1.14]

In the case of a non-polar phonon the effect of this deformation on the ion/electron interaction potential is usually considered to be proportional to this strain and modeled by introducing the deformation potential D, leading to

( )1/2

†ˆˆ ˆ ˆH S2

i ie ph D i D q a e a e

M Nω⋅ − ⋅

−⎛ ⎞

= = −⎜ ⎟⎜ ⎟⎝ ⎠

∑ q r q rq q

qq [1.15a]

which is the Hamiltonian of the electron/phonon coupling [RID 99].

In the case of a polar phonon, the quasi-electron/phonon coupling is essentially due to the interaction with the dipole moment induced by the phonon. The coupling Hamiltonian has a form similar to that of [1.15a] with a very different prefactor which contains the dielectric response of the solid [RID 99]

( )1/22

†2 2 0 00

1 1ˆ ˆ ˆH2

i ie ph

r

eqi a e a eVq q

ωε ε ε

⋅ − ⋅−

∞

⎡ ⎤⎛ ⎞⎢ ⎥= − −⎜ ⎟⎢ ⎥+ ⎝ ⎠⎣ ⎦

∑ q q r q rq q

q [1.15b]

where rε and ε∞ are, respectively, the relative low frequency and high frequency

semiconductor dielectric constants, 0q is the screening wave vector, and 0V is the volume of a unit cell.


For the two coupling types we have considered, and for the others considered for example in [RID 99], the Hamiltonian takes the general form

( ) ( )†ˆ ˆ ˆH i ie ph i F a e a e⋅ − ⋅− = −∑ q r q r

q qq

q [1.16]

where ( )F q is a real function. This is the expression that will be used in the following sections of the book. The two terms of this Hamiltonian can be interpreted easily. The first one represents a phonon annihilation, i.e. a phonon absorption by an electron. The second one corresponds to a phonon creation, i.e. a phonon emission. The exponential factors correspond to the momentum exchange between the lattice and the electrons, as will be clear later.

1.1.3. Quasi-electron/quasi-electron and quasi-electron/impurity scattering

In any electron device, the interaction between quasi-electrons plays an important role. To model, it is necessary to know the full wave function of quasi-electrons. This is still an extremely complex many-body problem. The simplest approach consists of considering a Hartree wave function. This is simply a product of one-electron wave functions associated with the individual electrons:

( ) ( ) ( ) ( ) ( )1 2 1 1 2 2, , ,e e N N Nψ ψ ψ ψ ψ= =r r r r r r r… … [1.17]

This wave function is not anti-symmetric to electron switching, which is not a priori physically acceptable for a fermion gas. However, this approach can be used if the quasi-electron gas is not degenerate, so that the particles can be considered as discernible (which is justified in Appendix C). In that case, exchange–correlation effects can be neglected and there is no need for anti-symmetry of the wave function [COH 06], [ASH 76].

A global Hamiltonian is then considered for the envelope function including the interaction between quasi-electrons and between quasi-electrons and ionized impurities:

( )2 2

0, ,

1ˆ ˆH2 ˆ ˆ4 ˆ4

e ii j i li i j i l

e eEπε πε

= + +− −

∑ ∑ ∑kr r r R

[1.18]

where the indices i and j label electrons and subscript l here labels ionized impurities (and not lattice ions). Theoretically, we should not use a Hartree wave function anymore to solve [1.18] since it is only justified for independent particles. However


it is still appropriate if we consider the quasi-electrons to interact weakly enough (in which case this interaction is acting as a perturbation). Then, if the doping density is assumed to form a continuum of charge (we’ll go beyond this approximation later) it can be shown (see e.g. [DAT 05]) that the problem becomes equivalent to N one-body Schrödinger equations:

( ) ( )ˆ ˆ ii i i iE eU i

tψψ ∂⎡ ⎤− =⎢ ⎥⎣ ⎦ ∂

k r [1.19]

and N Poisson equations

( ) ( ) ( ) 20i D j

j iU e N eε ψ

≠∇ ∇ + − =⎡ ⎤⎣ ⎦ ∑r r r [1.20]

where ( )DN r is the ionized impurity density. If the electron density is high enough one can omit to exclude the contribution of the electron i in the sum of [1.20] and the N Poisson equations can be transformed into a single equation including the total electron density

( ) ( ) ( ) 0DU e N e nε∇ ∇ + − =⎡ ⎤⎣ ⎦r r r [1.21]

The problem turns out to be a one-body problem coupled to a single Poisson equation. It should be noted that the latter approximation of the single Poisson equation, which artificially includes a self-interaction of the electron, has no effect in conventional devices containing a high number of electrons. However, in strongly confined nanodevices containing few electrons, this approximation does not allow the observation of Coulomb blockade effects [DAT 05] and may thus sometimes be unacceptable.

This mean-field approximation of the interaction between quasi-electrons is practically universally used in device physics, though it is theoretically valid only for a non-degenerate gas. If the electron density is high, the anti-symmetry of the wave functions gives rise to exchange and correlation effects which make the equation [1.20] non valid, and may induce additional interactions between electrons.

We now have all the ingredients to come to the study of transport. In all that follows, a simplified terminology commonly used in the literature of device physics will be used systematically: electrons for quasi-electrons, holes for quasi-holes, wave function for envelope function, momentum for quasi-momentum, and wave vector for quasi-wave vector. It should not be forgotten, however, that this simplified terminology hides a lot of formalism and approximation.


1.2. The semi-classical approach of transport

1.2.1. The Boltzmann transport equation

The difficulty of transport physics lies in the solution of the Schrödinger equation [1.12] for envelope functions including the effects of scattering, which play a fundamental role. Here we present the simplest approach: the semi-classical approach. It has been successfully used for a considerable number of device problems in the past.

This approach is based on the idea that electrons are in well-localized wave packets of space extension σ, formed from Bloch waves. If we consider the well-known Gaussian type wave packet, the envelope function can be written as

( )( )2

02

02 iN e eσψ−

−⋅=

r rk rr and ( )

( )22

2 iN e eσ

ψ− − ⋅′=

0

0

k-kr kk [1.22]

in the real and reciprocal spaces, respectively. N and N’ are here normalization constants. The dynamics of such wave functions is well known [KIT 96]. The real space velocity of the wave packet center is the group velocity, i.e.

0d Edt

∂= =∂g

r vk

[1.23]

The potential energy of the wave packet is

( ) ( ) ( )20PE d V Vψ= ≈∫ r r r r [1.24]

The approximation in [1.22] is correct if the potential ( ) ( )V eU= −r r varies slowly on the space extension σ of the wave function. It follows from the law of

energy conservation that PdE dE= − , i.e. 0 0CE Vd d

∂ −∂⋅ = ⋅∂ ∂

k rk r

, thus

0dV

dt⋅ = −∇ ⋅

kv v . In the absence of a magnetic field, the center of the wave

packet then propagates in k-space with a velocity

0dV e U

dt= −∇ = ∇

k [1.25]


The velocity equations [1.23] and [1.25] are similar to Newton laws for a point particle, which leads to the terminology “semi-classical” transport. From this point of view, electrons are considered as point particles moving with the velocity of their wave packet center. It is, however, well known that wave packets tend to spread during their propagation (i.e. their spatial width increases) [COH 06], so that [1.23] and [1.25] do not give a full picture of wave packet dynamics. We now consider that this warping is negligible and that wave packet extension is unchanged. These two assumptions are not obvious and will be discussed later in the context of the theory of decoherence. The fact that electron wave packets are spontaneously Gaussian will also be discussed.

The transport phenomena result from an average over the behavior of a large number of electrons. The model of semi-classical transport is thus based on the definition of the Boltzmann distribution function ( ), ,bf tr k which corresponds to the probability density to find a wave packet centered on r and k at time t (or equivalently to find an electron at positions r and k of the phase space at time t). The dynamic equation of this distribution function without collision is readily obtained from the probability conservation

( ) ( ) ( ) ( ), , , ,0 , , , ,b b

b bd f t f t d df t f t

d t t d t d t∂

= = + ∇ ⋅ + ∇ ⋅∂ r k

r k r k r kr k r k

which leads to the Boltzmann transport equation

( ) ( ) ( ) ( ) ( ), , 1, , , , , ,bb b b

f tf t V f t C f t

t∂

= − ⋅∇ + ∇ ⋅∇ +∂ r kr k

v k r k r k r k[1.26]

where a generic term ( ), ,bC f tr k has been added to include the collision or scattering effects on the Boltzmann distribution function. C is the collision operator. Within the semi-classical approach, this collision term is commonly evaluated using the Fermi golden rule, i.e. the first order perturbation theory. Collisions are considered as instantaneous scattering events which modify the wave vector. Between two consecutive scattering events, electrons experience ballistic free flights whose duration is determined from the rate of scattering ( ),s ′k k for an electron initially in state k to state ′k . The collision term for a non-degenerate semiconductor is then written as

( ) ( ) ( ) ( ) ( ), , , , , , , ,b i b i bi

C f t d s f t s f t′ ′ ′ ′= −⎡ ⎤⎣ ⎦∑∫r k k k k r k k k r k , [1.27]


where the subscript i labels the type of scattering process. For an interaction Hamiltonian int,iH the rate of scattering according to the Fermi golden rule is

( ) ( )2int,

2,i i f is H E Eπ ψ ψ δ′′ = −k kk k . [1.28]

The full derivation of most scattering rates can be found in many textbooks, e.g. in [JAC 89] and [LUN 00]. The two main approximations of the semi-classical approach (which will be discussed further later) should be remembered here:

– the potential is linear, quadratic or slowly variable on a distance σ,

– the electrons are in well localized wave packets of Gaussian typical form.

The semi-classical approximation (either using accurate or simplified treatments) has, until recently, been the foundation of most device simulations. It has shown tremendous success in explaining and modeling most phenomena seen in electron devices. The solution of the Boltzmann equation (using a direct solution or, more often, a statistical Monte Carlo approach) has been used for studies requiring detailed understanding of the transport. A modern application is the study of ballistic effects in ultra-small metal/oxide/semiconductors and Metal-Oxide-Semiconductor Field-Effect transistors (MOSFET) [SAI 04], [PAL 05] or High Electron Mobility transistors (HEMTs)[LUS 05a], [MIL 08]. Approximations of the Boltzmann equation (hydrodynamic or drift-diffusion equations) are now industry standards for device design and are also used for many fundamental studies. Modern examples are the study of variability [ROY 06] of current generation MOSFETs or of the impact of thermal effects in ultra-scaled and power transistors [POP 06].

1.2.2. Quantum corrections to the Boltzmann equation

However, the semi-classical approach is unable to explain all experimental results observed in modern down-scaled microelectronics devices. A well-known example is the under-estimation of the threshold voltage in Bulk MOSFET with a highly doped channel or in ultra-thin body transistors [DOR 91], [WU 96], [CHI 97], [MAJ 00]. In these situations, electrons occupy the first bound states of the quantum well which forms the nanometer inversion layer. Their energy is thus higher than estimated by semi-classical calculation, neglecting quantization effects. At a given gate bias, the electron sheet density is thus reduced and the peak density is shifted by typically 1 nm from the oxide interface [CAS 01], as illustrated in Figure 1.3, which increases the effective oxide thickness and reduces the transconductance. These effects contribute to enhancing the threshold voltage [DOR 91].


1016

1017

1018

1019

1020

-4 -3 -2 -1 0 1 2 3 4 5 6

Elec

tron

Den

sity

(cm

-3)

x (nm)

Poly-SiGate ox

ide

Figure 1.3. Electron density profile in conventional bulk MOS structure for VGS = 0.3 V using classical (dashed line) and wave-mechanical calculation (solid line). The gate oxide is 1 nm

thick and the doping levels are 5×1019 cm-3 and 1018 cm-3 in the N-Poly-Si gate and the Si substrate, respectively. Figure from [CAS 01]

To improve the semi-classical model the first idea is to correct the description and the dynamics of the wave packet by keeping the exact formulation [1.24] of its energy [FER 00], [JAU 06a]. If the wave packet is assumed to be Gaussian, as suggested in section 1.2.1, a Gaussian effective potential (GEP) can be used as the potential energy instead of the electrostatic potential. This effective potential produces

( ) ( ) ( )22

1expGEP

effx x

V x dx V xπ σ σ

⎛ ⎞′−⎜ ⎟′ ′= −⎜ ⎟⎝ ⎠

∫ [1.29]

This approach, however, raises an important question: what is the natural size σ of a wave packet1? This question is deeply non-obvious, and will be addressed again several times in this book.

In thermal equilibrium conditions, it is relevant to consider the energy bk T for electrons, and a simple analysis based on dimensional arguments leads to the following suggestion for this space extension

1 σ is defined in this book as the standard deviation of the wave function. It differs from that

defined in [FER 00a] or [FER 02] by a factor of 2 .


*bm k T

σ α= [1.30]

The problem is then to define the dimensionless proportionality coefficient α in [1.30] properly. Ferry tackled this question [FER 02] and his conclusion is very surprising: the calculation of σ gives very different results depending on the environment of electrons and the method used:

– 1 6α = using the formalism of Feynman path integrals for both cases of a slowly varying potential and a confined electron;

– 1 2α = using a Wigner-like derivation for a slowly varying potential;

– 3 4α π= using a Wannier functions approach for a free electron.

For a thermal electron, not too confined and not too free, one can thus assume the value of α to range between 0.4 and 1.5. The analysis is still to be done for a hot electron. In practice, in effective potential models the extension σ is used as an adjustable parameter to be calibrated according to more accurate calculations. With such adjusted effective potential, simulations can predict the correct threshold voltage of advanced MOSFETs [FER 00b].

However, the validity of Gaussian effective models is limited. In particular, it has been shown that the GEP cannot correctly reproduce at the same time the total inversion charge and the carrier density profile in the MOSFET channel because of overestimation of the repulsive effect at the oxide interface [JAU 06b]. It has also been shown that the GEP is not able to satisfactorily describe the tunneling effect [FER 02]. Actually, the limitation of this approach is related to two important approximations: the extension σ is considered a universal parameter for all electrons, and all wave packets are assumed to be Gaussian.

Some improvements to this approach make it possible to obtain better results. To describe the effect of interfaces on the wave packet properly, a non-symmetric Pearson IV distribution function can be used in place of the Gaussian function [JAU 07], [JAU 08]. As shown in Figure 1.4, the resulting Pearson Effective Potential (PEP) correction provides a density profile in MOS structures in much better agreement with Schrödinger-Poisson simulation than the conventional GEP correction. Other correction techniques consist of considering either a quantum force formulation based on the Wigner formalism [TSU 01], or a direct solution of Schrödinger’s equation [WIN 03].


However, if these correction techniques are able to mimic some first order quantum effects on the electrostatics, they are still based on a semi-classical transport formulation and cannot properly describe advanced quantum transport effects, such as resonant tunneling.

1023

1024

1025

1026

0 2 4 6 8 10

Schrodinger-PoissonGEP correctionPEP correction

Elec

tron

den

sity

(m-3

)

Distance x (nm)

Eeff

= 106 V.cm-1

Eeff

= 105 V.cm-1

Figure 1.4. Electron density in a double-gate n-MOSFET with Si-body thickness TSi = 10 nm and TOX = 0.5 nm using GEP and PEP corrections for two values of the effective field Eeff;

comparison with Schrödinger-Poisson solution. Figure from [JAU 07]

1.3. The quantum treatment of envelope functions

A first idea for performing a full quantum treatment of the envelope functions may be to solve their Schrödinger equation [1.19], which can be rewritten

( ) ( )ˆ Ê V itψψ ∂⎡ ⎤+ =⎣ ⎦ ∂

k r . [1.31]

However, a device is an open system which cannot be fully described by the Schrödinger equation [1.31]. The contacts and the interactions with the environment (phonons, doping impurities, other electrons, etc.) influence the electron states in a way that cannot really be described by [1.31]. Alternatively, different formalisms have been developed to model an open device. The key idea consists of using a formalism able to include the statistical physics of the device, and thus to make up an equivalent to the Boltzmann transport equation for the semi-classical approach, for quantum transport. We will focus here on three quantum formalisms able to


describe the statistical physics of a system. They are based on the density matrix, the Wigner function and Green’s functions, respectively. They are briefly presented in this chapter to capture the essential physics they contain and the relationship between them.

1.3.1. The density matrix formalism

1.3.1.1. Definition and properties of the density matrix

The most natural approach to rigorously modeling the statistics of a quantum system is the density matrix formalism. The density matrix (DM) is defined as follows: consider a basis iψ of the electron quantum states with an occupation probability ip related to the statistical physics of the system. The density matrix operator [COH 06] is defined by

ˆ i i ii

pρ ψ ψ=∑ [1.32]

or, in real-space representation,

( ) ( ) ( ) ( ) ( )*ˆ, , i i ii

t t p tρ ρ ψ ψ′ ′ ′= =∑r r r r r r . [1.33]

The occupation probability density ( )n r is obtained in a straightforward way from the diagonal elements of the DM

( ) ( ) ( ) ( ) ( )*, i i ii

n p tρ ψ ψ= =∑r r r r r . [1.34]

The off-diagonal elements are called coherences and characterize the real-space delocalization of electrons, as explained and illustrated below.

1.3.1.2. Example

To properly understand the meaning of this formalism, let us consider the following gedanken (“thought”) experiment. A Gaussian free wave packet, as defined in [1.22], is sent ballistically onto a tunneling barrier. A part of this packet is reflected and the other part is transmitted through the barrier but, without any other perturbative mechanism, the packet remains fully coherent: the electron is delocalized over the two sides of the barrier. A measurement is the only way to localize it on one of the two sides. Figure 1.5 illustrates the forming of this delocalized state and Figure 1.6 shows the DM corresponding to the final state of


Figure 1.5. It is compared with the DM for an electron having the same probability to be on the left side and on the right side of the barrier but not delocalized over the two sides. The probability density is the same in both cases, since the DM diagonal elements are the same. However, the two density matrices are very different: in the coherent case (Figure 1.6a) off-diagonal elements are not null and characterize the electron delocalization.

0

0.05

0.1

0 50 100 150 200

p(x)

(nm

-1)

x (nm)

t = 0 fs

0

0.05

0.1

0 50 100 150 200

p(x)

(nm

-1)

x (nm)

t = 0 fs

0

0.05

0.1

0 50 100 150 200

p(x)

(nm

-1)

x (nm)

t = 60 fs

0

0.05

0.1

0 50 100 150 200

p(x)

(nm

-1)

x (nm)

t = 60 fs

0

0.05

0.1

0 50 100 150 200

p(x)

(nm

-1)

x (nm)

t = 120 fs

0

0.05

0.1

0 50 100 150 200

p(x)

(nm

-1)

x (nm)

t = 120 fs

Figure 1.5. Time evolution of the occupation probability density for a Gaussian wave packet interacting with a tunneling barrier.

Result obtained from direct solution of the Schrödinger equation

This is the strength of the DM. The diagonal terms contain the uncertainties of quantum origin related to the electron delocalization and the classical uncertainties


as well (ignorance of the initial state, interactions with the environment). In contrast, the off-diagonal terms characterize the quantum uncertainties only. This formalism is thus able to describe the quantum physics of a system which is subject to statistics.

(b)

Figure 1.6. (a) Modulus of the density matrix elements for a wave packet after interaction with a tunneling barrier located at x = 100 nm, in Figure 1.5 at t = 120 fs; (b) density matrix elements for an electron having the same probability to be on both sides of the barrier but not

delocalized over the two sides. The quantities plotted are expressed in nm-1

1.3.1.3. Dynamic equation

To study actual problems of transport, a dynamic equation for the density matrix is required. First, consider the simple case of a closed system described by the Schrödinger equation [1.31]. By differentiating the density operator [1.32] with respect to time, we obtain

ˆ

1 ˆ ˆH H ,

i ii i i

i

i i i i ii

pt t t

pi

ψ ψρ ψ ψ

ψ ψ ψ ψ

∂ ∂⎡ ⎤∂ = +⎢ ⎥∂ ∂ ∂⎣ ⎦

⎡ ⎤= −⎣ ⎦

∑

∑ [1.35]

(a)


which leads to the Liouville equation

ˆ ˆ ˆH,did tρ ρ⎡ ⎤= ⎣ ⎦ [1.36]

For the Hamiltonian of equation [1.31] in the effective mass approximation m*, equation [1.36] becomes

( ) ( ) ( ) ( )( ) ( )2 2 2

1 21 2 1 2 1 2* 2 2

1 2

,, ,

2i V V

t m

ρρ ρ

⎛ ⎞∂ ∂ ∂= − − + −⎜ ⎟⎜ ⎟∂ ∂ ∂⎝ ⎠

r rr r r r r r

r r [1.37]

This looks promising. However, can we approach advanced topics with this formalism? When expressing the DM in the reciprocal space, this formalism may model the electron-phonon interaction accurately, including collisional broadening and retardation and an intra-collisional field effect [BRU 89], [JAC 92], [ROS 92a]. However, it does not allow the study of real space-dependent problems. DM-based device simulation is possible using the Pauli master equation that takes into account only the diagonal elements of the DM [FIS 98], [FIS 99]. However, in spite of recent improvements [GEB 04a], [GEB 04b], the modeling of terminal contacts in an open system is difficult within this formulation which is thought to be valid only for devices smaller than the electron dephasing length [FIS 99].

Nowadays, in most research on nanodevices, the density matrix tends to be replaced by the Wigner quasi-distribution function or by the Green’s functions whose formalisms make it possible to include the contacts and interactions in a more simple and rigorous way than in the DM approach.

1.3.2. The Wigner function formalism

1.3.2.1. Definition and basic properties of the Wigner function

For a statistical ensemble of particles described by a density operator ρ the Wigner formalism of quantum transport is based on the Wigner function fw originally introduced by Wigner in 1932 [WIG 32]. It is defined in the phase-space ( ),r k as a differential Fourier transform of the density matrix

( ) ( )ˆ, tρ ρ′ ′=r r r r , also called the Weyl-Wigner transform, [JAC 01], i.e.

( )( )

( )1 2, 22

iw df d e ρ

π′− .′ ′ ′, = + −∫ k rr k r r r r r [1.38]


where d is the real-space dimension of the transport problem. The fact that this quantum function depends on the same phase-space coordinates as the Boltzmann distribution function ( )bf ,r k of semi-classical transport is remarkable. It is all the

more surprising that the space probability density of electrons ( )n r can be obtained the same way as in the semi-classical case, i.e.

( )( )

( )

( )

31 ˆ

2 22

ˆ( ) .

iwd f d t d e

t n

ρπ

ρ

′− .′ ′′, = + −

= =

∫ ∫ ∫ k rr rk r k r r r k

r r r

[1.39]

Similarly, it is shown in Appendix A that the average value of an operator A depending only on operator r , such as we may define ( )ˆ Â V= r , is

( ) ( )A V wd d f= ,∫ ∫r k r r k [1.40a]

Similarly, the average value of an operator A depending only on operator k , so that we may define ( )ˆ Â= W k is

( ) ( )A W .wd d f= ,∫ ∫r k k r k [1.40b]

An average value for these two kinds of observables – which include many observables of interest in electron devices – can thus be computed the same way as in the semi-classical case. In the more general case, we show in Appendix A that the average value of A can be written as

( ) ( ), ,wd d f= ∫ ∫r k r k r kwA A [1.41]

where ( ) ( ) ( ), exp 2, 2d i A′ ′ ′ ′= − ⋅ − +∫r k r k r r r r rwA is the Weyl-Wigner

transform of the operator A .

The Wigner function thus shares many properties with the Boltzmann distribution function and in many cases it may be used as a distribution function. But shockingly it apparently violates Heisenberg’s uncertainty principle, which formally excludes the existence of a distribution function in quantum mechanics since it does not allow precise location within phase-space regions smaller than [COH 06]. How can this paradox be explained?


0 50 100 150 200x (nm)

1

0.5

0

- 0.5

- 1

K (n

m-1

)

0 50 100 150x (nm)

1

0.5

0

- 0.5

- 1

K (n

m-1

)0.2

0.1

0

- 0.1

- 0.2

- 0.3

0.15

0.1

0.05

0

(a)

(b)

0 50 100 150 200x (nm)

0 50 100 150 200x (nm)

1

0.5

0

- 0.5

- 1

K (n

m-1

)1

0.5

0

- 0.5

- 1

K (n

m-1

)

0 50 100 150x (nm)

0 50 100 150x (nm)

1

0.5

0

- 0.5

- 1

K (n

m-1

)

1

0.5

0

- 0.5

- 1

K (n

m-1

)0.2

0.1

0

- 0.1

- 0.2

- 0.3

0.2

0.1

0

- 0.1

- 0.2

- 0.3

0.15

0.1

0.05

0

0.15

0.1

0.05

0

(a)

(b)

Figure 1.7. Wigner’s functions corresponding to the density matrices of Figure 1.6; (a) delocalized electron wave packet; and (b) electron

localized on one of the sides of the tunneling barrier

In fact, it is easy to see that the Wigner function is real but not always a positive number. It can be illustrated by the previous example considered for the density matrix sub-section. Figure 1.7 shows the distribution functions associated with the density matrices of Figure 1.6. In the first case (coherent transport of delocalized electrons), the Wigner function exhibits oscillations centered on 0k = with positive and negative values which are the signature of quantum space coherence and interferences. It illustrates the fact that the Wigner function cannot be interpreted as a distribution function. In the second case, corresponding to a classical situation, the Wigner function does not take any negative values and can be interpreted as a distribution function.


Actually, regions of negative value are proved to be limited: they cannot extend to regions of the phase-space larger than a few , and disappear in the classical limit. They are the price to pay for defining a function at an exact location of the phase space and are finally less paradoxical than they appear to be at first glance. This intriguing feature is discussed in detail in [TAT 83].

1.3.2.2. Example of plane waves

It is interesting to consider the case of plane waves. The Wigner function associated with the typical plane wave ( ) ( )expx iK xψ = is

( ) ( )

( ) ( )

1, exp *2 2 21 exp exp

2

wx xf x k d x ik x x x

d x ik x iK x

ψ ψπ

π

′ ′⎛ ⎞ ⎛ ⎞′ ′= − + −⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

′ ′ ′= −

∫

∫

which leads to the δ function

( ) ( ), .wf x k K kδ= − [1.42]

This result is not surprising: the plane wave is made up of a spatially uniform component of wave vector K.

Now, consider the case of an electron in a state resulting from the superposition of the two plane waves ( ) ( )1 expx iK xψ = and ( ) ( )2 expx iK xψ = − , i.e.,

( ) ( ) ( )1 212

x x xψ ψ ψ⎡ ⎤= +⎣ ⎦ . The associated Wigner function is

( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( )( )

1, exp *2 2 21 exp exp exp exp

2

exp exp 2 exp 2

wx xf x k d x ik x x x

d x ik x iK x d x ik x iK x

d x ik x iK x iK x

ψ ψπ

π

′ ′⎛ ⎞ ⎛ ⎞′ ′= − + −⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

⎡ ′ ′ ′ ′ ′ ′= − + − −⎢⎣

⎤′ ′+ − + − ⎥⎦

∫

∫ ∫

∫

and, finally,

( ) ( ) ( ) ( ) ( )1, 2 cos 2 .2wf x k K k K k k K xδ δ δ⎡ ⎤= − + + +⎣ ⎦ [1.43]


Here, the coherence between the two components of the wave function manifests itself in the oscillating part ( ) ( )cos 2k K xδ around 0k = which can take positive and negative values as a function of x. We will see the consequences of this result in the chapter devoted to device analysis using the Wigner formalism.

If there is a 50-50 chance of finding an electron in one of the two plane waves propagating in opposite directions but not in both states at the same time, its Wigner function is just the sum of the Wigner functions associated with the two independent plane waves, weighted by the probability 1/2, i.e.

( ) ( ) ( )1,2wf x k K k K kδ δ⎡ ⎤= − + +⎣ ⎦ [1.44]

In this case of non-coherence between the two propagating states there is no longer oscillation around 0k = .

The interpretation of the Wigner function behavior is discussed several times throughout this book.

1.3.2.3. Dynamic equation: the Wigner transport equation

To derive the dynamic equation of the Wigner function, we first come back to a closed system without any scattering. The Liouville equation [1.37] is re-written in the so-called center-of masse coordinates ( )1 2 2= +r r r and 1 2′ = −r r r , i.e.,

2

*

,2 2 ,

2 2

,2 2 2 2

it m

V V

ρρ

ρ

′ ′⎛ ⎞∂ + −⎜ ⎟ ′ ′∂ ∂⎛ ⎞ ⎛ ⎞⎝ ⎠ = − + −⎜ ⎟ ⎜ ⎟′∂ ∂ ∂⎝ ⎠ ⎝ ⎠′ ′ ′ ′⎛ ⎞⎛ ⎞ ⎛ ⎞ ⎛ ⎞+ + − − + −⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎝ ⎠

r rr rr rr r

r r

r r r rr r r r

[1.45]

The differential Fourier transform of [1.45] gives, for a 3D problem, ( 3d = )

( )( )

( )

2

* 3

3

1 ,2 22

1 ,2 2 2 22

w i

i

fd e

t m i

d e V Vi

ρπ

ρπ

′− .

′− .

∂ , ′ ′⎛ ⎞∂ ∂ ⎛ ⎞′= − + −⎜ ⎟ ⎜ ⎟′∂ ∂ ∂ ⎝ ⎠⎝ ⎠

′ ′ ′ ′⎛ ⎞⎛ ⎞ ⎛ ⎞ ⎛ ⎞′+ + − − + −⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎝ ⎠

∫

∫

k r

k r

r k r rr r rr r

r r r rr r r r r [1.46]

We consider the two right-hand side terms of [1.46] separately. The first one is integrated by parts, which leads to


( )

( )( )

2

* 3

2

* 3

*

1 ,2 22

1 ,2 22

, ,

i

i

w

d em i

d em i

f t

m

ρπ

ρπ

′− ⋅

′− ⋅

′ ′⎛ ⎞∂ ∂ ⎛ ⎞′− + −⎜ ⎟ ⎜ ⎟′∂ ∂ ⎝ ⎠⎝ ⎠

′ ′∂ ∂⎛ ⎞ ⎛ ⎞′= − + −⎜ ⎟ ⎜ ⎟′∂ ∂⎝ ⎠ ⎝ ⎠

∂= − ⋅

∂

∫

∫

k r

k r

r rr r rr r

r rr r rr r

r kkr

[1.47]

By introducing the inverse Fourier transform of the Wigner function [1.38] in the second term of [1.46], we obtain

( )

( )( )

3

3

1 ,2 2 2 22

1 ,2 22

i

i iw

d e V Vi

d d e V V fi

ρπ

π

′− ⋅

′ ′ ′− ⋅ + ⋅

′ ′ ′ ′⎛ ⎞⎛ ⎞ ⎛ ⎞ ⎛ ⎞′ + − − + −⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎝ ⎠

′ ′⎛ ⎞⎛ ⎞ ⎛ ⎞′ ′ ′= + − −⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠⎝ ⎠

∫

∫ ∫

k r

k r k r

r r r rr r r r r

r rr k r r r k [1.48]

The Wigner dynamic equation is finally

( )ww w

f f Q ft m∗

∂ + ⋅∇ = ,∂ rk r k [1.49]

which includes the quantum evolution term for the potential V

( ) ( ) ( ) ( )w V w w wQ f Q f d V f′ ′ ′, = , = , − ,∫r k r k k r k k r k [1.50]

and the Wigner potential wV defined as

( )( )

12 22

iw dV d e V V

i π′− . ′ ′⎛ ⎞⎛ ⎞ ⎛ ⎞′, = + − −⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠⎝ ⎠∫ k r r rr k r r r . [1.51]

In the particular case of linear or quadratic potential V the Wigner potential becomes

( )( )

( )1 1 '2

iw dV V d e V

iδ

π′− .′ ′, = ∇ = − ∇∫ k rr k r r k [1.52]


The quantum evolution term is then

( ) ( )1w wQ f V f, = ∇ ∇ ,kr k r k [1.53]

which is nothing but the drift term of the Boltzmann equation!

In the case of a potential varying slowly with respect to typical values of 1 k , equations [1.52] and [1.53] are still a good approximation. Near thermal equilibrium

1 k is typically close to *2 bm k T which is also close to the “natural” delocalization length σ as discussed in section 1.2.2. Hence the Wigner equation tends to the Boltzmann equation in the case for which the Boltzmann equation was established, i.e. for linear, quadratic or slowly varying potential. The Boltzmann equation may thus be seen as the semi-classical limit of the Wigner equation.

One then immediately understands the potential advantage of the Wigner formalism to study the transport in nanodevices operating between the ballistic and the diffusive limits, and in particular to investigate the transition between the well-known semi-classical regime and the less understood quantum transport regime.

Finally, it should be noted that if the potential can be separated in rapidly and slowly varying parts, they can both be treated separately [NED 04], i.e. for

( ) ( ) ( )slow rapidV V V= +r r r [1.54]

The Wigner equation [1.49] may be rewritten as

1rapid

ww slow w V w w

f f V f Q f C ft m∗

∂ + ⋅∇ − ∇ ⋅∇ = +∂ r kk [1.55]

The Wigner transport equation (WTE) has been used in different domains of quantum physics, e.g. in atomic physics [LUT 97], in quantum optics [BER 02], [DEL 08], in time-varying signal processing [HLA 92], or in particle physics and cosmology [CAL 88]. The history and state of the art of its utilization in electronic nanodevices is also pretty rich and will be presented and discussed in the next chapter, that focuses on this formalism.


1.3.3. The Green’s functions formalism

The non-equilibrium Green’s functions have been built for many-body problems and make up a very rich formalism able to include electron correlations [MAH 90], [FER 97]. However, to model nanodevices they are generally used in a simplified one-body approach [DAT 95], [DAT 00], [LAK 92], [AKK 04], [DO 07a] which is presented here because they have become a fundamental element of the research on quantum device modeling.

Let us consider an isolated one-body system described by the basis of states iψ which are eigen-vectors of the Hamiltonian of eigen-energy iE . Using the same notations as in section 1.3.1.1, the “lesser” Green’s function of this system can be defined as

( ) ( ) ( )*, , , , ,2j j j

j

t tG t t i p t tψ ψ< ′+⎛ ⎞′ ′ ′ ′= ⎜ ⎟⎝ ⎠

∑r r r r [1.56]

Time/energy units can be introduced in the lesser Green’s function using the following transform

( ) ( )1, , , , , ,Ei

G t E d e G t tτ

τ τ τ−< <′ ′= − +∫r r r r [1.57]

Substituting [1.56] into [1.57] leads to

( ) ( ) ( ) ( ) ( )*, , , 2 j j j jj

G t E i p t E Eπ ψ ψ δ< ′ ′= −∑r r r r [1.58]

which is the form used for the simulation of devices. The relationship between this function and the density matrix is very clear, i.e.,

( ) ( ) ( ), , , , , , , ,2it iG t t dE G t Eρπ

< <′ ′ ′= − = − ∫r r r r r r [1.59]

The density matrix is thus obtained by integrating G< over energy. Similarly, the Wigner function can be easily obtained2 from G< as

2 There are several possible normalizations of the density matrix and of the Wigner function which lead to different expressions connecting density matrix, Wigner function and Green's functions. Remember that the choice here is ( ), 1d ρ =∫ r r r and ( ), 1wd d f =∫ ∫r k r k .


( )( )

( )

3

4

1, , ,2 22

, , ,2 22

iw

i

f t d e t

i d E d e G t E

ρπ

π

′− .

′− . <

′ ′⎛ ⎞′, = + −⎜ ⎟⎝ ⎠

′ ′− ⎛ ⎞′= + −⎜ ⎟⎝ ⎠

∫

∫ ∫

k r

k r

r rr k r r r

r rr r r [1.60]

It thus appears that the Green’s function G< contains more information than the density matrix or the Wigner function, which explains that some phenomena are simpler to model in this formalism, such as the presence of electrical contacts. To calculate G< it is useful to introduce the retarded rG and the advanced

†a rG G⎡ ⎤= ⎣ ⎦ Green’s function, defined by

( ) ( ) ( ) ( )*, , , , ,rj j

jG t t t t t tθ ψ ψ′ ′ ′ ′ ′= − ∑r r r r [1.61]

where θ is the Heaviside step function. It should be noted that rG includes only the information on accessible states and not their occupation (pj does not appear). By introducing the time/energy coordinates

( ) ( ), , , , , ,Eir rG t E d e G t t

ττ τ τ

−′ ′= − +∫r r r r [1.62]

one obtains

( ) ( ) ( )*, , , j jr

ijG t E

E i Eψ ψ

η′

′ =+ −∑r r

r r [1.63]

where the small parameter 0η +→ is inserted in the denominator to ensure the convergence of the Fourier transform. In practice, to calculate rG the expression [1.63] is generally rewritten as [DAT 95]

( ) ( )ˆ , ,rE i H G Eη δ⎡ ⎤ ′ ′+ − = −⎣ ⎦ r r r r [1.64]


One then has

( ) ( ) ( ) ( ) ( )

( )

*, , , , 2

, ,

r aj j j

ji G E G E E E

A E

π δ ψ ψ⎡ ⎤′ ′ ′− = −⎣ ⎦

′=

∑r r r r r r

r r [1.65]

which is nothing but the spectral density of states. If the occupation probability p of these states depends on the energy only, e.g. at thermal equilibrium, the lesser Green’s function can be calculated as [DAT 95]

( ) ( ) ( ), , , ,G E i A E p E< ′ ′=r r r r [1.66]

Today, the Green’s functions formalism is most used for quantum simulation of nanodevices. It was used in pioneering work by Lake and Datta in 1992 to model resonant tunneling diodes [LAK 92]. It is now widely used to simulate MOSFETs [SVI 02], [VEN 02], [SVI 03], [VEN 03], [VEN 04], [BES 04a], [JIN 06], [ANA 07], [WAN 04], [LUI 06], [BES 07], [BUR 08], [KHA 07], carbon nanotube transistors [GUO 04], [GUO 05], [ALA 07], [KOS 07a], [KOS 07b], [POU 07] and graphene nanoribbon transistors [FIO 07]. The technique originally raised difficult questions regarding the modeling of contacts and interactions. However, tremendous progress has been made in recent years to incorporate these real-life effects through appropriate formalism, as will be discussed later.

In Chapter 3, calculations based on the Wigner function will frequently be discussed in comparison to Green’s function calculations.

1.4. The two main problems of quantum transport

1.4.1. The first problem: the modeling of contacts

Electron devices are always connected to an external circuit through contacts. Surprisingly enough, the modeling of contacts in nanodevices is still a non-trivial and debated problem. Many articles have recently been devoted to this question. We have already mentioned that the treatment of contacts in the density matrix formalism is difficult. We focus here on the Wigner and Green formalisms. In these two approaches, the assumptions considered are different and we will try to explain the subtleties and difficulties related to all of them, since it is important to interpret correctly the quantum transport simulations. Additionally, the problem of the value of electrostatic potential at contacts of nanodevices will be discussed.


1.4.1.1. Wigner’s function formalism: semi-classical contacts

The treatment of contacts in the Wigner formalism is usually simple. It is based on the idea that, near the contact, the transport is semi-classical and under equilibrium. Hence, the Wigner function at the contact is assumed to be an equilibrium distribution function, either the Maxwell-Boltzmann function for a non-degenerate gas, or the Fermi-Dirac function for a degenerate gas. Considering a non-degenerate electron gas, the Wigner function at the contact/device interface of position Cx is then

( ) ( )( ),w C mbf f Eαx k k [1.67]

if k is directed outwards from the contact (this is illustrated in Figure 1.8).

It is thus essential to check that the transport is actually semi-classical, i.e. far enough from the regions of the device where quantum effects occur. This has important consequences which will be discussed later.

DeviceContact 1 Contact 2

( )2 2

, exp2 *w

kf x km

⎛ ⎞∝ −⎜ ⎟⎜ ⎟

⎝ ⎠

k > 0 k < 0

0 50 100 150x (nm)

( )2 2

, exp2 *w

kf x km

⎛ ⎞∝ −⎜ ⎟⎜ ⎟

⎝ ⎠

1.5

1

0.5

0

-0.5

-1

k(n

m-1

)

1412

86

20

(a.u.)

10

4

-2

DeviceContact 1 Contact 2

( )2 2

, exp2 *w

kf x km

⎛ ⎞∝ −⎜ ⎟⎜ ⎟

⎝ ⎠

k > 0 k < 0

0 50 100 150x (nm)

0 50 100 150x (nm)

( )2 2

, exp2 *w

kf x km

⎛ ⎞∝ −⎜ ⎟⎜ ⎟

⎝ ⎠

1.5

1

0.5

0

-0.5

-1

k(n

m-1

)

1412

86

20

(a.u.)

10

4

-2

1412

86

20

(a.u.)

10

4

-2

Figure 1.8. Illustration of the Wigner function in a device (here a resonant tunneling diode) and at the junction between the device and the two terminal contacts (For a color version of

this figure, see http://wigner-book.ief.u-psud.fr/)


Another important feature related to the modeling of contacts is the treatment of the Wigner potential [1.51], i.e.

( )( )

12 22

iw dV d e V V

i π′− . ′ ′⎛ ⎞⎛ ⎞ ⎛ ⎞′, = + − −⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠⎝ ⎠∫ k r r rr k r r r [1.68]

In a closed system the integral obviously extends over all space, but what are the relevant limits of integration for an open system?

A common approach is to cut the integration at a maximum length, chosen to be the coherence length, beyond which no quantum effects are expected [NED 04]. However, the results obtained are not insensitive to the choice of the coherence length, which is somewhat arbitrary. This choice can be made from physical considerations [NED 04] or for numerical reasons related to the real space discretization [FRE 90], [BIE 97a].

Another approach consists of considering that contacts are fully non-coherent with respect to the device [BER 03], which assumes the scattering rate in the contact to be high enough for an electron absorbed by a contact to immediately lose any memory of its history inside the device. Hence there are no off-diagonal terms of the density matrix connecting the contact to the device: ˆ( ) ' 0tρ =r r if r belongs to

the device and 'r to the contact3. Under these conditions, it can be shown that the integration in [1.68] should be limited to positions 'r for which both 2′+r r and

2′−r r belong to the device [BER 03], [FER 06]. (The derivation is presented in Appendix B).

The choice between these two possible integrations to calculate the Wigner potential, i.e. coherence length versus non-coherent contact, is subtle and should depend on the device being studied. Fortunately, in many real-life devices, this question is not always crucial.

1.4.1.2. Green’s function formalism: the self-energy concept

The treatment of contacts in the Green’s function formalism is very different from that in the Wigner approach. This brief description will be useful to understand the results of device simulation presented in Chapter 3. It is reviewed in much more detail in other textbooks, e.g. in [DAT 95].

3 One additional benefit of this hypothesis is avoiding aberrations that may emerge if we consider infinite coherent length, as illustrated in [TAJ 06].


Consider first the full system made up of a device and one contact, as represented in Figure 1.9. We can separate the retarded Green’s function rG into four parts:

– ( ), ,rDG E′r r if the two coordinates r and ′r are in the device;

– ( ), ,rcG E′r r if the two coordinates are in the contact;

– ( ), ,rcDG E′r r and ( ), ,r

DcG E′r r if one coordinate is in the device and the other in the contact.

For the sake of simplicity, the energy dependence of the Green’s functions will be omitted in the expressions below. In a discretized real-space the Green’s functions can take the form of matrices and [1.64] is rewritten as

( ) rE i I H G Iη+ − =⎡ ⎤⎣ ⎦ [1.69]

where I is the identity matrix. By separating the device and the contact the matrices may be rewritten by blocks and [1.69] becomes

( )( )†

r rc c c cD

r rc D Dc D

E i I H G GI

E i I H G G

η τ

τ η

⎡ ⎤+ −⎡ ⎤=⎢ ⎥⎢ ⎥

+ − ⎢ ⎥⎢ ⎥⎣ ⎦ ⎣ ⎦ [1.70]

where cH stands for the Hamiltonian of the contact, DH for the Hamiltonian of the isolated device, and cτ represents the device/contact coupling matrix. From the matrix formulation [1.70] one can extract the reduced equations

( ) [ ]r rc cD c DE i I H G G Iη τ+ − + =⎡ ⎤⎣ ⎦ [1.71]

( ) †r rD D c cDE i I H G G Iη τ⎡ ⎤+ − + =⎡ ⎤⎣ ⎦ ⎣ ⎦ [1.72]

Equation [1.71] can be transformed into

r R rcD c c DG g Gτ= − , [1.73]

where the Green’s function Rcg of the isolated contact has been introduced and

gives


( ) 1Rc cg E i I Hη −= + −⎡ ⎤⎣ ⎦ . [1.74]

Substituting [1.73] into [1.72] finally leads to

( )1r r

D DG E i I Hη−

⎡ ⎤= + − −Σ⎣ ⎦ [1.75]

where the retarded self-energy rΣ associated with the contact is defined by

†r Rc c cgτ τΣ = [1.76]

This self-energy plays the role of an effective Hamiltonian describing the impact of the contact on the device’s Green’s function. However, the retarded self-energy cannot be fully assimilated to a Hamiltonian since in the general case it is not

Hermitian. The advanced self-energy is thus defined as †a r⎡ ⎤Σ = Σ⎣ ⎦ . It should be

noted that the self-energy of the contact includes the Green’s function of the isolated contact, the determination of which is not trivial. In practice, the contact Green’s function is commonly assumed to be that of an idealized contact, as the semi-infinite contact [DAT 95].

In all that follows, the subscript D in the Green’s functions will be omitted since we are interested in the device itself, not in the contacts. We will now see that under the influence of contacts the energy levels in the device are shifted and, especially, broadened.

Indeed, consider expression [1.63] for the retarded Green’s function and assume the self-energy to be diagonal in the basis jψ and to take the unique value Σ .

The spectral density [1.65] is then

( ) ( ) ( ) ( ) ( )* *

*, , j j j jr a

jj jA E i G G i

E i E E i E

ψ ψ ψ ψη Σ η Σ

⎡ ⎤= − = −⎣ ⎦ + − − − − −∑

r r r rr r

( ) ( ) ( )[ ]

*2 2

2 Im, ,Re Im

j jj j

A EE E

Σψ ψΣ Σ

= −⎡ ⎤− − +⎣ ⎦

∑r r r r [1.77]

which is a sum of Lorentzian functions centered on energies jE shifted from ReΣ

and broadened by ImΣ . This broadening is associated with the contact-induced


finite lifetime of electrons on the device energy levels. A schematic illustration of the broadening effect on the spectral function centered on the level jE is shown in

Figure 1.9.

Finally, it should be noted that these derivations can be easily generalized to several contacts by summing up all the corresponding self-energies, provided that the contacts are independent. This is done in [DAT 95].

2 ImΣ

ReΣjE E

A

2 ImΣ

ReΣjE E

A

Figure 1.9. Schematic of the spectral density illustrating the broadening of the discrete level Ej induced by the coupling self-energy Σ

1.4.1.3. Comparison of the two approaches

The two approaches used for the modeling of contacts in Wigner and Green formalisms are not equivalent. It is important to understand that both have their own advantages and limitations with regards to device simulation.

The Wigner function treatment assumes a semi-classical contact, which is indeed the case of many nanodevices. However, it is a severe limitation for some devices, e.g. for devices in which the carriers are injected at contacts through a Schottky barrier, as in some carbon nanotube transistors [HEI 02]. Indeed, in that case the quantum character of transport appears immediately after injection in the device. A possible solution to simulating such devices with the Wigner formalism may consist of using an injection model, including tunneling through the barrier as developed for semi-classical simulations [GUO 05], [NGU 07].

The treatment commonly used within the Green formalism may seem more rigorous since it does not need to assume any semi-classical region near the contact. However, it requires knowledge of the electronic structure of the contact, which is


hard. The contact is thus generally replaced by a non-physical ideal contact. A consequence of this is that the contact-induced decoherence is thus neglected [BER 03], [KNE 08]. This phenomenon is, however, included in the Wigner formulation, as will be shown later in Chapter 3.

1.4.1.4. Discussion of the value of the electrostatic potential at contacts

The contact problem discussed above is related to the nature of the quantum coupling between the device and the contacts. Another problem related to the value of the electrostatic potential at contact will now be addressed.

In conventional devices this question is quite simple. Ohm’s law U = Rc I applies in the contacts of resistance Rc. In a well-designed device, this resistance is weak and the drop of potential through the contact can be neglected. However, in a nanodevice operating near the ballistic regime the problem is much more complex, as discussed for instance in [DAT 00], [DAT 05], [DO 07a], [BES 04b], [CAV 08].

Consider an ideal nano-FET operating in a ballistic regime under high drain bias DSV . As a consequence of ballistic transport, all energy dissipation takes place in

the contacts. Assume that the self-consistence of the transport with the Poisson equation ensures the charge neutrality at contacts. If the Fermi energy is Sμ in the source, it is D S DSVμ μ= − in the drain. In the OFF-state (Figure 1.10a) almost all incident electrons coming from the source are reflected by the potential barrier induced by the low-gate voltage. The charge neutrality at both contacts is thus obtained with the same difference 0μ between the Fermi energy and the bottom of the conduction band as under thermal equilibrium. All the drop of source-drain potential DSV takes place in the device.

In the ON-state (Figure 1.10b), the situation is very different. A large number of electrons are injected from the source and the number of electrons injected from the drain is weak because of the high drain bias voltage. Assume that the N electrons injected from the source have a probability T to be transmitted to the drain. There are thus ( )1R N T N= − electrons reflected to the source instead of N in the OFF-state, which means that T N electrons are missing on the source side and T N electrons are in excess on the drain side. The self-consistence must appropriately modify the potential to recover the neutrality at both contacts: compared to the OFF-state, the electrostatic potential decreases on the source side (to enhance the number of electrons) and increases on the drain side (to reduce the number of electrons). Hence, the effective bias voltage taking place within the device is smaller than the externally applied bias DSV and a significant part of the potential drop takes place in the contacts.


(a) OFF-State

(b) ON-State

( )cE x

Sμ

D S DSVμ μ= −DSV VΔ =

0μ

0μ

SeV−

DeV−

Sμ

DSV VΔ <

0μ>

0μ<

( )cE x

D S DSVμ μ= −

Sμ

DSV VΔ <

0μ>

0μ<

( )cE x

D S DSVμ μ= −

Figure 1.10. Schematic illustration of the drop of potential V and Fermi energy μ for a FET operating in ballistic regime in both (a) OFF-state (low-gate voltage VGS) and

(b) ON-state (high-gate voltage VGS)

It is difficult to model this effect correctly in a realistic device simulation. In a quantum transport model which neglects the scattering mechanisms in the active region of the device, this effect is very important and must be taken into account [DAT 00], as it is in most models based on the Green’s function formalism. Accordingly, the electric field at contacts is generally assumed to be zero and Neumann boundary conditions are applied at Ohmic contacts to solve the Poisson equation [DAT 00], [DAT 05], [DO 07a]. The Fermi energy at contact is fixed but the electrostatic potential is self-consistently floating to ensure neutrality according to transport conditions. The approximation of the zero-field is sometimes justified by the fact that the bias potential applied on the terminal contacts covers a large distance. Hence the field in the contacts can be neglected.

However, this ballistic model suffers some limitations. Nanodevices are usually connected to metallic contacts via very highly-doped semiconducting access regions where scattering events with impurities are numerous and the electric field is not zero [SAI 04], which is not necessarily consistent with zero-field boundary conditions. An apparently simple solution is to perform a simulation of the device with long enough access regions additional to the device itself. In that way, the supplementary potential drop of the potential in the access regions due to ballistic


transport in the device is modeled naturally, and we can apply a traditional boundary condition to the contacts. This requires a model that incorporates scattering and is efficient enough to deal with simulations of relatively big structures (the device itself and its access regions). This will be the preferred solution in this book, using the Wigner formalism.

1.4.2. The second problem: the treatment of collisions/scattering in quantum transport

Introducing the physics of the interaction of electrons with their environment in quantum transport formalisms is a difficult problem. In principle it is necessary to include the environment which originates the interaction in the system to be studied. This difficulty is not specific to solid-state and semiconductor device physics and is encountered also in all domains of quantum physics, such as in atomic physics and quantum optics [DAL 03], [PLE 98], or quantum information processing [CAL 96].

In a semiconductor device operating at room temperature the collisions/interactions play an important role and cannot be neglected; this is illustrated in the next chapters for many cases. Here we present in detail how the interactions can be introduced in the Wigner function formalism, since there is a very intuitive way to do so, and we briefly discuss the case of the Green’s function formalism that is very important in much current work. Scattering is, overall, harder to model with this formalism but a more exact level of description can be reached.

1.4.2.1. Interaction of an electron with a classical field in the Wigner formalism

We consider here the interaction between electrons and phonons. This is a difficult problem, and we first take a simplified approach that allows an understanding of how the interaction takes place and how it can affect the Wigner function. We imagine that the phonons act as a deformation of the potential seen by the electrons that propagate as a plane wave. The phonon of wave vector q and angular frequency ωq thus induces a supplementary potential which takes the form

( ) ( ), 2 cos

i t i t

V t U t

U e eω ω

ω⋅ − − ⋅ +

= ⋅ −

⎡ ⎤= +⎢ ⎥⎣ ⎦q q

q

q r q r

r q r [1.78]

where U represents the strength of the electron/phonon coupling.

This is a simplified approach with regard to the true electron/phonon coupling Hamiltonian seen in section 1.1.2. Electron/phonon interaction only modifies the potential experienced by electrons and its effect on the phonon field is neglected.


This can, however, serve as a good starting point. This potential [1.78] can indeed be introduced directly in the Wigner potential [1.51], which leads to

( ) 0

0

, ,2 2

2 2

i i tw

i i t

UV t ei

e

ω

ω

δ δ

δ δ

⋅ −

− ⋅ +

⎡ ⎡ ⎤⎛ ⎞ ⎛ ⎞= − − − −⎜ ⎟ ⎜ ⎟⎢ ⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦⎣⎤⎡ ⎤⎛ ⎞ ⎛ ⎞+ − − − −⎜ ⎟ ⎜ ⎟ ⎥⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦⎦

q r

q r

q qr k k k

q qk k

which simplifies into

( ) 0 0, ,

2 2

i i t i i tw

UV t e ei

ω ω

δ δ

⋅ − − ⋅ +⎡ ⎤= +⎣ ⎦

⎡ ⎤⎛ ⎞ ⎛ ⎞× − − − −⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦

q r q rr k

q qk k [1.79]

Substituting [1.79] into [1.51] gives the quantum evolution term

( ) 0 0, ,

, , , ,2 2

i i t i i tw

w w

UQ f t e ei

f t f t

ω ω⋅ − − ⋅ +⎡ ⎤= +⎣ ⎦

⎡ ⎤⎛ ⎞ ⎛ ⎞× − − +⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦

q r q rr k

q qr k r k [1.80]

Here, for the sake of simplicity, we assume the electrostatic potential to be uniformly zero. The case of the presence of non-uniform electrostatic potential is treated in the next sub-section devoted to the interaction with a quantum field. It could be similarly introduced in this derivation. The Wigner equation can then be written as

( ) 0 0* , ,

, , , ,2 2

i i t i i tw

w w

Uf t e et im

f t f t

ω ω⋅ − − ⋅ +⎡ ⎤∂ ∂ ⎡ ⎤+ = +⎢ ⎥ ⎣ ⎦∂ ∂⎣ ⎦⎡ ⎤⎛ ⎞ ⎛ ⎞× − − +⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦

q r q rk r kr

q qr k r k [1.81]

which is an implicit equation, difficult to integrate. It is convenient to make the Wigner function appear at times earlier than t to solve it. A crafty solution consists of making the well-known change of variable in the effective mass approximation [ROS 92b]


( ) ( ) ( )*t t t tm

t t

⎧ ′ ′ ′= + −⎪⎨⎪ ′ =⎩

kr r [1.82]

which gives the integral form of ( ), ,wf tr k

( ) ( )

( ) ( )

0 0

*

0

( , , ) , ,0

, , , ,2 2

w w

ti t i t i t i t

w w

f t f tm

U dt e ei

f t t f t t

ω ω′ ′ ′ ′ ′ ′⋅ − − ⋅ +

⎛ ⎞= −⎜ ⎟⎝ ⎠

⎡ ⎤′+ +⎢ ⎥⎣ ⎦

⎡ ⎤⎛ ⎞ ⎛ ⎞′ ′ ′ ′ ′ ′× − − +⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦

∫ q r q r

kr k r k

q qr k r k

[1.83]

Substituting [1.83] into [1.81] and defining ( ) ( ) ( ) ( ) *2t t t t m′ ′ ′= − − −r r k q

and ( ) ( ) ( ) ( ) *2t t t t m′′ ′ ′= − + −r r k q leads to

( ) ( ) ( )( )( )( ) ( )( )( )

( ) ( ) ( )( )( )( )

0 00 0

0 00 0

2*

0

'

0

( , , )

, , , ,

, ,

w

ti t i t i t i ti i t i i t

w w

ti t i t i t i ti i t i i t

w w

f t Ut m

dt e e e e

f t t f t t

dt e e e e

f t t f

ω ωω ω

ω ωω ω

′ ′ ′ ′ ′ ′⋅ − − ⋅ +⋅ − − ⋅ +

′′ ′ ′ ′′ ′⋅ − − ⋅ +⋅ − − ⋅ +

∂ ∂⎡ ⎤+ = −⎢ ⎥∂ ∂⎣ ⎦⎡⎢ ′× + +⎢⎣

′ ′ ′ ′ ′ ′× − −

′+ + +

′′ ′ ′ ′′× + −

∫

∫

q r q rq r q r

q r q rq r q r

k r kr

r k q r k

r k q ( )( )( ), ,t t

ic

⎤′ ′ ⎦+

r k

[1.84]

where the initial conditions ic gives

0 0

* *, ,0 , ,02 2 2 2

i i t i i t

w w

Uic e ei

f t f tm m

ω ω⋅ − − ⋅ +⎡ ⎤= +⎣ ⎦

⎡ ⎤⎛ ⎞ ⎛ ⎞⎛ ⎞ ⎛ ⎞× − − − − − + +⎜ ⎟ ⎜ ⎟⎢ ⎥⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠⎝ ⎠ ⎝ ⎠⎣ ⎦

q r q r

q q q qr k k r k k [1.85]


By choosing a time origin far enough from time t this term tends to vanish. We thus assume here that ic = 0. Consider the product of exponential functions in the first part of [1.84] which gives

( ) ( ) ( )

( )

( ) ( ) ( )

0 *00

0 * *

0

2

22

i t t i t ti t i ti i t m

i t tm m

i t tE E E

e e cc e cc

e cc

e cc

ωωω

ω

′ ′− + ⋅ − −′ ′ ′⋅ −− ⋅ +

⎡ ⎤′− − ⋅ +⎢ ⎥⎣ ⎦

′−− + −⎡ ⎤⎣ ⎦

+ = +

= +

= +

qq kq rq r

k q q

k k q

[1.86]

where cc stands for the complex conjugate, 0E is the phonon energy and

( ) 2 2 *2E k m= k is the parabolic dispersion relation of electrons. The similar term of the second part of [1.84] is

( ) ( ) ( ) ( )00 0i E E E t ti i t i t i t

e e cc e ccω ω ⎡ ⎤ ′+ − + −′′ ′ ′− ⋅ + ⋅ − ⎣ ⎦+ = +

k k qq r q r [1.87]

The other exponential terms cannot be factorized by [ ]' ( ')i E t t

e−

. They correspond to many-phonon processes.

To derive them, one should reinsert the integral form of the Wigner function into [1.84]. In the weak electron/phonon coupling approximation, these terms are neglected and only single-phonon processes are considered. The Wigner equation [1.84] now gives

( )

( ) ( )( )( ) ( )( )

( ) ( )( )( ) ( )( )

0

0

*

( )

0

( )

0

, ,

, , , ,

, , , ,

w

t i E E E t tw w

t i E E E t tw w

f t Ut m

dt e f t t f t t cc

dt e f t t f t t cc

′− + − −⎡ ⎤⎣ ⎦

′+ − + −⎡ ⎤⎣ ⎦

∂ ∂⎡ ⎤+ = −⎢ ⎥∂ ∂⎣ ⎦⎡⎢ ⎡ ⎤′ ′ ′ ′ ′ ′ ′× − − +⎣ ⎦⎢⎣

⎤⎥⎡ ⎤′ ′′ ′ ′ ′′ ′ ′+ + − +⎣ ⎦ ⎥⎦

∫

∫

k k q

k k q

k r kr

r k q r k

r k q r k

[1.88]

This equation is similar to the Levinson equation for a single phonon that is described for example in [NED 05]. It contains all the dynamics of the electron/phonon interaction including advanced effects of collisional broadening and


retardation. However, it is still difficult to solve and some approximations would be welcome to make the solution simpler.

A key approximation is to assume that the interaction process is fast with respect to all other phenomena involved in the electron dynamics. The frequency

( ) ( )0E E E+ − +⎡ ⎤⎣ ⎦k k q is thus higher than the variation frequencies of

( )( ) ( )( ), , , ,w wf t t f t t⎡ ⎤′′ ′ ′ ′′ ′ ′+ −⎣ ⎦r k q r k . The following approximation (see

[RIN 04]) can thus be made for any function ( )tΦ

( ) ( ) ( ) ( )'

0

1' 't

i t td t e t t i PVω Φ Φ π δ ωω

− ⎛ ⎞⎛ ⎞≈ + ⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠∫ [1.89]

where PV denotes the Cauchy Principal Value. All principal values vanish and only the Boltzmann-like term of [1.88] remains, i.e.

( )

( ) ( )( ) ( ) ( )( ) ( )( ) ( ) ( )

2*

0

0

, , 2

, , , ,

, , , ,

w

w w

w w

f t Ut m

E E E f t f t

E E E f t f t

π

δ

δ

∂ ∂⎡ ⎤+ =⎢ ⎥∂ ∂⎣ ⎦⎡× − + − − −⎡ ⎤⎣ ⎦⎣

⎤+ + − + + −⎡ ⎤⎣ ⎦⎦

k r kr

k k q r k q r k

k k q r k q r k

[1.90]

The four Wigner functions in [1.90] can be interpreted in an intuitive way, in terms of phonon emission and absorption as:

– the function ( ), ,wf t−r k q : phonon absorption by electron state −k q ;

– the first function ( ), ,wf t− r k : phonon emission by electron state k ;

– the function ( ), ,wf t+r k q : phonon emission by electron state +k q ;

– the second function ( ), ,wf t− r k : phonon absorption by electron state k .

This shows us that a strength of the Wigner function is that it can model scattering effect in a very natural way. However, let us now analyze the results in more detail. It is important to notice that the emission and absorption rates are similar in this calculation. This is actually surprising. It is indeed easy to show that, in this situation, the electron/phonon scattering cannot put the electron system to equilibrium. If the energy of the phonon mode 0E is not zero, the collision term is not zero for a Wigner function wf given by a Maxwell-Boltzmann distribution


function (as illustrated in Appendix D). However, much research has shown that optical phonons are very effective in relaxing hot electrons in devices [LUN 00], [SAI 04]. The present calculation is thus clearly not sufficient for optical phonons. What is missing in this physical description?

Actually, the very basis of the calculation is simplistic. The coupling between electrons and phonons cannot be seen as just a classical field acting on the electrons, and we need to account for the quantum nature of the phonons and the exact electron/phonon coupling Hamiltonian. This is the subject of the following sub-section.

1.4.2.2. Interaction of an electron with a quantum field in the Wigner formalism

To recover the difference between emission and absorption rates of phonons, the quantum nature of the field must be taken into account. After the scattering process, the state of the phonon bath is changed, which is fully neglected by the classical field approach. Here we consider an approach in which the phonon modes are quantized. Some parts of the derivation are inspired by that of Nedjalkov [NED 05]. Unfortunately, this requires the use of a more advanced formalism which can, however, still be approached in a relatively intuitive way.

1.4.2.2.1. Generalized Wigner equation for electron-phonon system

To go further, we have to consider the full system, made up of one electron and one phonon mode of wave vector q and energy 0E ω= . A basis of this system may be of the form nψ ⊗ , where ψ is the wave function of the electron and

n is the number of phonons of mode q. The density matrix of this system can be

written in the form ( ) ( ), , , , , ,n n t n t nρ ρ′ ′ ′ ′=r r r r . In the electron space coordinates the Wigner function then gives

( )( )3

1, , , , , , , ,2 22

iwf n n t d e n n tρ

π′− ⋅ ′ ′⎛ ⎞′ ′ ′= + −⎜ ⎟⎝ ⎠∫ k r r rr k r r r [1.91]

The full Hamiltonian of this system is the sum of three parts, i.e.

e ph e phH H H V −= + + [1.92]

which corresponds to:

– the Hamiltonian of free electrons


( )2 2

*2eH V

m= +k r [1.93]

– the Hamiltonian of free phonons

†0

12ph q qH E a a⎛ ⎞= +⎜ ⎟

⎝ ⎠ [1.94]

– the Hamiltonian of electron-phonon coupling

( ) ( ) †i ie ph q qV i F e a e a⋅ − ⋅− ⎡ ⎤= −⎣ ⎦

q r q rr q [1.95]

We now intend to derive the Wigner equation for this Hamiltonian.

For the free electron Hamiltonian (see [1.49]), the result is well known, i.e.

*( , , , , ) ( , , , , ) ( , , , , )w w wf n n t f n n t Q f n n tt m

⎡ ⎤∂ ∂′ ′ ′= − +⎢ ⎥∂ ∂⎣ ⎦

kr k r k r kr

[1.96]

For the phonons, the Liouville equation [1.36] gives

( )

† †0 0

0

1( , , , , ) , ,

1 ( , , , , )

q q q qn n t n E a a E a a nt i

E n n n n ti

ρ ρ ρ

ρ

∂ ⎡ ⎤′ ′ ′ ′= −⎣ ⎦∂

′ ′ ′= −

r r r r

r r [1.97]

which leads to the Wigner equation

( )0( , , , , ) ( , , , , )w wEf n n t n n f n n t

t i∂ ′ ′ ′= −∂

r k r k [1.98]

For the coupling Hamiltonian, one first derives the Liouville equation

( ) ( ) ( )( )

†

†

, , , , ,

,

i iq q

i iq q

n n t F n e a e at

e a e a n

ρ ρ

ρ

⋅ − ⋅

⋅ − ⋅

∂ ⎡′ ′ = −⎣∂

⎤ ′ ′− −⎦

q r q r

q r q r

r r q r

r


( ) ( ) ( )

( ) ( )( )

, , , , 1 , , 1, ,

, , 1, , , , , 1,

1 , , , 1,

i

i i

i

n n t F e n n n tt

e n n n t e n n n t

e n n n t

ρ ρ

ρ ρ

ρ

⋅

′− ⋅ ⋅

′− ⋅

∂ ⎡′ ′ ′ ′= + +⎣∂

′ ′ ′ ′ ′− − − −

⎤′ ′ ′+ + + ⎦

q r

q r q r

q r

r r q r r

r r r r

r r

[1.99]

From a Weyl-Wigner transform of [1.99], one obtains the Wigner equation

( ) ( ), , , , , , , ,w wf n n t C f n n tt∂ ′ ′=∂

r k r k [1.100]

where the collision term gives

( ) ( ), , , , 1 , , 1, ,2

, , , 1,2

, , 1, ,2

1 , , , 1,2

iw w

w

iw

w

C f n n t F e n f n n t

n f n n t

e n f n n t

n f n n t

⋅

− ⋅

⎡ ⎛ ⎛ ⎞′ ′= + − +⎜ ⎟⎢ ⎜ ⎝ ⎠⎝⎣⎞⎛ ⎞′ ′− + −⎜ ⎟⎟⎝ ⎠⎠

⎛ ⎛ ⎞′+ − + −⎜ ⎟⎜ ⎝ ⎠⎝⎤⎞⎛ ⎞′ ′+ + − +⎜ ⎟ ⎥⎟⎝ ⎠⎠⎦

q r

q r

qr k q r k

qr k

qr k

qr k

[1.101]

The full Wigner equation is finally written by collecting the three contributions

( ) ( ) ( )0* , , , , , , , ,w wi n n Q f n n t C f n n tt m

ω⎡ ⎤∂ ∂ ′ ′ ′+ + − − =⎢ ⎥∂ ∂⎣ ⎦

k r k r kr

[1.102]

This result may also be found in [BER 03] and [NED 05]. First, we are interested in the diagonal terms in phonon states, i.e. in the case 'n n= . For simplicity we will denote the corresponding Wigner function ( , , , ) ( , , , , )w wf n t f n n t=r k r k which obeys the equation


( ) ( )* , , ,

1 , , 1, , , , , 1,2 2

, , 1, , 1 , , , 1,2 2

w

iw w

iw w

Q f n t Ft m

e n f n n t n f n n t

e n f n n t n f n n t

⋅

− ⋅

⎡ ⎤∂ ∂+ + =⎢ ⎥∂ ∂⎣ ⎦⎡ ⎛ ⎞⎛ ⎞ ⎛ ⎞× + − + − + −⎜ ⎟ ⎜ ⎟⎢ ⎜ ⎟⎝ ⎠ ⎝ ⎠⎝ ⎠⎣

⎤⎛ ⎞⎛ ⎞ ⎛ ⎞+ − + − + + − +⎜ ⎟ ⎜ ⎟ ⎥⎜ ⎟⎝ ⎠ ⎝ ⎠⎝ ⎠⎦

q r

q r

k r k qr

q qr k r k

q qr k r k

[1.103]

The third line of [1.103] is the complex conjugate of the second line. In what follows, it will be noted as cc. We now have to determine the Wigner functions which couple several phonon numbers. Again, we make the weak coupling approximation, which means that ( , , , , ) 0wf n n t′ =r k if 1n n′− > . It amounts to neglecting the many-phonon processes. Equation [1.100] then leads to

( ) ( )0* , , 1, ,

1 , , , ,2

1 , , 1, 1,2

w

iw

w

i Q f n n t Ft m

e n f n n t

n f n n t

ω

− ⋅

⎡ ⎤∂ ∂+ + − + =⎢ ⎥∂ ∂⎣ ⎦⎡ ⎛ ⎞× − + +⎜ ⎟⎢ ⎝ ⎠⎣

⎤⎛ ⎞+ + − + +⎜ ⎟⎥⎝ ⎠⎦

q r

k r k qr

qr k

qr k

[1.104]

and

( ) ( )0* , , , 1,

, , 1, 1,2

, , , ,2

w

iw

w

i Q f n n t Ft m

e n f n n t

n f n n t

ω

− ⋅

⎡ ⎤∂ ∂+ + − − =⎢ ⎥∂ ∂⎣ ⎦⎡ ⎛ ⎞× − + − −⎜ ⎟⎢ ⎝ ⎠⎣

⎤⎛ ⎞+ −⎜ ⎟⎥⎝ ⎠⎦

q r

k r k qr

qr k

qr k

[1.105]

The system of implicit equations [1.103], [1.104] and [1.105] is difficult to solve. As in the case of a classical field, it is convenient to make the Wigner functions appear at times earlier than t. We thus again use the change of variables [1.82], i.e.

( ) ( ) ( )*t t t tm

t t

⎧ ′ ′ ′= + −⎪⎨⎪ ′ =⎩

kr r [1.106]


Equation [1.104] then becomes

( )( ) ( )

( )

( )

0 , , 1, ,

1 , , , ,2

1 , , 1, 1,2

w

iw

w

i Q f t n n t Ft

e n f t n n t

n f t n n t

ω

− ⋅

⎡ ⎤∂ ′+ − + =⎢ ⎥′∂⎣ ⎦⎡ ⎛ ⎞′× − + +⎜ ⎟⎢ ⎝ ⎠⎣

⎤⎛ ⎞′+ + − + +⎜ ⎟⎥⎝ ⎠⎦

q r

r k q

qr k

qr k

[1.107]

which may be integrated in

( ) ( )( )

( ) ( ) ( )

( )

0 0

0 0

0

'

0

, , 1, , , , 1, ,

1 , , ,2

1 , , 1,2

ti t i t

w w

ti ti t i t

w

w

f n n t ic e d t e Q f t n n t

e d t F e e n f t n t

n f t n t

ω ω

ω ω

′−

′ ′− ⋅−

′ ′ ′ ′+ = + +

⎡ ⎛ ⎞′ ′ ′ ′+ − + +⎜ ⎟⎢ ⎝ ⎠⎣

⎤⎛ ⎞′ ′ ′+ + − +⎜ ⎟⎥⎝ ⎠⎦

∫

∫ q r

r k r k

qq r k

qr k

[1.108]

where we can assume the initial condition to be 0ic = . Similarly,

( ) ( )( )

( ) ( ) ( )

( )

0 0

0 0

0

'

0

, , 1, , , , 1, ,

1 , , ,2

1 , , 1,2

ti t i t

w w

ti ti t i t

w

w

f n n t ic e d t e Q f t n n t

e d t F e e n f t n t

n f t n t

ω ω

ω ω

′−

′ ′− ⋅−

′ ′ ′ ′+ = + +

⎡ ⎛ ⎞′ ′ ′ ′+ − + +⎜ ⎟⎢ ⎝ ⎠⎣

⎤⎛ ⎞′ ′ ′+ + − +⎜ ⎟⎥⎝ ⎠⎦

∫

∫ q r

r k r k

qq r k

qr k

[1.109]

where, as in the previous sub-section, ( ) ( ) ( ) ( ) *2t t t t m′ ′ ′= − − −r r k q and

( ) ( ) ( ) ( ) *2t t t t m′′ ′ ′= − + −r r k q . Substituting [1.108] and [1.109] into [1.103] gives


( ) ( )

( ) ( ) ( )( )

( )( )

( ) ( )( )

( )( ) }

0 0

0 0

2*

0

0

, , ,

1 , , ,

, , 1,

, , 1,

, , ,

w

ti ti t i ti

w

w

ti ti t i ti

w

w

Q f n t Ft m

dt e n e e e f t n t

f t n t

dt e n e e e f t n t

f t n t cc ICF

ω ω

ω ω

′ ′− ⋅ ′−⋅

′′ ′− ⋅ ′−⋅

⎡ ⎤∂ ∂+ − =⎢ ⎥∂ ∂⎣ ⎦⎧⎪ ⎡′ ′ ′ ′× + −⎨ ⎣⎪⎩

⎤′ ′ ′+ − + ⎦

⎡′ ′′ ′ ′+ + −⎣

⎤′′ ′ ′− + +⎦

∫

∫

q rq r

q rq r

k r k qr

r k

r k q

r k q

r k

[1.110]

where an additional term ICF (intra-collisional field effect), including the effect of the external potential, gives

( ) ( )

( )

0 0

0 0

0

'

0

1 , , 1, ,2

, , , 1,2

ti t i ti

w

ti t i t

w

ICF F e n e dt e Q f t n n t

n e dt e Q f t n n t

ω ω

ω ω

′−⋅

−

⎡ ⎛ ⎞⎢ ′ ′ ′ ′= + − +⎜ ⎟⎢ ⎝ ⎠⎣⎤⎛ ⎞⎥′ ′′ ′ ′− + −⎜ ⎟⎥⎝ ⎠⎦

∫

∫

q r qq r k

qr k

[1.111]

In the first two terms of [1.110] the exponential functions are the same as for the classical field. We can transform them the same way, i.e.

( ) ( )

( ) ( ) ( )( ) ( )( )

( )( )( ) ( )

( )( )

( )( ) }

0

0

2*

'

0

( ')

0

, , ,

1 , , ,

, , 1,

, , 1,

, , ,

w

t i E E E t tw

w

t i E E E t tw

w

Q f n t Ft m

dt e n f t n t

f t n t cc

dt e n f t n t

f t n t cc ICF

− + − −⎡ ⎤⎣ ⎦

+ − + −⎡ ⎤⎣ ⎦

⎡ ⎤∂ ∂+ − =⎢ ⎥∂ ∂⎣ ⎦⎧⎪ ⎡′ ′ ′ ′× + −⎨ ⎣⎪⎩

⎤′ ′ ′+ − + +⎦

⎡′ ′′ ′ ′+ + −⎣

⎤′′ ′ ′− + +⎦

∫

∫

k k q

k k q

k r k qr

r k

r k q

r k q

r k

[1.112]


This equation is exact in the case of a weak coupling and includes the finite duration of the interaction. We can do the same approximation [1.89] of the fast interaction as in the case of the classical field

( ) ( ) ( ) ( )0

1t

i t td t e t t i PVω Φ Φ π δ ωω

′− ⎛ ⎞⎛ ⎞′ ′ ≈ + ⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠∫

All principal values vanish. The term ICF gives a contribution proportional to ( )0δ ω which can be ignored since we are interested in the case of a non-zero

phonon energy. Finally, only the Boltzmann term remains, i.e.

( ) ( )

( ) ( )( ) ( ) ( ) ( ){( ) ( )( ) ( ) ( ) }

2*

0

0

, , , 2

1 , , 1, , , ,

, , 1, , , ,

w

w w

w w

Q f n t Ft m

E E E n f n t f n t

E E E n f n t f n t

π

δ

δ

⎡ ⎤∂ ∂+ − =⎢ ⎥∂ ∂⎣ ⎦

× − + − + − + −⎡ ⎤⎣ ⎦

+ + − + + − −⎡ ⎤⎣ ⎦

k r k qr

k k q r k q r k

k k q r k q r k

[1.113]

The factors n and 1n + in [1.113] can easily be interpreted in terms of phonon absorption and emission processes, as illustrated in Figure 1.11

Electron: k + q

Phonons:n - 1

Emission:PhononCreation

n


n + 1

Absorption:Phonon

annihilationn

Electron: k

Phonons:n

Electron: k - q

Phonons:n + 1Absorption:

Phononannihilation

n + 1

Electron: k + q

Phonons:n - 1


n


n + 1

Absorption:Phonon

annihilationn

Electron: k

Phonons:n

Electron: k - q

Phonons:n + 1Absorption:

Phononannihilation

n + 1

Figure 1.11. Interpretation of equation [1.113] in terms of phonon absorption and emission

1.4.2.2.2. Reduced Wigner equation for electrons

The set of equations derived above describes the evolution of the full system of phonons and electrons. However, if we assume that the phonon system is effectively maintained at equilibrium by a thermostat, we should consider that the “electrons-phonons” system is still open and the generalized Wigner function is actually a reduced function. In other words, the phonon mode is coupled to an environment


which maintains it at equilibrium whatever the creations and annihilations due to interactions with electrons. Therefore the phonon distribution remains in equilibrium during the evolution. The mean equilibrium phonon number n is given by the Bose distribution and the probability ( )P n of finding n phonons of mode q [NED 05] is

( ) 01 exp1

P n nn kT

ω⎛ ⎞= −⎜ ⎟+ ⎝ ⎠

[1.114]

Hence the generalized Wigner function can be factorized as

( ) ( ) ( ), , , , , ,w wf n n t f t P n=r k r k [1.115]

where ( ) ( ), , , , , ,w wnf t f n n t=∑r k r k is the reduced Wigner function for electrons. Equation [1.113] is then rewritten in the form

( ) ( )

( ) ( )( ) ( ) ( ) ( ) ( ) ( )( ) ( )( ) ( ) ( ) ( ) ( )

2*

0

0

, , 2

1 1 , , , ,

1 , , , ,

w

w w

w w

Q f t Ft m

E E E n P n f t P n f t

E E E n P n f t P n f t

π

δ

δ

⎡ ⎤∂ ∂+ − =⎢ ⎥∂ ∂⎣ ⎦⎡× − + − + + − −⎡ ⎤⎣ ⎦⎣

⎤+ + − + − + −⎡ ⎤⎣ ⎦⎦

k r k qr

k k q r k q r k

k k q r k q r k

[1.116]

Now we just have to take the trace of [1.116] over the phonon modes to get the dynamic equation of the reduced Wigner function. By making use of the convenient equalities

( )

( ) ( )

( ) ( )

( )

1 1

1 1 1 1n n

n n

n P n n n P n n

n P n n n P n n

= + + =⎧ ⎧⎪ ⎪⎨ ⎨

+ = + − = +⎪ ⎪⎩ ⎩

∑ ∑

∑ ∑ [1.117]

we finally obtain

( ) ( )

( ) ( )( ) ( ) ( ) ( )( ) ( )( ) ( ) ( ) ( )

2*

0

0

, , 2

, , 1 , ,

1 , , , ,

w

w w

w w

Q f t Ft m

E E E n f t n f t

E E E n f t n f t

π

δ

δ

⎡ ⎤∂ ∂+ − =⎢ ⎥∂ ∂⎣ ⎦⎡× − + − − − +⎡ ⎤⎣ ⎦⎣

⎤+ + − + + + −⎡ ⎤⎣ ⎦⎦

k r k qr

k k q r k q r k

k k q r k q r k

[1.118]


With respect to [1.113] the prefactors n and 1n + have changed position. In particular, it should be noted that, in contrast to the case of a classical field, the prefactors of the absorption terms are n and that of the emission terms are 1n + . This slight asymmetry between absorption and emission rates changes everything. It is easy to show (see Appendix D) that, thanks to it, electron/phonon scattering can now bring the electron system to equilibrium. It is thus essential to understand it and to include it in device simulation in order to get accurate results.

It is worth noting that this result is the same as that obtained which includes phonon scattering in the Boltzmann equation [LUN 00]. This derivation generalizes the term used in the semi-classical approach to the quantum case, which gives a new depth to this scattering treatment, and shows how scattering may be included very easily in a Wigner function-based simulation.

1.4.2.3. Short-range interaction of an electron with ionized impurities in the Wigner formalism

To go beyond the continuum approach of doping impurities included in the Poisson equation, as described in 1.1.3, and to model the carrier relaxation due to Coulombic interactions, we here consider short-range scattering by ionized impurities.

For an assembly of dopant atoms j of position jr the short-range interaction

potential with electrons may be written in the form of a screened Coulomb potential

( )2 exp

4j

e iijj

eV

β

π ε−− −

=−∑r r

r r [1.119]

where ε and β are the dielectric constant and the screening factor, respectively. The corresponding Wigner potential gives

( )( )

( )( ) ( )

( )( ) ( )

2 2 2

3

2 22 233

2 2 23 2 2

1,42

2 2

1 242

1 14

j j

j j

j j

iw

j j j

ii i

j

i i

j

e e eV d ei

e e ee e di

e e ei

β β

β

π επ

π επ

ε βπ

′ ′− − − − + −

′− ⋅

′′−′′− ⋅− ⋅ − ⋅ −

− ⋅ − ⋅ −

⎛ ⎞⎜ ⎟⎜ ⎟′= −

′ ′⎜ ⎟− − + −⎜ ⎟⎝ ⎠

⎛ ⎞ ′′= −⎜ ⎟ ′′⎝ ⎠

⎛ ⎞= −⎜ ⎟⎝ ⎠ +

∑∫

∑ ∫

∑

r rr r r rk r

rk rk r r k r r

k r r k r r

r k rr rr r r r

rr

k [1.120]


which leads to the quantum evolution term

( ) ( )( )

( ) ( ) ( ) ( )

2

3 2 2

2 2

1, ,4

j j

w w

i i

j

eQ f d fi

e e

π ε β′ ′− − ⋅ − − ⋅ −

′ ′=′− +

⎛ ⎞× −⎜ ⎟⎝ ⎠

∫

∑ k k r r k k r r

r k k r kk k

[1.121]

With the same assumptions (external field is zero) and the same change of variables [1.82] as in previous sub-sections, [1.121] can be integrated in the form

( ) ( )( )

( ) ( )( ) ( ) ( )( )( )

2

30

2 22 2

, , , ,

1

4j j

t

w w

i t i t

j

ef t ic d t d f t ti

e e

π ε

β′ ′′ ′ ′ ′ ′′ ′ ′− − ⋅ − − ⋅ −

′ ′ ′′ ′ ′ ′′ ′= +

⎡ ⎤⎛ ⎞⎢ ⎥× −⎜ ⎟⎝ ⎠⎢ ⎥′− +⎣ ⎦

∫ ∫

∑ k k r r k k r r

r k k r k

k k

[1.122]

Still considering that the initial solution vanishes, substituting [1.122] into [1.120] leads to

( ) ( )( )

( ) ( ) ( ) ( )

( ) ( )( ) ( ) ( )( )

( )( ) ( )( )

4

2 6 20

2 2

2 2

1 12 22 2

, , ' ' , '', '

4 4

j j

j j

t

w w

i i

j

i t i t

eQ f t d t d d f t t

e e

e e

π ε

β β

′ ′− − ⋅ − − ⋅ −

′ ′′ ′ ′ ′ ′′ ′ ′− − ⋅ − − ⋅ −

− −

′ ′ ′′= −

⎡⎛ ⎞× −⎜ ⎟⎢⎝ ⎠⎣

⎛ ⎞× −⎜ ⎟⎝ ⎠

⎤′ ′ ′′× − + − + ⎥⎦

∫ ∫ ∫

∑ k k r r k k r r

k k r r k k r r

r k k k r k

k k k k

[1.123]

By developing the product of exponential functions, the non-cross terms give

( )( ) ( )( )( ) ( ) ( ) ( )( )

1 12 22 21

2 2

4 4

j ji i t

j

S

e e cc

β β− −

′ ′ ′′ ′ ′− − ⋅ − − − ⋅ −

′ ′ ′′= − + − +

× +∑ k k r r k k r r

k k k k


( )( ) ( )( )( ) ( ) ( ) ( )*

1 12 22 21

2 22

4 4

ji t t ii m

j

S

e e e

β β− −

′⎛ ⎞′ ′′ ′− − ⋅ − −⎜ ⎟ ′′− ⋅′− − ⋅ ⎝ ⎠

′ ′ ′′= − + − +

× ∑kk k r k k rk k r

k k k k

[1.124]

If the number of doping atoms in density DN is assumed to be large enough the discrete sum in [1.123] can be replaced by an integral that takes the form

( ) ( ) ( ) ( )( )2 2 32 2j ji iD j D

je N d e N π δ′′ ′′− ⋅ − ⋅ ′′≈ = −∑ ∫

k k r k k rr k k [1.125]

and then,

( )( ) ( )( ) ( )

( ) ( )( )*

1 12 2 22 21

23

4 4 i

i t tm

D

S e

e N

β β

π δ

− − ′− − ⋅

′⎛ ⎞′ ′′ ′− − − −⎜ ⎟⎝ ⎠

′ ′ ′′≈ − + − +

′′× −

k k r

kk k r

k k k k

k k

, i.e.

( )( ) ( )

( )*

2 '2 ' '3

1 2 24

4

i t tm

DS e Nπ δβ

⎛ ⎞− − ⋅ − −⎜ ⎟⎝ ⎠

⎡ ⎤′′⎢ ⎥≈ −

⎢ ⎥′− +⎣ ⎦

kk kk k

k k [1.126]

Similarly, the cross terms of the product of exponential functions in [1.123] may be written as

( )

( ) ( ) ( )( )

*2 2 '

432 2 2

1 2i t t

mDS e N ccπ δ

β

′′+⎛ ⎞′′− ⋅ − −⎜ ⎟⎝ ⎠

⎡ ⎤′ ′′⎢ ⎥≈ − + +

⎢ ⎥′′− +⎣ ⎦

k kk k

k k kk k

[1.127]

Substituting [1.126] and [1.127] into [1.123] gives


( ) ( )( ) ( )

( )( )( ) ( ) ( )

( )( )

*

2 2*

4 2 '

2 230

22 2

22 2 2

, , , ,

4

, ,

t i t tmD

w w

i t tmw

e NQ f t d t d f t e

cc

d f t e cc

επ

β

β

′⎛ ⎞′− − ⋅ − −⎜ ⎟⎝ ⎠

−

−′′ ′− −

⎧⎪′ ′ ′= − ⎨⎪⎩

′× − + +

⎫⎪′′ ′′ ′ ′′− − + + ⎬⎪⎭

∫ ∫

∫

kk k

k k

r k k r k

k k

k r k k k

[1.128]

The change of variable 2 ′ ′′= +k k k in the first integral of [1.128] leads to

( )( )

( )( ) ( )( ) ( )

( )( ) ( )

2 2*

2 2*

4

32 20

1222 2 4

124

, ,2

, ,

, , '

tD

w

i t tm

w

i t tm

w

e NQ f t d t

d f t e

f t e cc

επ

β⎛ ⎞′′ ′− −− ⎜ ⎟⎝ ⎠

⎛ ⎞′′ ′− −⎜ ⎟⎝ ⎠

′= −

⎧ ⎡⎪ ⎢′′ ′′ ′× − +⎨ ⎢⎪ ⎢⎣⎩

⎫⎤⎪⎥′′− + ⎬⎥ ⎪⎥⎦ ⎭

∫

∫k k

k k

r k

k k k r k

r k

[1.129]

In the limit of fast collisions (see previous sub-sections) we finally find

( )( )( )

( ) ( )( ) ( ) ( ) }

4 22 2* 2 2

( , , )2

, , , ,

Dw

w w

e Nf t dt m

E E f t f t

βεπ

δ

−⎧∂ ∂⎡ ⎤ ′′ ′′+ = × − +⎨⎢ ⎥∂ ∂⎣ ⎦ ⎩

′′ ′′× − −⎡ ⎤⎣ ⎦

∫k r k k k k

r

k k r k r k

[1.130]

This is exactly the same equation as that commonly used to model the electron/ionized impurity scattering in the Boltzmann equation (see e.g. [RID 99] equation (3.1.12)). Once again the Wigner function allows scattering to be modeled in an intuitive and familiar way that is ideal for electron devices.

1.4.2.4. Wigner-Boltzmann equation

From the three previous sections, one can conclude that, under some conditions, the collision term widely used in the Boltzmann transport equation (BTE) can also be used in the Wigner transport equation (WTE). It is a strong result and one of the main advantages of the Wigner function with regard to device modeling. All the


knowledge acquired in the past in the treatment of scattering in semi-classical transport can still be reused for quantum transport in the Wigner formalism. It makes it possible to study new problems such as scattering-induced decoherence and the transition from quantum to semi-classical transport regimes. The Wigner transport equation including the Boltzmann collision term is usually called the Wigner-Boltzmann transport equation (WBTE) [NED 04]. It gives, finally

( ) ( )

( ) ( )*

1, ,

, ,

w w slow w

w w

f t f t V ft m

Q f t C f t

∂ , + ⋅∇ , − ∇ ⋅∇ =∂

, + ,

r kr k k r k

r k r k [1.131]

where the Boltzmann collision term derives from the transition probabilities per unit of time ( ), 'is k k of each scattering process, calculated in the first order perturbation theory of the Fermi golden rule [JAC 89]

( ) ( )( , , ) , ( , , ) , ( , , )w i w i wi

C f t d s f t s f t′ ′ ′ ′= −⎡ ⎤⎣ ⎦∑∫r k k k k r k k k r k [1.132]

The quantum evolution term due to the potential

( ) ( ) ( ), , ,w w wQ f t d V t f t′ ′ ′, = , − ,∫r k k r k k r k [1.133]

where the non-local effects of the potential ( )V r appears in the Wigner potential, defined as

( )( )31

2 22i

wV d e V Vi π

′− ⋅ ′ ′⎡ ⎤⎛ ⎞ ⎛ ⎞′, = + − −⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦∫ k r r rr k r r r . [1.134]

Chapters 2 and 3 will focus on the solution of this formulation of the Wigner equation for application on real devices.

1.4.2.5. Modeling of scattering in the Green’s function formalism

The modeling of scattering in the Green’s function formalism is a very difficult problem. It is still a research topic, although considerable progress has been reported in recent years. It is usually included in self-energies, in a way similar to those we introduced to model contacts. Although this has roots in advanced quantum mechanisms, it can be understood intuitively using our previous discussions.

Scattering tends to thermalize electrons, i.e. change their momentum and energy towards thermal equilibrium. A very simplistic way to model the impact of


scattering can be to imagine fictitious contacts. Everywhere in the device we imagine contacts that absorb electrons, and re-emit the same amount of electrons, but thermalized. Such contacts are called “Büttker probes” and constitute a very simplistic model (the details of scattering physics are forgotten) [VEN 03]. They are, however, relatively natural to implement in a non-equilibrium Green’s function (NEGF) solver and can introduce some qualitative effects of scattering. We just need to introduce self-energies corresponding to the fictitious contacts, consistently with the derivation seen in section 1.4.1.2. The numerical value of these self-energies can simply be ( )2i τ− , where τ would be a relaxation time associated with the thermalization of electrons. It is typically chosen with regards to the electron electrical mobility μ so that *e mμ τ= .

The only difficulty is the choice of appropriate Fermi levels for the virtual contacts (this is essential for the Green’s functions computation). They have to be chosen such that all the Büttiker probes absorb as many electrons as they emit. Different techniques to achieve this goal have been proposed. They usually require iterative processes that unfortunately heavily burden simulation times with regards to ballistic simulation [REN 01], [VEN 03].

The Büttiker probe model has been widely used in first attempts to incorporate scattering in NEGF calculations. It is still a reasonable choice in contexts where scattering plays a relatively weak role, or where we only want “some scattering” in a simulation, without interest in its mechanism. However, in true electron devices, as has been shown by decades of device physics, the details of scattering play an essential role. This leads to the development of more accurate techniques for its inclusion in NEGF.

It is possible to understand how scattering may be modeled with NEGF in a more physical way with the previous derivations. It is possible to repeat the same process we performed in section 1.4.2.2.1 (for Wigner functions) with Green’s function: we define a super-system with electrons and phonons and reduce it to a subsystem with electrons only. In many ways, this is similar to the calculation we did in section 1.4.1.2 to go from a system including contacts and device to a subsystem with device only, and should also lead to the definition of self-energies. This process is well described in a recent book [DAT 05]. Unfortunately (and similarly to the case of the Wigner function), strong approximations are needed to make the process manageable.

Another currently used approach consists of deriving the self-energies expressions using equations from the many-body theory. It may, in principle, model scattering to an arbitrary precision (thanks to the many-body theory power) and is thus an extremely exciting research direction, which could make use of decades of


research on this theory. Unfortunately, once again, extremely strong approximations are needed due to the complexity of the original equations. However, if we manage to suppress some approximations, extremely good models could be derived and some new ideas may thus emerge from this approach.

Much work on devices now uses self-energy expressions beyond the Büttiker probe approach and progress is frequently being reported [SVI 02], [JIN 06], [DO 07a], [KOS 07a]. Description of the self-energies expressions is beyond the scope of this book which focuses on the Wigner formalism. However, readers are invited to find numerous details in the references. In the future, as the formalism matures, self-energies could be developed for all kind of phenomena occurring in nanodevices.

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The Wigner Monte Carlo Method for Nanoelectronic Devices (Querlioz/The Wigner Monte Carlo Method for...

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