Coherent electronic transport in nanostructures and beyond
Andrea Ferretti
INFMINFM Natl. Res. Center Natl. Res. Center SS33University of Modena and ReggioUniversity of Modena and Reggio EmiliaEmilia
Acknowledgments
People People @@ SS33::Andrea FerrettiArrigo CalzolariCarlo Cavazzoni
Rosa Di FeliceFranca ManghiElisa Molinari
External External CollaboratorsCollaborators::
Marco Buongiorno Nardelli (NCSU, US)Nicola Marzari (MIT, US)
Marilia J. Caldas (USP, Brazil)
National Research Center on nanoStructures and bioSystems at Surfaces
via Campi 213/A, 41100 Modena, Italy
Outline
The method
Motivations
Ab initio electronic transport from max. loc. Wannier Functions:
• Development• Implementation (WanT code)• Application to nanoscale systems
Inclusion of correlation in transport
Application to short range e-e interaction regime
♦ Novel systems for electronic devices (nanotubes, atomic chains, molecular systems,…)
C. Dekker et Al., Nature 429, 389 (2004)
H. Ohnishi et Al., Nature, 395, 780 (1998)
M. Reed et al., Science (1997)
♦ Semiclassical transport theory breaks down
♦ Full quantum mechanical approach Landauer Theory
D.Porath et al., Nature (2000)
Motivations
Ballistic transport
Conductance as transmission through a
nano-constriction
Landauer Formula
♦Ballistic transport: exclusion of non-coherent effects (e.g. dissipative scattering or e-e correlation).
♦Quantum conductance: depends on the local properties of the conductor (transmission - scattering) and the distribution function of the reservoirs
♦Transmittance from real-space Green’s functions techniques
♦Need for a localized basis set
Fisher & Lee formulation
The “WanT” approach♦ Create a connection between♦ab initio description of the electronic structure by means of state-of-the-art DFT- plane wave calculations
♦Real space Green’s function techniques for the calculation of quantum conductance.
♦Idea: Unitary transformation of delocalized Bloch-states into
Maximally localized Wannier functions
WanT methodA. Calzolari, PRB 69, 035108 (2004).
Wannier functions (WFs): definition
* N. Marzari, and D. Vanderbilt, PRB 56, 12847 (1997).
Single band transformation:
Generalized transformation:
Maximally localized Wannier functions*
Non-uniqueness of WFs under gauge transformation U(k)
mn
Goal: Calculation of WFs with the narrowest spatial distribution
Wannier functions: localization
Spread functional
Maximal localization given by the minimization of the spread wrt U:
WF advantages:orthonormalitycompletnessminimal basis setadaptabilitydirect link to phys. prop.
WF disadvantages:no analytical form
computational costalgorithm stability
Flow diagram
DFTDFTConductor (supercell)Leads (principal layer)
WFsWFs
GFsGFs
QCQCAll quantities on Wannier basis
Quantum conductance
Zero bias Linear response
www.wannier-transport.org
Features:
WanT Code
• Input from PW-PP, DFT codes.
• Maximally localizedWannier Functions computation.
• Transport propertieswithin a matrix GF’sLandauer approach.
• GNU-GPL distributed
A. Calzolari et al., PRB 69, 035108 (2004).
Zigzag (5,0) carbon nanotubewith a substitutional Si defect
♦Si polarizes the WF’s in its vicinity affecting the electronic and transport properties of the system
♦ General reduction of conductance due to the backscattering at the defective site
♦ Characteristic features (dips) of conductance of nanotubes with defects
Si
Nanotubes: Si defect
Beyond the coherent regime
♦Goal: Ab initio description of electronic transport in the presence of strong electron-electron coupling, from atomistic point of view.*
♦Landauer formalism breaks down:
Need for a novel theoretical treatment
♦ Evidences of strong e-e correlation effects:♦Kondo effect♦Coulomb blockade
T.W. Odom et al., Science 290, 1549 (2000)J. Park et al., Nature 417, 722 (2002)W. Liang et al., Nature 417, 725 (2002)
J. Park et. al., Nature 2002
* A. Ferretti et al., PRL 94, 116802 (2005)
Correlated transport in nanojunctions
Effective transmittance
Correlated transport Landauer + e-e correction to the Green’s functions
Formalism
A. Ferretti et al., PRL 94, 116802 (2005)
Generalized Landauer-like formula
• Conductor GF’s are interacting• Lambda is also given by:
Re-formulate the theory from more general conditions: Meir-Wingreen approach
Correlation effects:• Three Body Scattering (3BS) method*• Describes the strong short range electron-electron interaction• Based on a configuration interaction scheme up to 3 interacting
bodies (1particle + 1e-h pair) of the generalized Hubbard Hamiltonian
• Hubbard U is an adjustable parameter
* F. Manghi, V. Bellini and K. Arcangeli, PRB 56, 7159 (1997).
Implementation
Flow diagram
DFTDFTConductor (supercell)Leads (principal layer)
3BS3BS WFsWFs
Atomic pdos
GFsGFsΣnn’k(ω)
QCQCAll quantities on Wannier basis
Quantum conductance
Mean fieldCorrelation
NC
quasi-particle finite lifetimes
Pt PtPt
coherent comp.incoherent comp.total
CIT
CI T
Transport components
Correlated Pt chain
3 correlated atoms
Nanotubes: Co impurityEXP: Cobalt impurities adsorbed on metallic CNT
• Transition metal (TM) often present as catalyzers
• Interplay between CNT and TM physics
• changes on electronic and transport properties
T.W. Odom et al.,Science 290, 1549 (2000)
THEO: Co @ 5,0 CTN
Work in progress
Conclusions and outlook
Development of the freely available WanTWanT (Wannier-Transport) code.
Inclusion of electron correlation (incoherent, non-dissipative model), in the strong short-range regime (by 3BS method).
Application to Pt chains:renormalization effects on quantum transmittance and conductanceImportance of finite QP lifetimes
Computational
DFT calc. Transport calc.PW basis Wannier functions
basisWFs determination
Stability issues with hundreds of WFs
• Existing code optimization• PAW / USPP implementation• Variational functional redefinition,
minimization procedure
Strategies:
Molecular nanostructuresJ. Park et al., Nature 417, 722 (2002)
• Co coordination complex• Prototype for correlation effects
in molecular electronics• Computationally challenging
Free molecule
Device configuration
Work in progress
Pt PtPt
3 correlated atoms
Correlation within LDA+U
Static coherentdescription
Pt AuAu
Pt@Au chain
Tran
smitt
ance
3 correlated atoms
Interface effects do not suppress
correlation
Au chainMean field Pt@AuCorrelated Pt@Au
Transport: problem definition
L = left leadC = conductorR = right lead
Hypotheses:Hypotheses:
LL CC RR
Definitions:Definitions:Using a localizedbasis set:
Leads are non-interactingThe problem is stationary
Operators in block matrix form
Coupling to the leadsLL CC RRLeads selfLeads self--energiesenergies
• From the block inversion of the hamiltonian
• Allows to treat the coupling to the leads
• Computational interest
Retarted, Advanced SE
Coupling functions
♦Exact expression from Meir & Wingreen *
Expression for the current
In the interacting case
Ng-ansatz for G>,<
N. Sergueev et al., PRB 65, 165303 (2002)
Which results in
Equilibrium Green Functions
Time ordered
Various definitionsVarious definitions::Allows perturbation theory
(Wick’s theorem)
Correlation functions Direct access to observable expectation values
Retarded, Advanced Simple analitycal structure and spectral analisys
Analitycal properties
Time ordered GF
xxxxxxxxxxxxxx
Re ω
Im ω
Retarded GF
xxxxxxx xxxxxxxRe ω
Im ωAdvanced GF
xxxxxxx xxxxxxxRe ω
Im ω
Fermi Energy = 0.0
Just one indipendent GFJust one indipendent GF
Equilibrium Green Functions
General identity
Spectral function
Fluctuation-dissipation th.
Gr, Ga, G<, G> are enough to evaluate all the GF’s and are connected by physical relations
Non-Equilibrium GF’s
ContourContour--ordered perturbation theory:ordered perturbation theory:
Only the identity holds(no FD theorem)
Gr, Ga, G<, G> are all involved in the PT
• Electric fields (TD laser pulses)• Coupling to contacts at different chemical
potentials
2 of them are indipendentOrdering contour
Non-Equilibrium GF’s
Dyson Equation
Two Equations of MotionTwo Equations of Motion
Keldysh Equation
Computing the (coupled) Gr, G< allows for the evaluation of transport properties
In the time-indipendent limit
Gr, G< coupled via the self-energies
