Transport Calculations with Transport Calculations with TranSIESTATranSIESTA

Pablo OrdejónPablo OrdejónInstituto de Ciencia de Materiales de Barcelona - CSIC, SpainInstituto de Ciencia de Materiales de Barcelona - CSIC, Spain

M. Brandbyge, K. Stokbro and J. TaylorMikroelectronik Centret - Technical Univ. Denmark

Jose L. Mozos and Frederico D. NovaesInstituto de Ciencia de Materiales de Barcelona - CSIC, Spain

The SIESTA teamE. Anglada, E. Artacho, A. García, J. Gale. J. Junquera, D. Sánchez-Portal, J. M. Soler, ....

OutlineOutline

1. Electronic transport in the nanoscale: Basic theory

2. Modeling: the challenges and our approach:

• The problem at equilibrium (zero voltage)

• Non-equilibrium (finite voltage)

3. Practicalities

11. Electronic Transport in the . Electronic Transport in the Nanoscale: Basic TheoryNanoscale: Basic Theory

• Scattering in nano-scale systems:

– electron-electron interactions

– phonons

– impurities, defects– elastic scattering by the potential of the contact

d

L d m

RLd

• Semiclassical theory breaks down – QM solution needed

• Landauer formulation: Conductance as transmission probability

S. Datta, Electronic transport in mesoscopic systems (Cambridge)

Narrow constriction (meso-Narrow constriction (meso-nanoscopic)nanoscopic)

E

kx

Transversal confinement QUANTIZATION

2mk

)(kE2x

2

mxm

xik

mxe)(yy)(x,

Landauer formulation - no Landauer formulation - no scatteringscattering

bands

ig i),(v)(2eI kkk fd

kΩ12.9h

2eG

2

O

eVδμ

0GG N NVI

bandsbands

h2e2

E

k

BZ)ENV Fbands

(h

2e2

QUANTUM OF CONDUCTANCE

Landauer formulation - Landauer formulation - scatteringscattering

• Transmission probability of an incoming electron at energy :

• Current:

• Perfect conductance (one channel): T=1

εttTrεT †

transmission matrix:

εTεfεfdεh

2eI RL

2

I

eV

h2e

G2

O

inout ψtψ

22. Modeling: . Modeling: The Challenges and Our The Challenges and Our

ApproachApproach

• Model the molecule-electrode system from first principles:

No parameters fitted to the particular system DFT

• Model a molecule coupled to bulk (infinite) electrodes

• Electrons out of equilibrium (do not follow the thermal Fermi occupation)

• Include finite bias voltage/current and determine the potential profile

• Calculate the conductance (quantum transmission through the molecule)

• Determine geometry: Relax the atomic positions to an energy minimum

Restriction: Ballistic Restriction: Ballistic conductionconduction

• Effects not described: inelastic scattering – electron-electron interactions (Coulomb blockade)– phonons (current-induced phonon excitations)

RLd

• Ballistic conduction: consider only the scattering of the incoming electrons by the potential created by the contact

• Two terminal devices (three terminals in progress)

First Principles: DFTFirst Principles: DFT

• Many interacting-electrons problem mapped to a one particle problem in an external effective potential (Hohemberg-Kohn, Kohn-Sham)

• Charge density as basic variable:

• Self-consistency:

• Ground state theory: VXC

XCHpsexteff VVVVV

)]([ρV V effeff r

ρVeff

SSIESTAIESTA

• Self-consistent DFT code (LDA, GGA, LSD)• Pseudopotentials (Kleinman-Bylander)• LCAO approximation:

Basis set: Confined Numerical Atomic Orbitals

Sankey’s “fireballs”

• Order-N methodology (in the calculation and the solution of the DFT Hamiltonian)

http://www.uam.es/siesta

rμφ

Soler, Artacho, Gale, García, Junquera, Ordejón and Sánchez-PortalJ. Phys.: Cond. Matt. 14, 2745 (2002)

TTRANRANSSIESTAIESTA

Brandbyge, Mozos, Ordejón, Taylor and StokbroPhys. Rev. B. 65, 165401 (2002)

Mozos, Ordejón, Brandbyge, Taylor and StokbroNanotechnology 13, 346 (2002)

Implementation of non-equilibrium electronic transport in SIESTA

• Atomistic description (both contact and electrodes)

• Infinite electrodes• Electrons out of equilibrium• Include finite bias and determine the potential profile

• Calculates the conductance (both linear and non-linear) • Forces and geometry

The problem at EquilibriumThe problem at Equilibrium(Zero Bias)(Zero Bias)

Challenge: Coupling the finite contact to infinite electrodes

Solution:

Green’s Functions

iδεG Imπ1

ερ

H-zzG 1

rrr νμμν

FFμν φφμεnερdερ

μνD

Setup (zero Setup (zero bias)bias)

CRL

Contact: • Contains the molecule, and part of the Right and Left electrodes• Sufficiently large to include the screening

1

R

RRRC

CRCCL

LCLL

L

...-V

-VH-ε-V

-VH-ε-V

-VH-ε-V

-V...

1HεεG

Solution in finite system:

( ) = Selfenergies. Can be obtained from the bulk Greens functions

Lopez-Sancho et al. J. Phys. F 14, 1205 (1984)

1

RRRC

CRCCL

LCLL

HV

VHV

VH

G

Σε

ε

Σε

εBB R

CL

Calculations (zero Calculations (zero bias):bias):• Bulk Greens functions and self-energies (unit cell calculation)

• Hamiltonian of the Contact region:

• Solution of GF’s equations (r)

• Landauer-Büttiker: transmission probability:

PBC

SCF

1/2L

1/2R εΣImεGεΣImεt

εttTrεT †

The problem at Non-The problem at Non-EquilibriumEquilibrium(Finite Bias)(Finite Bias)

2 additional problems:

• Non-equilibrium situation: – current flow– two different chemical potentials

• Electrostatic potential with boundary conditions

e-

L

R

CRL

Non-equilibrium formulation:Non-equilibrium formulation:

• Scattering states (from the left)Lippmann-Schwinger Eqs.:

• Non-equilibrium Density Matrix:

oLo

o ψVεGψψ lll

RFRμνLF

Lμνμν μεnερμεnερdεD

μνt

LL

μν δiεGδiεΣImδiεGπ1

ερ

Electrostatic Electrostatic PotentialPotential

AuAu

12 au

Given (r), VH(r) is determined except up to a linear term:

( r): particular solution of Poisson’s equation

a and b: determined imposing BC: the shift V between electrodes

• (r) computed using FFT’s

• Linear term:

brarr HV

+V/2

-V/2

2L

zLV

33. . TranSIESTA PracticalitiesTranSIESTA Practicalities

3 Step process:

1. SIESTA calculation of the bulk electrodes, to get H, , and Self-energies

2. SIESTA calculation for the open system

• reads the electrode data

• builds H from

• solves the open problem using Green’s Functions (TranSIESTA)

• builds new

3. Postprocessing: compute T(E), I, ...

Supercell - PBC

• H, DM fixed to bulk in L and R

• DM computed in C from Green’s functions

• HC, VLC and VCR computed in a supercell approach (with potential ramp)

• B (buffer) does not enter directly in the calculation (only in the SC calc. for VHartree)

1

RRRC

CRCCL

LCLL

HV

VHV

VH

G

Σε

ε

Σε

ε

Contour Contour integrationintegration

Contour Contour IntegrationIntegration

SolutionMethod Transiesta

# GENGF OPTIONS

TS.ComplexContour.Emin -3.0 Ry

TS.ComplexContour.NPoles 6

TS.ComplexContour.NCircle 20

TS.ComplexContour.NLine 3

TS.RealContour.Emin -3.0 Ry

TS.RealContour.Emax 2.d0 Ry

TS.TBT.Npoints 100

# TS OPTIONS

TS.Voltage 1.000000 eV

TS.UseBulkInElectrodes .True.

TS.BufferAtomsLeft 0

TS.BufferAtomsRight 0

# TBT OPTIONS

TS.TBT.Emin -5.5 eV

TS.TBT.Emax +0.5 eV

TS.TBT.NPoints 100

TS.TBT.NEigen 3