Coherent electronic transport in nanostructures and beyond

Andrea Ferretti

INFMINFM Natl. Res. Center Natl. Res. Center SS33

University of Modena and Reggio EmiliaUniversity of Modena and Reggio Emilia

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 betweenab 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 method

A. 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

DFTDFT

Conductor (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 localized Wannier Functions computation.

• Transport properties within a matrix GF’s Landauer approach.

• GNU-GPL distributed

A. Calzolari et al., PRB 69, 035108 (2004).

Zigzag (5,0) carbon nanotube with 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 effectCoulomb 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

DFTDFT

Conductor (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

C

I

TC

I T

Transport components

Correlated Pt chain

3 correlated atoms

Nanotubes: Co impurity

EXP: 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 conductance

Importance of finite QP lifetimes

Computational

DFT calc. Transport calc.

PW basisWannier functions basis

WFs determination

Stability issues with hundreds of WFs

• Existing code optimization• PAW / USPP implementation• Variational functional redefinition,

minimization procedure

Strategies:

Molecular nanostructures

J. 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 coherent description

Pt AuAu

Pt@Au chain

Tra

nsm

itta

nce

3 correlated atoms

Interface effects

do not suppress

correlation

Au chain

Mean field Pt@Au

Correlated Pt@Au

Transport: problem definition

L = left lead

C = conductor

R = right lead

Hypotheses:Hypotheses:

LL CC RR

Definitions:Definitions:

Using a localized basis set:

Leads are non-interacting

The problem is stationary

Operators in block matrix form

Coupling to the leads

LL CC RRLeads self-energiesLeads self-energies

• 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 xxxxxxx

Re ω

Im ω

Advanced 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

Contour-ordered perturbation theory:Contour-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