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WanT - tutorial

Quantum ESPRESSO Workshop

June 25-29, 2012The Pennsylvania State University

University Park, PA

Marco Buongiorno NardelliDepartment of Physics and Department of Chemistry

University of North Texasand

Oak Ridge National Laboratory

an integrated approach to ab initio electron transport from maximally localized Wannier functions

www.wannier-transport.org

The WanT Project is devoted to the development of an original method for the evaluation of the electronic transport in nanostructures, from a fully first principles point of view.

This is a multi-step method based on: (a) ab initio, DFT, pseudopotential,

plane wave calculations of the electronic structure of the system;

(b) calculation of maximally localized Wannier functions (WF's)

(c) calculation of coherent transport from the Landauer formula in the lattice Green's functions scheme.

WanT code

WanT codean integrated approach to ab initio electron transport

from maximally localized Wannier functions

The WanT package operates, in principles, as a simple post-processing of any standard electronic structure code.

The WanT code is part of the Quantum-ESPRESSO distribution

WanT calculations will provide the user with:

- Calculation of Maximally localized Wannier Functions (MLWFs)- Calculation of centers and spreads of MLWFs- Quantum conductance and I-V spectra for a lead-conductor-lead geometry- Density of states spectrum in the conductor region

Credits

University of North Texas.

Credits

Outline

Quantum electron transport in nanostructureLandauer Formalism

Wannier functions for electronic structure calculations

definitions and problemsWanT - method implementationanalysis of chemical bonding

WanT method implementation

transport 3D system

Quantum Electron transport

Introduction

The standard approaches to electron transport in semiconductors are based

on the semiclassical Boltzman's theory.

The dynamics of the carriers and the response to external fields follow

the classical equations of motion, whereas the scattering events are

included in a perturbative approach, via the quantum mechanical

Fermi's Golden Rule.

The semiclassical description is unsuitable for nanodevices where the tiny

size requires a fully quantum mechanical theory for a reliable quantitative

treatment.

From micro- to nano-electronics

Deviations from Ohm’s law

General considerations

Quantum Electronic transport

Characteristic lengths

A conductor shows ohmic behavior if its dimensions are much larger than

each of the three characteristic length scales:

de Broglie wavelenght (le) related to kinetic energy of the electron

mean free path (lel, inel ) distance before initial momentun is destroyed

phase relaxation length (lφ) distance before initial phase is destroyed

If the dimensions of the conductor are smaller or equal to one of the

characteristic length the semiclassical Boltzmann approach breaks down


Coherent transport Given a generic conductor of dimension D, the electronic transport is said to

be

D < lφ & D> lel Coherent only elastic scattering (no dissipation)

D < lφ & D < lel Ballistic (no scattering)

The resistance is originated by the connection with the external contacts

The conduction properties depend on the coherence effect of the electronic

wavefunction (interference). The transport can be solved as a scattering

problem starting from the Schrödinger equation

The current that flows in a conductor is related to the probability that the

charge carrier may be transmitted throughout the conductor

The current that flows in a conductor is related to the probability that the

charge carrier may be transmitted throughout the conductor


Landauer approach

The Landauer approach provides a convenient and general scheme for the

theoretical description of electron transport at the nanoscale, in the framework

of scattering theory.

Hypotheses:

• Coherent transport

• Low temperature (ϑ0)

• Conductor connected to two external reflectionless leads, that act as

electron (hole) reservoir

• Each contact is fully described by its Fermi level


Landauer approach


Landauer approach

Let’s consider two semi-infinite one-dimensional leads (L, R) connected to one

point (C).

The expression for the current from the right through this point is

Where v is the velocity of the charge carrier and n(v) the charge density per unit

length and per unit velocity, with velocity between v and v+dv

L C R


Landauer approach

Using a wavevector representation we get

with 1/π the one-dimensional DOS per single spin in the wavevector interval k

and k+dk, and f(E) is the Fermi distribution at the actual temperature ϑ.

The current from one electrode is then:


Landauer approach

Assuming that the electrostatic potential of the left lead is zero, the total

current from both contacts for a given bias Φ is

Using the limits

We obtain the ideal quantum of conductance g0


Landauer approach

If the central point is replaced by a generic elastic scatterer, characterized by

its transmission and reflection functions

scatterer


Landauer approach

The expressions for (spin-unpolarized) current and conductance are modified

into:

LANDAUER FORMULALANDAUER FORMULA


Landauer approach

If the leads have many accessible transverse mode, the total contribution to

the transmission function (or transmittance) is given by:

The transmission coefficients are simply related to the scattering matrix Sij by

the relation:


Landauer approach From here on we focus on the ZERO BIAS REGIME with the exclusion of

non-coherent effects (e.g. dissipative scattering or e-e correlation).

The quantity that characterize the transport of a nano-restriction is the quantum-conductance

I-V characteristics may be obtained for low external bias in the linear regime.

Finite external bias may formally included within the full non-equilibrium Green’s function (NEGF) techniques [Datta, “Electronic transport in mesoscopic systems” Cambridge 1997] At present NOT IMPLEMENTED IN WanT CODE Critical problems in evaluation of current from first principles using NEGF


How to calculate transmittance

Instead of working in the basis of the exact solution of the total Hamiltonian

(i.e. using scattering states {i}), it is convenient use a new set of states {r}, {l},

{c} LOCALIZED IN REAL SPACE on the right and left electrode and on the

conductor.

We re-write the Hamiltonian as H = H0 + V

Where H0 is the sum the single hamiltonians of the electrode and of the

conductor, and V is the interaction term among them. The set {r}, {l}, {c} are

eigenstates of H0



We can generally define a TRANSMISSION OPERATOR as

G being the RETARDED GREEN FUNCTION of the total hamiltonian.

The direct coupling between L-R electrodes Vlr is usually neglected

the relevant matrix elements of the transmission operator are:



If we substitute in the expression for conductance:

couplingΓ function

Fisher – Lee formulaFisher – Lee formula


Lattice Green’s functions For an open system, exploiting the real space description of the system, we

can partition the total Green’s function into submatrices that correspond to the individual subsystems

conductor

semi-infinite leads

conductor-lead coupling

ε-HL,R

ε-HC

hLC,CR


Lattice Green’s functions

• From here, we can write the expression for the total GC as

where are the self-energy terms due to

the semi-infinite leads and gR,L are the Green’s functions of the semi-infinite

leads.

• The self energy terms can be viewed as effective Hamiltonians that arise from

the coupling of the conductor with the leads:


Lead self-energies

An open system, made of a conductor connected to two semi-infinite lead,

may be re-casted in a finite system by including non-hermitian self-energy

terms

conductorLead Lead

Lead

conductor

Self-energy terms

The self-energy terms can be viewed as effective Hamiltonians that

account for the coupling of the conductor with the leads.

The self-energy terms can be viewed as effective Hamiltonians that

account for the coupling of the conductor with the leads.


Principal layers

Any solid (or surface) can be viewed as an infinite (semi-infinite in the case of

surfaces) stack of principal layers with nearest-neighbor interactions [(Lee

and Joannopoulos, PRB 23, 4988 (1981)].

This corresponds to transform the original system into a linear chain of

principal layers.

For a lead-conductor-lead system, the conductor can be considered as one

principal layer sandwiched between two semi-infinite stacks of principal

layers.


Principal layers


Principal layers

• Express the Green’s function of an individual layer in terms of the Green’s function of the preceding or following one.

• Introduction of the transfer matrices

G10=TG00


Principal layers

In particular and can be written as:

where and are defined via a recursion formulas

and

For a detailed discussion see M. Buongiorno Nardelli, Phys. Rev. B, 60 , 7828 (1999)

The expressions for the self-energies can be deduced, using the formalism of

principal layers, as follows

where are the matrix elements of the Hamiltonian between the layer orbitals

of the left and the right leads, respectively, and and are the

appropriate transfer matrices, easy computable from the Hamiltonian matrix

elements via the interactive procedure outlined above.


Lead self-energies

Quantum electronic transport in nanostructuremodel calculation on a simple TB Hamitonianbulk conductance in linear chainstwo-terminal transport in nanojuctions

Practical examples

Outline





transport 3D system

-The electronic structure in periodic solids is conventionally described in terms of

extended Bloch functions (BFs)

- By virtue of the Bloch theorem, the Hamiltonian commutates with the

lattice-translation operator leading to a set of common set of

eigenstates (the Bloch states) for the Hilbert space.

- This allows to restricts the problem to one unit cell, and to recover the

properties of the infinite solid with an integral over the Brillouin zone (BZ), in

the reciprocal space.

- Wannier Functions (WFs) furnish an equivalent alternative in the real space

WFs for electronic structure calculations

From reciprocal to real space


Applications in solid state physicsPhysical problems:

- modern theory of bulk polarization

- development of linear scaling order-N and ab initio molecular dynamics approaches

- calculation of the quantum electron transport

- study of magnetic properties and strongly-correlated electrons

Physical systems:

- crystal and amorphous semiconductors

- ferroelectric and perovskites

- transition metals and metal-oxides

- photonic lattices

- high-pressure hydrogen and liquid water

- nanotubes, graphene and low-dimensional nanostructures

- hybrid interfaces.


Definitions & Properties

- A Wannier function , labeled by the Bravais lattice vector R is defined by

means of unitary transformation of the Bloch eigenfunction of the nth

band

- From the orthonormality properties of BFs basis set the orthonormality

and completeness of the corresponding WFs

WFs constitute a complete and orthonormal basis set for the same Hilbert

space spanned by the Bloch functions.

WFs constitute a complete and orthonormal basis set for the same Hilbert

space spanned by the Bloch functions.



- We rewrite the generic vector of the Hilbert space in real space and as

function of a finite mesh of N k-points as

- Any two WFs, for a given index n and different R1 and R2, are just translational

images of each other if we focus on the unitary cell R = 0



- A Bloch band is called ISOLATED if it does not become degenerate with any

other band anywhere in the BZ. A group of bands is said to form a COMPOSITE

GROUP if they are inter-connected by degeneracy, but are isolated from all the

other bands

For example the valence bandof insulators

Bandstructure of Si bulk



- For isolated bands, we define a WF for each band

- For composite bands, we define a set of GENERALIZED WANNIER

FUNCTIONS that span the same space as the composite set of Bloch

states.

- As the Wannier functions are linear combinations of Bloch functions with

different energies they do not represent a stationary solution of the

Hamiltonian

The WF's are not necessarily eigenstates of the Hamiltonian, but they may be

related to them by a unitary transformation

The WF's are not necessarily eigenstates of the Hamiltonian, but they may be

related to them by a unitary transformation


Fundamental drawback

- The major obstacle to the construction of the Wannier functions in practical

calculations is their NON-UNIQUENESS They are GAUGE

DEPENDENT

Infinite sets of WFs, with different properties, may be defined for the same

physical system.


Non-uniqueness

- For isolated bands the non-uniqueness arises from the freedom in the choice of

the phase factor of the electronic wave function, that is not assigned by the

Schrödinger equation.

- For composite group of bands additional complications arise from the

degeneracies among the energy bands in the Brillouin zone. This extends the

arbitrariness related to freedom of the phase factor to a gauge transformation

that mixes bands among themselves at each k-point of the BZ, without

changing the manifold, the total energy and the charge density of the

system.


Non-uniqueness

Starting from a set of Bloch functions , there are infinite sets of

Wannier Functions with different spatial characteristics, that are related by a

unitary transformation

A different gauge transformation does not translate into a simple change of

the overall phases of the WFs, but affects their shape, analytic behavior and

localization properties.

A different gauge transformation does not translate into a simple change of

the overall phases of the WFs, but affects their shape, analytic behavior and

localization properties.


Non-uniqueness

GOAL: Search for the particular unitary matrix that transforms a set of BFs

into a unique set of WFs with the highest spatial localization

MAXIMALLY LOCALIZED WANNIER FUNCTIONSMAXIMALLY LOCALIZED WANNIER FUNCTIONS

WanT is based on a specific localization algorithm proposed by Marzari and

Vanderbilt in 1997 [PRB 56, 12847, (1997)] and implemented in the code.

The formulation of the minimum-spread criterion extends the concepts of

localized molecular orbitals, proposed by Boys for molecules, to the solid state

case


Spread Operator

We define SPREAD OPERATOR W the sum over a selected group of bands of the

second moments of the WFs in the reference cell (R=0)

where

are the expectation values of the r and r2 operators respectively.

Wannier center


Localization condition

The value of the spread W depends on the choice of unitary matrices it is

possible to evolve any arbitrary set of until the minimum condition is satisfied.

At the minimum, we obtain the unique matrix that transform the first

principles into the maximally localized WFs :


Real-space representation

For numerical reasons it is convenient decompose the W functional as follows:

gauge invariant

off-diagonal component

band-diagonal component

band-off-diagonal component


Real-space representation

is gauge invariant it is invariant under any arbitrary unitary

transformation

of the Bloch orbitals

The minimization procedure corresponds to the minimization of the off-

diagnonal component

At the minimum, the elements are as small as possible, realizing

the best compromise in the simultaneous diagonalization, within the space of the

Bloch bands considered, of the three position operators x, y and z (which do not in

general commute when projected within this space).


Reciprocal-space representation

Following the expression proposed by Blount, the matrix elements of the position

operator between Wannier functions in the reciprocal space take the form

where is the periodic part of the Bloch function.

Reciprocal-space representation

If we restrict to the case of discrete k-point mesh calculations, we can use finite

differences in reciprocal space to evaluate the derivative /dW dU. For this

purpose we rewrite the operators r and r2 as:



Overlap matrix

Making the assumption that the BZ has been discretized into a uniform k-point

mesh, and letting b the vectors that connect a mesh point to its near neighbors, we

can define the overlap matrix between Bloch orbitals as

is the central quantity in this formalism, since we will express in its term

ALL the contribution of the to the localization functional


Overlap matrixWe can relate the overlap matrix to the expression of the differential

operators and we can express the expectation values of the

operators r and r2 as a function of

where are the weight of the b vectors and must satisfy the completeness

condition


Overlap matrix

From the expression of the the operators r and r2 we rewrite the different terms of

the spread functional as


Localization procedure

In order to obtain the minimum condition , we consider the first order change

in W arising from an infinitesimal transformation

Where is an infinitesimal antiunitary matrix

The gauge transformation rotates the wave functions into



After some algebra we obtain the final expression also for the gradient of the

spread functional G(k),in terms of overlap matrix M:

where



From the expression of the spread functional W and of its gradient G(k) in terms of

the overlap matrix, the minimum condition can be easily obtained via standard

steepest descent or conjugated gradient techniques in the reciprocal space,

while the transformation to real space is a post-processing step.

The minimization procedure is computationally inexpensive since it requires the

updating only of the unitary matrices (i.e. of the overlap) and NOT of the wave

function.


MLWFs: further properties The MLWFs are REAL, except for an overall phase factor.

The MLWFs are not truly localized, being instead artificially periodic with a

periodicity inversely proportional to the k-mesh spacing:

Even the case of Γ-sampling is encompassed by the above formulation


Entangled bands

The method described above works properly in the case either of isolated bands

or of groups of bands manifolds of bands entirely separated by an energy gap

from the others.

In many physical applications (e.g. in metals) the bands of interest are not

isolated, and one needs to compute WFs for a subset of energy bands that

are entangled or mixed with other bands.


Entangled bands

Consider, for example,

the separation of the five

d-bands of copper from

the s-band that crosses

them.

Image adapted from PRB 65 035109 (2001).

Since the unitary transformation mixes the energy bands at each k-point, the

choice of few of them from an entangled group may affect the localization

procedure, because it is unclear exactly which band to chose in those regions of

BZ where the bands of interest are hybridized with a few unwanted ones.


Disentangling procedure

The problem of entangled bands has been solved through the introduction of an

additional procedure, proposed by Souza, Marzari, and Vanderbilt in 2001 [PRB

65 035109 (2001)] that automatically extracts the best possible manifold of a

given dimension from the states falling in a predefined energy window.

This is the generalization to entangled or metallic cases of the maximally-

localized WF formulation.

By exploiting the same spread functional W and the same unitary

transformations

this method provides a set of optimal Bloch functions to be used in the

localization procedure described above.



Fix an energy window

that includes the N bands

of interest

At each k-point the

number Nk of bands that

fall in the energy

windows is equal or

greater than N

Image adapted from PRB 65 035109 (2001).

If Nk =N at eack k-point the manifold is isolated and there is nothing to do

Otherwise, we search the N-dimensional Hilbert space that minimize the

operator



is gauge invariant it is an intrinsic property of the band manifold

it heuristically measures the change of character of

the

states across the BZ

by minimizing we are selecting the proper N-dimensional Hilbert

subspace that changes as little as possible with k minimum spillage criterion


WanT flow diagram

DFT calculation Bloch Functions basis set ( N )

definition ofenergy window

Selection of a manifold of n band of interest( n ≤ N )

disentaglingprocedure

Minimization of

localization procedure Minimization of

real spacetrasformation

Maximally localized Wannier functionsbasis set ( n )

QE codepw.x

WanT codewannier.x

WanT codedisentangle.x


Analysis of the chemical bonding

Traditional chemistry is based on local concepts (Lewis like).

Covalently bonded materials are described in terms of bonds and electron

pairs, where the bonding properties are determined by its immediate

neighborhood.

The characteristics of the bond (distances, angles, strength, character, etc,..)

essentially depends on coordination number of each atom, while the second-

nearest neighbors and more distant atoms give only weaker contributions.


The standard electronic structure calculations typically do not provide a deep

insight into the localization properties of matter: the Bloch states, for

instance, being delocalized throughout the overall cell, describe the

electronic states of the overall crystal and not the single chemical bonds.

A set of MLWS, being localized, may give an insightful picture of the bonding

properties of the system



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

ALLOTROPIC CARBYNE CHAINSALLOTROPIC CARBYNE CHAINS

Linear chain of sp-hybridized carbon atoms* two possible forms - isomeric polyethynylene diylidene (polycumulene or cumulene)

- polyethynylene (polyyne)

Example*:

Cumulene form equidistant arrangement of C-atoms with double sp-bonds (= C = C =)n

Polyyne form dimerized linear chain with alternating single-triple sp-bonds (─C ≡ C ─)n

The calculated MLWFs allow us to investigate the effects of structural relaxation on the electronic properties of infinite carbyne chains.




cumulene form polyyne form

ALLOTROPIC CARBYNE CHAINSALLOTROPIC CARBYNE CHAINS

characterized by symmetric sp-bonds, uniformly distributed along the chain s states are localized in the middle of C=C bonds while p states are centered around single C-atoms.

Example*:

s orbitals are localized both on single C ─ C and on triple C ≡ C bonds, with a s state in the middle of each bond. The p orbitals are localized only on the C ≡ C bonds: two p orbitals in the middle of each triple bond, but no one around the single bonds.


Outline





transport 3D system


WanT implementation

By choosing the maximally-localized WFs representation, we provide

essentially an exact mapping of the ground state onto a minimal basis.

The accuracy of the results directly depends on having principal layers that do

not couple beyond next-neighbors, i.e. on having a well-localized basis.


Real-space hamiltonian

The Hamiltonian matrices HLR, HC, HCR that enter in the Landauer formula can

be formally obtained from the on site (H00) and coupling (H01) matrices between

principal layers.

In our formalism, and assuming a BZ sampling fine enough to eliminate the

interaction with the periodic images, we can simply compute these matrices

from the SAME unitary matrix obtained in the Wannier localization

procedure.



By definition of energy eigenvalues the Hamiltonian matrix

is diagonal in the basis of the Bloch eigenstates.

We can calculate the Hamiltonian matrix in the rotated basis

Next we Fourier transform Hrot(k) into a set of Nkp Bravais lattice vectors R

within a Wigner-Seitz supercell centered around R = 0, where Nkp derives from

the folding of the uniform mesh of k-points in the BZ



The real-space hamiltonian results to be

The term with R = 0 provides the on site matrix

and the term R = 1 provides the coupling matrix

These are the only ingredients required for the evaluation of the quantum

conductance.


Back to reciprocal space As a test of the accuracy of the WF transformation, we can compute back the

band structure of a system, starting from the Wannier-function Hamiltonian in

real space.

The Hamiltonian H(rot)(R) can be Fourier back-transformed in reciprocal space

for any arbitrary k-point

The resulting Hamiltonian matrix can then be diagonalized to find energy

eigenvalues and to recalculate the bandstructure.

The comparison between the original PW with the interpolated bandstructure

represents an important validation test, since it proves that the intermediate

transformations do not affect the accuracy of the first-principles PW

calculations.


WanT scheme of the methods

DFTConductor (supercell)Leads (principal layer)

1) Ab initio, DFT PLANE WAVEpseudopotential, calculations of theelectronic structure of the system.

Wannierfunctions

2) Real space transformation ofcalculated Bloch eigenstates intoMAXIMALLY LOCALIZED WANNIER FUNCTIONS and calculation of the HAMILTONIAN MATRIX on the WF basis set.

Green’sfunctions

QC

QUANTUM CONDUCTANCE

3) calculation of quantum conductance from the LANDAUERFORMULA in the LATTICE GREEN’S FUNCTIONS scheme.


Bulk-like transmittance

We consider a case in which leads and conductor are made of the same

material, and we compute the transmittace of the ideal and infinite

nanostructure. The corresponding conductance is given by the value of the

transmittace calculated at the Fermi energy.

In this case, it is not necessary to distinguish between conductor and lead

terms and the single layer H00 and the coupling H01 matrices are the only

necessary input


— Al — Al —

8 WFs from selected energy window – comparable to a TB with 2 atoms/cell and 4 orbitals per site.

Perfect agreement between original (grey dots) end interpolated (black lines) band structure WF localization procedure does not affect the accuracy of ab initio calculation.

Perfect agreement in conductance plots from different initial energy windows effect of the disentanglement procedure.

σ states localized on the bond, π states centered on the atoms.

Metallic behavior of the Al chain.

Good localization properties of the WF’s even in low dimensionality systems.

Perfect agreement between calculated band structure and quantum conductance: at a given energy, transmitting channels for charge mobility = number of bands.


Bulk-like transmittanceExample: infinite Al-chain


In the general case we need to compute the electronic structure and WFs for three different regions L, C, R. Very often one is interested in a situation where the leads are composed of the same material.

The conductor calculation should contain part of the leads in the simulation cell, in order to treat the interface from first principles.

The amount of lead layers to be included should be converged up to the local electronic structure of the bulk lead is reached at the edges of the supercell.

This convergence can be controlled taking a look at the hamiltonian matrix elements on Wannier states located in the lead region (e.g. nearest neighbor interactions).

This is a physical condition related to the need for a matching of different calculations and not to the peculiar use of WFs as a basis: nevertheless the smaller the WFs the more independent on the environment are the matrix elements, which leads to a faster convergence.

Two-terminal transmittance



Two-terminal transmittance

Si

L R

Zigzag (5,0) carbon nanotube in the presence of a substitutional Si defect

Example of two-terminal conductance calculation: leads = ideal nanotube, conductor = defective region.

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

Different conduction properties for isolated and periodically repeated defect

Different information from bandstructure and conductance plots in the presence of the leads importance of the proper inclusion of the leads in transport calculations.

DENSITY OF STATES

CONDUCTORLEAD LEAD

MODEL 3D SYSTEM Si bulk


Transport in 3D systemStandard Landauer theory describes truly one-dimensional systems BUT it is inadequate in the treatment of 3D system.

Van Hove singularities unphysical 1D behavior

Gc=Green’s function of the conductor

1D Green’s functions do not properly describe the electronic transport of the system the transport properties are also badly described


Transport in 3D system

LEAD LEADCONDUCTOR

Transport direction

Description of lateral interactions introduction of PARALLEL k-points k//

(supercell apprach)

z

x

y

k//

THREE-DIMENSIONAL QUANTUM CONDUCTANCE

NUMERICAL PROBLEM The finite number of k// of standard DFT calculations requires the introduction of the BROADENING OF THE ENERGY LEVELS through a smearing parameter d, where d~ 1/nk//


Broadening problem Conductor Green’s function with lead self-energies:

Green’s function general expression (SMEARING DEPENDENT*):

Green’s function expression with LORENTZIAN BROADENING

w

SLOW DECAY OF LORENTZIAN FUNCTION VERY SMALL d FOR ACCURATE ELECTRONIC STRUCTURE

SLOW DECAY OF LORENTZIAN FUNCTION VERY SMALL d FOR ACCURATE ELECTRONIC STRUCTURE

(standard expression)


NUMERICAL PROBLEM: small finite delta in lorentzian expression requires a huge number of k points

d~10-5 10+5 k// !!!!!!

RE-FORMULATION OF SMEARING EXPRESSION IN GREEN’S FUNCTIONS

Green’s function Spectral function

where

From Lorentzian To Gaussian

STRONG REDUCTION OF REQUIRED PARALLEL k-POINTSSTRONG REDUCTION OF REQUIRED PARALLEL k-POINTS

Broadening problem


Inclusion of parallel k-points

CONDUCTORLEAD LEAD

MODEL 3D SYSTEM Si bulk

GC

TRANSMITTANCE

32 parallel k-point +

Gaussian smearing

DENSITY OF STATES


Hybrid interfaces Example: Organic-Silicon interface: Si(111)/di-hydroxybiphenyl/Si(111)*

Important lateral interactions

I-V characteristic obtained by direct integration of T(E)

Simulation of doping effect by shifting the position of Fermi level.

Fermi level at the top of valence band corresponds to a doping

concentration = 1020 cm-3

* B. Bonferroni et al. Nanotechnology 19, 285201 (2008)


Hybrid interfaces Example: Organic-Silicon interface: Si(111)/di-hydroxybiphenyl/Si(111)*

I–V curve calculated for two systems, Si–S–C ((blue) thin solid line) and Si–O–C at different configurations, (b) I–V curve calculated for the Si–O–C system at different dopant concentrations: (c) interface transmittance near the VBM; the energy zero is set at the VBM; (d) k-resolved transmittance at the E1 energy indicated in (c), normalized to its maximum value.

* B. Bonferroni et al. Nanotechnology 19, 285201 (2008)


Practical examples

The transport Hamiltoninans

Main Hamiltonian blocks needed for transport calculation through a lead-

conductor-lead device

where

H00 L = NL × NL on-site hamiltonian of the leads L (from L-bulk calc.) H01 L = NL × NL hopping hamiltonian of the leads L (from L-bulk calc.) H00 R = NR × NR on site hamiltonian of the leads R (from R-bulk calc.) H01 R = NR × NR hopping hamiltonian of the leads R (from R-bulk calc.) H00 C = NC × NC on site hamiltonian of the conductor C (from C-supercell calc.) H LC = NL × NC coupling between lead L and conductor C (from C-supercell calc.) H CR = NC × NR coupling between conductor C and lead R (from C-supercell calc.)

lead L lead R

H01_L

conductor C

H00_L

H01_R

H00_RH00_C

H_LC H_CR


The H00_C term can be obtained directly from the conductor supercell calculation. The on-site block (R = 0) is automatically selected. The same is true in general for the lead-conductor coupling (consider for instance H_CR): here the rows of the matrix are related to (all) the WFs in the conductor reference cell while the columns usually refer to the some of them in the nearest neighbor cell along transport direction (say e.g. the third lattice vector).

The H_CR is therefore a NC × NR submatrix of the R = (0, 0, 1) block. In order to understand which rows and columns should enter the submatrix, we need to identify some WFs in the conductor with those obtained for bulk lead calculation.

This assumption is strictly correlated with that about the local electronic structure at the edge of the conductor supercell: the more we reach the electronics of the leads, the more WFs will be similar to those of bulk leads

The lead-conductor coupling matrices can be directly extracted from the supercell conductor calculation.


The missing Hamiltonians (H00_x, H01_x, where x=L,R) can be obtained from direct calculations for the bulk leads and are taken from the R = (0, 0, 0) and R = (0, 0, 1) blocks respectively.

All these Hamiltonian matrix elements are related to a zero of the energy scale set at the Fermi energy of the computed system (the top of valence band for semiconductors). It is not therefore guaranteed the zero of the energy to be exactly the same when moving from the conductor to the leads (which comes from different calculations. )

In order to match the hamiltonian matrices at the boundary, it is necessary to check that the corresponding diagonal elements (the only affected by a shift in the energy scale) of H00_L, H00_C and H00_R matrices are aligned. If not, a rigid shift may be applied.


The H00_x and H01_x matrices may be alternatively obtained from the conductor supercell calculation too.

We need to identify the WFs corresponding to some principal layer of the leads and extract the corresponding rows and columns. This procedure is not affected by any energy-offset problem, but larger supercells should be used in order to obtain environment (conductor) independent matrices.

Example : Au chain

SCF

NSCF

WANT pw_export, disentangle, wannier

Example : Au chain

bulk conductanceWe calculate the transmittance and the quantum conductance for an infinite Au chain.We calculate the 6 WFs corresponding to 5 double occupied and one half-occupied states

Isosurface of WF1

Example : Au chain

bulk conductance input

Example : Au chain

bulk quantum transmittance

Quantum conductance G=T(Ef)= G0

Step-like behavior The spectrum countschannel available for transport at a given energy.

Since the system is periodic the channel are simply the bands

Isosurface of WF1 it is gives the main contribution to transport at Fermi energy.

Proposed example: Al chain

bulk conductanceCalculate the WFs and the bulk quantum conductance for an aluminum atomic chain. The Al atoms have different electronic structure wrt gold (i.e. 3p instead 5d valence electrons) the system have different localization properties.Atomic chain with 5 Al atom per cell.

WF1

WF2


Practical examples

Example : Al-H junction

two-terminal nano-juctionWe consider the effect of an H impurity on the electronic and transport properties of the one dimensional Al chain.The system includes 10 Al atoms and one H impurity.

We define conductor C the region that includes the defect and the contacts with the leadsWe define L and R leads the external region, where the “bulk”-like behavior is recovered

lead L lead R

H01_L

conductor C

H00_L

H01_R

H00_RH00_C

H_LC H_CR

CL R


WFs calculationWe calculate the WFs for the valence band and the first unoccupied states 4 WFs per Al site + 1WF per H site 41 WFs

WF1

WF21

WF22

Far away from the defect we recover the “bulk”-like behavior we can extract the hamiltonian elements for the leads from the same supercell calculation


bulk-like conductance for leadsThe WFs states of the external atoms are sufficient to replicate the conduction properties of the clean Al chain

&INPUT_CONDUCTOR postfix = '_lead_Al.dat' dimC = 4 calculation_type = 'bulk' ne = 1000 emin = -7.0 emax = 2.5 datafile_C = "./alh_WanT.ham" transport_dir = 3 /

<HAMILTONIAN_DATA>

<H00_C rows="1-4" cols="1-4" /> <H_CR rows="38-41" cols="1-4" />

</HAMILTONIAN_DATA> These WFs completely describe the leads


two-terminal conductance input&INPUT_CONDUCTOR postfix = '_AlH‘ calculation_type = "conductor“ ! ordinary transport calculation for a

! leads/conductor/lead interface dimL = 4 ! number of sites in the left lead L (mandatory) dimC = 41 ! number of sites in the conductor C(mandatory) dimR = 4 ! number of sites in the right lead R (mandatory) ne = 1000 emin = -7.0 emax = 2.5 datafile_L = "./alh_WanT.ham" ! name of the file containing the Wannier Hamiltonian datafile_C = "./alh_WanT.ham“ ! blocks for the leads and conductor regions datafile_R = "./alh_WanT.ham" transport_dir = 3 /

<HAMILTONIAN_DATA>…</HAMILTONIAN_DATA>


two-terminal conductance input<HAMILTONIAN_DATA> ! if ( calculation_type = “conductor")

seven subcards are needed <H00_C rows="1-41" cols="1-41" /> <H_CR rows="1-41" cols="1-4" /> <H_LC rows="38-41" cols="1-41" /> <H00_L rows="1-4" cols="1-4" /> <H01_L rows="38-41" cols="1-4" /> <H00_R rows="1-4" cols="1-4" /> <H01_R rows="38-41" cols="1-4" />

</HAMILTONIAN_DATA>

CL R

H_LC H_CR

R=0reference cell in real space

R=1R=-1


two-terminal conductance output

The system is not periodicopen devices

The transmittance loses the step-like behavior characteristic of bulk conductance

Two-terminal conductance

Bulk –like conductance


current input

&INPUT filein = "./cond.dat“ ! the name of the input file containing the conductance fileout = "./current.dat“ ! the name of the output file containing the I-V curve Vmin = -1.0 ! minimum and maximum vales [eV] for the Voltage grid Vmax = 1.0 nV = 1500 ! the number of different voltage V values for which the

current has to be computed sigma = 0.05 ! thermal broadening parameter [eV] mu_L = -0.5 ! left and right *normalized* chemical potentials. mu_R = 0.5/

IMPORTANT NOTE: mu_(L,R) values are NOT the actual chemical potentials. The normalizazion is defined as:

mu_R - mu_L = 1 These parameters define the unbalance of resistance drop between the left and the right contact.

The true chemical potentials (depending on the actual bias value) are given by: mu_R(V) = V * mu_R mu_L(V) = V * mu_L


current output