Developing methods for transport calculations in heterojunctions with Coulomb interactionsFinal Full(Heb)

8/10/2019 Developing methods for transport calculations in heterojunctions with Coulomb interactionsFinal Full(Heb)

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Tel Aviv UniversitySchool of Chemistry

Subject:

Developing methods for transportcalculations in heterojunctions with

Coulomb interactions

Thesis submitted for the degree ofDoctor of Philosophy

by

Tal J. Levy

Submitted to the Senate of Tel-Aviv UniversityJuly 2014


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This work was carried out under the supervision of

Professor Rabani Eran

School of Chemistry, Tel-Aviv University


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Dedications and Acknowledgements

This thesis is dedicated to my parents, who throughout the years have given me much more

than I can ever express here; to my brothers, Gad and Ran, who always find the time for abeer and a laugh when I am down, and to my mischievous cats, who always remind me that

there is an easier way.

I also dedicate this thesis to my beautiful partner and companion through life, Ariel, who

recently gave me the best present of all - my daughter Phoenix.

Last but not least, I dedicate this thesis to my advisor, Prof. Rabani Eran, whose only

concern was my well-being (personally and scientifically). I really do mean it when I say that

without your help and guidance, all of this would have never happened. Thank you.

Finally, I would like to thank my past and present mentors, collaborators, group members,

teachers and friends. Every little thing you have said or done contributed to my success.


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Abstract

The demand for smaller, faster, and lower cost electronics has sent the industry and the scientific

community looking for different paradigms. It almost naturally led to the question of whether

it is possible to use single molecules and quantum dots as active elements in nanocircuits fora variety of applications. Developments in nanofabrication techniques have made possible the

old dream of contacting individual molecules and quantum dots to macroscopic electrodes and

explore their electronic transport properties. Moreover, it has been shown that molecules can

indeed mimic the behavior of some of the more commonly used microelectronic components

e.g., diodes, switches, and transistors. These achievements have given rise to what is now known

as Molecular Electronics.

There are still many experimental issues to overcome before molecular electronics will turn

from an idea into a realizable technology, regardless, the exploration of molecular-scale circuits

has already led to the discovery of many fundamental effects. In this sense, molecular elec-

tronics has become a new interdisciplinary field of science, in which knowledge from traditional

disciplines like physics, chemistry and engineering is combined to understand the electrical

and thermal conduction at the molecular scale. Nevertheless, theory still faces several impor-

tant challenges as the description of transport via an interacting nanoscaled region is a grave

problem.

Generally speaking, transport is a many-body, nonequilibrium phenomena, and its exact

treatment requires a formalism explicitly designed to work out-of-equilibrium. While the equi-

librium properties of systems similar to those to be treated in this thesis are quite well un-derstood and can be accurately solved for, the development of a general approach suitable for

the treatment of fully nonequilibrium, many-body systems still remains a formidable task. An

exact theoretical treatment of such systems is rather sparse to this day and includes only a

small class of over-simplified model problems.

We gather that in order to exploit the limitations and potential of the aforementioned

novel electronic it is compulsory to carry out a comprehensive theoretical study of the physical

properties that govern these systems. To this end, improvement of existing computational tools

and formalisms, and the development of new ones are mandatory. The limitations of todaysformally/numerically exact solutions are often too severe for them to be useful in practice

(usually, exponential scaling of computational resources and/or computational time with system

size), thus, most theoretical treatments of quantum transport rely on approximations of some

sort.

In this thesis we study, analyze, and develop two methods for transport calculations in

nanoscaled heterojunctions, described by model Hamiltonians (impurity models), with electron-

electron interactions. The methods are: (a) the equations-of-motion technique for the nonequi-

librium Green function and (b) a semiclassical approach.


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The nonequilibrium Green function formalism is considered as one of the pillars of current

approaches to quantum transport which have been implemented in both model Hamiltonian

formulations and first-principle methodologies. This approach is able to deal with a broad

variety of physical problems related to quantum transport at the nanoscale. It can deal with

strong non-equilibrium situations and it can also include interaction effects (e.g., electron-electron, electron-phonon) in a systematic way (diagrammatic perturbation theory).

In this work we elaborate on the equations-of-motion technique which is one of the more basic

and more powerful methods to calculate the Green functions of interacting quantum systems.

In spite of its simplicity, it gives the appropriate results for strongly correlated nanosystems,

describing qualitatively and in some cases quantitatively important transport phenomena, such

as, Coulomb blockade and the Kondo effect in quantum dots.

Be that as it may, we prove that Green functions calculated using the latter method may

break the symmetries and relations that the correlation functions should fulfill by definition.Consecutively, the various expectation values calculated with these objects may turn to be

unphysical. Thereafter, we suggest a strategy to restore the symmetries lost in the process

of deriving the equations-of-motion and advise to add this step as an essential part of the

equations-of-motion technique.

Illustrations are then provided for two impurity models: the Anderson model (which serves

as the simplest model to incorporates electron-electron interactions) and the double Anderson

impurity model (an extended Hubbard model which includes inter-site Coulomb repulsion on

top of the on-site electron-electron interactions). In addition, we develop two closures for the

equations-of-motion obtained for the double Anderson model based on physical justifications

in the regime studied, that outperform the commonly used closures found in the literature.

On a completely different note, we explore and develop a semiclassical approach, which al-

lows for the study of nonequilibrium quantum transport in molecular junctions using Hamiltons

classical equations-of-motion, where the number of equations scales linearly with the number

of the degrees-of-freedom. The dynamics of Fermions has always provided a challenge for semi-

classical methods, since Fermions have a particularly notable non-classical behavior, which is

the exchange antisymmetry of Fermions (which can be stated as the Pauli exclusion principle).

The key idea behind the three methods presented here is to transform a general second quan-tized many-electron Hamiltonian into a classical one by defining a prescription which maps the

electronic operators into classical functions that correctly account for the anti-commutativity

of the Fermion operators. The three different mappings depicted in this work are just a nat-

ural evolution of the same basic idea, where each transformation addresses the flaws of its

predecessor.


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Contents

1 Introduction 11

1.1 Molecular Electronics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.2 Theoretical treatment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.3 The model Hamiltonians . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.3.1 The single resonant level model . . . . . . . . . . . . . . . . . . . . . . . 16

1.3.2 The Anderson model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

1.3.3 The Double Anderson model . . . . . . . . . . . . . . . . . . . . . . . . . 17

1.4 Thesis outline and goal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2 Equations-of-Motion technique for the Nonequilibrium Green functions in

Quantum Transport 19

2.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.2 Nonequilibrium Green Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.2.1 Dyson equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.2.2 Analytic continuation and Langreth rules . . . . . . . . . . . . . . . . . . 28

2.2.3 Equations-of-motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.2.3.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.2.3.2 The equations-of-motion method . . . . . . . . . . . . . . . . . 31

2.2.3.3 The single resonant level model - EOM treatment . . . . . . . . 33

2.2.4 NEGF symmetry breaking within the EOM technique . . . . . . . . . . . 36

2.2.4.1 The Anderson model . . . . . . . . . . . . . . . . . . . . . . . . 36

2.2.4.2 The double Anderson model . . . . . . . . . . . . . . . . . . . . 40

2.2.5 NEGF symmetry restoration . . . . . . . . . . . . . . . . . . . . . . . . . 46

2.2.5.1 The Anderson model: symmetric-EOM approach . . . . . . . . 48

2.2.5.2 The double Anderson model: symmetric-EOM approach . . . . 49

2.3 Steady state conductance in a double quantum dot array:

Assessing the symmetric-EOM technique for the NEGF . . . . . . . . . . . . . . 52

2.3.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

2.3.2 Different closures of the EOM . . . . . . . . . . . . . . . . . . . . . . . . 53

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2.3.2.1 Approximation1 . . . . . . . . . . . . . . . . . . . . . . . . . . 55

2.3.2.2 Approximation2 . . . . . . . . . . . . . . . . . . . . . . . . . . 56

2.3.2.3 Approximation3 . . . . . . . . . . . . . . . . . . . . . . . . . . 56

2.3.2.4 Approximation4 . . . . . . . . . . . . . . . . . . . . . . . . . . 57

2.3.3 Master equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 572.3.4 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

2.3.4.1 Symmetric bridge . . . . . . . . . . . . . . . . . . . . . . . . . . 59

2.3.4.2 Asymmetric bridge . . . . . . . . . . . . . . . . . . . . . . . . . 66

2.4 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

3 Semiclassical approaches to quantum transport 73

3.1 overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

3.1.1 Action-angle mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

3.1.2 Cartesian mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

3.2 The Hubbard mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

3.2.1 Hubbard Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

3.2.2 Reformulating the Hamiltonian in terms of Hubbard operators . . . . . . 80

3.2.3 Classical mapping for the Hubbard operators . . . . . . . . . . . . . . . . 81

3.3 Quantum transport in impurity models:

Assessing the Hubbard mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

3.3.1 Resonant level model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

3.3.2 Anderson impurity model . . . . . . . . . . . . . . . . . . . . . . . . . . 853.3.3 Double Anderson model . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

3.4 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

4 Summary and Outlook 95

A Full derivation of the broken symmetry in the Anderson model 97

B Brief summary of the derivation of the NEGFs EOM: the double Anderson

model 103

C Calculating the poles of the retarded NEGF 109

D Complimentary information for the Hubbard mapping 113

Bibliography 117

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Chapter 1

Introduction

1.1 Molecular ElectronicsThe demand for smaller, faster, and lower cost electronics, is by no means new. Already back

in 1956 Arthur von Hippel formulated the basis of the bottom-up approach that he called

molecular engineering [1]:

Instead of taking prefabricated materials and trying to devise engineering applications con-

sistent with their macroscopic properties, one builds materials from their atoms and molecules

for the purpose at hand ...

Of course he was not alone, in 1959the physicist Richard Feynman discussed the possibility

of devices of extremely small dimensions in his lecture entitled Theres plenty of room at thebottom [2]:

I dont know how to do this on a small scale in a practical way, but I do know that

computing machines are very large; they fill rooms. Why cant we make them very small, make

them of little wires, little elements and by little I mean little. For instance, the wires should

be 10 or 100 atoms in diameter, and the circuits should be a few thousand angstroms across. [.

. .] There is plenty of room to make them smaller. There is nothing that I can see in the laws

of physics that says the computer elements cannot be made enormously smaller than they are

now.The concept of molecular engineering introduced by Von Hippel led to the first notion of

molecular electronics. It eventually resulted in a collaboration between Westinghouse Electric

and the U.S. Air Force at the end of the 1950s. At that time, the U.S. Air force was open to

new ideas and alternatives to the recently introduced integrated circuits and thus, organized

a conference on molecular electronics which included scientists and engineers from military

and private research labs. It was there that colonel C.H. Lewis, director of Electronics at the

Air Research and Development Command, coined the term Molecular Electronics:

Instead of taking known materials which wil l perform explicit electronic functions, and

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reducing them in size, we should build materials which due to their inherent molecular struc-

ture will exhibit certain electronic property phenomena. We should synthesize, that is, tailor

materials with predetermined electronic characteristic. Once we can correlate electronic prop-

erty phenomena with the chemical, physical, structural, and molecular properties of matter, we

should be able to tailor materials with predetermined characteristics. We could design and create

materials to perform desired functions. Inherent dependability might eventual ly result. We call

this more exact process of constructing materials with predetermined electrical characteristics

Molecular Electronics.

However, molecular electronics, as we understand it today, started at the end of the1960s

and the beginning of1970s. Thereupon, different groups started to investigate, experimentally,

electron transport through molecular mono-layers. For instance, Hans Kuhn and coworkers

studied new ways of fabricating mono-layers of organic materials (LangmuirBlodgett films) [3],

which they were able to sandwich between metal electrodes and measure the electrical con-ductivity of the resulting junctions. A few years later (1974), Arieh Aviram and Mark Ratner

published a now-famous paper on molecular rectifiers [4] in which they described how a mod-

ified charge-transfer salt could operate as a traditional diode in an electrical circuit. The idea

was considered for a long time a theoretical curiosity that could not be tested experimentally,

and in this sense, it did not have much impact in the scientific community back then.

Much has changed with the invention of the scanning tunneling microscope (STM) [5, 6]

in 1981, the introduction of mechanically controllable break junction (MCBJ) technique [7]

in 1985, and the development of controllable single molecule junctions [8, 9, 10, 11, 12, 13,

14, 15, 16, 17, 18, 19, 20, 21]. The STM was the first tool that provided a practical way

to see and manipulate matter at the atomic scale and very quickly it became clear that it

could provide a realistic way to address single molecules and to study their electronic transport

properties [22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35].

With these techniques a large amount of interesting phenomena have been observed. For

example: rectification [17], negative differential conductance [9, 33, 36], Coulomb blockade [22,

10, 11, 14, 15, 20], Kondo effect [11, 37], vibrational effects [24, 29, 30, 32, 34], and nanoscale

memory effects [38, 39, 40] just to name a few.

Many engineering challenges, such as robustness and stability of these molecular devices,still remain to be addressed before molecular electronics can be practical, however, these

results clearly show the potential of molecule-based electronics. Mechanical challenges are not

the only thing that needs to be attended to. In that regard the theoretical treatment of current

and heat flowing through a nanoscaled junction is far from complete.

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1.2 Theoretical treatment

The growing interest in the properties of both mesoscopic and nanometric devices and the new

physics and phenomena encountered in the various labs has raised fundamental and conceptual

questions. Theory faces several important challenges and the development of a general approachsuitable for the treatment of fully nonequilibrium many-body systems still remains a formidable

task. In this regard the main challenges facing theory are:

Understanding the coupling of individual molecules to macroscopic electrodes under

nonequilibrium conditions [41, 42, 43, 44].

Characterization of the temperature dependence on conductance as well as the role of

molecular vibrations are of crucial importance. This would facilitate understanding ofelastic and inelastic transport phenomena, where the nature of both coherent and inco-

herent transport is paramount for a complete picture [45, 46, 47, 48].

Electron-electron correlations in the macroscopic electrodes can often be neglected by the

application of an effective band-structure description of the leads, such correlations, on the

other hand, are important within the molecule and thus strongly affect the conduction [49,

50].

Electron-hole excitations (excitons [51]) formed in the heterojunction, can remarkably

affect the current through it. While in case of noninteracting electrons the electron-hole

interaction leads to reduction in the current (exciton blocking), in the case of strong

Coulomb repulsion, conduction exists even when the electronic connectivity does not

exist [52]. Also, electronhole pair creation processes can explain vibrational excitations

characteristic of a single-molecule contact [53].

Characterizing transport behavior in systems driven by weak and strong electromagnetic

fields and the optical properties of such junctions is another difficult issue which must be

overcome by theory [54, 55, 56, 57].

Although noteworthy advances have been made, there is still a discrepancy between experimen-

tally measured and theoretically calculated values [42, 18, 58]. While recent work has been more

encouraging regarding these discrepancies, the problem clearly remains unsolved. Thus, the de-

velopment of a practical, general approach suitable for the treatment of fully nonequilibrium,

many-body systems is desirable.

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In general, many-body, interacting, out-of-equilibrium problems cannot be solved exactly

but for a few simple cases [59, 60, 61, 62]. Excluding recent developments based on brute-

force approaches such as time-dependent numerical renormalization-group techniques [63, 64,

65, 66, 67, 68], iterative [69] or stochastic [70, 71, 72, 73, 74] diagrammatic techniques to

real- time path integral formulations, wave function based approaches [75], or reduced dynamicapproaches [76, 77, 78, 79], all suitable to relatively simple model systems, most theoretical

treatments of quantum transport rely on approximations of some sort.

From a more formal standpoint, there are roughly two main theoretical frameworks that

can be used to study quantum transport in nanosystems at finite voltage: the nonequilibrium

Green function (NEGF) formalism [80, 81, 82] otherwise known as the Keldysh NEGF or the

SchwingerKeldysh formalism [83, 84], and generalized master equations methods [85, 86].

In this work we also introduce an emerging paradigm suitable to tackle quantum transport

at the nanoscale; a semiclassical approach [62, 87, 88, 89, 90], where the quantum Hamiltonian,and the quantum operators are mapped onto classical functions and degrees-of-freedom that

follow Hamiltons equations of motion.

For the sake of completeness we mention two more emerging and improving theoretical

frameworks applicable to open nonequilibrium systems: (a) methods based on density functional

theory (DFT) [91, 92, 93, 94, 95, 96, 97, 98], and (b) time dependent multiconfigurational

methods [99, 100, 101].

In the next section we introduce the model Hamiltonians used throughout this thesis.

1.3 The model Hamiltonians

Up till 1970, in most treatments of electron tunneling through a metal-insulator-metal (MIM)

junction it was accustomed to transform an initial Hamiltonian (of the total system) into an

effective Hamiltonian of the form:

Heff= HL +HR +T , (1.3.1)

where HL and HR represents the metals in the (MIM) junction (as accurately as possible),

i.e., the left and the right electrodes, while the transfer term Tdescribes the probability of

an electron to tunnel through the junction. Although, the tunneling is energy dependent,

this dependency was neglected from the transfer term, thus, making this theory unsuitable to

describe many-body effects, such as electron-electron and electron-phonon interactions. The

main reason for using effective Hamiltonians was the belief that a description of a system as a

whole will not allow for the calculation of the current, which is a nonequilibrium process.

In1970Caroliet alproposed the prototype of model Hamiltonians [102, 103, 104, 105] used

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ever since in the context of transport. The Hamiltonian they proposed was designed to permit

direct calculation of the tunneling current through a MIM junction. This was achieved by

describing the system in terms of localized atomic functions instead of quasi-free electron wave

functions, which resulted naturally in a three-parts Hamiltonian describing the electrodes, the

system, and the interaction between these parts. The general form of the Hamiltonian is givenby:

H = Hsys +Hbath +Hint, (1.3.2)

where Hbath describes the macroscopic leads (left and right electrodes), Hsys describes the

system of interest, and Hint is the interaction Hamiltonian between the system and the leads.

The contacts (leads) are usually modeled as infinite noninteracting Fermion baths [106, 107,

108], they are assumed to be each at their own equilibrium characterized by chemical potentials

L and R, where the difference = LRe is the applied voltage bias across the junction(e is the absolute electronic charge of an electron), and are described by a grand canonical

distribution. The leads Hamiltonian in second quantization is given by:

Hbath=

K{L,R}

,kK

kckck , (1.3.3)

where k is the energy of a free electron in the left (L) or right (R) lead, in momentum state

kand spin , with the operators ck and ck being annihilation and creation operators of such

an electron. The density matrix of the left/right lead at equilibrium is given by:

L/R= e

,kL/R kckckL/RNL/R

Tr

e

,kL/R kckckL/RNL/R

, (1.3.4)

where= (KBT)1 is the reciprocal of the temperatureTand the Boltzmann constantKB, the

trace (Tr [ ])should be taken as the sum over a complete basis and NL/R=

,kL/R ckck

is the occupation operator of the left/right lead.

The interaction between the system and the contacts is simply given by the tight-binding

Hamiltonian [102]:

Hint=

K{L,R}

,kK

m

tkmckdm+ h.c.. (1.3.5)

The transition matrix element tkm represents the coupling strength between the system (site

m, spin ) and the leads (electron in state k and spin ). It depends on the specific contact

geometry between the bridge and the leads. The corresponding level width functions are given

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by:

L/Rnm () = 2

,kL/R(tkn)

tkm( k) . (1.3.6)

These functions describe the broadening of the energy levels on the molecular bridge due to

this coupling.The form chosen for Hsys depends on the system studied. In the following subsections we

describe the different system Hamiltonians in detail.

1.3.1 The single resonant level model

This simple model was introduced by Ugo Fano [109] in 1961when he worked on the effect of

auto-ionization on excitation spectra. In this model the system has one spin-less level:

Hsys = n. (1.3.7)

Here n= dd is the number operator of the electron occupying the level with free-energy .Since the system has only one level, and no spin, we can simplify the expressions of the bath

and interaction Hamiltonians:

Hbath=

K{L,R}

kK

kckck, (1.3.8)

Hint=

K{L,R}

kK

tkckd + h.c.. (1.3.9)

1.3.2 The Anderson model

The same year Fano introduced his single level Hamiltonian, Philip W. Anderson was seeking

for conditions required for the stability of localized magnetic moments like those of iron, cobalt

or nickel dissolved in non-magnetic metals [110]. In the model he proposed the system isrepresented by one electronic level that can accommodate up to 2 interacting electrons (spin

up and spin down electrons) [111, 112]. The systems Hamiltonian for the Anderson model is

given by:

Hsys=

{,}n+ U nn. (1.3.10)

Here n =dd is the number operator of the spin electron with energy and U is therepulsion energy between two electrons on the same site with opposite spins (on-site repulsion).

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The interaction Hamiltonian takes the form:

Hint=

K{L,R}

,kK

tkckd+ h.c.. (1.3.11)

1.3.3 The Double Anderson model

This model is an extension of the Anderson model to systems that include more than just

one electronic level. Here we have 2 electronic levels each of which can accommodate up to

2electrons. The model incorporates on-site and inter-site Coulomb repulsions and a hopping

term that allows electrons to hop from one site to the other. The Hamiltonian in second

quantization is given by [113, 114]:

Hsys =

,m{,}mnm+

m

Umnmnm

+,

V

nn +

hd

d+ h.c.

. (1.3.12)

The first two terms on the R.H.S are similar to the Anderson impurity model Hamiltonian

(extended to 2sites,and ),V

is the repulsion energy between two electrons on different

sites (inter-site repulsion), and his the coupling strength for electron hopping between the

two sites. In the calculations performed and results presented throughout this thesis the twosites are assumed to be in a serial configuration, thus, the interaction Hamiltonian takes the

form:

Hint=

,kLtkc

kd+

,kR

tkckd+ h.c.. (1.3.13)

That is, site is connected to the left lead while site is attached to the right lead.

1.4 Thesis outline and goal

The primary objective of the research program presented here has been to analyze and develop

existing and novel theoretical and computational methods tailored to investigate charge (and

spin) transport at nanometer length scales. The theoretical examination of transport at the

nanoscale requires the use of advanced and powerful techniques able to deal with the dynamical

properties of the relevant physical systems, to explicitly include out-of-equilibrium situations

typical for electronic transport as well as to take into account interaction effects. In particular,

we have been interested in techniques suitable for large electronic systems, i.e. low scaling tech-

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niques that describe the system at hand beyond a mean-field approximation. In that regard we

focused our attention on two methods for transport calculations in nanoscaled heterojunctions,

described by model Hamiltonians (impurity models), with electron-electron interactions:

1. The equations-of-motion technique for the nonequilibrium Green function.

2. A semiclassical approach.

The nonequilibrium Green functions formalism is introduced in chapter 2 where we elaborate on

the equations-of-motion technique. We prove that Green functions calculated using the latter

may at different levels of closures and decoupling schemes break the symmetries and relations

that the correlation function should fulfill by definition, consecutively, the various expectationvalues calculated with these objects may turn to be unphysical.

Thereafter, we suggest a strategy to restore the symmetries lost in the process of deriving

the equations-of-motion and advise to add this step as a mandatory part of the aforementioned

technique. Illustrations are then provided for two impurity models: the Anderson model (which

serves as the simplest model to incorporates electron-electron interactions) and the double

Anderson impurity model (an extended Hubbard model which includes inter-site Coulomb

repulsion on top of the on-site electron-electron interactions). In addition, we develop two

closures for the equations-of-motion obtained for the nonequilibrium Green function of the

double Anderson model based on physical justification in the regime studied, that outperform

the commonly used closures found in the literature.

In chapter 3, we present three mappings of a semiclassical approach developed in collabora-

tion with the Miller group at the University of California, Berkeley U.S.A. These methods allow

for the study of nonequilibrium quantum transport in molecular junctions using Hamiltons

classical equations-of-motion, where the number of equations scale linearly with the number

of degrees-of-freedom. The key idea behind these methods is to transform a general second

quantized many-electron Hamiltonian into a classical one. This is achieved by defining a pre-

scription which maps the electronic operators into classical functions that correctly accountfor the anti-commutativity of the Fermion operators. The three different mappings are just a

natural evolution of the same basic idea, where each transformation addresses the flaws of

its predecessor.

Out of the three mappings we center on the most recent Hubbard mapping and asses its

validity by employing the method on the three model Hamiltonians described in the previous

section.

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Chapter 2

Equations-of-Motion technique for the

Nonequilibrium Green functions in

Quantum Transport

2.1 Overview

In this chapter we outline the theory of the nonequilibrium Green function (NEGF) formalism,

which is widely used to describe transport phenomena in nanojunctions. As will be clear shortly,

the NEGF method is able to deal with a very broad spectrum of physical problems related to

transport at the nanoscale [80, 82, 59, 115, 116, 117]. For one, as it name implies, it is tailoredto take on out-of-equilibrium situations, and interaction effects can be dealt with in a well

defined manner (diagrammatic perturbation theory [118], functional derivatives technique [80,

119]). The NEGF formalism, initiated by Schwinger [83], Kadanoff and Baym [80] and later

Keldysh [84] allows one to:

Study the time evolution of a many-particle quantum system.

Calculate time-independent and time-dependent expectation values such as currents, den-sities, electron addition and removal energies and the total energy of the system.

Calculate the spectral function which gives access to the local density of states.

Describe dissipative processes and memory effects in transport that occur due to electron-

electron interactions and coupling of electronic to nuclear vibrations.

In a series of papers [102, 103, 105, 104] Caroli et al presented a general formalism for the

calculation of the current through an interacting system. An exact expression for the current

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was later reformulated by Meir and Wingreen [120] in terms of the systems NEGF and the

self-energies which represent the effects of the external baths on the system.

The NEGF formalism is widely used nowadays to describe electron and hole transport in a

large variety of devices and materials, such as: III Vresonant tunnel diodes [121, 122, 123,124, 125, 126], electron waveguides (i.e., electrons in 2D) [127], Silicon tunnel diodes [128, 129],carbon nanotubes [130, 131, 132, 133], metal wires [92, 134], organic molecules [135, 136, 137,

138, 139, 140, 141, 142], and magnetic leads [143, 144].

Physics that have been included are open-system boundaries [102], full-band structure [145,

146, 128, 129], the self-consistent Hartree potential [123, 147], exchange-correlation potentials

within a density functional theory (DFT) approach [148, 138, 149, 92], photon absorption and

emission [150, 151, 54, 152], energy and heat transport [153, 45], acoustic, optical, intra-valley,

inter-valley, and inter-band phonon scattering [125, 126, 154, 128, 129, 151], single-electron

charging and nonequilibrium Kondo systems [155, 156, 157, 158, 112], shot noise [159, 160, 161],alternate Current [121, 162, 163, 164], and transient response [165].

Excluding simple noninteracting cases, the calculation of the systems NEGF required to

obtain the current (or any other observable) is far from trivial. Most applications are based

on a perturbation expansion (and diagrammatic techniques) to obtain the systems NEGF.

Alternatively, one can use the equations-of-motion (EOM) approach, which allows to deduce

the systems NEGF by deriving the corresponding equations-of-motion.

This chapter continues with a brief description of the NEGF [117]. We shall focus on the

equations-of-motion technique, outline its advantages and disadvantages, and elaborate on the

single-particle NEGF symmetry breaking [166] due to the way this technique is carried out.

We show that this symmetry/relation breaking can lead to solutions which are not physical

and suggest a scheme to restore it [166]. Finally, we formulate a couple of closures, one of

which outperforms the commonly used closures found in the literature, for the EOM of the

NEGF for the double Anderson model and use the symmetric-EOM technique also developed

by us to calculate the current and differential conductance through the double quantum-dot

array [114, 167] (the double Anderson model).

2.2 Nonequilibrium Green Functions

In nonequilibrium theory one usually divides the full Hamiltonian into three parts:

H (t) = H0 +V H

+Hex (t) = H+Hex (t) , (2.2.1)

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where H0 is the single-particle, noninteracting part, Vcontains all the interactions (electron-

electron, electron-phonon, impurity scattering, etc.) and Hex (t) contains all external pertur-

bations (which may be time-dependent) driving the system out of equilibrium.

A standard device is to assume that the external perturbations are turned on at a certain

time t0, while prior to that time, the system is at equilibrium, i.e., Hex (t) = 0 for t < t0,and is described by the thermal equilibrium density operator which is a function of the time-

independent interacting Hamiltonian H:

= 1

ZeH, Z=Tr

eH

. (2.2.2)

Before introducing the single-particle, nonequilibrium Green function, which is a correlation

function between two Fermion/Boson field operators at different locations and times, we will

formulate the nonequilibrium problem. The task at hand, is the calculation of the expectation

value of an observable at time t > t0(associated to the quantum mechanical operator O):

O (t)

=Tr

OH (t)

, (2.2.3)

where the subscript H indicates that the time dependence of the Heisenberg operator is governedby the full Hamiltonian (equation (2.2.1)) and is given in equation (2.2.2).

The general plan of attack is to transform the extremely complicated time dependence of

the operator O (t)to a much simpler form, specifically, have its time evolution governed by the

noninteracting Hamiltonian H0. Furthermore, to allow a perturbation expansion and the use ofWicks theorem [168, 169] or the Feynman diagrams [169, 170, 118], which are just a graphical

way representing the results of Wicks decomposition, one also needs to transform the density

operator such that it becomes a single-particle density operator. This is usually achieved by

two transformations [115]. The first transformation is due to the following identity:

OH (t) = Sex (t0, t)OH(t)Sex (t, t0) , (2.2.4)

with

OH(t) = e ihHt Oe ih Ht , (2.2.5)

Sex (t, t0) =T

exp

i

h

tt0

dHexH ()

, (2.2.6)

and

HexH (t) = eihHtHex (t) e

ih Ht . (2.2.7)

The time-order operator T{}in equation (2.2.6) equals either the chronological time-orderoperator, Tf{}, which moves the operator with the later time to the left, if t > t0, or the

21


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anti-chronological time-order operator, Tb {}, which moves the operator with the later timeto the right.

At this point the time-contour (proposed by Schwinger [83] and later by Keldysh [84] and

Craig [171]) and contour-ordered quantities [82] are introduced. By joining the two exponentials

of equation (2.2.4) we get:

OH (t) = Sex (t0, t)OH(t)Sex (t, t0) ,

= Tb

exp

i

h

t0t

dHexH ()

OH(t) Tf

exp

i

h

tt0

dHexH ()

,

= TC

exp

i

h

t0t0

dHexH()

OH(t) ,

= TC

exp

i

h

C

dHexH ()

OH(t) . (2.2.8)

The integral in the exponent is now a contour integral along the contour C(depicted in fig-

ure 2.2.1) and TC{}is the contour time-order operator, which moves operators with timesfurther along the time-contour Cto the left. By defining:

SexC = TC

exp

i

h

C

dHexH ()

, (2.2.9)

equation (2.2.3) can be rewritten as:

O (t) =TrOH (t)

=Tr

TC

SexCOH(t)

. (2.2.10)

In order to define the fundamental object of nonequilibrium many-body theory, i.e. the

NEGF, one extends the notion of the time variable (t) to contour time variables (). We

Figure 2.2.1: The time-contourC, starts and ends at t0with contour variable1on theforward/upper branch(C+)and 2 on thebackward/lower branch(C). By definition, any point lying on the backward branch comesafter a point lying on the forward branch, meaning us, TC

A(1)B (2)

= B (2)A(1). The minus sign

originates from the permutation of two Fermion operators.

not in passing that the extension is not unique as a time t can be mapped into a contour

time variable on the forward or the backward branches. Finally, the contour-ordered Green

22


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function is given by:

G(r2, 2,r1, 1) = ih

TC

H (r2, 2)

H (r1, 1)

, (2.2.11)

where (r, ) / (r, ) is the annihilation/creation field operator which annihilates/createsa particle at position r and time . Henceforth, our discussion will only involve Fermion field

operators.

After applying the transformation described we get:

G(r2, 2,r1, 1) = i

h

TC

SexC

H(r2, 2)H(r1, 1)

TC

SexC

,

=

i

h

Tr

TC

SexC

H(r2, 2)H(r1, 1)

Tr

TCSexC . (2.2.12)The time-contour Cmay be elongated so that it would run beyond the latest time (t1 or t2);

usually the contour is elongated to t . In this form the contour-ordered Green function (GF)does not admit Wicks decomposition [172] and the use of Feynman diagrams, since the field

operators and the density operator depend on the (still complicated) interacting Hamiltonian

Hand not on the noninteracting Hamiltonian H0. To circumvent this problem one more

transformation is performed. The full derivation can be found in Ref. [82], but basically one

uses the identity:

=Z0

Z0S

v (i, 0) , (2.2.13)

0= 1

Z0eH0, Z0=Tr

eH0

, (2.2.14)

Sv (i, 0) =Ti

exp

i

h

i0

dtVH0 (t)

, (2.2.15)

with

VH0 (t) = ei

hH0tV e

i

hH0t. (2.2.16)

The operator Ti is the time-order operator, ordering the operators along the imaginary-time

segment [i, 0] (see figure 2.2.2 for more details) such that the time closest toiis to the

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left. Putting it all together one gets:

G(r2, 2,r1, 1) = i

h

Tr0TCi

SexH0,C

SvCiH0 (r2, 2)H0 (r1, 1)

Tr

0TCi

SexH0,CSvCi

,

= ih

Tr0TCi SCiH0 (r2, 2)H0 (r1, 1)

Tr0TCi

SCi

. (2.2.17)where

H0 (r, ) =eihH0 (r) e

ihH0,H0 (r, ) =e

ihH0 (r) e

ihH0,(2.2.18)

the contour time-order TCi operator, orders the operators along the contour Ci, and

SexH0,C= TCexp ih CdHexH0 () , (2.2.19)

SvCi= TCi

exp

i

h

Ci

dVH0 ()

, (2.2.20)

SCi=SexH0,C SvCi , (2.2.21)

with

HexH0 () = eih H0Hex (t) e

ihH0. (2.2.22)

The contour-ordered Green function in equation (2.2.17) accepts Wicks decomposition and

the perturbation expansion is possible. As already mentioned at the beginning of section 2.2,

Figure 2.2.2: The Kadanoff-Baym contour Ci, starts at t0 and ends at t0 i, with contour variable 1 onthe forward branch and2 on the backward branch. By definition any point on the vertical branch comes after

the points on either of the horizontal branches, thus, TCi {A(1)C(3)B (2)} = C(3)B (2)A(1).

all interactions are present in V. If one does not care about initial correlations (which is

a conceivable assumption when studying steady-state transport for example), one can take

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the limit t0 . With reference to the equilibrium theory, interactions are switched onadiabatically, such that at t0= the system is not interacting and H= H0. Under thisassumption = 0, therefore, the transformation in equation (2.2.13) is not called for and the

addition of the imaginary segment to the time-contour C is not required [173]. The resulting

time-contour is known as the Schwinger-Keldysh time-contour, which we will denote byC(seefigure 2.2.3). In this regard, the contoured-ordered Green function takes a somewhat simpler

form:

G(r2, 2,r1, 1) = ih

TC

H (r2, 2)

H (r1, 1)

, (2.2.23)

which can be reformulated in the interaction representation:

G(r2, 2,r1, 1) = ih

Tr0TC

SCH0 (r2, 2)

H0 (r1, 1)

Tr0TC SC , (2.2.24)

with

SC= SexC

SvC, (2.2.25)

SexC= TC

exp

i

h

C

dHexH0 ()

, (2.2.26)

SvC= TC

exp

i

h

C

dVH0 ()

. (2.2.27)

In this study we are interested in steady-state transport, therefore, in all that follows, we

confine ourselves exclusively to the Keldysh approach [174], i.e., we neglect initial correlations

and work on the Schwinger-Keldysh contour C. As already mentioned, by extending the time

Figure 2.2.3: The Schwinger-Keldysh contour C. When initial correlations are neglected, one ignores the

imaginary segment of the time-contour Ci and elongates the time-contour C from t0= to t = and back.

variables into contour time variables, the uniqueness of these variables is lost. Such being the

case one should keep track of which branch is in question. With two time variables that can

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be located on either one of the two branches, four real-time Green functions are introduced:

GT(r2, t2,r1, t1) = ih

Tf

H (r2, t2)H (r1, t1)

1, 2 C+,

G(r2, t2,r1, t1) =

ih H (r2, t2)H (r1, t1) 1 C+, 2 C,

GT(r2, t2,r1, t1) = ih

Tb

H (r2, t2)H (r1, t1)

1, 2 C,

(2.2.28)

where GT is the time-ordered Green function,GT is the anti-time-ordered Green function, Gis the greater Green function. As opposed to its equilibrium

counter-part, the single-particle NEGF is a 2 2tensor [175]:

G(r2, 2,r1, 1) = GT(r2, t2,r1, t1) G

(r2, t2,r1, t1) GT(r2, t2,r1, t1) . (2.2.29)

Usually one defines two other real-time Green function, the retarded Green function

GR

and

the advanced Green function

GA

:

GR(r2, t2,r1, t1) = GT(r2, t2,r1, t1) G(r2, t2,r1, t1)

= i

h (t1 t2)

H (r2, t2) ,

H (r1, t1)

, (2.2.31)

where { , } is the anti-commutator. With these definitions, the single-particle NEGF tensorcan also be presented as follows [84]:

G(r2, 2,r1, 1) =

GR(r2, t2,r1, t1) G are all connected via the fluctuation-dissipation

theorem [176] and only one Green function is needed to describe a system, this is not longer the

case in nonequilibrium situations. Nevertheless, one easily observes that the real-time NEGFs

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are not all independent and the following relations and symmetries hold by definition:

GR(r2, t2,r1, t1) =

GA (r1, t1,r2, t2)

,

G(r2, t2,r1, t1) =

G (r1, t1,r2, t2)

,

GR(r2, t2,r1, t1) GA(r2, t2,r1, t1) = G>(r2, t2,r1, t1) G(r2,r1; ) G


28/136

to interactions between the particle and the system it is part of, one obtains Dysons equation:

Gr, , r,

= G0

r, , r,

+

d1d2G0 (r, ,2) (2,1) G

1,r,

,

= G0 r, , r, + d1d2G (r, ,2) (2,1) G

01,r, ,(2.2.36)

where 1 (r1, 1)and

d1 dr1 C d1.Dysons equation can be easily expressed in matrix notation (see equations (2.2.32)):

Gr, , r,

= G0

r, , r,

+

d1d2G0 (r, ,2) (2,1) G

1,r,

,

= G0 r, , r, + d1d2G (r, ,2) (2,1) G01,r

, ,(2.2.37)

only now the contour integrals are transformed into real-time integration, i.e.,

d1 dr1 dt1.Of course, usually, one does not know the exact form of the self energy, and approximated ex-

pressions are used to approximate the Green function.

2.2.2 Analytic continuation and Langreth rules

When writing down the EOM for the NEGF or when considering the different terms in the

perturbation expansion we encounter contour quantities with the following structures:

F1 (2, 1) =

C

A (2, ) B (, 1) d, (2.2.38)

F2 (2, 1) =

C

A (2, ) B

,

C

, 1

dd, (2.2.39)

F3, = A, B , , (2.2.40)F4

,

= A

,

B

,

, (2.2.41)

where we suppressed all variables but the temporal ones for clarity and brevity. The contour

integration and contour variable are not practical for calculations and one can replace the

contour time variables and the contour integrals with real-time variables and integrals (analytic

continuation [115]) using Langreth theorem [81, 115, 117]. The rules provided by Langreth

theorem are summarized in table 2.1. For example, using Langreth rules and Dysons equation

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for the NEGF, the equations for the retarded and lesser Green functions are given by:

GRr, t,r, t

= GR0

r, t,r, t

+

d1d2GR0 (r, t,2)

R (2,1) GR1,r, t

,

= GR

0 r, t,r, t+ d1d2GR

(r, t,2) R

(2,1) GR

0 1,r, t ,(2.2.42)

G 1 on the contour

otherwise . (2.2.49)

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The EOM [180] for Gij (2, 1)can be written as:

ih

2Gij (2, 1) = C (2 1)

ai, a

j

+

TC

2ai (2) a

j (1)

,

= C (2 1)ai, aj+TC ih H (2) , ai (2)aj (1) ,= C (2 1) ij+

TC

i

h

H0, ai (2)

aj (1)

+

TC

i

h

V(2) , ai (2)

aj (1)

. (2.2.50)

Since H0is the time-independent, noninteracting, quadratic Hamiltonian, the term

H0, ai (2)

equals p pap (2), whereps are parameters of the noninteracting Hamiltonian, such as thekinetic energy. Using this and the definition in equation (2.2.48) we can rewrite:

TC

i

h

H0, ai (2)

aj (1)

=p

pGpj (2, 1) . (2.2.51)

The term

TC

ih

V(2) , ai (2)

aj (1)

results in higher order Green functions (depending

on the form ofV(t), these can be 2-, 3- or even higher particle Green functions). We shall

denote those as G(2, 1). Finally we can rewrite:

ih

2 i

Gij (2, 1) = C (2 1) ij+p=i

pGpj (2, 1)

+p

TpGp (2, 1) , (2.2.52)

where once again Tps are parameters of the interaction Hamiltonian

V(t)

, such as the

Coulomb repulsion energy. Define the Green functiongi (2, ):

ih

2 i

gi (2, ) =C (2 ) , (2.2.53)

and equation (2.2.52) now takes the form:

Gij (2, 1) = gi (2 1) ij+p=i

p

C

gi (2 ) Gpj (, 1)d

+p

Tp

C

gi (2 )Gp (, 1)d. (2.2.54)

Depending on the Hamiltonian, the newly generated Green functions (Gpj and Gp) can

involve lead (bath) operators as well as system operators. Except for very simple cases, where

an exact closure can be obtained (see sub-subsection 2.2.3.3), writing the EOM for Gpj

(2

, 1

)

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and Gp (2, 1) will produce new and/or higher order Green functions that need to be eval-

uated. This leads (in principle) to an infinite/ intractable set of equations. The idea of the

EOM method is therefore, to truncate this hierarchy of equations making a mean-field like ap-

proximation for the higher-order Green functions through lower order functions. This is the

Achilles heel of this method as there is no systematic way to close the equations and no theoryto back-up such a procedure. Usually the approximations have physical meaning within the

regime of the problem at hand [189, 190, 111, 167].

As an illustration of the method we next calculate the NEGF of a single resonant level [109,

62]described in subsection 1.3.1.

2.2.3.3 The single resonant level model - EOM treatment

Define the systems contour ordered Green function:

G

,

= ih

TC

d () d

. (2.2.55)

The equation of motion ofG (, )is:

ih

G

,

= C

+

TC

dd

d () d

= C

i

hTC d () d

ih

K{L,R}

kK

tk

TC

ck () d

= C

+ G

,

+

K{L,R}

kK

tkFk

,

, (2.2.56)

whereFk (, ) = ih

TC

ck () d ()

, is a new Green function generated in the procedure.

To obtain G (, )we now derive the EOM ofFk (, ):

ih Fk , = C ck, d+TCddck () d

= ih

k

TC

ck () d i

htk

TC

d () d

= kFk

,

+ tkG

,

. (2.2.57)

The last equation can be rearranged:

ih

k

Fk

,

= tkG

,

. (2.2.58)

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Define the leads noninteracting Green function through the following equation:

ih

k

gk ( 1) = C ( 1) , (2.2.59)

so equation (2.2.58) takes the form:

Fk

,

=

C

gk ( 1) tkG

1, d1. (2.2.60)

At this point the equations close. To finish our derivation, substitute equation (2.2.60) in

equation (2.2.56):

ih

G

,

= C

+

K{L,R} kKC

|tk|2 gk ( 1) G

1,

d1.

(2.2.61)

Define the levels noninteracting Green function through the following equation:

ih

g ( 1) =C ( 1) , (2.2.62)

so finally the equation for the systems NEGF is:

G, = g + K{L,R}

kK

C

g ( 2) |tk|2 gk (2 1) G1, d1d2.(2.2.63)

The Green function G (, ) contains the information on the resonant level in an explicitway (via g ( )), while the effect of all other (infinitely many) levels of the left and rightleads appear only in the sum:

K{L,R} kK|tk|2 gk (2 1) , (2.2.64)

which we recognize as the self energy:

(2 1) =

K{L,R}

kK

|tk|2 gk (2 1) . (2.2.65)

Applying Langreth rules one can obtain the retarded, advance, lesser and greater Green func-

tions:

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GR/A

t, t

= gR/A

t t

+

gR/A (t t2) R/A (t2 t1) GR/A

t1, tdt1dt2.

(2.2.66)

G

t, t

= g

t t

+

gR (t t2) R (t2 t1) G

t1, tdt1dt2

+

gR (t t2) (t2 t1) GA

t1, tdt1dt2

+

g (t t2) A (t2 t1) GA

t1, tdt1dt2. (2.2.67)

In steady state the NEGF is a function of one time variable [115] (the time difference). In

this case it is simpler to express the NEGFs in Fourier space. To simplify the notation we

denote the Fourier transform ofG (t)as G (), i.e., functions with an argument are Fourier

transforms of their time-domain counterparts. The resulting equations are:

GR/A () =gR/A () + gR/A() R/A () GR/A () , (2.2.68)

G () = g () + gR () R () G ()

+gR

()

() GA

() + g

() A

() GA

() , (2.2.69)

Here:

gR/A () = 1

h i , 0+, (2.2.70)

with the + signs corresponding to the retarded Green function,

g< () = 2i(h )

dd0

, (2.2.71)

g

>

() = 2i(h )1 dd0 , (2.2.72)R/A() =

R/AL () +

R/AR () =

K{L,R}

kK

|tk|2h k i

, 0+, (2.2.73)

< () = () = 2i K{L,R}

kK

|tk|2 (h k) (1 f(k K)) , (2.2.75)

wheref()is the Fermi-Dirac distribution and dd0represents the occupancy of the resonant35


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level in steady state.

An exact expression for the stationary current through a system coupled to large noninter-

acting metallic leads in terms of the systems Green function can be derived [120]:

I = ie

4h

d

Tr(f( L)L () f( R)R ())GR () GA ()

+Tr(L () R ())G< ()

, (2.2.76)

where L () = 2Im

RL()

, or equivalently

I = e

2h

dTr

L() >R ()

G< ()

. (2.2.77)

The resonant level model serves as a great toy model, nevertheless, it is too simple as it

involves no interactions. The NEGF and the self-energies can be calculated exactly, and noapproximations are needed. As already mentioned when interactions are involved, one needs

to truncate the infinite/intractable set of equations and decouple the higher order correlation

functions in terms of the lower order ones, the resulting single-particle NEGF does not nec-

essarily obey the symmetries and relations of equation (2.2.34). To demonstrate this we turn

to the Anderson model at the Kondo regime and the double Anderson model described in

subsection 1.3.2 and subsection 1.3.3 respectively.

2.2.4 NEGF symmetry breaking within the EOM technique

2.2.4.1 The Anderson model

Following the derivation in Refs. [158, 115] we define the following contour ordered Green

function:

G

t,

= ih

TC

d () d

, (2.2.78)

G, = i

h TC n () d () d , (2.2.79)

where is the opposite spin of. Various approximate decoupling procedures can be applied to

the many-particle Green functions [182] that are generated during the procedure of writing the

EOM for the single-particle NEGF. Here we follow the approximation scheme used in Ref. [158]:

1. All electronic correlations containing at mostone lead operator, are not decoupled and

their EOM are calculated.

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2. Higher order Green functions involving (opposite) spin correlations in the leads are set to

zero, that is:

ih

TC

Ak () Bq () Cs () d

= 0

where A and B are either creation or annihilation operators of the leads and Cs is asystem operator (dor dwith spin s).

3. The remaining higher order Green functions involving lead and system degrees of freedom

are decoupled such that

F

,

= i

h

TC

ck () d () cq () d

= kqf(k K) G

,

.

Given these approximations the EOM for G (, )can now close (full derivation can be foundin Ref. [115] pages 172-176). The resulting EOM (in steady state) are:

GR/A () =gR/A () + g

R/A () UG

R/A() , (2.2.80)

GR/A () = g

R/A2 ()n gR/A2 () R/A1 () GR/A () , (2.2.81)

G () = g () + g

R () UG

() + g


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gR/A2 () =

1

h U R/A4 (), (2.2.86)

g () = gR ()

0 () g

A() , (2.2.87)

g

2 () = gR

2 ()

4 () gA

2 () . (2.2.88)Also,

R/A0 () =

R/A0L () +

R/A0R () =

K{L,R}

kK

|tk|2h k i

, 0+, (2.2.89)

is the exact retarded/advanced self-energy for the noninteracting case. The self-energies due to

the tunneling of the electron are R/A1 ()and R/A3 ():

R/Aj () = R/AjL () + R/AjR () ,

=

k{L,R}A(j)k |tk |2

1

h + k U i

+ 1

h k + i

, j= 1, 3and 0+

(2.2.90)

withA(1)k = f(k K),A(3)k = 1. The lesser self energies are defined as in Ref. [186]:


39/136

where

PR/A = 1

1 + gR/A2

R/A1 g

R/A U

. (2.2.94)

Applying the principle of reductio ad absurdum we assume G


40/136

0 0.2 0.4 0.6 0.8 1

e/U

0

0.1

0.2

0.3

n

Re nIm n

Figure 2.2.4: One of the main flaws of the EOM approach for the Anderson impurity model, if the symmetrybreaking of the lesser Green function, which leads to a complex occupation numbers. Here we plot the value ofn

(occupation of the spin up electron) as a function of the bias voltage (e/U). The most notable effect is

the appearance of an imaginary portion ton

(red line) as the bias voltage is increased. To obtain the results

only the real part ofn was used to converge the self-consistent equations for the NEGF (equations (2.2.80)to (2.2.83)). Parameters used (in units ofU): L,= R,= 0.3, L,= R,= 0.05, = 0.2, = 0.2 and= 4. The wide band limit approximation was used for the calculation of the self-energies.

This simple approximation does not violet the symmetry relations of the single-particle Green

function (see appendix A), but as pointed above, it does not reproduce the Kondo peaks at low

temperatures.

2.2.4.2 The double Anderson model

For the double Anderson model, described in subsection 1.3.3, we follow the derivation given

in Ref. [114], and define the following contour ordered Green functions:

G

, =

i

h TC d () d

, (2.2.98)G

s

,

= i

h

TC

ns () d () d

, (2.2.99)

where , s = {,}and is the spin opposite to , and= {, }. In what follows we will usethe naming convention summarized in table B.1 in appendix B. The approximations used in

Ref. [114] are:

1. Neglect the simultaneous hopping of electron pairs to and from the system. i.e., neglect

40


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all2-particle Green functions Gqrst(, ), Fqrstk(,

)and Fqrstk(, ).

2. Assume that

Fsjk, =

i

h TC njs () ck () d

i

h

dt1gk ( 1)

TC

d (1) njs (1) d

,

wherenjs (t)is the number operator of the electron on site j and spins, and we assumed

k L, and ih

k

gk ( 1) = C ( 1) .

This is obtained by writing the EOM ofFsjk(, )and assuming that njs ()is constant,

i.e.

njs () = 0, which is the case in steady state. This assumption is equivalent totreating the coupling to the leads up to the second order with respect to tkm. It neglects

processes necessary to qualitatively capture the Kondo effect [191, 192, 193], yet results

are predicted to be reliable for temperatures above the Kondo temperature (TK) [158, 60].

3. Higher order Green functions (3-particle) of the form

ih

TC

nq() nr () d () d (

)

are decoupled to:

ih

nq()TC

nr () d () d

i

hnr (t)

TC nq() d () d

.

Here, = {, } .

These approximations lead to the following equations, written in Fourier space, where ()

was omitted for brevity (for a brief summary of the derivation of the EOM ofG(, ) see

appendix B):

h

R/A0

1

G

R/A= + h

G

R/A+ UG

R/A

+VG

R/A+ V

G

R/A,

(2.2.100)

G

R/A=

h U V

n

V

n

R/A0

1

hG

R/A+ n

V

G

R/A+ V

G

R/A,

(2.2.101)

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G

R/A=

h U

n

Vn V R/A0

1n + h

G

R/A+ n

U

G

R/A+ V

G

R/A,

(2.2.102)

G

R/A=

h U

n

Vn V R/A0

1n + h

G

R/A+ n

U

G

R/A+ V

G

R/A,

(2.2.103)

GR/A

= h U n V n V

R/A0

1

hG

R/A+

n

UG

R/A+ V,

G

R/A,

(2.2.104)

G

R/A=

h U n V

n

V R/A0

1

dd,

+ hG

R/A

+n,UGR/A + V,, GR/A ,(2.2.105)

G

R/A=

h U Vn Vn R/A0

1

n

+ hG

R/A+

n

V, G

R/A+ V,

G

R/A

.

(2.2.106)

We now show that given this set of equations, the symmetry relationG()

R=

G ()A

is not satisfied. For simplicity we prove that for the case where

42


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Vsij = 0. Define

gR/Ai () =

1

h i R/A0 (), (2.2.107)

(gui)R/A () = 1

h i Ui R/A0 (), (2.2.108)

guni

R/A() =

1

h i Ui ni R/A0 (), (2.2.109)

R/A0 () =

kL

|tk|2h k i

, 0+, (2.2.110)

R/A

0 () = kR tk

2

h k i , 0+

. (2.2.111)

Simple substitutions now yield the following set of equations:

G

R=

1 gR

h2 gR1

gRhg

R+ g

RU

G

R+ gRh

g

RU

G

R,

(2.2.112)

G

R=

1 (gu)R

gun

R h2

1

guR h2 gunR

1

h2 (gu)R guR gunR gunR nnUU

1

(gu)R h

gun

R n + (gu)R guR hgunR n

n

U

1

gu

R

h

2

gun

R1

, (2.2.113)

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G

R=

1

gu

R gun

R h2

1

gunR h2 (gu)R

1

h

2

(gu)R

guR

gunR

gunR nnUU

1

guR n+ h2 (gu)R guR gunR gunR n

n

U

1

gun

Rh (g

u)

R h

1, (2.2.114)

and

G

A=

1 gA

h2 gA1

gAhgA+ gAUGA

+ g

A

h

g

A

UGA ,

(2.2.115)

G

A=

1

gu

A gun

A h2

1 (gu)Ah 2gunA

1

h 2 guA (gu)A gunA gunAnnUU

1

guA

h gun

A

n

+

gu

A(gu)

A h gun

A

nnU

1 (gu)A

h 2 gunA1 , (2.2.116)

G

A=

1 (gu)A

gun

A h2

1

gunA h 2 guA

1

h 2 guA (gu)A gunA gunAnnUU

1

(gu)

A

n

+ h

2

guA

(gu)A gun

A

gunA

nnU

1

gun

A h2 guA1

, (2.2.117)

Taking the conjugate of equation (2.2.115) and using the fact that,

gRi=

gAi

i {, } , (2.2.118)

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we obtain:

G

A=

1 gR

h2 gR1

gR

h

gR

+ gR

U

GA

+ gR

h

gR

UG

A

,(2.2.119)

Comparing the result with equation (2.2.112) and since this should hold for any value ofU, it

is obvious that for the identity

G()R

=

G ()A

to be true, the following should

hold:

gRUG

R= gRh

g

RU

G

A, (2.2.120)

or equivalently:

G

R = gRh GA . (2.2.121)Using equations (2.2.113) and (2.2.117) and eliminating equal terms on both sides of the equal-

ity, we get:

L.H.S = h

gunR n + UguR hgunR nn ,

(2.2.122)

R.H.S = gRh

(n

+Uh2 guR gunR gunR nn

.

(2.2.123)

Note that by definition:

(gui)R =

(gui)

A

i {, } , (2.2.124)

guni

R= g

uni

A

i

{,

}. (2.2.125)

Obviously G

R = gRh

G

A, (2.2.126)

hence, the relation

G()R

=

G ()A

is not satisfied under these aforementioned

approximations and truncation of the EOM.

45


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The same can be done (not presented) to show that:

G()

= G ()

, (2.2.127)

and G()

R G()A = G()> G()< . (2.2.128)As a result the single-particle density matrix, which is obtain from the lesser Green function, is

not Hermitian (see figure 2.2.5), the occupation of levels is a complex number, and the value of

the stationary current calculated via equation (2.2.76) or via equation (2.2.77) yields different

results (figure 2.2.6).

0 0.1 0.2 0.3 0.4

e/U

realpart

0 0.1 0.2 0.3 0.4 0.5

e/U

imaginarypart

0 0.1 0.2 0.3 0.4 0.5

Coherences

imaginarypart

Figure 2.2.5: The single-particle density matrix should be Hermitian, i.e., =

, or in other words

Im

= Im

and Re

=Re

with = ih2

G ()

G< and B = GR GA. Definethe difference anti-Hermitian matrix C=AB, and redefine the retarded and advancedGreen functions GR = GR +C/2, and GA = GA C/2.

The resulting Green functions (GR, GA, G< and G>) obey all symmetry relations of equa-

tions (2.2.33) and (2.2.34) by construction. Note that if the original Green functions obeyed

the symmetry relations to begin with, our symmetrization procedure will not alter them in any

way.

We now return to our calculations for both the Anderson and double Anderson models. For

both models we use the closures described above. For each set of calculations we have applied

the above symmetrization scheme and compared the results to those obtained without restoring

symmetry.

2.2.5.1 The Anderson model: symmetric-EOM approach

The closure used in Ref. [158] is sufficient to describe the appearance of the Kondo resonances

at low temperatures, as seen in figure 2.2.7, where we plot the density of states as a function

of energy for several temperatures (all calculations are done with symmetry restoration). The

development of Kondo peaks in the density of states as the temperature decreases is clearly

evident, signifying a regime of strong correlations which is qualitatively captured by the simple

EOM approach when symmetry is restored.

In figure 2.2.8 we plot the value of

n

as a function of the bias voltage with and without

symmetry restoration. The most notable effect is the appearance of an imaginary portion ton

as the bias voltage is increased. By applying the symmetrization scheme proposed in

subsection 2.2.5 to the lesser Green function, we restore the relation G() =

G ()

.

This is sufficient to obtain a real value for

n

, as clearly shown in figure 2.2.8. All other

symmetry relation are not violated here and thus, our symmetrization procedure does not

affect them at all. Interestingly, taking only the real part of

n

provides identical results

48


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when compared to the results obtained after the full symmetrization procedure. However, this

is only true for the simple case of the single site impurity model and does not hold for more

complex systems.

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

/

0

0.1

0.2

DOS(1/)

Figure 2.2.7: Density of states (DOS) of the spin up electron as calculated using the symmetric-EOM approachfor different temperatures;= 200(red line),= 50(blue line) and= 5(black dashed line). The development

of Kondo peaks in the density of states as the temperature decreases is clearly evident. The DOS of the spin

up electron calculated from the unsymmetrized Green function is identical provided only the real part ofn

was used in the calculations. Parameters used are similar to those used in Refs. [158, 186], exceptU that was

chosen to be large (U= 10) but not infinite. Other parameters (in units of = L + R): L= 3/10,R= 0,

,= 2. The bands are modeled as a Lorentzian with a half bandwidth 100.

2.2.5.2 The double Anderson model: symmetric-EOM approach

While for the case of a single site Anderson model only the relation G() =

G ()

breaks down, in the double Anderson model we find that all 3symmetries described by equa-

tion (2.2.34) are violated. This can be traced to the more complex form of the Hamiltonian for

the double Anderson model, where each site is only coupled to one of the leads and transportin enabled by the direct hopping term between the two sites.

Similar to the case of the Anderson model, as a result of symmetry breaking the occupation

of the levels n is a complex number. In addition, the coherences,= ih2

G()

G()< is

preserved. Indeed, in the case of a single site Anderson model, even if symmetry is not restored,

this relation holds and the two calculations yield identical values for the current. However, in

the present case, all 3 symmetry relations are broken and thus, equations (2.2.76) and (2.2.77)

give different results for the current, as clearly evident in figure 2.2.10. More significantly is

the fact that equation (2.2.76) produces a finite value for the current even when the bias iszero, indicating the break down of the fluctuation dissipation relation. When symmetry is

restored (black line) the two calculations are identical, as they should be, and the violation of

the fluctuation dissipation relation is also resolved.

The symmetrization scheme proposed in subsection 2.2.5 is not a magic cure and, in fact,

does not resolve all issues of mater. It is well known that the lesser and greater Green functions

should obey a simple sum rule where the integral over the difference of their diagonal elements

should always sum to 1:

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imag

inarypart

0 0.1 0.2 0.3 0.4 0.5

e/U

0 0.1 0.2 0.3 0.4

e/U

realpart

Coherences

Figure 2.2.9: The imaginary (upper panels) and real (lower panels) parts of (blue line) and (red line)calculated before (right panels) and after (left panels) symmetry was restored. The solid black thin line in the

upper panels marks the zero axis. As expected, after symmetry restoration (left panels), Im

= Im

and Re

=Re

, while before symmetry restoration (right panels) these equalities are violated. See

figure 2.2.5 for parameters.

S=ih

2

d

(G ())

< (G ())>= 1. (2.2.129)

In figure 2.2.11 we plot the sum rule as given by equation (2.2.129) for the double Anderson

model where symmetry has been restored. A similar plot for the single site Anderson model

yields a value of 1 regardless of whether symmetry has been restored or not within the clo-

sure discussed above. However, in the case of the more evolved double Anderson model, even

when symmetry is restored and the Green functions obey all 3 relations described in equa-

tion (2.2.34), the sum rule is violated. Nonetheless, the sum m{,}

{,}

Sm=Ne, where

Ne= 4is the total number of electrons in the system at maximal occupancy, is indeed preserved

when symmetrization is restored.

In the remaining of the chapter, we compare 4 different closures for the EOM of the NEGF

for the double Anderson model, two of which are developed by us while the remaining two

can be found in the literature. We use the Green function obtained via the symmetric-EOM

approach to calculate the differential conductance of the model. We note in passing that this is

the simplest theoretical model that can describe transport phenomena in a donor-acceptor com-

plex or heterojunctions, which are the working principle of many electronic and optoelectronic

devices [194, 195].

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0 0.1 0.2 0.3 0.4e/U

0

2

4

6

8

10

12

Stationary

current

(e/h_)

Eq. (1)

Eq. (2)

Sym

10-2

Figure 2.2.10: IVcurves calculated using equations (2.2.76) and (2.2.77) before (blue and red lines respec-tively) and after (black line) applying the symmetry procedure suggested in subsection 2.2.5. See figure 2.2.6

for parameters.

2.3 Steady state conductance in a double quantum dot

array:

Assessing the symmetric-EOM technique for the NEGF

2.3.1 Overview

We study the role of different approximate closures to the EOM of the NEGF on steady state

properties (namely, the differential conductance) for a double quantum dot (QD) array, coupled

to two macroscopic leads [196] (the double Anderson model [113, 114]). Although general, this

model can be used to study transport through single diatomic molecules (where each atom is

represented by its conducting orbital [197] and even larger molecules and complexes [198, 199,

200] or structures [201].

Four closures are examined; two already proposed [114, 183] (we will refer to these closuresas approximations1and 4) and two developed by us (denoted as approximations 2and 3). The

results obtained from the different closures were compared to the results attained using a many-

particle Master Equation (ME) approach [202] adequate for weak hybridization (system-leads

couplings) and high temperatures [203, 204].

We find that in contrast to the simple case of a single site model (Anderson model) in which

different closures beyond the simplest Hartree approximation scheme [179] yield very similar

transport results [182] (steady state current and differential conductance as a function of the

applied bias voltage) at high temperatures, the double Anderson model yield very different

52


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0 0.1 0.2 0.3 0.4

e/U

0.9

1.0

1.1

S

S

Figure 2.2.11: S (blue line) andS (red line) calculated from the symmetrized lesser and greater Greenfunctions as a function of the bias voltage. The exact result should have been 1 (as marked by the solid black

line). Parameters used for the simulations in units ofU=U = U are: L=

L =

R =

R= 0.0025,

R= R=

L=

L= 0, h

=

h= 0.25, V = 0.1, = = 0.1, = = 0.175and = 80.

stationary currents and differential conductance curves for the different closures studied.

The performance of the different closures is analyzed in terms of the poles of the unperturbed

retarded Green function in comparison to the exact many-body result for the isolated system.

We find that one of the closures developed by us provides the most accurate description of the

poles and also the best overall agreement with the ME approach for all parameters studied inthis work. While these results are encouraging, a word of caution is in place. It is clear that

the conclusions drawn from the performance of the different closures for the single site model

cannot be extended directly to the two site model, in analogy, a suitable closure for the two

site model may fail in the larger systems. Thus, the study of larger arrays of QDs will require

either analysis along the lines sketched here or employ a different formalism. In what follows

we embrace the notation for the different Green functions described in appendix B.

2.3.2 Different closures of the EOM

The double Anderson model is described in subsection 1.3.3 and the relevant Green functions

can be found in appendix B. Here we chose U= U= U,V = V = V and h

= h

= h.

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The EOM in Fourier space of the retarded Green function is:

G()

R=

h (0 ())R

1+ h

G()

R+ V

G

()

R+VG()

R+ UG()

R

, (2.3.1)where 0/ ()(c.f. equations (2.2.110) and (2.2.111)) is the tunneling (from the left/right

lead) self energy (the label 0 refers to the limits U, V 0). Calculating the EOM for thehigher order Green functions will give rise to other 2-particle and 3-particle correlations:

G

()

R=

h V

0 ()

R1n + h

G

()

R,

hG

()

R+ h

G

()

R+ V

G()

R+ U

G()

R+

k

tk Fk()R tk Fk()R + tkFk()R .(2.3.2)

Henceforth, the superscript R and the implicit dependence on are omitted for brevity.

G = (h U 0)1

n + hG hG

+hG+ VG+ VG

+k

tkF

k tkFk+ tkFk

, (2.3.3)and

G =

h V 0

1 n

dd

+hG+ UG

+ VG

+ k tkF

k

tkF

k+ t

kF

k

. (2.3.4)

To continue, one has to formulate the equations for these new Green functions, which in turn

will lead to other (higher order) Green functions. This infinite hierarchy of equations needs to

be truncated at a certain level, a process which is referred to as closure. In general, closures

cannot be improved systematically. Furthermore, it is often difficult to assess, a priori, the

accuracy of a given closure. We now discuss several different closures which are physically

motivated, tractable, and some are commonly used in the context of transport.

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2.3.2.1 Approximation1

Following the derivation given in Ref. [183], the following approximations are made:

1. All3-particle Green functions set to zero.2. Simultaneous tunneling of electrons of opposite spins are neglected, i.e., all Fqrstk, F

qrstk

and Gqrstare also set to zero.

3. Remaining Green functions mixing leads and system operators are decoupled so

Fsk

,

= i

h

TC

ns () ck () d

,

ih

d1gLk( 1)

TC

d (1) ns (1) d

.

See sub-subsection 2.2.4.2, second assumption, for a short discussion regarding this ap-

proximation.

4. The remaining2-particle Green functions of the form Gs(, )are decoupled so Gs(,

) =ns ()G(, ).

The resulting equations (for the retarded Green functions) are given by:

G = (h 0)1

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