M
ODELING
M
OLECUL AR
J
UNCTIONS
- W
EAK
M
ODELING
M
OLECUL AR
J
UNCTIONS
- W
EAK
AND
S
TRONG
C
OUPLING
R
EGIMES
Proefschrift
ter verkrijging van de graad van doctor aan de Technische Universiteit Delft,
op gezag van de Rector Magnificus Prof. ir. K. C. A. M. Luyben, voorzitter van het College voor Promoties,
in het openbaar te verdedigen op maandag 17 december 2012 om 10:00 uur
door
Fatemeh M
IRJANI
Master of Physics, Isfahan University of Technology geboren te Isfahan, Iran.
Dit proefschrift is goedgekeurd door de promotor: Prof. dr. ir. H. S. J. van der Zant
Copromotor: Dr. J. M. Thijssen Samenstelling promotiecommissie:
Rector Magnificus, voorzitter
Prof. dr. ir. H. S. J. van der Zant, Technische Universiteit Delft, promotor Dr. J. M. Thijssen, Technische Universiteit Delft, copromotor Prof. dr. ir. L. M. K. Vandersypen, Technische Universiteit Delft
Prof. dr. M. A. Ratner, Northwestern University Prof. dr. H. Akbarzadeh Isfahan University of Technology Prof. dr. J. E. Inglesfield Cardiff University
Dr. F. C. Grozema, Technische Universiteit Delft
Prof. dr. ir. T. M. Klapwijk Technische Universiteit Delft, reservelid Prof. dr. M. A. Ratner heeft als begeleider in belangrijke mate aan de totstandkom-ing van het proefschrift bijgedragen.
Keywords: Molecular Electronics, Single-Molecule Junction, Green’s Function, Transition Voltage Spectroscopy, Self-assembled Monolayer, Raman Spectroscopy
Printed by: Gildeprint Drukkerijen - The Netherlands
Cover design: by Maryam Taghipour, Benzenedithiol and gold electrodes.
Copyright © 2012 by F. Mirjani
Casimir PhD Series, Delft-Leiden 2012-23 ISBN 978-90-8593-139-3
An electronic version of this dissertation is available at
When You And I Behind The Veil Are Past, Oh But The Long Long While The World Shall Last, Which Of Our Coming And Departure Heeds As The Sea’s Self Should Heed A Pebble-cast.
Robaiyat of Omar Khayyám Translated by Edward FitzGerald
PREFACE
While this preface includes no scientific content, it will probably be the most read part of this thesis. In spite of the fact that an extensive preface may be preferred by some readers, I would like to share only a few thoughts and feelings about my PhD journey.
First, during this journey I had the opportunity to contribute in the fast-growing and interesting field of Molecular Electronics. Like many PhD researchers, I en-countered real challenges coming one after another. Through many long week-ends and sleepless nights I struggled to face the scientific challeneges arising from my research, supported along the way by my supervisor. Beside these academic challenges for my mind, were physical challenges that resulted in Repetitive Strain Injury (RSI). However, this journey has been filled with many pleasant moments which will also remain with me. These moments range from the very simple, like finding a bug in the code after one week of searching, to the extraordinarily pleas-ant, of having the opportunity to work with the group of the father of Molecular
Electronics at Northwestern University. All in all, my true contribution to the field
may seem like a dent in the edge of the human knowledge circle. Although this dent may seem marginal, innovation never stops and other researchers may find this dissertation as a basis for their future work, which shall be the merit of this thesis.
The other feeling I wish to express is about people. Words cannot express my heartfelt gratitudes to those whose profound impacts deserve special acknowl-edgement; my supervisor, promoter, collaborators, friends, colleagues and of course
my family, devoted parents, beloved sister & brother and loving husband and his family. I would not have had the courage to embark on this journey without your
help, support and encouragement. I am indebted to all of you for all the guidance, kindness, patience, inspiration, and discussion. Thanks for listening to me, letting me make mistakes, sharing your expertise with me, exhilarating my life, support-ing me in the moments of frustration and teachsupport-ing me how to think and how to be a better person both in my professional and private life.
Fatemeh Mirjani Delft, December 2012
C
ONTENTS
1 Introduction 1 1.1 Molecular electronics . . . 1 1.2 Single-Molecule device . . . 2 1.3 Experimental approaches . . . 2 1.4 Theoretical approaches . . . 41.5 Energy scales of molecular junctions . . . 4
1.6 The strength of the coupling to the electrodes . . . 6
1.6.1 Strong coupling regime: Green’s function . . . 7
1.6.2 Weak coupling regime: constant interaction model . . . 9
1.7 Thesis outline . . . 12
References . . . 13
2 Many-body Green’s function approach 17 2.1 Introduction . . . 18
2.2 Single level inside the bias window . . . 18
2.2.1 Non-interacting system . . . 19
2.2.2 Tunneling coupling . . . 21
2.2.3 Interacting system . . . 22
2.3 Two levels inside the bias window . . . 24
2.4 The density and the current . . . 26
References . . . 27
3 DFT-based many-body tight-binding model 29 3.1 Introduction . . . 30 3.2 Model . . . 32 3.2.1 spinless fermions . . . 34 3.2.2 spin-1/2 fermions . . . 34 3.2.3 Method . . . 35 3.2.4 Effective coupling . . . 36
3.2.5 Outline of the method . . . 38
3.3 Results for a single level inside the bias window . . . 39
3.3.1 Spinless fermions . . . 39
3.3.2 Spin-1/2 fermions . . . 41 ix
x CONTENTS
3.4 Results for two levels inside the bias window . . . 43
3.5 Implementation of the method into a quantum chemistry code . . . 48
3.6 Conclusions . . . 49
References . . . 50
4 Transition voltage spectroscopy 55 4.1 Introduction . . . 56
4.2 Model and method . . . 58
4.3 Results and discussion . . . 59
4.3.1 Length dependence of Vmin . . . 59
4.3.2 (A)Symmetry . . . 65
4.4 Conclusions . . . 69
References . . . 70
5 Modeling the influence of molecular structure 73 5.1 Introduction . . . 74
5.2 DFT calculations . . . 76
5.2.1 DFT-NEGF based transport calculations . . . 76
5.2.2 Molecular orbitals . . . 78
5.3 Tight-binding Model . . . 81
5.4 Finding the tight-binding parameters . . . 84
5.5 Results for the tight-binding model . . . 86
5.5.1 Molecular orbitals . . . 86
5.5.2 Current . . . 89
5.6 Conclusions . . . 91
5.7 Appendix . . . 92
References . . . 93
6 Raman spectroscopy in molecular junctions 97 6.1 Introduction . . . 98
6.2 Method . . . 99
6.3 Results and discussion . . . 101
6.3.1 LUMO transport . . . 105
6.3.2 HOMO transport . . . 111
6.4 Conclusions . . . 116
References . . . 117
A Bethe-Ansatz solution for the Hubbard model 123 A.1 Spinless fermions . . . 123
A.2 L(S)DA-DFT for the Hubbard model . . . 124
CONTENTS xi B LDA-NEGF approach 129 B.1 LDA . . . 129 References . . . 131 Summary 133 Samenvatting 135 Curriculum Vitæ 139 List of Publications 141
1
I
NTRODUCTION
1.1
MOLECUL AR ELECTRONICS
T
he fast developments in the field of semiconductor technology in the recent past have proved that the efforts to miniaturize transistors will reach physical limits due to their fabrication methods [1]. Therefore, the trend in the miniatur-ization has raised the question whether it is possible to utilize organic elements in nanoelectronic structures. Since single molecules constitute the smallest possible stable structures, they are candidates for active elements in electrical circuits. In addition they offer some possibilities that are not found in conventional solid state devices. For instance, some molecules can provide new functionality due to their special structural properties. Lower cost, compatibility with flexible substrates and simpler packaging when compared to conventional inorganic electronics are other motivations to use molecules in electronic circuits.On the other hand, using molecules in electrical circuits may have distinct dis-advantages such as instability at high temperatures. Ultimately this means that molecules though offering great potential also introduce new complications, spe-cially from the point of view of fabrication and therefore it is not really clear whether individual molecular devices can be integrated into larger-scale computing cir-cuits. In any case, if such circuits will be realized, they will be very different from what we are used to in today’s systems. At this stage, a fundamental understanding of molecular devices, based on quantum mechanics, is not yet complete, but will be necessary for such a realization.
{ {
1
2 1. INTRODUCTION SourceGate
V
V
G DrainFIGURE1.1: Three-terminal device lay-out taken from [7]. The blue sphere represents the island, cou-pled to the source and drain electrodes. The gate voltage is shown by VGand the bias voltage, V is applied to the source and drain.
1.2
SINGLE-MOLECULE DEVICE
T
he first molecular electronic device was proposed by Aviram and Ratner in 1974 [2]. The structure of Aviram and Ratner’s rectifier was similar to a p-n junction. It consisted of an acceptor pi system (p-type) and a donor pi system (n-type), separated by a sigma-bonded (methylene −CH2) tunneling bridge to avoidstrong interaction of donor and acceptor. They proposed that a rectifier can be built based on the idea that electrons are allowed to pass from a cathode to an ac-ceptor site or from a donor site to an anode, but not in the other direction. It took a long time after this proposal before the dream of contacting molecules in molecular junctions was achieved in the mid-1990s [3–6]. In the next section, we introduce some experimental procedures for contacting single molecule with nano- electrodes.
1.3
E
XPERIMENTAL APPROACHES
D
uring the last two decades, different techniques have been used to fabricate molecular electronic devices, each with its own advantages and disadvan-tages. Some of these techniques can only provide a ‘two-terminal’ device in which a nanogap is created between the source and drain electrodes between which a molecule can be trapped. However, a third electrode, called gate electrode, can be added to such a device (shown in 1.1). Such a gate electrode can be used to shift the molecular energy levels up or down. Such a device is called ‘three-terminal’. The fabrication techniques differ in the way the nanogap or the molecularjunc-1.3. EXPERIMENTAL APPROACHES 3 { {
1
V Tip Sample Tunneling Current SubstrateFIGURE1.2: Schematic of STM experiment. When a conducting tip is brought very near to the surface to be examined, a bias voltage applied between the two metals can allow electrons to tunnel through the sample (molecule) between them.
tion is created.
For two-terminal devices, the common method is to use scanning tunneling mi-croscope (STM) [3,4,6]. The STM is based on the concept of quantum tunnel-ing. When a conducting tip is brought very close to a metal surface covered with a (mono) layer of molecules, a bias voltage applied between the two metals allows electrons to tunnel through one of the molecules between them (see Fig. 1.2). The resulting tunneling current reflects the properties of the molecular junction. In this setup, the coupling of the molecule to the surface is stronger than to the tip.
A popular technique to make three-terminal devices, is electromigration in which a voltage between two contacts, connected by a nanowire is ramped up [8]. The current then causes the wire to break and form a nanogap (see Fig. 1.3). The un-controllable nanogap geometry or size can be improved by applying a feedback loop. The gate coupling in this method can be large depending on the thickness of the oxide layer between the source/drain electrodes and the gate.
Another common technique is the mechanical controllable break junction (MCBJ) which has also provided the possibility to integrate a gate electrode in the junction (see Fig. 1.3) [9,10]. In MCBJ, a suspended metallic wire is patterned on top of a flexible substrate. By bending the substrate, the metallic wire can be broken into two electrodes. Although the gate coupling in this method is low compared to elec-tromigrated junctions, the important advantage of this method is the precise con-trol of the gap size. Other techniques such as angle evaporation technique, dimer contacting method, self-assembled monolayers and lithographic techniques have been also reported as fabrication techniques over the past years [11–13].
These achievements resulted in what is nowadays known as Molecular Electronics. Molecular electronics is an interdisciplinary field which combines physics, chem-istry, material science, electronic engineering and biology at the nanoscale.
{ {
1
4 1. INTRODUCTION
1.4
T
HEORETICAL APPROACHES
R
apid developments in the field of molecular electronics have induced a con-siderable theoretical effort to investigate the underlying physical mechanisms. Theoretical approaches in the field of molecular electronics can be divided up into two categories. The first one is to use theoretical models known as toy models in which the aim is to capture the essential aspects of the physics involved. The sec-ond category includes the great variety of electronic structure methods used in quantum chemistry. These can be either first-principles, or semi-empirical. An important quantum chemistry method is known as LCAO (linear combination of atomic orbitals). In this method, energies of different orbitals and the coupling be-tween them are parameters which are determined either by ab-initio calculations or by semi-empirical calculations. If the atomic orbitals are sufficiently localized, we speak of the tight-binding methods.A widely used ab-initio method is density-functional theory (DFT). To study the transport through molecules, first-principles calculations based on DFT combined with the non-equilibrium Green’s function (NEGF) formalism, have been success-fully used for understanding coherent transport through molecules in the strong coupling and off-resonant regimes. However, these techniques are usually compu-tationally expensive and due to inherent limitations in DFT (as we will discuss in the next chapter), these approaches do not provide qualitatively correct transport features for molecules where electron correlations are strong, which is the case when molecules are weakly coupled to the electrodes. Furthermore, although the chemical details are taken into account in these approaches, due to the fact that the details of the junction geometry in molecular electronic experiments are not known, it is often difficult to capture the observed experimental phenomena. In particular, the way the molecule is bonded to the contact can affect the results and this varies from sample to sample. To improve some of the shortcomings in the-oretical approaches, people sometimes combine DFT with other computational and analytical tools, such as rate equations [14], the Anderson impurity model [15], the dynamic mean field theory [16], the density matrix and numerical renor-malization group [17,18]. A many-body quantum chemistry approach is the GW approximation [19,20] which takes dynamical screening effects successfully into account.
1.5
E
NERGY SCALES OF MOLECUL AR JUNCTIONS
T
he title of the thesis shows that this thesis is devoted to the transport in differ-ent coupling regimes. To categorize differdiffer-ent regimes in molecular junctions we start with description of four energy scales. These energy scales are (1) charging1.5. ENERGY SCALES OF MOLECULAR JUNCTIONS 5
{ {
1
(a) (b)
FIGURE1.3: Schematics of two techniques to make three-terminal devices, taken from Ref. [21]. (a) Electromigrated thin metal wire on top of the gate electrode (b) Gated mechanical break junction.
energy, (2) level spacing, (3) tunnel coupling and (4) thermal energy.
The small size of a molecule implies a sizable amount of energy to be associated with adding a unit charge on the molecule. This energy partly has a classical and a quantum component. The classical (electrostatic) component, the so-called
charg-ing energy is given by
EC= Q2
2C. (1.1)
where C = CS+CD+CGis the total capacitance of the molecule with environment
(including source (S), drain (D) and gate(G) electrodes). For a metallic sphere with radius r , the capacitance is given by C = 4π²0r . For r = 1 nm, similar to the
dimen-sion of a molecule, the charging energy is about 1.4 eV. The effect of the source, drain and the gate is to reduce this energy.
The quantum component, called the level spacing,∆ results from the spatial con-finement of the electron into the small molecular volume (particle in the box prob-lem). The level spacing is the result of the interaction between the nuclei and the electrons. In addition, partial charge transfer due to coupling to an electrode, can shift the energy levels of the molecule, and image charges on the electrodes may lead to a reduction of the level spacing [22]. For molecular transistors, this energy is typically of the order of 0.1 eV [7].
It should be noted that the two parameters EC and∆ both decrease with the size
of the molecule. Furthermore, the capacitance of the device can depend on the molecular level and therefore the charging energy is level dependent. In single-molecule spectroscopy, one often refers to the addition energy which is a combi-nation of both these energy scales; Eadd= 2EC+ ∆ (see Section 1.6.2 for details).
The third energy scale is given by the electronic coupling of the molecule to the electrodes,Γ which is in fact the overlap of the molecular orbital with the conduc-tion electrons of the electrode, leading to a broadening of that orbital. The total coupling of the molecule to the source and drain is defined asΓ = ΓS+ ΓDwhere S
{ {
1
6 1. INTRODUCTION
and D refers to source and drain respectively. It is important to note that there is an uncertainty relation betweenΓ and time τ, Γτ = ħ which determines the rate at which electron escapes from the device.
The temperature determines the last energy scale, the thermal energy which is given by kBT . At 300 K, the thermal energy is about 25 meV. The comparison with
the typical charging energy and level spacing shows that kBT < EC,∆.
1.6
T
HE STRENGTH OF THE COUPLING TO THE ELECTRODES
D
epending on how strongly the molecule is coupled to the electrodes relative to the temperature and charging energies, different regimes are accessed. Usu-ally three coupling regimes are distinguished in the literature which are labeled as weak coupling regime, intermediate coupling regime and strong coupling regime each with its own characteristics.IfΓ << EC,∆, the molecule is in the weak coupling regime (known as Coulomb
blockade regime) where charging effects play an important role. This regime is characterized by integer charge transfer. A two-dimensional plot of the current as a function of bias and gate voltage (stability diagram) in this regime gener-ates diamond-shaped regions of suppressed current, which are traditionally called
Coulomb diamonds. The straight lines in the plot of bias and gate voltages separate
regions of suppressed current from those with finite current [7].
In the opposite limit whereΓ >> EC,∆,kBT , the electronic states of the molecule
and the electrodes are strongly hybridized and the signatures of the Coulomb block-ade are washed out. The charge of the molecule in this regime is usually fractional. Between the weak coupling and strong coupling regimes a third regime can be identified which is known as the intermediate coupling regime whereΓ ≈ EC,∆. In
this regime it is still possible to observe Coulomb diamonds, but higher-order pro-cesses lead to a non-negligible current inside the blockade regions. Different types of higher-order tunneling processes such as elastic cotunneling and the Kondo ef-fect, which is a particular elastic many-body effect can occur in this regime [21]. In molecular junctions, chemical anchoring groups determine the tunnel coupling Γ. In fact, one of the interesting aspects of molecular electronics is that the bond-ing between a molecule and the electrodes can be engineered. This bondbond-ing has a dramatic influence on the current flowing through the device. The most com-mon chemical anchoring groups to connect molecules to the electrodes are thiol (−SH) and amine (−NH2) linkers. Other anchoring groups such as cyano (−C ≡
N) [23,24], isocyano (−N ≡ C) [25], pyridyl (PY, −C5H4N) [23], nitro (NO2) [26],
1.6. THE STRENGTH OF THE COUPLING TO THE ELECTRODES 7
{ {
1
[29] have also been used. In addition, the direct coupling of carbon atoms to goldelectrodes has also been realized [30]. Different anchoring groups lead to differ-ent stability, and bonding formation probability. For instance, recdiffer-ent experimdiffer-ents revealed the following sequence for junction formation probability and stability: PY > SH > NH2> CN [23]. In addition, the tunnel couplings of these linker groups
to the metallic electrodes are different. For instance, the thiol linker is known to yield stronger coupling than amine, and the molecular level broadening (which is a measure for the tunnel coupling) is found to be 40% smaller for the cyano an-choring groups than for thiol coupling [25]. However, the role of anan-choring groups in the molecular junction is not only to control the strength of the coupling to the metal. They can also have strong influence on the chemical potentials of the fron-tier orbitals with respect to the Fermi level of the contacts. For example in short molecules like benzene rings or their derivatives, replacing the thiol or amine link-ers by nitrile (cyano), pyridine or nitro group can change the level closest to the Fermi energy of the contact and therefore, alter transport from HOMO to LUMO where the HOMO and the LUMO denote the highest occupied molecular orbital and the lowest unoccupied molecular orbital [26,31,32]. Hence, the anchoring groups can be utilized as chemical means to engineer the transport characteristic of these devices.
Although the level of understanding achieved in the field of molecular electronic is certainly remarkable, there are still many problems to be solved in all three regimes. From the experimental point of view, the reproducibility of the results, the stability of the contacts, etc. are still challenging problems. From the theoret-ical side, the interaction between the electrons complicates the description of the charge transfer process. For example, in the weak coupling regime, correlation ef-fects such as Coulomb blockade or the Kondo effect may appear which complicate the problem. The vibrational modes, in addition, can play an important role as they can influence the current-voltage characteristic of molecular junctions. To understand these issues, it is useful to use simple theoretical models or to use ap-proaches such as ab-initio quantum chemistry methods. Theoretical models can take correlations and effects of vibrations into account. However, ab-initio quan-tum chemistry methods make severe approximations or leave these effects out, but include the full chemistry.
1.6.1
S
TRONG COUPLING REGIME: G
REEN’
S FUNCTIONThe Green’s function is named after the British mathematician George Green, who first developed the concept in the 1830s. In mathematics, a Green’s function is an operator used to solve inhomogeneous differential equations. For a non-interacting
{ {
1
8 1. INTRODUCTION
Hamiltonian H , in the time domain, the Green’s function G reads (i ħ ∂
∂t− H)G = δ(t ) (1.2)
which after Fourier transforming, turns into
G = [ω1 − H]−1 (1.3)
where1 is the unit matrix. As an example for a one energy level device, the GF for this system is G(ω) =ω−ε1
1 whereε1denotes the energy of the single level. This function is undefined atω = ε1. The retarded and advanced Green’s functions are
well-defined on the real axis by adding an infinitesimal parameter to the equation,
G = [(ω ± i0+)1 − H]−1, (1.4)
where the term i 0+and similarly −i 0+in G†result in two opposite domains which
are known as retarded (Gr) and advanced (Ga) Green’s functions. These GFs in the time domain can be obtained by a Fourier transform which satisfies the following Schrödinger equation:
(iħ∂
∂t− H ± i0+)G(t ) = δ(t)1, (1.5)
which means that we can view the Green’s function as the impulse response of the Schrödinger equation. The difference between these Green’s functions (times i) is called the spectral function A(ω) = i(G −G †).
If the device is coupled to a reservoir, the Green’s function for the system reads
G = [(ω + i 0+)1 − H − Σ]−1, (1.6) whereΣ = τGRτ†is the self-energy that gives the effect of the coupling to the reser-voir andτ and GRare the coupling of the molecule to the reservoir and the Green’s function of the reservoir respectively .
For a system with a single level, we have (i ħ∂
∂t− ε1− Σ)G(t ) = δ(t ). (1.7)
Using the Fourier transform G(t ) =R∞
−∞e−iω/ħtG(ω)dω/ħ, one obtains
1.6. THE STRENGTH OF THE COUPLING TO THE ELECTRODES 9
{ {
1
whereθ(t) is the step function. The self-energy, Σ is in general a complex.There-fore the GF can be written as
G = e−iε0/ħte−Γ/(2ħ)tθ(t), (1.9) whereΓ = −2ImΣ and ε0= ε1+ Re(Σ).
This shows that the physical meaning of the real part ofΣ is an energy shift in the molecular level device. The imaginary part (times -2) is the couplingΓ. This parameter is related to the broadening of the level. Let’s calculate the density of states (DOS), D(ω) = A(ω)/(2π) to see the broadening. For G(ω) = ω−ε01+iΓ/2, the DOS can be obtained via the spectral function as
D(ω) = Γ/(2π)
(ω − ε0)2+ (Γ/2)2 (1.10)
which is exactly the Lorentzian broadening.
The physical meaning of i0+can be understood as follows: For a reservoir of finite size, the DOS consists of a series of delta functions (spikes). If the molecule is cou-pled to such a reservoir, every electron will return after some finite time to it – the reservoir therefore is reversible. If, on the other hand, the size of the reservoir is taken to infinity, the distance between the spikes vanishes and any positive broad-ening i0+will turn into a continuous DOS. In fact, i0+is the signature of a decaying state e−0+ton the molecule.
Once the Green’s functions are known, the current can be calculated by the Lan-dauer formula:
I =ie h
Z
( f (ω,µS) − f (ω,µD))T (ω)dω (1.11)
in which T (ω) = Tr{ΓLGrΓRGa} is the transmission and f is the Fermi function of
the source and drain electrodes. This formalism known as the non-equilibrium Green’s function method usually does not include the Coulomb interactions. In chapter 2, we extend this formalism to include systems with Coulomb interactions.
1.6.2
W
EAK COUPLING REGIME:
CONSTANT INTERACTION MODELIf the molecule is weakly coupled to the source and drain electrodes, i.e. Γ <<
EC,∆, the charge on the molecule is characterized by integer multiples of e.
Trans-port through such a system can be described within the constant interaction model [33], where it is assumed that the capacitances to the electrodes are independent of the charge state.
In the system consisting of molecule, source, drain and gate electrodes, elemen-tary electrostatics gives the following relation between the different potentials and
{ {
1
10 1. INTRODUCTION
the charge Q on the molecule
CV −CSVS−CDVD−CGVG= Q, (1.12)
where C = CS+ CD+ CG is the total capacitance of the device. Therefore, the
po-tential on the molecule is
V =Q C+
CSVS+CDVD+CGVG
C . (1.13)
Available states for transport can be characterized by their chemical potential. The chemical potential is the difference between the total energy of the system with different numbers of electrons:
µ(N + 1) = E(N + 1) − E(N). (1.14)
For a system with N + 1 electrons, the total energy can be calculated as
EN +1= ε1+ ε2+ ... + εN +1+ Z −(N +1)e 0 V dQ = N +1 X n=1 εn+(N + 1) 2e2 2C − (N + 1)e µC SVS+CDVD+CGVG C ¶ (1.15) whereεi denotes the energy of level i and the last term is the charging energy
which accounts for the Coulomb interaction between the electrons. The chemi-cal potential, i.e. the difference between EN +1and EN, can be found as
µ(N + 1) = EN +1− EN= εN +1+ (2N + 1) e2 2C− e µC SVS+CDVD+CGVG C ¶ (1.16) and consequently, the addition energy which we define as the difference between the chemical potentials of N + 1 and N electrons, is:
Eadd= ∆EN →N +1− ∆EN −1→N= εN +1− εN
| {z } level spacing,∆N + e 2 C |{z} 2EC (1.17)
The first term is the level spacing and the second term is twice the charging energy. In transport, we can either add or remove an electron to or from the molecule. In the former case, the associated chemical potential is E (N )−E(N +1), whereas in the latter, it is E (N )−E(N −1). Janak’s theorem states that the derivative of the total en-ergy with respect to the density, determines the enen-ergy of the orbitals, d E /d ni= ²i.
1.6. THE STRENGTH OF THE COUPLING TO THE ELECTRODES 11
{ {
1
leads to the identification of the HOMO level with the chemical potential for ‘holetransport’, i.e. when the electron needs to leave the HOMO to enable transport. On the other hand identifying the LUMO level with the chemical potential describes ‘electron transport’ in which the electron needs to jump from the lead onto the LUMO in order to move from one lead to the other. In practice, in molecules, there is a substantial difference between the frontier levels and the discrete derivatives, so that the HOMO and LUMO levels may be quite far from the discrete derivatives, which represent the ionization potential and addition energies as follows:
IP = EN −1− EN (1.18)
EA = EN− EN +1 (1.19)
Eadd= IP − EA = EN +1+ EN −1− 2EN (1.20)
Further details about the shortcomings of DFT will be discussed in section 3.1. In the weak coupling regime, at low bias, there is generally not enough energy to change the charge state of the molecule and the current is suppressed; there is a Coulomb blockade [21,34,35]. At higher biases, when a level is located within the energy window formed by the chemical potentials of the source and drain electrodes, the blockade can be lifted and electrons may tunnel on and off the molecule sequentially, leading to a sizable current. An expression for the current in this regime can be derived from a set of rate equations (also called master equa-tions) for the occupation probability of the molecular states. The evolution of the occupation probability of state |α〉 is given by
d P d t =
X
β
(Wβ→αPβ− Wα→βPα), (1.21)
where Wα→βis the transition rate from state |α〉 to state |β〉. This can be written in matrix form as
d P
d t = W.P (1.22)
with the following coefficients for the matrix W:
Wαβ= Wβ→α if α 6= β (1.23)
Wαα= − X
β6=α
Wα→β (1.24)
The rates reflect the processes in which an electron jumps to or from lead S or D. For example, if the addition energy associated with a process which involves an electron jumping from the source to the molecule isµαβ, the rate for that process is proportional to the occupation of levelµαβand to the hopping rateΓ to and from
{ {
1
12 1. INTRODUCTION
the source, yielding Wα→β,S = fS(µαβ)Γα→βS , where fS(µαβ) ≡ fαβS is the Fermi-Dirac distribution function of lead S which is kept at a a fixed chemical potential −V /2:
fαβS = 1
1 + exp((µαβ+ V /2)/kBT )
. (1.25)
Considering all such processes then leads to the total rate
Wβα=1
ħ(Γ
S
αβfαβS + ΓDαβfαβD), (1.26)
where for the drain we obviously have
fαβD = 1
1 + exp((µαβ− V /2)/kBT )
. (1.27)
The current can then be found as I = eP
αβWαβPα. We refer the reader to Refs
[33,36,37] for reviews on the rate equations.
1.7
T
HESIS OUTLINE
T
his thesis describes research in both categories of toy models and ab initio methods. First, the many-body Green’s function for a generic toy model is pro-posed and developed.Chapter 2, introduces this many-body method using a NEGF formalism in which the Coulomb interaction is fully taken into account. This enables description of the weak coupling limit using Green’s functions. In Chapter 3, the many-body method is applied to a one-dimensional tight-binding chain with Coulomb interaction on each site (Hubbard model).
In chapter 4, the model described in chapter 3 is applied to study the Transition Voltage Spectroscopy (TVS) method. This is a method which was proposed for an-alyzing experimental results in a way which would reveal the chemical potentials of transport levels before they actually enter the bias window. Chapter 4 focuses on two essential questions that were brought up recently: the length dependence of the transition voltage and the effect of asymmetric capacitive coupling of a molec-ular junction. Moreover a proposed novel method is shown to find both the dom-inant transport level energy and the asymmetry degree from TVS. The limitations of TVS as an analysis tool are also discussed.
In chapter 5, the influence of molecular structure on charge transport through self-assembled monolayers is studied, following a recent experiment by Yoon et al
REFERENCES 13
{ {
1
[38] on a series of self-assembled monolayers. In this chapter, we develop atight-binding Hamiltonian to analyze the surprising results of that experiment that sig-nificant variation of the chemical structure does not lead to substantial variation of the transmission.
In recent years, a few pioneering studies have appeared on combining molecular transport measurements with Raman spectroscopy and this burgeoning field is ex-pected to lead to many new exciting experiments in the near future. In chapter 6, the Raman spectroscopy in molecular junctions is studied. In particular, the Ra-man spectrum of different charge states of a series of molecules is investigated.
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[14] J. S. Seldenthuis, H. S. J. van der Zant, M. A. Ratner, and J. M. Thijssen, Vibra-tional Excitations in Weakly Coupled Single-Molecule Junctions: A Computa-tional Analysis, ACS Nano 2, 1445 (2008).
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[18] P. Schmitteckert and F. Evers,Exact Ground State Density-Functional Theory for Impurity Models Coupled to External Reservoirs and Transport Calcula-tions, Phys. Rev. Lett. 100, 086401 (2008).
[19] M. Strange, C. Rostgaard, H. Häkkinen, and K. S. Thygesen, Self-consistent GW calculations of electronic transport in thiol- and amine-linked molecular junctions, Phys. Rev. B 83, 115108 (2011).
[20] K. S. Thygesen and A. Rubio,Nonequilibrium GW approach to quantum trans-port in nano-scale contacts, The Journal of Chemical Physics 126, 091101 (pages 4) (2007).
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[22] K. Kaasbjerg and K. Flensberg, Strong Polarization-Induced Reduction of Addition Energies in Single-Molecule Nanojunctions, Nano Letters 8, 3809 (2008).
[23] W. Hong, D. Z. Manrique, P. Moreno-García, M. Gulcur, A. Mishchenko, C. J. Lambert, M. R. Bryce, and T. Wandlowski,Single Molecular Conductance of Tolanes: Experimental and Theoretical Study on the Junction Evolution De-pendent on the Anchoring Group, Journal of the American Chemical Society 134, 2292 (2012).
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[25] E. Lörtscher, C. J. Cho, M. Mayor, M. Tschudy, C. Rettner, and H. Riel,Influence of the Anchor Group on Charge Transport through Single-Molecule Junctions, ChemPhysChem 12, 1677 (2011), ISSN 1439-7641.
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[38] H. J. Yoon, N. D. Shapiro, K. M. Park, M. M. Thuo, S. Soh, and G. M. Whitesides, The Rate of Charge Tunneling through Self-Assembled Monolayers Is Insensi-tive to Many Functional Group Substitutions, Angewandte Chemie Interna-tional Edition 51, 4658 (2012), ISSN 1521-3773.
2
M
ANY
-
BODY
G
REEN
’
S FUNCTION
APPROACH
In this chapter, a many-body method is introduced to treat transport through a sin-gle molecule connected to two conducting leads in the weak and intermediate cou-pling regimes. These regimes are not accessible to standard non-equilibrium Green’s function (NEGF) calculations. The method is described for non-interacting and in-teracting systems. In addition, explicit expressions for occupation of the levels and the current are shown. This method provides the possibility to treat transport in the weak and intermediate coupling regimes using Green’s functions as will be described in chapter 3.Parts of this chapter have been published in Phys. Rev. B 83, 035415 (2011) [1].
{ {
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18 2. MANY-BODYGREEN’S FUNCTION APPROACH
2.1
INTRODUCTION
I
n the previous chapter, we discussed the limits of strong and weak tunnel cou-pling of a molecule to the source and drain electrodes. In the case of strong coupling, the standard non-equilibrium Green’s function can be combined with DFT to investigate the transport through molecules. In such a regime, the Lan-dauer formula is used to calculate the current flowing through the system. In the opposite limit, where Coulomb blockade occurs, transport is through individual orbitals, and the constant interaction model can be used to study the system. In this regime, rate equations combined with DFT can be applied to molecular junc-tions but the coupling of the molecule to the leads is usually considered as a fitting parameter in this approach [2,3]. Moreover it does not show the broadening due to the coupling.As was noted in chapter 1, for weak and intermediate couplings, the Coulomb interaction influences transport significantly and therefore it should be included in the theoretical description. In this chapter, we introduce a many-body non-equilibrium Green’s function formalism in which Coulomb interactions are included. This will provide the possibility to study the molecules in weak and intermediate coupling regimes using Green’s function in combination with DFT. We will investi-gate the validity of this approach for a toy model in chapter 3.
The organization of this chapter is as follows. The Green’s function for a Coulomb island with a single orbital level is described in section 2.2 for non-interacting sys-tems, systems with tunnel coupling and interacting systems. Then the transport for the case of two levels inside the bias window is discussed in section 2.3. It should be noted that in this chapter we use the term ‘level’ to indicate a chemi-cal potential corresponding to an energy resonance on the molecule.
2.2
S
INGLE LEVEL INSIDE THE BIAS WINDOW
I
n this section, the Green’s functions for a molecular device are derived where one single level exist inside the bias window [4]. The Hamiltonian for such a system is given byH = Hmolecule+ Hleads+ Hcoupling (2.1)
where the first term is the Hamiltonian of the molecule, the second that of the leads, and the molecule is coupled to the leads by Hcoupling(see Fig. 2.1).
We start with the simplest model, i.e. a non-interacting isolated system, and then its coupling to the leads is included. After that, the Coulomb interaction is added to the molecular Hamiltonian.
2.2. SINGLE LEVEL INSIDE THE BIAS WINDOW 19
{ {
2
FIGURE2.1: A molecule, coupled to the source and drain electrodes.
2.2.1
N
ON-
INTERACTING SYSTEMWe start with a non-interacting and uncoupled system and we explain how the GF can be derived from the equation of motion (EOM). In general, the Hamiltonian for a non-interacting, uncoupled system is given by
Hmolecule= X σ X i ²iσdi†σdiσ (2.2)
whereσ is a spin index on site i and di†and diare creation and annihilation
oper-ators acting on site i which satisfy the usual anticommutation relations. In order to find the current through a system with single level, we use non-equilibrium GF theory, which focuses on the one-particle GF on the molecule, defined as
Gαβ= −i〈T {dα(t )d†β(t0)}〉 (2.3) whereα and β denote two electrons of the single level device and T is the time-ordering operator
T {A(t )B (t0)} = θ(t − t0)A(t )B (t0) ∓ θ(t0− t )B(t0)A(t ) (2.4)
T always moves the operators with earlier time argument to the right.
The reason that we focus on the many-body GF to solve the problem is that exper-imentally relevant quantities can be extracted from the knowledge of the GF. We elaborate this subject in this chapter.
The first goal is to find the EOM for the Green’s function. To do this, we need to calculate i∂t∂Gαβ(t ) =∂t∂〈T {dα(t )dβ†(t0)}〉 which requires finding the derivative of d with respect to the time, ˙d .
Using the Heisenberg equation of motiond ˆd tA= −i [ ˆA(t ), H ] (for an operator ˆA) and
the following anti-commutation relations for the many-body operators ˆdαand ˆd† β
{ {
2
20 2. MANY-BODYGREEN’S FUNCTION APPROACH
{dα, dβ†} = δαβ (2.6)
We then obtain the derivative of ˆd for the non-interacting system as follows
i ˙dα= [dα(t ), H ] = εαdα (2.7)
Therefore the EOM for the Green’s function in the time frame can be written as id
d tGαβ(t − t
0) = δ(t − t0)δ
αβ+ εαGαβ(t − t0) (2.8)
After Fourier transformation of the time domain (using the convention f (ω) = R∞
−∞eiωtf (t ) d t ), the following equation for the GF is found :
(ω − εα)Gαβ(ω) = δαβ (2.9)
which yields the well-known form of the Green’s function (see section 1.6.1)
Gαβ(ω) = δαβ
ω − εα (2.10)
This Green’s function is undefined atω = εα. The retarded and advanced Green’s functions, Gr and Ga, are defined by adding the infinitesimal parameter,η, to the denominator (as shown in section 1.6.1):
Gr,aαβ(ω) = δαβ
ω − εα± i0+, (2.11)
which , in the time domain have the form,
Gr,a(t , t0) = ∓iθ(±(t − t0))〈{d(t),d†(t0)}〉 (2.12) The lesser and greater Green’s functions, G<and G>are defined as
G<(t , t0) = −i〈{d†(t0), d (t )}〉, (2.13) and
G>(t , t0) = i〈{d(t),d†(t0)}〉. (2.14) These Green’s functions obey the relation Gr− Ga= G>− G<. Each of these func-tions has its own properties and use. For example, G<,>are linked to the observ-ables such as density or current, whereas Gr,a contain the information about the spectral properties and scattering rates [4].
The spectral function formula introduced in chapter 1 (see section 1.6.1) still holds
A(ω) = i(Gr−Ga) = i(G>−G<). (2.15) Since (Gr)†= Ga, the spectral function is twice the imaginary part of the GF, A = −2Im(Gr).
2.2. SINGLE LEVEL INSIDE THE BIAS WINDOW 21
{ {
2
2.2.2
T
UNNELING COUPLINGWhen we consider the coupling to the leads, we must also take into account the terms generated by the tunneling Hamiltonian. In general, the Hamiltonian for the system with tunneling is given by
Hmolecule= X σ X i ²iσdi†σdiσ−X σ X i ti ,i +1[di ,†σdi +1,σ+ h.c.] (2.16)
Following the method of the previous section for obtaining the EOM and after Fourier transformation of the time domain, the following equation for the GF is found [4].
(ω − εα)Gαβ(ω) = δαβ+X
ηk
tηkαΓαβηk(ω), (2.17) whereα and β denote two electrons of the single level device and k labels the trav-eling wave states in the non-interacting leadsη = L,R where L and R stand for left and right and
Γαβηk(t − t0) = −i〈T {cηkσα(t )d †
β(t0)}〉 (2.18)
is the term added to the EOM due to the coupling to the leads.α denotes the spin-orbitalα and σαis the spin for thisα.
The time derivative of the c operator (for contacts) is found in a similar way as: i ˙cηkσα= εηkcηkα+X
α0
tηkαdα0. (2.19)
Using the EOM for cηkσα(t ), an equation forΓαβηk is found
(ω − εηk)Γαβηk(t − t0) = tηkαGαβ(ω). (2.20)
Using this to eliminateΓαβηk in Eq. 2.17, we arrive at
(ω − εα− Σ0(ω))Gαβ= δαβ, (2.21) where Σ0(ω) = X ηk |tη|2 ω − ²ηk (2.22)
is the self-energy. The self-energy has a real (Hermitian) part which has the effect of shifting the resonant energies. The imaginary (non-Hermitian) part broadens the resonances, reflecting the hybridization of the states of the central region with
{ {
2
22 2. MANY-BODYGREEN’S FUNCTION APPROACH
those of the leads.
Takingα = β and splitting the real and imaginary parts of the self-energy, Σr
0(ω) = Λ(ω) − iΓ(ω)/2, (2.23)
we can write
Gααr (ω) = 1
εα− ω − Λ(ω) + iΓ(ω)/2 (2.24) which forΛ and Γ independent of ω is a Lorentzian function.
2.2.3
I
NTERACTING SYSTEMNow, we consider an interacting system with spin-1/2 fermions. This means that the Coulomb interaction, U , is added to the Hamiltonian:
Hmolecule= − X σ ti ,i +1 X i [di ,†σdi +1,σ+ h.c.] +X i Udi ↑†di ↑di ↓†di ↓+X σ X i ²iσd†iσdiσ.(2.25)
The time derivative of the d operator (for the molecule) is (see [4]): i ˙dα= εαdα+X β6=α Uαβdαnβ+ X η=L/R q tηkα∗ cηkσα. (2.26)
This EOM leads to the following equation for the GF: (ω − εα)Gαβ(ω) = δαβ+X γ6=α UαγG(2) αγβ(ω) + X ηk tηΓαβ ηk(ω), (2.27)
while the equation forΓαβηk remains the same. We have introduced a new GF Gαγβ(2) , which is defined as Gαγβ(2) = −i 〈T {dα(t )nγ(t )dβ†(t0)}〉 with nγ(t ) = dγ(t )dγ†(t ). This GF satisfies an EOM: (ω − εα−Uαγ)G(2)αγβ= 〈nγ〉δαβ+X ηk (tη∗Γ(2)1,ηkαβ+ tηΓ(2)2,ηkαβ− tη∗Γ(2)3,ηkαβ) (2.28) where Γ(2)αβ 1,ηk = −i〈T {cηkσα(t )nγ(t )d † β(t0)}〉 (2.29a) Γ(2)αβ 2,ηk = −i〈T {cηkσγ(t )dα(t )dγ(t )d † β(t0)}〉 (2.29b) Γ(2)αβ 3,ηk = −i〈T {cηkσγ(t )d † γ(t )dα(t )dβ†(t0)}〉. (2.29c)
It should be noted that this EOM only holds for a single level. For two levels inside the bias window, it includes higher order Green’s function as we show in section 2.3.
2.2. SINGLE LEVEL INSIDE THE BIAS WINDOW 23
{ {
2
Hartree-Fock approximation
We neglect the correlation between the central region and the leads by setting Γ2= Γ3= 0 and keeping only Γ(2)1,ηkαβ which we approximate asΓ(2)1,ηkαβ= 〈nγ〉Γαβηk. This is known as the Hartree-Fock approximation. Note that this mean field ap-proximation only concerns the coupling between the central region and the leads, but not the Coulomb correlations within the central region. By substituting G(2) from (2.28) into (2.27) and eliminatingΓ by an equation like (2.20), it yields
Gαα(ω) = ω − εα− (1 − 〈nβ〉)U (ω − εα−U )(ω − εα) − Σr[ω − ε α− (1 − 〈nβ〉)U ] (2.30) whereΣr= ΣrL+ ΣrRand Σr jα(ω) = X ηk tηkα2 ω − εηk+ i0+ (2.31)
Since the eigenstates of the contacts for a linear chain can be written asεηk = −2tηkcos(k a), one could calculate the self-energy by doing the integration of
R 1/(ω + 2tηcos(k a) + i0+) d k. Using the Residue theorem to do this integration, one is left with
Σr jα(ω) = −t2j tη zj(ω) (2.32) and Im(zj) > 0, zj= −qj± q
q2j− 1, qj =ω−E2tF±V /2η , and EF is the Fermi energy of
the grounded lead.
The Keldysh equation
In order to calculate physical quantities such as charge densities and currents, we need the lesser Green’s function. This can be found from the lesser Green’s function of the leads and the advanced and retarded Green’s function on the molecule, using the Keldysh equation. The starting point of the derivation of the Keldysh equation is the Dyson equation which can be found if we invert G−1= G−10 − Σ,
G = G0+G0ΣG (2.33)
in which G0is the Green’s function of the isolated molecule without the Coulomb
potential and G is the Green’s function in the presence of interaction (i.e. the cou-pling to the electrodes and the Coulomb potential). In this equation the term G0ΣG
has the form of ABC . Using one of the so-called Langreth rules [5,6], for D = ABC we write
{ {
2
24 2. MANY-BODYGREEN’S FUNCTION APPROACH
and therefore [4],
G<= (1 +GrΣr)G<0(1 + ΣaGa) +GrΣ<Ga (2.35) The common Keldysh equation is known as a reduced form of the above equation, with only the last term on the right hand side, G<= GrΣ<Ga, as it can be shown for
transport through a single channel that the first term vanishes for the single parti-cle GF Gαβand one is left with the reduced form of Keldysh equation.
2.3
T
WO LEVELS INSIDE THE BIAS WINDOW
H
ere we explain how the transport calculation works for the case where we have two levels inside the bias window. We use a many-body approach which is a generalization of the technique described above. For the one-particle GF, similar to Eq. (2.27) we find : (ω − εα)Gαβ= δαβ+X γ6=α UαγG(2)αγβ+X α0Σαα 0Gα0β. (2.36)whereΣαα0is the self-energy
Σαα0= X
ηk
tηkα0tηkα∗
ω − ²ηk . (2.37)
For G(2)we obtain the EOM
(ω − εα−Uαγ)G(2)αγβ= 〈nγ〉δαβ+ X δ6=α,γ
UαδGαγδβ(3) +X
α0
Σαα0G(2)α0γβ (2.38)
whereα0denotes a particle with the same spin asα. Eq. (2.36) and (2.38) do not form a closed system as a new GF, G(3)αγδβ, is generated in deriving the equation for
Gαγβ(2) . The new GF, Gαγδβ(3) , is defined as
Gαγδβ(3) = −i 〈T {dα(t )nγ(t )nδ(t )dβ†(t0)}〉 (2.39) The EOM for G(3)αγδβreads
(ω − εα−Uαγ−Uαδ)Gαγδβ(3) = 〈nγnδ〉δαβ+ X ²6=α,γ
Uα²G(4)αγδ²β+X α0
Σαα0G(3)α0γδβ (2.40)
which introduces yet another new GF, G(4)αγδ²β
2.3. TWO LEVELS INSIDE THE BIAS WINDOW 25
{ {
2
for which the EOM is(ω − εα−Uαγ−Uαδ−Uα²)Gαγδ²β(4) = 〈nγnδn²〉δαβ+X α0
Σαα0G(4)α0γδ²β (2.42)
At this stage, the process of generating new GF stops, as the EOM for G(4)does not generate higher-order GFs.
We now must solve the set of equations (2.36), (2.38), (2.40) and (2.42) for the GFs
Gαβto G(4)αγδ²β. We organize these GFs into a 64 × 4 array GΛβ= (Gαβ,G(2)α0γβ,G (3) α00γ0δβ,G (4) α000γ00δ0²β) T (2.43)
As the indicesα and β run over four states, it is easy to see that the first index of this array runs over 4 + 12 + 24 + 24 = 64 values. The equation for G can be written in the form
G−1
0 G = 〈 ˜n〉 + ΣG (2.44)
Here,G0−1is a 64 × 64 matrix, which, in the frequency domain assumes the form
G−1 0 (ω) = ω−εα ω − εα0−Uα0γ ω − εα00−Uα00γ0−Uα00δ
ω−εα000−Uα000γ00−Uα000δ0−Uα000²
(2.45) and 〈 ˜n〉 is a 64 × 4 array including all many-body densities
(δαβ, 〈nγ〉δα0β, 〈nγ0nδ〉δα00β, 〈nγ00nδ0n²〉δα000β)T. (2.46) Finally,Σ is the 64 × 64 array with elements ΣαΛ,αΛ0, whereαΛdenotes the indexα of the composed indexΛ = (α,α0γ,α00γ0δ,α000γ00δ0²).
In order to find the lesser GFG<, from which 〈 ˜n〉 can be found, we should use a
Keldysh or Kadanoff-Baym equation. These equations are conveniently derived from the Langreth rules [7]. These rules apply to the GFG which is found from Eq (2.44). Therefore, using the notation of that equation, the Kadanoff-Baym equation can be written as
G−1
{ {
2
26 2. MANY-BODYGREEN’S FUNCTION APPROACH
whereGais found as
Ga
= (G0−1− Σ
a)−1〈 ˜n〉. (2.48)
For the case that the ‘higher’ GFs for instance G(3)etc. are included in the
calcula-tion, the Keldysh equation is,
G<= (〈 ˜n〉 + GrΣr)G<
0(〈 ˜n〉 + GaΣa) + GrΣ<Ga. (2.49)
It can be shown for transport through a single channel, the first term vanishes for the single particle GF Gαβ. However, this is not the case when the higher GFs are included (this was also pointed out by Song et al. [8]).
2.4
T
HE DENSITY AND THE CURRENT
I
n order to find the current, we need to calculate the density, n, which should be calculated self-consistently from G<,〈nα〉 = Z G<
αα(ω)
2πi dω. (2.50)
To calculate the lesser GF, G<(from the Keldysh or the Kadanoff-Baym equation), we first need to calculate the lesser self-energy which can be found asΣ<= i ΓLfL+ iΓRfR.
Once the retarded and advanced GF are known, the current can be calculated. The current from the left contact to the central region, can be calculated from the time evolution of the occupation number operator of the left contact,
IL= −e〈 ˙NL〉 = −ie/ħ〈[H, NL]〉 (2.51)
where NL=Pkck†ckand H is the Hamiltonian of the entire system. Calculating the
commutator and after some algebra [4], one arrives at
I =i e
2h Z
Tr{[ΓL(ω)−ΓR(ω)]G<(ω)+(f (ω,µL)ΓL(ω)−f (ω,µR)ΓR(ω))(Gr(ω)−Ga(ω))}dω
(2.52) for the total current where the symmetrized current is I = (IL+ IR)/2 andΓj = i (Σrj− Σrj†). Inserting
Gr−Ga= −iGr(ΓL+ ΓR)Ga (2.53)
to Eq. 2.54, and using the Keldysh equation for the lesser GF, G<= GrΣ<Ga =
Gr(iΓLfL+ i ΓRfR)Gaand making use of the cyclic property of the trace, one is left
with the Landauer formula
I = e h
Z
REFERENCES 27
{ {
2
in which T (ω) = Tr{ΓLGrΓRGa} is the transmission.REFERENCES
[1] F. Mirjani and J. M. Thijssen,Density functional theory based many-body anal-ysis of electron transport through molecules, Phys. Rev. B 83, 035415 (2011).
[2] C. W. J. Beenakker,Theory of Coulomb-blockade oscillations in the conductance of a quantum dot, Phys. Rev. B 44, 1646 (1991).
[3] J. S. Seldenthuis, H. S. J. van der Zant, M. A. Ratner, and J. M. Thijssen, Vibra-tional Excitations in Weakly Coupled Single-Molecule Junctions: A Computa-tional Analysis, ACS Nano 2, 1445 (2008).
[4] H. Haug and A. Jauho, Quantum Kinetics in Transport and Optics of
semicon-ductors (Springer, Berlin, 1995).
[5] D. C. Langreth and J. W. Wilkins,Theory of Spin Resonance in Dilute Magnetic Alloys, Phys. Rev. B 6, 3189 (1972).
[6] J. Sweer, D. C. Langreth, and J. W. Wilkins,Theory of spin resonance in dilute magnetic alloys. II, Phys. Rev. B 13, 192 (1976).
[7] D. C. Langreth, Linear and Non-Linear Response Theory with Applications ‘in
Linear and Nonlinear Electron Transport in Solids’, edited by J. T. Devreese and V. E. van Doren (Plenum Press, New York and London, 1976).
[8] B. Song, D. A. Ryndyk, and G. Cuniberti,Molecular junctions in the Coulomb blockade regime: Rectification and nesting, Phys. Rev. B 76, 045408 (2007).
3
DFT-
BASED MANY
-
BODY
TIGHT
-
BINDING MODEL
In this chapter, the many-body method described in chapter 2 is applied to a toy model with Coulomb interactions, i.e. the Hubbard model. The many-body param-eters required in the calculations are extracted from ground state DFT calculations. The current and density of the system are calculated for different systems such as spinless fermions, spin-1/2 fermions, a single level inside the bias window and dou-ble levels inside the bias window. The results of this method are compared with existing density matrix renormalization group (DMRG) calculations and the results of embedded-cluster approximation.
Parts of this chapter have been published in Phys. Rev. B 83, 035415 (2011) [1].
