theory of charge transport through vibrating molecules · theory of charge transport through...

Theory of Charge Transport Through Vibrating Molecules

by

Yelena Simine

A thesis submitted in conformity with the requirementsfor the degree of Doctor of Philosophy

Graduate Department of Department of ChemistryUniversity of Toronto

Copyright c© 2015 by Yelena Simine

Abstract

Theory of Charge Transport Through Vibrating Molecules

Yelena Simine

Doctor of Philosophy

Graduate Department of Department of Chemistry

University of Toronto

2015

In this Thesis we study several aspects of charge transport through single molecule junc-

tions. Focusing on the interaction of the electrons with vibrations we devise toy-models

which capture the consequences of many-body interactions for transport properties of

the junction. We begin with the study of heating, cooling and stability of the molecular

bridge. First, the dynamics of the vibrational mode are followed using the master equa-

tion. Using this approximate technique, we identify the regimes in which the electronic

current is pumping energy into the vibrational mode, or alternatively, absorbing energy

from it. An important conclusion of this study is that at large currents, the pumping of

energy into vibrations may result in uncontrolled heating, making the structure unstable;

Alternatively, at small biases, extraction of energy from vibrations is possible, resulting

in a cooling behavior. We extend the model to include a dissipative environment, such

as the vibrating backbone of the molecule or the solvent, and demonstrate that dissi-

pation may alleviate structural instability. From the comparison of the behavior of a

strongly anharmonic mode (two-level system) with a fully harmonic mode a surprising

conclusion is reached: the cooling/heating analysis in both cases produces qualitatively

similar results. Finally, we confirm the microscopic equivalent of the Second Law, the

charge and energy Fluctuation Theorem, at the level of second order markovian master

equation. In order to study exactly the vibration assisted electron transport through a

donor-acceptor junction we develop a numerical scheme based on the influence functional

ii

path integral (INFPI) and quasi-adiabatic path integral (QUAPI) methods. With this

tool, the results obtained with the perturbative kinetic master equation are confirmed

validating the approximations used there. We further extend the model to include direct

electronic tunneling. Encouraged by the similarity of transport behavior in harmonic

and strongly anharmonic junctions, we compare the non-equilibrium Anderson-Holstein

model of electron interacting with a harmonic mode to its variant, in which the infinite

ladder of the harmonic mode is truncated. In this case, interestingly, the two models show

significant differences: strong vibrational anharmonicity does not allow for the Franck-

Condon blockade or current hysteresis effects found in the harmonic model. Instead, it

introduces a novel many-body off-resonance blockade. We conclude with the study of

electronic decoherence due to the interaction with vibrations. To this end, we consider

three cases of electronic transport through a molecular junction: fully coherent (rigid

molecule), fully dephased and purely inelastic. We demonstrate that the three cases

produce characteristic energy profiles and may be distinguished from one another via the

measurement of the thermopower.

iii

Dedication

To Eugene and little Tali

iv

Acknowledgements

I thank my research advisor, Prof. Dvira Segal, for being a terrific mentor. I consider

myself lucky, and in the face of what my professional future may hold, I say “dayenu”.

I thank my committee members, Prof. Paul Brumer and Prof. Stu Whittington, for

involvement and advice, and Prof. Artur Izmaylov for tips on making presentations and

the fun visit to Scarborough. I thank Prof. Alex Hayat and Dr. Ilya Sutskever for their

teasing encouragement, and my fellow graduate students: Dr. Claire X. Yu, Dr. Salil

Bedkihal, Yaser Khan, Peter Colberg and Cyrille Lavigne for being there when I needed

to talk things over. Finally, I thank my family: Tali and Eugene who agreed to move to

Texas, and my parents, who thought all of this was a good idea. This thesis would not

be possible without you.

v

Contents

1 Introduction 1

2 Vibrational cooling, heating, and instability of conducting molecules 8

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.2 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.3 Anharmonic-mode Rectifier . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.3.1 Impurity dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.3.2 Resolved charge and energy equations . . . . . . . . . . . . . . . . 20

2.3.3 Fluctuation theorem . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.3.4 Currents, and measures for vibrational cooling, heating, or instability 25

2.3.5 Numerical results: isolated mode . . . . . . . . . . . . . . . . . . 26

2.3.6 Numerical results: dissipative mode . . . . . . . . . . . . . . . . . 31

2.4 Harmonic-mode Rectifier . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

2.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3 Numerically exact path-integral simulations: transport and dissipation 48

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.2 Path-integral formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

3.3 Iterative time evolution scheme . . . . . . . . . . . . . . . . . . . . . . . 55

3.3.1 Spin-boson-fermion model . . . . . . . . . . . . . . . . . . . . . . 55

3.3.2 Bosonic Influence Functional . . . . . . . . . . . . . . . . . . . . . 57

vi

3.3.3 Fermionic Influence Functional . . . . . . . . . . . . . . . . . . . 58

3.3.4 The iterative scheme . . . . . . . . . . . . . . . . . . . . . . . . . 59

3.3.5 Expectation values for operators . . . . . . . . . . . . . . . . . . . 62

3.3.6 Expressions for multilevel subsystems and general interactions . . 63

3.4 Application: Molecular Rectifier . . . . . . . . . . . . . . . . . . . . . . . 64

3.4.1 Rectifier Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . 64

3.4.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

3.4.3 Convergence and computational aspects . . . . . . . . . . . . . . 82

3.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

4 Electron transport with local anharmonic modes 89

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

4.2 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

4.2.1 N -state impurity . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

4.2.2 Case I: Harmonic oscillator . . . . . . . . . . . . . . . . . . . . . . 94

4.2.3 Case II: Two-level system . . . . . . . . . . . . . . . . . . . . . . 96

4.3 Kinetic equations for Γν < ω0, Tν . . . . . . . . . . . . . . . . . . . . . . 97

4.3.1 Unequilibrated impurity . . . . . . . . . . . . . . . . . . . . . . . 97

4.3.2 Thermally-equilibrated or dissipative impurity . . . . . . . . . . . 101

4.3.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

4.4 Adiabatic limit Γν > ω0 . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

4.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

5 Signatures of decoherence mechanisms in thermopower 125

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

5.2 Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

5.3 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

5.3.1 Coherent and fully dephased transport: Landauer-Buttiker formalism132

vii

5.3.2 Incoherent-inelastic transport: Master equation . . . . . . . . . . 135

5.4 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

5.4.1 Dependence of thermopower on the electronic hybridization Γ . . 137

5.4.2 Distinguishing between coherent and dephased transport . . . . . 139

5.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

6 Conclusions 143

A Full counting statistics in the dissipative anharmonic model 146

B Time-discrete Feynman-Vernon Influence functional 150

C Landauer formalism 151

C.1 Landauer formula for electronic and energy currents . . . . . . . . . . . . 151

C.2 Transmission coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

Bibliography 153

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List of Figures

2.1 Schemes of the two models considered in this Chapter. A biased donor-

acceptor electronic junction is coupled to either (i) a harmonic molecular

mode, or (ii) an anharmonic mode, represented by a two-state system. In

both cases the molecular mode may further relax its energy to a phononic

thermal reservoir, maintained at the temperature Tph. This coupling is

represented by a dashed arrow. Reproduced from Ref. [33]. . . . . . . . 9

2.2 Scheme of the vibrational mode excitation and relaxation processes. A full

circle represents an electron transferred; a hollow circle depicts the hole

that has been left behind. Reproduced from Ref. [33]. . . . . . . . . . . 15

2.3 (i) Charge current in a rectifying molecular junction. Inset: Energies

of the donor (full line) and acceptor states (dashed line). The dotted

lines correspond to the chemical potential at the left and right sides. (ii)

Damping rate Kvib. The junction parameters are Γν=0.2, 1/βν = 0.005,

α = 0.1, ω0 = 0.05 and ǫd(∆µ = 0) = −0.2, ǫa(∆µ = 0) = 0.4, all in units

of [eV]. Reproduced from Ref. [33]. . . . . . . . . . . . . . . . . . . . . . 27

2.4 Population of the two-state “vibration” as a function of bias voltage. Pa-

rameters are the same as in Fig. 2.3. Reproduced from Ref. [33]. . . . . . 28

ix

2.5 Damping rate in a rectifying junction for different broadening parame-

ters, Γν=0.2, βν = 200 (full) Γν=0.4, βν = 200 (dashed) Γν=0.2, βν = 5

(dashed-dotted). Other parameters are the same as in Fig. 2.3. Repro-

duced from Ref. [33]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.6 (i) Charge current and (ii) vibrational states population in a rectifying

junction with weak electron-phonon coupling and weak molecule-metal

hybridization strength, Γν=0.01, α = 0.01, ω0 = 0.2. Other parameters

are the same as in Fig. 2.3. Reproduced from Ref. [33]. . . . . . . . . . . 32

2.7 TLS population as a function of bias voltage for Γph = 0.001 (narrow

lines) and Γph = 0.1 (heavy lines). The excited (ground) state population

is presented by dashed (full) lines. Other parameters are the same as in

Fig. 2.3 with βph = 200. Reproduced from Ref. [33]. . . . . . . . . . . . . 35

2.8 Effective TLS temperature, Γph = 0 (dashed line); Γph = 0.001 and βph =

40 (full line) and Γph = 0.4 and βph = 40 (dashed-dotted line). The inset

zooms on the latter two cases. The dotted lines mark the values βph = 40

and β = 0. Other junction parameters are the same as in Fig. 2.3, with

βν = 200. Reproduced from Ref. [33]. . . . . . . . . . . . . . . . . . . . 36

2.9 Cooling of the molecular vibration for ω0 = 0.05 (dotted line), ω0 = 0.15

(dashed line), ω0 = 0.3 (full line). (a): Γph = 0. (b): Γph = 0.001. Other

junction parameters are ΓL = ΓR = 0.1, and βph = βν = 40. The levels

are shifted with the bias voltage as depicted in Fig. 2.3. Reproduced from

Ref. [33]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

2.10 Charge current in the harmonic-mode model for Γph = 0 (dashed line) and

Γph = 0.05 (full line). For comparison, we also present the current in the

TLS-mode model with Γph = 0.05 (dotted line). βν = βph = 40, other

parameters are the same as in Fig. 2.3. Reproduced from Ref. [33]. . . . 41

x

2.11 Population of the first four levels of the harmonic mode. Full lines: weak

interaction with the heat bath, Γph= 10−4. The population becomes un-

physical (negative) for ∆µ > 0.6. Dashed lines: strong interaction with

heat bath, Γph = 0.1, lifts the instability. βν = βph = 40, other parameters

are the same as in Fig. 2.3. Reproduced from Ref. [33]. . . . . . . . . . . 42

2.12 Stability diagram. The dark island and the narrow strip (at negative

bias) are the parametric region in which the junction becomes unstable.

Other parameters are the same as in Fig. 2.3, besides the temperatures,

βL = βR = βph = 40. Reproduced from Ref. [33]. . . . . . . . . . . . . . . 45

2.13 (a): Stability diagram for Γph = 0. (b): Stability diagram with Γph =

0.005. The dark region is the parametric region in which the junction

becomes unstable, Kvib < 0. Other parameters are the same as in Fig.

2.3, besides the temperatures βL = βR = βph = 40. Reproduced from Ref.

[33]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.1 Left Panel: Generic setup considered in this work, including a subsystem

(S) coupled to multiple fermionic (F ) and bosonic (B) reservoirs. Right

panel: Molecular electronic realization with two metals, L and R, con-

nected by two electronic levels, D and A. Electronic transitions in this

junction are coupled to excitation/de-excitation processes of a particu-

lar, anharmonic, vibrational mode that plays the role of the “subsystem”.

This mode may dissipate its excess energy to a secondary phonon bath B.

Reproduced from Ref. [34]. . . . . . . . . . . . . . . . . . . . . . . . . . . 50

xi

3.2 Study of the rectifier behavior using master equation calculations. Shown

is the steady-state charge current for α = 0.1, Kd = 0, and βν = 200.

The donor and acceptor energies mentioned here refer to the equilibrium

value; under voltage bias, these energies are linearly evolving with bias,

similarly to the trend depicted in Fig. 2.3. (a) Analysis of the role of

the metal-molecule coupling strength and the level organization, ǫd=ǫa=0,

Γν=0.05 (dotted); ǫd=ǫa=0, Γν=1.0 (dashed); ǫd = −0.2, ǫa=0.4, Γν=1.0

(full) ǫd=-0.4, ǫa=0.6, Γν=1.0 (dashed-dotted). (b) Study of the effect

of the vibrational frequency and metal-molecule coupling strength, ǫd=-

0.2, ǫa=0.4, ω0=0.2, Γν = 1.0 (full); ǫd=-0.2, ǫa=0.4, ω0=0.1, Γν = 1.0

(dashed); ǫd=-0.2, ǫa=0.4, ω0=0.2, Γν = 0.5 (dashed-dotted). (c) Same

as (b), zooming over the regime of low bias (semi-log scale). Note that

the full line and the dashed-dotted lines are overlapping in this region.

Reproduced from Ref. [34]. . . . . . . . . . . . . . . . . . . . . . . . . . 68

3.3 Absolute value of the dimensionless quantity πρηFl,r. The figure was gen-

erated by discretizing the reservoirs, using bands extending from −D to

D, D = 1, with NL = 200 states per each band, a linear dispersion rela-

tion, and a constant density of states ρ for the H0L,R reservoirs. Electron-

vibration coupling is given by α = 0.1. Reproduced from Ref. [34]. . . . . 69

3.4 Population dynamics and convergence behavior of the truncated and iso-

lated (Kd = 0) vibrational mode (TLS) with increasing Ns. (a)-(b)

Stable behavior at µL = −µR = 0.2. (c)-(d) Population inversion at

µL = −µR = 0.6. Other parameters are the same as in Fig. 2.3. In all

figures δt=1, Ns = 3 (heavy dotted), Ns = 4 (heavy dashed), Ns = 5

(dashed-dotted), Ns = 6 (dotted), Ns = 7 (dashed) and Ns = 8 (full). We

used Ls = 30 for the number of electronic states at each fermionic bath,

with sharp cutoffs at ±1. Reproduced from Ref. [34]. . . . . . . . . . . . 71

xii

3.5 (a) Independence of the population p0 on the initial state for different

biases, ∆µ = 0.4, 0.8, 1.2 top to bottom. Other parameters are the same

as in Fig. 2.3 and Fig. 3.4. (b) Extracting the relaxation rates from

the transient data of the population. Values close to zero, |∆µ| < 0.2,

should be taken with caution, see the discussion accompanying Fig. 3.7.

Reproduced from Ref. [34]. . . . . . . . . . . . . . . . . . . . . . . . . . 72

3.6 Population dynamics, p0(t). (a) Comparison between exact simulations

(dashed) and master equation results (dashed-dotted) at α = 0.2. (b)

Deviations between exact results and master equations for α = 0.1 (dot

and ◦) and for α = 0.2 (+ and ×). Other parameters are described in Fig.

2.3. Reproduced from Ref. [34]. . . . . . . . . . . . . . . . . . . . . . . 73

3.7 Converged data for the population of the isolated (Kd = 0) vibrational

mode in the steady-state limit with α = 0.1. Other parameters are the

same as in Fig. 2.3. We display path-integral data for p0 (◦) and p1 (�).

Master equation results appear as dashed line for p0 and dashed-dotted

line for p1. Reproduced from Ref. [34]. . . . . . . . . . . . . . . . . . . . 75

3.8 (a) Convergence behavior of the population p0 in the steady-state limit for

α = 0.1, all parameters are the same as in Fig. 2.3. Plotted are the steady-

state values using different time steps, δt = 0.8 (◦), δt = 1.0 (�), and δt =

1.2 (⋄) at different biases, as indicated at the right end. (b) Population

mean and its standard deviation, utilizing the last six points from panel

(a). (c) Current mean and its standard deviation, once averaged over data

points at large enough τc, where convergence sets. Reproduced from Ref.

[34]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

xiii

3.9 Charge current in the steady-state limit for α = 0.1, Kd = 0. Other

parameters are the same as in Fig. 2.3. Path-integral data (◦), master

equation results (dashed). The insets display transient results at ∆µ = 1.0

eV (top) and ∆µ = −0.5 eV (bottom). Reproduced from Ref. [34]. . . . 78

3.10 Charge current and vibrational occupation in the steady-state limit at dif-

ferent electron-vibration coupling. Path-integral data is marked by sym-

bols, α = 0.1 (◦), α = 0.2 (⋄) and α = 0.3 (�). Corresponding master

equation results appear as dashed lines. Inset: The population behavior

in the steady-state limit for the three cases α = 0.1 (◦), α = 0.2 (⋄) and

α = 0.3 (�), with empty symbols for p0 and filled ones for p1. Other

parameters are the same as in Fig. 2.3. Reproduced from Ref. [34]. . . . 79

3.11 Study of the contribution of different transport mechanisms. vda = 0 (◦),

with master equation results noted by the dashed line, and vda = 0.1

(�). The main plot displays the charge current. The inset presents the

vibrational levels occupation, with empty symbols for p0 and filled symbols

for p1. Other parameters are the same as in Fig. 2.3, particularly, the

vibrational-electronic coupling is α = 0.1. Reproduced from Ref. [34]. . 80

3.12 Equilibration of the molecular vibrational mode with increasing coupling

to a secondary phonon bath. Path-integral results, (full symbols for p1,

empty symbols for p0) with Kd = 0 (◦), Kd = 0.01 (⋄), Kd = 0.1 (�),

and, Kd = 0.1, α = 0 (⊳). Unless otherwise specified, α = 0.1, βph = 5

and the spectral function follows (3.42) with ωc=15. All other electronic

parameters are the same as in Fig. 2.3. Master equation results appear in

dotted lines. Reproduced from Ref. [34]. . . . . . . . . . . . . . . . . . . 83

xiv

3.13 Charge current for an isolated mode, Kd = 0 (◦), and an equilibrated

mode, Kd = 0.1, βph=5, ωc=15 (�). Other electronic parameters are given

in Fig. 2.3. Master equation results appear in dashed lines. Reproduced

from Ref. [34]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

4.1 1214.5=14.514.5Minimal modeling of nanojunctions with a single elec-

tronic level (energy ǫd) coupled to two metals. In the Anderson-Holstein

(AH) model the vibrational mode is displaced depending on the charge

number in the dot. The spin-fermion model (SF) is a truncated version of

the AH model. Its (nondegenerate) two states describe e.g., an anharmonic

mode or a magnetic impurity in an external magnetic field. Electrons re-

siding on the dot may flip the spin state. Reproduced from Ref. [35]. . . 90

4.2 Eigenenergies ǫn,q of the SF model (full) when (a) ǫd = 0, (b) ǫd = 1.5,

and (c) ǫd = −0.8. In panel (a) we also display low-lying (q = 0, 1, ..., 5)

eigenenergies of the AH molecular Hamiltonian (dashed). Reproduced

from Ref. [35]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

4.3 Dressing elements |Mq,q′ |2 in the AH model following Eq. (4.28) with q = 0

and q′ = 0, 1, 2 (dashed lines, left to right), and in the SF model following

Eq. (4.29), q, q′ = ±1 (full). ω0 = 1. Reproduced from Ref. [35]. . . . . 106

4.4 Dressing elements |Mq,q′ |2 for truncated harmonic impurities of N = 3 and

N = 5 states with Fq,q′ from Eq. (4.10). Reproduced from Ref. [35]. . . 107

4.5 Current-voltage characteristics of the AH and SF models at ǫd = 0 with

weak (a) and strong (b) electron-impurity coupling. (a) The inset zooms

on weak coupling features, demonstrating the similarity, and onset of devi-

ations, between the models, as coupling increases. Reproduced from Ref.

[35]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

xv

4.6 Current-voltage characteristics in a gated ǫd = 1.5 junction at weak (a)

and strong (b-c) coupling. Different types of blockade play a role: (a)

ORB at weak interactions, (b) FCB in the AH model, and (c) MB-ORB

in the strongly-interacting SF model. Reproduced from Ref. [35]. . . . . 112

4.7 Differential conductance plots of the SF (top) and the AH (bottom) models

as a function of gate (ǫd) and applied bias voltage ∆µ at weak and strong

coupling, as indicated in the figure. Reproduced from Ref. [35]. . . . . . 113

4.8 Differential conductance plots of the SF model as a function of electron-

spin interaction (α) and the bias voltage ∆µ at different gating, as indi-

cated in the figure. Reproduced from Ref. [35]. . . . . . . . . . . . . . . 114

4.9 Mild effect of mode equilibration on the current at ǫd > 0 in the (a)

weak coupling limit, and (b) at strong coupling; the inset zooms on the

region of interest. The legend describes all panels: (full) excluding a heat

bath, (dashed-dotted lines) including a dissipative spin bath at different

couplings, and (dashed) once enforcing impurity equilibration as in Eq.

(4.30). We used ǫd = 1.5 and Th = 0.05. Reproduced from Ref. [35]. . . 116

4.10 Strong influence of mode equilibration on the current at ǫd < 0 (a) linear

scale, (b) logarithmic scale, displaying steps at low temperatures. The

temperature of the electronic baths is (as before) Tν = 0.05. Th is indicated

in the figure, and we used α = 2 and ǫd = −0.8. Reproduced from Ref.

[35]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

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4.11 Electronic dot occupation in the SF model, quantum adiabatic limit, with

ǫd = 4.5, ∆µ = 0, and Γν = 0.25. The full line was generated from

Eq. (4.54). Eq. (4.55) provides the dashed (dashed-dotted) lines, based

on data consistent (inconsistent) with the derivation of Eq. (4.55); the

dashed-dotted line is included here for demonstrating that multiple solu-

tions can show only when ǫreorg/Γ > 1, deviating from the assumptions

leading to Eq. (4.55). Reproduced from Ref. [35]. . . . . . . . . . . . . 122

5.1 1214.5=14.514.5(a) Scheme of coherent transport through two closely spaced

electronic levels, (b) same as (a) but with a dephasing agent represented by

the observer, the eye, (c) inelastic transport: incoming electrons exchange

energy with a molecular vibrational mode . . . . . . . . . . . . . . . . . 128

5.2 1214.5=14.514.5Schematic of (a) generic model of a molecular island with

two electronic levels interacting with a vibrational mode (DES geometry)

and (b) Donor-Impurity-Acceptor (DIA) model. . . . . . . . . . . . . . . 130

5.3 1214.5=14.514.5Transmission functions of coherent DES (full), coherent

CON (dotted), and of fully dephased (dashed-dotted) models. Coupling

to electronic reservoirs is given in the figures (the inset zooms in on DES

transmission function). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

5.4 1214.5=14.514.5Thermopower of coherent D-A (red solid), fully dephased

D-A-Probe (punctured blue) and inelastic D-I-A (dashed green) models as

a function of electronic hybridization energy Γ. In the D-I-A model ∆ = 0,

ω0 = 0.02 eV, in in both dephasing and coherent DES models ∆ = 0.02

eV and ω0 = 0, α = 0; in all cases ǫ0 = 0.01 eV. The left panel (a) depicts

high temperature behavior of the thermopower S as a function of Γ. The

right panel (b) shows the low (but finite) temperature behavior of S. . . . 136

xvii

5.5 1214.5=14.514.5Critical electronic coupling Γc as a function of temper-

ature for coherent DES (solid dots), coherent CON (circles), and fully

dephased (diamonds) models (lines are there to guide to the eye). ǫ0 =

0.01eV, ∆ = 0.02eV, ω0 = 0.02eV, T = 17.5K . . . . . . . . . . . . . . . 141

xviii

Chapter 1

Introduction

Understanding charge and energy transfer in molecules is an important challenge on the

way to smart nano-materials and better use of energy. Owing to the effects of many-

body interactions, dissipation and quantum nature of charge carriers, nano-scale objects

present enormous space for exploration. In particular, the way charge travels through a

molecule is influenced by numerous effects: Coulomb repulsion of electrons and attraction

of electrons to the nuclei, properties of the solvent and the vibrating backbone of the

molecular structure, temperature and the extent to which charge carriers are prone to

decoherence. Control of these effects opens the possibility of engineering low-cost organic

materials for applications in electronic circuits [1], solar cells [2] and bio-medical devices

[3].

Interestingly, although motivated by technological applications, the problem of charge

transport in miniature devices presents a framework for exploration of fundamental ques-

tions in nonequilibrium physics, such as validity of the Second Law, manifestation of

many-body and quantum effects resulting in phenomena such as the Kondo effect and

Anderson localization. Specific to present work are the problems of validity of steady-

state Fluctuation Theorem for non-adiabatic electron transport, heating and cooling of

the molecule by means of electronic current, method development for numerically exact

1

Chapter 1. Introduction 2

description of charge transport in the presence of a dissipative environment, effects of

anharmonicity on Frank-Condon blockade and hysteresis, and signatures of decoherence

mechanism in thermopower.

In some nano-scale scenarios the separation between the different components is quite

intuitive. For example, in quantum-dot arrays, where the proximity of the dots allows

for quantum tunneling of electrons - the dots may be pulled apart, or brought closer

together, and there is no ambiguity about where one dot ends and another begins. In

contrast, in molecular setups the different components are held together by direct co-

valent bonding. This removes clear boundaries between sub-units of the structure, and

introduces some arbitrariness into the way charge flows are followed. The most intuitive

way to follow charge dynamics is to impose a spatial partition of the structure and follow

the populations of charge in each part as a function of time. For example, the partition

into donor and acceptor groups is often used for electron or proton transfer in biological

molecules. Research directed at electronic devices or materials for energy interconversion,

for example, solar energy antennae, usually focuses on source-bridge-sink configurations.

Particular identities of the source, the bridge and the sink may vary greatly depending

on the system. These may be small or large, flexible or crystalline, dielectrics, metals

or super-conductors. For example, recently, much attention has been devoted to energy

transfer in conjugated polymers and photosynthetic proteins [4, 5, 6]. In this case, for

the sake of following electronic excitation dynamics, the molecule is partitioned into the

initial excitation site, the final destination and the rest of the structure serving as a

bridge between the two. On the other hand, in molecular electronics scenarios [1] the

source and the sink are electrodes held at different chemical potentials and the bridge

is a small molecule. This list of possible configurations is not exhaustive, yet, although

the structures may differ in many aspects the discussion revolves around similar key con-

siderations: energy dissipation, electron-vibration interaction and decoherence. We will

keep our focus on scenarios relevant to molecular electronics, i.e., molecular junctions.


Experimentally molecular junctions are routinely made either using the tip of Scan-

ning Tunneling Microscope (STM) [7] or in mechanically controlled break junctions [9].

In the STM experiment, the tip and the metal sheet serve as two electrodes held at a

controlled distance away from each other. In break junctions a metallic (often gold) wire

is pulled until it breaks, exposing atom-sharp tips. Molecules of interest are then injected

and some get trapped between the tips. Although the idea is simple, operating at such

small scales is experimentally challenging. Early experiments in the late 90’s [10] were

able to capture only crude information about the junction: they measured current ver-

sus voltage and deduced positions of conducting orbitals from the steps of the resulting

curve. State of the art techniques today allow to measure shot noise [11], thermoelectric

effect [12], and signatures of electron-vibration interaction [13].

The early theory of electronic transport in molecular bridges relied on scattering the-

ory, in which dissipation was assumed to occur far away in the leads and the molecule

was treated as a simple scattering potential. Landauer’s formula [14], incredibly powerful

in spite of its simplicity, is often used in the analysis of experiments in which inelastic

scattering is suppressed. Alternatively, the master equation approaches [15, 16] follow

the evolution of the reduced density matrix of the molecule, capturing inelastic and inco-

herent effects, such as energy exchange with vibrations and thermally activated hopping.

The nonequilibrium Green’s functions technique invented independently by Keldysh [17],

and Kadanoff and Baym [18], provides perturbative solutions to minimal toy model sys-

tems at low temperatures and was successfully used to describe electron transport in

interacting systems by Galperin et al. [19, 20]. In addition to analytical approaches,

several numerically exact techniques based on a path integral formulation of the dynam-

ics exist. The different flavors of these may be roughly divided into two groups: those

based on Monte Carlo sampling [21] and those on deterministic iterative time propagation

[22, 23]. Monte Carlo methods suffer from the sign problem and allow for a short time

evolution only. The iterative methods allow for long time propagation, but become ex-


cessively computationally expensive in non-markovian situations (at low temperatures).

Here, using the perturbative quantum master equation, numerical path integral method

and scattering theory techniques, we investigate several aspects of molecular electronic

conduction.

To this end, we model the molecular junction with the general Hamiltonian

H = HM +Hel +Hph + VM−ph + VM−el. (1.1)

The molecular Hamiltonian HM = H0 + Hvib + HI , contains the electronic (H0) and

vibrational (Hvib) spectra of the molecular island along with vibronic couplings (HI).The

Hamiltonian Hel may comprise multiple fermionic baths forming the electronic leads,

and similarly, Hph may contain more than a single bosonic reservoir, which include the

secondary vibrational modes of the junction. The terms VM−el and VM−ph describe the

coupling of the molecular subsystem to the fermionic and bosonic environments, respec-

tively.

We now describe the elements of the molecular Hamiltonian HM . In our simple

description, the molecule is represented by two electronic levels: the donor with the energy

ǫd, and the acceptor with the energy ǫa, coupled to a single oscillator. In the common

Born-Oppenheimer approximation electronic potential surfaces Ua(d)(Q) are functions of

the displacement coordinate Q of the molecular oscillator. Assuming that displacement

from equilibrium remains small we may expand the potential about equilibrium position,

Q = 0, up to the linear term. This results in

HBO = Ua(Q = 0) + Ud(Q = 0) +∂Ua∂Q

∣

∣

∣

∣

Q=0

Q+∂Ud∂Q

∣

∣

∣

∣

Q=0

Q+P 2

2M+ω20M

2Q2.(1.2)

Here P represents momentum, M - the mass and ω0 - the frequency of the oscillator. In

second quantization the Hamiltonian takes the form

HBO = ǫac†aca + ǫdc

†dcd +

αa2c†aca(b

†0 + b0) +

αd2c†dcd(b

†0 + b0) + ω0b

†0b0 (1.3)


with αa(d) ∝ dUa(d)dQ |Q=0. The operators c†i (ci) create (annihilate) electrons at i

th site

or energy. The subscript i may take the form of “a” for acceptor, “d” for donor, or

in other situations, “1” and “2” in the energy basis. The indices l(r) run through

electronic wavenumbers in the metals. Along with these operators, electronic energies

are represented by the parameter ǫi. The operators b†k(bk) create (annihilate) phonons on

the kth oscillator. The zeroth mode (b0) is normally reserved for the molecular vibration,

whereas bath modes carry the subscript k. Phononic frequencies are typically denoted

by ωk’s, with ω0 representing the molecular mode, and ωk standing for bath modes of

kth energy. Note, although this is the dominant notation scheme adopted in the Thesis,

at times, considerations of clarity prompted us to deviate from it with new definitions

appearing as necessary. For simplicity, the charge of electron, Planck’s constant ~ and

Boltzman’s constant kB are taken to be unity.

In order to go beyond the Born-Oppenheimer approximation, we include linear vi-

bronic coupling between the donor and acceptor surfaces [24]

HI = αvib(c†dca + c

†acd)(b

†0 + b0), (1.4)

with αvib denoting the vibronic coupling strength (subscript is neglected where ambiguity

is absent), and write down the total molecular Hamiltonian

HM = HBO +HI . (1.5)

To complete the junction we need to connect the molecule to the metallic leads. These

are represented by collections of non-interacting electrons

Hel =∑

l

ǫlc†l cl +

∑

r

ǫrc†rcr, (1.6)

and are connected to the molecule through the coupling term

VM−el =∑

l

(vlc†l cd + v

∗l c

†dcl) +

∑

r

(vrc†rca + v

∗rc

†acr), (1.7)


where vl(r) are the electronic coupling strength parameters. Finally, a bosonic dissipative

environment described by a collection of free harmonic oscillators

Hph =∑

k

ωkb†kbk, (1.8)

is connected to the vibrational mode with strength determined by ζk through the inter-

action term

HM−ph = (b†0 + b0)

∑

k

ζk(b†k + bk). (1.9)

In order to make our work meaningful in connection with realistic systems, we pro-

vide an example of a molecular junction typical for such experiments: dimethylbiphenyl

(DMBP). The DMBP has the donor and acceptor orbitals with the energies ǫa = ǫd =

−1.2eV and the torsional rotation mode, which is responsible for the relative orientation

of the two benzene rings with the energy ω0 = 0.005eV [32]. The interaction of electrons

with this vibrational mode dominates the conduction properties of the junction.

Using this model Hamiltonian as our starting point, we consider three related scenar-

ios. In Chapter 2, in order to study stability of the junction, we focus on non-adiabatic

processes relevant for passage of the electrons from donor to acceptor. By neglecting

terms which depend on the curvature of electronic surfaces, i.e. we set αa(d) = 0, we

focus on processes which are unique to transport of electrons, and follow the dynamics

of the vibrational mode. Notably, the energies of the donor and the acceptor ǫd and ǫa

are expected to follow the chemical potential of the lead to which they are coupled. This

voltage bias dependence of the local orbitals is the root of the behaviour predicted in this

Chapter. Further, since heating is a limiting factor in practical applications of molecular

junctions in solid-state devices [25], it is important to consider ways to dissipate excess

energy; to this end we consider interaction of the electrons with harmonic and anhar-

monic vibrations as well as with nuclear spins, isolated or immersed in a dissipative bath.

We demonstrate that there exist situations in which heating sets the limit to how small

the electronic components may become due to the necessity to dissipate excess energy.


In order to justify the approximate results, obtained with a master equation in the limit

of weak vibronic coupling, in Chapter 3, we generalize the influence functional iterative

path integral scheme (INFPI) developed by Segal et al. [22, 26] to include a phononic en-

vironment. This is achieved by combining the INFPI scheme with the Quasi-Adiabatic

Path Integral scheme (QUAPI) developed by Makri et al. [27, 28]. Using this exact

method we confirm perturbative results of Chapter 2 and extend our study to include

direct electronic coupling between the donor and the acceptor.

Encouraged by the similarity of phenomena in harmonic and anharmonic junctions,

we continue our discussion in this direction in Chapter 4. Following the work of Koch

et al. [29] we study effects of many-body interactions on electronic transport through a

single electronic level coupled to either a harmonic vibrational mode (Anderson-Holstein

model) or a spin (spin-fermion model). Interestingly, although at a first glance the current

behaves quite similarly in both cases, closer analysis reveals that the underlying causal

mechanisms differ. This leads to significant deviations, and at times, to counter-intuitive

behavior at strong electron-vibration couplings in anharmonic systems.

We conclude the Thesis with the study of decoherence of electrons due to the interac-

tion with molecular vibrations. Building on the work of Solomon et al. [30] followed by

Hartle et al. [31] we look for signatures of decoherence in the thermopower. Focusing on

three crude limits of purely coherent, fully dephased and classical-inelastic electron trans-

port we study how the thermopower behaves, and demonstrate that in certain situations

is can be used to witness the mechanism of decoherence.

Chapter 2

Vibrational cooling, heating, and

instability of conducting molecules

2.1 Introduction

In this Chapter, we study the problem of bias-induced molecular cooling, heating, and

(potential) junction breakdown due to vibrational instabilities, using the Donor (D)-

Acceptor (A) Aviram-Ratner electronic rectifier setup [52], see Fig. 2.1. By coupling

electronic transitions within the junction to a particular internal molecular vibrational

mode, significant molecular heating can take place once the donor level is lifted above

the acceptor level, as the excess electronic energy is used to excite the vibrational mode.

This process may ultimately lead to junction instabilities and breakdown [25]. The model

can also demonstrate current-induced cooling at low bias, when tuning the junction’s

parameters.

Within this simple system, several issues are of fundamental and practical interest.

First, one would like to understand the role of mode anharmonicity in the transport

process and in the heating or cooling behavior. Second, the molecular vibration under

investigation, the one controlling junction stability, can be assumed to be well isolated

8

Chapter 2. Vibrational cooling, heating and instability 9

from other modes. Alternatively, this mode may be coupled to other phonons, allowing for

energy damping to a larger environment. These two situations should result in distinctive

cooling or heating behaviors. These issues will be explored here. Other relevant challenges

which are not considered here are the possibility to selectively excite vibrational modes

in the molecule, using voltage bias [53], or more generally, to drive molecular motion or

trigger chemical dynamics [54]. The discussion of this Chapter is based on Ref. [33].

D

A

R

TR

L

TL

(i)

(ii)

L

TLD

A

R

TR

Tph

Tph

Figure 2.1: Schemes of the two models considered in this Chapter. A biased donor-

acceptor electronic junction is coupled to either (i) a harmonic molecular mode, or (ii)

an anharmonic mode, represented by a two-state system. In both cases the molecular

mode may further relax its energy to a phononic thermal reservoir, maintained at the

temperature Tph. This coupling is represented by a dashed arrow. Reproduced from Ref.

[33].

Using a full counting statistics (FCS) approach, our analysis further contributes to

the institution of fluctuation relations in open many-body quantum systems. Fluctua-

tion theorems (FT) for entropy production quantifies the probability of negative entropy


generation, measuring “second law violation” [55, 125]. Such “anomalous” processes are

relevant at the nanoscale. While originally demonstrated in classical systems [57], recent

experimental efforts are dedicated to explore their validity within quantum systems [58].

From the theoretical side, the extension of the work and heat FT to the quantum do-

main has recently attracted significant attention [59, 60, 61, 62]. Specifically, a quantum

exchange FT, for the transfer of charge and energy between two reservoirs maintained

at different chemical potentials and temperatures, has been derived in Ref. [63] us-

ing projective measurements, and in Ref. [61] based on the unraveling of the quantum

master equation. It is of interest to test these relations in particular cases, e.g., for sys-

tems strongly coupled to multiple reservoirs, when the reservoirs cooperatively affect the

subsystem [64], including nonmarkovian reservoirs [65, 66, 67], and in models showing

coupled charge and energy transfer processes, yet the respective fluxes are not tightly

coupled. The system investigated here corresponds to the latter case.

Different flavors of the phonon-assisted-tunneling model have been analyzed in the

literature [46]. Among the various techniques adopted we list solution of the dynamics

as a scattering problem [68], extension of the basic nonequilibrium Green’s function

formalism to include molecular vibrations [69], or the use of master equation approaches

[151]. In this Chapter, we exploit the latter method, and present a full-counting statistics

of the system, allowing for the exploration of charge current, energy current and noise

processes at the same footing. Further, we analytically obtain the cumulant generating

function (CGF) of the model, allowing for the verification of the steady-state charge-

energy fluctuation theorem in this many-body quantum system.

In this Chapter our objectives are therefore twofold: (i) to analyze a simple model that

can elucidate cooling, heating and instability mechanisms in molecular rectifiers, specif-

ically, to understand the roles of mode anharmonicity and additional damping routes,

and (ii) to establish the steady-state entropy production fluctuation theorem within a

nonequilibrium quantum model, transferring charge and energy between the reservoirs in


a cooperative manner. Recent studies have analyzed the role of electron-vibration interac-

tion on the full counting statistics (FCS) within different approaches [187, 72, 66, 73, 188].

Complementing these efforts, our treatment offers an analytic structure for the CGF, al-

lowing for a clear inspection of the microscopic processes involved.

The plan of the Chapter is as follows. In Sec. 2.2 we introduce the D-A molecular rec-

tifier and its two flavors, either including a harmonic or an anharmonic internal vibration.

In Sec. 2.3 the anharmonic model is analyzed within a FCS approach, demonstrating

cooling, heating and instability dynamics at different parameter regions. The case with

an additional phonon bath is considered in Appendix A. Sec. 2.4 explores the harmonic

mode model. Sec. 2.5 concludes.

2.2 Model

Our model includes a biased molecular electronic junction and a selected internal vi-

brational mode which is coupled to an electronic transition in the junction. This mode

possibly interacts with other (reservoir) phonons, an extension presented in Appendix A.

For a schematic representation, see Fig. 2.1. Generally, this model allows one to investi-

gate the exchange of electronic energy with molecular (vibrational) heating. The model

has been utilized in Ref. [74] for studying the thermoelectric and thermal transport

of electrons in molecular junctions with electron-phonon interactions within the linear

response regime. The total Hamiltonian is given by the following terms,

H = HM +Hel +HM−el (2.1)

The first term, HM = H0 +Hvib +HI , stands for the molecular electronic part including

two electronic states

H0 = ǫdc†dcd + ǫac

†aca. (2.2)


Here, c†d/a (cd/a) is a fermionic creation (annihilation) operator of an electron on the donor

or acceptor sites, of energies ǫd,a. In our model the energies of the donor and the acceptor

shift in accord with the chemical potential of the lead to which they are coupled. This

voltage bias dependence of the local orbitals is the root of the behaviour predicted in this

Chapter. The molecular Hamiltonian further includes an internal molecular vibrational

mode of frequency ω0. The mode displacement from equilibrium is coupled to an electron

hopping in the system with an energy cost α, resulting in heating and/or cooling effects,

Hvib = ω0b†0b0,

HI = α[

c†dca + c†acd

]

(b†0 + b0). (2.3)

Here, b†0 (b0) represents a bosonic creation (annihilation) operator. Note that in our

construction, the donor and acceptor sites are coupled to each other only through the

interaction with the vibrational mode.

The second term in Eq. (2.1) describe the two metals, Hel = HL+HR, each including

a collection of noninteracting electrons

HL =∑

l∈Lǫlc

†l cl; HR =

∑

r∈Rǫrc

†rcr. (2.4)

The hybridization of the donor state to the left (L) bath, and similarly, the coupling of

the acceptor site to the right (R) metal, are incorporated into HM−el,

HM−el =∑

l

vl

(

c†l cd + c†dcl

)

+∑

r

vr(

c†rca + c†acr

)

. (2.5)

This model thus does not allow for coherent electron tunneling between the two metals,

as only inelastic processes, through the excitation or relaxation of the vibrational mode,

are allowed. A direct tunneling term has been included in other related studies, see e.g.,

[46, 47, 49, 50]. We could argue in favor of our simple model from several directions. First,

in the present study we are predominantly interested in the vibrational mode population.

Thus, electrons that are coherently transferred through the junction are irrelevant in that

respect: Including direct tunneling processes would definitely affect the magnitude of the


electron current in the system, however, the effect of the vibrational instability discussed

below should occur regardless. One could also slightly modify our model, and take the

two molecular states D and A to represent molecular orbitals, HOMO and LUMO, further

coupling each to both leads. Again, for calculating vibrational level population one needs

to only consider the contribution to the current due to vibrationally assisted electrons.

This situation is similar to that considered in Ref. [75], where current-induced-light-

emission in molecular junctions has been calculated. Indeed, our formalism could equally

describe such processes, see also Sec. 2.5. Another reason for selecting this model was

practical: It is a significant challenge to derive fluctuation relations for a model allowing

for both coherent and incoherent electron transmission effects, thus relaying on some

approximations is unavoidable [76, 64, 65]. In particular, Ref. [188] looked at the charge

transfer statistics of a model that in principle could allow for both effects. However, by

making the strong electron-phonon coupling approximation, the authors have practically

forced every single electron tunneling event to fully excite or de-excite the local phonon.

This situation is similar to that considered in our model.

We now diagonalize the electronic part of the Hamiltonian, HE = H0 +Hel +HM−el,

to obtain, separately, the exact eigenstates for the L-half and R-half of HE. Assuming

that the reservoirs are dense, their new operators are assigned energies same as those

before diagonalization. The donor and acceptor (new) energies are assumed to be placed

within the band of continuous states, excluding the existence of bound states. The old

operators are related to the exact eigenstates by [77]

cd =∑

l

λlal, cl =∑

l′

ηl,l′al′

ca =∑

r

λrar, cr =∑

r′

ηr,r′ar′ , (2.6)


where the coefficients, e.g., for the L set, are given by

λl =vl

ǫl − ǫd −∑

l′v2l′

ǫl−ǫl′+iα

ηl,l′ = αl,l′ −vlλl′

ǫl − ǫl′ + iδ. (2.7)

Similar expressions hold for the R set. It is easy to derive the following relation,

∑

l′

v2l′

ǫl − ǫl′ + iδ= PP

∑

l′

v2l′

ǫl − ǫl′− iΓL(ǫl)/2, (2.8)

with the hybridization strength ΓL(ǫ) = 2π∑

l v2l δ(ǫ− ǫl). The expectation values of the

exact eigenstates are

〈a†lal′〉 = δl,l′fL(ǫl), 〈a†rar′〉 = δr,r′fR(ǫr), (2.9)

where fL(ǫ) = [exp(βL(ǫ − µL)) + 1]−1 denotes the Fermi distribution function. An

analogous expression holds for fR(ǫ). The reservoirs temperatures are denoted by 1/βν ;

the chemical potentials are µν . With the new operators, the Hamiltonian (2.1) can be

rewritten as

HH =∑

l

ǫla†lal +

∑

r

ǫra†rar + ω0b

†0b0

+ α∑

l,r

[

λ∗l λra†lar + λ

∗rλla

†ral

]

(b†0 + b0). (2.10)

In this form, the model generally describes the process of an electron-hole pair excita-

tion by a molecular vibration. We denote it by HH , to highlight the vibrational mode

harmonicity. Anharmonicity can be introduced by replacing the harmonic mode by a

two-state system (spin), using the Pauli matrices,

HA =∑

l

ǫla†lal +

∑

r

ǫra†rar +

ω02σz

+ α∑

l,r

[

λ∗l λra†lar + λ

∗rλla

†ral

]

σx. (2.11)

The truncated harmonic spectrum imitates an anharmonic mode, as only several (two in

the present extreme case) states are bounded within the anharmonic potential [78]. We


denote this Hamiltonian by HA. The dynamics of this model should coincide with the

behavior dictated by HH , at low temperatures.

Charge and energy transfer dynamics in these models can be followed by studying

electronic properties [79, 73, 188]. In contrast, here we explore the junction response to an

applied voltage bias by studying the vibrational mode excitation and relaxation dynamics.

The analysis of the two-state model, Eq. (2.11), therefore turns out to be simpler than the

case when the vibrational mode has an infinite spectrum. In what follows, we derive in

details the CGF for the anharmonic-mode case. Appendix A generalizes this calculation

to include an additional dissipative thermal bath. We then extend these results and

discuss the model conveyed by Eq. (2.10).

1 0

L Rk

1 0

R Lk

0 1

L Rk

0 1

R Lk

Figure 2.2: Scheme of the vibrational mode excitation and relaxation processes. A full

circle represents an electron transferred; a hollow circle depicts the hole that has been

left behind. Reproduced from Ref. [33].


2.3 Anharmonic-mode Rectifier

2.3.1 Impurity dynamics

We explore the dynamics of an anharmonic mode, referred to as an “impurity”, or a

two-state-system (TLS), within an electronic rectifier, assuming a weak donor-acceptor

- mode interaction. We rewrite Eq. (2.11) as

HA =ω02σz + σxFe +

∑

l

ǫla†lal +

∑

r

ǫra†rar, (2.12)

by defining the electron-hole pair generation operator as

Fe = α∑

l,r

(λ∗l λra†lar + λ

∗rλla

†ral). (2.13)

The Hamiltonian (2.12) can be transformed to the spin-fermion model [80, 81] of zero

energy spacing, using the unitary transformation

U †σzU = σx, U†σxU = σz, (2.14)

with U = 1√2(σx + σz). The transformed Hamiltonian H̃A = U

†HAU is given by

H̃A =ω02σx + σzFe +

∑

l

ǫla†lal +

∑

r

ǫra†rar. (2.15)

In this form, the TLS dynamics can be simulated exactly using an influence-functional

path integral approach [34] as detailed in Chapter 3.

Back to (2.12), we denote the TLS ground state and excited state by |0〉 and |1〉,

with energies 0 and ω0, respectively. We express the Pauli operators by these states,

σz = |1〉〈1| − |0〉〈0|, σx = |1〉〈0| + |0〉〈1|. Next, using the quantum Liouville equation,

we obtain kinetic-rate equations for the states population pn (n=0,1) [82, 78]. This

standard derivation, valid for the harmonic-mode model as well, involves a second order

perturbation theory treatment with respect to α, the mode-molecule coupling parameter,

followed by a Markov approximation. The resulting equation for the reduce density


matrix ρS take the simple form

ρ̇S = −i[V (t), ρS(0)]

−∫ ∞

0

dτTrB {[V (t), [V (τ), ρS(t)ρLρR]]} . (2.16)

Here, V = σxFe represents the (mode-molecule) coupling term in Eq. (2.12). The

operators are written in the interaction representation, O(t) = ei(HA−V )tOe−i(HA−V )t and

we trace over the electronic degrees of freedom. The reservoirs ν = L,R are maintained

in a grand canonical state as ρν = e−βν(Hν−µνNν)/Zν ; Zν is the partition function of the

ν bath. Identifying the diagonal matrix elements as population, pn = [ρS]n,n, we obtain

the kinetic equation

ṗ1 = −ke1→0p1 + ke0→1p0, p1 + p0 = 1. (2.17)

It should be noted that for the specific model Hamiltonian (2.12), one obtains time-

convolutionless rate equations where the off-diagonal elements of the reduced density

matrix are decoupled from the population dynamics beyond second-order perturbation

theory in the system-bath coupling parameter, assuming a factorized (system-bath) initial

condition [83, 195]. In other words, the secular approximation is not invoked in the TLS

case [85]. However, we do need to use it once considering the harmonic mode model, see

Eq. (2.58). Within second-order coupling scheme, the excitation (ke0→1) and relaxation

(ke1→0) rate constants are given by Fourier transforms of bath correlation functions

ke1→0 =

∫ ∞

−∞eiω0τ 〈Fe(τ)Fe(0)〉dτ

ke0→1 =

∫ ∞

−∞e−iω0τ 〈Fe(τ)Fe(0)〉dτ, (2.18)

enclosing electron-hole pair excitation processes,

〈Fe(t)Fe(0)〉 = α2TrLTrR{

∑

l,l′

∑

r,r′

ρLρR ×[

λ∗l λra†l (t)ar(t) + λ

∗rλla

†r(t)al(t)

]

×[

λ∗l′λr′a†l′(0)ar′(0) + λ

∗r′λl′a

†r′(0)al′(0)

]}

.

(2.19)


The operators are given in the interaction representation, e.g., a†l (t) = eiHLta†l e

−iHLt. As

we separately trace over the L and R-baths’ degrees of freedom, it can be shown that

the rate constants can be decomposed into two contributions,

ke1→0 = kL→R1→0 + k

R→L1→0 ; k

e0→1 = k

L→R0→1 + k

R→L0→1 , (2.20)

satisfying

kL→R1→0 = 2πα2∑

l,r

|λl|2|λr|2fL(ǫl)(1− fR(ǫr))δ(ω0 + ǫl − ǫr)

kL→R0→1 = 2πα2 ×

∑

l,r

|λl|2|λr|2fL(ǫl)(1− fR(ǫr))δ(−ω0 + ǫl − ǫr).

(2.21)

Similar relations hold for the right-to-left going excitations. The energy in the Fermi

function fν(ǫ) is measured with respect to the (equilibrium) Fermi energy, placed at

(µL + µR), and we assume that the bias is applied symmetrically, µL = −µR. The

four rate constants describe distinct electron-hole excitation processes, depicted in Fig.

2.2. At forward bias, if we set the effective density of states (DOS) of the L bath to

lie higher in energy that the DOS of the right bath, we immediately note that the rate

kL→R0→1 should dominate over kL→R1→0 , potentially leading to ”population inversion” of the

vibrational mode. Utilizing electronic reservoirs with energy dependent DOS is thus the

basic ingredient of the instability formation here, as we show below. For convenience, we

define the spectral density for the ν bath as

Jν(ǫ) = 2πα∑

j∈ν|λj|2δ(ǫj − ǫ). (2.22)

Explicitly, using Eq. (2.7), we find that this function has a Lorentzian lineshape, and

that it is centered around either the D or A level,

JL(ǫ) = αΓL(ǫ)

(ǫ− ǫd)2 + ΓL(ǫ)2/4

JR(ǫ) = αΓR(ǫ)

(ǫ− ǫa)2 + ΓR(ǫ)2/4. (2.23)


Using the spectral density function, we express the terms in Eq. (2.21) as integrals

kL→R1→0 =1

2π

∫ ∞

−∞fL(ǫ) [1− fR(ǫ+ ω0)] JL(ǫ)JR(ǫ+ ω0)dǫ

kR→L1→0 =1

2π

∫ ∞

−∞fR(ǫ) [1− fL(ǫ+ ω0)] JR(ǫ)JL(ω0 + ǫ)dǫ

kL→R0→1 =1

2π

∫ ∞

−∞fL(ǫ) [1− fR(ǫ− ω0)] JL(ǫ)JR(ǫ− ω0)dǫ

kR→L0→1 =1

2π

∫ ∞

−∞fR(ǫ) [1− fL(ǫ− ω0)] JR(ǫ)JL(ǫ− ω0)dǫ.

(2.24)

The following relations hold (βL = βR and ∆µ ≡ µL − µR),

kR→L1→0kL→R0→1

= e−β∆µeβω0 ;kL→R1→0kR→L0→1

= eβ∆µeβω0 . (2.25)

In equilibrium, detailed balance is therefore maintained. Equation (2.17) and the rates

(2.21) generalize Ref. [86]. There, the damping of adsorbate vibration has been studied,

while assuming energy independent density of states. In our notation, this corresponds

to the case of flat spectral density functions. As we comment in Sec. 2.5, the fact that

the functions Jν(ǫ) do depend on energy is essential for the generation of vibrational

instability. The dynamics conveyed by Eqs. (2.17)-(2.24) is non-separable in terms of

the two metals, in contrast to simple linear interaction cases [78]. In other words, the

reservoirs cooperatively excite or damp energy from the impurity, thus their action is

non-additive.

It should be noted that while we assume a weak interaction limit, between electron-

hole pair generation and the vibrational mode, our scheme does not enforce weak metal-

molecule coupling; this part is exactly diagonalized to yield the reservoirs spectral func-

tion, peaked about the D or A levels. If one where to force weak metal-molecule interac-

tion, the spectral functions (2.23) would reduce to delta functions, JL(ǫ) = 2παδ(ǫ− ǫd)

and JR(ǫ) = 2παδ(ǫ − ǫa), and the resulting rates would be evaluated at the donor and

acceptor levels, e.g., kL→R1→0 = 2πα2fL(ǫd)[1− fR(ǫa)]δ(ǫd− ǫa+ω0). This also implies that

charge and energy currents are not “tightly coupled” here, such that for each transferred


electron not necessarily precisely one quanta of energy should be gained or drained at

either contact. In this aspect, our study complements the work reported in [47]. There,

using the small polaron transformation, the coupling of the molecular bridge to the leads

is assumed to be weak, while its coupling to the harmonic vibrational mode can be made

large. This study has further allowed for multiple molecular electronic states on the

bridge with electron-phonon coupling on each site, a situation is more complex than the

one considered here.

2.3.2 Resolved charge and energy equations

We write here a closed expression for the cumulant generating function, following the

approach developed in Refs. [64, 65]. It will allow us to obtain the current, its noise

power, and to confirm the FTs in this system. We define Pt(n,N, ω) as the probability

that by the time t the impurity (TLS) occupies the state n, N electrons have been

transferred from the L metal to the R side, and a net energy ω has been transferred,

L to R. Resolving Eq. (2.17) to its charge and energy components, we find that this

probability satisfies the following equation of motion [64, 65],

Ṗt(1, N, ω) = −Pt(1, N, ω)ke1→0

+

∫ ∞

−∞Pt(0, N − 1, ω − ǫ+ ω0)fL(ǫ)[1− fR(ǫ− ω0)]JL(ǫ)JR(ǫ− ω0)dǫ

+

∫ ∞

−∞Pt(0, N + 1, ω + ǫ)fR(ǫ)[1− fL(ǫ− ω0)]JR(ǫ)JL(ǫ− ω0)dǫ

Ṗt(0, N, ω) = −Pt(0, N, ω)ke0→1

+

∫ ∞

−∞Pt(1, N − 1, ω − ǫ− ω0)fL(ǫ)[1− fR(ǫ+ ω0)]JL(ǫ)JR(ǫ+ ω0)dǫ

+

∫ ∞

−∞Pt(1, N + 1, ω + ǫ)fR(ǫ)[1− fL(ǫ+ ω0)]JR(ǫ)JL(ǫ+ ω0)dǫ (2.26)

One could reason this rate equation as follows. In the first equation, the term

Pt(1, N, ω)ke1→0 stands for the decay rate of Pt(1, N, ω). The second line describes a

process where by the time t the TLS occupies the ground state, N − 1 excess electrons


have arrived at the R terminal, and an overall of ω − ǫ + ω0 energy has been absorbed

at the R bath. At the time t an electron-hole pair excitation generates an electron at

the R bath, leaving a hole at the L metal. This charge transfer process is accompanied

by an electronic energy transmission at the amount of ǫ− ω0: An electron leaving the L

bath has a total energy ǫ, however only ǫ− ω0 is gained by the R bath. The rest, at the

amount of ω0, is gained by the vibrational mode. A similar reasoning can explain other

terms in Eq. (2.26). For convenience, the factor (2π)−1 in Eq. (2.24) has been absorbed

into the definition of Jν(ω).

We Fourier transform the above system of equations with respect to both charge and

energy, to obtain the characteristic function Z(χ, η, t). It depends on the energy counting

field η and the charge counting field χ,

|Z(χ, η, t)〉 ≡

∑∞N=−∞ e

iNχ∫∞−∞Pt(0, N, ω)eiωη dω

∑∞N=−∞ e

iNχ∫∞−∞Pt(1, N, ω)eiωη dω

(2.27)

It satisfies the differential equation

d |Z(χ, η, t)〉dt

= −Ŵ(χ, η) |Z(χ, η, t)〉 , (2.28)

where the matrix Ŵ contains the following elements

Ŵ(χ, η) =

kL→R0→1 + kR→L0→1 −eiχF−1 (η)− e−iχF+2 (η)

−eiχF+1 (η)− e−iχF−2 (η) kL→R1→0 + kR→L1→0

(2.29)

Here,

F±1 (η) =

∫ ∞

−∞eiǫηfL(ǫ± ω0)[1− fR(ǫ)]JL(ǫ± ω0)JR(ǫ)dǫ

F±2 (η) =

∫ ∞

−∞e−iǫη[1− fL(ǫ± ω0)]fR(ǫ)JL(ǫ± ω0)JR(ǫ)dǫ (2.30)

The cumulant generating function is formally defined as

G(χ, η) = limt→∞

1

tln

∞∑

N=−∞eiNχ

∫ ∞

−∞Pt(N,ω)eiωηdω, (2.31)


where we introduced the short notation Pt(N,ω) = Pt(0, N, ω) +Pt(1, N, ω), that is the

probability to transfer by the time t, N electrons and an energy ω from left to right,

irrespective of the state of the TLS. The charge and heat currents can be readily derived,

by taking the first derivative of the CGF with respect to either η or χ,

〈Ie〉 ≡〈N〉tt

=dG(χ, η)

d(iχ)

∣

∣

∣

χ=0,η=0

〈Iq〉 ≡〈ω〉tt

=dG(χ, η)

d(iη)

∣

∣

∣

χ=0,η=0(2.32)

The quantity 〈ω〉t denotes the total energy ω transferred from L to R by the (infinitely

long) time t; 〈N〉t similarly counts the particles (electrons) transferred in the same di-

rection, by that time. The zero frequency noise current power density can be similarly

obtained,

〈Se〉 ≡〈N2〉t − 〈N〉

2t

t=d2G(χ, η)

d(iχ)2

∣

∣

∣

χ=0,η=0

〈Sq〉 ≡〈ω2〉t − 〈ω〉

2t

t=d2G(χ, η)

d(iη)2

∣

∣

∣

χ=0,η=0. (2.33)

The CGF can be expressed in terms of |Z〉 as

G(χ, η) = limt→∞

1

tln〈I|Z(χ, η, t)〉, (2.34)

with 〈I| = 〈11|, a left vector of unity. It is practically given by the negative of the

smallest eigenvalue of the matrix Ŵ ,

G(χ, η) = −w1,1 + w2,22

+

√

(w1,1 − w2,2)2 + 4w1,2(χ, η)w2,1(χ, η)2

. (2.35)

wi,j are the matrix elements of Ŵ , see Eq. (2.29).

2.3.3 Fluctuation theorem

We confirm next the following symmetry

G(χ, η) = G(−χ+ i(βLµL − βRµR),−η + i∆β), (2.36)


with ∆β = βR − βL. In order to prove this, we focus on the product D(χ, η) ≡

w1,2(χ, η)w2,1(χ, η) in Eq. (2.35),

D(χ, η) =[

eiχF−1 (η) + e−iχF+2 (η)

]

×[

eiχF+1 (η) + e−iχF−2 (η)

]

. (2.37)

Under the transformation χ → −χ + i(βLµL − βRµR) and η → −η + i∆β, using the

relation fν(ǫ) = [1− fν(ǫ)]e−βν(ǫ−µν), we find that

eiχF−1 (η) → e−iχe−βLµL+βRµR∫ ∞

−∞dǫe−iǫηe−∆βǫ [1− fL(ǫ− ω0)] e−βL(ǫ−ω0−µL)fR(ǫ)eβR(ǫ−µR)

×JL(ǫ− ω0)JR(ǫ) = e−iχeβLω0F−2 (η)

e−iχF+2 (η) → eiχeβLµL−βRµR∫ ∞

−∞dǫeiǫηe∆βǫ [1− fR(ǫ)] e−βR(ǫ−µR)

×fL(ǫ+ ω0)eβL(ǫ+ω0−µL)JL(ǫ+ ω0)JR(ǫ) = eiχeβLω0F+1 (η). (2.38)

Similarly, one could show that

eiχF+1 (η) → e−iχe−βLω0F+2 (η)

e−iχF−2 (η) → eiχe−βLω0F−1 (η). (2.39)

The extra factors e±βLω0 cancel, and we recover the symmetry

D(χ, η) = D(−χ+ i(βLµL − βRµR),−η + i∆β), (2.40)

confirming Eq. (2.36). We can now demonstrate the validity of a fluctuation relation for

this non-equilibrium system. The probability to transfer the energy ω by the long time

t, from L to R, is given by the inverse Fourier transform of Eq. (2.31),

Pt(N,ω) ∼1

2π

∞∑

−∞e−iNχ

∫ ∞

−∞C(χ, η)eG(χ,η)te−iωηdη, (2.41)

with limt→∞[lnC(χ, η)]/t = 0. Similarly, the quantity Pt(−N,−ω) represents the prob-

ability that N charged particles and an energy ω have been transmitted in the opposite

direction, right to left, up to time t. Based on the symmetry Eq. (2.36), one can show

that [61]

limt→∞

1

tln

Pt(N,ω)Pt(−N,−ω)

=ω∆β +N(βLµL − βRµR)

t, (2.42)


which is often written in a compact form as

P(N,ω)P(−N,−ω) = e

ω∆β+N(βLµL−βRµR). (2.43)

This expression goes beyond standard metal-molecule weak-coupling schemes as the en-

ergy and charge transfer and not tightly coupled, and the energy ω can take continuous

values, unlike Refs. [59, 60, 79].

It should be noted that the above derivation has assumed charge and energy conser-

vation between the two reservoirs. The full particle-energy counting statistics, without

such an assumption, would begin with the probability distribution Pt(n,NL, NR, ωL, ωR),

to find the system at time t in the spin state n = 0, 1, with Nν electrons and ων excess

energy accumulated at the ν bath. One can readily write an equation of motion for this

function, analogous to Eq. (2.26), to be Fourier transformed using four counting fields,

Pt(n, χL, χR, ηL, ηR) =∑

NL

eiNLχL∑

NR

eiNRχR

×∫ ∞

−∞eiωLηLdωL

∫ ∞

−∞eiωRηRPt(n,NL, NR, ωL, ωR)dωR. (2.44)

This quantity satisfies an equation of motion that is analogous to Eq. (2.28). It can be

readily proved that the negative of the smallest eigenvalue of the corresponding matrix

Ŵ(χL, χR, ηL, ηR) obeys the symmetry

G(χL, χR, ηL, ηR) = G(−χL + iβLµL,−χR + iβRµR,−ηL + iβL,−ηR + iβR), (2.45)

which can be translated into the FT for the probability itself,

Pt(NL, NR, ωL, ωR)Pt(−NL,−NR,−ωL,−ωR)

= e(NLβLµL+NRβRµR)e(βRωR+βLωL). (2.46)

Here, Pt(NL, NR, ωL, ωR) =∑

n=0,1Pt(n,NL, NR, ωL, ωR). Enforcing energy and charge

conservation, N = NL = −NR and ω = ωR = −ωL, we recover Eq. (2.43).


2.3.4 Currents, and measures for vibrational cooling, heating,

or instability

Currents. Analytical expressions for the charge and energy currents are obtained using

the definition Eq. (2.32), utilizing Eqs. (2.29) and (2.35). These currents are defined

positive when flowing L to R, and their closed forms are

〈Ie〉 = p1(kL→R1→0 − kR→L1→0 ) + p0(kL→R0→1 − kR→L0→1 ), (2.47)

and

〈Iq〉 = p1[

∫ ∞

−∞dωωfL(ω − ω0)[1− fR(ω)]JL(ω − ω0)JR(ω)

−∫ ∞

−∞dωω[1− fL(ω + ω0)]fR(ω)JL(ω + ω0)JR(ω)

]

+ p0

[

∫ ∞

−∞dωωfL(ω + ω0)[1− fR(ω)]JL(ω + ω0)JR(ω)

−∫ ∞

−∞dωω[1− fL(ω − ω0)]fR(ω)JL(ω − ω0)JR(ω)

]

. (2.48)

The TLS population is calculated in the steady-state limit,

p1 =ke0→1

ke0→1 + ke1→0

; p0 = 1− p1. (2.49)

The zero frequency noise current power is given by

〈Se〉 = −2

ke0→1 + ke1→0

〈Ie〉2 +4

ke0→1 + ke1→0

(kL→R0→1 kL→R1→0 + k

R→L0→1 k

R→L1→0 ). (2.50)

The energy current, directed towards the vibrational mode, is zero in the steady-state

limit, unless the mode is further coupled to a dissipative bath. Formally, it is given by

the expression

〈Ivib〉 = −ω0p1[

kL→R1→0 + kR→L1→0

]

+ p0ω0[

kL→R0→1 + kR→L0→1

]

. (2.51)

Measures for vibrational instability. The stability of the junction can be estimated,

against heating effects, by inspecting several measures. First, following Ref. [25], we


define the damping rate Kvib of the vibrational mode as the difference between relaxation

and excitation rates,

Kvib ≡ ke1→0 − ke0→1. (2.52)

Positive Kvib indicates on the “normal” thermal-like behavior, as relaxation processes

overcome excitations. In this case, the mode effective temperature (defined below) is

found to be either below (cooling) or above (heating) the environmental temperature,

yet the junction remains stable in the sense that the ground vibrational state population

is larger than the excited level population. A negative value for Kvib evinces on the

process of an uncontrolled heating of the molecular mode, eventually leading to junction

instability and breakdown. One can also directly inspect the TLS population: population

inversion reflects on vibrational instability.

Effective temperature. The TLS population can be further utilized as a measure

for the molecular vibration effective temperature, 1/βeff , defined using an equilibrium

relation,

p1p0

= e−βeffω0 . (2.53)

A negative value for βeff attests on population inversion, thus junction instability. When

βeff is positive, one should compare it to the reservoirs’ inverse temperature β: If βeff > β

the system demonstrates bias-induced cooling phenomena. For βeff < β the vibrational

mode is heated up relative to its environment. The latter typically occurs at an interme-

diate bias voltage, before instabilities take place.

2.3.5 Numerical results: isolated mode

We demonstrate cooling, heating and mode instability upon varying the bias voltage. A

generic mechanism leading to vibrational instabilities (and eventually junction rupture)

in D-A molecular rectifiers has been discussed in Ref. [25]: At large positive bias, when


0

0.05

0.1

0.15

〈 Ie〉

[1/fs

]

(i)

−2 0 2−2

0

2

∆ µ [eV]

Ene

rgy

[eV

]

εd

εa µ

L

µR

−2 −1 0 1 2−0.05

0

0.05

∆ µ [eV]

Kvi

b [fs

−1 ]

(ii)

Figure 2.3: (i) Charge current in a rectifying molecular junction. Inset: Energies of the

donor (full line) and acceptor states (dashed line). The dotted lines correspond to the

chemical potential at the left and right sides. (ii) Damping rate Kvib. The junction

parameters are Γν=0.2, 1/βν = 0.005, α = 0.1, ω0 = 0.05 and ǫd(∆µ = 0) = −0.2,

ǫa(∆µ = 0) = 0.4, all in units of [eV]. Reproduced from Ref. [33].


−2 −1 0 1 20

0.2

0.4

0.6

0.8

1

∆ µ [eV]

popu

latio

n

p1

p0

Figure 2.4: Population of the two-state “vibration” as a function of bias voltage. Param-

eters are the same as in Fig. 2.3. Reproduced from Ref. [33].


the D state is positioned above the acceptor level, electron-hole pair excitations by the

molecular vibration (TLS here) dominate the mode dynamics. This can be schematically

seen in Fig. 2.2, where the rate kL→R0→1 overcomes other rates once the donor spectral

function is positioned above the acceptor spectral function. As Kvib becomes negative,

population inversion is observed.

The junction setup is displayed in Fig. 2.3. D and A levels are positioned such

that in equilibrium, ∆µ = 0, the donor level is placed below the Fermi energy µ, while

the acceptor level is of a higher energy, ǫd(∆µ = 0) < µ < ǫa(∆µ = 0). Under an

applied bias, the levels are assumed to linearly follow the external potential drive (inset)

[87]. Therefore, at a particular positive bias the levels cross. Beyond that, the levels

exchange arrangement, and the D state is of a higher energy. Throughout the Chapter,

the parameters ω0, Γν , Γph, 1/β, α, ǫd,a and ∆µ are given in units of eV.

The junction’s current-voltage characteristics is displayed in Fig. 2.3 (i), manifesting

a substantial rectification effect. For negative polarity, ∆µ = µL − µR < 0, the current

is rather small. In contrast, for positive bias the current substantially increases once

∆µ > ω0, reaching a maximum when the energy levels satisfy ǫd − ǫa ∼ ω0. Level

broadening, Γν , affects the actual position of the maximum. The current scales with

α2, thus the value picked here, α = 0.1, does not affect the current characteristics. The

damping rate, Kvib, is displayed in Fig. 2.3(ii). It shows the following features: First,

for large negative bias, ∆µ < −0.2, Kvib is negative. This instability can be immediately

removed, once a very weak coupling to a phononic thermal reservoir is turned on, see

Figs. 2.7 and 2.12 below. Beyond that, the damping rate Kvib is positive between

−0.2 . ∆µ . 0.6, indicating on a stable mode of operation. However, for large enough

bias, ∆µ & 0.6, once ǫd > ǫa, uncontrolled TLS heating takes place, recognized by a sign

change in Kvib. It should be noted that the instability takes place in the parameter range

very relevant to the rectifier operation. It is thus important to understand how to tune

the system configuration so as to sustain junction functionality.


Fig. 2.4 depicts the corresponding population of the two levels. At zero bias, ke0→1 = 0,

thus the population of the excited state is identically zero. At low positive bias one finds

that ke0→1 < ke1→0, leading to the “normal” situation of p0 > p1. However, once the bias is

large and the donor state is positioned above the acceptor site, (∆µ ∼ 0.6) the excitation

rate ke0→1 exceeds the relaxation rate ke1→0 and population inversion takes place. We note

that for a negative bias, small population inversion is also observed, as electrons damp

energy to the TLS when crossing the junction. However, since 〈Ie〉 is rather small (Fig.

2.3), we do not expect molecular instability in this regime, see also Fig. 2.12.

The details of the damping rateKvib depend on the level broadening and the reservoirs

temperature as we show in Fig. 2.5. The position of the turnover, between positive to

negative damping, appears at a similar value for the bias, and it is generally independent

of the reservoirs temperatures and Γν . However, the width of the curve largely depends

on these parameters. We also display in Fig. 2.6 the charge current and the vibrational

mode population for a different set of parameters: The coupling of the electronic levels to

the respective leads is taken smaller, Γν = 0.01 (units are in eV). This choice represents

a multisite molecular chain, where the relevant electronic sites, those that couple to the

particular vibrational mode, are located within the chain center, few sites away from the

metals. We further assume a weaker electron-phonon coupling, α = 0.01 and a larger

vibrational frequency, ω0 = 0.2, in the range of typical stretching and bending molecular

modes. We then note that general features are left intact, but the charge current is

reduced by more than three orders of magnitude, bringing it to the nA scale. We also

note that the vibrational instability is still formed about similar values, at ∆µ = 0.4,

for positive bias. Our simulations therefore produce charge current values in the range

of 〈Ie〉 ∼ µA-nA, in agreement with experimental, e.g., Ref. [36, 42, 39], and other

theoretical values, for instance [46, 47, 50, 49, 53].

It should be noted that the development of the instability, as reported in Figs. 2.3,

2.4 and 2.5, does not depend on the concrete value of α, the strength of the molecule-


mode coupling, and the behavior persists in the limit of vanishing vibronic coupling,

α → 0. In the next section we allow the vibrational mode to thermalize with a phononic

environment at a rate Γph. In this case, the competition between α and Γph determines

the onset of instability, see Eq. (2.57).

−0.5 0 0.5 1 1.5 2

−0.04

−0.02

0

0.02

0.04

0.06

∆ µ [eV]

Kvi

b [fs

−1 ]

Γν=0.2, βν=200

Γν=0.4, βν=200

Γν=0.2, βν=5

Figure 2.5: Damping rate in a rectifying junction for different broadening parameters,

Γν=0.2, βν = 200 (full) Γν=0.4, βν = 200 (dashed) Γν=0.2, βν = 5 (dashed-dotted).

Other parameters are the same as in Fig. 2.3. Reproduced from Ref. [33].

2.3.6 Numerical results: dissipative mode

Up to this point, we have assumed that the molecular vibrational mode (TLS here) is well

isolated from other vibrations. In reality, internal modes typically exchange energy with

“secondary” reservoirs modes, either internal, or part of a larger environment, opening

up an additional route for energy dissipation. It is expected that in the presence of such


0

2

4

6

x 10−5

〈 Ie〉

[1/fs

] (i)

−2 −1 0 1 20

0.5

1

∆ µ [eV]

popu

latio

n

p1

p0

(ii)

Figure 2.6: (i) Charge current and (ii) vibrational states population in a rectifying

junction with weak electron-phonon coupling and weak molecule-metal hybridization

strength, Γν=0.01, α = 0.01, ω0 = 0.2. Other parameters are the same as in Fig. 2.3.

Reproduced from Ref. [33].


a thermal bath, the region of vibrational instability (Kvib < 0) would become limited.

A simple model that is capable of describing a hierarchy of energy transfer processes,

electronic energy → specific vibrational excitation → thermal bath, is given by an ex-

tension of the model (2.12),

HA+B =ω02σz + σx (Fe + Fb) +

∑

l

ǫla†lal +

∑

r

ǫra†rar +

∑

k

ωkb†kbk. (2.54)

The notation “HA+B” indicates that the anharmonic mode is coupled to a thermal bath

(B). The operator Fe describes electron-hole pair excitations as in Eq. (2.13). The

thermal bath operator, coupled to the TLS transitions, includes displacements of reservoir

modes,

Fb =∑

k

ζk(b†k + bk), (2.55)

with b†k (bk) as a bosonic creation (annihilation) operator for the kth phonon-reservoir

mode.

Derivation of the full counting statistics can be reiterated, while including energy

dissipation from the TLS to the phonon bath. For details, see Appendix A. We find

that the expression for the charge current stays intact, satisfying the formal expression

(2.47). However, the steady-state populations are corrected by a phonon relaxation rate

constant as

p1 =ke0→1 + Γph(ω0)nph(ω0)

ke0→1 + ke1→0 + Γph(ω0)[2nph(ω0) + 1]

. (2.56)

The electronic transition induced rates ken→n′ are those defined in Eq. (2.18); the phononic

relaxation rate constant is Γph(ω) = 2π∑

k ζ2kδ(ωk − ω). The function nph(ω) = [eβphω −

1]−1 stands for the Bose-Einstein distribution with βph as the temperature of the phonon

bath.

Fig. 2.7 presents the steady-state population for two choices of Γph. When this param-

theory of charge transport through vibrating molecules · theory of charge transport through...

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