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INFLUENCE OF MICROWAVE RADIATION ON
TRANSPORT PROPERTIES OF MESOSCOPIC
SYSTEMS
A D issertation
P resented to th e F aculty of th e G raduate School
o f C ornell U niversity
in P a rtia l Fulfillm ent o f th e R equirem ents for th e Degree o f
D o cto r of P hilosophy
by
M axim Georgievich Vavilov
August 2001
R e p r o d u c e d with p e r m i s s io n o f t h e co p y rig h t o w n e r. F u r th e r re p r o d u c tio n proh ibited w ithout p e r m is s io n .
© Maxim Georgievich Vavilov 2001
ALL RIGHTS RESERVED
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B iographical Sketch
The author was born in 1973 in Sverdlovsk, USSR, where he spent the first 17
years of his life. He obtained a B.S. degree in Physics (1995) with specialization
in Q uantum Optics at Moscow Institute of Physics and Technology (MIPT), and
a M.S. with honours in Theoretical Physics (1997). He performed research for his
M.S. degree a t Landau Institute for Theoretical Physics. He entered graduate school
at Cornell University in 1997, where he received a M.S. in Physics in 1999 and a
Ph.D. in 2001.
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To my Parents and Asya
IV
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A ck n ow led gem en ts
I would like to thank my thesis adviser, Prof. Vinay Ambegaokar. I am grateful for
his many valuable suggestions, ranging from the choice of problems to the presen­
tation of results. Prof. Vinay Ambegaikar allowed me to select my projects freely,
helping me to develop a degree of academic independence, but always reminded me
th a t the ultim ate goal should be the understanding of experimental phenomena.
Prof. Ambegaokar helped me not only by advising on scientific problems, but also
by arranging the financial support for more than three years of my study at Cornell.
I owe many sincere thanks to Prof. Igor Aleiner, who played a role of my second
thesis advisor. Igor’s dedication to Physics has been inspirational. I benefited from
the collaboration with him, especially from his detailed and patient explanations of
many topics of m odern condensed m atter physics.
I thank Prof. Piet Brouwer for the question he gave me for my A examination,
which grew into a separate project. I learned a lot from our numerous discussions
about methods of random m atrix theory.
I thank Prof. D an Ralph for reading my thesis, making useful comments and
attending my Ph. D. defence. I am also grateful to him and Erik Sm ith for their
explanations of techniques used for measurement in mesoscopic physics experiments.
I am grateful to Prof. Jeevak Parpia for serving on my committee, providing an
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p e r m i s s io n of t h e co p y rig h t o w n er. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
interesting question for my A examination and discussing with me various areas of
low tem perature physics.
I thank Prof. Vladimir Mineev for refining my research techniques and for di­
recting my scientific interest toward mesoscopic physics.
I also benefited from numerous discussions w ith Boris Altshuler, Jan von Delft,
Chris Henley, Alexei Kamenev, Dmitry Khmelnitskii, Vladimir Kravtsov, Kon­
stantin Matveev and Boris Narozhnyi. I also had the pleasure of working together
w ith Aashish Clerk, Mikhail Polianski, Tim ur Shutenko and Xavier W aintal.
T he staff of the Cornell Physics Departm ent, LASSP and CCMR, particularly
Debra Hatfield, Douglas M ilton and Sandy Sweazey, deserve many thanks for their
help.
The research reported here was supported by the Cornell Center for Materials
Research through Grant No. DMR-9632275 of the National Science Foundation.
I am grateful to Sergei Kiselev for his help and advice upon my arrival at Cor­
nell and to Marina Basina, Yulya Epifantseva, Nigar Hashimzade, Alexey Kisselev,
Sergei Kriminsky, Andrei Litvintsev, Leonid Malyshkin, Anatoly Olhovets, Mikhail
Polianski, Robert Sherbakov and Sergey Shpaner, who have made my years at Cor­
nell unforgettable.
I thank Asya, whose friendship enriched my life in many ways. She provided me
with encouragement and support in all my endeavors.
Finally, I thank my parents and my sister. Despite the distance between us,
they were always there for me, helping me to overcome all difficulties and sharing
all of my achievements. T heir inspiration and love have been a continuous source of
support th at I needed to carry me through the four years of my Ph. D. program.
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p e r m i s s io n of t h e co p y rig h t o w n er. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
Table o f C on ten ts
1
In tro d u c tio n
1.1 Experim ental P r o b l e m ...................................................................................
1.1.1 Mesoscopic C o n d u c to rs......................................................................
1.1.2 Quantum Dots ....................................................................................
1.2 Basic Phenomena in Small C o n d u c to rs ......................................................
1.3 Overview of the T h e s i s ...................................................................................
1.4 O utline of the T h e s i s ......................................................................................
2
M eta l W ires
10
2.1 Influence Functional for Disordered Interacting Electron Systems . . 13
2.2 Dephasing due to Classical Component of Electric Field ........................ 20
2.3 Q uantum Corrections ....................................................................................... 26
2.3.1 Uniform Electric F i e l d .......................................................................... 30
2.3.2 Electron-Electron I n te r a c tio n ..............................................................32
2.3.3 High Frequency C u t o f f ...........................................................................33
2.4 Dephasing by External Oscillating field
...................................................... 35
2.5 Discussion
...........................................................................................................38
3
O pen Q u an tu m D o ts
42
3.1 M o d e l .....................................................................................................................47
3.2 Photovoltaic E f f e c t ..............................................................................................53
3.2.1 Mesoscopic Fluctuations of the Photovoltaic Current at High
Tem perature ..........................................................................................57
3.2.2 Photovoltaic Effect as Pumping in Phase S p a c e ..............................62
3.2.3 Photovoltaic Effect at Low T e m p e r a tu r e ......................................... 65
3.3 Conductance of Open Quantum D o t s ............................................................. 69
3.3.1 Weak Localization Correction to C o n d u c ta n c e .............................. 71
3.3.2 Conductance F lu c tu a tio n s .................................................................... 75
3.4 Rectification of a.c. B i a s ....................................................................................81
4
C u rren t N o ise
85
4.1 C urrent Noise Correlation F u n c tio n ................................................................ 87
4.2 Bilinear Response ..............................................................................................92
4.3 Ensemble Averaged Noise ................................................................................ 95
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1
2
3
4
5
7
8
5
D iscu ssio n s and C on clu sion s
98
5.1 D is c u s s io n s .......................................................................................................... 98
5.1.1 Dephasing in W ires vs Dephasing in O pen Q uantum Dots . . 98
5.1.2 Observation of Dephasing in Open Q uantum D o t s ........................ 100
5.1.3 Adiabatic Charge Pum ping and a.c. R e c tific a tio n ...................... 102
5.2 C o n c lu s io n s.........................................................................................................105
A
W eak L ocalization E ffect in W ires
107
A .l Weak Localization Correction to C o n d u c tiv ity ......................................... 107
A.2 Cooperon Equation
........................................................................................ 109
A.3 Q uantum Correction to D e p h a s in g ..............................................................112
A.4 Q uantum Correction for the Nyquist Noise M o d e l ...................................115
A.5 Q uantum Correction for Electron-Electron Interaction ......................... 116
B
T im e D e p en d en t R a n d o m M a tr ix T h eo ry
119
B .l Keldysh Technique for Q uantum Dots ....................................................... 119
B.2 C urrent for Zero Bias .....................................................................................123
B.3 Basic Diagram E le m e n ts ................................................................................. 124
B.4 Photovoltaic Effect as Pum ping in Phase S p a c e ...................................... 129
B.5 C urrent Noise Correlation F u n c tio n ..............................................................130
B ib liograp h y
133
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List o f Figures
1.1
2.1
2.2
2.3
2.4
3.1
3.2
3.3
3.4
Various trajectories are formed as the result of scattering on: (a) impuri­
ties in bulk metals or (b) boundaries of quantum dots. Two time reversed
trajectories contribute to the interference correction to the electron prop­
agation through the system...............................................................................
6
Figure a shows maximally crossed diagrams which correspond to the weak
localization correction to conductivity. Open circles denote the position
for the Keldysh component of the electron Green’s function. The sum
of all possible diagrams is represented by the diagram in Fig. b. The
Cooperon C is defined in Appendix A.2
21
Four possible diagrams which contribute to the dephasing correction to
the conductivity. .................................................................................................29
T he weak localization correction to the conductivity is shown in
Fig. 2.3(A). Figure 2.3(B) represents the basic elements of the di­
agram m atic technique. T he renormalized vertex is the solution to
the diagram m atic equation, see Fig. 2.3(C)................................................... 34
T he diagram m atic equation for the Cooperon C in the presence of
electron-electron interaction. The basic diagram elements are defined
in Figs. 2.3(B) and (C )........................................................................................34
An open quantum dot is connected to two leads with applied voltages F[,rThe measured current through the dot has an offset I q at zero bias and a
linear response to small applied voltage V =V\ —Vr.......................................... 43
An experimental setup. The voltage applied to the gates Vj(£) and
V2(t) changes the shape of the dot, resulting in m otion of the energy
levels of the electrons in the d o t.......................................................................48
T h e diagram representing the contribution to the charge correlator
Qi at high tem perature T .................................................................................. 56
T he am plitude dependence of the pum ped charge for different values
of the frequency of the pum p. For u > 7esc the curves have the C 2
dependence at small values of C and the \ / C dependence at C ~^>
7esC- For small frequency (e.g. ui = 0.1'yesc) there is an intermediate
regim e......................................................................................................................58
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3.5
In figure (a) a loop is shown in the parameter plane. The grid divides
the plane onto pieces, so that parts of the loop in the different pieces give
uncorrelated contributions to the transported charge. Figure (b) shows
the loop in the phase space for strong pumping. In this case the loop can
be divided onto pairs, and the pairs are not correlated. On the other hand
parts of the loop of one pair are close to each other, so they are strongly
correlated................................................................................................................ 61
3.6 Diagrams, which contribute to the d.c.-current at low tem perature
limit. (We do not show diagrams, which can be obtained from the
above by omitting the upper or lower diffusons.) The last diagram
contains the Hikami box, which is presented in Fig. 3.7............................ 68
3.7 The Hikami box, introduced in Fig.3.6, can be obtained from these
diagram s and from their rotations. T he grey rectangle represent
averages of the type
and G ^ G ^ .................................................. 69
3.8 The diagram for the weak localization correction to the conductance. . . 72
3.9 Representative curves of F(y, z) as a function of z for two values of y.
It decreases linearly with z at small values of z. The inset shows the
y —dependence of the function F(y, z) for two values of z. It decreases
quadratically in y at small values of y and saturates at larger y......................74
3.10 Two diagrams, which contribute to the conductance correlation function
R .............................................................................................................................. 77
3.11 Functions Qc{x,y,z) and Qd{x,y) computed for x — 1 and z = 10. As
temperature y = T / j increases, function Qc{x, y, z) approaches frequency
independent function Qd(x, y).............................................................................. 79
4.1
4.2
Diagrams, which contribute to 7Z to the lowest order in l / N Ch with NCh 3>
1 in a bilinear regime............................................................................................. 94
Diagrams, which contribute to the ensemble averaged value of the noise
correlation function Q2 to the lowest order in 1/iVch f°r arbitrary strength
of the perturbation.................................................................................................96
5.1
Weak localization correction Sgw1 versus conductance fluctuations var g
of an open quantum dot with N\ = ATr 3> 1 for three values of frequency
uj: hu = 0.57esc (o); hui = 57eSc (v);
= 507esc (A). The temperature
for all lines was taken T = 107esc- The amplitude of the field Co varies
from 10- 27esc to 1027esc- Go = e2/irh is the quantum conductance. . . . 101
A .l
The Dyson equation for the Cooperon in the classical electric field V(t, r). 110
B .l
Correspondence between the sign of x and the direction of motion of
electrons with respect to the dot........................................................................120
x
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B.2
(a) Diagrams for the ensemble averaged G reen’s function. The sec­
ond term in the self-energy includes an intersection of dashed lines
and is as small as 1/M . (b) The Dyson type equation for the diffuson,
ty, 12 , t^)- (c) The Dyson type equation for the Cooperon,
.....................................................................................................125
xi
R e p r o d u c e d with p e r m i s s io n o f th e cop y rig h t o w n e r. F u r th e r r e p ro d u c tio n p rohib ited w ith o u t p e rm is s io n .
C h ap ter 1
In tro d u ctio n
This thesis focuses on the tran sp o rt properties of mesoscopic systems, such as m etal
wires or open quantum dots.
An im portant property of the transport through
mesoscopic devices is the interference phenomena, see [1]. One of the consequences
of the interference is the weak localization correction to the conductance.
Recent measurements of the weak localization correction to the conductivity of
m etal wires [2] exhibited a deviation of tem perature dependence from the power law
predicted by [3]. To verify th a t the deviation is not an artifact of the semiclassical approxim ation, a quantum mechanical correction to the semiclassical theory of
electron-electron interaction is calculated for m etal wires. This correction is shown
to be sm all in the regime of weak localization.
Sim ilar behavior of the weak localization correction to the conductance of open
quantum dots was observed in ref. [4]. A possible source of the deviation in both
cases could be an external electrom agnetic field of the environment. The theory
developed in this thesis aims to calculate the effects of the radiation on transport
properties of mesoscopic systems. The microwave radiation destroys coherence in an
electron system, suppressing quantum interference phenomena. The radiation also
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2
produces a num ber of new effects in open quantum dots, such as quantum pum ping
and rectification.
In order to model these phenomena, we have developed a time-dependent ran­
dom m atrix theory and a diagram m atic technique to calculate electronic transport
properties of open quantum dots w ith a large num ber of channels. This technique
is sim ilar to the one applied to electron disordered m etal systems.
T he technique described in this thesis can be applied to a number of sim ilar
problems of transport through chaotic quantum interacting systems.
1.1
Experimental Problem
Recent experim ents allowed the measurement of electron transport at low tem per­
ature, revealing discrepancies between theory and experim ent. These experiments
suggest th a t either new phenomena appear at low tem perature or the systems are
not sufficiently isolated from the external environment. In this thesis, we present
a model th a t estimates the influence of the environm ent on the systems.
If the
proposed model accurately describes the experiments, a new experimental design
should be developed, a design th a t would adequately isolate the systems from the
influence of the environment. Othenvise, the physical phenomena behind these sys­
tems require further study.
The experim ental measurements were performed for two types of electron sys­
tems: th in m etal wires and open quantum dots. B oth systems exhibited a num ber of
common features, such as weak localization corrections to the conductance and con­
ductance fluctuations. The quantitative difference between bulk mesoscopic systems
and quantum dots is determined by the Thouless energy E m , which is the inverse
ergodic tim e of the system r erg. Throughout this thesis, we will refer throughout
with p e r m i s s io n of t h e cop y rig h t o w n e r. F u r th e r r e p ro d u c tio n prohib ited w ith o u t p e rm is s io n .
3
the thesis to a system as a bulk m etal if the Thouless energy is the smallest energy
scale, and as a quantum dot if the Thouless energy is the largest energy scale.
1.1 .1
M eso sco p ic C o n d u c to r s
The quantum nature of electron transport appears at sufficiently low tem perature
when the phase coherence length becomes greater than the electron mean free path.
In this regime, the conductance of a m etal sample is sensitive to the applied mag­
netic field due to the quantum interference of electron wave functions. Particularly,
the weak localization correction to the conductivity (the conductivity peak at zero
magnetic field) is suppressed by inelastic processes, which destroy the interference
of electron wave functions. We refer to these processes as dephasing.
Examples of a dephasing process are the electron interaction with an oscillating
electric field or the electron-electron interaction. The effect of the electron-electron
interaction on the weak localization correction to the conductivity was developed in
refs. [3, 1], where the electric field is treated semiclassically. The experimental mea­
surements [5] were in agreement with the power law dependence of weak localization
on tem perature predicted by [3].
Recently Mohanty et. al.
[2] claimed that their m easurem ents deviate from
the theory of ref. [3] at low tem perature.
Their statem ent triggered a series of
independent measurements [6], which were performed w ith different materials and
for different sample geometries. Some experimental results agreed with the theory of
ref. [3] even at lower tem peratures than those for which the deviation was reported
in [2].
In this thesis, we do not discuss the experimental situation as regards weak
localization measurements. We will discuss the extent to which the semiclassical
with p e r m i s s io n of t h e co p y rig h t o w n er. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
4
calculations of ref. [3] are applicable to the weak localization regime and also show
how the quantum corrections can be taken into account.
The system s considered as bulk m etals have at least one large length parameter,
but they can be confined in one or more directions. We can further distinguish
these systems as three, two and one dimensional, according to the Thouless energy
corresponding to different directions. If the Thouless energy for two perpendicular
directions is a large energy scale with respect to the tem perature of the system or
the frequency of an interaction, while the Thouless energy for the third dimension
is small, the system is treated as one dimensional (wires). If the Thouless energy
associated with only one direction is large, then the system is two dimensional. Three
dimensional system have a negligibly small Thouless energy for all three directions.
1.1.2
Q u a n tu m D o ts
Recently, another object for the study of quantum effects appeared — ballistic
quantum dots. An open quantum dot is a small conductor, com parable in size to the
electron mean free path, well connected with two leads. Electron transport between
leads through a quantum dot is characterized by the conductance. A detailed study
of the conductance properties of open quantum dots was perform ed in refs. [7, 8, 4, 9].
The electron motion is chaotic inside the dot due to the scattering from the
dot boundaries. The shape of the dot is random. Sm all perturbations of the dot
significantly transform the electron configuration of the dot. At low tem peratures,
the electron motion in the dot is coherent and exhibits quantum mechanical inter­
ference phenomena. The experiments allow the study of the statistics of electron
transport with respect to different realizations of the dot. Specifically, conductance
through open quantum dots has been studied. The conductance, at sufficiently low
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n er. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
tem peratures, exhibited the weak localization correction. This correction enabled
the authors of [4, 9] to investigate the electron dephasing tim e in the system. The
conductance fluctuations with respect to the shape of the dot, studied in [8, 9], give
independent measurements of th e dephasing time. T he results of both experiments
are in agreem ent. Particularly, the dephasing time satu rates at low tem perature.
1.2
Basic Phenomena in Small Conductors
Q uantum phenomena of electron transport through mesoscopic systems result from
the interference of electron trajectories. The sim plest m anifestation of a quantum
transport is the weak localization effect produced by the interference of two tim ereversed trajectories. This interference increases the return probability and leads to
the weak localization correction to the conductance.
A comm on illustration of th e weak localization effect (backscattering) is shown in
Fig. 1.1. Figure 1.1(a) represents two time-reversed trajectories resulting from scat­
tering on impurities. Each im purity creates a scattered wave. A fter many scattering
processes, electron may return to the starting point. T he time-reversed trajectory
repeats the original trajectory, but an electron moves in the opposite direction. T he
electron acquires the same phase when it moves along these two trajectories and
there is non-zero interference term . We notice th a t due to interaction, the phase
difference for two time-reversed trajectories is finite and the interference term de­
creases after averaging.
A sim ilar process of backscattering exists in open quantum dots, as shown in
Fig. 1.1(b). The only difference from the discussed above case for disordered m etals
is th a t the scattering is produced by the dot’s boundaries.
An im portant property of mesoscopic systems is the random reproducible be-
p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p ro d u c tio n prohibited w ith o u t p e rm is s io n .
6
Figure 1.1: Various trajectories are formed as the result of scattering on: (a) impurities in
bulk metals or (b) boundaries of quantum dots. Two time reversed trajectories contribute
to the interference correction to the electron propagation through the system.
havior of its observable properties, such as conductance. These fluctuations appear
as a result of the small changes in the magnetic field or the shape of the system. At
the same tim e, if the system returns to its original configuration, the quantities are
reproduced. Since mesoscopic systems are extrem ely sensitive to external param e­
ters, the scope of mesoscopic physics is the statistics of observable quantities rather
than their values for a particular realization.
We study the statistics of transport param eters of open quantum dots for small
voltages applied across the dot, such as the conductance. The ensemble average
conductance has an inverted peak at zero m agnetic field, known as the weak lo­
calization correction to the conductance. The conductance itself fluctuates around
its ensemble averaged value. The value of the fluctuations around the ensemble
averaged value of the conductance is of the order of Go, where G q = e2/irh is the
quantum conductance, see [10] and references therein.
Bulk mesoscopic systems behave in exactly the same way. Namely, their con­
ductance has a weak localization peak at zero m agnetic field and fluctuates from
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n er. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
sample to sample. The variance of these fluctuations is of the order G q, see [11, 12].
For open quantum dots another interesting phenomenon can be observed, known
as the photovoltaic effect [14. 15, 17, 16]. Even if the voltage across the dot is
zero, the external oscillating field can generate a direct current through the dot.
The direction and the m agnitude of the current are random and fluctuate from
sample to sample. This effect has to be taken into account to adequately define the
conductance of the system.
1.3
Overview of the Thesis
The experiments mentioned above have demonstrated th a t the quantum transport is
affected by electron interaction. We will discuss how this interaction can be treated
to describe experimental observations. The deviation of the experimental results
from the theoretical predictions [3] for the weak localization measurements of the
conductivity [2] raised the question of whether the semiclassical theory [3] is valid in
the regime of [2], To answer this question, we consider an exact quantum mechanical
system of electrons interacting with a quantum field and explicitly determine the
semiclassical contribution studied in [3]. The remaining contributions are consid­
ered as quantum corrections and will be treated perturbatively. We will show that
the semiclassical treatm ent of [3] allows to cure the infrared divergence for the weak
localization correction. On the other hand, the exact quantum mechanical consid­
eration limits the contribution of electromagnetic modes interacting with electrons
with a frequency greater th an the system tem perature. We will conclude th at for the
weak localization regime the semiclassical theory of interaction [3] is well justified.
Next we consider the effect of microwaves on tran sp o rt properties of open quan­
tum dots. Formally, we should repeat the analysis done for m etal wires in C hapter 2
p e r m i s s io n of t h e co p y rig h t o w n er. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
8
of the thesis and tre a t the electron interaction with an electrom agnetic environment,
taking into account the exact quantum mechanical form of the interaction. Both
mesoscopic wires treated in C hapter 2 and quantum dots are described by the iden­
tical equations of motion (with the only difference being the particular form of
electron wave functions). The separation into classical and quantum fields for open
quantum dots can be done sim ilarly to the case for mesoscopic m etal wires.
On the other hand, the quantum p art of the interaction is im portant only for
the frequency of electromagnetic modes comparable or greater th an the equilibrium
tem perature of the system. For the intensive external radiation (out of equilibrium
with electrons) or for modes w ith low frequency (sm aller th an the tem perature)
the quantum corrections can be neglected. Thus, to study the effect of microwave
radiation on transport through open quantum dots, we will only use the semiclassical
theory.
1.4
Outline of the Thesis
In C hapter 2 we discuss the electron-electron interaction in bulk m etals and demon­
strate th a t the quantum nature of this interaction gives a small correction to the
semiclassical result of ref. [3] in the regime of weak localization. We explain how
the quantum mechanical electron-electron interaction can be treated to avoid in­
frared divergences in the theory and show that the electron-electron interaction can
be treated as an interaction with microwave radiation and an infrared regularized
quantum field. In Chapter 3 we study the influence of microwave radiation on low
bias tran sp o rt through open quantum dots. We discuss the suppression of the weak
localization correction to the conductance of the dot, the conductance fluctuations
and the photovoltaic effect in open quantum dots. We use the model developed in
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n er. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
9
C hapter 3 to calculate the current noise correlation function in C hapter 4. Chapter
5 contains a com parison of our theoretical results with experim ents and conclusions.
In appendix A we present auxiliary calculations for C h ap ter 2. Appendix B
contains a detailed and extensive introduction to the tim e-dependent random m atrix
model and the diagram m atic technique for a quantum dot w ith m any open channels.
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n er. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
C h ap ter 2
M eta l W ires
Experim ental and theoretical studies of weak localization have given considerable
insight into the physics of small disordered conductors. A review of this research can
be found in [1]. Since interference between time-reversed electron trajectories[l, 19]
is the root cause of weak localization, its strength depends on phase coherence
between such paths. Dephasing can be due to extrinsic causes, such as applied mag­
netic fields, or to intrinsic mechanisms th at remove phase information, for example
scattering against localized spins, or the electron-phonon and electron-electron in­
teractions, the last being the more im portant at low tem peratures.
In this chapter we reconsider the effect of electron-electron interactions on weak
localization. O ur aim is to understand quantum corrections to a theory [3] in which
the influence of these interactions is modeled as a fluctuating classical electromag­
netic field. T he subject is topical because of experim ental work [2] suggesting th at
the dephasing rate saturates to a finite value as the tem perature approaches zero,
whereas ref. [3] predicts the rate to vanish in this lim it. Now, one certainly expects
quantum corrections to the last mentioned theory. The relevant question, which
we address here, is whether the corrections are im portant in the region of weak
10
p e r m i s s io n of t h e co p y rig h t o w n er. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
11
localization. O ur conclusion, in brief, is th a t they are not.
It is worth comparing our approach to quantum effects w ith a preprint [20] in
which very complete calculations of the quantum mechanical correction to the Drude
conductivity of a weakly disordered m etal are carried out to second order in the
screened electron-electron interaction. In the absence of dephasing, the weak local­
ization correction for narrow wires contains an infrared divergence. This divergence
cannot be cured in finite order perturbation theory. For this reason, a magnetic field
is posited in ref. [20], providing a low frequency cut-off . The cut-off dependence
of the correction to weak localization is found to be exactly as expected from [3].
Although we find this argument completely convincing, it is nonetheless true that
the results of ref. [20] are, strictly speaking, only valid when the electron-electron
interaction produces a small correction to the effect due to some other mechanism.
[See, however, Sec. 2.5 below.] Now, the calculation of [3] has the considerable
merit of treatin g classical fluctuations exactly, thereby being free of low frequency
divergences. By building on this calculation, we are able to test its validity in the
absence of extrinsic influences.
According to paper [20], the first order diagrams in the interaction can be divided
onto two parts: A ^ V = A a deph -h A cTcwt,
A
deph is called the dephasing term and
ct
the other A cr^i is the interaction correction to weak localization. Although they
both originate from the same type of diagrams they have different dependences on
the param eters of the system.
For example, it was shown in [20], th at for one
dimensional wires
e2s/ D r„
Trh
3C (3/2)
1/2 3/2 e2T
H 4/i’a'i
27Th
2irhai
( 2 . 1)
V 27xT
Here rH is the dephasing time due to an external magnetic field, which appears in
with p e r m i s s io n of t h e cop y rig h t o w n e r. F u r th e r r e p ro d u c tio n prohib ited w ith o u t p e rm is s io n .
12
the calculations of [20] as a low frequency cutoff. We see th a t the most singular
term is the dephasing correction which is proportional to rfj, while A
~ \/th-
We consider here only the dephasing term of the weak localization correction to
conductivity.
The physical meaning of both of these term s was explained in [20]. We would
like to note only th a t the dephasing term is the result of the interference of two
time reversed paths (see below), while the interaction correction term is the result
of interference for more complicated trajectories.
It is appropriate here to mention some other recent work. In e—print [22] by us we
suggested th a t the saturation in dephasing rate [2] can be explained in the framework
of the Caldeira-Leggett model [23]. This paper is simply wrong, because it treats
the phase as a single particle and loses the physics associated w ith the exclusion
principle. We also remark that Golubev and Zaikin [24] have a calculation which
claims to trea t interaction effects to all orders in perturbation theory and obtains
a finite dephasing rate from the electron-electron interaction at zero tem perature.
We believe th a t this surprising result is due to uncontrolled approxim ations, and
comments on it can be found in [25, 26].
In outline, the plan of this chapter is as follows. In Section 2.1 we introduce a
model in which the dephasing environment is modeled by a set of harmonic oscil­
lators, in the m anner of Feynman and Vernon [27] and Leggett and Caldeira [23],
which are coupled to the charge density of an electron gas in a random potential. By
a suitable choice of the spectrum of oscillators, we obtain after integrating out the
oscillators the influence functional corresponding to the diffusively screened electron
interaction. Up to this point our calculation is identical to th a t in refs. [20] and
[24]. We deviate from previous work by now separating the influence functional
with p e r m i s s io n of t h e co p y rig h t o w n er. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
13
into a classical part and a rem ainder. In Sec. 2.2 we show th a t the effect of the
classical fluctuations on the back scattering (or weak localization) correction to the
Drude conductivity can be w ritten as a path integral, thereby exactly reproducing
the result obtained in ref. [3]. T he improvement over th a t work is th at we have
an explicit expression for the rem ainder. At this point it is convenient to specialize
the model to the case of N yquist noise which yields an sim pler p ath integral, and
allows further calculations to be done analytically. T his m odel is used in Sec. 2.3.1
to explicitly calculate the quantum corrections to second order. In Sec. 2.3.2 we
give sem iquantitative argum ents for how these calculations are modified when the
more realistic spectrum corresponding to the screened electron-electron interaction
is used, and infer the structure of the corresponding quantum corrections up to nu­
merical coefficients. The final section, Sec. 2.5 contains a discussion of the results
and conclusions.
2.1
Influence Functional for Disordered Interact­
ing Electron Systems
As the sta rtin g point for our calculations we use a variant of the Feynman-Vernon
[27], Leggett-Caldeira [23] m ethod in which a dissipative environm ent is described
by a set of harmonic degrees of freedom. In this way we are able to construct a for­
malism general enough to accom m odate different models for the electron interaction
with an electromagnetic field, referred to as the environm ent.
Consider a closed system described by a H am iltonian consisting of three parts
U{t ) = Ho(t) + Henv (t) + 'Hint{t) ■
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p ro d u c tio n p rohibited w ith o u t p e r m is s io n .
(2-2)
14
Here the first term describes a disordered free electron subsystem ,
J
n Q{t) =
dTilj+(t,r)
fx + U ( r ) j
(2.3)
where p, is chemical potential and U (r) is a random im purity potential. The next
term is the H am iltonian for harmonic electromagnetic modes,
Henv(t) = ^ 2
(a+av -+- ^
,
(2.4)
where v labels different, corresponding to spatial wave functions 0„(r) which can be
chosen to be real. The interaction between the field and the electron system is
Uintit) =
J
dTip+(t,r)v(t,r)ip(t,r) =
J
d rv (t,r )q (t,r ),
(2.5)
where q(t, r) = •0+ (t, r)-0(f, r) and
«(*>r ) = S„ vZMyU,,
/oTT —= (&
v '(r )a*(*) + </^(r K ( * ) )'
(2-6)
is an operator in the space of the electromagnetic field quantum states. Above,
the Fermion and Boson field operators, ^ ( t,r ) and
are in the Heisenberg
representation.
The electron G reen’s function is defined according to
f G(t, r, £', r') = <
{
,
r ) ^ +(£', r')),
if t > K ?,
r')^ (f, r)),
if t < K ?-
(2.7)
Here the average ( ...) is understood as a trace over the quantum state of the whole
system, taken with the density m atrix p of the system:
/
\ _ Tr (pTk (• • • exP(iS[ip+, V])))
T v ( p T K {exp(iS[tp+,i},V])))
(2 gx
The action introduced in Eq.(2.8) has the form:
S [^ +, ^ V] = -
f
JK
dt (U0 + U env + Hint) ■
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n er. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
(2.9)
15
In the above formulas K refers to the Keldysh contour, which runs from —oo to
+ 00
and then backward to —00 , (see, e.g. [28]), and the subscript K refers to ordering
along this contour. W ithout loss of generality we may take the t =
—00
initial
therm al state of the system to be a product of the noninteracting electron density
m atrix pei and the environment density m atrix penv, where pei obeys Fermi-Dirac
statistics and penv obeys Bose-Einstein statistics. [29]
Since the electromagnetic field is described by a set of Harm onic oscillators, the
trace over its quantum states can be worked out exactly. T his yields an influence
functional (cf. ref. [27]) for the electrons:
F[ q, 0.'] = ex p (-< % , <?']),
(2.10)
duJ dTlJ rfr2
(2-U)
with
<%
,?'] = \
x
(( 9 (5, Ti) - q'is^Ti^fCiis - u , T l,T2)(q(u,T2) - q ' { u , r 2))
+ (g (s,r0 - g '( s ,r l))/C2(s - u , r i , r 2) ( ? ( u ,r 2) + g '(u ,r2)))
where we have introduced the notations
JC^s - u , r t , r 2) = J U r r
2, y
7
I Mv U Jy
c o th ^ c o s ( w i/( s - u ) ) 0 i/( r 1)^ I/(r2)
2J
(2.12)
and
A:2(s - u , r i , r 2) = - i ® ( s - u)
e2
——
sin(o;t/(s —u ))0 „ (r1)0 I/(r2).
(2.13)
Above q (s ,r) = ■0+ (s, r)^ (s , r) is the electron density taken on the forward part of
the Keldysh contour, and q'{s, r) = i>+{s, r)i/'(s, r) is taken on the backward part of
the contour.
i
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n er. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
16
Note th a t a result of the very sam e form can be obtained for the Coulomb
interaction
(2.14)
where V^r) = e 2/ |r |.
The usual way to achieve this can be found in, e.g., [30].
One performs the H ubbard-Stratanovich transform ation to decouple the density
operators q(t, r') in Eq.(2.14), introducing a fluctuating electric field. Then the
effective action is expanded to second order in the fluctuating fields, and the screened
random phase approxim ation is used for the electronic polarization. Finally one
integrates over the H ubbard-Stratanovich variables to obtain an effective action for
electrons in the form of Eq.(2.10), w ith the specific forms of K.\ and fC2 given by the
choice of mode density in Eq.(2.26) below.
The influence functional Eq.(2.11) is very sim ilar to that obtained for a quantum
particle coupled to the environment of harm onic oscillators. Here, however, q(s, r)
is not a particle coordinate but fermion density operator. E xpanding the exponent
of the influence functional F[q, q'\ we can reproduce the Keldysh diagram technique
for electrons coupled to a Bose field w ith a propagator, which can be expressed in
terms of /Ci and 1C2. One finds th at the Keldysh component of the Bose field is K,i,
and the retarded component is K,2. It is convenient to use the electronic Keldysh
Green’s function in the ‘rotated’ form [28]
(
G ^ (* ,r ,£ ',r ')
^
0
G ^(t,
r, £', r') N
G(£,r,f',r') =
G (A )( i , r , ^ r ' )
(2.15)
j
One can check th a t in this representation the vertices corresponding to the the
coupling of electrons via /Ci and the vertex a t tim e s of the electron coupling via
/C2 in Eq.(2.11) are proportional to the unit m atrix in the Keldysh space, whereas
the vertex corresponding to the electron scattering by the field fC2(s —u, r i , ^ ) at
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
17
time u is proportional to the first Pauli m atrix rx, because of th e plus sign between
q and q' at this vertex in Eq. (2.11).
The structure achieved thus far is formally identical to the sta rtin g points of refs.
[20] and [24]. At this stage, we make a new departure by explicitly distinguishing
between the classical and quantum effects of the ‘environm ent’. Note th a t at high
tem perature JCi contains the huge factor c o th u ;/2 T % 2 T / uj, divergent as u —>0, so
th at the second term /C2 is small by comparison. We represent the effective action
F[q, q'] as a product of classical Fc[q, q'] and quantum Fq[q,q'] parts.
q'] = Fc[q, q']Fq[q, q'].
(2.16)
In Fq we include all of /C2 from Eq.(2.10) and the part of ICi chosen as
(2.17)
The remainder of fZ1, namely Eq.(2.12) w ith co th cj/2 T replaced by 2T / u is in­
cluded into Fc. O ur strategy will be to trea t Fc to all orders, and Fq as a perturbation
to a finite order.
We now introduce a new H ubbard-Stratanovich transform ation to decrease a
power of the fermion operators in the exponent of the classical p a rt of the influence
functional Fc and obtain
(2.18)
where
(2.19)
4>u(r) cos(cjut + tyv)
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
( 2 . 20 )
18
= (Wfj., 0P) is a two component H ubbard-Stratanovich variable for each electro­
m agnetic field mode v and
j Vvr... = Ilu J
(2.21)
After these operations, the average in Eq.(2.8) reduces to
* = Tr {pTK ( ... Fc[j)+, ip]Fq[ip+, ib] exp{iS0[i/;+, t/>])))
T r (pTK (Fc[t//+, %jj\Fq{^+, ip] exp(z'50[^+ , ip])))
,
,
and S Q['ip+,'ip\ = JK dt'HQ.
As we have already mentioned, we consider finite order perturbation theory
in Fq[ip+,il}] keeping all orders in Fc[ip+,ip\. For this purpose it is convenient to
introduce Green’s function Gc(£, t', r, r ;) of electrons, interacting w ith the classical
component of the electric field. It is still defined by Eq.(2.7) where the average is
taken in the sense of Eq.(2.22) with Fq[ip+,-0] = 1. In the Keldysh representation
(2.23)
Gc(£ ,r,£ ',r') =
and
G {cr) = G[r) + G l0R)V G iR) + . . .
G^ck) = G[ k) + G[r) V G [ K) + G {qk )V G \? ] + . . .
(2.24)
G[ a) = G^a) + G iA)V G {QA) + . . . ,
where Go is the Green’s function of electron in m etal without interaction for a given
im purity realization (no disorder averaging).
To conclude this section we discuss two physically m otivated choices for the
density of environmental modes. Instead of the sum m ation over m ode index v we
integrate over
uj
and q, where
? s
i
a;
is a frequency and q a wave vector, and write
*> = / £
/ i f - /(w - * > * " ■ 9)-
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(2-25)
19
The choice
(2.26)
corresponds to the low frequency and small momentum spectral function of the
screened Coulomb interaction in a disordered metal, with cq the one dimensional
conductivity of the wire.
Both for its own interest and because it gives an analytically sim pler structure,
we shall also consider the dissipative effect of Nyquist noise associated with an
external resistor. We note that the spectral function of the electric field responsible
for the Nyquist noise can be described by
(2.27)
Here R c is determined by resistances of the circuit and q* is a wave vector of the
electric field in the wire (limit q*L —>■0 corresponds to the uniform electric field).
When J{u,q) is thus defined, the voltage fluctuations obey the fluctuationdissipation theorem, see [32], appropriate to Nyquist noise. To see this, we calculate
the average value of the voltage U = v ( L ) —v( 0) at the ends of the wire, where v(x)
is defined by Eq.(2.6):
{Ul) = ((u(0) - v{L))l) = e2u R c coth
(2.28)
For a wire with resistance R w connected with resistor R q one has, see [3]
RlRo
(i ?0 + Rw)2
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(2.29)
2.2
Dephasing due to Classical Component of Elec­
tric Field
In this section we calculate the weak localization correction to the conductivity in­
cluding the classical part of the influence functional, defined by Eqs. (2.2), exactly
but neglecting the quantum part. We shall show th a t the environm ent now be­
haves like a classical electromagnetic field, so th a t the calculation— repeated here
for completeness—is very sim ilar to the one done in [3]. A small difference is th a t
in our representation the fluctuating field is described by a scalar potential V (t , r)
instead of a vector potential.
T he current operator is given by
=
( 2 ' 30)
where A (r) is the vector potential corresponding to an external field which produces
an average current given by
j ( r) = j(r) /
r, r W
(2.31)
To calculate the linear response to the vector potential A (r) it is sufficient to find
the Keldysh component of the electron Green’s function to the first order:
G[K)(e,r,r')
= Gm ( e ,r ,r ') + ^ d r 1G (/z)(e, r, r i) e A ( r 1)j(ri)G (/^)(e - weit,r i , r')
+
y 'd r iG (* )(eIr ,r i ) e A ( r 1)j(ri)G *A)( e - a ; ea:t>r i , r ') .
(2.32)
We have supposed th a t the external field oscillates with a finite bu t small fre­
quency ojext, so th at A = cE/iujext. Because we are dealing with a non-superconducting
disordered system, the diamagnetic (i.e. the second) term of the current operator
on the right hand side of Eq.(2.30) cancels a ui~Ji contribution to the conductivity
permission of the copyright owner. Further reproduction prohibited without permission.
21
3( )
4 ( )'
O
2
F ig .lb
Figure 2.1: Figure a shows maximally crossed diagrams which correspond to the weak
localization correction to conductivity. Open circles denote the position for the Keldysh
component of the electron Green’s function. The sum of all possible diagrams is repre­
sented by the diagram in Fig. b. The Cooperon C is defined in Appendix A.2
from the first term . Note th a t G(e, r, r') is the exact electron Green’s function of the
electron system, defined by Eq.(2.7). Since in this section we consider Fq[ip+, iJj\ = 1,
we can replace G (e,r, r') by G c(e ,r ,r ') , defined in Eq.(2.23).
We shall treat the im purity potential U( t ) and electric field V(t, r) defined in
Eq.(2.20) as perturbations, keeping all orders in them . T he corresponding electron
vertices are proportional to the unit m atrix in the Keldysh space. From this ob­
servation it follows th a t there is only one Keldysh com ponent in every conductivity
diagram.
The weak localization correction to the d.c. conductivity is determ ined by the
diagrams in Fig. (2.1) and is given by (see Appendix (A .l) for derivation)
& cTu>i {t ) =
7T
f
J0
P ( t , rf)drj,
(2.33)
where P ( r , rj) has the meaning of the probability of returning to a startin g point at
time r after a time
77,
and is given by the Cooperon evaluated at coinciding space
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
22
and tim e points,
(2.34)
P ( t , v ) = C (t, , +V, - v , r >r)
For a closed system (no external fields) P ( r , 77) does not depend on time r.
To evaluate the Cooperon, we can use the following p ath integral form (see
Appendix A .2 for more details).
with
V(r, C, r(C)) = V ( t + C/2, r(C) ) - V ( r - C/2, r(C)),
(2.36)
and the electric field potential V'(t, r) defined by Eq.(2.20). In performing the inte­
gration over the magnitude and the phase of the electric field according to Eq.(2.18),
we note th a t the time variable r can be absorbed into the phase (j)u of the field. Con­
sequently, the correction to the conductivity is independent of r. As the result of
the integration we get for a wire, in which case r is one dimensional,
r r (+ r i)= r
C (+ „, - 7), r, / ) = f r(_ ^
f
r+T]
V r ( t ) exp ( -
\\
f r~(t)
dt ( —
+ U (r, o ) ) ,
(2.37)
where
(r(t) — r ( —£))2,
for the Nyquist noise model,
(2.38)
U (r, t ) — < 2e2T
|r(t) —r ( —£)|,
for the electron-electron interaction.
I cr\
Here we have used the specific time variables rji = r) and
772
= —77, needed for the
weak localization correction to the conductivity.
We see th a t the interaction with the environment produces friction in the diffu­
sion process, described by the Cooperon, see Eq. (2.37). The effect of the friction is
to reduce contribution of long trajectories to the return probability P(r}).
F irst we discuss how the result of ref. [3] can be obtained by qualitative argu­
ments. T he weak localization correction to conductivity is still given by Eq.(A.7)
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
23
and the Cooperon can be represented in the form of Eq.(2.35). Although Eq.(2.35)
was derived for small momentum q of the electric field V ( uj, q) — see Appendix
A.2 — we can use it for the Coulomb interaction with unbounded spectrum in the
momentum space, since the m ain contribution comes from the long wavelength part
of the interaction. In higher dimensions D = 2, 3, the m om entum integration has
to be cut to satisfy conditions a t which the Cooperon has been derived.
We can rewrite the exponent of Eq.(2.37) in terms of dimensionless variables of
space y = r f L ee and time C =
71
. Then L ee and
7 -1
determ ine the characteristic
space and tim e scales at which the electron-electron interaction becomes im portant.
At smaller scales the interaction is not im portant and the diffusion can be considered
without interaction. In the opposite case the diffusion process is suppressed by the
interaction. According to Eq.(2.37) and Eq.(2.38) L ee and
7 -1
are given by
(2.39)
This result was found for the first time in [3].
This intrinsic cutoff may be introduced into the expression for the weak local­
ization correction for noninteracting electrons in the time representation, by cutting
off the tim e integral with the exponential function exp (—2^7 ).
(2.40)
This expression is in agreement with the exact result found in [3]. [See also Eq.(2.52)
below.]
Now we perform analytical calculations.
It is convenient to introduce new
variables for the path coordinates. Let us define R(t) = (r(t) + r ( —t ) ) / \ / 2 and
x(t) = (r ( t ) — r ( —t))/y/2. T hen we can eliminate the integration over negative
time. Also this change of variables separates R(t) and x ( t ) in the exponent. The
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
24
motion described by R ( t ) is an ordinary diffusion and a direct integration gives
unity for the whole integral. The p a th integral over x ( t ) can be done easily for the
Nyquist noise model, because the m otion corresponding to x(t) is th at of a particle
in a harmonic oscillator which is at the position of m inim um potential energy a t the
starting and final points. Thus, Nyquist noise yields the following simple explicit
result for the Cooperon
Q
im p
—------------------------6 XD
87tD sinh Qt]
coth Qt]
where
= 16TT,
(2.42)
and T, = D T e 2R „ „ / L 2.
We thus have
87i D
sinh Qrj ’
(2.43)
from which it follows that in this model
(2.44)
where
(2.45)
So far we have assumed th at the interaction with the environment is the only
source of the dephasing. If the sample is in a magnetic field H we have to consider the
competition between the dephasing produced by the interaction and the magnetic
field. The magnetic field exponentially suppresses the Cooperon (Eq.(2.35)) as a
function of
772 —771.
The magnetic characteristic tim e is r Hl = e2D H 2a2/3c2, where
a is the thickness of the wire, see [1]. W hen both sources of dephasing are present
the weak localization correction to conductivity is given by
(2.46)
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
25
This formula allows us to reproduce in a different way some results of [3] for
A a wi in two lim its.
For
th Q.
< 1 we can expand (sinhO,T])~1^2 = 1/y/firj — {Q.rf)z/2/V2 and obtain
A<7„, =
(l -
In the opposite lim it, r HVt
1
■
(2.47)
we expand exp(—tj/ t h ) ~ 1 — v / th and integrate
over rj to get
=
( 2 -4 8 )
where
roc
r
I x = / - 7 ==— dC « 5.84.
Jo v s m h ^
(2.49)
In the case of the electron-electron interaction, the functional integral has been
related to a Schrodinger-like differential equation in ref. [3] w ith the result
a
*
= ^ C h A
i d T ; ^ -
<2 -50>
where Ai(x) is the Airy function and
'e*rVD_y /3
Tee —
<Tl
(2.51)
J
compare to Eq. (2.39).
In weak m agnetic field L h 3> L ee with L h =
A oS =
wl
s/D th
(2-52)
35/ e r 2(2/3)'
In the opposite lim it L h <C L ee we use
"<*> ~ 2
^
^
' ’
(2 '53)
to get
e2L
7T
- (l -
j (t« 7 )3/2)
•
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(2.54)
26
Note, th a t the result of Eq.(2.51) differs from one found in [3] by a factor of 2. One
the other hand the expansion Eq.(2.54) is consistent with the result of [20]. Also
the result of E q.(2.44) has an extra 2 -I/4 to the numerical factor r (l/4 ) /2 7 r r ( 3 /4 )
found in [3] for the weak localization correction to conductivity in the presence of
the Nyquist noise.
We have thus seen th a t our ’classical’ action reproduces known high tem perature
results for the intrinsic dephasing effect in weak localization w ithout the need for
an external infrared cut-off. In the rem ainder of this C hapter we build on these
calculations to examine quantum corrections.
2.3
Quantum Corrections
In this section we consider the contribution of the quantum p a rt of the influence
functional to the weak localization correction to conductivity.
described here are similar to those in ref.
The calculations
[20]. The difference is th at ours are
free from infrared divergences, because the low frequency modes of interaction have
already been taken into account exactly.
Expanding
?/>], the second term in Eq.(2.16), to the first order we obtain
1 r+oo
F ,[ ^ + ,^ ] =
1
ds J
r+ o o
r
du J d n
J
r
-
-J
x
( £ {K) ( s
+
£ (R)(s —u,
+
C{A)(u - s , r 1, r 2)(g(u,r2) + q ' ( u , r 2))) ,
-
dr2(q{s,Ti) - q (s ,ri))
(2.55)
u, r ! , r 2) (q(u, r 2) - q'(u, r2) )
r 2)(q(u, r 2) + q'{u, r 2))
where Keldysh labels have been assigned according to the definitions
£<*>(<, r „ r 2)
=
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(2.56)
27
£ (A)(£,r1?r2) =
2 0 i/(ri) 0 „(r2)
exp( —
- © ( - * ) X X ~ 4Muu„
(2-57)
V
C^K \ t ,
r l5 r2) =
As we have already mentioned in the previous section, the conductivity is deter­
mined by the Keldysh component of the electron Green’s function (see Eq.(2.31)).
The Keldysh component can be represented in terms of the G reen’s function defined
in Eq.(2.7):
(2.59)
where the / —index of time argum ent means that it is taken a t the forward part
of the Keldysh contour, and the b— index for the backward p art, so th at tf < k tbThe average in Eq.(2.7) is represented by Eq.(2.22) with respect to the classical part
of the effective action Fc[^+,^ j, see Eq.(2.18) and the quantum part in the form
of Eq.(2.55). The first term in Eq.(2.55), which is unity, provides the expression
for G^K\ t , £', r, r') considered in the previous section and the remaining part of
Eq.(2.55) gives the first order quantum correction. Using the W ick’s theorem for
the electron operators we get after cumbersome calculations the quantum correction
to the Keldysh component of the G reen’s function
1xint
(2.60)
where x = (£, r). Subscript i means th at the Green’s functions should be calcu­
lated up to the first order in the external electric field A(r), similarly to what has
been done in Eq.(2.31). Thus, the Keldysh component is given by Eq. (2.32) and
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28
G{l*’j4)(iI«',r,r') are
G(1R’-4)(M ',r,r')
=
G i R’A\ t , t ' ,
+
J
(2.61)
r,r')
, r . r l ) e A ( t i , r i ) i ( r l )G[R’A)(tu t' , r^r'),
Equation (2.60) is in agreement with the expression for the first order correction
to the Keldysh component of the electron Green’s function obtained in the stan d ard
technique. [See [28] for a general discussion and [20] for the explicit form.]
It was shown in paper [20] th a t to second order in the interaction the weak
localization correction to the conductivity can be represented as a sum of two term s,
called the dephasing term and the cross term. The dephasing term contains the
contribution from the electron interaction with Boson modes having energies sm aller
than tem perature of the system. On the other hand the cross term has contribution
from the whole spectrum , and corresponds to the electron scattering from the Friedel
oscillations. According to [20] [c.f. Eq.(2.1) above], the interaction correction term
has a weaker divergence, A c r^ ~
the dephasing term A adeph ~
t *.
(where r^ is the decoherence tim e), than
Keeping this in mind, we restrict our calculations
to this latter term , which is expected to give the most im portant correction to the
semiclassical calculations discussed in the preceding section.
The derivation of the dephasing term of the weak localization correction is de­
scribed in A ppendix A.3. The corresponding diagrams are shown in Fig. 2.2. In gen­
eral the result may be represented as a product of three Cooperons, see Eqs. (2.62)
and (2.64). T he whole process corresponds to the diffusion of an electron from an
initial point to some other point, where it emits a boson. T hen it travels to th e sec­
ond point and absorbs the same boson, after that it goes back to the initial point.
In the case when the propagation of the em itted boson is described by the
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F ig .3 a
F ig .3 b
F ig .3 c
F ig .3 d
Figure 2.2: Four possible diagrams which contribute to the dephasing correction to the
conductivity.
Keldysh component of the boson Green’s function, the Cooperons have the same
tim e argum ents at intermediate points so th a t the diffusive m otion is continuous in
time, see Eq.(2.62). Terms which have the retarded or advanced components of the
boson G reen’s function have a discontinuity a t the points which represent the non­
diagonal vertices in the Keldysh space. T his discontinuity can be explained by the
fact th a t the energy of an electron is not conserved along its trajecto ry because of
the interaction with the classical field. We disregard these discontinuities. Then the
integrals over r are trivial. It is possible to show th a t in this case we disregard terms
which are of the same order as higher order quantum corrections, see Appendix A.3.
The dephasing correction to conductivity is
(2.62)
where
P 2 ( ,) = / I
f
(2.63)
/
is the correction to the return probability [cf.
Eq. (2.33)] due to the quantum
part of the influence functional. The diffusion process is described by the following
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30
expression, containing a product of three Cooperons:
x
e- ‘-<7( n - r 2)C(T) r?) Clj r) r i )C(T, Cl, C2, ri, r 2)C(T, Cz, 77, r 2, r). (2.64)
w ith the therm al function
Note th a t f ( x ) vanishes at x ^ 0, i.e in the infrared region, because of the inclusion
of classical fluctuations in the factors C.
We can elim inate the integrals over r i i2 and represent C2(T, 77, r) as
( 2 . 66 )
X
exp (-*q(r(C i) - r(C 2))) C(T,
+ 77, - 77,
r, r),
and C(T, + 77, —77, r, r) is given by Eq. (2.35).
Now we can easily perform averaging over the fluctuations of the classical field
according to Eq. (2.18). As a result we find, th at we can substitute the Cooperon
defined by Eq. (2.37). The path integral can be simplified by using the variables
R ( t ) and x(t) introduced below Eq. (2.36). Since the integrals over R(t) and x{t)
are defined only a t positive values of £, we have to rewrite Eq.(2.66) as an integral
over positive values of Ci and £>.
2 .3 .1
U n ifo rm E le c tr ic F ield
In this section we consider dephasing correction to conductivity due to the Nyquist
noise. In this case th e calculations can be done analytically. After some algebra (see
Appendix A.4), we obtain
(2.67)
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
where I i was defined by Eq.(2.49),
( 2 .68 )
and
(•2.69)
(2.69)
°° ^2Z - 1)4 2' — ccothzdz
o th zd z « 7.54.
x/sinhz
The th ird term of the right hand side of Eq. (2.67) is of the second order in Q / T
and can be omitted.
We see th a t the quantum corrections are not im portant, unless T < Tq, with
Tq being defined by Eq.(2.42). Let us evaluate dephasing rate £l(Tq) at T ~ Tq.
According to Eq.(2.42) we have
(2.70)
Here we use dimensionless resistance in terms of /i/e 2. We substitute Q(Tq) to
Eq. (2.44), see also Eq. (2.29), and get
(2.71)
The weak localization becomes strong localization already at T
R q)2/ R 0 > 1. In the opposite case (R„, + R q)2/ R q
Tq, if (Rw +
1 we have R e/ f
1, con­
sequently £l(Tq) <gi D / L 2 and the wire becomes zero-dimensional at higher tem­
perature th an Tq. We conclude th a t in both limits the weak localization correction
to conductivity deviates from the result of [3], see Eq.(2.33, at higher tem perature
than Tq due to other reasons rather than due to the quantum corrections.
The plus sign before the second term in Eq.(2.67) m eans th a t the dephasing rate
found in [3] is overestimated, and the quantum correction suppresses it.
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32
2 .3 .2
E le c tr o n —E lectro n In te r a c tio n
Now we discuss how the result obtained in the previous section can be generalized
to the case of the real electron-electron interaction. The screened Coulomb spectral
function of a dirty metal is given by Eq.(2.26), where cq is the one dimensional
conductivity of the wire and the m om entum integral in Eq.(2.25) runs from —oo
to -l-oo.
[33] We will use the effective action in the form of Eq.(2.10) with the
appropriate choice of mode density Eq.(2.26) to make a sem i-quantitative calculation
of the quantum correction to the semiclassical result Eq. (2.50).
Calculations, described in Appendix A .5, give
( 2 -7 2 >
~
where
x2
J
=
<2 -7 3 )
Let us introduce the dimensionless conductance of the wire at length L :
S(L) = f r -
( 2 -74)
where <q is the Drude (bare) conductivity of the wire.
T hen Eq.(2.39) can be
rew ritten in the form:
7 = j h
Since in the weak localization regime g ( L ee)
-
( 2 -75)
1, Eqs.(2.40) and (2.72) mean th at
the quantum correction to the classical result obtained in [3] is proportional to a
small quantity [g(Lee)]~l/2. At sufficiently low tem perature g ( L ee) approaches unity,
but in this case the classical correction to the conductivity becomes comparable with
the Drude conductivity cq, and the perturbation treatm ent of localization breaks
down. The conclusions of Sec. 2.3.1 and 2.3.2 are thus the same: in the region where
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33
the correction to the conductivity may be described by the interference of two time
reversed trajectories, neglecting more com plicated interference term s, the quantum
fluctuations of th e dissipative environment are unim portant.
2 .3 .3
H ig h F req u en cy C u to ff
The above analysis was aimed to study low frequency interaction w ith electromag­
netic field. We saw th at the main contribution to the weak localization correction to
the conductivity come from the electrom agnetic modes w ith frequency ui ~
7
<C T,
see Eq. (2.75). In this Section we show th a t the high frequency contribution to the
Cooperon is exponentially suppressed for the electromagnetic m odes w ith frequency
greater than tem perature T of the electron system.
The weak localization correction to the conductivity
(27r)3 (27r)d 2T cosh 2 e / 2 T
(2.76)
is determined by the diagram shown in Fig. 2.3, where the C ooperon C is defined
in Fig. 2.4. Here crd is the conductivity of a d-dimensional sam ple and 1/ is the
density of states at the Fermi surface.
Terms (a) - (f) in Fig. 2.4 describe the
interaction effect on the Cooperon. T he role of terms (a) in Fig. 2.4 is to cure the
unphysical infrared divergence at small energy-momentum transfer (Q < l / L ^ ) and
thus to restore the gauge invariance [3, 20, 21], see also Sec. 2.3. These terms do
not diverge ultravioletly and thus are not im portant for our discussion.
For rem aining term s (b) - (f) we have
(2.77)
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34
1
R,A
£~^(p)±i/2T
= l/ 2 nvx
“
R,A
2i S \ S \ S \
=
l m
K
-2i S \ S \ S \
= r
R
(A)
4
<C)
Figure 2.3:
(B)
<J
=
l/(i(ti + D Q 2 )
The weak localization correction to the conductivity is shown in
Fig. 2.3(A). Figure 2.3(B) represents the basic elements of the diagram m atic tech­
nique. The renormalized vertex is the solution to the diagrammatic equation, see
Fig. 2.3(C).
Figure 2.4: The diagram m atic equation for the Cooperon C in the presence of
electron-electron interaction. The basic diagram elements are defined in Figs. 2.3(B)
and (C).
i
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35
The dephasing time
energy at eL =
62
is determined by r ~ l = E(e, e, Q = 0) the Cooperon’s self
= e and Q = 0. Expansion over Q leads to an infrared divergent
correction to th e the diffusion constant. (See Ref. [20] for details).
Taking only term (b) in Fig. 2.4 into account,
wb)
t , „ f d u d *Q I m £ A(u;,Q )
s
= R e i 27T (27r)d ( —iui 4- DQ2)
u
2T’
^
(2'78)
we encounter an ultraviolet divergency. Here we used the following relation be­
tween the Keldysh component, £ K(u;,Q ), and the retarded (advanced) components
£R(A)(a;, Q) 0f th e screened interaction propagator: £ K(o;, Q) = c o th (o ;/2 T )(£ R(u;, Q )£ A(u,Q)).
Nonetheless, the problem is artificial[34]: the contribution
V(c)-(f) _ Re f ^ d*Q Im £ A(a.,Q)
J 2ir (in f ( - . w + D Q 2)
£ -u
2T
1 '
'
exactly cancels out the ultraviolet divergency. Thus, the self-energy £ = E b+ £ (c)_(f)
is determined by [o;| < T (according to Eq. (2.76) e < T ).
2.4
Dephasing by External Oscillating field
In this section we consider the effect of external microwave radiation on the weak
localization correction to the conductivity. In this case electrons are no longer in
equilibrium w ith the radiation. We assume th a t the intensity of radiation is large, so
th at we can neglect quantum corrections, discussed in the previous sections. From
Eq. (2.33) we obtain the following expression for the d.c. conductivity:
&crwl =
4p2 D
7T
where the retu rn probability P(r,
t])
/
r°°
JO
dr]
r2 n /u
P ( t , r])dr,
(2.80)
Jo
is determ ined by the Cooperon, see Eqs. (2.34)
and (2.35) w ith V ( t , r ) = eE0r(t) cos uit.
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36
The path integral can be done analytically for this form o f the potential, see [31].
We have:
= - - V
7T
d
r
JO
ri))dq,
(2.81)
first obtained in [1]. Here
p2 F 2
a = D— ±
u.r
and
(2.82)
f 2ui t'
if u t <C 1 ;
45 ’
’ .
(2.83)
. t,
if u t
1.
We introduce dephasing rate 7 ^ into Eq. (2.81) to take into account dephasing from
. o
,/
x
1 . ^
sin cj£
4>{u,t) = t + — sin 2u t — 2 --- =— =
2u
u 2t
other sources, such as magnetic field or finite size of the system.
In the latter
case the dephasing rate coincides w ith the electron escape rate from the system.
The dephasing rate
7^
is necessary to avoid divergence of the weak localization
correction to the d.c. conductivity, which is not removed by the external oscillating
electric field. T he divergence comes from moments of time, when the electric field is
symmetric w ith respect to these m oments of tim e and the effect of the field vanishes
and the weak localization correction to the conductivity diverges, unless it is cut off
by other sources of dephasing.
We conclude from Eq. (2.81) th a t the Oi(j>{ut) determ ines suppression of the
interference term , which is responsible for the weak localization correction to the
conductivity. This suppression can be found from the following analysis. We assume
th a t the uniform electric field E q penetrates into the wire. An electron in the wire
experiences different electric fields a t tim e t, when it moves along two time reversed
trajectories for tim e t. We denote the difference of the electric field as E(t). For a
low frequency of the perturbation, so th a t u t <C 1, E(t) % Eout and in the opposite
limit of high frequency field, u t »
1,
the characteristic value of the electric field
difference is E ( t ) = E q. The electron moves along diffusive trajectories produced by
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37
scattering on impurities and acquires the difference in energy e E ( t ) L for those two
tim e reversed trajectories. Since the energy determines the velocity of the phase,
the electron phase difference for the same p a rt of the tim e reversed trajectories is
Sip = eE{t)Lt. Then, the fluctuations of the electron wave function’s phase can
be estim ated as Sip2. We also use the relation between the characteristic size of a
diffusive trajectory and tim e: L 2{t) = D t , where D is the diffusion coefficient, to
complete estim ation for the phase fluctuations. For a small tim e interval, u t <C 1,
wre obtain Sip2 = De2E 2u 2t 5. For the opposite lim it of high frequency, u t »
1,
we
first estim ate the phase fluctuations for a tim e, equal to one period of the electric
field: Sip2 = De2EQU~z and then multiply it by the number of periods, u t, so th at
we find Sip2 = De2E l u ~ 2t. T he above presented arguments lead to th e result, which
is consistent with Eq. (2.83), provided that Sip2 « a<f>{ut).
Assuming that the phase difference over each straight p art of th e tim e reversed
trajectories is independent, we conclude th at the phase difference for electron motion
along two time reversed trajectories obeys (Sip2(t)) = e2E 2(t)L2t / E Th- Using the
correspondent approxim ations for E(t) for the low and high frequency of the external
field, we obtain (Sp2(t)) = Co4>(z,t), in agreem ent with Eq. (3.68), where C0 =
e2E%L2/ E ThSu see Eq. (3.12).
Now we consider different lim its of Eq. (2.81)and compare them w ith the results
for bulk systems [3, 1]. In a weak field we expand the integrand of Eq. (2.81) to the
lowest order in a and obtain
(2.84)
for slow external field, u <C 7 ^, and
(2.85)
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
38
for fast external field.
For the limit of strong high frequency electric field we obtain:
23/,2e2 I
= --------------—
7r2 \ j a + 2 j cfi
/
a
I\ a + 2'yVj 1
(2-86)
and in the opposite lim it of low frequency of the electric field
Acr«"
24/s3 i/5 r ( l/1 0 ) r ( 2 / 5 ) e2 \[D
59/10
r ( 9 / 10 )
TraViOo;2/5'
We notice th at for the la tte r lim it the external dephasing rate
since the result does not diverge even at
2.5
7^
=
7^
(2’8 ^
is not necessary,
0.
Discussion
In this chapter, we have constructed a bridge between semiclassical calculations [3]
which keep the interaction to all orders of p erturbation theory and exact quantum
mechanical calculations to first order in the interaction.[20] While the semiclassical
calculations are self consistent and do not require an external cutoff, the finite
order perturbation theory is infrared divergent, and requires a low frequency cutoff.
By contrast, our calculation of quantum corrections is intrinsically regularized at
low frequencies and the param eter of the perturbation theory is the ratio of the
dephasing rate
7
and tem perature T,
7 /T ,
see Eq. (2.72).
These differences notw ithstanding, we shall now show th at when the cutoff in­
troduced in [20] is treated as a param eter to be determined self consistently, one
obtains agreement w ith our results.
The weak localization correction to conductivity found in [20], Eq. (4.13a):
A a wl = Acr^0/ + A adeph -I- Aa'deph -I- A
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(2. 88)
The first term A a®} comes from the m axim ally crossed diagram w ithout electronelectron interaction with finite dephasing rate
7
. The result for Acr^/ is:
Act™ = - —
f^ C (u ,= 0 ,k )= -7T J I'K
7T 1
(2.89)
where C(u, k) is given by
-TiW
( 2
' 9 0 )
The first order correction due to the interaction is given by an equation similar
to Eq.(2.62), see [20]
J 5^ / ^ f
^ ^ F ( w / 2T)J'(w,<!)C 2(a .,k ,q ),
(2.91)
where
C 2(w, k , q ) = ( C 2(0,
k) [ C{ - uj : k - q) +C( uj , k - g ) ] -
C ( - u , k - q)C{oj , fc - <7) [ C ( 0 , k) + C (0 , k -
2q)})
^
.
Here F(x) = rz:2/s in h 2 a; is the thermal function, which differs from f ( x ) defined
by Eq.(2.65)by unity. [Recall that the unity in Eq.(2.65) elim inated the classical
component in Eq.(2.62), which we treated to all orders of perturbation theory.]
W ith the spectral function of the Coulomb interaction J(w , q) given by Eq. (2.26),
the integrals in Eq.(2.91) can be done analytically in the lim it
integral over
uj
converges at
uj
~
7
7
T, since the
and one can use the approxim ation F ( x ) «
1.
The result is [20]:
Acrdep/l =
e4D T
.
47T7^crI
(2.93)
Assuming th a t the electron-electron interaction is the only mechanism of decoher­
self-consistently. Minimizing the sum A o®} + A adepfl with
ence, let us determ ine
7
respect to
in agreement with Eq.(2.51).
7,
wo get
7
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The third term in Eq.( 2 .88) corresponds to the high frequency contribution to
the dephasing term Eq.(2.91) beyond th e approxim ation F ( x ) =
1
and has the form:
(2.94)
The last term is the interaction correction to weak localization and it was presented
in Eq.( 2 . 1).
Let us substitute
7
from Eq.(2.51) into Eq.(2.88). The first two term s now
become of the same order of m agnitude and they correspond to the result of [3].
The third term becomes
(2.95)
in agreement w ith our Eq. (2.72).
The interaction correction of ref. [20], which we have not considered
A cr^, - -3C(3)
e 2\ / D th e 2
2-kH 2irh<Ji
The last equality comes from
(2.97)
Note th at this correction is even sm aller th an Aa^dlJph.
The reason for the smallness of these corrections is worth emphasizing: the
contribution of electromagnetic modes w ith frequencies greater than th e tem perature
is exponentially suppressed because detailed balance requires th a t dephasing be
produced by the available electronic excitations which have energies sm aller than
the tem perature of the system.
At sufficiently low tem perature th e semiclassical result of [3] breaks down, be­
cause the phase breaking length
becomes large and the dimensionless conduc­
tance on th a t length scale g ( L ~ 1. In this lim it the weak localization picture
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41
for a disordered electron system with Coulomb interactions is not applicable and
the notion of the dephasing rate becomes irrelevant to the problem of the quantum
correction to the conductivity, which is no longer determined by the interference of
two tim e reversed paths.
In summary, we have dealt with interactions as a mechanism for electronic de­
phasing by explicitly separating the low and high frequency com ponents. The low
frequency, or classical p art, was shown to reproduce the previously obtained results
[3]. Then we treated the quantum component perturbatively and showed th at it
contributes negligibly to the conductivity in the regime of weak localization. Fi­
nally, we showed th a t a recent cutoff dependent calculation of quantum corrections
[20], originally intended to apply only in the presence of extrinsic phase-breaking
effects, is in agreement with ours if the cutoff is interpreted self consistently.
We conclude, th a t the electron-electron interaction via Coulomb force cannot
explain the saturation of the dephasing rate in the weak localization regime. We
believe th a t another mechanism is needed to explain the results of the experimental
paper [2],
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C hapter 3
O p en Q u an tu m D ots
The semiconductor quantum dot is a conducting region inside a semiconducting
host. [35, 4] The size and shape of the quantum dot are controlled by electrostatic
potentials applied to m etallic gates near the dot. Typically, the quantum dot is
connected to two or more conducting unbounded regions, called leads. Changing the
shape of the dot makes it possible to study numerous realization of the quantum dot
using a single device. Particularly, one can observe th e cross over from an isolated
quantum dot, separated by tunnel barriers from the leads, to an open quantum dot
with completely open channels. In this section we study the statistical properties of
d.c. transport through open quantum dots in the presence of microwave radiation.
We consider the dependence of current through the dot to the linear order in the
applied bias V between two leads connected to the dot. In general, we have
I dc = I0 + </V + O ( V 2),
(3.1)
The conductance of the system g determines the linear dependence of the current
with respect to the applied bias V. We will study the effect of an external time
dependent perturbation on the conductance of the dot. We will show th a t both weak
42
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43
O
Vd
V
Figure 3.1: An open quantum dot is connected to two leads with applied voltages Vj r.
The measured current through the dot has an offset I q at zero bias and a linear response
to small applied voltage V = V\ —Vr.
localization correction and conductance fluctuations are suppressed by external time
dependent fields.
Another consequence of an external time dependent field is a direct current I0
through the dot even if the d.c.-bias across the dot is zero. Although reproducible
for any given dot shape, this current varies random ly as the shape is changed. The
current is due to either the photovoltaic effect [14, 36, 17] or the rectification of an
a.c.-bias across the dot, see [37, 38].
The photovoltaic current appears when both d.c. and a.c. bias voltages across
the dot are zero.
Thus, the photovoltaic current is the result of interaction of
electrons in the dot. On the other hand, the oscillating bias across the dot may also
result in a finite d.c. current through the dot. This happens when the conductance
i
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44
of the dot varies in time with the same frequency as the frequency of the a.c. bias.
This effect is called rectification.
We would like to emphasize th a t current rectification is due to the finite a.c.
voltage between two leads and is a consequence of inequilibrium between leads. The
photovoltaic effect exists at zero bias across the dot and is the result of inequilibrium
produced by the perturbation inside the dot.
The photovoltaic current vanishes if the perturbation frequency is zero as one
can expect for a stationary system in equilibrium. Consequently, the expansion
Eq. (3.1) for the current through the dot starts from the second term gV. This
term was extensively studied in the literature, see [10, 8] and references therein.
Distribution of the conductance with respect to various realizations of quantum
dots was studied experimentally in ref. [8].
The average value of the conductance is determined by the num ber of channels,
connecting the dot and the leads. For an ensemble with tim e reversal symmetry
(f3 =
1)
or w ith completely broken tim e reversal symmetry (/? =
2 ),
the ensemble
average value of the conductance g is given by (see [43] and [10] for a review)
- - r —
J W L _ _
9
° N y + Nr - l + 2/P'
fo 9\
'
where N^T is the number of left (right) channels and G0 = e2/irh is the quantum
conductance.
The difference between the ensemble average values of the conductance for a sys­
tem with and without time reversal symmetry' is called weak localization correction
to the conductance and is equal to
i g « = Go N * " £ + T y
( 3
' 3 )
Since magnetic field destroys tim e reversal symmetry, the ensemble averaged con­
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45
ductance of quantum dots varies as m agnetic flux through th e dot changes. The
difference of th e ensemble averaged values of the conductance a t zero magnetic field
and at strong m agnetic field (when the ensemble averaged conductance no longer
changes w ith m agnetic field) can be m easured experimentally. T he measured value
should equal Sgwi, given by Eq. (3.3). The observed values of weak localization
correction to th e conductance are usually smaller than Eq. (3.3) predicts.
The discrepancy between experim ent and theory of the weak localization cor­
rection to the conductance is attrib u te d to interaction processes, called dephasing.
The dephasing m ay result from numerous sources: electron-electron interaction,
electron-phonon interaction or interaction with external fields.
In this thesis we study the effect of external microwave radiation on the weak
localization correction to the conductance.
[39] We will show th a t the effect of
external field is not described by the exponential phase coherence decay. Our result
depends on system param eters in a different m anner from the result of [1] for the case
of long wires due to the finite space available for electrons in th e dot, see Sec. 3.3.1.
We would like to notice that Eq. (3.3) is valid for a rb itrary num ber of open
channels. T he diagram m atic technique, used in this thesis allows us to perform
calculations for a large number of open channels, iVch
1.
In this technique we are
not able to reproduce the unity in the denom inator of Eq. (3.3). Thus, the results
of the diagram technique may be inaccurate for quantum dots w ith a few open
channels. At the same time, the electron-electron interaction becomes im portant as
the number of open channels decreases. We assume th at the qualitative picture is
captured by our large N ch expansion, and below we discuss only a large lVch limit.
The conductance of open quantum dots fluctuates from sam ple to sample. The
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variance of the conductance at zero tem perature is given by
/V2 /V2
vaTg = g * - g 2 = = 2 G l - ^ - ,
(3.4)
where, again, /3 = 1(2) for a system w ith (without) time reversal symmetry and at
tem perature higher than the escape rate T
j esc:
Go
<*i
= 6p - x T T
(33)
In contrast to the weak localization correction to the conductance, which is not af­
fected by the therm al width of the electron distribution function, the conductance
fluctuations are suppressed as tem perature increases. The other source of the sup­
pression of the conductance fluctuations is inelastic scattering, which also changes
th e weak localization correction to the conductance. It is the purpose of this C hapter
to study how external time dependent fields (microwave radiation) affect both the
weak localization correction to the conductance and the variance of the conductance
[40, 41],
Following the experimental paper [9], we plot the weak localization correction
to the conductance and the conductance fluctuations as param etric functions of
intensity of external field. This procedure allows us to avoid unknown param eters,
such as the intensity of the perturbation, and present our theoretical result in a form
easy for experimental verification.
O ur calculations demonstrate, see Sec. 3.3.2, that the tem perature dependence
of the conductance fluctuations, Eq. (3.5), is governed by the w idth of the electron
distribution function in the leads ra th e r th an the width in the dot. This statem ent
means th a t the conductance variance is not a reliable quantity to determine the
electron tem perature in the dot.
We also discuss the effect of rectification of a.c. voltage. T he rectification appears
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47
in the case when the conductance of the dot changes with the frequency of the
a.c.
voltage due to connection between the source of the a.c.
voltage and the
induced electric field inside the dot in the experimental setup. The induced field
varies conductance of the dot and effectively results in non-linear electron transport
through the dot.
Brouwer [37] argued that the rectification effect might be mistakenly treated as
adiabatic charge pumping (low frequency photovoltaic effect). The crucial difference
between the rectification and photovoltaic effect is the m agnetic field symmetry of
the rectified charge in contrast to the pumped charge, which has no magnetic field
symmetry.
3.1
Model
We consider the following experimental situation [35, 4]. G ates near a two-dimensional
electron gas (2DEG) form the shape of the dot. An oscillating voltage is applied to
the gates Vi(£) and V2(t), see Fig. 3.2. As a result of m otion of electron energy' levels
in the dot, the system is out of equilibrium and its transport properties are changed.
We study the statistics of the d.c.-current with respect to different realizations of
the dot.
Calculations will be performed for an open quantum dot in the limit of a large
num ber of open channels (occupied transverse quantum states) N ch connecting the
dot to the leads. This condition allows us to neglect the electron-electron interac­
tion, which gives corrections of the 1/A(?h order, see Ref. [42]. The same condition
perm its the use of a diagramm atic technique, similar to th a t described in ref. [44],
to calculate ensemble averaging.
We also assume th at the quantum dot is small and the Thouless energy E t ~
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48
v=u
Tv2(t)
Figure 3.2: An experimental setup. The voltage applied to the gates Vi(f) and V2(£)
changes the shape of the dot, resulting in motion of the energy levels of the electrons
in the dot.
1/ Terg is much greater than all other energy scales of the problem. In this lim it,
one can use random m atrix theory (RMT) to study transport and therm odynam ic
properties of the system, see Ref. [10]. All corrections to the RMT are assmall
as iVch/ ^ dot, where £dot = ET / S L,
is the mean level spacing. We neglect the
fluctuations of the time dependent perturbation from sam ple to sample, created by
the voltages
since they depend on the small param eter <7dolt <C 1.
The Ham iltonian of the system is [10, 39, 17, 45]
= 'H.d (£) + "Rl + T£lD)
(3.6)
where 'HD(£) is the Hamiltonian of the electrons in the dot, determined by the M x M
m atrix Ho(t):
#d =
M
H ^ nH DnTn(t)il>m + E cn \
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(3.7)
49
ijjn corresponds to the states of the dot and the therm odynam ic lim it M
assumed, E c is the charging energy, and n = J2m=i
oo is
the last term in Eq. (3.7)
is the largest contribution to the interaction effects in quantum dot, see Ref. [45] for
the discussion of the statu s of this approxim ation. M atrix Hv>{t) is given by
H 0 {t) = H + Y , V m ( t ) .
i
(3.8)
Here the tim e independent part of the Ham iltonian H is a random realization of a
M x M m atrix, which obeys the correlation function
— ?r2l
4JVf)
+
4iV/)
m n ' u nmr
(3.9)
where 6i is the m ean level spacing of the dot and param eters Nd,c describe the
effect of the m agnetic field on the dot [10, 45]. These param eters can be estim ated
as ATd>c ~ gdot (4>!
<E>2)2 m
where $
1,2
is the magnetic flux through the dot and
$ 0 = hc/e is the flux quantum. The tim e dependent pertu rb atio n is described
by symmetric M x M matrices Vjfil and functions cPi(t) of tim e t. We assume
th at the perturbation is harmonic with single frequency u), ipi{t) — cos cot and
ip2(t) = cos (cut + 0), even though m ost of the consideration is valid for arbitrary
functions
The effect of the perturbation on the system is determined by
parameters,
C* =
* =
Param eters
(3.10)
have a meaning of the average velocity of the energy levels of the
dot under the external perturbation V* and can be om itted from our consideration
due to screening [see below]. Param eters Cij characterize the variance of the level
velocity [46] due to the Hamiltonian perturbation XiVi\
28 1
7r
_
lj
_ de^ de^
dxi dxj
dxj dxj
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. ^
50
Since all physical responses of the system are characterized by the same param eters,
the values of Cij can be eliminated by an independent measurement. Below, in the
case of a single perturbation, we will use the following notation Co = C u .
For a rath e r unrealistic case of homogeneous electric field E q introduced into a
dot of linear size L , we can estim ate
C0 « (eE0L)2/ E r h5l .
(3.12)
The electron spectrum in the leads near Fermi surface can be linearized:
UL =
(3.13)
a ,k
where il>Q(k) denotes different electron states in the leads, k labels the continuum of
momentum states in each channel a , hvp = l/2irv is the Fermi velocity and u is the
density of states per channel at the Fermi surface. We put h = 1 in all interm ediate
formulas below.
The coupling between the dot and the leads is
^LD = E
( ^ n a ^ ( fc )^ n + H.C.) .
(3.14)
a ,n , k
For reflectionless point contacts, the coupling constants, W na, in Eq.(3.14) are given
by [10, 45]:
W nn
____
MSl
— ,
7l2U
=
i f n —a < i Vc h ,
_0,
otherwise.
For open dots with a large num ber of open channels Ach
(3.15)
1 the interaction
term can be treated within the m ean field approxim ation, so th at the Ham iltonian
(3.7) takes the form
M
=E
r i [ H v n m { t ) + e V d { t ) 6 n m ]i(>m,
(3-16)
n ,m = l
eVd(t)
=
2 E c(n)q,
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(3.17)
51
where (n)q stands for the quantum mechanical (but not ensemble) average of the
electron num ber in the dot. Corrections to the mean-field treatm ent (3.16) were
calculated in ref. [42], see also ref. [45], and are small as l/iV^h.
In the m ean field approximation (3.16), one can introduce one-particle S - m atrix
Sotp(ti,t2) as
S„a (t, f ) = SaS6(t - t') - 2 w i v W ^ G ™ ( t , t')W mS,
(3.18)
and the Green functions G(R'A) (t , t') are the solutions of:
~~
W “ eVd{t) ±
t') = S(t - t'),
(3.19)
where the m atrices Ho and W are defined by Eqs. (3.8) and (3.15).
The current through the dot is given in terms of the scattering m atrices S ( t , t r)
by the following expression, see Appendix B .l:
(/(£)) = e £ AQtt J d t , d t 2 j x : S Qp(t,
~ t 2 ) S l a (t2, t) - / o (+0) |
(3.20)
where f Q(t) is the Fourier transform of the electron distribution function in the
leads,
=T t
— OO
h t n - } = tL rv
5
2
(3-21)
(...) stands for the quantum mechanical and thermodynamic averages for a given
ensemble realization (no ensemble averaging) and
Nr-,
A q/3 — 5a0 <
1,
N Ch
if 1 < a < iVi;
"
if A, < a < N ch.
(3.22)
To complete the theory one needs an equation for the averaged number of par­
ticles (n)q, see Eq. (3.17). It is found from the continuity relation as
d( n( t))q
=
dt
- J d t ldt2T v { f ( t x - t 2) { s \ t 2, t ) S ( t H { ) - 5 ( t 2 - t l ) ) } (3.23)
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52
Equations (3.20) — (3.23) are sim ilar to those used in Ref. [47] for studying the
frequency dependence of the conductance of mesoscopic systems.
In order to find the characteristic value of the m atrix elem ents describing the
external p erturbation Vnm acting on electrons one has to take into account th at
the num ber of electrons in the dot 5n may be changed by such perturbation. This
change of the charge, in its turn, leads to the change in the effective potential
which is experienced by an electron.
To take this effect into account, wre use the leading approxim ation in small pa­
ram eter l/iV ch. The fluctuations of the conductance are sm aller than its average
value and we can use instead of sam ple specific Eq. (3.23) its ensemble averaged
counterpart:
d (n(t))q = _ N ± f ( j m u _
dt
7T \
2
+
\ + eA W +
J
7T
here we lim it ourselves to a single perturbation Z\ — Z and <^i(t) =
Equation
(3.24) is nothing but a discrete form of the diffusion equation for the bulk system and
the last two term s correspond to the divergence of the drift current. Substituting
Eq. (3.17) into Eq. (3.24), solving th e resulting differential equation, we find
4 eE c
j_
y
iV,V[ + N rVr
N,ch
<51 + (2ir/l\rch)dt
<y1 + 4 E c + (27r/lV ch) a t ^
(3.25)
h
We notice from Eq. (3.25) th at the characteristic energy scale governing charge
dynamics is E,cArch/ 2tz. Usually, this scale is of the order of the Thouless energy,
E t - Because the random m atrix theory is only able to describe energy scales smaller
than E t , we can consider only the u> <§C E t — E cN c^/2 tv case. Moreover, for small
quantum dots E c
£i, so that Eq. (3.25) gives
„
!V,Vi + lVrUr
Vd =
Nch
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(3.26)
53
and the tim e dependent perturbation (3.10) can be considered as traceless, Z = 0.
3.2
Photovoltaic Effect
Adiabatic charge pum ping through open quantum dots was studied recently in the
literature both experimentally [35] and theoretically [16, 48, 49]. Such pumping
occurs in a system described by a H am iltonian periodic in tim e with a period Tp
larger than all other characteristic tim e scales of the system .
After one period,
the system returns to its initial form; however finite charge Qi can be transm itted
through a cross-section of the system:
Qi = IdcTp =
fJoT (I{t))dt,
(3.27)
where (I ( t )) is given by Eq. (3.20) and the subscript i m eans th a t the charge Q i
is the quantum mechanical and therm odynam ical average of the transported charge
QTo obtain a finite transm itted charge a t low frequencies, the Ham iltonian should
depend on a t least two parameters. In refs. [16, 48, 49] the tim e dependence of
the H am iltonian was replaced by a dependence on param eters and the system was
considered quasistationary for each param eter value. The tran sp o rted charge during
one period of the Ham iltonian was calculated as an integral in param eter space. The
theory [49] shed some light on the recent experiments [35], nam ely on the amplitude
dependence of the root mean square fluctuations of the tran sm itte d charge, averaged
over different realizations of the Ham iltonian.
A very sim ilar phenomenon was considered previously by Falko and Khmelnitskii,
who theoretically studied the photovoltaic effect in mesoscopic micro junctions [14].
The experim ental observation is described in Ref. [50]. T he photovoltaic effect is
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54
a generation of d.c.-current by radiation of a finite frequency. (It is obvious that
this effect can only be non-linear in the oscillating field.) T he bilinear regime of
adiabatic pum ping, [16, 48, 49] is precisely the circular photovoltaic effect introduced
in Ref. [15] and applied to a mesoscopic system in Ref. [14]. The results of Ref. [14]
are not directly applicable to quantum dots because in micro junctions the Thouless
energy E t ~ 1/ rerg is of the same order as the inverse escape tim e l/resC, whereas
for quantum dots 1/ r esc <C E t - (Here r erg is the characteristic tim e for a classical
particle to cover all of the available phase space in the dot and we put h = 1.)
Therefore the considerations of refs. [16, 48, 49] have their own physical significance.
On the other hand, the theory of Ref. [14] is not restricted to the adiabatic regime,
the results being valid in a broad interval of frequencies.
The purpose of the present section is to go beyond the adiabatic approximation
for d.c.-current generation in open quantum dots, see [17]. One can identify two
contributions to the d.c.-current — reversible and irreversible. To make connection
with the terminology of the photovoltaic effect, used in Ref. [14], we consider the
bilinear d.c.-current response through the dot, generated by several time-dependent
perturbations ^Piit) =
.
The direct current Idc can be w ritten a t
uj
—v 0 as
+ 0 ( u 3),
/ dc =
(3-28)
where Cy and l ij are reai sample specific coefficients. They are not fixed by any
symmetry (a sample does not have any!) except the condition th a t /q c is a real
number, which gives the requirements
tij = -Cji,
l i j = Iji-
(3-29)
These relations m ean th a t as the direction of the contour in the param eter space
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55
{<£>*(£)} is reversed
-> <Pi(—t) or <piiU, —>
■
the first term changes its sign
whereas the second term remains intact. In the language of the photovoltaic effect,
the first term is the circular photovoltaic effect, and the second term is the linear
effect [14]. Equation (3.28) makes an explicit connection between the bilinear re­
sponses of d.c.-current generation through open quantum dots and the photovoltaic
effect.
We will see th a t the separation onto reversible and irreversible parts goes far
beyond the bilinear and low frequency expansions. We will find th a t the reversible
part vanishes at high frequency whereas the irreversible part saturates. We will also
show th a t the irreversible contribution may be interpreted in a manner similar to
adiabatic pum ping. In adiabatic pum ping the transm itted charge was determined
by a contour in param eter space. The irreversible current is determined by the
contour in the extended phase space
cpi(t) , ...} , see Sec. 3.2.2.
The first term in Eq. (3.28) vanishes for a single pump due to the antisymmetry
of £ij, Eq. (3.29). The contour in param eter space degenerates to a line in this case,
while in phase space the contour encompasses a finite area. We will see th at the
current is proportional to this area. Note, th at the contour is invariant with respect
to time inversion.
Before we proceed, we present an equivalent form of Eq. (3.20) for the case when
the chemical potentials in both leads are equal, see Appendix B.2:
(/(£)) = 2eiiru I f d t i d h f i h - < 2)tr { W ' G w {t,t,) (fffa ) - H { t 2)\ G(A\ t 2, t ) W \ } ,
(3.30)
which is more convenient for our calculations.
The ensemble average of the transm itted charge
defined by Eq.(3.27), is zero:
= 0. We calculate the second order correlator with respect to an ensemble of
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56
A
—
*2
= f ( t r t 2) [ H ( t J - H ( t 2 )]
h
Figure 3.3: The diagram representing the contribution to the charge correlator Q\
a t high tem perature T.
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57
random m atrices:
3 .2 .1
TpTp
Q \= j jdtdt'(I(t))(I(t)).
0 0
(3.31)
M e s o s c o p ic F lu c tu a tio n s o f th e P h o to v o lta ic C u rren t
a t H ig h T em p era tu re
In this section we will consider the high tem perature lim it, in which the frequency of
the p erturbation is much smaller th an the tem perature. (More accurate definition of
the high tem perature limit is given in Sec. 3.2.3.) In this case th e only contribution
to the charge correlation function Q \ is given by the diagram shown in Fig. 3.3. The
corresponding analytical expression is
Q\
=
x
„
x
2tt
+oo 0
4e2g JixdyTZ(x, y) J d d J d r F 2(r)
V 2u j
+
2ui
+
u j
(3.32)
J
^ (x + V , „ x + y
_ x -y \
P
+
“ a T " T’ ~ J ’
where T)(t, t ' , r ) is the diffuson, defined by Eq. (B.40) in A ppendix B.3,
7Z(x,y)
=
C n sin x sin y + C 2 2 sin(x -f- 4>) sin(y -f- $)
+
C \ 2 (sin(x + (fj) sin y + sin x sin(y + 0) ) ,
(3.33)
9 =
(3.34)
is the dimensionless conductance through the dot from the left to right leads and
FM = S E S S v
(3'35)
Expression (3.32) can be computed for different values of param eters. In Fig. 3.4 we
present the result of computation of Q\ for two frequencies
u j
= 0.l7esc and
uj
= yesc,
where y^c = iVch51/27r. Both those curves exhibit C f and \fC[ dependences at
t
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10'
cn
=
10 '
CO
-Q
CO
O
-V- “ =0-1Yesc
W=Yesc
co»Y‘esc
10'-5
10',0
C/y•tesc
10‘,2
.4
10
Figure 3.4: T he am plitude dependence of the pumped charge for different values
of the frequency of the pump. For
uj
> 7esc the curves have the C 2 dependence at
small values of C and the \[C dependence a t C » 7esc- For sm all frequency (e.g.
uj
= 0.l7esc) there is an intermediate regime.
weak and strong pum ping respectively. We also show the analytical curve given by
Eq.(3.43) for the
uj
-> oo limit. Below we discuss different lim its of Eq.(3.32)
B ilin e a r re s p o n s e
First we consider the weak perturbation and perform an expansion of the diffusons up to term s linear in C^. As a result we obtain:
Q\
r+9
~ C 2{2uj6 — sin ‘2aid) + C 2 sin 2u6
—4'K2e Lg [ dde 2Nch0 f
d r F 2(r)UJ
Jo
J-o
(3.36)
where we have introduced the linear and circular pumping am plitudes
C\
=
Cc =
C\\ + 2 C n cos (j>+ C 2 2 ,
(3.37)
2 sin <fj\JC 1 1 C 2 2 — C22 -
(3.38)
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59
la the case of tem perature T larger th an the escape rate 7esc, we find
(3.39)
The second term of Eq.(3.39) survives the limit u —> 0, thus reproducing the
known result for adiabatic pumping [16, 49]. On the other hand, this term vanishes
at high frequency. The linear term is quadratic in frequency at sm all frequency and
tends to a constant a t large frequency.
The linear pum ping amplitude C\ in the case of two pum ps has the form of
Eq.(3.37), which implies th a t the am plitude is just a vector sum of different pumps
in the param eter space. On the other hand the circular am plitude is related to the
area in the param eter space, covered by the pumps.
L o w fr e q u e n c ie s
Equation (3.32) in the adiabatic limit w —> 0 is in an agreem ent with the results
of Ref. [49]. Namely, this expression gives the same asym ptotic behavior for the
limits of weak and strong pumping. To dem onstrate this we consider the special
case of the C m atrix having the form C u = C 22 = C and C 12 = 0. In this case we
obtain
(3.40)
As tem perature drops down to T ~ 7esc = ■Nch51/27r, the variance of the transm itted
charge, Eq. (3.32) saturates to
(3.41)
Note, th a t the d.c.-current through the dot /dC = Q / T p ~ uiQ vanishes as u —> 0,
as expected for a system in equilibrium.
The authors of Ref. [49] showed th a t for strong pumping the mean square trans­
ported charge is proportional to the length of the contour in the parameter space,
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60
and does not depend on the particular shape of the contour. Equations (3.40) and
(3.41) support this statem ent, since for C
7esc they reproduce a \ f C dependence
on th e pum ping amplitude. In the opposite case of weak pum ping Eqs. (3.40) and
(3.41) give C 2 dependence in accordance with [16, 48]. To understand the strong
pum ping dependence, we consider a loop in the param eter space. See Fig. 3.6(a).
We notice (see Eq.(3.48) below ) and refs. [16, 48], that adiabatic pumping can be
related to a contour integral in the param eter space. At sufficiently strong pumping
the system at distant points of this space is uncorrelated and the total contribution
to th e pum ped charge, which comes from the uncorrelated pieces of the loop, is
proportional to the number of pieces.
In the limit of low frequency and zero C c (single pum p), the mean square fluc­
tu atio n of the charge per cycle is quadratic in frequency. For weak pumping the
am plitude of charge fluctuations is determ ined by Eq.(3.39) w ith 0 = 0 for arbitrary
frequency
uj.
(but still
uj2C[
(For a single pum p, C \ is the only param eter.) For strong pumping
<C 7 ^ c) we find
(3.42)
We explain this dependence on the am plitude of the perturbation in the next section.
H ig h fre q u e n c ie s
In the limit of high frequencies, T » u » 7esc, the variance of the transm itted
charge is given by
(3.43)
In the lim it of strong pumping this expression has the \fC\ asym ptotic behavior.
The function of the right hand side of Eq.(3.43) is represented in Fig. 3.4 by the
solid line.
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61
(b)
Figure 3.5: In figure (a) a loop is shown in the parameter plane. The grid divides
the plane onto pieces, so that parts of the loop in the different pieces give uncorrelated
contributions to the transported charge. Figure (b) shows the loop in the phase space for
strong pumping. In this case the loop can be divided onto pairs, and the pairs are not
correlated. On the other hand parts of the loop of one pair are close to each other, so
they are strongly correlated.
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62
3 .2 .2
P h o to v o lta ic E ffect as P u m p in g in P h a s e S p a ce
In this section we discuss the mechanism of charge transport by a single pump at
finite frequencies (i.e. the irreversible contribution to the d.c.-current). We show
the sim ilarity with the mechanism of adiabatic charge pumping, discussed in [16].
In the adiabatic approxim ation the system ’s m otion is considered in a parameter
space. For finite frequencies the param eter space has to be extended to phase space,
which contains not only the perturbation param eters but also th eir time derivatives.
According to Eq. (3.20), the transported charge for one period is determined by
(3.44)
x
tr
e
t + if
t + t'
r
We u se the W igner transform for the scattering m atrix:
(3.45)
We consider charge pum ping at high tem perature (T
cu). In this case the
integration over r is lim ited by the inverse tem perature 1 /T . O n the other hand,
the scattering m atrix S(e, t) in the Wigner representation varies slowly with respect
to its tim e argum ent t. T his allows us to expand the scattering m atrices in Eq.(3.44)
to linear order in r . Using the unitarity of the scattering m atrix we finally obtain
(3.46)
This equation was used by Brouwer in [16], see also [47]. T he scattering m atrix
in the W igner representation is a function of the perturbation itself and its higher
order derivatives with respect to time. (See Appendix B.4.) In th e adiabatic approx­
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im ation the derivatives are neglected as being small to higher orders in frequency.
Beyond the adiabatic approximation, we have to include the derivatives.
We dem onstrate th a t the analysis of Ref. [16] can be applied to the high frequency case. We assume th at there is a single function of tim e <p(t). Then, following
Brouwer, Ref. [16], we introduce a vector field
(3.47)
where X i = dlip{t)/dtl and i is a non-negative integer.
In these notations Eq.(3.46) for the transported charge Q\ is given by
(3.48)
The loop integral in the above equation can be rewritten as a surface integral using
Stoke’s theorem. We develop our analysis for the transported charge to the lowest
order in frequency, so th a t the scattering m atrix depends only on X 0 = ip(t) and
Xi =
According to Stoke’s theorem for this two dimensional space, we obtain
J
d(pd(pU(e),
(3.49)
In appendix B.4 we present a formal derivation of dS/dtp from the equation of
m otion, Eq.(3.19), in term s of the Green’s functions of the dot to lowest order in
W/ Tesc-
Now we interpret results found in the previous section using Eq.(3.49).
We
consider one param eter pumping (linear photovoltaic effect) characterized by a har­
monic perturbation cp(t) = cosu t with frequency cj. In this case the strength of
perturbation C, defined by Eq. (3.10), is a single number, which we denote as C q.
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64
For weak pum ping we keep the tim e dependent perturbation to lowest order to cal­
culate the derivatives with respect to ip and ip. We assume 11(e) to be a constant
and the transported charge per period is proportional to the area of the contour in
phase space. The contour is an ellipse with semiaxis \/C q and ujy/Co and area
ttuiC q.
For the variance of the transm itted charge, we expect Q\ oc oj2C o /7 ^ c, which is in
agreem ent with Eq.(3.39) for u; <£L 7esCIn the limit of low frequency but strong pumping, we can apply Eq.(3.49) to
understand Eq.(3.42). The power dependence [C ^ 2] is different from the adiabatic
case [ C ^ 2\. The loop in phase plane is long along the <p axis b u t narrow in the cp
direction because the frequency is small. [See Fig. 3.5(b).] T he charge variation is
determ ined by a sum of independent contributions from pieces of the contour along
the <p axis. As can be seen from Fig. 3.5(b) the number of the independent pieces is
Arind = \JC q/7esc- In the <p direction the system is correlated inside each piece of the
contour since all points along the ip direction are separated by a distance, smaller
th an the correlation length. The characteristic area Sc of each p a rt is proportional
to uiyjC'o7esc- The variance of the transported charge can thus be estim ated as
, ,2 /
Q\ oc e2N indS 2 oc e2^ -
ft
\
3/ 2
.
(3.50)
/esc V /e s c /
W hen the am plitude of the field C or the frequency u increases further, so th at
u)2C0 >
7 ^ c,
this picture is no longer valid. The trajectory does not have parts
close to each other and each p a rt gives an independent contribution. The situation
is sim ilar to the case of strong adiabatic pumping, as shown in Fig. 3.5(a) and
discussed in [49]. The variance of the transported charge is proportional to the total
num ber of uncorrelated parts, so th a t Q \ oc y/Co, see Eq. (3.43).
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65
3 .2 .3
P h o to v o lta ic E ffect a t Low T e m p e r a tu r e
The previous discussion of d.c.-current generation is quite general. However, it does
not take into account the heating of electrons by an external field which becomes
im portant at low tem perature. In this regime, the electron distribution function
in the dot changes and acquires a w idth larger th an the electron tem perature in
the leads. For simplicity we lim it our discussion to one param eter pum ping (linear
photovoltaic effect). The external perturbation is determ ined by a harmonic function
cp(t) = cosojt and the strength of the perturbation is Co = C u , see Eq. (3.10).
T he new w idth
of the distribution function can be estim ated from the fol­
lowing picture. An electron has random transitions betw een different energy levels.
T he tim e between consecutive transitions t lr is determ ined by the Fermi golden rule:
m
The first equality sign follows from the Fermi golden rule, the ~ sign represents an
estim ate of the characteristic value of the m atrix elem ents \Vnm\ 2 and the density
of states l/5 i, the last equation is the definition of Co, cf. Eq.(3.10). Since an
electron stays in the dot for a time r esc =
7 “^
= 27r/iVch51, it performs N tr =
l/£ tr7esc = C q/ 7esc transitions. Each transition changes the energy of the electron
by
o j.
is oc o j
As in the random walk problem, the displacement of electrons in energy space
T esc •
T his analysis gives a new tem perature scale T^:
(3.52)
T his scale has a meaning only for strong fields, Co
7esCj so that the diffusion
picture in energy space is valid. Otherwise, electrons experience few transitions
w ith change of energy
o j.
Now we consider low tem peratures, so th at T
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is
66
not valid. We calculate the fluctuations of d.c. current for a system with a single
pum p. As we know from Sec. 3.2.1, a t high frequency the num ber of pumps is not
im portant and the result depends on their linear combination.
Unlike the diagram , shown in Fig. 3.3, diagrams presented in Fig. 3.6 have
additional diffusons dressed on the distribution functions / ( r ) . Collecting diagrams
in Fig. 3.3 and Fig. 3.6 we obtain the following expression for the variance of the
pum ped charge:
TU
r+ g
Q\
=
4e2C N chg JJdtdt' J d6 J
d r F 2(r)
,t-ej
x
r+oo
(3.53)
r+co
x
/ d?D{t, t - <£, 2r) / d£ V ( t', t' Jo
Jo
x
|2 C sin2 Lu{t —
sin2 uj(t' — £') sin2 u
2r)
jt
+ iVch sin uit sin ojt'}
where g is the dimensionless conductance of the dot (see Eq.(3.34)) and
“ i S
At high tem perature T
oj
?
'
( 3
' 5 4 )
Eq.(3.53) reduces to Eq.(3.32). Indeed, all three
diagram s in Fig. 3.6 are smaller th an the diagram in Fig. 3.3 a t least by one factor
cj/ T .
Now we discuss the limit of high frequency
allows us to perform integration over
and
oj
max{7esC)C}- This inequality
in Eq.(3.53). We can replace sin2 u>(t —
£) by 1/2 and use the approximation
Na, + 2 C s J ? - - r -
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( 3
- 5 5 )
67
In this lim it the product of the diffusons in the second and th ird lines of Eq.(3.53)
does not depend on r.
Energy of an electron in the dot changes due to the external field resulting in the
redistribution of the electrons in the energy space. The new distribution function
becomes wider th an th a t of electrons in the leads at tem perature T. Consequently,
the Fourier transform of the electron distribution function becomes narrower. The
right hand side of Eq.(3.55) represents the effect of heating. T his function appears in
the integral over r in Eq.(3.53) along with the function F (r), defined by Eq.(3.35).
At sufficiently low tem perature the convergence of the integral over r is determined
by the ’heating factors’, Eq.(3.55) rather than by F (r). We note, that the shape
of the new distribution function is not a Fermi function with a higher tem perature.
Instead, its Fourier transform has the form of the right hand side of Eq. (3.55).
To be more specific, we consider the strong pumping lim it C
7esc, when the
electron distribution function is determined by the new scale Th, see Eq.(3.52). We
find
Q'l = — e2g — .
^
16
uj
(3.56)
The same param eter dependence can be found from Eq.(3.43) replacing tem perature
T by the new energy scale Th.
Equation (3.53) has two terms. One term contains factor sin cut sin out'. This term
survives the high tem perature limit, see Sec. 3.2.1. Nonetheless at low tem perature
the heating modifies the results of Sec. 3.2.1, so that the tem perature dependence
saturates at the characteristic tem perature scale ThT he second term was completely neglected in the previous sections. The origin
of this term is sim ilar to that of the therm oelectric effect in a conductor out of
therm odynam ic equilibrium. Although electrons are in equilibrium in the leads, the
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A
A
Figure 3.6: Diagrams, which contribute to the d.c.-current at low tem perature lim it.
(We do not show diagrams, which can be obtained from the above by om itting the
upper or lower diffusons.)
The last diagram contains the Hikami box, which is
presented in Fig. 3.7.
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69
l^ = y
t^ r 3
^ ^vyvyv y v
yryyyyyy
Figure 3.7: The Hikami box, introduced in Fig.3.6, can be obtained from these
diagram s and from their rotations. The grey rectangle represent averages of the
type G(*>G(R) and G ^ G ^ .
heating changes their distribution function in the dot, producing a non-equilibrium
distribution. T hen non-equilibrium electrons escape from th e dot. The direction
of each escape is determined by the realization of the dot. An unbalance between
electrons escaping through the left or right leads gives current.
(sin2 cut term in
Eq. (3.53) reflects the electron-hole asymmetry, necessary for therm oelectric effects.)
3.3
Conductance of Open Quantum Dots
The purpose of the present Section is to extend the results of Ref. [1] to describe the
effect of the external microwave radiation on conductance of open quantum dots.
The ultim ate goal is to identify observable features which allow one to distinguish
the effect of the external field to the dot from simple heating [4, 9].
However, there is a significant difference in the calculation of the mesoscopic
conductance fluctuations [40] and the averaged conductance [39]. Because, the dot
is subjected to the external classical radiation which produces non-equilibrium in
the dot, the d.c.-current I0 through the dot is finite (though random ly changing from
one configuration to another) even if the d.c.-voltage V = V] — Vr across the dot is
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70
zero, see Fig. 3.1. This current I q is due either to the photovoltaic effect[14, 36, 17]
or to rectification of a.c.-bias across the dot, see [38, 37]. We are interested in the
linear response to the applied d.c.-voltage across the dot, see Eq. (3.1), which is
described by the conductance of the dot, g.
T he characteristic energy scale governing charge dynamics is E cN c^/2-k , see
Eq. (3.25). Weassume th a t this scale is of the order of the Thouless energy, ET .
Moreover, for the small quantum dot E c
„
Si, so th at Eq.(3.25)
gives
N M + N rVt
=
Nch
( 3
' 5 7 )
and the time dependent perturbation, defined by Eq. (3.10) can be considered as
traceless, Z = 0. This tim e independent component of the electric voltage on the
dot can be removed from Eq. (3.19) for the Green function by the following gauge
transform ation:
G(t,lf) = G Vd=0
.
(M ')e~ lWd(t_t,)-
(3-58)
S ubstituting Eq. (3.58) into Eq. (3.20) and expanding up to the first power in
V = V\ — Vr, we find
_
Tp +oc
g = ? - p - = g c l + G0 f d t J d t ^ n n u - f2))Tr{,S(£, tO A S tfe , £)A}.
0 —oo
(3.59)
Here
N\Nr
gc\ = G o - r y —
f
m
( 3 .6 0 )
•**ch
is the classical conductance of the dot, G 0 = e2/ n h is the quantum conductance and
F (x) is the Fourier transform of the derivative of electron distribution function:
F(x) =
smh 7tx
(3.61)
As we mentioned before, the photovoltaic current exists even a t zero bias across
the dot. One can expect th a t when a finite bias is applied to the dot, the photovoltaic
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71
current m ay change so th at the linear in the bias V contribution to the current comes
from two sources: (?) the non-equilibrium of the electrons in the leads and (ii) the
change in the photovoltaic current, corresponding to a different realization of the
dot because of the finite bias. We would like to emphasize, th a t Eq. (3.59) does
not contain Vd- This result demonstrates, th at the finite bias does not change the
photovoltaic current to the linear order, and the linear part of the current is due to
the non-equilibrium across the dot.
3.3.1
W ea k L o c a liz a tio n C orrection to C o n d u c ta n c e
The first order correction in l/A^h to Eq. (3.59) is given by the diagram in Fig. 3.8 It
represents the weak localization correction to the conductance, and has the analytic
expression in term s of the Cooperon:
27T/ U!
A j„, =
^
ch 0
oo
/ C(T .
0
->->*•'
<3-62)
where, according to Appendix B.3, Eq. (B.39), the Cooperon has the form
C(T, r , —r) = exp
^7 C+ 4C0 sin2 ^
sin2 ojT ^
(3.63)
In the absence of the time dependent perturbation ip = 0, one obtains [10, 63]
from Eq. (3.62):
^ “-v J I im
( 3
-6 4 )
The solution to Eq. (3.62) gives the weak localization correction to the conductance
A g wi in the presence of the time dependent field. It can be expressed in terms of
the unperturbed correction (3.64) as
A gwl = A jS ’f ( - , - ) •
\ 7c
7cj
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(3.65)
72
Figure 3.8: The diagram for the weak localization correction to the conductance.
Here the dimensionless function F(x, z) is given by
TOO
F(x, z) = / dre ~T- x^ z'T)I0 (x0(z, r ) )
Jo
(3.66)
with dimensionless parameters
Cq
x = — ,
co
z = —
7c
(3.67)
7c
and I0(O is the modified Bessel function and
sin z r
f r’
0 ( z ,r ) = r -^ ^ = { 23
^
ifzr»l
(3.68)
ifzrCl.
Some curves of this function are plotted in Fig. 3.9.
Equations (3.65) - (3.66) are the m ain results of this subsection. They give the
universal description of the effect of the external field on the weak localization cor­
rection. T he term x<f)(z, t ) in Eq. (3.66) suppresses long tim e contribution to the
Cooperon due to the external field and can be treated as dephasing. We assume
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73
th a t the uniform electric field
E q
penetrates into the dot. An electron experiences
different electric fields a t time t, when it moves along two tim e reversed trajecto­
ries. We denote the difference of the electric field as E(t). For a low frequency of
the perturbation, so th a t ojt <C 1, E{t) « EoOJt and in the opposite limit of high
frequency field, out
E{t) =
E q.
1, the characteristic value of the electric field difference is
T he electron moves along straight lines between collisions with impu­
rities and boundaries of the dot. Between collisions it acquires the difference in
energy e E ( t ) L for two time reversed trajectories. Since the energy determines the
velocity of the phase, the electron phase difference for the same p a rt of the time re­
versed trajectories is 5(p = eE(t)Lrcou, where r con is the tim e between collisions with
the boundaries and is the inverse Thouless energy Erh- Assum ing th a t the phase
difference over each straight part of the tim e reversed trajectories is independent,
we conclude th a t the phase difference for electron motion along two time reversed
trajectories obeys (5ip2(t)) = e2E 2(t)L2tfE^h- Using the correspondent approxi­
m ations for E{t) for the low and high frequency of the external field, we obtain
{5<p2(t)) = CQ<p(z,t), in agreement w ith Eq. (3.68), where
C q
= e2E l L 2/ E ^ S i , see
Eq. (3.12).
Below we will discuss different asym ptotic regimes and com pare them with the
results for bulk systems [3, 1].
For weak external field x <C maar(l, y~2) we find
%
AgW = 1
9-
t t w
2 + 5\9Z
(»■“ )
In this regime the correction is quadratic in the frequency for slowly oscillating
field, sim ilarly to the bulk system result at u smaller than the dephasing rate l/r ^ .
However, the frequency dependence saturates at large frequency. It is different from
the result for bulk systems, where a characteristic spatial scale shrinks as 1/y/Uj,
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74
1.0
1.0
0.9
0.8
^N_ 0.7
*T
^ 0.6
0.9
x=0.3
"N 0 . 8
x’
uT 0.7
x=l
0.6
^ 0.5
0.5
z=2
0.4
z=5
0.3
X
Figure 3.9: Representative curves of F(y,z) as a function of z for two values of y. It
decreases linearly with z at small values of z. The inset shows the y —dependence of the
function F(y, z) for two values of z. It decreases quadratically in y at small values of y
and saturates at larger y.
whereas in our case it is determined by the size of the dot.
Beyond the bilinear response, x » m a:r(l, y ~2), vve can consider the limit of fast,
y
1, and slow, y <^1 field oscillations. In the first case we find
Aff„, = AffS 1
1
v' 1 + 2 C „ / 7 c
(3.70)
The linear dependence of the quantum correction on l/v^Co is sim ilar to that for
the bulk system and can be qualitatively understood. The weak localization correc­
tion to the d.c. conductance is determined by averaged value over a period of the
external perturbation, see Eq. (3.62). For strong enough field, th e weak localization
correction is exponentially suppressed, except certain moment of tim e, when the ef­
fect of the perturbation to the Cooperon vanishes. [See Eq. (3.63). For T = 0, 7t/oj,
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75
the perturbation does not affect the Cooperon.] The weak localization correction
to the conductance comes from a short time interval S T ~ 6j - l ^/7c/Co, when the
effect of the external perturbation on the Cooperon is com parable with the effect of
electron escape rate 7C. The above qualitative argument gives Agwi = A <7^ / \Jyc/ C q.
Contrary to the bulk systems, the result does not depend on the frequency u,
because of the finite length scale L of the problem, as we have already discussed.
In the case of slow field y < l , but still z y 2
1 (strong field) we obtain
A gwl _ r ( l / 6 ) ( 2Sjgl \ 1/3
Asl°>
* r(5 /6 )
1 '
1
i.e., the dependences both on the am plitude and frequency are different from the
bulk case.
Finally, we note th at therm al fluctuations of the gate potentials may induce
dephasing by virtue of the mechanism considered in this subsection. However, the
spectral density of such fluctuations is model dependent and thus not universal.
3 .3 .2
C o n d u c ta n c e F lu c tu a tio n s
In this subsection we study the effect of external radiation on conductance fluctua­
tions, following ref. [40]. We note th a t ref. [40] has a certain overlap with the recent
paper by W ang and Kravtsov [41], where the conductance fluctuations were calcu­
lated for open quantum dots subjected to a periodic ac pumping. Our treatm ent is
different in several aspects. Firstly, our results are applicable for the frequencies of
the external radiation u smaller than the Thouless energy of the dot, E r , whereas
treatm ent of Ref. [41] is valid in the opposite regime. Secondly, unlike Ref. [41], we
will restrict ourselves to the case of the monochromatic radiation acting on the dot.
Finally, we highlight the role of the electro-neutrality requirement in a separability
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76
of the photovoltaic effect and the mesoscopic conductance fluctuations, which was
not done in Ref. [41].
The correlation function of the conductance
R ( * u * 2) = g @ i) g & 2 ) - sfWi1
9
W 2j
(3.72)
is given by the diagram s shown in Fig. 3.10 and can be found from the following
analytical expression
Equation (3.73) deserves a little bit of additional discussion. We notices th at
the diagrams in Fig. 3.10 do not contain any piece corresponding to the classical
distribution function in the dot, compare to Sec. 3.2.3. We can trace it into the
expression for conductance (3.59) th a t contains traceless vertices A, which can not
be dressed by the dashed line. On the other hand, any vertex w ith finite trace
corresponds the modified distribution function of electrons in the dot and represents
the effect of heating [17], see also Sec. 3.2.3. Since the distribution function is not
dressed in the expression for conductance fluctuations, we conclude, th a t the effect
of heating is not relevant for the conductance fluctuations and the tem perature
dependence of the conductance fluctuations is uniquely determ ined by the electron
tem perature in the leads. T hat means th a t contrary to the common belief, see e.g.
Ref. [9] the am plitude of the mesoscopic fluctuations can not be used for the study
of the distribution function of electrons in the dot. The same conclusion for the
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77
R
Figure 3.10: Two diagrams, which contribute to the conductance correlation function R.
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78
conductance fluctuations of open quantum dots was reached in ref. [51]. From the
theoretical side, it is im portant to emphasize, th a t the appearance of the traceless
vertices is determined solely by the electro-neutrality condition (3.57), any other
choice of the dot bias would lead to the change in the photovoltaic current [17]. We
find the reasoning of ref. [51] imprecise, since the electroneutrality condition is not
invoked there.
L im itin g c a s e s
Below we consider the lim it of high (hco
For the high frequencies, cu
4?r- \7 d
C ) and low (hcu
7d,c) frequencies.
C, we obtain
\ 7d 7d
J
7c
V 7c 7c 7c
JJ
(3.74)
where the dimensionless Q —functions are given by
Q d(x,y)
=
Qc(x, y, z)
=
°r
o
2r exp ( —r f l 4- 2x sin 2 C/ 2)) dC
J d r F (yr) J
1 + 2xfAn* u 2
2^
(3J5)
o
o
°r
o
2F
exp ( —r ( l + 2x sin 2 C /2))
dC
j d r F (yr) J i + ^ + sin2(c /2 + z r f 2)) ^ (3-76)
Let us now consider the dependence of the functions Qd,c and Q c ( x ,y ,z) from
Eqs. (3.74) and (3.75) on tem perature y. For the limit of high tem perature, y^> 1,
we obtain
7T2
1
Q c( x , y, z) « Qd(x, y) « — ~ j = = = .
(3.77)
The equality between functions Qc and Qd means the magnetic fieldsymmetry of
the
conductance [52].Indeed, using Eqs.(3.74) and (3.77) we observe, th at
R ( $ u $ 2) = f l ( * i , - * 2).
(3-78)
However, in low tem perature lim it y <C 1, we obtain for x ^ 1
“
Qc(%, 0, z)
57S-
ss
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<3 7 9 >
(3.80)
79
0.25
0.2
0.15
0.1
0.05
Temperature, y=v 1T
Figure 3.11: Functions Qc{x,y,z) and Qd(x ,y) computed for x =
1
and z = 10. As
temperature y = T f 7 increases, function Qc(x -y, z) approaches frequency independent
function Qd(x,y).
Comparison of Eqs.(3.79), (3.79) and (3.74) reveals an im portant fingerprint of
the dephasing by the external radiation — violation of the Onsager relation
R($ u - Q 2)
R (& 1, $ 2 )
where
7
=
7 d($i,
$ 2) = 7c($i,
- $ 2 ),
27
(3.81)
Co’
provided that 7c( $ i, $ 2) = 7d($i»
- $ 2) »
7-
This breakdown of the Onsager relation is a simple m anifestation of the lifting of
the tim e reversal sym m etry in the system with time dependent Hamiltonian, see
Fig. 3.11.
In the limit of low frequency hoj <C 7d,o the contribution from the Cooperon
and diffuson parts are described by the same function, so th at the conductance
correlation function can be represented in the form
R =
cl
47T2
(3.82)
7d
\7 d
7d/
7c
V 7c
7c,
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80
so the Onsager relation (3.78) holds. Here,
Q ( x , y) = t
Jo
r F 2 ( y r ) eXp(~ (1 + 4 1 Sf j / 2 t 2 j / 2 )T)dr.
4tt2 Jo
^ '
1 + 4a; sin 2 ^ / 2 sin 2 C/2
This expression in the lim it of high tem perature T
7 d,c
(3.83)
v
'
has an asym ptotic behav­
ior
Q (x ,y)= ^-K (-4 x ).
3y
At zero
(3.84)
tem perature Q(x, y ) is given by the expression
q (x , o)
=
+ a + ^ )g (--to )|
7T
(385)
1 + 4x
where fv(a;) and £ ( 2;) are the elliptic integrals of the first and second kind respec­
tively
K(x)
E (x)
=
=
r
^
■/0
v —^sin
J
- -
/•tt/2-- /--------------y l —xs i n 2 ipd(p.
(3.86)
(3.87)
We conclude th at the conductance fluctuations are suppressed by external ra­
diation even in the lim it of the low frequency, see Eq. (3.82). Indeed, during one
period of time, the system goes along a closed loop in the param eter space and the
contribution to the d.c.-conductance is effectively determ ined by the equilibrium
conductance, correspondent to each point of the loop. T he equilibrium conductance
fluctuates along this loop. Thus, the observed d.c.-conductance is already partially
averaged over some realizations of the quantum dot and its fluctuations decrease.
The perturbation strength is related to the length of the contour in the param eter
space and effectively determ ines how many different dot configurations contribute to
the d.c.-conductance. Consequently, the stronger perturbation, over the larger num­
ber of the realizations the d.c.-conductance is averaged and the smaller fluctuations
of the d.c.-conductance.
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81
This should be contrasted with the suppression of the averaged m agnetoresistance, discussed in Sec. 3.3.1, see also [3, 39]. There, the stationary field does not
do anything because the result is already ensemble averaged. In order to suppress
the average quantum correction, the field should have change on the tim e scale of
the order of 1/ 7eSc, where 7esC = 5iNQh/2% is the escape rate from the dot. T h at is
why the effect of the low-frequency radiation on conductance fluctuations and weak
localization corrections are significantly different.
At high frequency hui
7 esc
the d.c.-conductance no longer can be represented in terms of the statio n ary con­
ductance and the suppression of both the conductance fluctuations and the weak
localization correction to the conductivity can be interpreted as dephasing.
3.4
Rectification of a.c. Bias
Although some of the properties of the pum ped charge, discussed in Sec. 3.2 were
observed experimentally, the pumped charge behavior with respect to the applied
magnetic field contradicts theoretical results. Specifically, experim entally measured
voltage t^(<£), generated by pumps exhibit the m agnetic field sym m etry Vr(<h) =
V {—<£), where
is the magnetic flux through the dot, while the theory predicts
th a t the generated voltage V'(<&) for two opposite orientations of the m agnetic field
is uncorrelated: Q($>)Q{—$) -* 0 as $ »
§o/s/gZx- Also>
m agnitude of the
mesoscopic fluctuations of the generated voltage decreases by a factor of two after
the magnetic field is turned on, while the theory predicts no difference between zero
and finite m agnetic fields.
Brouwer [37] proposed th a t the observed d.c. voltage in experim ents of ref. [35]
can be explained as a result of rectification of a.c.-bias applied across the dot.
In this section we present calculations of the transm itted charge through the dot
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82
for one period of external perturbation [53]. We assume th at the a.c.-bias across
the dot is V ( t ) =
£>{t), where cp(t) = cos ust, and th at the dot is subjected to
the external perturbation with the same frequency, but shifted by phase <p, and
m agnitude of the perturbation is Co, defined by Eq. (3.10). For the experimental
realization suggested in ref. [37], <j>=
tt/
2 and the transm itted charge vanishes for
a single pump experim ent. Somehow, our calculations can be applied to the two
pum p experiment after some simple modifications.
For the transm itted charge Qx for one period Tp due to the a.c.-bias V (t ) is given
by
F(t\ —t 2 )dtdtidt2.
Qr = G0f V(t)
The first term in the right hand side of Eq. (3.88) vanishes.
(3.88)
T he second term
represents the quantum correction to the conductance. Taking the ensemble average
Q r, we find th a t the second term corresponds to the weak localization correction
to the conductivity. In the adiabatic limit of slow external field (co <C 7eSc)i this
correction is not modified by the field and is equal to GoN\Nr/ N ^ . Consequently,
Q~r= 0 .
Next, we consider the variance of the transm itted charge Q2:
Q2
=
x
J dtdt'dt]_dt\dt 2 dt'2 V (t) V {t')F{ti — t 2 )F{t'l — t'2)
tr |«S(t, t i) k S ^ { t 2, t)A} tr |<S(t',
(3.89)
t')A}-
We notice th at the second line of Eq. (3.89) can be represented by th e same diagrams
which were studied in Sec. 3.3.2 in connection with the mesoscopic conductance
fluctuations. Using Eq. (3.73), we immediately obtain
Q2 = GoJ
dtdt’V(t) V ( t1)
d r F 2 (r)
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(3.90)
Equation (3.90) is sim ilar to Eq. (3.82) in Sec. 3.3.2 for the low frequency limit.
For one param eter perturbation of the dot shifted by the phase difference 0 with
respect to the a.c. bias V(t) = Vac cos(uit + (f>) we obtain
=
where Z(C q,T,
7)
l “ r VacCOS2 0
\
z
(Co, T , 7d) +
L7d
\
z
(Co, T \ 7c)
(3.91)
7c
is an analogue of the Q function in Eq. (3.82):
r
=
s^ )2
(3 -92)
exp ( - r ( l + 4 C 0/ 7 Sin2 £/2 sin2 C/2))
1
+ 4Co/ 7 sin 2 <f/ 2 sin 2 C/2 )
Equation (3.92) can be evaluated for low and high tem perature limits. As tem­
perature goes to zero, T
Z(Cq,T =
0,7)
7,
=
Z ( C q, T ) saturates to
4 tt ■1
± AC o / y ) K ( - 4 C o p f ) - E ( - 4 C 0 / j )
1 + 4C0/ 7
^ g3)
As tem perature increases, Z{C q, T ) decreases according to
Z (C 0, T =
0, 7 )
=
47r^
——1
~ Ei-ACo/jr)'
(3
g4)
First, we assume th a t the am plitude of a.c. bias is given by V-^ = w trcK ) and Co oc
1/2
Vq2, where t r c Is a constant with dimension of tim e. Using Z(Co,T) oc C 0
at strong pum ping, we conclude, th a t Q* oc Vo In Vq.
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In Co
84
Brouwer proposed [37]. th a t in experiments of ref. [35] th e a.c. bias is generated
by capacitance coupling between wires connected to th e gates and the electron
reservoirs. We denote the a.c. voltage at the gates as Vs (t). T hen, the generated a.c.
sources across the quantum dot, V'(£) = T^cdVg(t)/dt, where -rRC = R C characterizes
coupling between the gate and reservoir voltages, R is th e resistance of the circuit
p ath , containing the current m eter and C is the m utual capacitance between the
gate and reservoir wires. In the case of a single pump Vg(t) = Vo cos cut, the phase
difference between the d o t’s perturbation and the a.c. bias is 0 = 7r/2, and the
rectified current vanishes.
E quation (3.92) can be easily extended to the case of several pumps with the
sam e frequency but shifted w ith respect to each other by th e phase difference
0.
In summary, the transported charge Q has the following features: (z) For sm all
am plitude of the perturbation, Q 2 oc Vq, while for strong perturbation Q 2 oc Vo In Vo;
(zz) the ensemble average Q =
0;
(Hi) Q2(<F) = Q ( $ ) Q ( —*$); (iv) For an open
quantum dot with large num ber of open channels, Q2($ = 0) = 2Q2(<& = oo). These
properties were observed in experiments of ref. [35].
i
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C h ap ter 4
C urrent N o ise
In the previous chapter we discussed the properties of the d.c.-current through open
quantum dots. We treated current averaged with respect to quantum mechanical
and therm odynam ic fluctuations and studied statistical properties of the current
w ith respect to different realizations of the quantum dots. O n the other hand, the
current through a quantum dot fluctuates due to therm al and quantum nature of
the system. In this chapter we consider these fluctuations of the current through an
open quantum dot at zero bias across the dot.
The therm al fluctuations of electron occupation num bers n(e) are the source of
current fluctuations at zero bias across the dot. T he therm odynam ic average of the
electron occupation num ber is the Fermi distribution, (n(e)) = /(e ). The occupation
num ber n(e) itself is either zero or one and, consequently, n 2(e) = n(e). The la tte r
equality allows us to determ ine the fluctuations of electron occupation number
((«(«) - (n(£) ) f ) = /(«)(1 - /(£ )).
(4.1)
T his result dem onstrates th a t at zero tem perature therm al fluctuations in a
fermion system disappear. At finite tem perature th e fluctuations of the electron
85
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86
occupation num ber lead to the equilibrium current fluctuations known as NyquistJohnson noise [59, 60]. The relation between the occupation number fluctuations
and the equilibrium current fluctuations is described by the fluctuation-dissipation
theorem. T he only system specific param eter for the relation is the conductance of
the system. We conclude th at the statistical properties of the equilibrium current
noise due to therm al fluctuations w ith respect to different realization of the quantum
dot are described by the statistical properties of the conductance.
We will consider the influence of external time dependent perturbation of an
open quantum dot on the current fluctuations. [54] We assum e that the bias across
the dot is zero and two leads connected to the dot are a t the same tem perature.
We may expect th a t the external field changes the electron occupation number in
the dot and produces correction to the Nyquist-Johnson noise. We consider namely
this correction in this chapter.
The interest to the problem appeared after the photovoltaic effect in open quan­
tum dots was extensively studied in the literature [16, 55, 48, 49, 17]. The current
noise due to the external tim e dependent perturbation was first studied by Levitov
and Ivanov [56]. Recently the current noise for the experim ental setup for adiabatic
quantum pum p through open quantum dots was studied by Kamenev and Andreev,
[57] and Levitov [58]. References [57, 58] demonstrate th a t the external perturba­
tion produces current fluctuations as well as the adiabatic pum ping (low frequency
photovoltaic effect). In both papers the zero tem perature lim it was considered, and
the current noise through the dot is completely determ ined by the contribution due
to the external perturbation.
Our prim ary goal is to study statistical properties of the current noise with
respect to different realizations of the quantum dots. We also consider arbitrary
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87
tem perature of electron system.
The physical interpretation of the noise due to the external perturbation is as
follows. The external field modifies the electron occupation numbers in the dot.
As a result, the occupation number fluctuations are different from those given by
Eq. (4.1), and change the current fluctuations. Indeed, we know th at the external
perturbation broadens the electron distribution function in the dot. Characteristic
energy scale, which the external field affects, is determ ined by heating tem perature
Th, see Eq. (3.52). We find th a t at finite tem perature and weak external field, the
contribution to the noise due to the external field is represented as an expansion
in
T h /T ,
and can be treated as a result of broadening of the electron distribution
function in the dot.
4.1
Current Noise Correlation Function
The current correlation function Q 2 represents fluctuations of the charge transported
through the dot over the observation time interval
T0
Q 2 = f T° { ( m m ) - ( I ( t ) ) </(£')» dtdt'.
Jo
(4.2)
In Appendix B.5 we derived the following equation for the charge fluctuation
correlation function:
Q2
=
J
dtdt' J dtidt 2 dt[dt '2
x
tr | ( ^ ( £ 2, £)A«S(£, t[) —AS(t 2 - t)S(t —£ 'j) f(t[ — t2)
x
( S \ t ' 2, t ') k S ( t ', ti) - A8 {t '2 - £> (£' - £1)) ( i St u t 2 - f ( h - £2))} •
(4.3)
Here / Q(g(£) = 5Q^ / Q(£), and / Q(£) is the time representation of the Fermi distribution
function of electrons in channel a:; 5tit>=
8
(t — £').
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88
In this chapter we consider the case of zero voltage across the dot f a (t) = /(£)We represent th e variance of current fluctuations in the form:
0.2 =
Qn + Q'2t
(4-4)
where the second term Q 2 is chosen in such a way, th a t in the absence of any time
dependent perturbation Q 2 =
0.
In this stationary case the current fluctuations are
known as N yquist-Johnson noise and are a consequence of the thermal fluctuations
of electron occupation numbers in the reservoirs across the dot.
The first term has the same structure as the Nyquist-Johnson current-current
correlation function:
= 2e2 J
x
dtdt'
J
dtidt 2 f ( t x —t ' ) f ( t ' —t2)
(4-5)
tr \ k 2 8t,tl5tte - AS(£, £l)A«St (£2,£)} •
Here we introduced the following notation /(£) = 5{t) — /(£)The Nyquist-Johnson contribution to the current is due to the therm al fluctua­
tions of the electron occupation numbers, and the fluctuation - dissipation theorem
can be applied to calculate the current fluctuations, as we have already mentioned
at the beginning of this Chapter. Equation (4.5) supports this statem ent. Indeed,
the second line in Eq. (4.5) is the conductance of the dot [compare to Eq. (3.59)] and
the product of two electron distribution functions /(£) determines the fluctuations
of the electron occupation numbers [see Eq. (4.1)].
We present Qn for a stationary system (no time dependent field). Using the
following property of the Fermi distribution function, / ( e ) /( e ) = —T /'(e ) , we obtain
Qn = 2Tg,
(4-6)
where g is the conductance through the dot. An im p o rtan t observation is th a t Qn
vanishes a t zero tem perature.
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89
The external field changes the conductance of the dot, see C hapter 3. Conse­
quently, we can expect th a t the Nyquist-Johnson contribution to the current noise
wall also be modified due to the external field. In particular, the ensemble aver­
age
Q
n
will increase (at zero magnetic field), since the weak localization correction
will be suppressed by the oscillating field. Similarly, fluctuations of Q n with re­
spect to different ensemble realizations of the dot will be suppressed, because the
conductance fluctuations are suppressed by external field.
For the equilibrium between left and right leads the current through the dot
is determ ined by internal processes of the dot, see Sec. 3.2. The external radiation
modifies the electron distribution function inside the dot, producing non-equilibrium
between electrons in the dot and in the leads. Due to this non-equilibrium, a pho­
tovoltaic current flows through the dot. A lthough the photovoltaic current has a
random sign and its ensemble average is zero, its is finite for each p articu lar real­
ization of the dot, as it was shown in Sec. 3.2. We consider the photovoltaic current
correlator, represented by the second term of the right hand side of Eq. (4.4):
Q2 =
x
e2
J
dtdt'
J
dt]_dt2 dt,l dt 2 f ( t i ~ t 2 ) f ( t \ — t'2)
(4.7)
tr { S ( t 2, t)AS*(t, ti) S ( t 2, t')AS^(t', t[) — A 25t>
2'tSt'tl St2,t,tit',t\ } ,
This term of the current correlator was studied recently, see [57, 58] in th e adiabatic
limit, when the frequency of the external noise and system tem perature are much
smaller th a n the escape rate 7 ^ . In this lim it th e scattering m atrix is instantaneous
S ( t , t1) = S{t)S(t — £') (the width of the S function is the electron escape time
T’esc =
7 “c
Q 2 (p )
from the dot) and Eq. (4.7) reduces to (see [57, 58]):
= e2 j " pTp dtdt't r {A2 - & ( t ) A S ( t ) S H t ’) A S (t') } f { t - t') f ( t ' - t ).
(4.8)
Papers [57, 58] considered zero tem perature lim it, when the electron distribution
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90
function in tim e representation does not vanish fast. Since the system is observed
for a finite tim e T0 = iVpTp, only discrete energies are involved to the observable
processes. (Here Tp is the period of th e external field oscillations and N p is the
number of periods, during which the system is observed.) As a result of the “quan­
tization” w ith frequency step ui/Np, a t zero tem perature the distribution function
takes the form /( f ) =
1/T 0 sm^cuf/A/p).
In this regime, the contribution to Q2 comes
from distant m oments of tim e f and t'.
In the energy representation the distribution function is sharp, and we have to
treat each energy value and take into account discreteness of the energies in the
system. At high tem perature, the distribution function is sm ooth in the energy
representation and the discreteness of the energy states is not im portant.
The lower tem perature limit of Eq. (4.8) means th at the frequency u of the
external field is much larger than the tem perature of the system . On the other
hand, Eq. (4.8) is derived for the adiabatic limit, when the frequency
uj
is smaller
than the m ean levels spacing of the dot. Indeed, if the characteristic energy scale
in Eq. (4.7) is larger th an the mean level spacing c^, the energy dependence of the
scattering m atrix should be involved, which leads to finite delay tim e t — t1 for the
scattering m atrix <S(f, f'). Consequently, we have the following inequality for the
tem perature T <C ui <C Si. This inequality makes Eq. (4.8) unapplicable for the
present m easurem ents on open quantum dots [35].
At higher tem perature, T ~ w, but still T <C 7esC5 so th at Eq. (4.8) is applicable,
we use Eq. (4 .8 ) and the distribution function /(f ) = T / sin h 7rT f, see Chapter 3.
We can neglect the discreteness of energies as Np —> oo.
It is w orth mentioning, that as tem perature increases, the contribution to the
noise correlation function Q2 decreases, since only close moments of time f and f',
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91
such th a t |£—£'] ~
1/ T ,
contribute to the noise. On th e other hand, for slow external
field with frequency c j < T we have S(t)S^(t') &
1
and the num erator of Eq. (4.8)
vanishes.
We have form ulated the problem in terms of th e scattering m atrix approach.
Following Sec. 3.2 we can depart from the scattering m atrix approach and rewrite
the correlation function of the photovoltaic current in terms of the electron Hamil­
tonian in the dot. Sim ilarly to the transformation of Eq. (3.20) to Eq. (3.30), using
Appendix B.2, we obtain from Eq. (4.7):
Q2 =
qP
+
qP
,
(4.9)
where we introduced two terms
Q2l) =
-27rive2
dtdt'
J
d txdt2 { /(ri - t2) f { t ' - t) + f ( t - £')/(£i — t2) }
- <p(*2))tr
f2"o
Q2 } =
-
47r2u2e2
,
r+ oo
(4.10)
_
/ dtdt' /
dtidt2dt[dt2f ( t i — t 2)f(t[ - t’2)
J0
J —00
- v ( t 2))(ip{t\) ~ <p(t2))
(4.11)
x tr {AW t & R)(£, £1) V & A)(£2, £') W A .W * & R)(£', t[) V & A)(£(,, t) W ) .
Here we explicitly substituted the time dependent p a rt of the Ham iltonian in the
form V<p(t).
Equations (4.10) and (4.11) have a similar structure to the structure of Eq. (3.30).
The electron distribution function /(£ —£') is dressed by the external field V ( y ( t ) —
<£>(£')), and thus corresponds to a new distribution function of electrons subjected to
external radiation. The w idth of the electron distribution function increases, as we
already discussed in Sec. 3.2.3. Below we calculate the ensemble averaged Q2 and
find th at the characteristic w idth of the distribution function (tem perature due to
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92
heating) is governed by the same param eter T& = u>yJCo/jesc, which first appeared
in Sec. 3.2.3, see Eq. (3.52).
Since the fluctuations of the photovoltaic current Q2 are due to the heating of
electrons in the dot, Q2 does not vanish at zero tem perature, when the NyquistJohnson term, Q n, is zero.
4.2
Bilinear Response
Following ref. [58], we consider an adiabatic limit, when the frequency of the external
perturbation
7 eSC of
uj
and system tem perature T are much sm aller than the escape rate
electrons from the dot. We calculate the noise of the photovoltaic current to
the lowest order in the external perturbation.
The scattering matrices in Eq. (4.8) can be param eterized by a time dependent
function X ( t ) = X Q(p(t): S ( t ) = S ( X ( t ) ) . To the second order in X 0 we obtain
Q2 = - X l l Z f T° f i t - t')f{t' - t) (<p(t) - cp(t'))2 dtdt',
Jo
(4.12)
where 1Z is a factor, determ ined by a realization of the quantum dot and given by:
(4.13)
■R. = -RSl>
and we introduced
(4.14)
We will discuss the statistical properties of 1Z with respect to ensemble realizations
below. Now we consider tem perature and frequency dependence of the current noise
Q2First, we consider the high frequency limit, so th at
uj/ N q
<§; T , where cu is the
frequency of the harm onic perturbation <p(t) = cos cut . In this case we can neglect
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93
’quantization’ of the energy for our system due to th e finite observation tim e during
N 0 periods of th e external field. Then, the d istrib u tio n function f { t) is given by
Eq. (4.12). T he integral in Eq. (4.12) over t — t' converges fast and we obtain
02
= t (coth I f - f t )
n
(415)
-
Note, th at as tem perature increases, the current correlation function decreases as
co / T for T
cj.
As tem perature goes to zero we have to take into account the discreteness of
energy levels and substitute f ( t ) = (T0 simct/T0) ~ l into Eq. (4.12). Then, after
some algebra we obtain
q
(4.16)
2 = ^ n,
which coincides w ith the low frequency limit of Eq. (4.15).
Now we discuss the statistical properties of the configuration dependent factor 1Z.
We would like to make a connection between the scatterin g m atrix and H am iltonian
presentations of the problem, for this purpose we consider Eqs. (4.10) and (4.11)
in the adiabatic approxim ation and substitute
t') = G(K’A)(X (t))5(t — t'),
where X ( t ) describes the strength of the V(t) p ertu rb a tio n according to V ( t ) =
X {t) v and, again, X (t) = X Q(p{t). Then, to the lowest order in the p ertu rb ation
X Q, both <32^ and Q ^ are bilinear in the interaction. Com paring Eqs. (4.10) and
(4.11) with Eq. (4.13) we conclude th at
?e(1) =
27rii/tr{A2W * & R)v ( G (R) - & A)) v & A)W } ,
(4.17)
n {2) =
- ± K 2v H v { k W ' G {R)vG{A)W k W ^ G {R)vGtA)w ) .
(4.18)
The charge fluctuations Q 2 are again given by Eq. (4.13).
We calculate the average value and variance of the 1Z factor with respect to
different realizations of quantum dots. For simplicity, we consider only a u n itary
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94
R
A
(a )
\ 'A.
(b)
Figure 4.1: Diagrams, which contribute to 7Z to the lowest order in 1/A^h with Nch S> 1
in a bilinear regime.
ensemble.
First we notice that the ensemble average of TZ^1^
are given by the
diagrams in Fig. 4.1. Figure 4.1a represents the ensemble averaged value of TZ^
and Fig. 4.1b contains TZ^. Their sum is given by
n = 2^ ^ ^ - ( i _
-N ch
7 esc
\
.
ch
(4.19)
1/
The lowest order term for TZ^ is two orders in l/iVch smaller for dots with iVCh
1
and we can om it it.
Next, we consider the variance of the 1Z factor. After some cumbersome cal­
culations, we find th a t the lowest order in l/iVCh contribution to var 7Z is equal
to:
Comparing the ensemble average and variance of the photovoltaic current noise,
we conclude th a t the fluctuations are smaller by factor 1 / N ^ th an the ensemble
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95
averaged current noise. Consequently, fluctuations of the photovoltaic current with
respect to different realizations of quantum dots can be neglected for dots with a
large number of open channels N ch
4.3
1.
Ensemble Averaged Noise
In this section we discuss statistical properties of the current correlation function
with respect to various realizations of quantum dots. We calculate the ensemble
average value of the current correlation function Q 2 for arbitrary am plitude Co of
the perturbation.
We use Eq. (4.9) to calculate the ensemble averaged value of the current corre­
lation function. Diagram s representing the ensemble average are shown in Fig. 4.2.
The corresponding analytical expression is
q
2
=
f d td t'f 2(t —
Nc.h
'c h Jo
f
^,ft~\~t?t-\-t'
e2
X [Jo
2
xCjf((p(t -
2
t
)
(4.21)
t')
t\
(t
/
tr t + 1'
V 2
- <p{t' - r ) ) 2(<^(t -
t
ro c
+ 2C0 Jq (ip(t - r) - p t f - r))~V
2
t
1
r',t-t'
- ip(tf -- r ') ) 2
' )) -
( ft +■ t '
2
t -t- t '
'
2
— T . t — t'
The structure of this expression is as following. It contains a product of two
distribution functions of electrons in the dot, which is similar to Nyquist Johnson
noise expression. E quation (4.21) also contains diffusons T> (T, T ' , t
dressed on the electron distribution function
f ( t — t 1) .
— t'),
which are
This structure is similar to
one, considered in Sec. 3.2.3. There, we argued th at an external tim e dependent
perturbation changes the electron distribution function of electrons in the dot, and
called this effect heating.
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96
t- X
V- x
A
(a)
t-
A
X
X
t ’-
><
N /"
a '’
R
X
x’
t-
R
r- x
(b)
Figure 4.2: Diagrams, which contribute to the ensemble averaged value of the noise corre­
lation function Q2 to the lowest order in 1/Ach for arbitrary strength of the perturbation.
We limit ourselves to the limit of short escape tim e of electrons from th e dot,
so th a t the escape rate
7 esc
is much bigger than the frequency of the external field.
For the present purpose, the relation between the tem peratute and the escape rate
is irrelevant.
We rewrite Eq. (4.21) for the harm onic perturbation ip(t) = coseut in the form:
Q2 =
e2
N iN r
N ch
f
Jo
.2
/
(4.22)
dtdt' f 2(t —t')
sin
t — t' . 2 t + f
sin a;—-—
oj—-—
8 C0-
t + t'
t —t'
iVch + 4Co sin 2 o j —- — sin 2 oj
2
sin 4 oj
t-t!
sin 4 oj
t -f-1'
+ 16 C 2
•
2
+ 4C0 sim oj
t
~
t '
^ ■
■
2
s ir w
t
+
— —
t
'
Y
J
^
The above equation can be further simplified in the limit of long observation time
T0, when T0T
1. We change integrals over t and t' to new variables (t + t') /2 and
t - t 1. The integral over t —tl converges fast at \t — t'\ ~ T ~ l <C Ta and we perform
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97
the integration from —oo to + 00 . After integration over (t + 1')/2 we obtain:
Tesc
(1 + 4 A
(4.23)
sin 2 U r )3/2
/
'■v__
Tesc
We consider th e strong case of strong perturbation Co
uj
7 esc
at finite frequency
and expand s in x functions to the lowest order in x. We also limit ourselves to
the low tem perature lim it, so th a t T
Th, where
Eq. (3.52). T his inequality allows us to substitute sin h x ss x. After integration
over r we obtain:
(4.24)
Equation (4.24) is the m ain result of this Chapter. It has a form similar to the
expression for the Nyquist noise, see Eq. (4.6): the current noise correlation function
is determined by the conductance of the dot, given by Eq. (3.60), and the electron
tem perature. Due to the heating by a strong perturbation, the electron distribution
function is broadened and the new energy scale for the electron distribution function
is given by Th- E quation (4.24) supports our initial statem ent th at the m ain effect
of the external radiation on the current noise is heating of electrons in the dot.
We conclude th a t fluctuations of electron occupation num bers are the m ain source
for the current noise. Consequently, the current noise due to the external field has
exactly the sam e origin as the Nyquist noise.
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C hapter 5
D iscu ssio n s and C on clusions
5.1
5 .1 .1
Discussions
D e p h a s in g in W ires v s D e p h a sin g in O p en Q u a n tu m
D o ts
In this section we compare the weak localization correction to the conductivity of
long metal wires (Sec. 2.4) and open quantum dots (Sec. 3.3.1). In both cases,
we considered a sim ilar mechanisms of dephasing.
Namely, we assumed th at a
uniform electric field E 0 penetrates the systems and changes electron wave functions,
destroying interference of time-reversed paths. Nevertheless, these two cases are
different by the Thouless energy E?h- For open quantum dots we assumed that
the Thouless energy is the biggest energy scale, on the other hand, the result for
wires is derived for vanishing Thouless energy. The difference can be seen from the
dephasing rates for these two cases.
In open quantum dots, param eter C0 plays a role of the dephasing rate, see
98
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99
Sec. 3.3.1. Here we repeat Eq. (3.7a) for convenience:
(5.1)
The above result can be interpreted as the second order perturbation theory expres­
sion, where the m atrix elements between different diffusion modes are represented
by eEoL, and the energy difference between the modes is E m - The diffusion process
is described by the lowest energy mode, and higher energy diffusion modes appear
as the result of the perturbation by the external field.
Now, we consider the dephasing rate in metal wires, which is given by a , see
Eq. (2.82). To make a connection to Eq. (5.1), we introduce the length of the wire
L, so th a t the Thouless energy is E Th = D / L 2. This definition allows us to rewrite
Eq. (2.82) in the form:
We can treat the spectrum of the diffusion modes in wires as continuous, since the
energy difference between the modes is determined by the smallest energy scale,
which is the Thouless energy. Then, according to the second order perturbation
theory, we write
u>
(5.3)
where, again, eE0Ld is the m atrix element, and u is the energy difference between
diffusion modes. Next, we notice th a t if the frequency of the external field is larger
than the Thouless energy, the m atrix element is determ ined by the length scale
Ld = \J D/td.
We observe that the dephasing rates in metal wires and in open quantum dots are
significantly different in terms of their dependence on the Thouless energy. In wires,
the dephasing rate is proportional to the Thouless energy, while in open quantum
dots, the dephasing rate is inversely proportional to the Thouless energy.
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100
T he m ain distinction between our theory [39] and the theory of ref. [41] is the
relation between the frequency of the perturbation
In our case,
oj
and the T houless energy £rh-
a;, and this lim it is realized in experiments [4, 9]. The authors
of ref. [41] considered the opposite limit and found a different dependence of the
dephasing rate on system’s param eters, given by Eq. (5.3).
Another disrepancy between m etal wires and open quantum dots is the short
tim e dependence of the interference suppression, determined by the function
0 (cu,
t ),
com pare Eqs. (2.83) and (3.68). In open quantum dots 4>{oj,t) increases faster with
tim e, since the length scale is fixed by the system size L. O n th e other hand, in
wires the length scale itself depends on the time \f~Dt.
5 .1 .2
O b serv a tio n o f D e p h a s in g in O p en Q u a n tu m D o ts
O ur results still contain an unknown param eter Co, defined by Eq. (3.10), char­
acterizing the strength of the perturbation. There is a way, however, to represent
results in a form not depending on this param eter, thus elim inating the need for
additional fitting. Following Ref. [4], we represent the param etric dependence of
the weak localization correction Sgwi versus var g at strong m agnetic field, so th at
the weak localization effects are suppressed. The weak localization correction to
the conductance is given by Eq. (3.65), see [39], and the conductance variance is
determ ined by Eq. (3.73) with j c —> oo for the broken time-reversal symmetry case.
Figure 5.1 shows the param etric dependence for various values of the parameters
C0,
oj
and T = 107eSc-
We observe that the shape of the curves depends on the frequency of the ex­
ternal radiation. Particularly, in the lim it of low frequency,
Tio j
7 esP ,
the weak
localization correction is not changed by the radiation, while the conductance fluc-
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101
0.3
0.25
0.1
0.05
var g/G'
X
10
-3
Figure 5.1: Weak localization correction 5gw\ versus conductance fluctuations var g of an
open quantum dot with. N\ = Nr
1 for three values of frequency u>: hu = 0.5jesc (o);
huj = 57esc (v ); hu = 507esc (A). The temperature for all lines was taken T = 107escThe amplitude of the field
Co
varies from 10“27eSc to 1027eSc-
Go
=
e2/iz h
is the quantum
conductance.
tuations may be significantly suppressed. At high frequency, hut
C0, j esc. the
curves become non-sensitive to the radiation frequency.
The authors of Ref. [4] found that the radiation applied to their device produced
curves in var g vs Sgwi plane identical to the curve produced by increasing temper­
ature of the device for a wide range of frequencies. This observation demonstrates
th a t the radiation produces the heating of electrons and the effect of dephasing
w ithout heating, see Ref. [18], is not observed in experiments [4]. The authors of
ref. [4] suggest th at the main mechanism is the increase of the tem perature in the
dot due to the Joule heat by induced ac source-drain bias.
Although the present d a ta of ref. [4] support the heating mechanism of suppres­
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102
sion of the weak localization correction to the conductance and the conductance
fluctuations, we believe th a t a more detailed analysis is required. According to our
theory, see Eq. (3.59) and the paragraph below Eq. (3.73): (i) mesoscopic fluctua­
tions are sensitive only to the tem perature in the leads, and therefore, the electron
heating in the dot is not responsible for the tem perature dependence of the meso­
scopic fluctuations [if there is a heating, it manifests itself only through the tempera­
ture of the leads]: (ii) high-frequency curves of our theoretical Fig. 5.1 quantitatively
agree with the data in Fig. 3 of Ref. [4], for frequencies / =
1,
10 and 25 GHz. An
exception is the lowest frequency curve ( / = 100 MHz) represented in th at plot, for
the dot w ith ^ = 2Ap,eV, iVch = 2 corresponds to h f j j esc ^ 0.5. According to our
Fig. 5.1 it should have observable deviations from the high frequency curves, which
are not seen. However, taking into account the uncertainty in determ ination of the
level spacing <5i from the geometrical area of the dot, the param eters do not rule
out the microwave dephasing mechanism.
We believe that the “smoking gun” evidence for the mechanism considered in
this thesis is the violation of the Onsager relation [52] in high frequency regime,
huj
7 Cid,
T . The dependence of this violation on the amplitude of the field Co is
the main prediction of our theory.
5 .1 .3
A d ia b a tic C h a rg e P u m p in g a n d a.c. R e c tific a tio n
Now, we will estim ate w hether the experimental results published in ref. [35] corre­
spond to equations, derived in Section 3.2. First, we notice that in the experimental
setup used in ref. [35], d.c. voltage across the dot was measured at zero current
through the dot. To make a connection between our expressions for the pumped
current I and the voltage across the dot V measured in [35], we assume, th a t the
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103
relation between I and V is given by the “O hm ’s law”
I = 9dV,
(5.4)
where gci is the conductance of the dot, given by Eq. (3.60). Equation 5.4 is not
exact and valid only for an open quantum dot with a large number of open channels
iVCh, see ref. [61] for more details.
Measurements in ref. [35] were performed at tem perature T = 330 mK for a dot
with m ean level spacing
= 13 jj.eV (150 mK) and two open channels for each of
the two leads, i.e. T % 7 esc- To make our estim ations we will use the expressions of
Section 3.2 derived for the T = 0 case.
From Fig. 3 in ref. [35] we conclude that the crossover from bilinear pumping to
strong pum ping occurs when the voltage applied to the dot-forming gates is Ac s; 80
mV. As we already discussed in Section 3.2, the crossover happens at Co(.4c) ^
7 esc.
This observation allows us to estim ate the relation between the a.c. voltage across
the gates and the strength of the perturbation C0. According to Eq. (3.41), the
num ber of transm itted electrons nc during one period of the perturbation is of the
order of one, nc «
1.
On the other hand, from Fig. 3 in ref. [35] we find that at the crossover the
am plitude of the d.c. voltage across the dot is V « 5 - 10-7 V, that gives / ^ 4-10_u
A, or n 0bs ~ 24 electrons, transm itted through the dot per cycle. This result shows
th a t the observed current is larger than what we can expect for the current due to
the adiabatic pumping.
The other im portant difference between the adiabatic charge pum ping and the
observed experimental effect is the magnetic field symmetry, see [35] and [49]. These
two facts lead us to conclude th at the effect observed in ref. [35] is unlikely to be
the adiabatic charge pumping. Below, we discuss the other possible source of the
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104
current observed in the experim ent [35]. Namely, following Brouwer, see ref. [37],
we assume th at the a.c. voltage applied to the gates generates current through the
dot:
(5.5)
where
are the voltages applied to the gates, and
2
are the stray capacitances
between the gates and the leads. The measured d.c. voltage is given by
[ Tp R {t)I( t)dt.
Jo
(5.6)
where R(t) = g~l (t) is the d o t’s resistance, which is the inverse conductance g{t).
To estim ate V in Eq. (5.6) at the crossover, we represent g(t) in the form g(t) =
gci + 5g(t) and suppose th a t the characteristic value of 5g(t) ~ Go with G 0 = e2/irh.
Indeed, a t the crossover, the dot goes over various uncorrelated realizations of the
ensemble and the conductance varies significantly. Correspondingly, the oscillating
part S R ( t ) of the resistance R(t) can be approxim ated by G0/ g
Using this as­
sum ption and Eqs. (5.5) and (5.6), we find th at for V\^ ~ 80 mV and V « 5 • 10” '
V (the voltage at the gates and the measured d.c. voltage across the dot at the
crossover), the expected values of the capacitances C \ yi are C 1;2 ~ 10-6 F.
The tem perature dependence, found in ref. [35], also differs from the one, pre­
dicted by the theory, see ref. [49] and Section 3.2. For the mechanism discussed
above, we expect th a t the tem perature dependence of the generated d.c. voltage
across the dot should coincide with the tem perature dependence of the conductance
fluctuations, because the generated d.c. voltage is determ ined by the variation of
the conductance with respect to the applied gate voltages.
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105
5.2
Conclusions
In this thesis we studied the influence of microwave radiation on transport proper­
ties of m etal wires and open quantum dots. Recent weak localization measurements
on wires [2] showed a discrepancy with theoretical calculations th a t considered the
electron-electron interaction semiclassically. We dem onstrated th a t quantum me­
chanical treatm ent of the electron-electron interaction cannot explain this discrep­
ancy, since the quantum corrections to the semiclassical calculations are small in
the regime of weak localization. To understand the observed behavior of the weak
localization correction to the conductivity, other sources of dephasing have to be
taken into account. Possible explanations of the experim ent [2] were presented in
refs. [18, 64].
Furtherm ore, the m easurem ents published in ref.
[9] for open quantum dots
also exhibit low tem perature saturation of the dephasing tim e. Following ref. [18]
we developed a theory, which treats the effect of tim e-dependent perturbation on
transport properties of open quantum dots. We dem onstrated th at both the weak
localization correction and conductance fluctuations are suppressed by external os­
cillating perturbation. We showed th a t a t low frequency, conductance fluctuations
are still suppressed by the external field, while the weak localization correction is
not sensitive to the low frequency perturbation.
The other effect of the tim e-dependent perturbation of open quantum dots is
photovoltaic current th at flows through the dot even a t zero bias across the dot.
We studied the dependence of the photovoltaic current on the frequency and the
intensity of the perturbation. We also showed th at at low tem perature a new energy
scale appears - heating tem perature. This energy scale characterizes the width of
the electron distribution function in the dot due to the perturbation, which produces
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106
electron heating. The heating effect does not change the conductance fluctuations,
which depend on the tem perature of electrons in the leads, but it changes the pho­
tovoltaic current through the dot.
We also calculated the noise of the photovoltaic current through an open quan­
tum dot. We found th a t the oscillating perturbation produces therm al fluctuations
of the photovoltaic current sim ilar to the Nyquist-Johnson fluctuations of the cur­
rent for a system in equilibrium . At a strong perturbation, the current fluctuations
are determ ined by the heating tem perature, which is the same as that appears in
the low tem perature lim it of the photovoltaic effect.
The effects considered in this thesis are difficult to observe and, presently, only
a few relevant experiments have been performed. Perhaps, with the future develop­
ment of nanotechnology and electronics, the effects will be experimentally studied
in greater detail.
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A p p en d ix A
W eak L ocalization E ffect in W ires
A .l
Weak Localization Correction to Conductiv­
ity
We consider here the weak localization correction to conductivity which is given by
a maxim ally crossed diagram, see Fig. 2.1. The first term in Eq.(2.32) does not
contribute to this correction and two other terms do not vanish only if the Keldysh
component stands at the places marked by circles in Fig. 2.1a. Otherwise we get
integrals
/ ^ jG < ''> (€ ,p )G < '41( £ ',P + k ) = / ^ j G < R)(€,p)G <R)(«'.P + k) = 0 .
(A.l)
The G reen’s functions in Eq.(2.24) are also im purity renormalized so th a t
G
( R , A ) ( e
p )
=
------------------
1
—
--------------
,
(A.2)
and e(p) = p 2/2 m — /z is the energy of a free electron with momentum p .
In
equilibrium the Keldysh component satisfies the equation
G W (e , p) = A(e) (G <s>(£, p) - G<A>(€, p )) .
107
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(A.3)
108
where h(e) = ta n h e /2 T.
Using Eq.(A.3) we note th a t the diagram, which contains the Keldysh component
at the position marked by 1 is canceled by a part of the diagram with the Keldysh
component a t position 2.
T here is a similar cancellation of some term s of the
remaining two diagrams.
The result can be represented in the form, shown in
Fig. 2.1b. We will keep the leading term s in u r imp and D k 2r imp, where u and k are
the characteristic energy and m om entum changes due to the fluctuating field.
The corresponding analytical expression for the weak localization correction to
the conductivity is given by
Ac rmt (a/)
=
e2/ ( 1 ^ / ( 1 ^ : p -p '.M (e,€ ',p ,p ')
(A -4>
X
(h(e + ujext / 2 ) — h(e — wex*/2))
^eit
x
C(c' + u //2 , e —^ext/2, e + Wext/2, e — cj /2, p -F p ).
Here
M ( e , e ’, p , p ' )
=
G ('R)(e + u ext/ 2 , p ) G ('A){e - u ext/ 2 ,p )
(A.5)
x
G (/e) (d + w'/2, p')GM)(e' - u ' / 2, p').
(A.6)
is known as a Hikami box, C(ei, e2,
e4, p + p') is the Cooperon, u ' is the current
frequency. Since the Cooperon is singular function of p + p', we can use the equality
p r; —p ' everywhere, except as an argument of th e Cooperon.
Performing the
integration over p and keeping the Cooperon dependence on p + p' = k, we get
,
^ 2 2
f
r im ,
X
J
~
C
( c
'
+
J
j
&
f de
—
!
2,
d
e
-
]
u
„ t/ 2 ,
(h(e + u}ext/ 2 ) - h ( e - u ext/ 2 ))^x ^
---------------------- —
€+
u
„
t/ 2 ,
r '
-
----------------------
u '„ J 2 ,
k).
(A.8)
Further calculations are m ore convenient in the tim e representation. We define
I
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109
the Fourier transform of the Cooperon by the equation:
r,r').
x
(A.9)
Not all four time variables of the Cooperon are independent and we introduce a
new notation
where 2T = t f + t l , and
this function for rji =
771,2
—772
=
= tfa ~~ *1,2- For the present purposes one only needs
77
producing the constraint 5(r]l +
This is the result of integration over e' in Eq.(A.T)
772)
and allowing one tim e integral to be done. Then
one can also complete the integration over e, since the only remaining dependence
on e is in the difference of the electron distribution functions h(e + uiext/ 2 ) and
h(e — oJext/2). After these steps one reaches Eq. (2.33).
A.2
Cooperon Equation
We discuss the derivation of the Cooperon in the presence of classical electric field
V(t, r). It is convenient to work in space-tim e representation. The diagram equation
for the Cooperon is shown in Fig. A .l. The corresponding analytical equation has
the form:
(A. 11)
-t-
J
x
( S + (£+ fj,£+ ,£4 ,r 3,r 4) + £ - ( # , £^t;f,£4 ,r 3,r 4))
d t ^ d t 4 dtz dtAdr3dr4C (0) (£*, t x , tz , f3 , r, r3)
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Figure A .l: The Dyson equation for the Cooperon in the classical electric field V ( t,r).
We introduced notations:
J dt 5J dr 5G{R)(4 - t 5,r 3- T 5)V(t 5,r5)
S +(i3 , * 4 , ^ * 4 , r 3, r 4) =
x G w (t 5 - t+, r 5 - r 4)G(A)(iJ - t ^ , r 3 - r 4)
£ " ( # , # , ^ , * ^ 3, ^ )
= J dt 5 J dr 5 G { R ) (
4
-
r 3 - r 4)G (A)( ^ - t5, r 3 - r 5)
x V{t 5l r 5)G(A)(tT - t - , r 5 - r 4),
Here C'(0)(£+, *7 ,
(A.13)
(A.14)
r, r') is the Cooperon without interaction. In the Fourier
representation it is given by a ladder shown in Fig. A .l:
C
(w,k) — 2-KVTimp 1 - (27r^rt'mp )-l n 0(n;,k)
2'KUTirnp £>k2 - iu ’
^ 15^
where
n„K k)
=
/ ^ j C < R>(e + u/,p + k)G<'4>(e.P)
—-
^TTUTiYYip( 1 “f" ibJT~iTnp
D \c
'^'irrvp) •
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(A. 16)
I ll
Using th e definition of the Fourier transform of the Cooperon, see Eq.(A.9), it
is easy to prove the following relation:
C < » > (tr,(r,iJ,i2- , r , r ')
=
dk. dui
* ± c '° > ( o ;,k ) eik't - r'>
(27-)d 27T*
|
x
+t2
(A.18)
+ t~[ —t j —^2 )
Now consider the self energy of the Cooperon, which is the product of G reen’s
functions given by Eq.(A.13) calculated to the lowest order in uirimp and D q 2r!mp:
= 2iruiTfmpV ( t t , r A)6(tt -
)*(*3
» C ^ r . r 3, r 4) = -2 7 r^r2 mpU ( ^ , r 4)^ 3 +
- t^)S{r 3 - r 4)/A.19)
- t 4- )^ (r 3 - r 4)(A.20)
We see th a t the form of the Cooperon without interaction given by Eq.(A.18)
allows us to find the interacting Cooperon:
C(t^,
, t2
i ri *') =
“b ty ,
—ti ,
—12 , r, r )£(£i -F ^ —^ ~
^2
)• (-'^■-21)
Using Eqs. (A .19) - (A.21) we obtain the integral equation for the Cooperon:
C{T,m,r] 2 ,T, r')
=
*fc,r,r')
+
27Tuirfmp f
x
(V'(T +
j
(A.22)
dr]3 dr 3 C{0) (T, rjv, 77j , r, r 3)C(T, rj3, tj2, r 3, r')
773/ 2 , r 3)
- U (T - t?3/2 , r 3)).
This equation can be considered as a Dyson equation for the Cooperon in the clas­
sical field.
Since the noninteracting Cooperon satisfies
^
.
D^
y
^
r ) = _ j _ SM S{Ih
(A .23)
we get the Shrodinger type equation for the interacting Cooperon:
( |_ _
D V >_
iV(T,
r, r')) C(T,
r) =
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(A-24)
112
where V"(T, 77, r) = V ( T + rj/2,r) —V ( T —77/ 2 , 1:). Note th a t a constant electric field
does not influence the Cooperon.
The solution to Eq.(A.24) can be represented in path integral form, given by
Eq.(2.35).
A .3
Quantum Correction to Dephasing
To the lowest order in the interaction with the quantum field, the conductivity cor­
rection can be represented in the form of two terms, see Eq. (2.1). In the present
paper we consider only the dephasing term . The dephasing correction to the con­
ductivity can be represented in term s of the retarded and advanced components of
the electron G reen’s function:
A a depfl =
x
. rdeide2 de2 du; f
i f (27_)3 — (coth
f
f
uj
\
2
in d r 2dr3 ( £ £ r4 M - C* r3 (w))
x
{ G ^ (e, d ) G ^
+
2G[^l(e,e2)G[.^ll(e2 — cu, e i ) G
d"
T
,
J - — + ta n h
2(ei, e2)G ^ 3
e2 —
u i\
J
.
(. - o )
ta n h ^
(e2 - w, e3 - w )G ^ r}(e3, e)
^ 3( e i , e 3 — w ) G ^ ( e 3 , e )
e2) G ^ 3 ( e 2 — u ; , e 3 — c u J G ^ (e3 , e i ) G ^ ( e i , e ) | .
The only difference of this expression from the similar equation in ref. [20] is th a t
now the electron G reen’s functions depend on two energy variables, because the
classical part of the field changes electron energy.
All diagrams which contribute to the dephasing term, see [20], can be classified
according to Fig. 2.2 into four groups.
We perform averaging over im purities for the last term in Eq. (A.25), the pro­
cedure can be repeated for the other terms. The corresponding diagram Fig. 2.2
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
113
can be represented as a com bination of Cooperons and Hikami boxes. One vertex
depends on electron energy e2, scattered by the quantum p a rt of the electric field.
In this case we cannot perform integration over th a t energy, and we have a n oninstantaneous vertex in term s of the Cooperon coupling to the electric field. We
find:
~
JdT]dQ\,dQ2 d r { C ^ ( u j,X i — r 2) — £ (i4)(w,r i —r 2))
^
r deide 2 f
, / u> \
2T
, e2 —uA
,,
x / “4^ " ( COth( 2 r ) " 17 + tanh-2 T " )
x
x
-2- tan h
(X€.\
(A-26)
rj, Ci, r, r i)C(T, Ci, C2, r t , r 2)
2 1
C{T 4- r / 2 , C2 +
t. t
— 77, r 2, r)ela,(<’2~^l)ez(e2_£l)'r .
The conductivity of the system is determined by the average value with respect to
the fluctuations of the
electric field, see Eqs. (2.18)-(2.20), we consider the averaged
value of the right hand
side of Eq.(A.26). Substituting the Cooperons inthe form of
Eq.(2.35), we obtain for the interaction term of the Cooperon action, see Eq. (2.38):
C.
r (f\]
Stnt [ r , m \
__
T
f
J £2
+
f
J
—
d q
~
x (J
+
r
dti J
dti f
1 Ci
dti f
~T}
1 tq(r(tL)-r(t2))
ai J {2 ir)d q2
J-T)
J
dt2(5(ti —t2) —S(ti -+-12 -F 2t ))
dt 2 (S(ti — t2) — S(ti - F 12))
dt 2 (5(ti -
2
t — t2) + S(ti — t2))
J C2
dti [ dt 2 (5(ti
f
fA .9 ?)
-F
t2)
+
S(ti
-F
t2
+ 2 t))
J C2
J
J
We replace S int[ri, r , r (t)\ by 5 int[7?, r — 0, r (t)} and consider the difference Afnt(r7) =
Sintlv,T, r(t)] - Sint[r], t = 0, r(t)] as perturbation. We discuss it in more details.
F irst we perform integration over e3 in Eq.(A.26). We obtain
f ^
e ' “' r i ; U a h w
= - ^ P r -
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(A'28)
114
This function vanishes fast at
and AfiU(rf) for rj ~
tv ,
tT
» 1. Now we can compare S int[r], r = 0, r(i)]
when the interaction w ith the electromagnetic field becomes
im portant [ Sint[v,T = 0 ; r(i)]
1. ] Let us consider the term in the second line
of Eq. (A.26), which contains r . It produces one of three parts of A \nt(r]). In case
of electron-electron interaction, when the m om entum integration is unbounded we
have:
= —
C Ti
[ <2
J-T)
dt (|r(f) - r ( —£)| - |r(£) - r ( - £ - 2 r ) |) .
(A.29)
For the diffusion process with the diffusion constant D, we can evaluate |r(£t ) —
r(£2)|2 = D\ti — t2\- We have
A ';1, = ^ \ / 2 D £
dt (v F T - v / | F m ) ~ A
(A.30)
We conclude th a t
Sint[r],T = 0,r(£)]
~ — = — -— .
Tp
g(Lv)
(A.31)
This result is easy to understand. We are interested in time scales of the order
of dephasing tim e r^. On the other hand the discontinuity of the Cooperon is of the
order of inverse tem perature. Since r^T
1, those discontinuities can be neglected.
So we can neglect the dependence on r of the right hand side of Eq. (A.27), and
perform integration over r . After that integrals over energies e2 and e3 can be done.
The rem aining three diagrams can be sim ilarly calculated. Collecting all dia­
grams together, we get Eq.(2.62).
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115
A.4
Quantum Correction for the Nyquist Noise
Model
The averaged C2(T, 77, r) has the form
C2(T,r),r) = 4 J
d£i J
dQ (cosu;(Ci - C2 ) - coscu(Ci + C2 ))
The integral over i?(£), written in the second line of Eq.(A.32), can be done:
r ° ° d R l / ' K<,)=r/^ P « ( i ) e x p ( - r & ^ - r r )
J—co
JR(0 )=Ri
\
Jo
=
4D J
(A.33)
The integral over the odd part of trajectories x is more complicated. We will
represent the th ird line in the form:
f M=0c l ( t )exp f - r d t i 2 m + 0 V ( t ) \ ei„ <C!)/^ sin n s ( f i ) =
ix(o)=o
\ Jo
4.D
y
v2
/ -roo
r-f oo _
_
~
' / /o
Q
dxt j
dx2C(0,X2, C2 )C(^2 , i i, C i - C2)C(I1,0,I) - Ci)e‘"1" s i n —;=Xi,
“(A.34)
where
C(xu x 2 , 0 =
/
n
/
r CC x2(£)
;
'■
+ Q2 x 2 (t) \
Vx(t) exp I — / dtJx{0)=l2
\ -£0
rx(C ) = n
(
o
4D
V 4tt£> sinh QC eXP ( ~ 4 g m h n c ^
\
(A'35)
+ ^
is the Green’s function of a harmonic oscillator in imaginary time.
The integrals over x x and x 2 are Gaussian and can be done exactly for arbitrary
value of q. In our case q is bounded by qc <C 1/L. It allows us to perform the
expansion in powers of q. Keeping the first non-vanishing term we find
f
^ /(w /2 T )
dr,I(v, W),
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(A.36)
where
« -= r « r
«•«>
1
f
0 .T ]
Q(uj 2 - Q2)
sin2a;77 Q.2
sin 2 0777 cosh ^ 77^
Q.2 + u 2 ^2\/sinhfi77 + 4cj(uj2 4- Q2) y/sinh Qtj
u 2 + Q2
y^sinh3 fir)
J
We perform integral over frequency
uj.
The second and third term s in Eq.(A.36)
converge fast and we can use an approximation for the therm al function at small
values: f ( x ) = —x 2 /3. The first term can be evaluated to the lowest order in Q/T.
As the result we obtain Eq. (2.67).
A.5
Quantum Correction for Electron-Electron In­
teraction
The analysis of A ppendix A.3 can be directly applied to the case of electron-electron
interaction with unbounded momentum integration. It follows th a t for the dephas­
ing term of the weak localization correction to the conductivity we get Eq.(2.62)
with C2(T, 77, r) given by Eq.(2.64), which again can be represented in the form
of Eq.(2.66). Averaging over the fluctuations of the electric field V ( t, r) gives an
expression for C2( T , 77, r), sim ilar to that in Eq.(A.31):
Ci(T, Tj, r) = —2 i J **dQ J
dC, (cosuj(Ci - C2) - cosw(G + C2))
x p d H , ( RM=" ' A V R ( t ) e x p { J - 00
JR(Q)=Ri
\
j0
4
U )
(A.37)
C„,
We have changed variables for the path integral, introducing even and odd parts, as
discussed in Sec 2.2. The integral over the even p art of trajectories is not disturbed
by the interaction and the result of integration is given by Eq.(A.32). The path
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117
integral over x(t) can be rew ritten in the form of Eq.(A.34), where C ( t , x 1, 2:2) is a
solution to:
(A _ DVlt +
This equation can
C ( t , x u x 2) =
6
{t) 6 (Xl - x 2).
(A.39)
be rew ritten in the dimensionless variables, iftim e and space
coordinates are divided by
7 -1
and L ee. After this the solution can be found nu­
merically.
The solution to Eq (A.39) has the following im portant property: a t small time
t <C 7 _l it is very sim ilar to the Cooperon w ithout interaction, but a t tim e scale
greater th a n Tee it is suppressed. To evaluate the right hand side of Eq.(A.38), we
can use Cooperon w ithout interaction, see Eq.(A.15), but introduce the upper cutoff
for tim e integrals at
7 -1 .
More exactly, we introduce exponent weight factor, which
vanishes a t tim e greater than Tee, and substitute C(t, aq, x%) by the expression given
by
C ( u , k ) = -.iuj
- + D k l -b 7
(A'4°)
In this case calculations can be easily completed. The dephasing p a rt of the weak
localization correction to conductivity is given by an analogue to Eq.(A.38):
/ I ; / ( “V2T) j f ” drjlin, u),
(A.41)
where
7 (77, 0; ) =
r+n
r+v-x
J x (t} — x )
/
dx
dy (cosuix —cosuiy) -------------- exp(—2777).
JO
J—T]+X
77
(A.42)
The factor e~Tn produces the upper cutoff of the tim e integral. An analytical ex­
pression for / 0°° I{r})dr) can be found. This expression is cumbersome an d we present
only a term which has the weakest frequency dependence. We would like to remind,
i
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118
that now we consider high frequency contribution ui ~ T , since the low frequency
part is taken into account in term s of the classical field. Thus, we have:
roo
L
tr cos ( | arctan ^
=
~ 2
V ^ ( o ; 2 + 7 2 )3 /4
^
+
• *• ~
( 2 7 W )3 /2 -
S ubstituting this expression into Eq.(A.41), we obtain Eq. (2.72).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
^ A ‘4 3 )
A p p e n d ix B
T im e D ep en d en t R andom M atrix
T h eo ry
B .l
Keldysh Technique for Quantum Dots
We define the wave function of electrons in channel a moving towards the dots by
ipa (t,x) w ith x < 0, where [a:| determines the distance from the dot boundary, see
Fig. B .l. T hen tpa (t,x) for x > 0 represents the outcoming electrons. The boundary
x = 0 is described by a superposition of the incoming and outcoming electron states
and we denote it by yja (t,0). The wave function of electrons in sta te i is denoted
by ipi{t).
We introduce the Keldysh Green’s functions
(B.l)
(B.2)
119
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120
L
A
f
y
1
L
\
n
„
R 1
----
1
R
R
)
/
—
i
\ _____ /
X<0
X>0
incoming
0 outcoming
X>0
x< 0
0
0
Figure B .l: Correspondence between the sign of x and the direction of motion of electrons
with respect to the dot.
which are defined in term s of
- i Q { t - t'){[i;a {t,x),i)l{t',x')}+),
,X,x')
i&(t' - t)([ipa ( t , x ) , i j l { t \ x ' ) } +),
g {ap \ t , t ' , x , x ' )
K
p{t,
x')
where [-, -]± denote com m utator and anticom m utator respectively and ( ...) means
the quantum mechanical and statistical average. Similar expressions can be written
down for Gia (t, t1, x') Green’s function, with ipa ( t , x ) replaced by ipi(t).
We assume th a t electrons do not interact in the reservoirs and the G reen’s func­
tion of the incoming electrons (a;; x' < 0) is given by the Keldysh structure:
(
g Qp(t ,t' ,x,x' )
—
(R}
,
.
.
(t - t ' , x - x')
,n
G % ](t - t ' , x - x ’)
\
=
0
\
(B.3)
G (aV (t - t', x - x') /
where
liR),
Ta0 <
iQ(t)
8
(BA)
ap6 (vFt - x ) ,
G 'fl& x )
- i Q ( - t ) 5apS (vpt —x ) ,
G{af ( e , x)
ha(e) (< % $(€,x) - G % > (e,x))
(B.5)
= i 2 7 r u e ^ h a (e).
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(B-6)
121
Here hQ(e) is the electron distribution function in the channel a , vp is the Fermi
velocity at channels and v is the electron density of states at the Ferm i surface. In
equilibrium at tem perature T
ha{e) = tanh
e - eVn
2T
(B.7)
where Va represents the voltage applied to a lead, containing channel a.
The equations of m otion for the Green’s functions defined by Eqs. (B .l) and
(B.2) have the form:
d
dt
d
Qaf } { t , t \ x , x ') = 6 (x)Waig i0 ( t , t \ x ' ) + S(t - t')5{x — x ') i,
Vf dx
.d
rT
l di ~
gjQ(t,t',x') = wJ0g0Q(t1t',o,x').
(B.8)
(b.9)
We notice th a t from its definition Eq. (B.5), G ^ ( t , tf, 0, x') = 0 for x' < 0
(causality relation). This observation significantly simplifies further calculations.
Indeed, we can represent the Keldysh component of the Green’s function in the left
hand side of Eq.(B.9) in the form
=
r a t ,
J —oc
•a/a
i d /d t — H (t) J
( . ^ t i ) w j 0 ga0 ( t l ,trt oJx ,) t (b .io )
T he corresponding advanced component is zero. Here 1/ { i d / d t — H { t )) is the re­
tarded component of the electron Green’s function in the dot. T his definition is
different from th a t given in the m ain part of the thesis, see Eq.(3.19). The latter
will appear naturally in the end of this derivation. T he additional term ~ WtyV in
Eq. (3.19) takes into account the escape from the dot through the leads.
The next step is to represent Eq.(B.8) as
S i f ( t , t ' , x : x')
=
G'ahP ( t - f , x - x ' ) + J d h d t ^ K t - t u x )
w-i d / d t - H ( t ) W f]J 7,j
t 2 ) g Sl P ^(^2) t , 0 , x )
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(B .ll)
122
In the above equation we take x = 0. Using G ^ ( t — t',0) from Eq.(B.4) and
G a p ( t ~ t ' i x ) from Eq.(B.6) we find
g£\t,fJ0,zf)
=
r')
(B.12)
-1
+
IW
fd tJ l-W
i7ru,- w*
J
i d / ad t -— H (i t )
aS
50
where x ’ < 0.
S ubstituting this expression to E q .(B .ll) and taking x =
—t 0, we obtain
for x' < 0:
g (af ( t , t ' , + \ 6 \,x') = f d
U
S
^
t J
G
^
f i t , (B.13)
where the scattering m atrix S a0 (t,t') is given by Eq.(3.18).
E quation (B.13) is valid for x' < 0. We have to repeat the procedure described
above to calculate the electron Green’s function in the leads for x ’ > 0. Since the
equations which determ ine evolution of the Green’s function from x' < 0 to x' > 0
are complex conjugated to those for x, we conclude, that
S $ \ t , f , +|<J|, +|<S|) =
I
f d t vd t 2S „ ( t ,
( h - «2 , 0 ) 4 , ( i 2, t')
(B .1 4 )
The currents in the left and right leads are given by
m
=
m
=E
1
where 5 —> 0.
(B.lo)
Q=1
*
( 5 < 5 > ( t , t , + l < 5 | , + H ) - 5 < 2 )( f , i . - | ' 5 | , - | < 5 | ) ) ,
(B.1 6)
Q=Ari+l
This limit is just a rem inder th at G ^ { t , t , —\5\,—\5\) istaken for
incoming electrons and is given by Eq.(B.6). Consequently,
e £ , (‘ . ‘, . - | < S | . - | * l )
=
AM
=
G<
£ ’( t - t ' , 0 ) = i 2 K v h ( t - t ' ) ,
f-+0°
, ,duj
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(B .1 7 )
Since th e charge is conserved, I\(t) = —/ r(t). We rewrite the current through
the dot as
m
= N , I '{ t ) ~ ‘v ' -r ( - = m
^vch
= - m -
( b .is )
Substituting Eqs. (B.14) and (B.17) into Eqs. (B.15) and (B.16) and using Eq.(B.18)
we obtain Eq.(3.20).
B.2
Current for Zero Bias
In this appendix we derive Eq.(3.30) for the case of zero voltage between the leads
from general Eq. (3.20). The only assum ption we are using here is that the distri­
bution function of electrons is the same in all
channels, i.e. f a (t) = /(f).
S ubstituting the explicit form of the scattering m atrix from Eq. (3.18),we obtain
S ( t u t 2 ) & ( t 2 ,t[)
=
lS(ti - t 2 )5(t[ - t 2)
+
2mvW* (G (A)( 4 W h - t 2) - G(R)( t i , t 2)5(f; - t'2)) W
+
Aiz2 u 2 W^G^R){tu t 2 ) W W ^ G {A){t'2, t[)W .
(B.19)
Equation of motion (3.19) for the G reen’s function allows us to write:
0
(B.20)
t2
+iKisG(R}(tl , t 2 ) W W f G (A)(t'2 ,t[)
=
S(tl - t 2 )G(A>(t'2 ,t'l ),
i ^ r & R\ t u t 2 ) G ^ \ t ' 2 ,t\)
-
& R)(tu t 2 )H(t' 2 ) G ^ ( t ' 2 ,t[) (B.21)
-iw G W itu t^ W W tG W feX )
=
t i ( t i - t 2)G(R)(t'2,t[).
Now we subtract Eq. (B.21) from Eq. (B.20) and substitute the result into
Eq. (B.19). After simple algebra we find:
5 ( f 1,*2)«5t & , f ,1) =
i<5(t! - t2)5(f'i - *2)
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(B.22)
124
+
2™
-
27rivW G {R) (£i, t2) (H ( t 2) - H(t' 2 j) G (A)(t'2, t[)WK
U L
+ JL \
We use Eq. (B.22) to rewrite expression for the photovoltaic current at zero bias
across the do t given by Eq. (3.20). We notice th at the first term of the right hand
side of Eq. (B.22) vanishes after taking the trace with a traceless m atrix. The second
term is a full derivative with respect to t 2 -F t '2 and also vanishes after integration
with respect to tim e variables. The only remaining third term gives contribution to
the photovoltaic current at zero bias, given by Eq. (3.30).
B.3
Basic Diagram Elements
First we calculate ensemble average G reen’s function. According to Eq. (3.19) the
Green’s function in the energy representation has the form
G R{e) = ------ .---e - H + i-KvWW*
(B.23)
Using the correlation function for the m atrix elements of the H am iltonian H given by
Eq. (3.9) we conclude th at the contribution to the electron G reen’s function comes
from the
diagram s
w ithout intersection of dashed lines, seeFig.
diagrams w ith the intersection of
dashed lines are smaller by at
(B.2).Indeed, the
least one factor of
1/M.
The rem aining diagram s for the ensemble averaged G reen’s function are diagonal
and can be represented in the form of the following equation:
5closed(e) =
5open(e)
=
c- E
( € ) + zO’
(B'24)
e — 12(e) + i M S i / i v ’
( B ' 25)
with permission of the copyright owner. Further reproduction prohibited without permission.
125
I
8+
n
iiz v W + W
»
m'
<3>
4
=
m
n
n
m
n
m
n
(a)
V ____
O'nn,
OL,
r,
(c)
v nm Vmn
V =
%J- + 3 L
(d)
Figure B.2: (a) Diagram s for the ensemble averaged Green’s function. The second
term in the self-energy includes an intersection of dashed lines and is as small as
1/M . (b) The Dyson type equation for the diffuson, V ( t f , ij", i f ,
type equation for the Cooperon, C ( t f , if? ^ f ?£/)•
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)- (c) The Dyson
126
where Gciosed(e) is the ensemble averaged G reen’s function for energy levels without
overlap w ith electron states in the leads, while Gopen(e) is the ensemble averaged
Green’s function for electron states extended to the leads. The self energy is deter­
mined by self-consistency equation:
U/>2 M
£(*) = — r £ & • ( * ) ,
T 1= 1
(B-26)
and Gi(e) = <20pen(e) for 1 < i < iVch and Gi{e) = Gciosed(^) for all other values of
i. Com bining Eqs. (B.24)- (B.26) we obtain an equation for the self energy in the
form
S(e)
=
—
M IS
1
7r e —£(e) + tO
.N chM 2 52
1
t7r2
(e —E(e) + z’0)(e —£(e) -f- i M S i / i r )
(B.27)
The second term of the right hand side of Eq. (B.27) is as small as N Ch/M . First
we solve Eq. (B.27) keeping only the first term of the right hand side and obtain
e - i J l i V P S l / n 2 - e2
£o(e) = ------ "----- 2 ^ ---------- •
(B-28)
then we substitute £o(e) into Eq. (B.27) and find the lowest order correction due to
the second term . As a result we have
R ,A t \
l( 0
i_
=
± 7
Sn
umn
VXM
X+ N ± ± t e
N
< n < M;
(B.29)
^2
1 < n < Ach.
Above we introduced the dimensionless energy e measured in units of 5i/27r. We
expanded these Green’s functions in e / M and N Ch/M , since only those terms survive
the therm odynam ic lim it M
oo. For the same reason, in the expression for Gn(e)
for n < N ch one has to neglect such terms, since their contribution to the final result
is already of the order of N ch/M.
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127
Before we proceed, we would like to notice th a t using S 0(e) for the closed system
(jVch = 0) gives
_2
G cio sed (e) =
( e -
i \ l ± M 2P ji T 2 -
e2)
(B.30)
,
and the density of states
—
/i 2*2
6
rtf) = - h m t r G = I j l is given by the semicircle law [62],
The other elements of the diagram technique used in this thesis are am putated
averages of the product of two Green’s functions, called diffuson and Cooperon and
defined by
(B.32)
x
i ^2 )lamp =
X
5(t± —
4-1^)
4JV/AC ^£^ —t l , t J —t 2 ,
8
~
(B.33)
( t f +ty - q - t ^ )
We can use these relations since the time arguments of the diffuson satisfy
£+ = £- -
and the Cooperon do
=
t t
-
. Introducing new variables
+
£u2 = (£+2 + t ~[ 2 ) / 2 and r = *1 , 2 ~~ *r,2 we obtain the following equations for the
diffuson and The Cooperon:
d
-x— + /Ca(fi, t ) V ( t i , t 2 ,T)
(jt\
Q
2—---- 1- /Cc(t, r j C ( n , T 2 ,t)
on
=
6
(tl - t 2),
(B.34)
=
2S(ti —t 2),
(B.35)
where
(B.36)
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128
7d,c =
p - (N ch + Nd,c) ,
Z'K
( B .3 7 )
(B.38)
and param eters N ^ z describe the effect of the magnetic field, see Eq. (3.9).
The solution to the above equations are
(B.39)
(B.40)
Equations (B.34)-(B.40) are w ritten in dimensionless variables, so th at energy
and time are measured in units of S i / 2 tt and 2w/5i respectively.
The non trivial elements of the diagram technique are unconventional averages U
in the form
and G ^ G ^ A\ which are similar to the diffuson and Cooperon,
defined by Eqs. (B.39) and (B.40) and shown in Fig. B.2.
These type averages
vanishes in th e diagram technique for the bulk disordered m etals due to Eq. (A .l)
but survive in the random m atrix diagram technique. We evaluate these averages to
the lowest order in N Ch / M neglecting the corrections due to the external field. We
are allowed to do so since the averages are already of the order of iVch/M sm aller
than the conventional Cooperon and diffuson.
We have the following equation for U
71
To the leading order in N & /M , X)
= —1/A (compare to Eq. (A .l)), and
we obtain
(B.42)
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129
B.4
Photovoltaic Effect as Pumping in Phase Space
The equation for the G reen’s function in the W igner representation for a tim edependent H am iltonian is
2 e G (e , t)
-
[H
q
- mvW W \
G (e ,
t)}
(B.43)
+
Here G(e,T) is the G reen’s function in the W igner variables e and t , cf. Eq. (3.45).
We represented H (t) in the form H(t) =
H
q
-+- Vip(t).
In the adiabatic lim it only the fc = 0 is taken into account. T his approximation
is crucial in the case of a single pump. By appropriate choice of the beginning of
the cycle the pum p moves for the second half of the cycle along the same trajec­
tory as for the first half, but in the opposite direction. As a consequence, the total
transported charge Q, Eq.(3.44), vanishes in the adiabatic approxim ation. To re­
move this symmetry, we can add another pum p oscillating with the same frequency,
but with different phase shift. Also, the higher order term s in Eq.(B.43) break this
symmetry.
We consider contribution to the lowest order in frequency to the transported
charge Q. For this purpose, we neglect all term s with k > 2, keep the k = 1 term
to the first order and include all orders in k = 0 term. The solution is
6
— Hq — Vif{t) + ITTUWW^
(B.44)
To the lowest order in y(t), the scattering m atrix S has the form
(B.45)
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w here
S 0 (e, <p(t)) = 1 - 27ruWf G0(e, t)W ,
(B.46)
and
A ( e ,t )
B.5
s
||=
Z
- ( G 0 (e ,t)V G l(e ,t)-G l(e ,t)V G 0 (e,t)).
(B.47)
Current Noise Correlation Function
In this Appendix we derive Eq. (4.3) for the current noise correlation function
through a quantum dot.
We will sta rt from the quantum mechanical operator for current through the dot
Ni
li(t)
=
evp
+£) - 0* (£, - 6 )ij;a (t, -<f)) ,
(B.48)
Q=1
/ r(£)
=
evF
( y l ( t , + 6 )Tpa( t , + 6 )
,
(B.49)
q=AT!+ 1
where a denotes channels, connecting the dot with the left (right) lead, vf is the
±<5") is the operator for outcoming (+£) or incoming (—6 )
Fermi velocity and
electrons through channel a.
We consider a slowly oscillating external field, so th at the period of one oscillation
is much longer than the charging time r Rc of the dot. In this limit charge of the dot
is conserved and we have the charge conservation equation: I \ + I r = 0. This equation
allows us to rewrite the current operator in term s of the A m atrix, introduced by
Eq. (3.22).
Substituting the expression for the current operator, Eqs. (B.48) and (B.49), into
Eq. (4.2), we obtain the following expression for the current correlation function
(below, 5 -* +0):
Q 2 = e2vp j
'T
dtdt'{^ tr
|A ^ <(t/, £ ,+ 5 ,+ 5 )A ^>(t, £ ',+ 5 ,+ 5 )}
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(B.50)
131
Here
-
tr
{Ag < { t ' , t , + 6 , - 5 ) k g > ( t , t r, - 5 , + S ) }
-
tr
{A ^< (t/, t , - ^ + 5 ) A ^ >(t; t,,+ ^ -< 5 )}
+
tr
- £ ,-£ ) } ) .
weintroduced the electron Green’s
following definitions (for a review of the
functions in the leads according
to the
Keldysh Green’s function formalism see
[28]):
,x ,x ')
=
g>p{t,t',x,x')
=
(B.51)
-i('ipa (t,x)ipl{t',x')).
(B.52)
The above introduced Green’s functions g <*> can be represented in terms of the
retarded, advanced and Keldysh Green’s functions:
+ g w (t,t',x,x')),
(B.S3)
f , x, x')) . (B.54)
= i ( g ^ ( t , t ' , x , x ‘) + S < B>(M ',a:,x') -
Next step is to represent the Green’s functions as a product of incoming electron
G reen’s functions, Eqs. (B.2)-(B.5), and the scattering m atrix, Eq. (3.18). The
procedure is sim ilar to one, described in Appendix B .l.
We have the following
relations:
& R\ t , t ' , - 8 , + 8 )
g^ (t,t',+ 6,+ 8)
gW (t,t\+5,+6)
= gW {t,e,+6,-6)=Q
(B.55)
= G [R'A)( t - t ' , 0),
(B.56)
= J s ( t , t l ) G iK)( t l - t 2, 0 ) S * ( t 2 , t , ) d h d t 2,
(B.57)
& R'K\ t , t ' , + 5 , - 5 )
=
f S { t i t l )G<R'K){tl - t ' t + 6 ) d t u
(B.58)
& A-K\ t , t ' , - 8 , + 5 )
= j G ^ K\ t - t x, - 8 ) & { t u t ' ) d t x,
(B.59)
where, again,
8
+0.
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132
Now the derivation of Eq. (4.3) reduces to simple algebraic calculations. Indeed
using Eqs. (B.55)-(B.59), we rewrite Eq. (B.50) in terms of the retarded, advanced
and Keldysh com ponents of the Green’s function. We represent these components
of the Green’s function scattered from by the dot as a product of scattering matrices
and the Green’s functions of the incoming electrons, given by Eqs. (B.3)-(B.5). The
resulting expression is Eq. (4.3).
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