Electronic scattering and spin statistics in nanostructures

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Elec

tronic Scatter

ing and Spin Statistics in NanoStructure

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Electronic Scattering and Spin Statistics

in NanoStructures

Proefschrift

ter verkrijging van de graad van doctor aan de Technische Universiteit Delft,

op gezag van de Rector Magnificus prof.dr.ir. J.T. Fokkema, voorzitter van het College voor Promoties,

in het openbaar te verdedigen op dinsdag 30 Mei 2006 om 10.30 uur door

Gabriele CAMPAGNANO

Dottore in Fisica

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Dit proefschrift is goedgekeurd door de promotor: Prof. dr. Yu. V. Nazarov

Samenstelling promotiecommissie:

Rector Magnificus, voorzitter

Prof. dr. Yu. V. Nazarov, Technische Universiteit Delft, promotor Prof. A. Tagliacozzo, Universiteit Napels, Itali¨e

Prof. dr. C. W. J. Beenakker, Universiteit Leiden, Nederland Prof. dr. J. M. Ruitenbeek, Universiteit Leiden, Nederland Prof. dr. ir. T. M. Klapwijk, Technische Universiteit Delft Prof. dr. ir. L. P. Kouwenhoven, Technische Universiteit Delft

Dr. Ya. M. Blanter, Technische Universiteit Delft, adviseur Het onderzoek beschreven in dit proefschrift is financieel ondersteund door de stichting voor Fundamenteel Onderzoek der Materie (FOM).

Published by: Gabriele Campagnano Cover design: Daniele Campagnano

An electronic version of this thesis is available at: http://www.library.tudelft.nl/dissertations/ Casimir Ph.D. Series, Delft-Leiden, 2006-07

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Acknowledgments

I always wonder why I like Rotterdam, after four years in the Netherlands I guess I know the answer. In a way this City looks a bit like me and I feel at home here. If you take a walk in the city you can see many things, many buildings, some are beautiful, some other not. And if they aren’t you can’t always blame who made them. The city has its own soul... With me it is the same, many people tried to ”build” something. Someone succeeded, someone else not. I want to thank all of them.

It was an honor for me to work with Arturo Tagliacozzo, Yuli Nazarov, Yaroslav Blanter and Domenico Giuliano, not only professionally, but also on the human side. Dear Yuli and Yaroslav, after four years in Delft, though in the Netherlands, I hope I absorbed a bit of russian pragmatism. Thank you for helping me reaching this goal in my life. I will be always grateful.

Many persons have been important for me in these years, for sure all the people at the Physics department, old and new friends: Jo¨el, Daniel, Sijmen, Siggi, Wouter, Wataru, Omar, Oleg, Alberto, Silvano, Antonio, Milena, Saverio, Babak, Fabian, Jeroen, Xuhui, Jos, Jens, Alexey, Rashid, Miriam, Inanc, Izak, Moosa, Henry, Vitaly. The order is quasi-random, don’t ask me the logic. Some of you I know better, some other less but thank you all for the good time to-gether. Gerrit and Yvonne thank you for being the ”core” of the group. Special mention goes to my Music friends here in the Netherlands: Joeri, Suze, Benny, Simon, Joep and Pieter, who enjoyed (tolerated?) my trumpet performances. Among old friends a very special thanks goes to my friend, mentor, brother Alioscia Hamma. Without you my life would have been really different! Thank you for all. Again about cities and buildings, it is very difficult to change some-thing or someone when all places are taken. You changed me a lot and some of those buildings are wavering... Talea, voor mij zal je altijd familie zijn.

Dedico questa tesi alle persone che pi´u amo, la mia famiglia, mio padre Gianfranco, mia madre Virginia e mio fratello Daniele. Anche se sono lontano siete sempre con me, vi voglio bene.

Gabriele Campagnano Rotterdam, May 2006

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1. INTRODUCTION

Mesoscopic physics describes systems whose size lies between the microscopic and macroscopic scale. Even though they consist of a large number of atoms these systems are interesting because they exhibit quantum behavior. This hap-pens because electrons can propagate without inelastic collisions over distances comparable to the sample size. To observe such a behavior experiments must be carried out at very low temperature, to prevent thermal energy from washing out quantum coherence.

A typical experiment is the measurement of electrical conductance, but other quantities, like current fluctuations, can also provide important information. Theoretically, many relevant quantities can be calculated from the scattering matrix, which describes electron propagation through the sample. For instance, the sum of the transmission eigenvalues, obtained from the scattering matrix, gives the conductance in units of the conductance quantum GQ. On the other

hand an exact calculation is almost never feasible because of the large number of uncontrollable parameters, i.e. the impurities configuration. Instead, one has to take a statistical approach and calculate average properties.

In this thesis we mostly address the problem by a method developed by Nazarov and collaborators (Circuit Theory) [1]. The method is based on the use of semiclassical Green’s function and allows to calculate the transport prop-erties of compound systems (circuits). The theory is valid when the conductance is large compared to the conductance quantum GQ. Nevertheless it is known

that, even for systems with conductance G GQ, sample to sample

fluctua-tions of the conductance of order GQare present. These are known as universal

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2 1. Introduction

Circuit Theory.

This introductory chapter is organized as follows: In section 1.1 the use of the scattering matrix and its link to the conductance is introduced, other quantities like full counting statistics(FCS) and distribution function (DF) of transmission eigenvalues are defined. In section 1.2 the standard calculation of conductivity and its weak localization correction by the Kubo formula of the linear response theory is presented in order to introduce Diffusion and Cooperon modes which we need to discuss GQ corrections. Section 1.3 is a very short introduction to

the ideas of random matrix theory in quantum transport, finally section 1.4 presents an introduction to the circuit theory method with some examples.

1.1 Scattering Approach to Quantum Transport

The idea to relate the conductance of a mesoscopic system to its scattering properties is due to the pioneer work of Landauer[2], later works gave the theory a more solid ground, generalizing it to the multi-terminal case, and clarifying the role of the contact resistance[3, 4] (see ref.[5] for a review). In this section we report a derivation of the Landauer formula. For simplicity, we consider only the two-terminal case, a mesoscopic sample connected to two reservoirs by ideal leads. In the reservoirs inelastic processes are assumed to take place in order to restore thermal equilibrium. In the sample only elastic processes are considered; in the leads no scattering event occurs at all. Here, the electrons are treated as non-interacting particles; in a real system, on the other hand, electrons are not free but subject to mutual interactions. It would then be more correct to say that we look at the quasi-particles of a Landau Fermi-liquid. This picture is valid as long the temperature is low enough and small voltages are applied to measure the conductance. It is also important that the system is almost homogeneous, otherwise non Fermi-liquid behavior could be relevant. For instance, when the electrons are confined in small regions, i.e. in a Quantum Dot, the large electron-electron interaction plays an important role giving rise to effects like the Coulomb blockade not described in the scattering formalism.

1.1.1 Definition of Scattering Matrix

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1.1. Scattering Approach to Quantum Transport 3

to the left and to the right lead as ψ(xL, yL, zL) = X n 1 √ 2π~vn Φn(yL, zL) h aLneik (n) x xL_{+ b} Lne−ik (n) x xLi_, _(1.1) and ψ(xR, yR, zR) = X m 1 √ 2π~vm Φm(yR, zR) h aRme−ik (m) x xR_{+ b} Rmeik (m) x xRi_. (1.2) The normalization of the wave functions is such that each mode carries an unit current. Here the index n (m) label the modes of the left (right) lead. Let E be the total energy and En the energy of the wave function Φn. The longitudinal

wave vector is then k(n)x =p2m(E − En)/~, real for En < E and imaginary

otherwise. In the first case the mode contributes to the current. In the second case the mode, being evanescent, does not contribute to the current. As a result we have a finite number of transport channels NLand NRwhich depends on the

total energy E. The coefficients aLnand aRmare the amplitudes of the waves

coming from the left and right reservoir, bLn and bRm are the amplitudes of

the waves coming from the sample. These coefficients are not independent but related by linear equations which define the (NL+ NR) × (NL+ NR) scattering

matrix ˆS. We have bαl= X β=L,R X l0 Sαl,βl0a_βl0, β = L, R, l = n, m. (1.3)

It is convenient to explicitly write down the block structure of the scattering matrix as ˆ S = ˆ r ˆt0 ˆ t rˆ0 . (1.4)

The NL× NL matrix ˆr describe the reflection of waves coming from the left,

the NL× NR matrix ˆt describes the transmission from the left to the right. In

the same way the matrices ˆr0 and ˆt0 describe reflection and transmission from the right. Because of flux conservation the scattering matrix is unitary, namely

ˆ SS = ˆ1.

1.1.2 Landauer-B¨

uttiker Formula

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4 1. Introduction

interested in linear conductance the difference between left and right chemical potential is assumed to be small. We proceed to calculate the current through a cross section in the left lead (calculation in the right one would give the same result). Electrons coming from the left lead have wave vector kx> 0 and have

filling factor equal to fL(E). Electrons with kx< 0 either come from the right,

being transmitted by the sample, and have filling factor equal to fR(E), or, are

electrons which were reflected by the sample, so they have filling factor fL(E).

We can write the current as I = 2eX n Z ∞ 0 dkx 2πvx(kx)fL(E) + Z 0 −∞ dkx

2π vx(kx) [Rn(E)fL(E) + (1 − Rn(E))fR(E)] (1.5) = 2eX n Z ∞ 0 dkx

2πvx(kx)(1 − Rn(E)) [fL(E) − fR(E)] .

Rn(E) =Pn0|rnn0|2 is the probability for the electron to be reflected into the

channel n starting from any channel n0. We can use the fact that the scattering matrix is unitary and write

1 − Rn=

X

m

|tmn|2= (ˆtˆt)nn

By changing the integral over the momentum to an integral over the energy we have I =2e h Z ∞ 0 dE Tr_ˆ tˆ_{t [f}_L(E) − fR(E)] . (1.6)

Integration over energy gives the Landauer Formula G = GQ

X

p

Tp(EF). (1.7)

where we have introduced the eigenvalues Tp of the matrix ˆtˆt and quantum of

conductance GQ = 2e2/h, in this case the 2 account for spin degeneracy. For

systems where the spin degeneracy is broken one would take GQ= e2/h with a

double number of transmission eigenvalues.

1.1.3 Electron Counting: FCS

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1.1. Scattering Approach to Quantum Transport 5

Let us give some definitions; let us suppose we want to measure some random quantity, typically the charge Q transmitted in an interval of time ∆t between two reservoirs through a scattering region. If the measure is repeated many times the average value we obtain is a quantity which depends only on the properties of the system under examination. The average charge transmitted is, of course, given by the Landauer formula: the charge transmitted is just the current times the time interval ∆t. Besides the average transmitted charge in the interval we could address the variance (or second moment) and so on. To be more more specific let us introduce the probability PN that after a time ∆t

the number of events (charge counts in our example) gives as result N , then the average value would be

hN i =X N PN, the variance hhN2ii =X N N2PN− ( X N N PN)2,

and similar expression for higher moments of the distribution. It is convenient to introduce the characteristic function of the probability distribution

Λ(χ) =X

N

PNeiχN.

We have that cumulants are given by the compact relation hhNk_{ii = (−i)}k dk

dχkln Λ(χ).

An important property of the characteristic function is that if many indepen-dent types of events contribute to the random process then the characteristic function is the product of the characteristic functions relative to each type of event. From the Ladauer formula we know that the first cumulant is linear in the measurement time. The same is true for higher cumulants. Let us divide the time interval ∆t in two shorter intervals (but still long enough to allow for a sta-tistical description), the events in the first interval are not correlated to those in the second one. We can conclude that the logarithm of the characteristic function for an interval of time ∆t is the sum of the logarithms of the charac-teristic functions associated to each subinterval, therefore all the cumulants are proportional to ∆t.

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6 1. Introduction

the time interval ∆t in very short intervals dt. The probability to transfer one electron in this short time is then Γdt 1, 1 − Γdt is the probability that no electron is transferred. The characteristic function for this short interval is hence

Λdt(χ) =

D

eiχQ/eE= (1 − Γdt) + (Γdt)eiχ.

The characteristic function for the whole interval ∆t is obtained as product of the characteristic function of each small interval since the transfer in each interval are not correlated.

Λ∆t(χ) = (Λdt(χ))∆t/dt= exp Γ∆t(eiχ− 1) = exp ˜N (eiχ− 1)

. (1.8) Here ˜N ≡ Γ∆t is the average number of electrons transferred, ˜N = hQi/e. Taking the inverse Fourier transform, we find for the probability PN for N

particles to be transferred during the time ∆t,

PN = Z 2π 0 dχ 2πΛ(χ)e −iN χ_≈Z 2π 0 dχ 2πe −iχN + ˜N (eiχ₋₁₎ = ˜ NN N ! e − ˜N ∆t_. _(1.9)

Eq. (1.9) is recognized as the Poisson distribution. This situation of uncorre-lated electron transfer occurs in tunnel junctions where all transmission eigen-values are small. In this case, the currents are small implying that the time intervals between successive transfers are large. Therefore it is easy to under-stand why they do not correlate. As second example we consider a Quantum Point Contact.

A QPC is a constriction realized on a very clean 2DEG where the mean free path is much larger then the constriction itself. The constriction is realized by applying negative voltage to electric gates (for an introduction see [8]). It was observed experimentally [9] that the conductance increases as function of the constriction size but not linearly as one would naively imagine but stepwise. The steps are of the size of the conductance quantum. The reason of these steps is the following: the system is very clean so that the channels are either completely open or completely closed. Making the constriction wider allows new transport channels to join the transport, each of them carries a quantum of conductance GQ. The QPC experiment was the key experiment to show that

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fluc-1.1. Scattering Approach to Quantum Transport 7

tuate in time. The distribution is then PN = δ( ˜N − N ) where ˜N is

propor-tional to the number of open channels, note that in this case the time depen-dence is contained in ˜N . The characteristic function in this case simply reads Λ(χ) = exp[iχ(N − ˜N )]. For many transport channels and arbitrary values of the transmission coefficients to compute the characteristic function is not an easy task, it was found by Levitov and Lesovik [10] that the characteristic function is given by the following expression (Levitov formula)

ln Λ(χ) = 2∆t

Z _dE

2π~ X

p

ln1 + Tp eiχ− 1 fL(E) [1 − fR(E)]

+ Tp e−iχ− 1 fR(E) [1 − fL(E)] . (1.10)

1.1.4 Statistics of Transmission Eigenvalues

In the previous section we have seen the the conductance of a mesoscopic system can be expressed by the sum of the transmission eigenvalues Tp, there are

quan-tities, like the FCS, which do not depend just on the sum of the eigenvalues. A simple example is the shot noise, at zero temperature it reads[6]

S = 2GQ|eV |

X

p

Tp(1 − Tp),

where V potential difference applied. We cited these two examples to explain that to fully characterize a mesoscopic system the knowledge of all transmission eigenvalues is needed and not just the sum.

Let us suppose that a mesoscopic system is realized experimentally, its shape is the same for many realization, we call it design of the structure which define the kind of structure, i.e., a Quantum Point Contact, Tunnel Junction, etc. It is natural to assume that some impurity potential is always present. This potential is inherently random, meaning that each realization of the sample has a slightly different impurity distribution. The value of the transmission eigenvalues Tp is a function of the design of the mesoscopic structure and of

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8 1. Introduction

the distribution function is defined as ρ(T ) = * X p δ(T − TP) + . (1.11)

The brackets mean ensemble average over the distribution of the impurity po-tential. If the conductance of the system is large compared to the conductance quantum GQ the transmission eigenvalues are dense. In this case we can

cal-culate the average of any function of the transmission eigenvalues f (T ) as an integral

hf i = Z 1

0

dT f (T ) ρ(T ).

Moreover, if the conductance is large compared to the conductance quantum, the fluctuation of transport properties are small compared to their average value. In this sense we say that self-averaging occurs. As an example of distribution function we consider the case of a disordered wire of lenght L and mean free path l. From Ohm’s law we know that the conductance is proportional to the mean free path l so that one could conclude that the transmission eigenvalues are all of the order of l/L. This result, after the work of Dorokhov[7] was proved to be wrong. The correct result is obtained by looking at the distribution function of transmission eigenvalues, it reads

ρD(T ) =

hGi GQ

1

T√1 − T. (1.12)

The above expression can be obtained in different ways; it is universal in the sense that depends only on the total conductance and on the assumption of diffusive motion of the electrons. From Eq.1.12 we observe that the distribution function is non zero only for values of the transmission eigenvalues close to 0 or to 1 ( channel almost closed or almost open.) This means that the transport is due to few open channels whose number is of the order of L/l.

As second example we consider the distribution function of the QPC. The distribution function for an ideal QPC, i.e., when the channels are either open (T = 1) or closed (T = 0) is very simple

ρB(T ) = N δ(1 − T ) + ∞ δ(T ), (1.13)

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1.2. Quantum Transport 9

1.2 Quantum Transport

In this section we outline the calculation of the electrical conductivity and its correction due to coherent back-scattering for a system of non-interacting elec-trons in a random potential. The geometry considered is that of a cube in d dimension; the size is set to infinity at the end of the calculation. More details can be found in [11, 12, 13, 14, 17] on which this section is based.

1.2.1 Drude Conductivity

The system is described by the following Hamiltonian

H = −~

2

2m∇

2_{+ u(r)}

where u(r) is the random potential assumed to be Gaussian. We can also choose

hu(r)i = 0, (1.14)

hu(r)u(r0)i = wδ(r − r0). (1.15)

The angular brackets denote ensemble average. For this choice of random po-tential random popo-tential we have

hu(r1)u(r2) · · · u(r2k+1)i = 0,

and

hu(r1)u(r2) · · · u(r2k)i

= hu(r1)u(r2)ihu(r3)u(r4)i · · · hu(r2k−1)u(r2k)i

+ all possible parings. (1.16) The advanced and retarded Green’s functions in the energy representation are defined by the following equation

[E ± i0+− H]G(±)_{(r, r}0_{, E) = δ(r − r}0_).

In matrix notation it reads

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10 1. Introduction = + + + + I II III IV +...

Fig. 1.1: Ensemble averaged Green’s function, the thin line represent the free propagator. Diagrams like III are disregarded in the non-crossing ap-proximation.

We proceed now to the calculation of the averaged Green’s function, this is done by standard diagrammatic tecnique [15]. Let G±₀ be the unperturbed Green’s function, i.e. when u = 0, then the exact Green’s function reads

G(±)= G(±)₀ + ∞ X n=1 u G(±) n

By making use of Eq. 1.15 and 1.16 we can average the previous expression, this is done graphically by joining impurity lines (see Fig. 1.1). By introducing the self energy Σ we come to the Dyson equation

hG(±)_{i = G}(±)

0 + G

(±)

0 Σ

(±)_hG(±)_i. _(1.17)

Of course is practically impossible to calculate the averaged Green’s function, we limit ourself to the non crossing approximation[16] which mean to retain only the simplest diagram in the self energy. This approximation is valid for weak disorder potential (1/kFle 1), condition that we assume always satisfied.

Going to momentum representation the approximate self energy reads Σ(±)(k; E) = w

Z _dq

(2π)dG (±)

0 (q − k; E)

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1.2. Quantum Transport 11

in a shift of the energy, the averaged Green’s function reads hG(±)_{(k; )i =} 1

− ξk± i~/2τ

, (1.18)

where τ = 1/2πwν, ν being the density of states. The energy is measured from the Fermi level, = E − EF and ξk= k− EF. Our starting point to calculate

the conductivity is the Kubo formula. The zero temperature d.c conductivity of an hypercube of size L reads[17]

σ(0) = e 2 ~3Ld 2πm2 Z _dk 1 (2π)d dk2 (2π)dk1xk2x D G(+)_F (k1, k2)G (−) F (k1, k2) E , (1.19) where the Green’s functions are evaluated at the Fermi energy. In the simplest approximation the averaged product of advanced and retarded Green’s functions can be approximated by the product of the averages

σ(0) = e 2 ~3 2πm2 Z _dk (2π)dk 2 x D G(+)_F (k)E DG(−)_F (k)E (1.20) Substituting Eq.1.18 in Eq.1.20 we obtain the Drude conductivity

σDrude(0) =

ne2_τ

m ,

where n is the electron density. With this approximation we have disregarded diagrams with impurity lines connecting the advanced and the retarded Green’s functions. Next step would be to include diagrams with vertical impurity lines (see Fig. 1.2), this is know to give no contribution to the conductivity because of the vector nature of the conductivity vertex [15]. The basic unit of these diagrams with one impurity line is called Diffusion.

1.2.2 Weak Localization

In the seminal works [18, 19] it was shown that there is a class of diagrams whose contribution to the conductivity is not negligible (maximally crossed diagrams, see Fig.1.3.) To calculate their contribution we have to evaluate the kernel X. To do that it is sufficient to notice that by inverting the direction of one Green’s function we obtain an uncrossed ladder diagram which can be easily summed up (the elementary unit in this case is called Cooperon.)

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12 1. Introduction

L

k1 k1 k2 k2

L

=

+

+...

Fig. 1.2: Diffusion Diagrams

where K(k1, k2) = w Z _dq (2π)d 1 k1+q− F− iη 1 k2−q− F+ iη (1.22)

with η = πνw. By contour integration one can check that K(k, −k) = 1 making X(k1, k2) divergent for k1+ k2= 0, this is why one has to sum the complete

series of diagrams. Let us introduce the total momentum Q = k1+k2we obtain

for the Cooperon

K(k1, k2) = 1 −~ 2_v2

FQ2

4η2_d + · · ·

Considering only maximally-crossed diagrams and noticing that the relevant contribution comes from small values of Q we obtain the weak localization correction to the conductivity

δσwl= σ(0) − σDrude(0) = −2e 2 π~η 2_wZ dk (2π)d[hG (+) F (k)i] 2_[hG(−) F (−k)i] 2Z dQ (2π)d 1 Q2 (1.23)

The second integral is divergent, we have to introduce wave vector cut-offs Qmax and Qmin. The cut-offs are set to Qmax = 1/le and Qmin = 1/lφ, with

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1.3. Random Matrix Theory 13

X

=

+

+...

X

k1 k1 k2 k2

Fig. 1.3: Inclusion of the back scattering to the conductivity bubble.

account a factor 2 due to the spin degeneracy we obtain

δσ =    −(1/π)(e2 /~)(lφ− le) d = 1, −(1/π2_)(e2 /~) ln(lφ/le) d = 2, −(1/π3_)(e2 /~)(1/le− 1/lφ) d = 3. (1.24)

1.3 Random Matrix Theory

Originally developed as method to investigate the nuclear resonances random matrix theory (RMT) was proved to be later an efficient method to investigate the statical properties of transmission eigenvalues in mesoscopic systems. For a review we refer the reader to ref.[20]. In this section we present some basic ideas of RMT, in particular the definition of statistical ensemble which we will use in chapter 4.

As discussed earlier the scattering matrix of a mesoscopic system satisfies the unitary condition

ˆ S ˆS= ˆ1.

In absence of other symmetries unitarity is the only condition the scattering matrix must obey. In presence of time reversal and spin-rotation symmetry the scattering matrix is also symmetric

ˆ S = ˆST.

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14 1. Introduction

and self dual: ˆS = ˆSR_{. These three cases are usually labeled through an index}

β according to Wigner classification [21]: β = 1 corresponds to a system with both time reversal-simmetry and spin-rotation symmetry, β = 2 corresponds to broken time-reversal symmetry, and β = 4 to a system where time-reversal symmetry is present but spin-rotation symmetry is broken. The basic idea of the RMT method is to assume that the transport properties, in particular the fluctuations around average values, are universal, not related to the specific form of the sample or to the distribution of impurities there present. This means that one can calculate all the quantities of interest assuming that the scattering matrix is distributed in an opportune ensemble fixed only by the symmetries of the system.

1.3.1 Quantum Cavity

A typical setup where the RMT is proved to be successful is the 2D chaotic cavity with attached ideal leads. Chaotic means that the scattering matrix is highly sensitive to the momentum of the incoming electrons or to the shape of the cavity itself. The assumption is that, by changing slightly these quantities, the scattering matrix ˆS explores the complete matrix space available (ergodic hypothesis) with uniform probability. The notion of uniform distribution over the space is related to the invariant measure associated with each symmetry. There are three symmetries classes: Circular Orthogonal, C. Unitary, and C. Symplectic Ensemble for β = 1, 2, 4 respectively.

Le us assume that in each lead there are N transport channels and consider for β = 1, 2 as examples. The average conductance obtained by averaging the transmission elements of the scattering matrix over the appropriate group

h|tab|2i =

Z

tabt∗badµ(S),

where dµ(S) is the invariant measure. Averages of product of ˆS and ˆS∗ matrix elements where calculated by Mello[22] by exploiting the invariant properties of the matrix space measure. For β = 1, 2 on finds

h|tab|2i(β)= 1 2N + 2 − β, h|tab|2|tcd|2i(β)= 2(N + 2 − β)(1 + δacδbd) − δac− δbd 2N (2N + 1)(2N + 7 − 4β) .

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1.3. Random Matrix Theory 15 hGi GQ =N 2 − δ1β N 4N + 2 hhG2_ii G2 Q = N (N + 1) 2 (2N + 1)2_{(2N + 3)} (β = 1) (1.25) = N 2 4(4N2_{− 1)} (β = 2) (1.26)

This result has been later generalized to the case of non-ideally transmitting leads in [23].

1.3.2 Disordered Wire

In the case of a disorder wire the previous scheme is not valid, the electrons moving through the wire do not have enough time to explore ergodically the phase space of the wire giving rise to non vanishing elements of the scattering matrix upon averaging. The scattering matrix approach for such a system ad-dresses the evolution of the transmission eigenvalues of the wire as function of its length. The idea is to study the change of the transmission eigenvalues of a wire of length L when a stripe of length δL is added. The stripe is assumed macroscopic, its size exceed the Fermi wave length λF, but short compared to

the mean free path l. Let assume that there are N transport channels, it is convenient to introduce the transfer matrix ˆM . This 2N × 2N matrix relates the amplitude of the wave function on the left to the amplitude to the right. It is obvious that this matrix can be obtained from the scattering matrix ˆS. The transfer matrix of a compound system is the product of the transfer matrices of its elements (this property is not satisfied by the scattering matrix.) Because of this multiplicativity property the resulting transfer matrix ˆML+δL is simply

ˆ

ML+δL= ˆMLMˆδL.

Assuming that the two pieces are statistically independent the probability to have a transfer matrix ˆM is given in terms of the individual probabilities

pL+δL( ˆM ) =

Z

pL( ˆM ˆMδL−1)pδL( ˆM δL)dµ(MδL) (1.27)

where dµ(M ) is the invariant measure of the group of transfer matrices. The sta-tistical distribution for the building block is assumed on the basis of a maximum-entropy criterion [24], the only constrain being the mean free path l. Let us in-troduce the joint probability density to have the eigenvalues T1, ..., Tn, w

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16 1. Introduction

The integral equation 1.27 can be transformed in a Fokker-Plank differential equation taking the limit δL → 0 and using the ansatz of maximum entropy for the distribution probability pδL one gets (for β = 1, 2)

∂ ∂sw (β) s (T ) = 2 βN + 2 − β N X a=1 T_n2 ∂ ∂Tn (1 − Tn)Jβ(T ) ∂ ∂Tn w(β)s (T ) Jβ(T ) (1.28)

where s = L/l and Jβ(T ) = Qi<j|Ti− Tj|β. The boundary condition for the

solution is w0(T ) = δ(1 − T ). Here we report the result for the conductance and

its variance as calculated form the solution of Eq. 1.28 in the metallic regime (l < L < βN l.) hGi GQ = N 1 + N/l+ β − 2 3β + O(N −1_), _(1.29) hhG2_ii G2 Q = 2 β15+ O(N −1_). _(1.30)

1.4 Circuit Theory Of Quantum Transport

In this section we illustrate how quantities like Distribution Function and FCS can be addressed by a semi-classical theory which we call Circuit Theory of quantum transport. This approach allows to calculate all the quantities of interest of a nanostructure starting from the properties of its components.

1.4.1 From Green’s Function to Usadel Equation

Let us consider a matrix Green’s function, to keep the discussion as general as possible we do not specify precisely the matrix structure (example of such a structure are the Bogoliubov-de Gennes equations1_{.) Let be the energy}

parameter and let us work in the energy representation. We have

[ ˇE − ˆHr] ˇG(r, r0; ) = δ(r − r0), (1.31)

where ˆHr is the one particle operator

ˆ H = −~

2

2m∇

2_{+ U (r)}

1_{For superconductivity we would have ˇ}_{E = η}

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1.4. Circuit Theory Of Quantum Transport 17

We outline here how to get a semiclassical equation for the Green’s function, this allows for some simplification in the calculation of the transport properties. Let us write Eq. 1.31 in its conjugated form

ˇ

G(r, r0; )[ ˇE − ˆHr0] = δ(r − r0) (1.32)

By taking the difference of Eqs 1.31 and 1.32 we obtain _ˇ E, ˇG − ~2 2m(∇ 2 r0− ∇2_r) + U (r) − U (r0) ˇ G(r, r0; ) = 0. (1.33) Le us introduce the Wigner representation of the Green’s function

ˇ G(r, p ; ) = Z dy e−~i(p·y)G(r +ˇ y 2, r − y 2; ) (1.34)

Assuming a smooth dependence on the position of the Green’s function at scale of the Fermi wavelength λF, ∂/∂r kF, and substituting the Wigner

repre-sentation in Eq. 1.33 we obtain i ~ _ˇ E, ˇG + v∂ ˇG ∂r + F ∂ ˇG ∂p = 0, (1.35)

where F = −∂U (r)/∂r. Transport properties are determined by the values of the Green’s function at Fermi energy EF, moreover the Green’s function has a

strong singularity for = ±p2_{/2m. Traditionally to get to a more simple, and}

regular function, the Green’s function is integrated over ξ = p2_{/2m − E}

f, so we

define a energy integrated Green’s function ˇ

G(r, n ; ) = i π

Z

dξ ˇG(r, p ; )

The Green’s function obtained is dimensionless and still satisfies Eq. 1.35. Let us now include the effects of the disorder potential u(r) defined in 1.2.1. The theory we are considering allows for perturbative expansion in terms of the disorderd potential. We can include its effect by adding the impurity self-energy ˇ

Σ to the original hamiltonian. With our choice of disordered potential we have ˇ

Σ = 1

2τh ˇG(n)i

were τ is the averaged scattering time. The equation 1.33 becomes (not consid-ering the potential U (r))

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18 1. Introduction

The previous equation is the Eilenberger equation, traditionally found in su-perconductivity. This equation has an important integral of motion. We have ∇( ˇG(n, r))2_{= 0, calculating the integral at the boundaries, where the Green’s}

function takes the equilibrium values, gives [25, 26] ˇ

G2= ˇ1 (1.37)

This normalization condition is important in practical calculation. Next step is to move to what in superconductivity is called dirty limit. The idea is that due to the disorder potential the Green’s function is almost isotropic, in this case some simplification occurs and a diffusion equation for the Green’s function is obtained. We write the Green’s function as the an ˇG(n) = ˇG+G(1)_{(r, p), where}

the latter term is a small anisotropic term, h ˆG(1)i = 0. With the brackets we indicate the angle average. Because of the normalization condition the matrices

ˇ

G and ˇG(1)anticommute. Averaging over the angle Eq. 1.36 and subtracting it from the original equation, assuming 1/τ ˆE one obtains the Usadel equation [27] ie2_ν ~ [ ˇE, ˇG] +∂ˇj ∂r = 0, ˇ j = −σ(r) ˇG∇ ˇG. (1.38) with σ(r) the conductivity of the system and ν the density of states at Fermi energy. It is important for us that the previous equation defines a current which is almost conserved.

1.4.2 Finite Element Approach

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should not be mistaken as a real current which flows out of the nanostructure, it just describes some decoherence processes, i.e. in super-conducting systems the decoherence between holes and electrons. We need now to give the expression for the matrix current flowing between two nodes. A connector is a scattering region described by a set of transmission eigenvalues {Tc

n}. The matrix current

associated to the current density ˇj between the nodes α and β is given by [28] ˇ Ic = GQ X p Tc p( ˇGαGˇβ− ˇGβGˇα) 2 +Tp 2( ˇGαGˇβ+ ˇGβGˇα− 2) (1.39) The leakage current associated the node α is obtained by spacial integration of the commutator in Eq.1.38, it gives

ˇ Ileak,α=

ie2_νV α

~ [ ˇE, ˇGα] (1.40)

where Vα is the volume of the node α. With these definitions we write the

Kirchhoff-like equation X c ˇ Ic+ X α ˇ Il,α = 0. (1.41)

An example of multi-terminal circuit is illustrated in Fig. 1.4 where ˇG1, ˇG2and

ˇ

G3 are fixed in the reservoirs and ˇG4, ˇG4 are determined by Eq.4.6.

The Kirchhoff rules can be also obtained by introducing a functional of the Green’s matrices and requiring that the solution of the problem is such to minimize this functional upon variation of the Green’s matrix in the nodes. We call this functional action S. The action is presented as sum of contribution from each connector and each node

S =X c Sc( ˇG1c, ˇG2c) + X α Snod( ˇGα) (1.42)

Where connectors and nodes are labeled by latin and greek letters respectively. Variation of ˇG are not arbitrary because of the normalization condition ˇG2_{= 1.}

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20 1. Introduction

Fig. 1.4: Example of multi-terminal structure

1.4.3 Full Counting Statistics

The relations illustrated in the previous section can be used to access the charge transfer through a connector between the reservoirs, the technique is also suit-able for multiterminal structure[29] but in this subsection we confine ourself to the most simple case of a two-terminal system and show how to reproduce Levi-tov formula. For more details we refer the reader to ref.[30]. Let us consider non interacting electron and define the following 2 × 2 Green’s function in Keldish space[31]

(E − ˆH − τ3χ ˆI) ˇG(r, r0) = δ(r − r0) (1.45)

where ˆI is single-particle operator of the full current through a certain cross-section of the connector. Since the current is conserved we can choose the cross section to be in one of the reservoir. The counting field χ can be incorporated in the boundary condition by gauge transform the Green’s function.

ˇ

G(χ) = exp(−iχτ3/2) ˇG(χ = 0) exp(iχτ3/2), (1.46)

ˇ

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the following relation[32]

ln Λ(χ) = −∆t Z _dE

2π~S( ˇGL, ˇGR(χ)) (1.47) The equilibrium Green’s function in the leads are

ˇ GL(R)= 1 − 2fL(R)(E) −2fL(R)(E) −2 + 2fL(R)(E) 2fL(R)(E) − 1 (1.48) Substituting this last expression in Eq.1.47 we obtain Levitov formula.

1.4.4 Examples of Circuit Theories

We have seen that a circuit theory for matrix Green’s functions relying on a solid microscopic calculation can be found. In this subsection we show how to use Circuit Theory (CT) to access the physical information of the nanostructure under exam. As explained previously the circuit rules can be applied to a large class of problems, for instance to calculate the Distribution Function of transmission eigenvalues, Full Counting Statistics and transport properties in superconducting systems. We illustrate in the following how to use CT to access DF and FCS.

Distribution Function of Transmission Eigenvalues

The Green’s function must satisfy the normalization condition ˇG2 = 1 and Tr ˇG = 0, this means that eigenvalues of ˇG can only be ±1. The second condition implies that there are an equal number of positive and negative eigenvalues. We can choose the Green’s matrices to be 2 × 2 matrices of the form

ˇ G(φ) = 0 e−iφ eiφ 0 .

This form satisfies the above normalization conditions. The Green’s matrices so defined depend only on the parameter φ which we call phase. Let assume to have two reservoirs with phase φ1 and φ2. Let {Tn} be the set of transmission

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22 1. Introduction

This last equation can be rewritten introducing the distribution of transmission eigenvalues ρ(T ) I(φ) = GQ Z 1 0 dT ρ(T ) T sin φ 1 − T sin2φ/2. (1.50)

Using the identity Im[x − i0+] = πδ(x) and taking the analytic continuation of the current ρ(T ) = 1 2πGQT √ 1 − TRe I[π − 0 +_{+ 2 cosh}−1 _√1 T] (1.51)

In reference [33] was shown that the current for a disorder conductor is an universal function, it reads

I(φ) = Gφ,

where G is the conductance. Substituting the previous expression in Eq.1.51 we get the distribution function 1.12. Let us suppose we want to study the DF of a two terminal compound system, Eq. 1.51 still holds but now we need to know the expression for the current I(φ). The characteristic I − φ is obtain by dividing the system in its components. Next step is to ascribe phase 0 to the one terminal and φ to the other one. Then in each node the phase takes values θi,

these phases (which parametrize the Green’s matrix) are to be determined by the conservation of the current (in the case of DF no leakage current is needed.) The current for each connector is given by Eq.1.39. There are special circuit elements for which the current take a simple form

• Diffusive conductor I(θ) = GDθ

• Tunnel junction I(θ) = GTsin θ

• QPC I(θ) = 2GBtan(θ/2)

As example we discuss the DF of two quantum point contacts in series. Let GB1 and GB2 the conductance of the first and in the second quantum point

contact. In order to apply circuit theory we have to exclude the possibility that the electrons are transmitted directly from one QPC to the other. In that case indeed the system would behave as a single QPC with GB = min{GB1, GB2}

open channels. Let be θ the value of the phase between the two point contacts, it values is fixed by the current conservation

GB1tan

θ

2 = GB2tan φ − θ

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1.5. This thesis 23

This gives the total current

I(φ) = (G1+ G2) cot(φ/2){

s

1 + 4G1G2 (G1+ G2)2

tan2(φ/2) − 1}. (1.52)

By analytic continuation we get the transmission distribution

ρ(T ) = 2(G1+ G2) πGQT r T − Tc 1 − T at T > Tc; Tc= ( G1− G2 G1+ G2 )2, (1.53) ρ(T ) = 0 otherwise.

1.5 This thesis

In the previous section we have illustrated three different approaches to the prob-lem of transport in nanostructure, in this thesis we mostly address the probprob-lem by the Circuit Theory outlined in section 4.2, the advantage of this method is its applicability to a wide class of problems where the other two methods are either technically too complicated or not applicable. This thesis is organized as follows: In Chapter 2 we derive the distribution function of a Quantum Point Contact in presence of a disorder potential. For one open channel the DF is possonian, for many open channels the circuit theory is applicable and we find that only transmission eigenvalues close to one are allowed. In the intermediate regime we perfom a numerical simulation. In both cases the result is universal, in the sense it depends only on the average reflection eigenvalues hRi. To calculate hRi we perform a quantum-mechanical calculation to treat the weak disorder case, and a classical calculation based on the Boltzmann equation for the general case. In the limit of weak disorder the two calculations give the same result. Chapter 3 is devoted to the study of the Full Counting Statistics (FCS) of non-commuting variables. We consider the concrete case of spin counts in a two terminal sys-tem. The spin detectors are oriented in orthogonal directions. The main result is that by measuring non-commuting variables one should take into account the internal quantum dynamic of the detectors. This last point is important when the decoherence time of the detector is larger then the average time between two counts, making the back action of the detectors relevant to the FCS. We illustrate the difference between a naive application of the projection postulate and a quantum calculation which takes into account the detectors dynamic. In Chapter 4 we extend Circuit Theory to include corrections of order GQ to the

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24 References

in terms of Cooperon and Diffusion ladders of the matrix Green’s function. The method is valid for arbitrary matrix structure of Green’s function. This allows to consider a large class of problems like transport properties of normal nanos-tructures, super-conducting systems, non-equilibrium problems, FCS. We also consider the effect of decoherence due to external magnetic field and spin orbit scattering; this allows us to discuss transition between ideal RMT ensembles. In the last part examples are presented, we consider two by two matrix structure which allow to calculate GQ correction to the distribution function of simple

nanostructures.

References

[1] Yu. V. Nazarov in Handbook of Theoretical and Computational Nanotech-nology, American Scientific Publishers (2006).

[2] R. Landauer, IBM J. Res. Dev. 1, 223 (1957); R. Landauer, Philos. Mag. 21, 863 (1970).

[3] Imry, Y., in Directions in Condensed Matter Physics, edited by G. Grinstein and G. Mazenko (World Scientific, Singapore), p. 101 (1986).

[4] B¨uttiker, M., Phys. Rev. B 57, 1761 (1986);IBM J. Res. Dev. 32, 63 (1988). [5] C. W. J. Beenakker and H. van Houten, Solid State Phys. 44, 1 (1991). [6] For review, see Ya. M. Blanter and M. B¨uttiker, Phys. Rep. 336, 1 (2000). [7] O.N..Dorokhov, Solid State Comm. 51, 381 (1984).

[8] C.W.J. Beenakker and H. van Houten, Physics Today, July 1996, page 22. [9] B. J. van Wees, H. van Houten, C. W. J. Beenakker, J. G. Williamson, L. P. Kouwenhoven, D. van der Marel, and C. T. Foxon, Phys. Rev. Lett. 60, 848 (1988); D. A. Wharam, T. J. Thornton, R. Newbury, M. Pepper, H. Ahmed, J. E. F. Frost, D. G. Hasko, D. C. Peacock, D. A. Ritchie, and G. A. C. Jones, J. Phys. C 21, L209 (1988).

[10] L. S. Levitov, H. W.Lee and G. B.Lesovik, Journ. Math. Phys. 37, 4845 (1996);L. S.Levitov and G. B. Lesovik, JETP Lett.58, 230 (1993).

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References 25

[12] B. L. Altshuler and A.G. Aronov, in: Electron-Electron Interactions in Disordered Systems p.1, eds., A.L. Efros and M.Pollak (North Holland, Amsterdam, 1985).

[13] B. L. Altshuler, A.G. Aronov, D.E Khmelnitskii, and A.I. Larkin, in: Quan-tum Theory of Solids p.130, ed., I.M. Lifshits (Mir Publisher, Moscow, 1982)

[14] S. Datta, Electronic Transport in Mesoscopic Systems, Cambridge Univer-sity Press, Cambridge, 1997.

[15] A.A. Abrikosov, L.P. Gorkov and I.E. Dzyaloshinski, Methods of Quantum Field Theory in Statistical Physics (Dover, 1963).

[16] S. F. Edwards, Phyl. Mag., 3 1020 (1958).

[17] P. A. Mello and N. Kumar, Quantum Transport in Mesoscopic Systems, Oxford University Press, New York, 2004.

[18] L. P. Gor’kov, A. I. Larkin and D. E. Khmel’nitskii, Pis’ma Zh. Eksp. Teor. Fiz. 30, 248 (1979) [JETP Lett. 30, 228 (1979)].

[19] E. Abrahams, P. W. Anderson, D. C. Licciardello, and T. V. Ramakrishnan, Phys. Rev. Lett. 42, 673 (1979).

[20] C. W. J. Beenakker, Rev. Mod. Phys. 69, 731-808 (1997).

[21] E. P. Wigner, in Proc. Canadian Mathematical Congress (Univ. of Toronto Press, Toronto, 1957)

[22] P.A. Mello and T.H.Seligman, Nucl. Phys A 344 (1980) 489; P.A. Mello, J.Phys.A 23 (1990) 4061.

[23] P.W. Brouwer and C.W.J. Beenakker, J. Math. Phys. 37, 4904 (1996). [24] P.A. Mello and J.-L. Pichard, Phys. Rev. B 40, 52765278 (1989). [25] A. I. Larkin and Yu. V. Ovchinninkov, Sov. Phys. JETP 41,960 (1975). [26] A. I. Larkin and Yu. V. Ovchinnikov, Sov. Phys. JETP 46, 155 (1977). [27] K.D. Usadel, Phys. Rev. Lett. 25, 507 (1970).

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26 References

[29] Yu. V. Nazarov and D. A. Bagrets Phys. Rev. Lett. 88, 196801 (2002). [30] W. Belzig in Quantum Noise in Mesoscopic Systems, ed. Yu. V. Nazarov

(2003).

[31] J. Rammer and H. Smith, Rev. Mod. Phys. 58, 323 (1986). [32] Yu. V. Nazarov, cond-mat/9908143.

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2. STATISTICS OF TRANSMISSION

EIGENVALUES FOR A DISORDERED

QUANTUM POINT CONTACT

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28 2. Statistics of Transmission Eigenvalues for a Disordered Quantum Point Contact

2.1 Introduction

A quantum point contact (QPC) is one of the reference systems of meso-scopic physics. The experimental discovery of conductance quantizaton [1] trig-gered further research which contributed much to our modern understanding of nanoscience. QPC is a constriction defined in a 2DEG by gates. The width of the constriction can be changed by the voltage applied to these gates. In the adi-abatic regime, if the distance between the gates changes slowly compared to the wavelength of an electron, the theoretical description is readily obtained [2, 3]. The two-dimensional motion of an electron confined between the gates is equiv-alent to one-dimensional scattering of an electron at a potential barrier. The height of the barrier is different for different transport channels. Semi-classically, the electron is fully transmitted if its energy exceeds the top of the barrier in a given channel, and is fully reflected otherwise. Thus, semi-classical transmis-sion eigenvalues of a QPC are strongly degenerate: One has a finite number of transmission eigenvalues equal to one and an infinite number of transmission eigenvalues equal to zero. This picture would manifest experimentally in the precise quantization of conductance as a function of gate voltage.

In real experiments, this degeneracy is lifted. A brief glance at any of many available experimental studies shows that conductance does not rise in ideal steps. The question whether the transmission eigenvalues are degenerate is also important for a number of other reasons. For instance, if a QPC is prepared in a superconducting material, discrete subgap (Andreev) states develop [4]. These states describe quasiparticles localized around QPC. The number of these states equals the number of transport channels, and their energies are expressed via transmission eigenvalues Tn, En= ∆

q

1 − Tnsin2ϕ/2, with ∆ and ϕ being the

superconducting gap and the phase difference across the QPC. If the transmis-sion eigenvalues are degenerate, the Andreev levels are also degenerate. Thus, any small perturbation would lift this degeneracy and produce a number of states with very close energies. Such a perturbation would then drastically affect properties of the system.

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2.2. Model of QPC with impurities 29

In this chapter, we study how the degeneracy of transmission eigenvalues is lifted by the scattering on impurities, which are always present in and around the QPC. Properties of a disordered QPC have been investigated (see Refs. [5, 6, 7, 8, 9]), mostly in relation to the disorder smearing of conductance steps or evolution of conductance fluctuations in ballistic regime. In contrast to the previous literature, we investigate the case when the conductance of the QPC is only slightly modified by the impurities, or, in other words, the impurity-related splitting of transmission eigenvalues is much less than one. This regime is realized for low concentration of impurities. In this situation we can disregard quantum effects like resonant tunneling through impurity states or Kondo effect. Our main result is that in this regime, reflection amplitudes are Gaussian distributed with zero average and second-order correlation function which does not depend on the channel index. This provides us with a new class of random matrix theory. The results for the distribution function of transmission eigen-values are universal — they only depend on the number of transport channels and on the average reflection eigenvalue. All other information can be extracted from these two parameters.

The chapter is organized in the following way. In Section 2.2 we treat a disordered QPC in the adiabatic approximation. In Section 2.3 we introduce the scattering matrix and show that in the expansion up to the second or-der in disoror-der potential closed channels do not contribute to the properties of transmission eigenvalues of open channels. Section 2.4 finalizes the quantum-mechanical calculation of reflection coefficient and conductance of a disordered QPC. We then turn to the classical (Boltzmann equation) consideration, which facilitates the consideration of the diffusive regime (Section 2.5).

In Section 2.6 we discuss noise properties of disordered QPC. Finally, Section 2.7 is devoted to the distribution function of transmission eigenvalues. For one open transport channel, we calculate this distribution function analytically by performing the disorder averaging directly. In the limit of large number of open channels, we obtain the distribution function by means of the circuit theory [10], which presents a disordered QPC as a pure QPC and a diffusive resistor connected in series. For intermediate numbers of open channels, we perform a numerical simulation based on random matrix theory.

2.2 Model of QPC with impurities

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into account that the transverse profile is not sharp [3]. Since here we employ semi-classical approximation (do not discuss the rounding of conduction steps), the results do not depend on the details of the potential profile. For this reason, we use the simpler model. The Schr¨odinger equation,

−~ 2 2m∇ 2_{+ V (x, y)} ψ(x, y) = Eψ(x, y), (2.1)

is supplemented by the boundary conditions, ψ(x, y = ±d(x)/2) = 0. Here

V (x, y) =X

i

v(x − xi, y − yi),

with v being the single impurity potential, and the sum is taken over impurity positions.

If the width of the constriction d(x) changes smoothly, we can employ the adiabatic approximation and separate the transverse motion,

ψ(x, y) =X

n

φn(x)ϕ(x)n (y).

The transverse wave functions ϕ(x)n (y) that satisfy the boundary conditions are

ϕ(x)_n (y) = s 2 d(x)sin _nπ d(x) y +d(x) 2 .

Substituting this into Eq. (2.1) and disregarding the terms containing the derivatives of d(x), we obtain a one-dimensional equation for the longitudinal wave function, −~ 2 2m d2 dx2 + n(x) − E φn(x) = − X m Vnm(x)φm(x), (2.2)

with the channel-dependent effective potential barrier

n(x) = ~ 2_π2_n2

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2.3. Correlators of the scattering matrix elements 31

and the matrix element of the disorder potential,

Vnm(x) =

Z d(x)/2

−d(x)/2

dy ϕ(x)_n (y)V (x, y)ϕ(x)_m (y).

Eq. (2.2) is the generalization of the equations previously written in Refs. [2, 3] to the case of disordered QPC.

In the semi-classical (WKB) approximation, in the absence of disorder, for each transport channel, electrons with the energies above (below) the top of the barrier are perfectly transmitted (reflected). This approximation breaks down if the energy of an electron coincides with the top of the barrier. Here, we do not consider this case. The wave function of an ideally transmitted electron is

φ(0)_n (x) = s pn(∞) pn(x) exp i ~ Z x pn(z)dz , (2.3)

with the channel-dependent momentum

pn(x) = [2m(E − n(x))]1/2.

2.3 Correlators of the scattering matrix elements

We proceed by introducing the scattering matrix, ˆ S = ˆ r ˆt ˆ tT _r_ˆ0 ,

which is unitary, ˆSS = 1, due to the current conservation requirement. At zeroˆ temperature, conductance of the system is expressed via Landauer formula,

G = GQTr ˆtt = Gˆ Q

X

n

Tn,

where Tn are the eigenvalues of the matrix ˆtt, and Gˆ Q = e2/π~ is the

con-ductance quantum. Without impurities, the matrix ˆtˆt is diagonal, with the elements describing the transmission of an electron in the same open transport channel equal one and all others equal zero. In this case, the conductance is G0= GQN , with N being the number of open transport channels.

To treat the effect of disorder, it is more convenient to investigate the matrix ˆ

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we calculate the correction to the transmission eigenvalue Rn due to disorder.

For this purpose, we consider the perturbation expansion of the reflection matrix ˆ

r up to the second order in the disorder potential, ˆ

r = ˆr(0)+ ˆr(1)+ ˆr(2). Let us now separate open and closed channels,

ˆ r = ˆ roo ˆroc ˆ rco rˆcc ,

where the submatrices rαβ, α, β = o, c describe reflection from/to channels of

different type (o and c stand for open and closed). The reflection eigenvalues are found from the secular equation, det ˆrr − Rˆˆ 1 = 0, or, equivalently,

det ˆ r_oo rôo+ ˆrco rˆco− Rˆ1 rôorôc+ ˆrco rˆcc ˆ r_ocˆroo+ ˆrccrˆco rôcrôc+ ˆrccrˆcc− Rˆ1 = 0. (2.4)

We now expand this equation in powers of the disorder potential. For open channels, the reflection eigenvalues R are expected to be of the second order in disorder. We use now the identity

det A B C D = det(A − BD−1C) det D,

and expand the first determinant taking into account that ˆroc(0)= ˆrco(0)= ˆrcc(0)= 0.

The terms of zeroth order in the disorder potential cancel, since ˆr(0)ccrˆ(0)cc =

1. Terms of the first order do not appear, and in the second order one has det(ˆroo(1)r(1)oo − Rˆ1) = 0. Thus, closed channels have no effect on transmission of

open channels. In the rest of the chapter, we drop the subscript oo and operate only with the quantities related to open channels.

Elements of the matrix ˆr are random quantities. In the next Section, we char-acterize their statistical properties. We show that they are Gaussian distributed with zero average. Thus, it is enough to specify the correlation functions

hˆr∗(1)_ij ˆr_kh(1)i,

where the average is performed with respect to disorder. We show that the only non-zero correlator isD|r(1)_ij |2E_{, which is the probability for an electron coming}

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2.3. Correlators of the scattering matrix elements 33

L

d φ 0 θ b) G G a) QPC D

Fig. 2.1: (a) Layout of a disordered QPC. (b) An equivalent circuit representing a disordered QPC as a clean QPC and a diffusive resistor in series.

the result does not depend on the indices i and j. We then relate this reflection probability with the correction to the conductance.

Thus, the set of eigenvalues Rn describing open channels is obtained by

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2.4 Scattering matrix approach

To calculate the correction to the reflection amplitudes we consider the Green’s functions of Eq. (2.2), −~ 2 2m d2 dx2 + n(x) − E Gn(x, x0) = −δ(x − x0).

Solving this equation, we find the Green’s functions for open channels, Gn(x, x0) = i~ mppn(x)pn(x0) exp " i ~ Z x0 x dz pn(z) # . (2.5)

The formal solution of Eq. (2.2) takes the form φn(x) = φ(0)n (x) + Z dx0Gn(x, x0) X m Vnm(x0)φm(x0), (2.6)

with φ(0)n being the solutions in the absence of disorder (Eq. (2.2) with the zero

right-hand side.) In the first order in V , we obtain φn(x) = φ(0)n (x) + X m Z dx0Gn(x, x0)Vnm(x0)φ(0)m(x 0_). _(2.7)

Substituting Eqs. (2.3) and (2.5) into Eq. (2.7), we find rnm = Z ∞ −∞ dx0 i~ mppn(x0)pm(x0) × exp " i ~ Z x0 dz (pn(z) + pm(z)) # Vnm(x0). (2.8)

For the Gaussian distribution of disorder, the reflection amplitudes rnm are

also Gaussian distributed. This distribution is fully characterized by the pair correlation function. If the impurities are not located in the constriction, the momenta pn(x) and pm(x) in Eq. (2.8) can be replaced by their values taken at

x → ∞, which is pF independently of the channel index. The impurity averaging

is straightforward. As anticipated, all the averages of the type hˆr_ij∗ˆrkhi turn to

zero due to the oscillating behavior, except for the term h|rnm|i2,

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2.5. Boltzmann equation 35

where ˜v(π) is the Fourier transform of the single impurity potential with the momentum transfer 2kF, and niis the concentration of impurities per unit area.

The correction to the conductance reads

hδGi = −GQ N

X

n,m=0

h|rnm|2i,

and in the case of large number of open channels N can be written as hδGi ' − e 2_n i π~3_v2 F L d|˜v(π)| 2_N2_. _(2.10)

For further reference, we identify the average reflection eigenvalue hRi by means of Landauer formula, hδGi = −GQN hRi,

hRi = ni ~2v2F

L d|˜v(π)|

2_N.

The correlation function of the reflection amplitudes can then be expressed via only one parameter hRi,

h|rnm|2i = 1 N hRi 1 + δnm 2 . (2.11)

2.5 Boltzmann equation

In this Section we analyse transport properties of a disordered QPC in the framework of the Boltzmann equation. This approach allows for an extension of the analysis to the diffusive case, when the mean free path becomes much smaller then the length of the system. In this case the second order perturbation expansion breaks down, and the treatment of previous Sections can not be applied any more.

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the cross-section x = 0 (to model the position of QPC). Electrons with higher angles are reflected from this cross-section. The Boltzmann equation reads

v·∇rf (r, p) = I[f ], (2.12)

where f (r, p) is the distribution function of electrons, and I[f ] is the collision integral [11], I[f ] = 2πni ~ Z _d2_p0 (2π~)2[f (r, p 0_{) − f (r, p)]} ×|˜v(p − p0)|2δ ((p0) − (p)) .

Since only electrons with energies betwen EF and EF + eV , with V being

the applied voltage, contribute to the net current, the absolute value of the momentum p is fixed to lie at the Fermi surface. We are only interested in the angular dependence of the distribution function, f (x, α). The Boltzmann equation (2.12) is supplemented by the boundary conditions,

f (x, α) = 1, cos α > 0, x < 0 f (x, α) = 0, cos α < 0, x > L,

which state that electrons coming from the reservoirs are in thermal equi-librium. To take into account reflection of electrons at the QPC, we introduce the further boundary condition,

f (0, α) = f (0, π − α), | cos α| < cos α0.

The distribution function does not depend on the transverse coordinate y, and we thus rewrite Eq. (2.12) as

vFcos α ∂xf (x, α) (2.13)

= nim 2π~2

Z

dα0cos α0|˜v(α − α0)|2[f (x, α) − f (x, α0)] .

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2.5. Boltzmann equation 37 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 d L 0 α1 α2

Fig. 2.2: Disordered QPC modeled classically. Electrons with the angle α lower than the critical angle α0 are transmitted through the cross-section

x = 0, others are reflected from this cross-section.

2.5.1 Non-diffusive regime

For low impurity concentration, we replace the function f (x, α0_{) in the rhs of}

Eq. (2.13) with the distribution function of the pure system. The remaining differential equation is readily solved, yielding for the distribution function at x = 0 and with the direction of propagation | cos α| > cos α0,

f (0, α) =    1, cos α > 0 −α0Lnim|˜v(α)| 2 vFπ~3cos α , cos α < 0 .

Now we calculate the current density, j = (eν(EF)vF/2π)R dα cos αf (0, α),

with ν(EF) being the density of states at the Fermi energy. Only the angles

| cos α| > cos α0 contribute, and we obtain the conductance,

G = e2vFν(EF) 2α0− 2α20 nim 2π~3_v F L|˜v(π)|2 d, (2.14)

where we have taken into account that α0 1. The second term represents the

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of open channels to the critical angle α0, α0 = 2N π/kFd, we find that this

correction is identical to Eq. (2.10), which we obtained from the scattering matrix approach in the limiting case N 1.

2.5.2 Diffusive regime

For high impurity concentration, we introduce the transport relaxation time τ , defined as [11] τ−1= nim 2π~3 Z 2π 0 dα|˜v(α)|2(1 − cos α).

In the linear regime, the collision integral takes the form I[f ] = −(f − ¯f )/τ , where ¯f is the distribution function averaged over the angles,

¯ f = 1 2π Z 2π 0 dαf (α).

Solution of the Boltzmann equation in this case is given by

f (x, α) = πl 2Lα0 + 1 −1 1 − x L + l Lcos α ,

with l = vFτ being the mean free path, l L. Calculating the current, we find

that the conductance is

G = e 2_v Fν(EF)d 2π ₁ 2α0 + L πl −1 . (2.15)

Identifying the conductance of the diffusive region,

GD=

e2vFν(EF)ld

2L ,

and of the clean QPC,

GQP C =

e2_v

Fν(EF)d

2π 2α0= GQN,

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2.6. Noise 39

2.6 Noise

In a two-terminal system, zero-temperature shot noise [12] can be expressed in terms of reflection eigenvalues in the following way,

S = e 3_{|V |} π~ * X n Rn(1 − Rn) + .

In a clean QPC, all transmission eigenvalues in the semi-classical regime are either zero or one, and the current is noiseless. Quantum tunneling away from the conductance steps only brings exponentially small contribution. Thus, we expect a drastic effect of disorder on the shot noise of a QPC. Indeed, up to the second order in the disorder potential, one writes

S = −2e

3_{|V |}

π~ X

n

δRn= −2e|V |δG = 2eGQN |V |hRi. (2.16)

Eq. (2.16) corresponds to the notion of Poissonian stream of reflected particles, which is, indeed, expected in the case of good transmission.

2.7 Distribution function

Now, we turn to the calculation of the distribution function of transmission eigenvalues, ρ(T ) = * X n δ(T − Tn) + ,

where the sum is taken over all open channels, and disorder averaging is per-formed. The distribution function is normalized so that its integral is the num-ber of open channels N . For a clean QPC, ρ(T ) = N δ(1 − T ).

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2.7.1 One open channel

For one open channel, we perform a “brute force” disorder averaging. The reflection probability is R = |r|2_{, where we have suppressed the channel indices.}

In its turn, the reflection amplitude r is related to the potential V by Eq. (2.8), with m = n = 1. It is a complex quantity, and we first calculate the distribution function of the transmission amplitude, defined as

P(r) = hδ (Re(r − r[V ])) δ (Im(r − r[V ]))i . Representing delta-functions as integrals, we write explicitly

P(r) = Z ∞

−∞

dω1dω2

(2π)2

× exp {iω1Re [r[V ] − r] + iω2Im [r[V ] − r]}i . (2.17)

Performing the averaging with the Gaussian distribution ˜P[V ], and introducing the short-hand notation

r[V ] = Z drF (r)V (r), we obtain P(r) = Z ∞ −∞ dω1dω2 (2π)2 exp −1 2 Z

dr1dr2[ω1ReF (r1) + ω2ImF (r1)] hV (r1)V (r2)i [ω1ReF (r2) + ω2ImF (r2)]

−iω1Re r − iω2Im r} .

Now we use Eq. (2.8) and disregard the terms containing rapidly oscillating functions. Calculating the integrals over dω1 and dω2, we finally obtain

P(r) = 1 πh|r|2_iexp − |r| 2 h|r|2_i . (2.18)

Since dR = 2|r|d|r| = π−1d Rer d Imr, we can rewrite Eq. (2.18) in terms of the reflection eigenvalue,

ρ(R) = 1 hRiexp − R hRi ; ρ(T ) = ρ(1 − R). (2.19)

Electronic scattering and spin statistics in nanostructures

Electronic Scattering and Spin Statistics

in NanoStructures

Proefschrift

Gabriele CAMPAGNANO

Acknowledgments

CONTENTS

1. INTRODUCTION

1.1

Scattering Approach to Quantum Transport

1.1.1

Definition of Scattering Matrix

1.1.2

Landauer-B¨

uttiker Formula

1.1.3

Electron Counting: FCS

1.1.4

Statistics of Transmission Eigenvalues

1.2

Quantum Transport

1.2.1

Drude Conductivity

1.2.2

Weak Localization

L

L

=

+

+...

X

=

+

+...

X

1.3

Random Matrix Theory

1.3.1

Quantum Cavity

1.3.2

Disordered Wire

1.4

Circuit Theory Of Quantum Transport

1.4.1

From Green’s Function to Usadel Equation

1.4.2

Finite Element Approach

1.4.3

Full Counting Statistics

1.4.4

Examples of Circuit Theories

1.5

This thesis

References

2. STATISTICS OF TRANSMISSION

EIGENVALUES FOR A DISORDERED

QUANTUM POINT CONTACT

2.1

Introduction

2.2

Model of QPC with impurities

2.3

Correlators of the scattering matrix elements

L

2.4

Scattering matrix approach

2.5

Boltzmann equation

2.5.1

Non-diffusive regime

2.5.2

Diffusive regime

2.6

Noise

2.7

Distribution function

2.7.1

One open channel