PHYSICAL REVIEW B

VOLUME 62, NUMBER 15

15 OCTOBER 2000-I

Andreev conductance of chaotic and integrable quantum dots
A. A. Clerk, P. W. Brouwer, and V. Ambegaokar
Laboratory of Atomic and Solid State Physics, Cornell University, Ithaca, New York 14853
共Received 20 April 2000兲
We examine the voltage V and magnetic field B dependent Andreev conductance of a chaotic quantum dot
coupled via point contacts to a normal metal and a superconductor. In the case where the contact to thesuperconductor dominates, we find that the conductance is consistent with the dot itself behaving as asuperconductor—it appears as though Andreev reflections are occurring locally at the interface between thenormal lead and the dot. This is contrasted with the behavior of an integrable dot, where for a similar strong
coupling to the superconductor no such effect is seen. The voltage dependence of the Andreev conductancethus provides an extremely pronounced quantum signature of the nature of the dot’s classical dynamics. For the
chaotic dot, we also study nonmonotonic reentrance effects that occur in both V and B.

I. INTRODUCTION

Though the actual process of Andreev reflection is simple
to describe—an electron in a normal metal incident on asuperconductor is reflected back as a hole1—it serves as thefundamental basis for some of the most striking effects
known in mesoscopic physics.2 In particular, Andreev reflection may be viewed as the phenomenon underlying the proximity effect, in which a superconductor is able to strongly
influence the properties of a nearby normal metal. Andreev
reflection is also the process responsible for the unique conductance properties of normal-superconducting 共NS兲 junctions, such as the conductance enhancement observed in NS
point contacts.
Attention has recently turned to so-called Andreev billiards as ideal systems in which to study the proximity
effect.3–10 Such systems consist of an isolated normal metal
region, small enough that electrons remain phase coherent
within it 共i.e., a quantum dot兲, coupled weakly through a
point contact to a superconducting electrode. Calculations of
the density of states in such structures have shown, rather
remarkably, that the proximity effect is sensitive to the na˜ in the specture of the classical dynamics—a gap of size ⌬trum is present in the case of a chaotic billiard, whereas in
the integrable case the density of states vanishes linearly at
the Fermi energy.5,7In the present work we also focus on the Andreev billiard,
but now add a second point contact leading to a normal
electrode and investigate the Andreev conductance of theresulting structure. The Andreev conductance is the differential conductance dI/dV at voltages smaller than the superconducting gap in the S electrode, where Andreev reflection
at the interface between the normal metal dot and the superconductor is the only current carrying mechanism. Unlike
previous studies,11,12 here we calculate the full voltage 共V兲
and magnetic field 共B兲 dependence of the conductance.
Two questions are of particular interest in this study.
First, does the sensitivity of the proximity effect to chaotic
versus integrable dynamics, as seen in the density of states,
also manifest itself in the Andreev conductance? We findthat indeed it does. In the case where the contact to the0163-1829/2000/62共15兲/10226共12兲/$15.00

PRB 62superconducting electrode is much wider than that to thenormal electrode, the V and B dependent conductance of achaotic dot is consistent with assuming that the dot has itself
become superconducting. The conductance is doubled withrespect to the normal state, and remains bias independent
˜ in the Andreev billiard.
until eV reaches the induced gap ⌬This is in sharp contrast to what is found in exact calculations for integrable dots. For both rectangular and circular
dot geometries, the conductance drops off linearly with voltage without any plateau near V⫽0. We argue that this difference is ubiquitous for integrable versus chaotic systems.
The second motivation for the present study is the question of reentrance effects—is the behavior of the conductance simply monotonic in V and B? In the case of a diffusive NS junction, it is well known that this is not thecase.13,14 At zero voltage and also at voltages large enough tobreak electron-hole degeneracy, the conductance of the junction is the same as in the normal state. However, a conductance enhancement does occur at intermediate voltages. We
examine reentrance effects in the conductance of chaotic Andreev billiards, both in the case of ballistic contacts and in
the case where both point contacts contain opaque tunnel
barriers.
The remainder of this paper is organized as follows. In
Sec. II, we present our model for the chaotic dot and introduce the technique used to calculate the voltage and fielddependent Andreev conductance. In Sec. III, we discuss results in the case where the contact to the superconductor
dominates the contact to the normal electrode, and contrast
the results for a chaotic billiard with exact quantum mechanical calculations for two integrable systems. In Sec. IV, we
discuss reentrance effects. We conclude with a synopsis of
our key results in Sec. V.

II. MODEL AND TECHNICAL DETAILS
A. Formulation of the problem

We consider a chaotic quantum dot coupled via point contacts to a normal metal and a superconductor having, respectively, N N and N S propagating modes at the Fermi energy E F
共see Fig. 1兲. We assume that the ergodic time is much shorter10 226©2000 The American Physical Society

ANDREEV CONDUCTANCE OF CHAOTIC AND . . .

PRB 6210 227

冉冊

1
2⫺T n ⫺2 冑1⫺T n 1/2
1/2
W mn ⫽ ␦ mn 共 2M ␦ 兲␲Tn⫽ ␦ mnFIG. 1. Schematic drawing of a dot coupled via point contacts toa normal metal and a superconductor.

than other relevant time scales of the dot 共i.e., the inverse
superconducting gap ប/⌬ and the dwell time兲, so that random matrix theory15 共RMT兲 may be used to describe its
transport and spectral properties. In RMT, the Hamiltonianof the dot is represented by an M ⫻M Hermitian matrix H,
which, at zero magnetic field, is real symmetric and a member of the Gaussian orthogonal ensemble:

冉冊

1
P 共 H 兲 ⫽exp ⫺ M ␭ ⫺2 tr H 2 .
4

P 共 H 兲 ⬀exp⫺冉

M 共 1⫹ ␥ 2 兲冊兺 关共 Re H i j 兲 2 ⫹ ␥ ⫺2共 Im H i j 兲 2 兴 .

i, j⫽1

共2.2兲

As ␥ increases from 0 to 1, the distribution evolves from one
with complete time-reversal symmetry 共Gaussian orthogonalensemble兲 to one with no time-reversal symmetry 共Gaussianunitary ensemble兲. The parameter ␥ may be related to themagnetic flux ⌽ through a two-dimensional dot having area
A 17,18
M ␥ 2 ⫽C

冉 冊冑⌽ 2 បvF
,
⌽0
A␦

共2.3兲

where ⌽ 0 ⫽hc/e is the flux quantum, v F is the Fermi velocity, and C is a constant of order unity.
The normal (N N ⫹N S )⫻(N N ⫹N S ) scattering matrix S(␧)
of the system at an energy ␧ above E F can be expressed interms of the matrix H:19
S 共 ␧ 兲 ⫽1⫺2 ␲ iW † 共 ␧⫺H⫺i ␲ WW † 兲 ⫺1 W⫽

冉r NN 共 ␧ 兲t NS 共 ␧ 兲t SN 共 ␧ 兲r SS 共 ␧ 兲

共2.5兲

S⫽

冉

r ee 共 ␧ 兲r eh 共 ␧ 兲r he 共 ␧ 兲r hh 共 ␧ 兲

冊

共2.6兲

.

Here, r he (␧) is the N N ⫻N N matrix describing the Andreev
reflection of an incoming electron in the N point contact toan outgoing hole in the same lead; r eh (␧), r ee (␧), andr hh (␧) are defined analogously. These matrices may be written in terms of the submatrices of the normal scattering matrices S(␧).2 For r he (␧) one has* 共 ⫺␧ 兲 M ee 共 ␧ 兲 ␣ 共 ␧ 兲 t SN 共 ␧ 兲 ,r he 共 ␧ 兲 ⫽t NS

共2.7兲

with

* 共 ⫺␧ 兲兴 ⫺1 ,M ee 共 ␧ 兲 ⫽ 关 1⫺ ␣ 共 ␧ 兲 2 r SS 共 ␧ 兲 r SS
␣ 共 ␧ 兲 ⫽exp关 ⫺i arccos共 ␧/⌬ 兲兴 .

4␭ 2

M

⫻

1/2
␭
Zn
.
␲

The T n are the transmission probabilities for each mode,
which we take simply as T N for modes coupled to the normal
electrode and T S for those coupled to the superconducting
electrode 共this in turn defines Z N and Z S ).
For voltages below the excitation gap ⌬ of the superconductor, electrons and holes incident on the dotsuperconductor interface may be Andreev reflected. In thiscase scattering from the dot, as seen from the normal metal
contact, can be represented by a 2N N ⫻2N N scattering matrix S:

共2.1兲

The matrix size M is sent to infinity at the end of the calculation. The energy scale ␭ is related to the mean level spacing 2 ␦ by ␭⫽2M ␦ / ␲ . More specifically, 2 ␦ is the meanlevel spacing for particlelike excitations in the absence of
coupling to the superconductor; with the superconductor, therelevant excitations are of mixed particle-hole type, and havea level spacing ␦ for excitation energies much larger than the
gap energy.
In the case of a nonzero magnetic field, the Hermitianmatrix H is a member of the Pandey-Mehta distribution:15,16

冉 冊

冊

,

For voltages below the gap ⌬ in the S electrode, the zerotemperature conductance of the system is given by the
Tabikane-Ebisawa formula20
G 共 eV 兲 ⫽4e 2 †
tr r he 共 eV 兲 r he 共 eV 兲 .
h

共2.8兲

We wish to calculate the ensemble-averaged Andreev
conductance of the dot for arbitrary values of voltage V andmagnetic field B; previous studies focused exclusively on thecases where V,B were 0 or large enough to completely breakthe symmetry between electrons and holes. To this end, we
first rewrite Eq. 共2.6兲 in a manner that is formally equivalent
to the normal state expression 共2.4兲. Letting ⍀ S ⫽ ␲ W S W †S
and ⍀ N ⫽ ␲ W N W N† , we define the 2M ⫻2M effective
particle-hole Hamiltonian4,8
H⫽

冉

H

0

0

⫺H *

冊

共2.9兲

,

and the self-energies from the leads
共2.4兲

where W is an M ⫻(N N ⫹N S ) matrix representing the coupling between the point contacts and the dot, having elements⌺ N0 共 ␧ 兲 ⫽⫺i⌺ 0S 共 ␧ 兲 ⫽⫺

⌬

冑⌬ ⫺␧
2

2

冉

冉

⍀N

0

0

⍀N

冊

,

共 ␧/⌬ 兲 ⍀ S⍀S⍀S共 ␧/⌬ 兲 ⍀ S

冊. 共2.10兲

A. A. CLERK, P. W. BROUWER, AND V. AMBEGAOKAR

10 228

FIG. 2. 共a兲 Definition and diagrammatic representation of theunaveraged Green function. This is a 2M ⫻2M matrix; n,m refer tomode indices, while i,i ⬘ are electron-hole indices which can takeone of two values (e or h). 共b兲 The effective Hamiltonian H isrepresented by a cross; the dashed lines indicates averaging. 共c兲
Diagrammatic Dyson equation; the double line is the averaged
Green function 具 G典 . 共d兲 Self-energy to leading order in 1/M ; theneglected higher order terms correspond to graphs with crossed
dashed lines. The intermediate index m is to be summed over. Notethat ⌺ and hence 具 G典 are diagonal in mode space but have offdiagonal terms in particle-hole space.

With these definitions in place, a direct algebraic manipulation shows that the particle-hole scattering matrix S in Eq.
共2.6兲 can be expressed in terms of the effective retarded
Green function GR ,GR 共 ␧ 兲 ⫽ 关 ␧⫺H共 ␧ 兲 ⫺⌺ N0 共 ␧ 兲 ⫺⌺ 0S 共 ␧ 兲兴 ⫺1 ,by
S共 ␧ 兲 ⫽1⫺2 ␲ i

冉

W N†

0

0

W N†

冊 冉
GR 共 ␧ 兲

WN

0

0

WN

共2.11兲

冊.

1/N N ,1/N S , we need only sum diagrams that have no crossed
lines 关lines indicate the averaging of matrix elements of H
using the rule 共2.13兲; see Fig. 2 above兴. Note that ␭ will besent to infinity only at the very end of the calculation; thisprocedure corresponds to first taking the matrix size to infinity while keeping a finite bandwidth, and then independently
taking the bandwidth to infinity. This ordering of limits isnecessary to generate a well-defined perturbative expansion.
The first step in this framework is to calculate the averaged matrix Green function 具 G典 . As in Refs. 5 and 8, we findthe following self-consistent Dyson equation:
共2.14a兲where the self-energy from averaging ⌺ is given by
共2.12兲

B. Details of the calculation: BÄ0

We proceed to average Eq. 共2.8兲 for the Andreev conductance over the Gaussian orthogonal ensemble defined by Eq.
共2.1兲, which is the appropriate ensemble for zero magnetic
field. We make use of the relation
␭.
M

FIG. 3. 共a兲 Direct contribution to the Andreev conductance. The
trace appearing in Eq. 共2.8兲 is represented by the bubble diagram.
The double lines indicate the averaged Green function 具 G典 . The
index n corresponds to a mode coupled to the normal contact andshould be summed over. 共b兲 Diffusion contribution to the Andreev
conductance. 共c兲 Dyson equation for the 4M ⫻4M matrix D. Notethat upper and lower branches need to have matching mode indices,
but not matching particle-hole indices.

具 G共 ␧ 兲 典 ⫽ 关 ␧⫺⌺ N0 共 ␧ 兲 ⫺⌺ 0S 共 ␧ 兲 ⫺⌺ 共 ␧ 兲兴 ⫺1 ,The major simplification offered by Eq. 共2.12兲 comparedto Eq. 共2.6兲 is that it allows one to compute the conductancein a manner analogous to that used in the normal state, as we
now demonstrate. From this point onward, we focus on thecase ⌬Ⰷ␧,eV. In this regime, the properties of the system
become independent of the specific details of the superconductor. Calculations retaining a finite ⌬ will be presented
elsewhere.212

具 H i j H kl 典 ⫽ 共 ␦ ik ␦ jl ⫹ ␦ il ␦ jk 兲

PRB 62共2.13兲

By expressing the scattering matrix in terms of the Greenfunction G, the trace appearing in the conductance formula
may be represented as a standard ‘‘conductance bubble’’ diagram 共see Fig. 3 below兲. Further, at this stage of the calculation ␭ 2 /M is taken to be a small parameter, meaning thatthe usual diagrammatic technique for impurity averaging
may be used. To obtain the leading-order result in⌺⫽1 M 丢

冉

⌺ ee

⌺ eh

⌺ he

⌺ hh

冊

⫽1 M 丢

冉

␭ 2 具 tr Gee 典
M ⫺ 具 tr Ghe 典⫺ 具 tr Geh 典

具 tr Ghh 典

冊.

共2.14b兲

In this equation, 1 M denotes the M ⫻M unit matrix, and 丢
denotes a direct product; it is pictured diagrammatically in
Fig. 2. Having performed the summation of diagrams, we
now let M ⬁, keeping ␧, ␦ ,N S , and N N fixed.
Three of the equations thus obtained relate the components of the self-energy to one another:⌺ ee ⫽⌺ hh ,⌺ eh ⫽⌺ he ,

2
2
⫺⌺ ee
⫽␭ 2 .
⌺ eh

共2.15a兲共2.15b兲

The last equation allows us to parametrize ⌺ in terms of apairing angle ␪ (␧):␭ sin ␪ 共 ␧ 兲 ⫽⫺⌺ eh 共 ␧ 兲 ,共2.16a兲␭ cos ␪ 共 ␧ 兲 ⫽i⌺ ee 共 ␧ 兲 .

共2.16b兲

The remaining self-energy equations now take the formtan关 ␪ 共 ␧ 兲兴 ⫽N SQ S,N N Q N ⫺i ␲ ␧/2␦共2.17a兲PRB 62

ANDREEV CONDUCTANCE OF CHAOTIC AND . . .

Q N⫽

TN
兵 2⫺T N 关 1⫺cos ␪ 共 ␧ 兲兴 其 ⫺1 ,2

共2.17b兲

TS
兵 2⫺T S 关 1⫹sin ␪ 共 ␧ 兲兴 其 ⫺1 .
2

共2.17c兲

Q S⫽

To obtain a unique solution of the self-energy equations wehave imposed the boundary condition ⌺ ee (␧)⫺i␭ as␧⬁; this represents the physical condition that we recovera normal metal with a constant density of states at large
energies. The pairing angle ␪ can is related the density ofstates of the dot:

␳ 共 ␧ 兲 ⫽⫺Im
Re兵 cos关 ␪ 共 ␧ 兲兴 其tr 具 G共 ␧⫹0 ⫹ 兲 典 ⫽; 共2.18兲␲␦␪ ⫽ ␲ /2 corresponds to a fully superconducting state, while␪ ⫽0 corresponds to the normal state. It is interesting to notethe analogy between ␪ (␧) and the pairing angle used in thequasiclassical theory of dirty normal-superconducting
interfaces22 and in the circuit theory of Andreev
conductance.23 In the present case, the normalization condition 共2.15b兲 does not need to be externally imposed, but is adirect consequence of the averaging procedure.
While the leading-order solution of the self-energy ⌺ issufficient if one is interested only in the density of states 共asin Ref. 8兲, the conductance calculation requires that N S /M ,N N /M , and ␧/M ␦ corrections to Eq. 共2.15b兲 be calculated.
This correction may be expressed in terms of the leadingorder self-energy solution:
2
2
⫺⌺ ee
⌺ eh

␭

2

⫺1⫽⫺

冉

冉 冏 冏
冉 冏 冏

⌳ 共 ␧ 兲 ⫽2N N 1⫺

QN 2
兩 1⫹Z N cos共 ␪ 兲 兩 2
ZN⫹2N S 1⫺冊

QS 2
兩 1⫹Z S sin共 ␪ 兲 兩 2
ZS

冊⫺2 Re兵 N N Q N 关 Z N ⫹cos共 ␪ 兲兴
⫹N S Q S 关 Z S ⫹sin共 ␪ 兲兴 其 ⫹␲␧␦Im 关 cos共 ␪ 兲兴 ,共2.22a兲⍀ 共 ␧ 兲 ⫽2N N 兩 Q N sin共 ␪ 兲 兩 2 ⫹2N S 兩 Q S cos共 ␪ 兲 兩 2 ,共2.22b兲

⌸ N 共 ␧ 兲 ⫽2N N 兩 Q N 兩 2 兵 共 1⫹6Z N2 ⫹Z N4 兲关 1⫹ 兩 cos共 ␪ 兲 兩 2 兴
⫹8Z N 共 1⫹Z N2 兲 Re关 cos共 ␪ 兲兴 其 ,共2.22c兲⌸ S 共 ␧ 兲 ⫽2N S 兩 Q S 兩 2 兵 共 1⫹6Z 2S ⫹Z 4S 兲关 1⫹ 兩 cos共 ␪ 兲 兩 2 兴
共2.22d兲where Z N and Z S were defined below Eq. 共2.5兲. With thesedefinitions, we find

冊

␲␧
.
2␦
共2.19兲

Having computed the self-energy from averaging ⌺ andthus the average Green function 具 G典 , we can proceed to sum
diagrams for the conductance. In analogy to the usual impurity technique,24 two sets of contributions arise 关see Fig.
3共a兲兴 to leading order: G⫽(4e 2 /h)(g Dir⫹g Diff). The first is adirect contribution which is completely determined by 具 G典 ,that is, by the ensemble-averaged probability amplitude
具 r he 典 :g dir⫽ 共 2 ␲ 兲 2 tr W N† 具 G典 W N W N† 具 G典 † W N ⫽tr 具 r he 典具 r he 典 † .
共2.20兲
This contribution may be interpreted as arising from Andreev reflections that effectively occur at the interface of thenormal lead and the cavity. The second contribution is due tothe fluctuations of r he :g diff⫽ 具 共 r eh ⫺ 具 r eh 典 兲共 r eh ⫺ 具 r eh 典 兲 2 典 .

We ignore here quantum corrections, which are formallysmaller by a factor of max(1/N N ,1/N S ). Note that in computing the diffusion sum each individual graph is of order 1/M ;this means that terms of order 1/M must be retained when
computing the 4⫻4 matrix inverse arising from summing
the series. As with the calculation of the average Green function 具 G典 , we let M tend to infinity only after having performed the partial summation of diagrams.
To present the results of the conductance calculation, we
first define the following kernel functions:⫹8Z S 共 1⫹Z 2S 兲 Re关 cos共 ␪ 兲兴 其 ,1
N S Q S 关 sin共 ␪ 兲 ⫹Z S 兴M⫹N N Q N 关 cos共 ␪ 兲 ⫹Z N 兴 ⫹i

10 229共2.21兲

It describes the current carried by quasiparticles in the dotthat are Andreev reflected at the interface of the dot and thesuperconductor. Diagrammatically, it is equivalent to a diffusion ladder, where the averaging links the upper and lower
branches of the conductance bubble 关see Figs. 3共b兲–3共d兲兴.25

g dir⫽2N N

T N2
兩 2⫺2T N sin2 共 ␪ /2兲 兩 2

冏 冏

g diff⫽8N N2

Q N2 2ZN

兩 sin共 ␪ 兲 兩 2 ,共2.23a兲共 ⌳ 2 ⫺⍀ 2 兲 ⫺2 关 ⌸ N 兩 sin共 ␪ 兲 兩 2⫹ 共 1⫺Z N2 兲 2 ⌳ 兩 sin共 ␪ 兲 兩 2 ⫹⌸ S 兩 cos共 ␪ 兲 兩 2 兴 .
共2.23b兲
Equations 共2.22a兲–共2.23b兲, along with Eq. 共2.17兲 for the selfenergy ⌺, determine the ensemble-averaged B⫽0 Andreev
conductance through the dot to leading order in 1/N N ,1/N S
for all voltages such that eVⰆ⌬. The extension of theseformulas to nonzero B is presented in the Appendix.
III. PROBING INDUCED SUPERCONDUCTIVITY VIACONDUCTANCE
A. Chaotic dot

In this section, we consider the situation where the coupling between the dot and the superconductor is much stronger than the coupling between the dot and the normal lead,
N N T N ⰆN S T S . In this limit, the normal metal will only
weakly perturb the properties of the dot-superconductor system; thus, we may view the conductance through the struc-10 230

A. A. CLERK, P. W. BROUWER, AND V. AMBEGAOKAR

ture as being a probe of the induced superconductivity in thedot. As mentioned in the Introduction, previous studies haveexamined the density of states of such Andreev billiards inthe absence of a normal lead.5 It was found for the case of a˜ opens up on a scale set bychaotic dot that an energy gap ⌬the inverse of the time needed for a particle to escape fromthe dot to the superconductor:
˜ ⫽c 共 E S 兲 ,
⌬
E S⫽

冉 冊冉
ប

␶ S esc

⫽

共3.1兲

冊

N ST S␦
.
2␲

共3.2兲

The parameter c is of order unity and a monotonic functionof T S ; it varies from 0.6 in the case of no tunnel barrier to 1
in the case of an opaque tunnel barrier. It was also found that˜ was vastly differthe shape of the density of states above ⌬ent in these two limits: in the tunnel regime T S Ⰶ1, the density of states was BCS-like, having a square-root singularity
˜ , whereas in the case of no tunnel barrier the density ofat ⌬˜ . Note that the gap ⌬˜ doesstates gradually increased above ⌬not depend on the gap ⌬ in the bulk superconductor, which
was taken to infinity in the calculations.
The question that naturally arises here is how the induced
superconductivity seen in the density of states manifests itself in the conductance. Some insight may be obtained byconsidering Eqs. 共2.23a兲 and 共2.23b兲 to lowest order in thesmall parameter ␩ ⫽N N T N /N S T S . Without the normal lead,˜.
we have a gap in the dot density of states up to an energy ⌬As the density of states is proportional to Re兵 cos关␪(␧)兴其, this˜ . Using this,
implies that Re关 ␪ (␧) 兴 ⫽ ␲ /2⫹O( ␩ ) for ␧⬍⌬
˜
the ‘‘subgap’’ 共i.e., ␧⬍⌬ ) direct contribution to the conductance takes the formg⫽g dir共 ␧ 兲 ⫽2N N T N2
共 2⫺T N 兲 2 ⫺4 共 1⫺T N 兲 „Im兵 sin关 ␪ 0 共 ␧ 兲兴 其 …2

⫹O 共 ␩ 兲 ,共3.3兲

while the diffusion contribution g Diff is of order ␩ 2 and thus
negligible. As the energy is increased above the gap, thedensity of states returns to its normal state value, and conse˜ the direct term
quently ␪ (␧)0 to leading order. For ␧Ⰷ⌬
is negligible 关being proportional to sin(␪)], while the diffusion term gives the normal state conductance:g⫽g diff共 ␧ 兲 ⫽N N T N ⫹O 共 ␩ 兲 .

共3.4兲

The above considerations become extremely suggestive
when one considers the case T N ⫽1. There is a perfect con˜,
ductance doubling for voltages below the effective gap ⌬while at higher voltages the conductance drops to its normal
state value:
g⫽

再

2N N

˜
for ␧⭐⌬NN˜.
for ␧Ⰷ⌬共3.5兲

This is precisely what would be expected if the normal point
contact were in perfect contact with a bulk superconductor
˜ —conductance doubling would be expectedhaving a gap ⌬PRB 62

below the gap due to Andreev reflection, whereas above thegap the normal state conductance would be recovered as quasiparticles would now carry the current. In this respect, note˜ both the averaged reflectionthat below the induced gap ⌬probability 具 兩 r eh 兩 2 典 and the averaged reflection amplitude
˜ , the latter vanishes. Nonethe兩 具 r eh 典 兩 are nonzero; for eVⰇ⌬˜ ),less, in both cases 共i.e., for voltages above and below ⌬Andreev reflection is the only current carrying process. For
˜ , the dot itself appears as if it were superenergies below ⌬conducting and Andreev reflection effectively occurs at the
˜ , current
normal-lead–dot interface. For energies far above ⌬through the dot is effectively carried by quasiparticles in the
dot 共which are either electron- or holelike兲 which Andreev
reflect at the dot-superconductor interface.
The quantum dot continues to act as if it were a superconductor in the case where the contact to the normal lead is nolonger perfect 共i.e., T N ⫽1); one need only note that, if
˜ , Eq. 共3.3兲 becomes identiIm兵 sin关␪0(␧)兴其 is replaced by ␧/⌬
cal to the Blonder-Tinkham-Klapwijk 共BTK兲 formula forthe subgap conductance of a normal-point-contact–
superconductor junction, where there is a tunnel barrier at theinterface having transmission T N . 26 The BTK formula is derived assuming there is no spatial separation between thesites of Andreev scattering and normal scattering; the only
energy dependence thus comes from the Andreev reflection
phase ␣ (␧), not from the lack of electron hole degeneracy at
˜ by Im兵 sin关␪0(␧)兴其 infinite voltages. The replacement of ␧/⌬
Eq. 共3.3兲 means that the Andreev reflection phase ˜␣ in thepresent case is not necessarily the usual ␣ (␧) given in Eq.
共2.7兲, but is rather defined throughIm兵 sin关 ␪ 0 共 ␧ 兲兴 其 2 ⫽1⫹Re关 ˜␣ 共 ␧ 兲 2 兴
.
2

共3.6兲

This notwithstanding, the overall implication is still that for
˜ we can effectively consider all Andreev reflections aseV⭐⌬occurring at the N-dot interface rather than at the dot-S
interface—the dot is indeed acting as though it were itself asuperconductor.
We have numerically solved Eqs. 共2.17a兲–共2.17c兲 for thepairing angle ␪ (␧), and used this to compute the Andreev
conductance versus voltage. In Figs. 4 and 5, we plot thecalculated conductance vs voltage curves for various valuesof T N , and compare to what would be expected from the
BTK theory for a simple N-S interface. We find an excellent
agreement in the case of T S Ⰶ1 共see Fig. 4兲 even for voltages
˜ in the dot. Such an agreement is not unabove the gap ⌬likely, as for T S Ⰶ1 the dot density of states is BCS-like, andconsequently the effective Andreev phase ˜␣ is just equal tothe usual Andreev phase ␣ . In the opposite case of a transparent contact to the superconductor 共Fig. 5兲, clear deviations from the BTK line shapes are seen. These deviations
result completely from the fact that ˜␣ ⫽ ␣ , which is to be
expected as the dot density of states in this case is quite
different from the BCS form.
The ‘‘induced superconductivity’’ effect in the dot also
manifests itself in other ways. A straightforward calculationin the limit T N →0 shows that the Andreev conductance becomes proportional to the dot density of states, i.e.,PRB 62

ANDREEV CONDUCTANCE OF CHAOTIC AND . . .

10 231

冉 冊 冑
⌽C
⫽C
⌽0

FIG. 4. Conductance versus voltage for N S ⫽104 N N , T N ⫽0.1,
illustrating the ‘‘local Andreev reflection’’ effect, in which the dot
˜ . Solid lines are calitself acts as a superconductor having a gap ⌬culated curves for, from top to bottom, T N ⫽1,0.9,0.75. The dashedcurve shows the BTK result for T N ⫽1.0; close agreement is foundfor all values of T N . The dot density of states 共shown in the inset兲
in this case resembles a BCS density of states. The voltage is measured in units of E S , the inverse of the escape time to the superconductor 关cf. Eq. 共3.2兲兴.

Re关 cos(␪)兴; thus, the Andreev conductance becomes equivalent to a conventional superconductor tunneling density ofstates measurement. We have also calculated the magnetic
field dependence of the Andreev conductance in the limit ofsmall ␩ 共see the Appendix兲; here too, our results are consistent with a picture in which the dot itself acts as a superconductor. The conductance enhancement at zero field remains
constant until a critical flux ⌽ C given by

˜
⌬
⫽C
E erg

冑

˜ ␶ erg
⌬
,
ប

共3.7兲

where ␶ erg is the ergodic time and C is a geometry dependent
constant of order unity. This is the same critical field re˜ . Note that, unlike a conventionalquired to close the gap ⌬BCS superconductor, where the critical field is proportional
to the gap, in the present case Eq. 共3.7兲 implies that the
˜.
critical field is proportional to 冑⌬
The conclusions reached here are markedly different fromwhat would be expected from a naive trajectory-based semiclassical analysis.27 Consider the situation of a ballistic dot
with no tunnel barriers, where N S ⰇN N . In the semiclassicalpicture, electrons entering the dot from the normal lead typically bounce off the walls of the dot several times before
hitting the superconductor, where they Andreev reflect. At
V⫽0, holes are the time-reversed partners of electrons; thus,
when an Andreev reflection occurs, the hole will retrace thepath of the incoming electron and cancel its acquired phase.
Andreev trajectories will thus interfere constructively, leading to a large conductance enhancement at V⫽0 and a nonvanishing average reflection amplitude 具 r he 典 . This enhancement should be lost, however, as the voltage is increased—atfinite V, electrons and holes are no longer degenerate, andthe phase acquired by the hole will not precisely cancel that
acquired by the electron. The presence of a residual phase␦ ␾ ⫽eVL/(ប v F ) 共where L is the length of the trajectory兲
will lead to destructive interference and a consequent decrease in the conductance. In this picture, even a small voltage should impair the conductance enhancement seen at zerovoltage.
To make this picture quantitative, we may use the factthat our exact random matrix theory indicates that the con˜ is proportional to 兩 具 r he 典 兩 2 关see Eq. 共3.3兲兴.
ductance below ⌬Using the semiclassical Andreev phase ␦ ␾ ⫽eVL/(ប v F ) andthe fact that path lengths in chaotic systems have an exponential distribution function P(L)⫽exp(⫺L/L̄)/L̄,28 we estimate the average of the semiclassical reflection amplitude as

具 r he 典 SC⯝冕

dLe ieVL/ប v F P 共 L 兲 ⫽1
1⫺ieVL̄/ប v F, 共3.8兲

where L̄ is the mean path length to the superconductor; for aballistic dot, we have L̄⫽ v F ␶ S esc . Hence,冋 冉 冊册

eV
兩 具 r he 典 兩 ⫽ 1⫹
˜c⌬2

FIG. 5. Conductance versus voltage for N S ⫽104 N N , T S ⫽1,illustrating the ‘‘local Andreev reflection’’ effect. Solid lines arecalculated curves for, from top to bottom, T N ⫽1,0.9,0.75; dashedcurves show BTK result for same values of T N . The differences inthe dot density of states 共shown in the inset兲 from a BCS density ofstates lead to large differences from the BTK line shapes.

2 ⫺1

,

共3.9兲

˜ ⫽cប/ ␶ S esc 关see Eq. 共3.1兲兴 with cwhere we have used ⌬⯝0.6 in the case of a perfect contact to the superconductor.
The semiclassical approach thus predicts that 兩 具 r he 典 兩 2 共andhence the Andreev conductance兲 will fall off with voltage as
˜ . This is clearly at odds with our fullya Lorentzian for eV⬍⌬quantum mechanical calculation at T N ⫽1, which shows that˜ . We return to the dis兩 具 r he 典 兩 2 ⫽1 for all voltages below ⌬crepancy between the semiclassical and quantum mechanicalcalculations in the next subsection.

10 232

A. A. CLERK, P. W. BROUWER, AND V. AMBEGAOKAR

FIG. 6. Conductance 共scaled by N N ) vs voltage for a rectangular
dot and a circular dot, each with N N ⫽10, N S ⫽30N N 共shapes shownin the inset, with the size of the normal lead exaggerated兲. Each plot
was produced by averaging over small variations in E F . Unlike achaotic dot 共dashed line兲, the Andreev conductance of the integrable
˜ . The conductance of the rectangle
dots has no ‘‘plateau’’ for eV⬍⌬drops below N N due to a weak localization correction. The energy
scale E S is defined in Eq. 共3.2兲.
B. Integrable Dot

In this subsection, we examine the conductance of a
N-dot-S system where the classical dynamics of the dot are
integrable. Previous studies5,7 have indicated that the proximity effect in the density of states is very different for chaotic and integrable billiards; in the latter, there is no induced
gap, but rather the density of states tends to zero linearly at
E F . We argue here that the proximity effect’s strong sensitivity to chaos versus integrability also manifests itself in the
Andreev conductance.
We consider two different integrable systems: a rectangular dot and a circular dot. In each case, the dot is coupled via
narrow leads to both a normal metal and a superconductor;
there are no tunnel barriers. As in the previous subsection,
we consider the case where the width of the normal contact
is much smaller than the width of the superconducting contact 共see the inset of Fig. 6兲, so that the normal contact serves
as a probe of the proximity effect in the quantum dot. The
scattering matrix of the system is computed by numericallymatching wave functions across the structure, and the conductance then follows from Eq. 共2.8兲. The number of modesin the normal lead was fixed at N N ⫽int(k F W/ ␲ );29 we averaged results over small variations in k F , which did notchange N N . Our results are displayed in Fig. 6; similar results are obtained if one changes the positions of the twopoint contacts.
The results for the two integrable systems are similar to
one another, and differ significantly from what was found fora chaotic dot—as opposed to a flat subgap region followed
by a rapid drop-off, we have instead a gradual linear reduction of the conductance with voltage. We argue that the difference exhibited here is generic for integrable systems. The
reason is the same as that given in Ref. 5 to explain the
difference in density of states, namely, that the distributionof path lengths is very different for integrable and chaotic

PRB 62

dynamics.28 In the first case, there is a power law distributionof path lengths, meaning that there is an appreciable probability of finding long paths. Even for small voltages, theselong paths will quickly acquire a random phase, leading todestructive interference and a reduction of the conductance.
In contrast, chaotic systems have an exponential distributionof path lengths—long paths are quite rare. In this case, small
voltages will not be able to cause any significant phase randomization, and thus there will be no resulting destructive
interference of Andreev trajectories.
Although the semiclassical reasoning in terms of path
lengths presented here provides a qualitative account for thedifference between the chaotic and integrable Andreev conductance line shapes, attempts to translate it into a quantitative theory have not been successful. As was demonstrated in
the previous section, a simple semiclassical theory for the
chaotic case fails to recover the correct line shape. A similar
problem is encountered when one tries to do the analogous
calculation for the integrable case—such a calculation predicts that the Andreev conductance should fall off quadratically with voltage, albeit at a faster rate than in the chaotic
case. This behavior is clearly at odds with our exact calculation showing a linear dependence on voltage. In many
ways, the failure of a semiclassical approach is not surprising. It is known that semiclassical approximations are unreliable in superconducting systems, as the usual diagonal approximation is worse than for normal systems.5,7Finally, we note that there is a striking connection between the Andreev conductance’s sensitivity to the nature of
the dot’s classical dynamics and weak localization. It has
been shown both theoretically30 and experimentally31 that themagnetic field dependence of the weak localization correction of a quantum dot with two normal leads is very different
for chaotic versus integrable dots. In the chaotic case, one
finds a smooth Lorentzian field dependence, while in the
integrable case a much sharper profile is found, with a cusp
at zero magnetic field. The similarity between the Andreev
conductance effect and that in weak localization is not coincidental; both effects rely on the interference of timereversed paths.
Despite this strong similarity, it is worth noting that theeffect in the Andreev conductance is much more pronounced. Here, the difference between the chaotic and integrable line shapes is more severe than in the weak localization effect 共i.e., the chaotic case is much flatter than a
Lorentzian兲. The magnitude of the effect is also much larger
in the Andreev case; the signature of chaos versus integrability is the entire conductance itself, not a quantum correction like weak localization. For this reason, the effect in the
Andreev conductance should be observable in a single
sample, whereas an ensemble average is required in the weak
localization case.
IV. REENTRANCE EFFECTS

We shift focus in this section, and examine so-called reentrance phenomena in the Andreev conductance of a chaotic
quantum dot. These effects are loosely defined by nonmonotonic behavior of the conductance in either voltage or magnetic field. They are well known in the case of diffusive NS
systems,13,14 where the word ‘‘reentrant’’ is used because theANDREEV CONDUCTANCE OF CHAOTIC AND . . .

PRB 62Andreev conductance is the same as the normal conductanceat zero voltage and magnetic field, and at high voltage orfield, but not for intermediate values. The theory developed
here allows us to address this behavior using a scattering
approach, whereas previous approaches almost exclusively
made use of the quasiclassical Green function technique. In
what follows, we discuss the cases of ballistic contacts (T N
⫽T S ⫽1) and tunnel contacts (T N ,T S Ⰶ1) separately.

g diff共 ␧ 兲 ⫽N N2
2

冏 冉 冊冏
cos

␪共 ␧ 兲
2⫺4

共 2N S 兲共 N N 兲.
2N S ⫹N N

N sum⫺2N S ,
NN

In the absence of tunnel junctions, the equations determining the conductance simplify considerably. We findg dir共 ␧ 兲 ⫽2N N tan2

冉 冊

␪共 ␧ 兲
,2⌳ 共 ␧ 兲 2 ⫺⍀ 共 ␧ 兲 2

共4.3兲

It represents a contribution from Andreev quasiparticles inthe dot, and is thus sensitive to the dot’s density of states. In
the present case of ballistic contacts, the density of states
does not have any BCS-type peak and thus the diffusion termrises steadily with applied V or B.
Given that the direct and diffusion contributions react in
opposite fashions to an increase in V or B, it is not surprisingthat a nonmonotonic V or B dependence of the total conductance can be found if the relative strengths of these two terms
are varied. The latter can be achieved by tuning the ratio of
N N /N S . If N N ⰆN S , the direct term will be dominant at
V,B⫽0, and we expect a monotonic decrease of G as amagnetic field or finite bias are applied. In the opposite limit
N N ⰇN S , the diffusion term dominates at V,B⫽0, and G is
expected to increase with V or B. A nonmonotonic V or B
dependence can thus be anticipated in the intermediate regime where N N and N S are comparable.
To quantify the competition between the direct and diffusion contributions, we examine these terms at V⫽B⫽0. Thisis done by solving the self-energy equations 共2.15a兲–共2.17c兲
to determine ␪ (␧), and then substituting this into Eqs. 共4.1兲
and 共4.2兲. Letting N sum⫽ 冑N N2 ⫹6N N N S ⫹N 2S , we obtaing dir⫽ 共 N sum⫺N S ⫺N N 兲

A. Ballistic contacts

N N 兩 tan关 ␪ 共 ␧ 兲 /2兴 兩 2 ⫹N S 兩 cos关 ␪ 共 ␧ 兲兴 / 兵 1⫹sin关 ␪ 共 ␧ 兲兴 其 兩 2

As discussed in the previous section, these two contributions to the Andreev conductance can be interpreted as representing two distinct physical consequences of the proximity effect. The direct term g dir represents processes in which
the dot mimics a bulk superconductor, with Andreev reflections effectively occurring locally at the N-dot interface. It
decreases monotonically with voltage and magnetic field,
going to zero at large voltages or magnetic fields. This reflects the fact that the induced superconductivity effect issensitive to the averaged amplitude for Andreev reflection,
which is largest at V⫽B⫽0. On the other hand, the diffusion
term g diff increases monotonically with V and B, tending to
the classical result for two conductances in series:

g diff→g class⫽

10 233

共4.4a兲

g diff⫽ 共 N sum⫺N S ⫺3N N 兲

共4.1兲

共4.2兲

.

N S 4N N N S
⫹
,
NN
N sum

共4.4b兲

with the total conductance given by12冉冊

N N ⫹N S
.
共4.5兲
N sum
Plotted in Fig. 7 as a function of N N /N S are the zero fieldvalues of g dir , g diff , and g, normalized by the high field
conductance given by the classical formula 共4.3兲. As N N /N S
is increased from zero, the total conductance initially decreases with g dir as the local Andreev reflection effect issuppressed, while at larger values it starts to increase withg diff as the density of states at E F returns to its normal statevalue. At N N ⯝ 21 N S , we find that the total conductance atzero field is the same as the classical result. Not surprisingly,
we find that reentrance effects in both magnetic field andvoltage are maximized here 共see Fig. 8兲.
It is instructive to make a comparison at this point to thenonmonotonic reentrance behavior of diffusive NS systems
共i.e., a diffusive normal metal in good contact with a superconductor兲. In the diffusive case, the zero field and high field
conductances are the same, being equal to the normal stateg⫽g dir⫹g diff⫽ 共 N S ⫹N N 兲 1⫺

FIG. 7. Plot of the direct and diffusion contributions to the V⫽0,B⫽0 conductance as a function of N N /N S for transparent point
contacts (T N ⫽T S ⫽1). At N N ⯝N N /2, the total V⫽0,B⫽0 conductance is the same as the classical result g class .

10 234

A. A. CLERK, P. W. BROUWER, AND V. AMBEGAOKAR

FIG. 8. Plot of the excess conductance 关 g(B,V)⫺g class兴 /g class asa function of B at V⫽0, showing nonmonotonic behavior. From top
to bottom, curves are for N N /N S ⫽0.39,0.45,0.54,0.68. The inset
shows the excess conductance at B⫽0 as a function of voltage;
from top to bottom, curves are for N N /N S ⫽0.49,0.51,0.57,0.6. In
general, reentrance effects are maximized at N N /N S ⯝0.54, where
the V⫽0,B⫽0 conductance is the same as at large V or B.

conductance. The lack of any change due to superconductivity at V⫽B⫽0 is usually explained as an exact cancellationof the conductance doubling effect of Andreev reflection by
a suppression of the density of states at the Fermi energy. A
conductance enhancement is, however, found at finite voltage, roughly at eV⯝E T ⫽បD/L 2 , where D is the diffusion
constant and L is the length of the normal metal. Quasiclassical calculations find maximum conductance enhancementson the order of 10% of the normal state conductance.13
In the present system, we can tune the relative significance of the Andreev reflection enhancement and density ofstates suppression terms by varying N N /N S . We find that
nonmonotonic effects are maximized when we adjust thisratio to mimic the diffusive system, by insisting that the zero
field and high field conductances are the same; this occurs at
N N ⯝1/2N S . The conductance maximum in voltage issmaller however, being on the order of 0.01g class , and occurs
roughly at eV⫽E S ⫽N S ␦ /2 ␲ . This is the inverse of the timeneeded to escape to the superconductor, and is thus the analog of E T in the diffusive system, which represents the inverse of the time needed for a particle to diffuse across thenormal metal and reach the superconductor. Note also that if
˜ in
the normal lead is removed the size of the induced gap ⌬the dot density of states is ⬃E S .
Finally, we also find pronounced nonmonotonic behavior
in magnetic field for this range of N N /N S , with the magnitude of the effect being larger than that in voltage 共Fig. 8兲.
The maximum conductance occurs roughly at a flux ⌽ Cgiven by Eq. 共3.7兲, the same flux that would be required toclose the gap in the density of states in the absence of thenormal lead.
B. Tunnel regime

We turn now to the case where both point contacts contain opaque tunnel barriers (T S ,T N Ⰶ1). In this regime, it is

PRB 62

sufficient to consider the conductance to lowest nonvanishing order in T N and T S . Of particular interest here is thewell-known ‘‘reflectionless tunneling’’ effect27—the V⫽0,B⫽0 Andreev conductance of a N-I-N-I-S structure
共where I is an insulating region having transmission T) is
found to be proportional to T, not to T 2 as one has for asingle barrier. It is as though the Andreev reflected hole is
not reflected at all by the tunnel barriers. A similar effectoccurs for a N-I-S system where the normal region is sufficiently disordered. The origin of this striking behavior is
now understood to be the constructive interference of trajectories that reflect many times between the two barriers, leading to a fraction ⬃T of the conductance channels being open
共i.e., having a transmission probability close to unity兲.32
In the present case, we are able to examine the effects offinite voltage and magnetic field on reflectionless tunnelingwhen a quantum dot separates the barriers. At large voltages
or magnetic fields, the pairing angle ␪ (␧) tends to its normal
state value of 0, and we find that the conductance is given toleading order by the classical series addition formula冉冊1
N T 2 共 N NT N 兲
2 S S.
g class⫽
1
2
N T ⫹N N T N
2 S S

共4.6兲

For N S T 2S ⰆN N T N , this simplifies to g class⫽ 21 N S T 2S . Unlike
the case at V⫽0,B⫽0, there is no order T reflectionless tunneling contribution here, as the necessary constructive interference is lost when electron-hole degeneracy or timereversal symmetry is broken. Below we show thatreflectionless tunneling does survive at small values of B andV, and describe how this contribution evolves as B and V are
increased.
We begin by solving Eqs. 共2.17a兲–共2.17c兲 for the selfenergy to lowest order in T N ,T S for B⫽0; we find the pairing angle is given simply by␪ 共 ␧ 兲 ⫽arctan

冉

冊

ES
.
E N ⫺i␧

共4.7兲

E S and E N are the inverse escape times to the superconductor
and normal metal leads, respectively, and are defined by
E N⫽

N NT N␦,2␲

E S⫽

N ST S␦
.
2␲

共4.8兲

Using this result for ␪ (␧), we next write Eqs. 共2.23a兲 and共2.23b兲 for the conductance to lowest order in T S ,T N . The
direct contribution g dir corresponds to Andreev reflection at
the N-dot interface and is of order T N2 . The only order T
contribution is found in the diffusion term g diff , which yields
g diff共 ␧ 兲 ⫽N N T N

E N E 2S
Ẽ 共 ␧ 兲2

冑2
Ẽ 共 ␧ 兲 ⫹E N2 ⫹E 2S ⫺␧ 2
2

⫹O 共 T 2 兲 ,共4.9兲

where we have definedẼ 共 ␧ 兲 ⫽ 关共 E 2S ⫺E N2 ⫺␧ 2 兲 2 ⫹4 共 E S E N 兲 2 兴 1/4.

共4.10兲ANDREEV CONDUCTANCE OF CHAOTIC AND . . .

PRB 6210 235have also found that adding strong tunnel barriers makes adiffusive normal wire with many channels equivalent to aquantum dot.
V. CONCLUSIONS

FIG. 9. Conductance vs voltage in the tunnel regime T N ⫽T S
⫽0.01, exhibiting the enhancement of reflectionless tunneling at
finite voltage. The parameter q is defined as the ratio N N /N S . The
inset shows the V⫽0 conductance as a function of magnetic field
for the same parameter choices.

Equation 共4.9兲 gives the complete voltage dependence ofthe reflectionless tunneling effect. At V⫽0, it reduces tog diff共 0 兲 ⫽共 N NT N 兲 2共 N ST S 兲 2
关共 N N T N 兲 2 ⫹ 共 N S T S 兲 2 兴 3/2,

共4.11兲

which is similar to the formula found in Ref. 32, generalized
to the case where the point contacts have different widths.
Depending on the relative magnitudes of E S and E N , theconductance drops monotonically with voltage, or shows amaximum around eV⯝E S 共see Fig. 9兲. The location of the
maximum is at eV⫽ 冑7/6E S if E S ⰇE N . Similar behavior isfound for the magnetic field dependence of the conductance共see the inset of Fig. 9兲.
It is easy to understand the origin of this nonmonotonic
behavior within our theory. When N S becomes larger thanN N , the effect of the superconductor on the dot density ofstates becomes significant. As we have seen in Sec. II, theinduced density of states will have a sharp peak at E S when
T S Ⰶ1; it is this peak that is manifesting itself in the conductance.
It is interesting to note that Eq. 共4.9兲 has the same form as
was found for a diffusive N-I-N-I-S system using a quasiclassical Green’s function approach.33 In that system, thequantum dot is replaced by a diffusive normal wire of lengthd, width W, and mean free path l, and the normal and superconducting leads are not attached via point contacts, but arealso wires of the same width. In the limit T N ,T S Ⰶl/d theresistances of the tunnel barriers dominate, and an expression
identical to Eq. 共4.9兲 is obtained, but now the energies
E N ,E S are given by
E N ⫽T NបvF,
4d

E S ⫽T S

បvF,
4d

共4.12兲

where v F is the Fermi velocity in the normal interbarrier
region. This is identical to the definition of E N and E S in Eq.
共4.8兲 if one takes ␦ to be the level spacing in the wire. Notethat this correspondence is not surprising; previous studies

We have studied the voltage 共V兲 dependence of the Andreev conductance of chaotic and integrable quantum dots;
we have also examined the magnetic field 共B兲 dependence inthe chaotic case. In the regime where the contact to the superconductor dominates, we find that the voltage dependenceof the Andreev conductance is extremely sensitive to thenature of the dot’s classical dynamics—in the chaotic case,the dot itself mimics a superconductor and the conductanceis initially flat in voltage, whereas in the integrable case theconductance falls off linearly with voltage. This effect islarge in that the entire conductance forms the signature ofchaos vs integrability; also, it does not require any ensemble
averaging to be done. Both these facts make it particularly
amenable to experiments using semiconductor quantum dots,
where a good contact between dot and superconductor can beachieved in samples having a fixed dot shape.34,35 These experiments typically employ Nb for the superconducting contact, while the quantum dot is defined in a two-dimensional
electron gas residing in an InAs layer.
We have also studied nonmonotonic reentrance phenomena in the V and B dependence of the Andreev conductanceof a chaotic dot. We find that such behavior is ubiquitous,
and is the result of two competing processes—Andreev reflection at the dot–normal-lead interface, which decreases as
V or B increases, versus quasiparticles being injected into thedot, which increases with V and B.
ACKNOWLEDGMENTS

We thank C. W. J. Beenakker for a useful discussion.
A.C. acknowledges the support of the Olin Foundation and
the Cornell Center for Materials Research. This work was
supported in part by the NSF under Grant No. DMR9805613.
APPENDIX: CALCULATING THE ANDREEVCONDUCTANCE FOR BÅ0

In this Appendix, we outline the method used for calculations at nonzero magnetic field B. Formally, the magnetic
field enters the model in a different fashion from a voltage
difference. The latter is dealt with by the fact that the Andreev scattering matrix S defined in Eq. 共2.12兲 is an explicit
function of energy. The field dependence, however, does notappear directly in the expression for S, but rather only
emerges in the averaging procedure—as we use the PandeyMehta distribution defined in Eq. 共2.2兲, the ensemble of random matrices is itself a function of field.
As in the calculation at B⫽0, the first step in obtaining
the conductance is to calculate the averaged matrix Greenfunction 具 G典 defined in Eq. 共2.14a兲. The diagrams used are
the same as in the B⫽0 case, but now use of the distribution
共2.2兲 leads to a different self-energy ⌺ in the Dyson
equation:A. A. CLERK, P. W. BROUWER, AND V. AMBEGAOKAR

10 236⌺ 共 ␧, ␥ 兲 ⫽1 M 丢⫽1 M 丢冉⌺ ee

共 1⫺2 ␥ 兲 ⌺ eh

共 1⫺2 ␥ 兲 ⌺ he

⌺ hh

冉

具 tr Gee 典
␭2
M 共 2 ␥ ⫺1 兲 具 tr Ghe 典

冊

2
2
⌺ eh
⫺⌺ ee

共 2 ␥ ⫺1 兲 具 tr Geh 典

具 tr Ghh 典

冊

␭

2

⫹i

共A1兲

tan关 ␪ 共 ␧, ␥ 兲兴 ⫽N S Q S ⫺2 ␥ sin共 ␪ 兲,
N N Q N ⫺i ␲ ␧/2␦

共A2兲

where Q N and Q S are functions of ␪ defined in Eqs. 共2.17b兲
and 共2.17c兲. The 1/M corrections to the self-energy in thepresence of a magnetic field read冊␲␧
⫹2 ␥ sin2 共 ␪ 兲 .
2␦

共A3兲

The next step in the calculation is to sum diagrams for the
conductance. The necessary diagrams are the same as thoseretained in the B⫽0 calculation 共i.e., direct and diffusion
terms兲, although their evaluation is different. The direct term
is still given by Eq. 共2.23a兲, but with ␪ (␧, ␥ ) now determinedfrom Eq. 共A2兲.
The diffusion term acquires a form different from Eq.
共2.23b兲, as factors of (1⫺2 ␥ ) now appear in graphs where
the particle-hole indices of upper and lower branches do notmatch. These factors lead to 1/M corrections both to thematrix inverse that arises when summing the diffusion ladder, and to the matrix prefactors of the ladder. As discussed
earlier, such corrections are important when calculating the
conductance. The result is冏 冏

g diff⫽8N N2

冉

1N S Q S 关 sin共 ␪ 兲 ⫹Z S 兴M⫹N N Q N 关 cos共 ␪ 兲 ⫹Z N 兴,

where ␥ is a function of magnetic field 关see Eq. 共2.3兲兴. The
additional factors of (1⫺2 ␥ ) here reflect the fact that breaking time-reversal symmetry suppresses off-diagonal superconducting correlations. Solving the Dyson equation 共2.14a兲to leading order in 1/M , we find that relations 共2.15a兲 and共2.15b兲 continue to hold, meaning that we may still parametrize the self-energies in terms of a pairing angle ␪ (␧, ␥ )through Eq. 共2.16a兲. The equation determining ␪ (␧, ␥ ) now
takes the form

⫺1⫽⫺

PRB 62˜ 兩 sin共 ␪ 兲 兩 2 ⫹ 共 ⌸ N ⫹ ␥ ⌸ B 兲 兩 sin共 ␪ 兲 兩 2 ⫹⌸ S 兩 cos共 ␪ 兲 兩 2
Q N2 2 共 1⫺Z N2 兲 2 ⌳˜ 2 ⫺⍀̃ 2
⌳

ZN

共A4兲

,

where
⌸ B 共 ␪ 兲 ⫽2„共 1⫹Z N 兲 2 兵 1⫹2 Re关 sin2 共 ␪ 兲兴 其 ⫹4Z N 共 1⫹Z N 兲 Re关 cos共 ␪ 兲兴 ⫹4Z N2 兵 兩 cos共 ␪ 兲 兩 2 ⫺2 Re关 sin2 共 ␪ 兲兴 其 …,

共A5兲˜ 共 ␪ , ␥ 兲 ⫽⌳ 共 ␪ 兲 ⫺12␥ sin2 共 ␪ 兲 ,
⌳

共A6兲

冉 冏 冏

˜ 共 ␪ , ␥ 兲 ⫹8 ␥ N N⍀̃ 共 ␪ , ␥ 兲 ⫽⍀ 共 ␪ 兲 ⫹4 ␥ 兵 1⫺ 兩 cos共 ␪ 兲 兩 2 ⫹4 Re关 sin2 共 ␪ 兲兴 其 ⌳Y N共 ␪ 兲 ⫽冉冏 冏 冊

冏 冏

冊

QN 2
QS 2
Y N 共 ␪ 兲 ⫹N S
Y S共 ␪ 兲 ⫹ ␥ Y 2共 ␪ 兲 ,
ZN
ZS

共A7兲

ZN 2
⫺1 兵 1⫺ 兩 cos共 ␪ 兲 兩 2 ⫹4 Re关 sin2 共 ␪ 兲兴 其 ⫺2Z N 兵 5 Re关 sin2 共 ␪ 兲兴 Re关 cos共 ␪ 兲兴 ⫹Im关 sin2 共 ␪ 兲兴 Im关 cos共 ␪ 兲兴 其
QN

⫹Z N2 兵 1⫺ 兩 cos共 ␪ 兲 兩 2 ⫺2 Re关 sin2 共 ␪ 兲兴关 1⫹2 兩 cos共 ␪ 兲 兩 2 兴 其 ,Y S共 ␪ 兲 ⫽

冉冏 冏

冊

ZN 2
⫺ 兩 1⫺Z S sin共 ␪ 兲 兩 2 兵 1⫺ 兩 cos共 ␪ 兲 兩 2 ⫹4 Re关 sin2 共 ␪ 兲兴 其 ⫺Z 2S 兩 sin共 ␪ 兲 cos共 ␪ 兲 兩 2 ,QN

Y 2 共 ␪ 兲 ⫽1⫺ 兩 cos共 ␪ 兲 兩 2 兵 1⫹4 Re关 sin共 ␪ 兲 2 兴 其 ⫹Re关 3 sin2 共 ␪ 兲 ⫹4 sin4 共 ␪ 兲兴 ⫹4 兩 sin共 ␪ 兲 兩 4 .

共A8兲
共A9兲
共A10兲

Here ⌳( ␪ ), ⍀( ␪ ), ⌸ N ( ␪ ), and ⌸ S ( ␪ ) are given by Eqs. 共2.22a兲–共2.22d兲. The above equations, together with Eq. 共A2兲 for␪ (␧, ␥ ), determine the Andreev conductance for arbitrary voltage and magnetic field.

A. F. Andreev, Zh. Éksp. Teor. Fiz. 46, 1823 共1964兲 关Sov. Phys.
JETP 19, 1228 共1964兲兴; 关 49, 455 共1966兲兴.
2
C. W. J. Beenakker, Rev. Mod. Phys. 69, 731 共1997兲.
3
I. Kosztin, D. L. Maslov, and P. M. Goldbart, Phys. Rev. Lett. 75,
1735 共1995兲.
4K. M. Frahm, P. W. Brouwer, J. A. Melsen, and C. W. J. Beenakker, Phys. Rev. Lett. 76, 2981 共1996兲.
1

5

J. A. Melsen, P. W. Brouwer, K. M. Frahm, and C. W. J. Beenakker, Europhys. Lett. 35, 7 共1996兲; Phys. Scr. T69, 223 共1997兲.
6
A. Altland and M. R. Zirnbauer, Phys. Rev. Lett. 76, 3420
共1996兲; Phys. Rev. B 55, 1142 共1996兲.
7
A. Lodder and Yu. V. Nazarov, Phys. Rev. B 58, 5783 共1998兲.
8
P. W. Brouwer and C. W. J. Beenakker, Chaos, Solitons Fractals
7, 1249 共1997兲.

PRB 62
9

ANDREEV CONDUCTANCE OF CHAOTIC AND . . .

H. Schomerus and C. W. J. Beenakker, Phys. Rev. Lett. 82, 2951
共1999兲.
10
W. Ihra, M. Leadbeater, J. L. Vega, and K. Richter,
cond-mat/9909100 共unpublished兲.
11
N. Argaman, Europhys. Lett. 38, 231 共1997兲; N. Argaman and A.
Zee, Phys. Rev. B 54, 7406 共1996兲.
12
P. W. Brouwer and C. W. J. Beenakker, J. Math. Phys. 37, 4904
共1996兲.
13
Yu. V. Nazarov and T. H. Stoof, Phys. Rev. Lett. 76, 823 共1996兲.
14
P. Charlat et al., Phys. Rev. Lett. 77, 4950 共1996兲.
15
M. L. Mehta, Random Matrices 共Academic, New York, 1991兲.
16
A. Pandey and M. L. Mehta, Commun. Math. Phys. 87, 449
共1993兲.
17
O. Bohigas, M.-J. Giannonia, A. M. Ozorio de Almeida, and C.
Schmit, Nonlinearity 8, 203 共1995兲.
18
K. M. Frahm and J.-L. Pichard, J. Phys. I 5, 847 共1995兲.
19
J. J. M. Verbaarschot, H. A. Weidermuller, and M. R. Zirnbauer,
Phys. Rep. 129, 307 共1985兲.
20
Y. Takane and H. Ebisawa, J. Phys. Soc. Jpn. 61, 1685 共1992兲.
21
A. A. Clerk 共unpublished兲.
22
For a recent review, see W. Belzig, F. K. Wilhelm, C. Bruder, G.
Schön, and A. D. Zaikin, Superlattices Microstruct. 25, 1251
共1999兲.
23
Yu. V. Nazarov, Phys. Rev. Lett. 73, 1420 共1994兲.
24
See, e.g., A. D. Stone, in Physics of Nanostructures, edited by J.
H. Davies and A. R. Long 共Institute of Physics, Edinburgh,
1992兲.
25
A similar decomposition can be made for transport through anormal dot. In this case, the first term corresponds to direct

10 237

reflection off tunnel barriers in the contacts, while the second
contribution is for processes that explore the dot ergodically.
See, e.g., Ref. 2.
26
G. E. Blonder, M. Tinkham, and T. M. Klapwijk, Phys. Rev. B
25, 4515 共1982兲.
27
B. J. van Wees, P. de Vries, P. Magnée, and T. M. Klapwijk,
Phys. Rev. Lett. 69, 510 共1992兲.
28
W. Bauer and G. F. Bertsch, Phys. Rev. Lett. 65, 2213 共1990兲.
29
Note that in the case of a circular dot we make use of wedgeshaped leads for technical convenience. The leads serve both todefine scattering modes and to provide boundary conditions forthe dot, and thus their particular shape does not affect the conclusions reached here. In the case of wedge-shaped leads, thenumber of propogating modes in the leads depends on the distance from the center of the circle. We may truncate the numberof scattering modes retained because higher order modes 共withclassical turning points outside the billiard兲 do not couple to thedot, and thus do not contribute to the Andreev conductance.
30
H. U. Baranger, R. A. Jalabert, and A. D. Stone, Phys. Rev. Lett.
70, 3976 共1993兲; Chaos 3, 665 共1993兲.
31
A. M. Chang, H. U. Baranger, L. N. Pfeiffer, and K. W. West,
Phys. Rev. Lett. 73, 2111 共1994兲.
32
J. A. Melsen and C. W. J. Beenakker, Physica B 203, 213 共1994兲.
33
A. F. Volkov, A. V. Zaitsev, and T. M. Klapwijk, Physica C 210,
21 共1993兲.
34
A. F. Morpurgo, S. Holl, B. J. van Wees, and T. M. Klapwijk,
Phys. Rev. Lett. 78, 2636 共1997兲.
35
E. Toyoda, H. Takayanagi, and H. Nakano, Physica B 284, 569
共2000兲.