Quasiclassical fluctuations of the superconductor proximity gap in a chaotic system

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by  M. C. Goorden

Quasiclassical fluctuations of the superconductor proximity gap in a chaotic system
M.C. Goorden, Ph. Jacquod∗, and C.W.J. Beenakker
Instituut-Lorentz, Universiteit Leiden, P.O. Box 9506, 2300 RA Leiden, The Netherlands
(Dated: 30 June2003)
4 Wecalculatethesample-to-samplefluctuationsintheexcitationgapofachaoticdynamicalsystem
0 coupledbya narrowlead toasuperconductor. Quantumfluctuationson theorderof magnitudeof
0 the level spacing, predicted by random-matrix theory, apply if τE ≪¯h/ET (with τE the Ehrenfest
2 timeandET theThoulessenergy). ForτE >∼¯h/ET thefluctuationsaremuchgreaterthanthelevel
spacing. Wedemonstratethequasiclassical natureofthegapfluctuationsinthelarge-τE regimeby
n
correlating them to an integral over theclassical dwell-time distribution.
a
J
PACSnumbers: 74.45.+c,03.65.Sq,05.45.Mt,74.78.Na
4
1

Theuniversalityofstatisticalfluctuationsisoneofthe inelastic scattering take over before the effect of a finite
]
l most profound manifestations of quantum mechanics in Ehrenfesttimecanbeseen. Inacomputersimulationin-
l
a mesoscopicsystems[1]. Classically,the conductanceg of elasticscatteringcanbeexcludedfromthemodelbycon-
h a disordered metal (measured in the fundamental unit struction, so this seems a promising alternative to inves-
- 2e2/h) would fluctuate from sample to sample by an tigate the crossoverfromuniversalquantumfluctuations
s
e amountoforder(l/L)3/2 1,with l the meanfree path to nonuniversal quasiclassical fluctuations. Contrary to
m and L the length of the≪conductor [2]. Quantum me- what one would expect from the disordered metal [2],
. chanical interference increases the fluctuations to order where quasiclassical fluctuations are much smaller than
t
a unity, independent of disorder or sample length. This the quantum value, we find that the breakdown of uni-
m is the phenomenonofuniversalconductance fluctuations versality in the chaotic system is associated with an en-
- [3,4]. Thesameuniversalityappliestoavarietyofother hancement of the sample-to-sample fluctuations.
d propertiesofdisorderedmetalsandsuperconductors,and The quantity on which we choose to focus is the exci-
n
random-matrixtheory(RMT)providesaunifieddescrip- tation gap ε of a chaotic system which is weakly cou-
o 0
tion [5]. pled to a superconductor. We have two reasons for this
c
[ Chaotic systems (for example, a quantum dot in the choice: Firstly, there exists a model (the Andreev kicked
shapeofastadium)sharemuchofthephenomenologyof rotator) which permits a computer simulation for sys-
2 tems large enough that τ > τ . So far, such simula-
v disordered systems: The same universality of sample-to- E D
tions, have confirmed the mi∼croscopic theory of Ref. [11]
1 sample fluctuations exists [6, 7, 8]. What is different is
for the average gap ε [13]. Secondly, there exists a
3 the appearance of a new time scale, below which RMT 0
h i
7 breaks down [9, 10]. This time scale is the Ehrenfest quasiclassical theory for the effect of a finite Ehrenfest
6 time ontheexcitationgapanditsfluctuations[14]. This
time τ , which measures how long it takes for a wave
E
0 allows us to achieve both a numerical and an analytical
packet of minimal size to expand over the entire avail-
3 understandingofthemesoscopicfluctuationswhenRMT
ablephasespace. Ifτ islargerthanthemeandwelltime
0 E
breaks down.
/ τD in the system (the reciprocal of the Thouless energy
t We summarize what is known from RMT for the
a ET = h¯/2τD), then interference effects are inoperative.
m Achaoticsystemwithconductanceg 2e2/h,levelspac- gap fluctuations [15]. In RMT the gap distribution
× P(ε ) is a universal function of the rescaled energy
ing δ, and Lyapunov exponent λ has τ = 2π¯h/gδ and 0
- D
d τ =λ−1ln(gτ /τ ),withτ thetimeofflightacrossthe (ε0 Eg)/∆g, where Eg = 0.6ET is the mean-field en-
E 0 D 0 −
n system [11]. The defining characteristic of the Ehrenfest ergygapand∆g =0.068g1/3δdeterminesthemeanlevel
o time is that it scales logarithmically with h¯, or equiv- spacing just above the gap. The distribution function
c
: alently, logarithmically with the system size over Fermi has mean hε0i = Eg +1.21∆g and standard deviation
v wavelength [12]. ε2 ε 2 1/2 δε given by
i h 0i−h 0i ≡ RMT
X The purpose of this paper is to investigate what hap- (cid:0) (cid:1)
r pens to mesoscopicfluctuations ifthe Ehrenfesttime be- δεRMT =1.27∆g =1.09ET/g2/3. (1)
a
comes comparable to, or larger than, the dwell time, so
The RMT predictions for P(ε ), in the regime τ
one enters a quasiclassical regime where RMT no longer 0 E
≪
τ , were confirmed numerically in Ref. [13] using the
holds. This quasiclassical regime has not yet been ex- D
Andreev kicked rotator.
ploredexperimentally. The difficulty isthatτ increases
E
We will use the same model, this time focusing on
so slowly with system size that the averaging effects of
the gap fluctuations δε in the regime τ > τ . The
0 E D
Andreev kicked rotator provides a strobosco∼pic descrip-
tion (period τ ) of the dynamics in a normal region of
0
∗Presentaddress: DepartmentofTheoreticalPhysics,Universityof phasespace(areaM¯heff)coupledtoasuperconductorin
Geneva,24QuaiErnestAnsermet,CH-1211Geneva4,Switzerland. amuchsmallerregion(areaN¯heff,1 N M). Were-
≪ ≪
2
Sinceweworkinmomentumrepresentation,the leadde-
fined by Eq. (4) is a strip in phase space of width N
parallel to the coordinate axis. One could alternatively
consider a lead parallel to the momentum axis, if one
would work in coordinate representation. We do not ex-
pect any significant differences between the two alterna-
10
tives. Putting all this together we arrive at the Floquet
operator of the Andreev kicked rotator [13],
T
M
F 0
R = 1/2 1/2. (5)
ε F P (cid:18)0 F∗ (cid:19)P
δ
/
0 The matrix can be diagonalized efficiently using the
ε
δ LanczostechFnique incombinationwiththe Fast-Fourier-
Transform algorithm [17]. This makes it possible to cal-
1
culate the quasi-energies ε and eigenfunctions ψ for
m m
systems of sizes up to M = 5 105. The gap value ε is
2 3 4 5 · 0
10 10 10 10 given by the eigenphase of closest to zero.
F
The Floquet operator (5) provides a stroboscopic de-
M scriptionof the electron and hole dynamics, which is be-
lievedtobeequivalenttothetrueHamiltoniandynamics
FIG.1: Root-mean-squarevalueδε0ofthegapdividedbythe on long time scales t τ0. The support for this comes
≫
RMTpredictionδεRMT,asafunctionofthesystemsizeM for from two sides: (i) In the absence of superconductivity,
dwell time M/N =5andkickingstrengthK =14. Thedata and for varying parameters K and h¯ , the 1-D kicked
eff
points result from the numerical simulation of the Andreev rotator correctly reproduces properties of localized [18],
kicked rotator. The solid line has slope 2/3, indicating that diffusive[19],andevenballistic[20]quasiparticlesindis-
δε0 depends only on M/N and not on M or N separately in orderedmedia. (ii) In the presence of superconductivity,
thelarge-M regime.
the kickedAndreev rotator,and extensions thereof, ade-
quately describe quantum dots in contact with a super-
fertothiscouplingasa“lead”. TheeffectivePlanckcon- conductor [13], and give a proper description of quasi-
stant is h¯ =1/M. The mean dwell time in the normal particles in dirty d-wave superconductors [21]. Since we
eff
region (before entering the lead) is τ = M/N and the will be giving a classicalinterpretationof our results, we
D
correspondingThoulessenergyisE =N/2M. We have also describe the classical map corresponding to the An-
T
setτ andh¯ equalto1. Thedimensionlessconductanceof dreev kicked rotator. The map relates the dimensionless
0
the lead is g = N. The product δ = 4πET/g =2π/M is coordinatexn ∈(0,1)andmomentumpn ∈(0,1)attime
the mean spacing of the quasi-energiesεm of the normal (n+1)τ0 to the values at time nτ0:
region without the coupling to the superconductor. The p
n
phase factors eiεm (m = 1,2,..,M) are the eigenvalues pn+1 = pn (K/2π)sin[2π(xn )], (6a)
± ± 2
of the Floquet operator F, which is the unitary matrix p p
n n+1
x = x . (6b)
thatdescribesthe dynamics inthe normalregion. Inthe n+1 n
± 2 ± 2
model of the kicked rotator the matrix elements of F in
The upper and lower sign correspond to electron and
momentum representation are given by [16]
hole dynamics, respectively. Periodic boundary condi-
F = e−(iπ/2M)(n2+m2)(UQU†) , (2a) tionsholdbothforxandp. Thequasiparticlereachesthe
nm nm
Unm = M−1/2e(2πi/M)nm, (2b) sthuepecrecnotnedruocfttohreifle|apdn+.1A−t tphleeadn|e<xtNit/e2raMtio,nwhtheereelpelceatdroins
Qnm = δnme−(iMK/2π)cos(2πn/M). (2c) is converted into a hole and vice versa.
WestudyasystemwithkickingstrengthK =14(fully
The coupling to the superconductor doubles the di-
chaotic,Lyapunovexponentλ=1.95)andvarythe level
mension of the Floquet operator, to accomodate both
spacing δ = 2π/M at fixed dwell time τ = M/N = 5.
electron and hole dynamics. The scattering from elec- D
Sample-to-sample fluctuations are generated by varying
tron to hole, known as Andreev reflection, is described
the position p of the lead over some 400 locations.
by the matrix lead
The resulting M-dependence of δε is plotted in Fig. 1
0
1 (1 1√2)PTP i1√2PTP onadoublelogarithmicscale. Wehavedividedthevalue
1/2 = − − 2 − 2 ,(3)
P (cid:18) −i21√2PTP 1−(1− 21√2)PTP (cid:19) δε0 resultingfromthesimulationbytheRMTprediction
δε from Eq. (1). The numerical data follows this
with the projection operator prRedMicTtionforM <103,butforlargerM thefluctuations
1 if L n L+N 1, arebiggerthanpr∼edictedbyRMT.ForM >104theratio
PTP nm =δnm×(cid:26) 0 othe≤rwise≤. − (4) δε0/δεRMT grows as M2/3 (solid line). S∼ince δεRMT
(cid:0) (cid:1) ∝
3
-4 -3 -2 -1
10 10 10 10 1 0 0.25 0.5 0.75 1
1
p
0
1
p
0
0 x 1 0 x 1
FIG.2: Leftpanels: Husimifunction(7)fortheelectroncomponentoftheground-statewavefunctionψ0 oftheAndreevkicked
rotator, for two different positions of the lead. The parameters are M = 131072, τD = M/N = 5, K = 14. The calculated
valuesarescaled byafactor 0.019 (0.017) in thetop(bottom) panel,so thattheycovertherange (0,1),indicated bythegray
scale at the top. Right panels: The corresponding classical density plots of all trajectories which have a time t > 7 between
Andreevreflections. Thecalculated values are rescaled bya factor 0.30 (0.32) in thetop (bottom) panel.
M−2/3, this means that δε is independent of the level In Fig. 2, left panels, the Husimi function of ψ is
0 0
spacing δ =2π/M at fixed dwell time τ =M/N. This shown for two lead positions. Shown is a logarithmic
D
suggests a quasiclassical explanation. grayscale density plotofthe Husimifunction, withlight
Torelatethe fluctuationsofε tothe classicaldynam- (dark) areas corresponding to low (high) density. The
0
ics, we firstexamine the correspondingwavefunctionψ . lead is visible as a light strip parallel to the x-axis. It is
0
In the RMT regime the wavefunctions are random and clear that these wavefunctions are not random. We ex-
showno featuresofthe classicaltrajectories. Inthe qua- pectthatthe structurethatoneseescorrespondsto long
siclassicalregimeτ >τ weexpecttoseesomeclassical classical trajectories, since the wavefunctions are for the
E D
features. Phase spac∼e portraits of the electron compo- lowestquasi-energy. Totestthisexpectation,weshowin
nents ψe of the wavefunctions are given by the Husimi the right panels (on a linear gray scale) the correspond-
m
function ing classical density plots for all trajectories with dwell
time t > t∗. A total of 3 105 initial conditions (x ,p )
(n ,n )= ψe n ,n 2. (7) · 0 0
H x p |h m| x pi| for these trajectories are chosen uniformly in the lead.
Each new iteration of the map (6) gives a point (x ,p )
The state n ,n is a Gaussian wave packetcentered at n n
x p
| i in phase space, which is kept if the time of return to the
x = n /M, p = n /M. In momentum representation it
x p lead is greater than t∗. We take t∗ =7, somewhat larger
reads
thantheEhrenfesttimeτ =λ−1ln(N2/M)=4.4. The
nn ,n e−π(n−np)2/Me2πinxn/M. (8) E
x p
h | i∝
4
More precisely, it is a plot of
0.084
∞
1 P(t)dt
0.54 = t∗ , (9)
< hti∗ Rt∞∗ tP(t)dt
R
with P(t) the classical dwell time distribution. We see
thatthe sample-to-samplefluctuationsinthe gapε cor-
0
0.52 0.082 relate very well with the fluctuations in the sample-to-
sample mean dwell time of long trajectories. Again, the
T correlationis not sensitive to the choice t∗ >τ . Such a
E > E
/ * correlationis in accordwith recenttheoreticalwork[14],
0 >
ε t inwhichaneffectiveRMTdescriptionisexpectedtohold
<
0.5 1/ forthe partofphasespacewith dwelltimes greaterthan
the Ehrenfest time. But we should emphasize that the
agreement is only qualitative. In particular, the relation
0.08
ε 1.5/ t 0.07 that we infer from Fig. 3 is different
0 ∗
00 00..55 11 ≈ h i −
from the relation ε = 0.3/ t that would be expected
0 ∗
h i
pp fromRMT.While the theory ofRef. [11]has been found
lleeaadd
to be in good agreementwith the averagegapvalue ε
0
h i
[13], itis notclearhowit comparesto the data ofFig.3.
FIG. 3: The data points (left axis) are the quantum me-
chanical gap values ε0 of the Andreev kicked rotator as a Inconclusion,wehaveinvestigatedthetransitionfrom
function of the position plead of the lead, for parameter val- quantummechanicaltoquasiclassicalgapfluctuationsin
ues M = 131072, τD = M/N = 5, K = 14. The solid thesuperconductorproximityeffect. Thetransitionisac-
line (right axis) is the reciprocal of the mean dwell time companiedby a lossof universalityanda substantialen-
∞ ∞
hti∗ = t∗ tP(t)dt/ t∗ P(t)dt of classical trajectories longer hancement of the fluctuations. Our numerical data pro-
∗
than t R=7. R
vides qualitative support for an effective random-matrix
theory in a reduced part of phase space [14], as is wit-
plotisnotparticularlysensitivetothevalueoft∗,aslong nessed by the precise correlation which we have found
as t∗ >τ . There is a clear correspondence between the between the value of the gap and the dwell time of long
E
classical trajectories (see Fig. 3). It would be of interest
quantum mechanical Husimi function and the classical
to investigate to what extent quasiclassical fluctuations
density plot. We conclude that the wavefunction of the
of the conductance in a ballistic chaotic system are sim-
lowestexcitationcoverspredominantlythatpartofphase
ilar or different from those of the superconducting gap
space where the longest dwell times occur.
studied here.
To make this more quantitative we show in Fig. 3 the
gapvalue fromthe quantum simulationsas a function of We have benefitted from discussions with J.
theleadposition. Thesolidcurveresultsfromaclassical Tworzyd lo. This work was supported by the Dutch
calculation of the mean dwell time of those trajectories Science Foundation NWO/FOM and the Swiss National
with t > t∗, for the same value t∗ = 7 used in Fig. 2. Science Foundation.
[1] Y. Imry, Introduction to Mesoscopic Physics (Oxford [11] M.G. Vavilov and A.I. Larkin, Phys. Rev. B 67 115335
University,Oxford, 1997). (2003).
[2] Yu.M.GalperinandV.I.Kozub,Europhys.Lett.15,631 [12] G.M. Zaslavsky, Phys. Rep. 80, 157 (1981).
(1991). [13] Ph. Jacquod, H. Schomerus, and C.W.J. Beenakker,
[3] B.L. Altshuler, JETP Lett. 41, 648 (1985). Phys. Rev.Lett. 90, 207004 (2003).
[4] P.A. Lee and A.D. Stone, Phys. Rev. Lett. 55, 1622 [14] P.G. Silvestrov, M.C. Goorden, and C.W.J. Beenakker,
(1985). Phys. Rev.Lett. 90, 116801 (2003).
[5] C.W.J. Beenakker, Phys.Rev. B 47, 15763 (1993). [15] M.G. Vavilov, P.W. Brouwer, V. Ambegaokar, and
[6] C.W.J. Beenakker, Rev.Mod. Phys. 69, 731 (1997). C.W.J. Beenakker, Phys.Rev.Lett. 86, 874 (2001).
[7] T. Guhr, A. Mu¨ller-Groeling, and H.A. Weidenmu¨ller, [16] F.M. Izrailev, Phys.Rep. 196, 299 (1990).
Phys.Rep.299, 189 (1998). [17] R.Ketzmerick,K.Kruse,andT.Geisel, PhysicaD 131,
[8] Y.Alhassid, Rev.Mod. Phys. 72, 895 (2000). 247 (1999).
[9] A. Lodder and Yu. V. Nazarov, Phys. Rev. B 58, 5783 [18] S. Fishman, D. R. Grempel, and R. E. Prange, Phys.
(1998). Rev. Lett.49, 509 (1984).
[10] I.L. Aleiner and A.I. Larkin, Phys. Rev. B 54, 14423 [19] A. Altland and M. R. Zirnbauer, Phys. Rev. Lett. 77,
(1996). 4536 (1996).
5
[20] J. Tworzydl o, A. Tajic, H. Schomerus, and C.W.J. [21] I˙. Adagideli and Ph. Jacquod, cond-mat/0307159.
Beenakker,cond-mat/0304327.