Table Of ContentDensity of States below the Thouless Gap in a Mesoscopic SNS Junction
P. M. Ostrovsky,M. A. Skvortsov,and M. V. Feigel’man
L. D. Landau Institute for Theoretical Physics, 117940 Moscow, Russia
Quasiclassical theory predicts an existence of a sharp energy gap Eg ∼~D/L2 in theexcitation
spectrumofalongdiffusivesuperconductor–normalmetal–superconductor(SNS)junction. Weshow
1
that mesoscopic fluctuations remove the sharp edge of the spectrum, leading to a nonzero DoS for
0
all energies. Physically, this effect originates from the quasi-localized states in the normal metal.
0
2 Technically,weuseanextensionofEfetov’ssupermatrixσ-modelformixedNSsystems. Anon-zero
DoS at energies E <Eg is provided by theinstanton solution with broken supersymmetry.
n
a
J When a small piece ofa normal(N) metal is placedin that their conjecture holds for sufficiently narrow junc-
4 contact with a superconductor (S), paired electrons en- tionsclosetoEg. Technically,ourapproachiscompletely
ter the N region, changing its excitation spectrum. How differentfromRef.[11]asweemploythefullymicroscopic
] drastic are these changes? Recent studies [1–5] demon- method based on the supermatrix σ-model method [13]
l
l strate that the answer depends crucially on the type of for mixed NS systems [14], which is free from any as-
a
h dynamicsintheNregion: Inthecaseofintegrable classi- sumptionabouttherandom-matrixuniversality. Wefind
- caldynamics,thedensityofstates(DoS)ofexcitationsis a nontrivial instanton solution to the saddle-point equa-
s
e suppressedatlowenergiesandvanishesnearlylinearlyat tions of the supermatrix σ-model and show that it is re-
m the Fermilevel. Contrary,inNsystems withchaotic dy- sponsible for the non-zero DoS at energies E <E . Our
g
. namics, coupling to a superconductor produces a gap in method is similar to the one used in [15]to calculate the
at the DoSofthe orderof~/τc, whereτc is the typicaltime DoS in the quantum Hall regime. The relevance of the
m needed to establish contact with the superconductor. nontrivialsaddle point solutions for the formationof the
- To illustrate this statement, consider a generic exam- DoS tail in NS systems was pointed out in Ref. [14].
d
n ple of a chaotic NS system: an SNS junction made of a We consider an SNS junction with the N region being
o disorderedconductor of size L connected to the S termi- a rectangular bar of size L Ly Lz, coupled to the S
× ×
c nals. When the N metal is diffusive, with the mean free terminals by the ideal contacts situated at x = L/2.
[ ±
path l L, and sufficiently long, with the Thouless en- We neglect superconductivity suppression in the bulk
2 ergyET≪h =~D/L2 ∆(hereD =vFl/3isthediffusion terminals, provided that they are sufficiently large, and
≪
v constant, and ∆ is the superconductive gap in the ter- assume zero superconductive phase difference between
8
minals), then τ is given by the diffusion time across the them. The dimensionless conductance of the N region,
c
7 N region, τ ~/E . Thus, the energy gap E E G(L,L ,L ) = 4πνDL L /L, where ν is the normal-
4 c ∼ Th g ∼ Th y z y z
2 develops in the DoS [3,4]. metal DoS per a single projection of spin.
1 However, all the results mentioned above are based Our results can be summarized as follows. The
0
either on the quasiclassical theory of superconductivity behavior of the subgap DoS ρ(E) depends on the
0 h i
andproximityeffect[6–8]oronthemead-fieldtreatment relation between L ,L , and the effective transverse
/ y z
at of the random-matrix theory (RMT). Although usually length L⊥(E), with the latter scaling as L⊥(E
m this is a good approximation, it is interesting to check Eg) L(1 E/Eg)−1/4 and L⊥(E Eg) →
∼ − ≪ ∼
whether some effects, which are beyond quasiclassics, L. In the vicinity of the quasiclassical gap, at
-
d may lead to qualitative changes in the above picture. E E , we find an intermediate asymptotic behav-
g
n ior→ln ρ(E) G(L, (E), (E))(1 E/E )3/2
co atiIonntshsimsLeaertttehrewheasrhdowgatphiant,tihnedeqeuda,simpaesrotisccloepsipceflcutrcutum- (1 hE/Egi)(6∼−d)−/4,wheLrye y,zL(Ez)=min−(Ly,z,Lg⊥(E)∼),
− − L
v: of dirty SNS junctions, producing a tail of the subgap and d is the effective dimensionality of the junction:
Xi sdtuaetetsowthitehexenisetregniceesoEfq<uaEsgi-.loTcahleiszeedloswta-ltyeisn[g9,s1t0a]teins tahree fdor=w0idfoerr njuanrcrotiwonjsunwcittihonLsywithLL⊥y(,EL)z ≪LLz⊥,(aEn)d,dd== 21
≫ ≫
r Npartofthe junctionwhichareweaklycoupledto theS for films with Ly,Lz L⊥(E). In the low-energy limit,
a ≫
E E , the behavior of the DoS is log-normal in 0D:
terminals, thereby having a larger τ . The magnitude of g
c ln ≪ρ(E) Gln2(E /E), and is a power law in 1D:
this mesoscopic effect is controlled by the dimensionless g
h i ∼ −
normal-state conductance G h/e2R of the junction, ln ρ(E) ln(Eg/E). These results are similar to
n h i ∝ −
≡ those for the distribution of relaxation times in 1D and
and is small (except for the vicinity of E ) at G 1.
g
≫ 2D diffusive conductors [9], although with an important
A related problem was considered recently in Ref. [11],
difference: the basic energy scale in our case is given by
where a hypothesis of universality and some nontrivial
E ,whereasinRef.[9]itwasthelevelspacingδ =1/νV.
results from the RMT [12] were used to find an expo- g
nentially small DoS below the mean-field gap in the N As a warm-up, we recall the standard quasiclassical
dotweaklyconnectedtoasuperconductor. Wewillshow approach [3,4] to diffusive SNS junctions based on the
1
Usadel equation [8] for the quasiclassicalretarded Green ˆ =τ p2 E +U(r) +∆(r)τ , (5)
function GˆR(E,r). For the latter we will assume the an- H z(cid:20)2m − F (cid:21) x
gular parametrization, GˆR = τ cosθ + τ sinθcosϕ +
z x
τ sinθsinϕ, where τ are the Pauli matrices in the where∆(r)=∆θ(x L/2),andU(r)istherandompo-
y i | |−
Nambu space. In the absence of the phase difference be- tential. In the absence ofmagnetic field, one has to dou-
tweenthe Sterminals,onecansetϕ 0,andtheUsadel ble the field space in order to account the time-reversal
equation acquires the form (hereafter≡~=1) symmetry, introducing the “time-reversal” (TR) space
according to Φ = (φ,iτ φ∗)T/√2. This definition coin-
y
L π
D 2θ+2iEsinθ =0, θ x= = . (1) cides with the one used in [16]; it differs from the nota-
∇ (cid:18) ±2(cid:19) 2 ∗
tions of [13,14] by the factor iτ in the φ sector. Pauli
y
The boundary conditions at the free edges of the N film matrices operating in the TR space are denoted by σi.
are of von Neumann type. The local density of states is After this definitions the derivation of σ-model is
given by ρ (r,E) =2νRecosθ(r). The substitution straightforward [13,14,16]. One has to carry out (i) av-
local
h i
θ(x) = π/2 + iψ(x) leads to the real equation for the eraging over the random potential with the correlator
function ψ(x) with ψ( L/2)= 0. This equation can be U(r)U(r′) =δ(r r′)/2πντ,(ii)Hubbard-Stratonovich
± h i −
easilyintegratedyieldingtherelationbetweenE andthe transformation introducing an 8 8 matrix Q acting in
×
magnitude of ψ(x) at x=0: the product of FB, N, and TR spaces, (iii) expansion to
the leading terms in Q, E, and ∆. The result is
ψ(0) ∇
E dψ
rETh = Z0 sinhψ(0)−sinhψ. (2) hρlocal(r,E)i= ν4 ReZ DQstr{kΛQ(r)}e−S[Q], (6)
p
πν
Eq. (2) has real solutions for ψ(0) only for E ≤ Eg = S[Q]= 8 Z dr str D(∇Q)2+4iQ(iτx∆+ΛE) . (7)
cETh, with c=3.12. At E >Eg, ψ(0) becomes complex (cid:2) (cid:3)
resulting in the square-root singularity [3] in the DoS
Here Λ = σ τ , the matrix Q = U−1ΛU with the
(hereandbelowweprovideresultsfortheDoSintegrated z z
proper set [14] of matrices U is subject to the condition
over the whole N region, ρ(E) = dr ρ (r,E) ):
h i h local i Q=CQTCT,whereC = τ σ [(1+k)σ +(1 k)iσ ]/2,
R − x z x − y
hρ(E)iquasicl. =2.30δ−1qE/Eg−1. (3) banedpakr=amdeitargiz(e1d,−b1y)F8Br.eaTlhaendma8nGiforaldssomfaQnn-mvaatrriiacbelsesc.an
Below the mean field gap, at E < E , Eq. (2) has The next step is to find the saddle point solutions to
g
two real solutions, ψ (0) and ψ (0) > ψ (0), merging the action(7). We startfromthe simplestcasewhenthe
1 2 1
at E = E . They determine the corresponding solu- Q matrix does not contain Grassmann variables. Then
g
tions θ (x) = π/2+iψ (x) to Eq. (1). Having real it splits into the Fermi-Fermi (FF) and Bose-Bose (BB)
1,2 1,2
ψ(x), both of them do not contribute to the DoS at sectorswhichcanbeparametrizedindependentlybyfour
E < E . Usually the solution ψ (x) is chosen by the variables in each sector:
g 1
continuity argument, as it obeys the natural condition
limE→0ψ1(x,E)=0,whileψ2(x,E)divergesinthelimit QFF =τzcosθF[σzcoskF+sinkF(σxcosχF+σysinχF)]
ofvanishingE. Wewillseehoweverthatitisthissecond +sinθ (τ cosϕ +τ sinϕ ), (8a)
F x F y F
solutionwhichisresponsibleforthe finiteDoSbelowthe
Q =[σ cosk +τ sink (σ cosχ +σ sinχ )]
BB z B z B x B y B
quasiclassicalgap.
[τ cosθ +σ sinθ (τ cosϕ +τ sinϕ )]. (8b)
In order to extend the quasiclassicalsolution and take × z B z B x B y B
intoaccountmesoscopicfluctuations,wewillusethenon-
In terms of the angular parametrization (8), the action
linear supermatrix σ-model similar to that derived in
(7)acquirestheformS =(πν/2) dr( ),where
[14]. Asweareinterestedintheaveragedensityofstates LFF−LBB
R
only, it is sufficient to calculate the retarded single par-
=D ( θ )2 +sin2θ ( ϕ )2
ticle Green function. The derivation of σ-model starts LFF ∇ F F ∇ F
with representingGˆR in terms of the functional integral: +cos(cid:2)2θF( kF)2+ cos2θFsin2kF( χF)2
∇ ∇
GˆR(r,r′,E)= i φ (r)φ+(r′)e−S[φ]Dφ∗Dφ, (4) +4iEcosθFcoskF−4∆sinθFcosϕF, (cid:3)
−2Z F F BB =D ( θB)2 +sin2θB( ϕB)2
L ∇ ∇
S[φ]= drφ+(r)(E+i0 ˆ)φ(r). +( (cid:2)kB)2+ sin2kB( χB)2
∇ ∇
Z −H +4iEcosθ cosk 4∆sin(cid:3)θ cosk cosϕ .
B B B B B
−
Inthisexpressionφisthe4-componentsuperfieldconsist-
ing of commuting (complex) andanticommuting (Grass- The variables θ and ϕ coincide (at k =0) with
F,B F,B F,B
mann) parts. The Hamiltonian ˆ is a matrix in the the standard Usadel angles, whereas their counterparts
H
Nambu-Gor’kov (N) space: k and χ are new ingredients of the field theory.
F,B F,B
2
Minimizationoftheactionforauniformsuperconduc- ρ(E) in the energy range G−2/3 1 E/E 1.
g
h i ≪ − ≪
tor at E ∆ gives θ = π/2 and k = 0, that Letusstartwiththesupersymmetricsaddlepoint(1,1,1)
F,B F,B
≪
provides the boundary conditions for Q in the N region. and look at fluctuations around it. Almost all of them
In the absence of the phase difference between the S ter- are hard, having a mass of the order of (or larger than)
minals, one obtains ϕ = k = 0 and χ = const at E . There are only 8 (corresponding to 4 commuting
F,B F F,B g
the saddle point in the N part of the structure. Intro- and 4 anticommuting variables) soft modes whose mass
ducing new variables α =θ +k and β =θ k in vanishes at E = E . Half of them transform the saddle
B B B B B B g
−
the BB sector, one obtains for the saddle-point action: point (1,1,1) to the instanton (1,1,2), and the other half
transform it to the instanton (1,2,2). Below we will con-
S[θF,αB,βB]=2S0[θF] S0[αB] S0[βB], (9) sider the case when∆S 1 that allowsto disregardthe
− − ≫
πν contribution of the instanton (1,2,2) and to take Gaus-
S [θ]= dr D( θ)2+4iEcosθ . (10)
0 4 Z ∇ sian integrals over the corresponding soft and all hard
(cid:2) (cid:3)
fluctuations. As a result, we end up with 2 commut-
Varying with respect to θ, one recovers Eq. (1) as the ing (q and χ ) and 2 Grassmann (ζ and ξ) variables
B
saddle-point equation for the action (10). parametrizing the relevant soft degrees of freedom in
The Usadel equation (1) possesses, apart from the x- the matrix Q = e−Wc/2e−Wa/2ΛeWa/2eWc/2. Here the
dependentsolutionsdiscussedabove,solutionswhichde- matrix Wa contains anticommuting variables: Wa =
FB
pend on the transverse (y,z) coordinates. The role of (if (x)/4)[(ζ +ξ)(iτ +τ σ )+(ζ ξ)(iτ σ +τ iσ )],
0 y z x y z z y
−
the transverse dimensions will be discussed later, while Wa = τ σ (Wa )Tτ iσ , while in the absence of Wa,
BF x x FB x y
nowwewillconsiderthe0Dcase,relevantforsufficiently Q reduces to the form (8) with θ = β = θ (x),
F B 1
narrow junctions with Lx,Ly ≪ L⊥(E). Then, accord- αB = θ1(x)+iqf0(x), kF = ϕF = ϕB = 0. The func-
ing to the previous analysis, the Usadel equation has tion f (x) is the normalized difference δψ(x) = ψ (x)
0 2
−
two solutions, θ1(x) and θ2(x). Therefore, the full ac- ψ1(x) at Eg: f0(x) = limE→Egδψ(x)/||δψ(x)||, where
tion (9) has in total 8 different saddle point solutions: F(x) 2 = (1/L) L/2 F2(x)dx. Evaluating the action,
(θ ,α ,β ) = (θ ,θ ,θ ), with i, j, k = 1,2, that will || || −L/2
F B B i j k integrating over thRe cyclic angle χ , and performing the
be refered to as (i,j,k). However, only 4 of them with B
x-integration, one obtains
θ = θ can be reached by a proper deformation of the
F 1
integration contour. q3
S =G˜ √εq2 +ζξ(2√ε q) , (11)
The simplest is the supersymmetric saddle point
(cid:20) − 3 − (cid:21)
(1,1,1). In this case, Gaussian integrations over com-
πc E cc 2c E E
muting and anticommuting variables near it cancel each G˜ = 2 g = 2G, ε= 1 g− , (12)
2δ 8 c E
other,andthecontributionto ρ(E) reducestotheform 2 g
h i
(3) with the vanishing DoS below Eg. Thus, the saddle- wherec = L/2 coshψ (x)f2n−1(x)dx/L,andψ (x)=
pointapproximationfortheσ-model(7)restrictedtothe n −L/2 0 0 0
ψ (x,E );Rc = 1.15, c = 0.88. The measure of in-
supersymmetric saddle point is equivalent to the quasi- 1,2 g 1 2
tegration (including the Berezenian) is given by DQ =
classical treatment based on the Usadel equation (1).
qdqdζdξ. Wetakethecontourofintegrationoverqalong
To get a non-zero DoS below E it is necessary to
g
the imaginary axis to get the convergent integral. Fi-
take saddle points with broken supersymmetry into ac-
nally, substituting str(kΛQ)dV =2ic V(4√ε q) into
count. Sucha solutionwith the lowestactionis givenby 1 −
Eq. (6), we arrive aRt the one-instanton correction to the
(1,1,2) [actually, a whole degenerate family of the sad-
quasiclassicalresult (3):
dle points, and, in particular, (1,2,1), can be obtained
kfreoympoitinbtyisrotthaattioGnaounsstiahne flanugctleuaχtiBon∈s n[e0a,r2πt)h].is sTahde- ρ(E) = c1 Im i∞+0exp G˜ √εq2 q3 dq. (13)
dle point have a negative eigenvalue which leads to an h i 2δ Z (cid:20)− (cid:18) − 3 (cid:19)(cid:21)
additional imaginary unity in the preexponent and, con- 0
sequently, to the nonzero contribution to the DoS. This Eq.(13)is valid providedthat ε G˜−2/3 when the con-
contribution is suppressed by the factor e−∆S, where ≫
tributionoftheinstanton(1,2,2)canbe neglected. Then
∆S = S [θ ] S [θ ] > 0 is the difference between
0 1 − 0 2 the integral over q can be calculated by the saddle point
the actions of the solutions θ1,2(x). Finally, the sad- method yielding at G˜−2/3 ε 1 in the 0D case:
dle point (1,2,2) has the action 2∆S and its contribu- ≪ ≪
tion can be disregarded at ∆S 1. Thus, the sub- c π 4
≫ ρ(E E ) = 1 exp G˜ε3/2 . (14)
gap DoS can be estimated with exponential accuracy as h → g i0D 4δrG˜√ε (cid:18)−3 (cid:19)
ρ(E) δ−1e−∆S(E). Below we will calculate ρ(E) in
h i∼ h i
the limiting cases E E and E E . This result can be generalized for a normal dot of an
g g
→ ≪
The solutions θ (x) and θ (x) merge at E . This fact arbitrary shape coupled to a superconductor (cf. [1,11]),
1 2 g
can be used to find the asymptotically exact result for providedthatthe numbersc, c aredefinedwith the use
n
3
of the exact solutions θ1,2(r) of the Usadel equation in 1 E 4π2νDLz
a given geometry: cn = (1/V) coshψ0(r)f02n−1(r)dr. hρ(E ≪Eg)i1D ∼ δ (cid:18)Eg(cid:19) . (19)
The functional form of the resultR(14) coincides with the
RMT conjecture of Ref. [11]. An exact correspondence
with the RMT prediction [11] is expected in the case of The DoS at E Eg in the 2D case can be calculated in
≪
weakly transparent NS interfaces. the same way, and appears to be vanishingly small.
Coming back to the planar SNS junction, we turn To conclude, we have shown that mesoscopic fluctua-
to the limit of small energies, E Eg. Here one has tionssmearthequasiclassicalgapintheDoSofadiffusive
≪
ψ1(x) 0 and ψ2(x) A(1 2x/L), where, according SNS junction. The tail in the DoS is due to the states
≈ ≈ − | |
to Eq. (2), A ln(Eg/E) (cf. Ref. [9]). Thus, the action anomalously localized in the N part of the junction and
≈
of the instanton (1,1,2) becomes ∆S(E) S0[θ2] = weakly coupled to the S terminals. Technically, the tail
≈ −
πEThA2/δ, and the result for DoS reads is described by instantons with broken supersymmetry.
1 G E Atthefinalstageofthepreparationofthismanuscript
ρ(E E ) exp ln2 g . (15)
h ≪ g i0D ∼ δ (cid:18)−4 E (cid:19) we became aware of Ref. [17] where a similar instanton
solution is found for the problem of tail states below E
g
Now let us consider the role of the saddle-point solu-
in a superconductor with magnetic impurities.
tions which depend on the transverse coordinates y,z.
We are grateful to C. W. J. Beenakker, D. A. Ivanov,
AtE E ,onehastoretainonlysoftmodesassociated
g
→ A.D.Mirlin,Yu. V.NazarovandA.V.Shytovforuseful
with the instanton (1,1,2). As a result, the action (11)
discussions. This research was supported by the NWO-
acquires a gradient term:
Russia collaborationgrant, Swiss NSF-Russia collabora-
S =G˜ dy dz L2 ( ⊥q)2+√εq2 q3 , (16) tiongrant,RFBRgrant98-02-16252,andbytheRussian
Z Ly Lz (cid:18)4cc2 ∇ − 3 (cid:19) Ministry of Science within the project “Mesoscopic elec-
tron systems for quantum computing”.
where the Grassmann variables are discarded as we are
not interested in the preexponent. Comparing the first
and the second term in Eq. (16), one extracts the char-
acteristic transverse scale L⊥(E) = L(cc2)−1/2ε−1/4
Lε−1/4, which determines the effective dimensionality o∼f
the system. If Ly or Lz is shorter than L⊥(E), then
it costs too much energy to have gradients in that direc- [1] J.A.Melsenetal,Europhys.Lett.35,7(1996); Physica
tion,andthecorrespondingdimension“freezesout”. The Scripta 69, 223 (1997).
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≪
L⊥(E). Otherwise,aninstantonwillappearinthetrans- (1998).
verse direction to minimize the total action. In the 1D [3] A.A.GolubovandM.Yu.Kupriyanov,Sov.Phys.JETP
case (Ly L⊥(E) Lz), the action (16) achieves its 69, 805 (1989).
stationary≫point atq≫(y)=3√εcosh−2(y/L⊥), leading to [4] F. Zhou et al, J. Low Temp. Phys. 110, 841 (1998).
[5] S.Pilgram,W.BelzigandC.Bruder,cond-mat/0006222.
1 12π [6] G. Eilenberger, Z. Phys. 214, 195 (1968).
ρ(E E ) exp √cc νDL ε5/4 . (17)
h → g i1D ∼ δ (cid:18)− 5 2 z (cid:19) [7] A. I. Larkin and Yu. N. Ovchinnikov, Sov. Phys. JETP
26, 1200 (1968).
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≫
quasiparticle DoS tail has the form [9] B. A. Muzykantskii and D. E. Khmelnitskii, Phys. Rev.
B 51, 5480 (1995).
1
hρ(E →Eg)i2D ∼ δ exp(−24.4νDLε). (18) [10] V. I. Fal’ko and K. B. Efetov, Europhys. Lett. 32, 627
(1995).
The length L⊥(E) diverges at Eg, indicating that any [11] M. Vavilov et al, cond-mat/0006375.
junction becomes effectively 0D close to the quasiclassi- [12] C. A. Tracy and H. Widom, Comm. Math. Phys. 159,
cal gap. However, different parts of the DoS tail may 151 (1994); 177, 727 (1996).
exhibit different exponents, from 1 to 3/2 [cf. Eqs. (14), [13] K. B. Efetov, Supersymmetry in Disorder and Chaos
(Cambrigde University Press, NewYork,1997).
(17), and (18)], with some cross-overin between.
[14] A.Altland,B.D.Simons,D.Taras-Semchuk,Adv.Phys.
The value of L⊥(E) is getting shorter as E decreases, 49, 321 (2000).
and becomes of the order of L at E E . The solution
≤ g [15] K.B.EfetovandV.G.Marikhin,Phys.Rev.B40,12126
of the 1D problem at E E can be found following
≪ g (1989).
Ref. [9]. The function ψ (x,y) has a sharp peak at the
2 [16] M. A. Skvortsov, V. E. Kravtsov, M. V. Feigel’man,
centeroftheinstantonandwiththelogarithmicaccuracy JETP Lett. 68, 84 (1998).
is given by ψ2(x,y) = 4ln(2 x2+y2/L). The result [17] A. Lamacraft, B. D. Simons, Phys. Rev. Lett. 85, 4783
−
for the DoS then reads p (2000).
4
