Voltage fluctuations on a superconductor grain attached to a quantum wire

5 Pages·0.14 MB·English

by  Masaki Oshikawa

Voltage fluctuations on a superconductor grain attached to a quantum wire
Masaki Oshikawa1 and Alexandre M. Zagoskin2
Department of Physics, Tokyo Institute of Technology1, Oh-oka-yama, Meguro, Tokyo 152-8511 JAPAN
Department of Physics and Astronomy, University of British Columbia2, Vancouver, BC V6T1Z1 CANADA
(October 15, 1998)
9
9
9
1 ations will translate into fluctuating Andreev boundary
Whenafinitesuperconductorisincontactwitha1Dnor- condition [8].
n
mal conductor, superconducting phase fluctuations lead to
a First let us present a heuristic picture of the phe-
power-lawresponseofthenormalsubsystem. Asaresult,the
J nomenon at zero temperature. The effect of a sud-
charge fluctuations on the superconductor at zero tempera-
3 den change in a boundary condition is a slow, power-
ture have logarithmic correlator and a 1/ω power spectrum
1 law relaxation of the normal system to its new ground
(1/f-noise). At higher temperatures 1/ω is pushed to the

high frequency region, and 1/ω2 behavior prevails. state, that is, Fermi edge singularity. It is conveniently
]
n described in terms of boundary conformal field theory,
o which allows a unified description of FES and Ander-
c
son orthogonality catastrophe [11]. Both are character-
-
r ized by scaling dimension x of the boundary condition
p Correlated character of transport in mesoscopic sys-
changing operator , so that the correlation function
u tems strongly influences character of current and charge †(τ) (0) τ−2Ox, and the dimension x itself is re-
s
fluctuations there. Examples are non-Poissonian shot O O ∼
t. (cid:10)latedtothe(cid:11)finite-size(O(1/L))correctiontotheground
a noise in quantum point contacts and NS junctions( [1] state energy shift due to the action of :
m and references therein) and giant current noise in super- O
- conducting quantum point contacts [2–4]. Noise mea- L
d surements may provide information about quantum cor- x= π∆E1/L. (1)
n
relations unattainable by any other means [5].
o Hereafterweset¯h=1andthevelocityoftheexcitations
In this paper we will show that equilibrium fluctua-
c (Fermi velocity in the non-interacting case) v =1 unless
[ tions in a mesoscopic NS system reveal the many-body
statedotherwise;Listhecharacteristicsizeofthesystem
reactionofaconductingsystemtoasuddenexternalper-
3 (wire length in our case). This relation holds both for
v turbation, thus linking the noise to such phenomena as
usual potential scattering and Andreev scattering [9].
3 Fermi edge singularity (FES) [6] and Anderson orthog-
Theboundaryconditionchangingoperatorinourcase
0 onality catastrophe [7]. Unlike the standard situation
3 when the perturbation is due to creation of a scatterer, changes the phase of superconductor by Φ: OΦ =
0 exp[Φ ∂ ]=exp[iΦnˆ].Hereweusedtherelationbetween
here the normal subsystem reacts to the change in the ∂φ
1 φ and the Cooper pair number operator on the grain,
8 off-diagonal pairing potential in the superconductor, to nˆ = 1 ∂ , whichholds if n 1[10]. Inagreementwith
9 which it is coupled via Andreev reflection processes on i∂φ h i≫
general CFT argument, the Green’s function of is
/ the NS boundary [8,9]. OΦ
t
ma ItMcoonrseisstpseociffiacamllye,socoscnospidicersuthpeerscyosntedmuctsihnogwgnraininFSig.i1n. GΦ(τ)=DOΦ†(τ)OΦ(0)E≡De−iΦnˆ(τ)eiΦnˆ(0)E=(τ/τ0)−2x,
d- contact with a 1D normal wire N. We assume that the (2)
grain contains a large number of Cooper pairs, n 1,
n
h i≫
o and its capacitance C is large enough (so we are in the with some cutoff parameter τ0. Though by itself GΦ has
c oppositelimittotheCoulombblockaderegime[10]),but no direct physical meaning, it is related to the correla-
v: the contact region is smaller than the Josephson screen- tor of charge fluctuations on the grain: KQ(τ)/(4e2) =
i ing length λ , so that both the modulus and phase of [δnˆ(τ),δnˆ(0)] /2, where δnˆ =nˆ nˆ . Obviously,
X J + −h i
the order parameter ∆ = ∆eiφ can be taken constant (cid:10) (cid:11)
ar tqhuearsei.paTrteimclpeeerxactiutraetiiosnlsowin|etnh|oeusguhp,eTrco≪nd|u∆ct|,ort.oFniengallelcyt, KQ(τ)4−e2KQ(0) ≡ 12 [δnˆ(τ),δnˆ(0)]+ − δnˆ2 =
(cid:10) (cid:11) (cid:10) (cid:11)
we assume that the 1D conductor is scattering free, and 1 [nˆ(τ),nˆ(0)] nˆ2 = lim 21(GΦ(τ)+GΦ(τ)∗)−1. (3)
its length L ξ0, where ξ0 is the superconducting co- 2 + − Φ→0 Φ2
≫ (cid:10) (cid:11) (cid:10) (cid:11)
herence length.
Chargefluctuationsaremeasuredbye.g. monitoringthe
If isolated, the superconducting phase in the grain
electrostatic potential of the grain.
wouldfluctuate, the system being in the state with fixed
The dimension x is easily found [9] if notice that the
particle number (see e.g. [10], Ch.7). To the 1D conduc-
energy correction entering (1) is simply the Josephson
torbroughtincontactwiththegrain,thesephasefluctu-
energy of a ballistic 1D SNS junction of length L, with
1
phase difference Φ [12,13]at zerotemperature: E (Φ)= √2e∂θ(x,t)
J J(x,t)= . (5)
Φ2/(4πL), so that x = LE (Φ)/π = [Φ/(2π)]2. Sub- − π ∂t
J
stituting this and (2) into (3), we find for the charge
Therefore the electric charge of the superconductor at
fluctuations on the grain
time t is given by
K (τ) 1 τ
Q = δnˆ2 ln . (4) t √2e
4e2 − 2π2 τ Q(t)=Q( ) J(0,t)dt= θ(0,t), (6)
(cid:10) (cid:11) 0 −∞ −Z π
−∞
For the spectral density of charge fluctuations, S (ω)=
Q
∞ where we have defined Q and θ so that Q = θ(0,t) = 0
2 K (τ)cos(ωτ)dτ, suchlogarithmicbehaviorofcor-
0 Q corresponds to the neutral superconductor grain. Simi-
reRlator translates into SQ(ω) 1/ω (1/f-noise) in the
∝ lar identification was used in studies of Coulomb Block-
corresponding frequency interval; it is often observed in
ade [18], although our application is to quite different
spin glasses, magnetic polycrystals and other systems
regime.
with wide distributionofrelaxationtimes [14,15]. Inour
IncludingthechargingenergyassociatedwithQ,E =
case the source of slow relaxation times is the normal Q
system with its power-law approach to the new ground Q2C2 = πeC22θ(0,t)2, we arrive at the effective action of the
state. Theideaofinvokinginfraredsingularitytoexplain problem:
scaling that underlies current 1/f-noiseobservedin nor-
1 u
malconductorswasexploitedine.g. modelof“quantum S = dtdx (∂ Θ)2 dt Θ(0,t)2. (7)
µ
Z Z 2 −Z 2
1/f-noise” due to Bremßtrahlung, but it encountered fa-
tal difficulties (see [15] and references therein). In con-
Here Θ = θ/√πK and u = 2e2K/(πC) (u =
trast, we see no fundamental difficulty in the present
2e2K/(π¯hC) in conventionalunits). This is a free boson
mechanism, which will be shown to be consistent with
action with a mass term only on the boundary, and this
fluctuation-dissipation theorem.
effectivetheoryisexactlysolvable. Thecalculationofthe
Expression (4) is obviously only an intermediate
charge fluctuation in the superconductor is reduced to
asymptotics for 1 τ/τ exp[2π2 δnˆ2 ]. Therefore
≪ 0 ≪ the calculation of the correlation function of the bound-
we needamoreconsistentapproach,ta(cid:10)king(cid:11)intoaccount
aryfieldΘ(0,t). Thephysicsofthe problemiscontained
effectsofchargingenergyandfinitetemperature. Wewill
in a proper choice of the boundary condition at x = 0
seethatthe regionof1/f-fluctuationsis pushedbyther-
(on the NS boundary) [19].
mal effects to high frequencies, and 1/ω2-type spectrum
The Matsubara Green’s function of Θ (for τ = it)
prevails except at very low temperature.
(x,τ)= T Θ(x,τ)Θ(0,0) , isthe Green’sfunctionof
We will describe the wire in terms of Tomonaga- G −h τ i
theLaplacianontheplane,withtheboundaryconditions
Luttingerliquid(TLL),whichallowsustotreatsimulta-
∂ Θ(x=0,τ)=uΘ(x=0,τ)andΘ(x,τ+β)=Θ(x,τ),
neously non-interacting (1D Fermi-liquid) and interact- x
where β = 1/T is the inverse temperature. As a conse-
ing case of such S-TLL system [16]. Though as we will
quence, the Green’s function for the source at (x ,0)
see interactions do not change qualitative picture of the G 0
satisfies
phenomenon, they may lead to interesting side effects.
Spin and charge degrees of freedom in TLL separate. (∂ 2+∂ 2) (x,τ)=δ(x x )δ(τ) (8)
x τ 0
Since spin is totally reflected by NS boundary both in G −
∂ (x=0,τ)=u (x=0,τ). (9)
the process of ideal Andreev reflection and ideal normal xG G
reflection, it is irrelevant and will be dropped. For the
Because we are interested in the correlation function of
charge, the Lagrangiandensity is =1/(2πK)[(∂ θ)2
t the boundary, we take the limit x 0. Integrating
L − 0
(∂ θ)2],whereK isthe couplingconstant,whichisunity →
x (8) over x from 0 to x +0, we see that the boundary
0
for non-interacting electrons.
condition (9) must be now modified:
Physically, the boson field θ(x,t) corresponds to the
phase of Charge Density Wave (CDW), while its dual ∂ (x=0,τ)=u (x=0,τ) δ(τ). (10)
x
G G −
φ(x,t) to the superconducting phase of the Cooper pair
[6]. For a perfect NS interface, the boundary value of For Fourier components we obtain an ordinary differen-
the superconducting phase of the TLL should be equal tialequation: ∂2 (x,ω )=ω 2 withtheboundarycon-
xG n n G
to that of the superconductor, which in the absence of dition∂ (x=0,ω )=u (x=0,ω ) 1.Imposing the
x n n
G G −
phase fluctuations gives a Neumann boundary condition asymptotic condition lim (x,τ) 0, we find
x→∞
G →
forthefieldθ(x,t)inthewire[17],equivalenttoaDirich-
1
let boundary condition on φ: φ(0,t)=Φ=const. (x,ω )= − e−|ωn|x. (11)
n
G u+ ω
ThechargeQonthegraincanberelatedtothebound- n
| |
ary value of the “CDW” field θ. Indeed, the current in
The Matsubara Green’s function for the boundary field
TLL can be written as
is (x=0,ω )= 1/(u+ ω ).
n n
G − | |
2
Thecorrelationfunctions inthe realtime areobtained istheGreen’sfunctionoftheLaplacianwiththesourceat
by ananalytic continuationfromthe MatsubaraGreen’s x . The1/ωfluctuationofthegrainvoltagetranslatesto
0
function. In a standardway, we obtain the retarded and anω-linearfluctuation of the current,althougha similar
advanced Green’s functions, GR(A)(ω) = 1/(u iω). behaviorpersists evenfor the currentfar fromthe grain.
− ∓
Their mismatch at the real axis, which is pure imagi- So far, we have considered a semi-infinite wire con-
nary, gives the spectral density of the system: ρ(ω) = nected to the grain. In the remainder of this paper, we
2ImGR(ω) = 2ω/(u2+ω2). The spectrum of electro- discuss how our results are affected by wire being finite.
−
static potential fluctuations on the grain is given by: Instead of manufacturing a very long wire, it would
be easier to connect the other end of wire to a normal
S (ω) ∞ 1
S (ω)= Q = Q(t)Q(0)+Q(0)Q(t) eiωtdt reservoir. Weassumethatthejunctiontothereservoiris
V C2 Z−∞ 2C2h i adiabatic(nobackscattering),andthephaseinformation
2Ke2 1 βω 2Ke2 ω βω is immediately lost in the reservoir (perfect thermaliza-
= ρ(ω) coth( )= coth( ). (12)
πC2 2 2 πC2 u2+ω2 2 tion). Then the situation is not qualitatively different
fromthecaseofthesemi-infinitewire,sincetheparticles
This result has the form expected from fluctuation-
can leave to infinity (reservoir), and the usual reason-
dissipation theorem (FDT) [20].
ing of Landauer formalism applies [21]; the transport of
In the low temperature/high frequency limit βω 1,
≫ the chargefrom/tothe graincontrolledbythe geometric
the spectrum is proportional to ω /(u2 +ω2). At fre-
| | restriction (narrow 1D channel) implies that our theory
quency much higher than the “infrared cutoff” u, the
basedon1Dholds,eventhoughthe systemis notpurely
spectrum is indeed 1/ω as expected from the heuristic
1D. Thus we expect our results for the semi-infinite wire
argument (4). On the other hand, at smaller ω fluctua-
applies to the more realistic case of the finite wire con-
tionsaresuppressed,becausefinitecapacitanceobviously
nected to a normal reservoir, if the interaction effects in
excludeslargefluctuationofthechargeinthegrain. The
the wire can be neglected.
1/ω spectrum, which is independent of the interaction
When the interaction in the wire is not negligible, the
parameterK. Itautomaticallyarisesfromthefreeboson
problem is more subtle, since a quasiparticle excitation
fieldtheory oncethe bosonfielditself is identifiedwith a
inside the wire is a collective motion of many electrons
physical observable, because the correlation function of
andcannotbe simply transferredto the reservoir. While
thebosonfieldislogarithmic. Wehavethusshownanex-
arigoroustheoryofjunction betweenaTLL andnormal
ample of “quantum 1/f noise” naturally derived within
reservoir is not yet established, it has been studied by
the framework of equilibrium statistical mechanics and
modelling the reservoir with a (semi-infinite) 1D Fermi
fieldtheory. Althoughthe 1/ω behavioris onlyexpected
liquid. As a simplest model, we study a finite wire of
at very low temperature and high frequency, we believe
lengthL,inwhichthecouplingconstantisK,attachedto
this presentscertaininterestbothfromthe pointofview
semi-infinite wire with coupling constant K′, K αK′.
of translationof infrared divergences into 1/f-spectrum, ≡
Tomodelthereservoirbya1DFermiliquid,K′shouldbe
and of its destruction by thermal fluctuations.
unity but we shall obtain the solution for general values
At higher temperature/lower frequency βω 1, the
≫ of K′ because it can be applied also to other interesting
spectrum is proportional to T/(u2+ω2). The spectrum
cases. It has been found that normal and Andreev-type
is(restoringthefundamentalconstants)4kTKe2/(π¯hω2)
reflections occur at the boundary if K =K′ [22].
for frequency larger than inverse capacitance ω u. In 6
≫ Byasimilarlineofargumentsasbefore,weobtainthe
the opposite limit of small frequency ω u, (but still
≪ Matsubara Green’s function of the boundary field Θ as
ωβ 1),thefluctuationissimplyawhitenoisewiththe
≫
magnitude 2R kT/(4K) where R = h/e2, again in a
H H (0,ω )=
n
complete agreementwithFDT.The transitionfromlow- G
[cosh(ω L)+αsinh(ω L)]
to high- temperature behavioroccurs at βω 1, namely − n | n| (14)
∼ (u+αω )cosh(ω L)+(ω +αu)sinh(ω L)
kT hf (see Fig.2). For T = 10mK, the boundary is | n| n | n| n
∼
at f 200MHz. This would be within the capabilities
Since the temperature effect is given by the same FDT
∼
of modern experiment; we expect the crossover between
factor coth(βω/2)/2 as in the semi-infinite wire case,
1/ω2 and 1/ω behavior could be directly observable.
we concentrate on how the spectral density ρ(ω) =
The charge fluctuation in the grain means the current 2ImGR(ω) is affected.
flowing through the wire also fluctuates. The correla- −
For α = 1, the result reduces to the semi-infinite wire
tion function of the currentcan be obtained by a similar
case, as it should be. For α 1, the overall shape of
method. Forexample,theMatsubaraGreen’sfunctionof ≈
thespectraldensityremainsthe same. However,avaria-
ofthecurrentatJ(x )isgivenby(2e2ω2/π2) (x ,ω ),
0 n Gx0 0 n tion with the period 1/L is superimposed on the depen-
where
dence for semi-infinite wire (Fig.3). This is the resonant
1 u ω structuredue to (normalandAndreev)reflectionsatthe
(x ,ω )= 1 −| n|e−2|ωn|x0 (13)
Gx0 0 n −2ω (cid:20) − u+ ω (cid:21) boundarybetweentwowires. Ifthejunctionbetweenthe
n n
| | | |
3
wire and the reservoir is smooth and not abrupt, we ex-
pectthattheresonantstructureissmearedout;thenthe
observedspectrumwouldbequiteclosetotheprediction
for the semi-infinite wire.
If α far from 1, the resonant structure dominates. In
[1] M.D.M. de Jong and C.W.J. Beenakker, in: ”Meso-
particular,thecaseK′ 0(orK′ )isequivalentto
scopic Electron Transport,” edited by L.L. Sohn, L.P.
→ →∞
a finite wire with open end (or a finite wire attached to
Kouwenhoven, and G. Schoen, NATO ASI Series Vol.
a very large superconductor reservoir), respectively. In 345(KluwerAcademicPublishers,Dordrecht,1997),pp.
these limits α , (α 0) the spectral density devel- 225-258.
→∞ →
ops into a set of discrete δ-functions with peak spacings [2] A. Mart´ın-Rodero, A. Levy Yeyati, and F.J. Garc´ıa-
of order 1/L, rather than a continuous function of ω. Vidal, Phys.Rev.B53, R8891 (1996).
This reflects the finiteness of the system. In the L [3] D. Averin and H.T. Imam, Phys. Rev. Lett. 76, 3814
limit the δ-functions become dense and converge t→o th∞e (1996).
[4] P. Dieleman et al., Phys. Rev.Lett. 79, 3486 (1997).
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[5] M.Bu¨ttiker,Phys.Rev.Lett.68,843(1992);G.B.Leso-
wire.
vik and L.S. Levitov, Phys. Rev. Lett. 72, 538 (1994);
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wire L 0. For the open end case (K′ 0), the reso-
2763 (1998).
→ →
nancesarefoundatω (2n 1)π/(2L)wherenisanat-
[6] G.D. Mahan, Many-particle physics. New York: Plenum
∼ −
uralnumber. Forthewireconnectedtothesuperconduc- Press (1990).
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found at ω nπ/L, n = 1,2,.... The resonance which Rev. 164, 352 (1967).
would corres∼pond to n=0 is actually affected by the fi- [8] A.F. Andreev,Sov. Phys.JETP 19, 1228 (1964).
nitecapacitance,anditslocationisgivenbyω =1/√LC. [9] A.M. Zagoskin and I.Affleck,to be published.
This can be simply understood from the following ar- [10] M. Tinkham, Introduction to superconductivity (2nd
ed.), New York: McGraw Hill (1996), Ch.7.
gument: from the conjugate relation between the phase
[11] I. Affleck and A.W.W. Ludwig, J. Phys. A: Math. Gen.
and the charge, the effective Hamiltonian of the system
27, 5735 (1994).
is givenby (see e.g.[10])Hˆ =−21C(∂∂Φ)2+21L(Φ−Φ0)2, [12] G. Ishii, Progr. Theor. Phys. 44, 1525 (1970).
where Φ is the superconducting phase of the grain, the
[13] A.V. Svidzinski˘ı, T.N. Antsygina, and E.N. Bratus’,
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Inconclusion,wehaveshownthatquantumphasefluc- and F.W.J. Hekking,ibid.,6653.
tuations in a superconducting grain in contact with a [17] M. Oshikawa and A.M. Zagoskin, unpublished.
[18] K.A. Matveev, Phys.Rev. B 51, 1743 (1994).
normal wire are translated into equilibrium charge fluc-
[19] When thereare more than one channels in thewire, the
tuations in the system, with specific spectrum demon-
action isasum offree boson ones, asfar asthewire can
strating transition from 1/ω2 to 1/ω behavior at very
be treated as collection of independent TLL’s. The volt-
low temperature. The mechanism of fluctuations is re-
agefluctuationisalsogivenbythesumofsingle-channel
action of the normal subsystem to sudden change of the
contributions.Inparticular,ifthereareN identicalchan-
off-diagonalpairingpotentialontheNSboundary(Fermi nels, the fluctuation is multiplied by N, and it suffices
edgesingularity),leadinginthelow-temperaturelimitto to consider the single-channel case. (If interchannel in-
quantum1/f noise. Similareffectcanbeexpectedtoex- teractions cannotbeneglected ,thissimplepictureisno
istalsoinsystemswithsuperconductinggrainincontact longerapplicableforwireswithN >1duetoappearance
with 2D conductors under appropriate conditions. of collective interchannelexcitations [Ya.M. Blanter and
While the experimental observation of this effect M.Bu¨ttiker, Europhys.Lett. 42, 535 (1998)].)
[20] E.M. Lifshits and L.P. Pitaevski˘ı, Statistical Physics,
presents a certain challenge, we hope it to be possible
Part 2, Pergamon Press: Oxford et al. (1980).
in near future. Concerning the experiments, impurity
[21] Y.Imry,Introductiontomesoscopicphysics(OxfordUni-
scatterings and physical effects of probing will require
versity Press, 1997).
further theoreticalstudy, whichis howeverbeyondscope
[22] I.SafiandH.J.Schulz,cond-mat/9803326andreferences
of the present paper.
therein.
WearegratefultoI.Affleck,N.KokuboandS.Okuma
for valuable discussions andto M. Bu¨ttiker and R. Fazio
for helpful comments on the manuscript.
4
FIG.1. SuperconductinggrainS incontactwitha1Dcon-
ducting wire N. Tip P probes the electrostatic potential of
thegrain.
103
3
102
2
10
1
1
_S 0
g
l _
-1 T = 0.1
-2 1/w_2 1/w_
-3
-4
-1 0 1 lgw_ 2 3 4
FIG. 2. Reduced spectral density of voltage noise in a
grain, S¯(ω) = SV(ω)/[Ke2/πc2] (Eq.(12)), as a function of
ω¯ ≡ω/ufor different values of T¯≡T/u.
1
0.5
0
-0.5
-1
-1.5
-1 -0.5 0 0.5 1
FIG.3. Resonantstructureofspectraldensityforα=0.1
(dots), α= 1 (solid line), and α= 10. Dimensionless length
ofthewireL=1; T¯=1. Dashedstraightlineillustrates1/ω¯
dependence.
5