Table Of ContentParamagnetic Instability at Normal-Metal – Superconductor Interfaces
Alban L. Fauch`erea, Wolfgang Belzigb, and Gianni Blattera
aTheoretische Physik, Eidgen¨ossische Technische Hochschule, CH-8093 Zu¨rich, Switzerland
bInstitut fu¨r Theoretische Festk¨orperphysik, Universit¨at Karlsruhe, D-76128 Karlsruhe, Germany
(December15, 1998)
9
9 We study the proximity coherence in a mesoscopic normal-metal film (N) in contact with a
9
superconductor (S). Accounting for a repulsive interaction between the electrons in the normal
1
metal, we find an enhanced local density of states close to the NS interface. The sharp peak in
n thedensityispinnedtotheFermienergyandleadstospontaneousparamagneticinterfacecurrents.
a Theinducedorbitalmagneticmomentsexhibitthecharacteristicfeaturesofparamagneticreentrance
J observed in normal-metal coated superconducting cylinders[Phys. Rev.Lett. 65, 1514 (1990)].
2
1
]
n
A normal metal in contact with a superconductor ex- finite order parameter ∆(x) is induced in the metal, op-
o
c hibits the phenomenonofproximity—the superconduc- posite in sign as compared to ∆S in the superconduc-
- tor exports its coherent state across the interface into tor, see Ref. [6]. The NS junctions then behaves like a
r
p the normal metal. On a microscopic level, this phe- Josephson junction with a phase difference π, trapping
u nomenon is described through the Andreev reflection of quasi-particle states at the Fermi energy close to the NS
s
the normal-metal quasi-particles at the normal-metal– interface. The local density of states N(E,x) exhibits
.
t superconductor(NS)interface,convertingnormal-tosu- a peak at zero energy on top of the Andreev density of
a
m percurrent. Proximity superconductivity exhibits a rich states, as shown in Fig. 1. This peak involves a macro-
phenomenology and has attracted considerable interest scopic number of states with density n ∼ k2/d, which
- p F
d recently [1]. A particularly puzzling finding is the ultra- in the following we call the π-states.
n
low-temperature reentrance observed in normal-metal The change in the DOS crucially affects the response
o
coated superconducting cylinders [2], where, contrary to of the proximity metal. The linear current responsej[A]
c
[ expectation, the fully diamagnetic cylinder develops a canbedividedintotwocontributionsj =jdia+jpara,the
1 paramagneticresponseatlowtemperatures. Recently,it diamagneticcurrentjdia =−(e2n/mc)Agivingthe rigid
has been speculated that a novel kind of persistent cur- response of the bulk density n = k3/3π2 and the para-
v F
2 rent states circling the cylinder might be responsible for magnetic currentjpara followingfromthe deformationof
1 this phenomenon [3], but closer inspection of the experi- the wavefunction at the Fermi surface [7],
1 mentallymeasurablequantitiesrevealsthatthepredicted
1 effectistoosmall[4]. InthisLetter,wedemonstratethat e2n ∂f N(E)
0 jpara = A dE − (1)
the presence of a repulsive electron-electron interaction mc Z (cid:18) ∂E(cid:19) N
9 0
9 inthe normalmetalnaturallyleads tothe appearanceof
for slowly varying fields A (f is the Fermi occupation
/ a paramagnetic instability at very low temperature, of-
t number). While in a bulk superconductor the param-
a fering a possible explanation of the reentrance effect in
m agnetic current is quenched by the energy gap at low
the NS cylinders.
temperatures producing a net diamagnetic response, the
-
d To be specific, we shall consider a clean normal-metal paramagnetic current of a bulk normal metal cancels
n
slabof thickness d (0<x<d), inperfect contactwith a the diamagnetic current exactly. In the non-interacting
o
bulk, conventional superconductor. The proximity ef- metal under proximity, the linear density of states sup-
c
: fect is mediated by the Andreev reflection at the in- pressionN(E)∝E isstillsufficienttosuppressthepara-
v
terface with the superconductor, which transforms inci- magnetic current at zero temperature [8]. Including a
i
X dent electronsinto back-reflectedholes, thus binding the repulsive interaction places the system in the opposite
r quasi-particles states to the normal layer for E < ∆S. limit: ThesharpDOSpeakattheFermilevelproducesa
a In the usual free electron gas description of the normal paramagneticsignalwhichovercompensatesthe diamag-
metal, the Andreev bound states are found at E = netic response. Such a paramagnetic response naturally
n
¯hv (2n+1)π/4d (n = 0,1,..; v = v cosϑ) producing leads to an instability: The free energy δF =−cjδA<0
x x F
a linear suppression of the DOS [5] N(E) ∼ N Ed/¯hv canbeloweredviaanon-zeromagneticinductioninduced
0 F
close to the Fermi level E = 0 (N = mk /¯h2π2). In by spontaneous currents along the NS interface. The in-
0 F
the following we assume that the electron-electron in- terface currents are associated with an orbital magneti-
teraction in the normal layer, which follows from the zation M(T) producing a low-temperature reentrance in
delicate balance between the phonon-mediated- and the the magnetic susceptibility.
Coulomb-interaction, is repulsive. As a consequence, a In the following we present a quantitative analysis
1
of the paramagnetic instability induced by the π-states. from a value ∼ −|V /V |∆ at the NS interface, to ∼
N S S
The magnetic induction B (x) parallel to the surface is −V N ¯hv /dattheouterboundary. ClosetotheNSin-
z N 0 F
describedby the vectorpotentialAy(x) whichdrivesthe terface, the local DOS N(E,x) =N0Re[hg−iE+δ(x,vx)i]
currents j (x). The electron-electron interaction in the exhibits a pronounced peak at zero energy, as shown in
y
superconductor is accounted for by an effective coupling Fig. 1. In order to proceed with analytical results, we
constant V < 0 and similarly V > 0 in the normal approximate the order parameter by a step function,
S N
metal, see also Refs. [9,10]. Two self-consistency prob-
lems have to be solved: First, we evaluate the order pa- ∆(x)= ∆S, x<0,
rameter ∆(x) accounting for the different coupling con- (cid:26)−∆N, 0<x<d,
stants in the superconductor and the normal metal, and
obtain the local DOS N(E,x). Second, we determine where ∆N ∝ VN enters as a parameter. The Green’s
the current functional j[A] which we solve together with function in the normal layer x > 0 can be determined
Maxwell’sequationtofindthespontaneousinterfacecur- exactly and takes the form
rents.
¯hωsinh[χ(d)−γ]+∆ cosh[χ(d−x)]
N
6 gω(x,vx)= ,
¯hΩcosh[χ(d)−γ]
101
1
(4)
N(E=0,x) N
(E
N0 4 ∆∆(x)/S 0 ∆(x) 1100−01 =0,x) o∆wnhNde/rt¯heeΩrχm((xwi)ne=(c4o)2nΩdsexids/cevrrxib,theh¯se2tΩlhim2e=iπt-s∆∆ta2NSte≫+sh¯a∆2tωNth2,,e¯haNωnS)d.itnTathneerhfaγsecc=e-.
)/ The poles of the Green’s function ath¯ω =−iE+0 yield
E 0 d
N( x the bound state energies. While for E ≫ ∆N the An-
dreev states of the free electron gas are down-shifted by
2
δE ≈ −2∆ /(2n + 1)π (n = 0,1,..), below the gap
n N
E <∆ we find the π-states at
N(E,x=0) N
N(E,x=d)
2 ∆2 −E2d
E =∆ /cosh N ∼∆ e−2∆Nd/h¯vx, (5)
0 N p ¯hv N
0.0 0.5 x
E / E
A exponentiallyclosetoFermienergy. Alltrajectorieswith
FIG.1. Local DOS N(E,x) at theNS interface x=0 and v =v cosϑ≪∆ d/¯h possess a bound state at E ≈0,
x F N
at the metal boundary x = d (EA = ¯hvF/d), as it follows thusproducingthemacroscopicweightofthezeroenergy
from the self-consistent solution of the Eilenberger equation, DOS peak: For ∆ >¯hv /d the number of π-states per
N F
Eqs. (2) and (3), for a thickness d = 10¯hvF/∆S and the unit surface N is equal to the number of transverse
coupling constants VS = −0.3 and VN = 0.1. Inset: Spa- levels N ∼ks2ur,fwhile for ∆ <¯hv /dit is reducedto
tial dependence of the order parameter ∆(x) and local DOS N ∼sku2rf(∆ dF/¯hv )2 viathNereducFtionoftheavailable
N(E =0,x) at thepeak energy (S: x<0, N:x>0). surf F N F
solid angle cosϑ<∆ d/¯hv .
N F
We derive the current-field relations at low tempera-
We use the quasi-classical description following from
tures,assumingk T ≪¯hv /d,∆ . This implies a ther-
the Eilenberger equation for gˆ, B F N
mal length ξ (T) = h¯v /2πk T larger than the thick-
N F B
−¯hv ∂ gˆ=[{¯hω+iev A (x)/c}τˆ +∆(x)τˆ ,gˆ], (2) ness d and no thermal smearing on the scale ∆ . Only
x x y y 3 1 N
the trajectorieswithcosϑ<∆ d/¯hv contribute to the
N F
where the 2×2 matrix gˆ contains the Green’s functions
current at low temperatures. We describe them in the
g (x,v ) and f (x,v ) (ω Matsubara frequency, v =
ω x ω x x limit h¯v /∆ d→0 by
x N
v cosϑ, τˆ Paulimatrices,seeRef.[11]). Eq.(2)iscom-
F i
pletedbythe self-consistencyrelationforthe pairpoten- ω ω∆
g (x,v )= + N e−χ(x). (6)
tial (h..i is the angular average), ω x
Ω Ω(¯hΩ−∆ )
N
∆(x)=−VN πT hf (x,v )i. (3)
0 ω x The current in the presence of a slowly varying vector
ωX>0
potential A, follows from Eq. (6) after replacing ω by
The self-consistent numerical solution of Eqs. (2) and ω + iev A/¯hc and inserting it into the current expres-
y
(3) is shown in the inset of Fig. 1. The course of the sion j (x) = ieN 2πT hv g(x,v ,v )i. In addition
y 0 ω>0 y x y
order parameter in the normal layer is asymptotically to the usual diamagnPetic current jdia = −(c/4πλ2)A
givenby ∆(x)∼−V N ¯hv /x, as expected from the f- [λ = (mc2/4πne2)−1/2 denotes the London length], we
N 0 F
functioninthenon-interactingcaseV =0. ∆(x)decays obtain the paramagnetic current
N
2
j ≈ c e−x/αξN0 α 3Φ0 arctan evFA , (7) fectthequalitativenatureofthefinalresults,aswehave
para 4πλ2 2πξN0 cπkBT checked numerically).
ThemagnetizationcurvesM(T,H),whichfollowfrom
in the limit evFA/c,kBT ≪ ∆N. Here, Φ0 = π¯hc/e de- Eq. (7) are shown in Fig. 2. Approaching from large
notesthefluxquantum,ξN0 =h¯vF/2∆N givestheextent fields, Fig. 2 shows two paramagnetic branches P± with
of the π-states, and under the assumption ∆N > ¯hvF/d alineardiamagneticslopeexhibitingaspontaneousmag-
we have α = 1. At temperatures kBT ≫ evFA/c the netization in zero field. They result from the superposi-
paramagnetic current jpara ∼ (c/λ2)(∆N/kBT)A is lin- tion of the paramagnetic magnetization M0(T) and the
earinAand∝1/T,asignatureofthethermallysmeared Meissner response to the applied field H. As the field is
zero energy DOS peak, and competes with the diamag- decreased (increased) past H = 0, the branch P+ (P−)
netic current on the scale ξN0 . At T →0, Eq. (7) is non- becomes meta-stable. The spontaneous magnetization
linear in the field and generates the spontaneous para- M (T)appearsbelowasecondordertransitionpointTM
0 c
magnetic current. This paramagnetic interface current and saturates at low temperatures, as shown in the in-
results from the energy splitting of the π-states in the set of Fig. 2. The magnetization curve includes a dia-
field, E ≈ ±evFA/c, allowing the system to gain en- magnetic branch D, which arises from the competition
ergy by shifting the DOS below the Fermi surface. For between the paramagnetic instability and the thermal
∆N < ¯hvF/d, the paramagnetic current is reduced by smearing and is thermodynamically unstable.
the factor α = (∆Nd/¯hvF) < 1 in Eq. (7). The surface In the following we give a semi-quantitative analysis
current I = jdx ∼ α2cΦ0/λ2 is in agreement with the of the magnetization M = dxM(x)/d, first at zero
current estimRate Iπ ∼ NsurfevF based on the number of temperature and field [M0], pRroceeding to finite temper-
π-states at zero energy. Eq. (7) thus always produces a atures [M (T)], and finally including an applied mag-
0
netparamagneticresponseatlowtemperatureandfields. netic field H [M(T,H)]. The boundary conditions are
given by A(0) = 0 and ∂ A(d) = H. We concen-
x
trate on the most relevant limit where ξ0 ,d ≫ λ. At
N
T = 0, according to Eq. (7), the paramagnetic interface
T=0.05
1 TT==00..24 P cmuartrcehnetdjb∼y jαcΦ∼0/−λ2cξAN0/λr2e,mpariondsuucninscgraeevneecdtournptoiltebnetiniagl
+ dia
A∼αΦ /ξ0 on the scale λ. The vector potential A sat-
0 D 0 N
Φ urates beyond λ, as the para- and diamagnetic currents
/
2 canceleachother. Assumingthatthe π-statesextendup
) 0
0
ξN k T / ∆ to the outer metal surface (α < 1), the induced magne-
M ( P− 0.0 B0.2 N 0.4 tization M =A(d)/4πd is given by
π
−4 −1 20Φ)/0N0.4 M0 ∼αξΦN00d ∼ (ξΦN00)2. (8)
ξ
M (00.0 Wenotethatalthoughthespontaneouscurrentsincrease
as ∆ >¯hv /d (α=1) they are screened exponentially
N F
0 1 2 beyond the extent of the π-states in this limit, giving
H (ξ 0)2/Φ a magnetization M ∼ (Φ /ξ0 d)exp[−(d−ξ0 )/λ]. We
N 0 0 0 N N
assume α<1 in the following.
FIG. 2. Magnetization M(T,H) curve at various temper- At finite temperature, the spontaneous magnetization
atures (in units of ∆N/kB), for λ = 0.3d and ξN0 = d. The is suppressed by the factor arctan(ev A/cπk T), which
F B
two meta-stable branches P± exhibit a spontaneous magne- itself depends on the magnetization via A∼Md, imply-
tization in zero field, the diamagnetic branch D is unstable.
ing the implicit equation
Inset: Zero field magnetization M0(T).
M (T) M (T)α∆
0 0 N
The evaluation of the induced magnetization re- ∼arctan . (9)
M M k T
0 0 B
quires the self-consistent solution of Maxwell’s equation
−∂2A(x) = 4πj(x)/c together with the current func- The spontaneous magnetization appears below a second
x
tionalj[A(x)]. The solutionof the screeningproblemre- ordertransitionatk TM ∼α∆ ,saturatingatlowtem-
B c N
quires the full dispersive relationbetweenj(q) andA(q), peratures,asshowninthe insetofFig.2. The transition
which in the proximity effect typically features a non- temperatureisequalinmagnitudetotheenergysplitting
local current-field dependence on the scale ξ0 [11]. Eq. of the DOS peak E ∼ev A/c∼α∆ .
N F N
(7) represents the long wavelengthlimit q →0. For sim- Under an applied magnetic field H, the Meissner cur-
plicity,hereweuseEq.(7)undertheassumptionoflocal rent j screens both the spontaneous interface current
dia
response(accountingforthefullnon-localitydoesnotaf- and the applied field. At zero temperature we deal with
3
a linear problem and the magnetization is given by the counting for the nonlocality of the current-field relation
superpositionM(H)=M +χHofthespontaneousmag- and its sensitivity to disorder.
0
neticmomentM andtheMeissnerresponseχH. Asthe
0 Inconclusion,wehavedemonstratedthattheinclusion
temperature increases, M (T) decreases and the meta-
0 of a finite electron-electronrepulsionin a proximity cou-
stableregimeshrinks. AtT >TM thespontaneousmag-
c pled normal-metal layer naturally produces spontaneous
netization in zero field has disappeared, the signature
interfacecurrentsleadingtoaparamagneticreentrancein
of the paramagnetic currents remains, however, reduc-
the magnetic response: The sign change in the coupling
ing the diamagnetic susceptibility χ at small fields. At
across the NS interface leads to the trapping of π-states
largetemperature T≫TM we recoverthe pure Meissner
c at the Fermi energy. The frustrated NS junction relaxes
response.
throughthegenerationofspontaneousinterfacecurrents,
Note that the two meta-stable branchesP+ and P− in inducingaparamagneticmoment. Anon-trivialissuere-
the magnetization curve, see Fig. 2, imply a first order
mainstherequirementthattheelectron-electroninterac-
transition with changing field at H = 0. The first order
tion be repulsive at the low energy scales involved. In-
transitionissimilartothemagneticbreakdownoccurring
teresting consequencesof this assumptionhavebeen dis-
in the same system at large fields between the fully dia-
cussedinthecontextoftheproximityeffect[6]andmost
magnetic phase and a field penetration phase [12]. The
recently in relation to the low temperature transport in
rotation of the magnetic moments to the energetically
mesoscopicNS structures[9,17]. Infact,the noblemetal
more favorable polarization will show the hysteretic be-
coatingsusedintheexperimentsofMotaandco-workers
havior typical for a first order transition. The transi-
[2]appeartobethemostplausiblecandidatesforarepul-
tion from P+ to P− implies a paramagnetic slope in the sive electron-electroninteraction. Turning the argument
thermodynamic dc-magnetization hM(T,H)i, which will
around, in the light of our findings the experimental ob-
link the meta-stable solutions P± in Fig. 2 and cross the servation of a paramagnetic reentrance can be taken as
origin at M(H = 0) = 0. In summary, we find that
an indication of the presence of a repulsive interaction.
on approaching TM from above, the diamagnetic sus-
c We acknowledge fruitful discussions with C. Bruder,
ceptibility χ = hM(T,H)i/H is reduced, exhibiting a
dc M. Dodgson, V. Geshkenbein, C. Honerkamp, A. Mota,
low-temperaturereentrance. BelowTM,thespontaneous
c B. Mu¨ller, M. Rice, G. Scho¨n, and M. Sigrist.
interface currents produce a net paramagnetic suscepti-
bility χ .
dc
Inthefollowingwediscussourresultsinthecontextof
theexperimentsbyMotaandco-workers,whohavemea-
sured the magnetic response of normal-metal coated su-
perconducting cylinders at low temperatures [2,13]. Re-
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[12]. The Nb-Ag and Nb-Cu cylinders show an anoma-
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(1963).
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[7] J. R. Schrieffer, Theory of Superconductivity, Addison-
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(1997).
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