Nonequilibrium transport in mesoscopic multi-terminal SNS Josephson junctions

0.65 MB·English

by  M. S. Crosser

Nonequilibrium transport in mesoscopic multi-terminal SNS Josephson junctions
M.S. Crosser,1,2 Jian Huang,1,∗ F. Pierre,1,† Pauli Virtanen,3
Tero T. Heikkila¨,3 F. K. Wilhelm,4 and Norman O. Birge1,‡
1Department of Physics and Astronomy, Michigan State University, East Lansing, MI 48824-2320, USA
2Department of Physics, Linfield College, 900 SE Baker Street, McMinnville, OR 97128, USA
3Low Temperature Laboratory, Helsinki University of Technology, P. O. Box 2200, FIN-02015 TKK, Finland
4Department of Physics and Astronomy and Institute for Quantum Computing,
University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada
8 (Dated: February 2, 2008)
0
We report the results of several nonequilibrium experiments performed on superconduct-
0
ing/normal/superconducting (S/N/S) Josephson junctions containing either one or two extra ter-
2
minals that connect to normal reservoirs. Currents injected into the junctions from the normal

n reservoirs induce changes in the electron energy distribution function, which can change the prop-
a erties of the junction. A simple experiment performed on a 3-terminal sample demonstrates that
J
quasiparticle current and supercurrent can coexist in the normal region of the S/N/S junction.

9 When larger voltages are applied to the normal reservoir, the sign of the current-phase relation of
the junction can be reversed, creating a “π-junction.” We compare quantitatively the maximum

] criticalcurrentsobtainedin4-terminalπ-junctionswhenthevoltagesonthenormalreservoirshave
l
l the same or opposite sign with respect to the superconductors. We discuss the challenges involved
a
in creating a “Zeeman” π-junction with a parallel applied magnetic field and show in detail how
h
- theorbitaleffectsuppressesthecriticalcurrent. Finally,whennormalcurrentandsupercurrentare
s simultaneouslypresentinthejunction,thedistributionfunctiondevelopsaspatiallyinhomogeneous
e
component that can be interpreted as an effective temperature gradient across the junction, with
m
a sign that is controllable by the supercurrent. Taken as a whole, these experiments illustrate the
. richness and complexity of S/N/S Josephson junctions in nonequilibrium situations.
t
a
m PACSnumbers: 74.50.+r,73.23.-b,85.25.Am,85.25.Cp
-
d
n I. INTRODUCTION maydependcriticallyonthe ratesofelectron-electronor
o electron-phonon scattering. A pioneering work in this
c area was the demonstration by Baselmans et al.5 that
[ Whenasuperconductingmetal(S)andanormalmetal
the current-phase relation of a S/N/S Josephson junc-
(N) are placed in contact with each other, the proper-
1 tioncanbe reversed,producing aso-called“π-junction”.
ties of both metals are modified near the S/N interface.
v This effect is produced by applying a voltage that suit-
4 This effect, called the superconducting proximity effect, ably modifies the form of the distribution function.
2 was widely studied in the 1960’s.1 Our microscopic un-
Thispaperpresentsresultsofseveralexperimentsper-
3 derstanding of the proximity effect underwent dramatic
formed on S/N/S Josephson junctions with extra leads
1 progress in the 1990’s as a result of new experiments
. connecting the N part of the devices to large normal
1 performedonsubmicronlengthscales,coupledwiththe-
reservoirs. Samples are made from polycrystalline thin
0 oreticalideasaboutphase-coherenttransportfrommeso-
filmsofaluminum(S)andsilver(N)depositedbythermal
8 scopic physics. It is now understood that the conven-
evaporation. Electrical transport is in the diffusive limit
0
tionalproximityeffectinS/NsystemsandthedcJoseph-
: –i.e. theelectronmeanfreepathismuchshorterthanall
v son effect in S/N/S junctions arise from the combina-
otherrelevantlengthscalesintheproblem,includingthe
i tion of three ingredients: Andreev reflection of electrons
X sample length and the phase coherence length. In these
intoholes(andviceversa)attheS/Ninterface,quantum
experiments, the two superconductors are usually at the
r
a phasecoherenceofelectronsandholes,andtime-reversal same potential, referred to as ground. Different voltages
symmetry in the normal metal. Our new understanding
areappliedto the normalreservoirs,whichinmostcases
of the proximity effect in equilibrium situations and in
cause the distribution function in the structures to devi-
linear response transport is demonstrated by a wealth
ate strongly from a Fermi-Dirac distribution.
of beautiful experiments2 and is summarized in several
Severalofthe experimentshavebeen analyzedquanti-
theoretical reviews.3,4
tativelywithintheframeworkoftheUsadelequations,4,6
In the past several years,research in S/N systems has which are appropriate for S/N samples in the diffusive
increasingly focused on nonequilibrium phenomena. Un- limit. The equilibrium component of the Usadel equa-
derstanding nonequilibrium situations is more difficult tion is a diffusion equation describing pair correlations
than understanding near-equilibriumsituations, because in N and S. The nonequilibrium, or Keldysh,component
the electron energy distribution function in nonequilib- consistsoftwocoupledBoltzmannequationsforthespec-
rium may be quite different from a Fermi-Dirac func- tral charge and energy currents. Incorporating inelastic
tion. Insuchsituations,thebehaviorofaspecificsample scattering into the Keldysh equations involves inserting
2
theappropriatecollisionintegrals;butthisprocedurehas bution function is measured by a tunnel probe near one
so far been followed fully in only a few cases. More- of the S/N interfaces. The discussion provides informa-
over, the effect of inelastic scattering on the equilibrium tionthat wasnotincluded inourprevious reportonthis
component of the Usadel equation or the proximity ef- experiment.12 Together these experiments demonstrate
fect on the collision integrals have never been included the richness of phenomena present in S/N/S Josephson
self-consistently to our knowledge. More commonly, re- junctions under nonequilibrium conditions.
searchers analyzing nonequilibrium phenomena solve ei-
ther the Keldysh equation without collision integrals, or
thestandardBoltzmannequationwithcollisionintegrals
II. EXPERIMENTAL TECHNIQUES
but without superconducting correlations, depending on
which aspect of the problem is more important. At the
A. Fabrication
endofthispaperwecomparetheseapproachesasapplied
to the last experiment discussed in the paper.
All samples in this work were fabricated using e-beam
lithography. Abilayerofresistwasdepositedontoanun-
doped Si wafer covered only with its native oxide layer.
The bilayer was formed by first depositing a copolymer
P(MMA/MAA), followed by a second layer of PMMA.
The bilayer was exposed by 35-keV electrons and then
developed to make a mask for evaporation. With the
resist bilayer, it is possible to fabricate undercuts in
the mask, allowing angled evaporation techniques to be
used.13 Therefore, multiple layers of different metals (ei-
ther 99.99% purity Al or 99.9999% purity Ag) were se-
quentially deposited without breaking vacuum.
These techniques were used to prepare the sample
shown in Fig. 1. To create the tunnel probe (TP), 30
nm of Al was deposited while the sample was tilted 45
degrees, creating an actual thickness of about 21 nm of
Al on the surface. Next, a mixture of 90%Ar-10% O
2
FIG.1: SEMimageofsamplewithtwosuperconductingreser- gas was leaked into the vacuum chamber to a pressure
voirs, labeled S1 and S2, and normal reservoir labeled N. A of 60 Torr. After 4 minutes, the chamber was evacuated
tunnelprobe,labeledTP,consistsofthinAloxidizedpriorto
again, in preparation for the following depositions: For
deposition of Ag wire.
the silver wires; labeled R , R , and R ; 30 nm of Ag
1 2 N
was deposited with the plane of the wafer perpendicu-
The paper is organizedas follows: SectionII describes
lar to the evaporation source. For the superconducting
thesamplefabricationandmeasurementtechniques. Sec-
reservoirs, S1 and S2, the sample was tilted 45 degrees
tion III describes a simple experiment, called the “dan-
and rotated 180 degrees in order to deposit 90 nm of
gling arm”, involving a 3-terminal S/N/S device with a
Al (for a 60 nm thickness). Finally, the sample was ro-
singleextraleadtoanormalreservoir. Thedanglingarm
tatedanother140degreesinpreparationforafinal,thick
experiment was first reported by Shaikhaidarov et al..7
layerofAg to be deposited overthe normalreservoir,N.
We include it here because it provides a clear demon-
ThesampleinFig.6followedasimilarprocedure,except
strationofthe superpositionofquasiparticlecurrentand
foregoing the first Al deposition and oxidation steps.
supercurrent in a S/N/S junction, an essential result for
the remainder of the paper. Section IV describes the π-
junction experiment in 3- and 4-terminal devices. The
B. Experimental setup
4-terminalsampleallowsadirectcomparisonofthesitu-
ationspresentinthe3-terminalπ-junction8 andtheorig-
inal4-terminalπ-junctionofBaselmanset al.5 SectionV Samples were measured inside the mixing chamber of
discusses the behavior of the critical supercurrent as a a top loading dilution refrigerator. All electrical leads
function of magnetic field applied parallel to the plane to the sample passedthroughcommercialLC π-filters at
of the sample, and shows the difficulty involved in try- thetopofthecryostatandcoldRCfiltersinthecryostat
ing to achieve a π-junction by Zeeman splitting of the consistingof2.2kΩresistorsinseriesand1nFcapacitors
conduction electrons.9,10 The theoretical calculation rel- coupled to ground.
evant to this geometry is given in the appendix. Section Current-voltage characteristics (I-V curves) were ob-
VI discusses an experiment in which supercurrent and tainedthrough4-probemeasurementsacrossthesample.
quasiparticle current are independently controlled in a The current was swept using a triangle wave and several
3-terminal S/N/S junction, leading to an effective tem- cycles were collected and averaged together. Measure-
perature gradientacrossthe junction.11 The local distri- ments of dI/dV were obtained by adding a slow ( 1
∼
3

mHz) triangle wave pattern to the sine output of a lock-
in amplifier. The lock-in amplifier was operated at low
40
frequencies (less than 100 Hz) to allow for extrapolation
of the system response to zero frequency. Both the in- 30
phase and out-of-phase components of the signal were
20
recorded and utilized in the analysis.
V) 10

(
V 0
35
III. DANGLING ARM EXPERIMENT )
-10
( 30
dI
The dangling arm experiment was first proposed in -20 V/ 25
the Ph.D. thesis of S. Gueron,14 although a related ge- -30 d 20
ometry was discussed by Volkov two years earlier.15 The 15
experiment is performed on a 3-terminal S/N/S Joseph- -40 -3 -2 -1 0 1 2 3
I ( A)
sonjunctionsamplesimilartotheoneshowninFig.1,in
which the tunnel probe in the lowerleft was unused. We -2 -1 0 1 2
labelthethreeterminalsofthesampleS1,S2,andN,and I ( A)
the resistances of the three arms R , R , and R . (We
1 2 N
neglect for the moment the variationof these resistances
FIG.2: VoltageversuscurrentmeasuredbetweenreservoirN
due to proximity effect.) One measures the resistance
andS1withS2floating,atT =51mK.Dottedlinesrepresent
from N to S1, while leaving S2 open (dangling). Naively, slopesof20.7and24.6Ω,whichcorrespondtotheresistances
onemightexpectthemeasuredresistancebetweenNand RP and RN +R1, respectively. Inset: Differential resistance
S tobeequaltoR +R . Thatresultwouldimplythat vs. current under similar conditions, showing agreement be-
1 N 1
the current travels directly from N to S , which in turn tween the two measurement techniques.
1
implies that S and S are at different voltages. Given
1 2
that S and S are coupled by the Josephson effect, the
1 2
relative phase φ between S and S then accumulates at cancelled by the supercurrent, one should expect that
1 2
the Josephson frequency, dφ/dt = 2eV /~. If, however, INS = ISNS(R +R )/R . The data in Figure 3 show
12 c c 1 2 1
the injected current I splits into a piece I through R that this expectation is fulfilled at relatively high tem-
1 1
and a piece I through R , such that I R =I R , then peratures, but that at lower temperature INS falls well
2 2 1 1 2 2 c
S andS willbe atthe same potential. To avoidhaving below this value.
1 2
a net current flowing into the dangling arm, the sample Two reasons for the small values of INS at low tem-
c
must then provide a supercurrent I from S to S that perature were given by Shaikhaidarov et al..7 Those
S 2 1
exactly cancels the quasiparticle current I . In this sce- authors solved the Usadel equation analytically in the
2
nario,R andR areeffectivelyactinginparallel,andthe limitwheretheS/Ninterfaceshavehighresistance(poor
1 2
measuredresistancewillbeR R +R R /(R +R ). transparency), so that proximity effects are small and
P N 1 2 1 2
≡
the Usadelequationcanbe linearized. They pointedout
Figure 2 shows the 2-terminal I V curve taken at that ISNS is suppressed below its equilibrium value due
− c
T = 51 mK from a sample similar to the one shown to the applied voltage at N, a result we will reinforce
in Fig. 1, with nominal resistance values R = 7.0 Ω, below. They also argued that the phase-dependence of
1
R = 7.0 Ω, and R = 16.9 Ω. The inset shows dV/dI the resistances R , R , and R due to proximity effect
2 N 1 2 N
vs. I, providing a clearer view of the effective resis- causesthe measuredvalue ofINS to be smallerthan the
c
tance. Either plot shows that the resistance is about nominal value INS =ISNS(R +R )/R .
c c 1 2 1
20.7 Ω when the applied current is less than about 0.94 We believe that the effect related to the phase depen-
µA. This resistance is very close to the nominal value of denceoftheresistancesissmallandtherelativedecrease
R =20.4Ω. Whenthe currentexceeds0.94µA,there- in INS at low temperature is due predominantly to the
P c
sistanceincreasestothe value24.6Ω,whichisveryclose decreaseinISNS asafunctionofU. Thiseffectisdemon-
c
toR +R =23.9Ω. (Resistancedifferenceslessthanan stratedgraphically in the inset to Fig. 3. There the crit-
N 1
Ohmare attributedto the finite size ofthe “T-junction” ical current of the Josephson junction, ISNS, multiplied
c
in the middle of the sample.) These data confirm the by the constantratio(R +R )/R , is plotted asa func-
1 2 1
idea outlined in the previous paragraph, that supercur- tion of the voltage U applied to the normal reservoir.
rent and quasiparticle current can coexist in the normal As can be seen in the inset, ISNS decreases rapidly as
c
region of a S/N/S Josephson junction. a function of U. The straight line through the origin in
The transition at INS where the resistance increases the inset represents the current injected into the sam-
c
to R + R occurs when the supercurrent across the ple from the N reservoir, U/R , where the resistances
N 1 P
S/N/S Josephson junction exceeds the S/N/S critical areevaluatedatphasedifferenceπ/2betweenS andS .
1 2
current, ISNS. However, since only the fraction I = The intersectionofthe twocurvesshowsthe valueofthe
c 2
IR /(R +R ) I/2 of the injected current must be danglingarmcriticalcurrentINS (ordinate)atthe criti-
1 1 2 ≈ c
4
a major discovery in nonequilibrium superconductivity
by Baselmans et al.5 in 2000. Those authors showed
2.4 that I first decreases as a function of U, but then in-
c
2.0 creases again at higher U. The explanation17,18 for this
2.0 A) 1.6 counter-intuitive result consists of two pieces. First, one
( 1.2 can view the supercurrent in the sample as arising from
S
N
1.6 S C 0.8 the continuous spectrum of Andreev bound states in the
I 0.4 normalmetal,19,20 whichcarrysupercurrentin either di-
)
A
0.0 rection, depending on their energy. Second, in the pres-
1.2 0 10 20 30
( ence of the applied voltage U the electron distribution
c
I U ( V) function in the junction is not a hot Fermi-Dirac distri-
0.8
bution,butisclosertoatwo-stepdistribution-aslongas
the shortsample lengthdoes not allowelectronthermal-
0.4 ization within the sample.21 The two-step distribution
functionpreferentiallypopulatestheminorityofAndreev
boundstatesthatcarrysupercurrentinthedirectionop-
0.0
0 50 100 150 200 250 300 350 positetothemajority,henceitreversesthecurrent-phase
relation in the junction.17,18 Such a Josephson junction
T (mK)
is called a “π-junction”, because the energy-phase and
current-phase relations are shifted by π relative to those
FIG. 3: Critical current measured between N and S1 (•), of standard Josephson junctions.
between N and S2 (N), and between S1 and S2 ((cid:3)) – the
lattermultipliedbytheratio(R1+R2)/R1 ≈2–fordifferent
temperatures. The three data sets are in close agreement at
temperaturesaboveabout250mK.Inset: Graphicalapproach
to calculation of low-temperature critical current between N Theoriginalπ-junctionexperimentofBaselmanset al.
and S1. The dots are measurements of the critical current was performed in a 4-terminalsample, where voltagesof
acrossS1-S2,againmultipliedby(R1+R2)/R1,asafunction opposite sign were applied to the two normal reservoirs.
of applied voltage U between N and S1. The critical current Later,Huang et al.8 demonstratedthat a π-junction can
decreases rapidly with increasing U. The line through the
also be obtained in a 3-terminal geometry with a single
originrepresentstheinjectedcurrentfromN.Theintersection normal reservoir,a result predicted by van Wees et al.22
givesthecritical currentINS at thecritical valueof U. Note
thatallcriticalcurrentvalcuesintheinsetare15−20% larger 10 years earlier.
thaninthemainpanel,duetoasmallmagneticfieldB=125
G present when thelatter data were obtained.
Figure 4a shows results of a 3-terminal π-junction ex-
cal voltageUNS (abscissa). The figure demonstrates the
c periment performed on a sample similar to the one in
large reduction in S/N/S critical current due to the ap-
Fig. 1, where the tunnel probe in the lower left portion
pliedvoltageU,whichexplainswhyINS ismuchsmaller
c of the figure is not used. We measure the I V curve of
than ISNS(R +R )/R at low temperature. At high −
c 1 2 1 the S/N/S Josephson junction using a 4-probe current-
temperatures T & eU/kB, the relative reduction is less bias measurement, while a dc voltage is simultaneously
significantdue to two reasons: First, increasingthe tem-
applied to the normal reservoir via a battery-powered
peraturedecreasesthecriticalcurrentIcSNS,andthereby floating circuit. Figure 4a shows a series of I V curves
also UNS. Moreover, to observe a sizable reduction in −
c at different values of the voltage U applied to the nor-
ISNS(U), eU has to exceed k T.
c | | B mal reservoir. Figure 4b shows the criticalcurrent Ic vs.
U. Notice that I initially decreases with an increasing
c
U, as shown in the inset to Fig. 3. But as U increases
IV. S/N/S NONEQUILIBRIUM π-JUNCTION
further, I reaches a minimum value (indistinguishable
c
fromzerointhisexperiment23)atU =U 34µV,then
c
A. Three-terminal π-junction grows again to reach a second maximum a≈t U 63 µV.
≈
The minimum in I separates the standard Josephson
c
Figure 3 shows, not surprisingly, that the critical cur- junctionbehavioratlowvalues ofU fromthe π-junction
rent I of an S/N/S Josephson junction decreases when behavior at higher U. If instead of plotting the critical
c
quasiparticle current is injected into the junction from currentI (whichisbydefinitionapositivequantity)one
c
a normal reservoir. Indeed, if the only effect of the in- were to plot the supercurrent I at a fixed phase differ-
s
jected current were to heat the electrons in the junc- ence φ = π/2 across the junction then the graph would
tion, then one would expect the critical current to con- show a smooth curve passing through zero at U = U ,
c
tinue decreasing monotonically as a function of the ap- reaching a local minimum at U 63 µV and gradually
plied voltage U.16 That this is not the case represents returning to zero at large U. ≈
5

120
(a)
) 80
V
(
V
40
0
-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5

I ( A)
1.2 (b)
FIG.5: a)Depictionofelectronflowinfour-terminalconfigu-
ration. b)Depictionofelectronflowinthree-terminalconfig-
) uration. c)Schematicrepresentationofthedistributionfunc-
A 0.8
tion on the path between a normal (N) and superconducting

( (S)terminal in thestructure(a) underhigh biasU >>kBT.
Ic d) Distribution function in the T-structure (b) undersimilar

conditions. Due to Andreev reflection f(ε) is discontinuous
0.4
at theN-Sinterface as explained in the text.
0.0 of the supercurrent (in the absence of electron-electron
0 40 80 120 interactions),suggestingthatthevoltage-dependentcrit-
ical current, I (U), would be identical in 3-terminal and
c
U ( V) 4-terminal samples with identical dimensions and resis-
tances. However, there are three reasons why this is not
FIG. 4: a) Voltage vs. current across theS/N/S junction for quite true: First, Joule heating is more prevalent in the
selected voltages, U,applied tothenormalreservoir. Graphs 4-terminal device, which rounds the distribution func-
for different U are offset for clarity, with U = 0, 17, 29, 35, tions more than in the 3-terminal device. Second, the
41,63,92,and114µVfrombottomtotop. Thehysteresisin
spectral supercurrent density, j (E), evaluated at the
E
the U =0 data is probably due to heating of the Ag wire in
junction point will be slightly smaller in the 4-terminal
thenormal state. b) Critical current vs. U.
sample than in the 3-terminal sample due to the pres-
ence of the additional arm connecting the sample to a
normal reservoir.20 Finally, f(E) will be slightly more
B. Comparison of four-terminal π-junctions with rounded in the 4-terminal sample due to the increased
symmetric and antisymmetric bias
phase space available for electron-electron interactions.
Roughly speaking, the rate of e-e interactions at a given
The physical explanation of the π-junction in the 3- energy is proportional to f(E)[1 f(E)], which is max-
−
terminalsample is the same as in the 4-terminalsample, imized when f(E) = 1/2. Each of these effects serve
with the differences arising only from the distribution to increase I in the 3-terminal geometry relative to the
c
functions. Figure 5 shows a schematic drawing of the 4-terminal geometry.
distribution function f(E) along a path from a reser- It is not practical to compare critical currents from
voir N to S for both 4-terminal and 3-terminal samples two different samples, since they will never have identi-
for U >> k T, assuming weak electron-electron inter- cal dimensions nor electrical resistances. Instead, it was
B
actions in the N wire, and neglecting the proximity cor- proposed in Ref. 8 to compare the critical currents in a
rections. Notice in figures c and d that f(E) consists single 4-terminal sample under conditions of symmetric
of a double-step function, the step height within the en- and antisymmetric voltage bias of the two normal reser-
ergy range eU to eU changing with the location along voirs. Figure 6 shows the sample we fabricated for this
−
the wire. As we will show in the next section, the even experiment, with the superconductive reservoirs labeled
(in energy) part of f(E) has no effect on the magnitude S1 and S2 and the normal reservoirslabeled N1 and N2.
6
By applying a positive potential U to N1 and a nega-
tive potential U to N2, one reproduces the experiment
performed by−Baselmans et al. We call this situation 30 (a) 30 (b)
abnytiasynmemgaettirviec sbiigans., siInncethtihsectawseo athpepliqeudasvioplatartgiecsledciffuerr- V (V) 1200 V (V) 1200
rent overlaps with the supercurrent only at the crossing
0 0
point of the sample where the electrostatic potential is
equal to zero. In contrast,applying the identical voltage -10 -10
-1.0 -0.5 0.0 0.5 1.0 -1.0 -0.5 0.0 0.5 1.0
U on both N1 and N2 (with ground defined at one of
I ( A) I ( A)
the superconducting electrodes), called symmetric bias, 1.0 1.2
willproduceasituationmimickingthatinthe3-terminal (c) (d)
0.8 1.0
experiment of Huang et al.8 Notice that by mimicking a
) 0.6 ) 0.8
3-terminalsample with a 4-terminalsample, geometrical R ( R (
differences between the two experiments are eliminated, 0.4 0.6
so any observeddifference in the critical currents will be 0.2 0.4
due either to e-e interactions or to Joule heating.
0.0 0.2
-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 -1.0 -0.5 0.0 0.5 1.0
I ( A) I ( A)
FIG. 7: Data showing voltage drops across segments of wire
eitherinantisymmetricorsymmetricarrangementswhilecur-
rent flows from S1 to S2. Each line (offset for clarity) corre-
sponds to a different value of U. a) Voltage across S1 to S2
for the antisymmetric measurement. Applied voltages U are
fromthebottom: 19,28,38,52,and71µV b)Voltageacross
S1to S2 for thesymmetric measurement. Applied U:17, 25,
37, 49, 72, and 131 µV. c) Resistance across N1 to N2 for
theantisymmetricmeasurements. d)ResistanceacrossN2to
S2forthesymmetricmeasurements,takenfromvoltagemea-
surementsinwhichaconstantresistancewassubtractedfrom
thegraph.
datafollowthesametrendobservedinFig.4,namely,the
criticalcurrentfirstdecreaseswithincreasingU,thenin-
creasesagainbeforefinally disappearingaltogether. Fig-
FIG. 6: SEM image of S/N/S Josephson junction. The ‘x’-
ure8bshowsthiscriticalsupercurrentasafunctionofU.
shape is deposited Ag that connects to (difficult to see) Al
reservoirsaboveandAgreservoirsbelowpatternedbyangled In this figure we have plotted the critical currentas neg-
evaporation. ThefeatureintheAgwirenearN2islikelydue ativeintherangeofvoltagesafterIc disappearsinitially,
to a near burnin thesample. to signify that the junction is in the π-state as discussed
earlier.
The preceding description of symmetric and antisym- The transition from the 0-state to the π-state can be
metric biases holds strictly only if the resistances of the confirmed directly by experiment,5 without recourse to
two lower arms are identical. Otherwise, application of the theoretical explanation. The resistances of the Ag
antisymmetric bias will result in a nonzero potential at arms of the sample vary with the phase φ due to the
thecrossandsomequasiparticlecurrentwillflowintothe proximity effect.24 The phase φ, in turn, varies between
superconducting reservoirs. In that case, f(E) will take π/2 as a function of the supercurrent I passing be-
S
±
a form intermediate between those depicted in Figures tween S and S ; hence, one observes a variation of the
1 2
5c and d, which decreases the measurement contrastbe- resistances as a function of I . This effect is shown in
S
tween the two biases. In our experiments we took care Figs. 7c and d, in which the resistances between N and
1
to measure the resistances of all the arms and to ensure N (N andS )weremeasuredversusI fortheantisym-
2 1 2 S
thatthevoltagesatthetwonormalreservoirswereindeed metric and symmetric bias configurations, respectively.
equal(forsymmetricbias)oropposite(forantisymmetric Each curve in the lower two figures has the same value
bias). of U as the corresponding I V curve in the upper fig-
−
Figures 7a and 7b show I V curves measuredacross ures. One can see that proximity effect induces a local
−
the S/N/S junction at T =170 mK, for several different minimumintheresistanceatI =0whenthejunctionis
s
valuesofU. CurveswithincreasingvaluesofU areoffset in the 0-state, because that is where φ=0. In contrast,
upward for clarity. Figure 7a shows the data for anti- theresistanceexhibitsalocalmaximumintheresistance
symmetricbiaswhileFig.7bshowssymmetricbias. The at I = 0 when the junction is in the π-state, because
s
7
C. Calculation of the Critical Current in an S/N/S

Josephson Junction
1.5
(a) TheamountofsupercurrentpassingthroughanS/N/S
Josephson junction may be calculated by17
1.0 σ A ∞
N
) I = dE[1 2f(E)]j (E) (1a)
A S 2 − E
Z−∞
( 0.5 ∞

Ic = σNA dE fL(E)jE(E) (1b)
Z0
0.0
whereσ andAarethe conductivityandcross-sectional
N
area of the normal metal, respectively. f(E) is the dis-
-0.5 tribution function within the normal wire, and f (E)
L
≡
f( E) f(E) is the antisymmetric component of f(E)
− −
20 40 60 80 100 120 140 withrespecttothepotentialofthesuperconductors. The
0.8
spectral supercurrent density, jE(E), is an odd function
of energy, and describes the amount of supercurrent at
0.6 (b)
a given energy travelling between superconductors with
) relative phase difference φ. In the samples considered
A 0.4
in the present work, it is generally sufficient to calculate
(
the supercurrent using the the distribution function at
c
I 0.2
the crossing point of the wires.
To determine j (E), we solvethe Usadelequationnu-
E
0.0 merically using the known physical dimensions and elec-
tricalresistancesofthevariouswiresegmentsofthesam-
-0.2 ple. We then look for consistency with the measured
temperature dependence of the equilibrium critical cur-
0 20 40 60 80 100 120 140
rent, I (T) shown in Fig. 9. The j (E) used to fit these
c E
U ( V) data,evaluatedatφ=π/2,isshownintheinsetofFig.9.
SincethelengthLofthejunctionismuchlongerthanthe
superconducting coherence length ξ of the S electrodes
FIG. 8: (color online) Critical current of a 4-terminal S/N/S s
for all the samples studied in this work, the damped os-
Josephson junction versus voltage U applied to the normal
reservoirs. The voltages are applied either antisymmetrically cillations in jE(E) occur onan energyscale givenby the
((cid:4)) or symmetrically (N) to the two reservoirs. Solid lines Thoulessenergy,ETh =~D/L2,whereD isthediffusion
representsimultaneous best fitstodataat differenttempera- constant in the wire. E characterizesthe temperature
Th
tures. Fittingmethodsarediscussedinthetextfordatataken scaleoverwhichthe equilibriumcriticalcurrentdropsto
atbathtemperatures(a)35mKand(b)170mK.Thedashed zero, and also determines the voltage scale U needed to
line represents thebest fit when Joule heating is excluded. create a nonequilibrium π-junction. The transition from
the 0-state to the π-state occurs at eU 8E . The fit
Th
≈
shown in Fig. 9 was obtained with E =4.11 µeV.
Th
Nextwecalculatef(E)inthenonequilibriumsituation
withantisymmetricbias,i.e. withvoltagesU and U ap-
−
φ=π. Interestingly, the top curve in Fig. 7d shows that plied to reservoirsN and N . Because we are interested
1 2
at large enough values of U, the system returns to the inthedistributionfunctionfarfromthesuperconducting
0-statesincethe resistanceagainshowsalocalminimum reservoirs, we consider f(E) using the Boltzmann equa-
at Is = 0. This high-U transition from the π-state back tion. Let us first ignore the supercurrent and proximity
to the 0-statewasnot visible inthe S/N/SI V curves. effects, although inclusion of those effects will be dis-
−
cussed in detail in Sec. VI. In a reservoir at voltage U,
Figure 8 shows the behavior I vs. U at two different f(E) is a Fermi-Dirac distribution displaced by energy
c
temperatures. The squaresrepresentantisymmetric bias eU,f(E)=fFD(E eU)=[exp((E eU)/kBT)+1]−1.
− −
while triangles representsymmetric bias. Both bias con- Intheexperimentwithantisymmetricbias,wethenhave
figurations appear similar in that the samples cross to f(E) = fFD(E eU) at reservoir N1 and f(E) =
−
the π-state at nearly the same value of U. It should be fFD(E + eU) at reservoir N2. If we neglect inelastic
noted,however,thatthemaximumπ-currentislargerfor electron scattering, then in the middle of the wire (at
symmetric bias than for antisymmetric bias. That result the intersection of the cross) f(E) has the double-step
isconsistentwiththequalitativeargumentsmadeabove. shape:
In the next section we present a quantitative analysis of 1
the results. f(E)= [fFD(E+eU)+fFD(E eU)] (2)
2 −
8
wire. (The electron-phonon interaction, in contrast, is

much weaker, and need be considered only in the mas-
0.8 sive normal reservoirs. See the discussion below.) To
2.0 incorporate electron-electron interactions, we solved the
0.4
Boltzmann equation in the wire numerically, following
1.6 jE 0.0 previous work by Pierre.26,27 The results of this numeri-
-0.4 calcalculationoff(E)inthe situationswitheither sym-
)
A metric orantisymmetric biaswereextremelysimilar. In-
-0.8
I (C 1.2 -200 -100 0 100 200 tdieseydm,mtheetriscligchatseadcdouitlidonnaoltroaucncoduinngt ofofrf(tEhe) dinifftehreenacnes-
observed in the experiment, shown in Fig. 8.
0.8 E ( eV)
To account for the difference in the observed I (U)
c
between the two experiments, we next consideredthe ef-
0.4
fect of Joule heating on the temperatures of the normal
reservoirs. (Due to Andreev reflection at the N/S inter-
0.0 faces, there is no heat transport into the superconduct-
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 ing reservoirs.) Although we intentionally fabricated the
T (K) normal reservoirs much thicker than the wires, this was
not enough to eliminate the effects of Joule heating al-
together. The heat current in a reservoir at a distance r
FIG. 9: Critical current Ic at several temperatures for the
from the juncture with the wire is given by:
sample shown in Fig. 6. The line is the best fit by solving
equation 1b with a Fermi-Dirac distribution function. Inset:
P
Solution for spectral supercurrent used to fit data. ¯jQ = £σT T rˆ (4)
− ∇ ≡ θrt
where P =I2R is the total power dissipatedin the wire,
In fact, the odd part of f(E) is the same everywhere in
σ andt=310nmarethe conductivity andthe thickness
the Ag wire:
of the reservoir, respectively, and £ π2/3(k /e)2 =
B
1 E eU E+eU 2.44 10−8 V2/K2 is the Lorenz num≡ber. (We neglect
fL(E)= tanh − +tanh (3) thesm×alladditionalJouleheatgeneratedinthereservoirs
2 2k T 2k T
(cid:20) (cid:18) B (cid:19) (cid:18) B (cid:19)(cid:21) themselves.) The spreading angle θ π if we consider
≈
This conclusionholdsalsointhe presenceofthe proxim- the combination of the two normal reservoirs shown in
ity effect. At energies E < eU, the even part, f (E), Fig. 6.
T
varies linearly with dist|an|ce between the two reservoirs Using (4) as a boundary condition, one can find an
(the lower two arms of the sample), but is zero every- effectivetemperatureatthewire-reservoirinterfaceequal
where along the direct path connecting the two super- to28
conductors(inthe idealcasewherethe resistancesofthe
T = T2+b2U2 (5)
two lower arms are equal). eff
To calculate f(E) in the experiment with symmetric p
The temperature far away in the normal reservoir is as-
bias, we need the boundary conditions at the interfaces
sumed to be T, the bath temperature. The factor b is
betweenthenormalwiresandthesuperconductingreser-
given by29
voirs. For energies below the superconducting gap, ∆,
these conditions are fT = 0 and ∂fL/∂x = 0, where b2 = R(cid:3) lnr1, (6)
fT(E) 1 f(E) f( E). These boundary condi- θ£R r
≡ − − − 0
tionsassumehigh-transparencyinterfaces,no chargeim-
balance in the superconductors, and no heat transport where R(cid:3) 1/(σt) is the sheet resistance of the normal
into the superconductors.25 (Note that f (E) is discon- reservoir,r≡ thewirewidth,andr isthedistanceover
L 0 1
≈
tinuous at the N-S interface for energies below the gap, whichtheelectronsinthereservoirthermalizetothebath
and returns to the standard form tanh(E/2k T) in the temperatureviaelectron-phononscattering. Theparam-
B
S electrodes.) The solution for f(E) at the N/S inter- eter b varies inversely with the thickness of the metal
face is identical to Eq. (2), but the symmetric compo- reservoirandthe electricalresistanceencounteredbythe
nentf (E) isnonzeroelsewhereinthe wire. Notice that quasiparticle current in the wire. Because the voltage
T
the odd component of the distribution function, f (E), drop U in the antisymmetric bias situation occurs en-
L
is identical in the two cases everywhere in the sample. tirely betweena normalreservoirand the crossingpoint,
The proximity effect induces a small feature in f (E) the resistance R is smaller than in the symmetric bias
L
discussed in Sec. VI, but it is zero at the crossing point situation where U drops fully from the N reservoirs to
in the middle of the sample. theN/Sinterfaces. Thelargercurrentinthe formercase
Calculation of f(E) in the realistic situation requires causes more Joule heating, and hence a larger reservoir
consideration of electron-electron interactions in the Ag temperature. For our sample, the values of b needed to
9
fit the data (see solid lines in Figs. 8a and 8b) are 2.7 ferromagnet. In the standard elemental ferromagnets,
and 3.2 K/mV,respectively,for the symmetric and anti- E is large ( 0.1 meV), hence ξ is extremely short –
ex F
≈
symmetric bias experiments. Their ratio of 1.2 matches ontheorderof1nm. Controlofsamplethicknessunifor-
the ratio calculated from the sample parameters. Their mity at this scale is difficult, hence several groups have
magnitudes, however, are nearly three times larger than used dilute ferromagnetic alloys, with reduced values of
what we calculate based on the total reservoirthickness. E , to increase ξ . The advantage of the “Zeeman”
ex F
The experiment seems to suggest that heat was trapped π-junction is that it is fully tunable by the field. The
in the 35-nm Al layer at the bottom of the reservoirs, disadvantage is that the sample must be thin enough to
ratherthanimmediatelyspreadingthroughoutthewhole minimize the effects of orbital pair-breaking in both the
reservoir thickness.30 superconductingelectrodesandinthenormalpartofthe
junction.
V. APPLICATION OF A PARALLEL

MAGNETIC FIELD, AND THE “ZEEMAN”
0.5
π-JUNCTION 0.5
0.4
)
There is a long history of applying magnetic fields 0.4 A
0.3
perpendicular to the direction of current flow in super- (

C 0.2
conductor/insulator/superconductor (S/I/S) Josephson I
junctions, to observe the famous Fraunhofer pattern in ) 0.3 0.1
A
the critical current. In S/N/S junctions, the Fraunhofer 0.0
pattern is observed only in wide junctions, whereas nar- (C 0 20 40 60
I 0.2
row junctions exhibit a monotonic decrease of the crit- U ( V)
ical current with field due to the orbital pair-breaking
effect.31 0.1
In this section we discuss the effect of a magnetic field
applied parallel, rather than perpendicular, to the cur-
rent direction. In this geometry there should never be a 0.0
0.0 0.1 0.2
Fraunhofer pattern. And because the samples are thin
films, one expects the orbital pair-breaking effect to be B (T)
muchweakerthanforafieldappliedperpendiculartothe
plane. In the case of an extremely thin sample the Zee-
FIG. 10: Critical current across S/N/S junction as a func-
man (spin) effect should dominate over the orbital effect
tion of external magnetic field applied parallel to the cur-
of the field. rentdirection. Intheabsenceoforbitalpair-breakingeffects,
The effect of Zeeman splitting on an S/N/S Joseph- the transition into the π-state would be expected near 0.35
son junction was studied theoretically in 2000 by Yip10 T. Markers indicate experimental results and solid line the
and by Heikkila¨ et al..9 Their idea is that the electronic scaling (7) obtained from the Usadel equation. Inset: Criti-
structure of a normal metal in a large applied magnetic cal current versus applied voltage for same sample, showing
transition to theπ-state at U =20 µV.
field resembles that of a weak ferromagnet, in that the
up and down spin bands are displaced by the Zeeman
energy. They also showed how the Zeeman-split junc- Figure 10 shows a plot of I vs. B in an S/N/S sam-
c
tion behaves analogously to the nonequilibrium S/N/S ple whose normal part had length L = 1.4 µm, width
junction, with the Zeemanenergyplaying the roleof the w = 50 nm and thickness t = 33 nm. The criti-
voltage in Eqs. (1b) and (3). Josephson junctions made cal current decreases monotonically to zero, over a field
with real ferromagnetic materials (S/F/S junctions) are scale of 0.1 T. This result might appear surprising at
≈
the subject of intense current interest, as they can also first glance: At a field B = 0.1 T, the magnetic flux en-
show π-junction behavior.32 Unlike the π-junctions dis- closed in the cross-section of the wire perpendicular to
cussed earlier in this paper, however, the π-junctions in the field is only Φ 1.6 10−16 Tm2 = 0.08Φ , where
0
≈ ·
S/F/Ssystemsoccurinequilibrium. Theyappearonlyin Φ = h/2e is the superconducting flux quantum. Fur-
0
particular ranges of the F-layer thickness, due to spatial thermore, separate tests of the Al banks confirm that
oscillations in the superconducting pair correlations in- they remain superconducting to fields of order 0.85 T.
ducedinthe Fmetalnearthe F-Sinterfacebyproximity A quantitative understanding of the data in Figure 10
effect. Thoseoscillations,inturn,arisefromthedifferent can be obtained from a solution to the Usadel equation.
Fermiwavevectorsofthespin-upandspin-downelectrons The analysis discussed in Appendix A shows how a par-
in the F metal. In diffusive S/F/S junctions, the sign of allel magnetic field can be absorbed into a spin-flip rate
the coupling between the two superconductors oscillates Γ in the equations. This allows us to apply the scaling
sf
over a distance scale ξ = (~D/E )1/2, where D is the I (B)/I (B = 0) exp( 0.145Γ /E ) for the zero-
F ex c c sf Th
≈ −
diffusion constantand E is the exchange energy in the temperaturesupercurrentofanS/N/Sjunction foundin
ex
10
Ref. 33 and find this reason,the resultwasdubbed a “Peltier-likeeffect.”
I (B)/I (B =0) e−(B/B1)2, (7) Although a tiny cooling effect does occur, observing it
c c ≈ in a real electron temperature would require a slightly
~√w2+t2 modified experimental setup.35 In the present case, one
B 6.43 0.10T. (8)
1 ≈ eLwt ≈ should view this effect mostly as a redistribution of the
Joule heat generated in the sample by the applied bias
Our numericalcalculations confirmthatthis scaling also
U.
applies in our multi-probe experimental geometry. This
InSec.IVCitwasdiscussedhowthedistributionfunc-
prediction is in a good agreement with the experiment,
tions behave in the absence of proximity effects and su-
as seen in Fig. 10.
percurrent. Includingtheseeffects,butignoringinelastic
In the limit w t, the characteristic field scale B
≫ 1 scattering, results in the kinetic equations:36
varies as Φ /Lt, rather than the more intuitive result
0
Φof0/thwetwwierempigehrpteenxdpieccutlabrasteodtohne tfiheeldc.roTssh-esepchtiyosnicaallareexa- ∂jT =0, jT DT(x)∂fT +jEfL+T(x)∂fL; (9a)
∂x ≡ ∂x ∂x
planation for this result was given by Scheer et al.34 in
a paper discussing universal conductance fluctuations as
a function of parallel field in normal metal wires. As an ∂jL =0, j D (x)∂fL +j f T(x)∂fT; (9b)
electron travels down the length of a long diffusive wire, ∂x L ≡ L ∂x E T − ∂x
its trajectory circles the cross-section of the wire many
times – on order N (L/w)2. Because diffusive mo- where jT(E) and jL(E) are the spectral charge and en-
≈ ergy currents, respectively. The energy-dependent coef-
tion can be either clockwise or counterclockwise as seen
ficients D , D , j , and T can be calculated from the
lookingdownthewire,thestandarddeviationinnetflux T L E
Usadel equation6,36, and vary with the superconducting
and accumulated phase between different trajectories is
approximately proportional to Bwt√N = BLt, which phase difference φ between S1 and S2. In general, these
equationsmustbesolvednumerically;however,they can
gives the scaling for dephasing.
besolvedanalyticallybyignoringtheenergydependence
It is instructive to ask what constraints on the sam-
inD andD andneglectingtheTterms. Onecanshow
ple geometry would have to be met to enable observa- T L
that, in the presence of both the applied voltage U and
tion of the Zeeman π-junction. We estimate the Thou-
a nonzero supercurrent between S and S , f along the
less energy of the sample discussed in Fig. 10 to be 1 2 L
horizontalwire connecting the two superconductorscon-
E 2.5 µeV both from the temperature dependence
Th
≈ tainsaspatiallyantisymmetriccontributionproportional
of I (not shown), and from the voltage-induced tran-
c
toj (E). IntheexactnumericalsolutiontoEqs.(9),the
sition to the π-state at U = 20 µV (inset to Fig. 10). E
c
feature is distorted due to the rapid evolution of the dif-
According to theory, the Zeeman π-junction should oc-
cur when gµ B 16E ,9,10, or B = 0.35 T. Attempts fusioncoefficients DT andDL near the N/S interfaces.11
B Th
≈ Theantisymmetricfeatureinf(E)canbemeasuredby
to make thinner samples in order to increase the field
performingtunnelingspectroscopywithalocalsupercon-
scale B in Eq. (7) were unsuccessful, due to the ten-
1
ducting tunnel probe, which has been demonstrated by
dency of very thin Ag films to ball up. According to
Pothieretal.21torevealdetailedinformationaboutf(E)
the theory, much thinner films, with t/L of the order of
ina metal under nonequilibriumconditions. Inour case,
0.2gµ /eD 0.001, will be required to enable observa-
B
∼ thelocalprobemustbeplacedclosetotheN/Sinterface
tion of the Zeeman π-junction.
where the predicted feature in f(E) has its maximum
amplitude. This location introduces a new difficulty in
VI. ENGINEERING THE DISTRIBUTION our experiment because the density of states (DOS) of
FUNCTION the Ag wire near the N/S interface is strongly modified
by proximity effect. Hence we must first determine the
modified DOS at equilibrium before we measure f(E)
The discussion in Section IV.B. implied that the 3-
under nonequilibrium conditions.
terminal and 4-terminal π-junctions are similar, with
only minor differences due to a slight decrease in the The current-voltage characteristic of the probe tunnel
phasespaceavailableforelectron-electroninteractionsin junction is
the 3-terminal case. But that oversimplified discussion
1
misses some important physics. Heikkila¨ et al.11 showed I(V )= dEn (E) dεP(ε) (10)
Al
−eR
thatthesuperpositionofquasiparticlecurrentandsuper- T Z Z
current in the horizontal wire in the 3-terminal sample [fAl(E)nAg(E eV ε)(1 fAg(E eV ε))
× − − − − −
induces a change δf(E) in the distribution function at (1 f (E))n (E eV +ε)f (E eV +ε)],
Al Ag Ag
− − − −
energiesoforderE . The new feature is antisymmetric
Th
inspaceandenergy(seeFig.16forthetheoreticalpredic- where R is the normalstate tunnel resistance,n and
T Ag
tionandourexperimentalresultswhichfolloweachother n are the normalized densities of states, and f and
Al Ag
nicely), andcanbe interpretedasa gradientinthe effec- f are the electron energy distribution functions on the
Al
tiveelectrontemperatureacrosstheS/N/Sjunction. For AgandAlsidesofthe tunneljunction,respectively. The