Table Of ContentStabilizing Superconductivity in Nanowires by Coupling to Dissipative Environments
Henry C. Fu1, Alexander Seidel2, John Clarke1,2 and Dung-Hai Lee1,2,3
1 Department of Physics,University of California, Berkeley, CA 94720-7300, USA
2 Materials Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA and
3 Center for Advanced Study, Tsinghua University, Beijing 100084, China
6
(Dated: February 4, 2008)
0
0 Wepresentatheoryforafinite-lengthsuperconductingnanowirecoupledtoanenvironment. We
2 showthatintheabsenceofdissipation quantumphaseslipsalwaysdestroysuperconductivity,even
at zero temperature. Dissipation stabilizes the superconducting phase. We apply this theory to
n
a explain the “anti-proximity effect” recently seen by Tian et. al. in Zincnanowires.
J
9
The effect of dissipation on macroscopic quantum co- (a) (b) (c)
1 I I
herence is currently a subject of considerable interest[1]. I
] In the context of quantum bits (”qubits”), a dissipative Sn/In R/2
n
r-co eOjunnnvcirttohionenmoeitsnhteernahlhwaanancydes,disnbucypreedracsisoesnsipdtauhtceitoinvdi[et3cy]o.hineIrneantcJheoissreappthaesp[2oe]nr. L 1 mmV C/2(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)C/2 V RC (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) V
p we examine a particularly striking example of the lat- I R/2
u ter phenomenon: the stabilization of superconductivity
I I
s
in a nanowire by dissipation in its environment.
.
at In one-dimensional superconducting wires quantum FIG. 1: Schematic diagrams of experiment by Tian et al.
m phase fluctuations can destroy long-range phase coher-
(a)Experimentalconfiguration: poresinpolycarbonatemem-
- ence even at zero temperature; however, finite super- branecontainnanowires,withInorSnelectrodesmakingcon-
d fluiddensitycansurvivethroughtheKosterlitz-Thouless tact to the wire(s). In this case, only one wire is contacted.
n (KT) physics[4]. The quantum action for a supercon- Current I and voltage V are measured as shown. (b)Circuit
o ducting wire at zero temperature is equivalent to that of representation showing contact resistances R/2 to each elec-
c trode and a capacitance C/2 from each half of the parallel
[ the two-dimensional classical XY model[4] at finite tem-
platecapacitor, assumingthesingle wire isplaced symmetri-
1 perature. The two phases of the latter correspond to cally. (c) Simplified model. For the purpose of damping, R
v the superconducting and insulating phases of the wire. and C are connected in series across thenanowire.
7 In particular KT vortices in the XY model correspond
5 to phase slip events in the wire in which the phase gra-
4 dient unwinds by 2π. The resistance of real nanowires lel boundary lines that can screen the vortex-antivortex
1
can display both insulating and superconducting behav- interaction in the XY model. This screening destroys
0
ior as temperature is decreased[5, 6, 7]. Both thermal[8] the superconducting phase even at T = 0. When the
6
0 and quantum[6, 9] phase slips play an important role ends of the wire are coupled to a dissipative environ-
/ in generating resistance and destroying superconductiv- ment,the screeningbecomesincomplete. As aresult,for
t
a ity. However, it is not settled what determines which sufficientlylargedissipationthesuperconductingphaseis
m
nanowiresaresuperconducting[7]. Inparticular,itisun- stabilized. The importance of the boundary dissipation
- clear whether or not dissipation is important in deter- has been suggested by Bu¨chler et al.[10].
d
mining the low-temperature phase of nanowires[10]. We begin by summarizing the experimental findings
n
o In a recentexperiment[11] Tianet al. observedanun- reported in Ref.[11]. Tian et al. prepared Zn nanowires
c expected effect when a 2-µm long, 40-nm diameter Zn in the pores of porous polycarbonate or porous alumina
v: nanowireis sandwiched between two bulk superconduct- membranes [Fig. 1(a)]. They pressed In or Sn wires
i ing electrodes. Under zero applied magnetic field, when on each side of the membrane to form circular disks ap-
X
the electrodes are superconducting, the Zn nanowire ex- proximately 1 mm in diameter that made contact to the
r
hibitsresistivebehaviordowntothelowestmeasurement ends of a single nanowire or, more generally, a num-
a
temperature. However after a sufficiently strong mag- ber of nanowires. By applying a magnetic field above
netic field B has suppressed the superconductivity of the critical field of the electrodes but below the criti-
the electrodes, the nanowire becomes superconducting cal field of the Zn nanowire (which is enhanced by its
at about 0.8 K. Tian et al. dubbed this phenomenon smalldiameter[12]),theysuppressedthesuperconductiv-
the“antiproximityeffect”(APE).Herewepresentathe- ity of the electrodes. They measured the resistance and
ory suggesting that this surprising effect is due to the current-voltage(I V)characteristicsoftheirsamplesus-
−
dissipation at the boundary between the nanowire and ingthefour-terminalarrangementindicatedinFig. 1(a).
electrodes. We show thatwhen the nanowirehas a finite In this paper we focus on the sample Z4, the behavior
length, the ends of the wire are mapped onto two paral- of which is shown in Fig. 3(b) of Ref.[11]. This sam-
2
ple had In electrodes and is believed to have contained V
a single nanowire with length L = 2 µm. When the In
electrodes were driven normal by the magnetic field, the I RN I
Zn nanowire exhibited a superconducting transition at a
R C
temperature that decreased as the field was further in-
creased. Incontrast,withzeroappliedfield(hencesuper-
conducting electrodes), the resistance showed a drop of FIG.2: Phasefluctuationsultimatelydrivethequantumwire
about 20 Ω as the temperature was lowered through the normal via depairing. The normal state resistance of the
transition temperature of the In (about 3.4 K), but the quantumwire is RN.
nanowire did not go superconducting downto the lowest
measurement temperature (0.47 K).
ture,0.47K.Thus inthe relevantfrequency/temperature
Our model of the experiment is shown in Fig. 1(b).
range the external circuitry behaves as a pure resistor
When the electrodes are in the normal state, the
(i.e. 1/2πfC 0). As we show later, if this shunt-
nanowireisconnectedtoeachofthemviaacontactresis- →
ing resistance is smaller than the quantum of resistance
tance R/2. When the electrodes become superconduct-
h/4e2 6.4kΩ it damps the superconducting phase fluc-
ing, we assume that this resistance is eliminated by the ≈
tuationssufficientlytostabilizesuperconductivity. Ifthe
proximityeffect; forsampleZ4[Fig. 3(b)]ofRef.[11], we
measurement temperature is lowered below 0.04K we
estimate R 20Ω from the drop in resistance when the
≈ expect the residual quantum phase fluctuations to de-
electrodesbecomesuperconducting. Weestimatethere-
stroy superconductivity and cause a reentrant behavior.
sistance of the electrodes themselves in the normal state
The fact that dissipation can stabilize superconductiv-
to be on the order of 1mΩ , which is negligible for our
ity is rather similar to the behavior of the “resistively
present discussion. The nanowire and its contact resis-
shunted (Josephson) junction” (RSJ)[3]. However, un-
tance are in parallel with the capacitance of the parallel
like the RSJ, a quantum wire can undergo depairing
plate capacitor formed by the two electrodes and the in-
when the phase fluctuations are severe. As a result the
tervening dielectric layer. This model can be simplified
normal state resistance of the non-superconducting wire
to a resistance R and capacitance C in series connected
does not have to exceed h/4e2. Our theory is consis-
acrossthenanowire,asdepictedinFig. 1(c). Thisfigure
tent with the observation that the APE practically van-
makes it clear that phase fluctuations in the nanowire
ished when one of the electrodes was replaced with a
at frequencies above f = (2πRC)−1 induce currents
0 non-superconducting metal.
through the shunting resistance and capacitor and are
In the remainder of the paper we presenta theoretical
thereby damped. Using the area A = π(0.5mm)2 of the
analysisofhowdissipationsuppressesphaseslipsinasu-
capacitor, the dielectric thickness L = 2µm and assum-
perconducting wire. Our main results are: (1) Through
ingadielectricconstantforpolycarbonateǫ=2.9[13],we
adualitytransformation,weestablishtheconnectionbe-
find C = ǫǫ A/L 10pF. Here, ǫ 8.85 10−12Fm−1
0 ≈ 0 ≈ × tweenquantumphase slips andKT vortices(instantons)
is the vacuum permittivity.
in 1+1 dimensions. Our theory can be viewed as an ap-
Whentheelectrodesaresuperconductingthestaticre-
propriate generalization of the RSJ model to quantum
sistance R in Fig.1(c) vanishes. Thus for energy (fre-
wires. (2) At T=0 an isolated superconducting wire of
quency) less than the bulk superconducting gap of the
finite length is equivalent to a classical two dimensional
electrodesthequantumwireisshuntedbyacapacitor. In
electrostatic problem where bulk charges (vortices) in-
thelatterpartofthepaperweshowthatundersuchcon-
teractwithtwometallicboundarieseachrepresentingthe
ditions the quantum fluctuations of the superconducting
(imaginary-time)world-lineoftheendpointsofthequan-
phase drive the superfluid density of the zinc nanowire
tum wire (Fig.3). Due to screening, vortices separated
to zero even at zero temperature. When the superfluid
sufficiently far apart in the imaginary time direction al-
density vanishes Cooper pairs disassociate and the wire
ways unbind, and the wire is normal even at T = 0. (3)
becomes normal. In that case the equivalent circuit of
WithshuntresistanceR,thescreeningisincomplete. For
Fig.1(c) becomes that of Fig.2 where the quantum wire
R<h/4e2,andwith1/C =0,thevorticesremainbound
acts as a normal resistor with resistance R . This ex-
N and the quantum wire is superconducting at T =0.
plainsthe ohmicbehavioratzeroappliedmagneticfield,
The imaginary-time action describing the quantum
andtheconstantdifferentialresistancedV/dI intheI V
− fluctuation of the superconducting phase of a quantum
curve from zero voltage to about 500µV.
∞ β
wire is given by S = dx dt , where
When the electrodes are drivennormalby B 30mT, −∞ 0 L
≥
the boundary resistances between the Zn wire and the =(K/R2)∂ φR2+(1/2u)∂ φ2. (1)
x t
electrodes become non-zero. Using the parameters R L | | | |
≈
20Ω, C 10pF we estimate f =(2πRC)−1 0.8 GHz. In Eq. (1) φ(x,t) =eiθ(x,t) is the phase factor of the su-
0
≈ ≈
Thisgiveshf 0.04K,anorderofmagnitudelowerthan perconducting order parameterat position x and time t,
0
≈
the thermal energy at the lowest measurement tempera- K isthesuperfluiddensity,anduistheinversecompress-
3
(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1) 1
= [σ (t)+σ (t)] F(t t′) [σ (t′)+σ (t′)]
diss L R L R
L 2 −
= i[χ(L/2,t)σ (t)+χ( L/2,t)σ (t)]. (4)
ends R L
L − −
−
InEq.(4)ρ (x,t)= Q δ(x x )δ(t t )isthe vortex
v i i − i − i
density in space time, and
σ
P
h L
σ (t)= iφ¯( L/2,t)∂ φ( L/2,t). (5)
k T σ R,L ± ± t ±
B R
Afterwehaveintegratedoutχ,thefirstterminEq.(4)is
the standardvortexCoulombgasaction, andthe second
+
termis due tothe couplingto the environment. The last
L term is a boundary term arising from the finite spatial
extent of the quantum wire.
FIG. 3: A finite-length quantum wire is mapped onto a 1+1 Equation (4) is quadratic in χ, so χ is integrated out
dimensional Coulomb gas sandwiched between two metallic exactlyviathe saddle pointsolutionδS/δχ=0,yielding
lines(theworld-lineoftheendpointsofthewire). Intheab-
sence of dissipation, the “surface” vortex density σL and σR (1/K)∂2χ+u∂2χ= iρ , u∂ χ = iσ .(6)
completely screens the logarithmic interaction between bulk t x − v x ±L/2 ± R,L
vortices. As a result, vortex-antivortex pairs unbind and the (cid:12)
(cid:12)
wireisnormal. Inthepresenceofdissipation,thelogarithmic If we define Φ(x,t) iχ(x,t) Eq. (6(cid:12)) becomes
≡−
interactionisnotcompletelyscreened,andthesuperconduct-
ing phase is stable for sufficiently small shunt resistance. (1/K)∂2Φ+u∂2Φ= ρ , u∂ Φ = σ .(7)
t x − v x ±L/2 ± R,L
(cid:12)
(cid:12)
ibility. At zero temperature (β ), depending on K Equation(7)takestheformofa2Dele(cid:12)ctrostaticproblem
→ ∞ with bulk charge density ρ and surface charge density
and u there are two possible phases: a superconducting v
σ . In terms of Φ, and become
phase and an insulating phase. In the superconducting L,R D ends
L L
phasethe topologicalsingularityinφ, i.e. the vorticesin
1 u
space and time, are bound. In the insulating phase, the D = [∂tΦ(x,t)]2 [∂xΦ(x,t)]2+Φ(x,t)ρv(x,t)
L −2K − 2
space-timevortices(orinstantons)proliferate. Fromthis
=[Φ(L/2,t)σ (t)+Φ( L/2,t)σ (t)]. (8)
ends R L
point ofview ofthe quantumwire a space-timevortex is L −
a quantum phase slip event. Substituting the solution Φ (x,t) of Eq. (7) into
c
In the presence of the external circuitry in Fig.1(c) Eqs.(8) and (4) we obtain an action,S [σ ,σ ,ρ ], that
D L R v
the action in Eq. (1) acquires an extra boundary term depends only on ρ and σ :
v L,R
(S S+S ) with S = β βdtdt′ given by
→ diss diss 0 0 Ldiss 1 L2 β β β
InLEdqi.ss(2=)−η(1/φ2)([Lη¯/(t2),∂tt)ηφ¯((t)]LFR/(2t,−Rt)t′i)s[η¯t(hte′)r∂et′laηt(itv′)e].pha(2se) SD = 2Z−L2 dxZ0 dtΦc(x,t)ρv(x,t)+Z0 dtZ0 dt′Ldiss
β
≡ −
factor of the two ends of the quantum wire and + dt[Φ (L/2,t)σ (t)+Φ ( L/2,t)σ (t)]. (9)
c R c L
−
Z0
F(t t′)=(π/β ) ω −1exp[iω (t t′)]. (3)
n n
− R | | − To obtain a final action that depends only on the vortex
Xωn density ρ we integrate out σ ,σ . Since Eq. (9) de-
v R L
Here is the dimensionless resistance = R/(h/4e2), pends on σ ,σ only quadraticallywe canagainuse the
R R R L
and ωn = 2πn/β is the Matsubara frequency. We omit saddle point method, solving δS/δσL,R = 0 and substi-
the capacitance of the external circuitry because we are tuting the solutionback into Eq.(9). In a wire of length
interested in the temperature range kBT >> hf0 ≈ L, the result is that vortices with spatial coordinates xi
0.04K where the reactance of the capacitance is negli- ( L/2 x L/2) interact via
i
gible. Clearly, this assumption would be invalid if one − ≤ ≤
were to repeat the experiment of Tian et al.[11] at tem- S = Q G(x ,x ;t t )Q . (10)
D i i j i j j
−
peratures below 0.04K;in particular,this treatment can
i6=j
X
not predict the expected re-entrant behavior.
Next we perform the standard duality In Eq. (10) G(xi,xj;tij)= β1 ωn(G1+G2)eiωntij with
transformation[14] keeping track of the finite
P
spatial extent of the wire to obtain SD = G = K cL−|xi−xj|−cxi+xj
L2 dx βdt + βdt βdt′ + βdt where 1 ru ωnsL
−L2 0 LD 0 0 Ldiss 0 Lends −1
K c c sgn(ω ) s
R R 1 R R u R G = xi xj 1+ n R L . (11)
LD = 2K[∂tχ(x,t)]2+ 2[∂xχ(x,t)]2−iχ(x,t)ρv(x,t) 2 ru ωnsL 4πs2L/2 !
4
Here c = cosh(ω x/√Ku) and s = sinh(ω x/√Ku). To conclude, in this paper we present a theory for the
x n x n
ExpandingEq.(11)forsmallω showsthatvorticeswith quantumphase slipsofasuperconducting nanowire,and
n
time separation much greater than L/√uK interact via the effect of environmental dissipation on them. We ap-
ply this theory to explain the anti-proximity effect re-
S = Q Q (1/2 )ln(t /τ )+J (t ) . (12) cently observed by Tian et al. We attribute the recur-
D i j ij 0 ij ij
− R | |
renceofsuperconductivitywhentheelectrodesaredriven
Xi6=j h i
normal by a magnetic field to the onset of dissipation
In Eq. (12) Jij is a short-range interaction (in time). fromthe boundary resistancebetweenthe quantumwire
It is important to note that the long-range logarithmic and the electrodes. This dissipation suppresses phase
interaction is controlled only by the resistive dissipa- fluctuation in the wire and stabilizes superconductivity.
tion. Aside from the irrelevant short-range interaction
Acknowledgement: We thank MingLiang Tian and
J ,Eq.(12) is identicalto the phase slipactionofa sin-
ij Moses Chan for providing very useful information on
gle RSJ (we identify Q with the phase slip). For <1
j R theirmeasurements. WealsothankZ-YWeng,X-GWen,
thephaseslipandanti-phaseslipformboundpairs,while
and A. Viswanath for useful discussions. This work was
for >1thephaseslipandanti-phaseslipunbind[3,15].
R supported by the Directior, Office of Science, Office of
The former correspondsto the superconducting phase of
Basic EnergySciences, Materials Sciences and Engineer-
the quantum wire, while the latter corresponds to the
ing Division, of the U.S. Department of Energy under
non-superconducting phase.
ContractNo. DE-AC02-05CH11231(AS, JC and DHL).
For an isolated quantum wire a similar calculation
leads to a short-range action for the vortices [Eq. (10)]
with G(xi,xj;ti −tj) = β1 ωnG1eiωn(ti−tj). Since the
interaction is short-ranged, phase slips and anti-phase
P
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