## Edge Tunneling of Vortices in Superconducting Thin Films

Edge Tunneling of Vortices in Superconducting Thin Films

Roberto Iengo

International School for Advanced Studies, Via Beirut 4, 34014 Trieste (Italy) and INFN, Sezione di Trieste, 34100 Trieste (Italy)

arXiv:cond-mat/9608059v1 13 Aug 1996

Giancarlo Jug

INFM and Istituto di Scienze Matematiche, Fisiche e Chimiche Universit` a di Milano a Como, Via Lucini 3, 22100 Como (Italy) and INFN, Sezione di Pavia, 27100 Pavia (Italy) (12 June 1996)

Abstract

We investigate the phenomenon of the decay of a supercurrent due to the zero-temperature quantum tunneling of vortices from the edge in a thin superconducting ?lm in the absence of an external magnetic ?eld. An explicit formula is derived for the tunneling rate of vortices, which are subject to the Magnus force induced by the supercurrent, through the Coulomb-like potential barrier binding them to the ?lm’s edge. Our approach ensues from the non-relativistic version of a Schwinger-type calculation for the decay of the 2D vacuum previously employed for describing vortex-antivortex pair-nucleation in the bulk of the sample. In the dissipation-dominated limit, our explicit edge-tunneling formula yields numerical estimates which are compared with those obtained for bulk-nucleation to show that both mechanisms are possible for the decay of a supercurrent. I. INTRODUCTION Recently there has been much renewed interest in the physics of vortices in high-temperature superconductors in the presence of applied magnetic ?elds [1]. Phenomena involving vortex dynamics and their hindrance by means of added pinning centres pose well-de?ned physics problems of great relevance for the technological applications of the new high-Tc materials. Some of the lesser understood issues are related to the problem of the residual resistence due to the thermal or quantum motion of the vortices in the presence of an externally applied supercurrent, but in the absence of the applied magnetic ?eld. Even in the absence of an external ?eld, vortices can be dragged into the bulk under the action of the Lorentz-like Magnus force which they experience sitting at the edge of the sample. This phenomenon, leading to an electrical resistence in the presence of dissipation, has been investigated

1

recently by relatively few authors [2,3] considering its importance from both the conceptual and the practical points of view. In addition, in a recent article [4] we have pointed out how such a residual resistence may arise, in the absence of an applied ?eld, also due to the spontaneous homogeneous nucleation of vortex-antivortex pairs in the bulk of the sample. These are created as ?uctuations of the electromagnetic-like Magnus ?eld acting on the quantised vortices and antivortices thought of as electron-positron-like pairs. By means of this analogy with quantum electrodynamics (QED), we have set up a powerful “relativistic” quantum ?eld theory (QFT) approach to study vortex nucleation in the two-dimensional (2-D) geometry. This study has been conducted in the presence of quantum dissipation and for the cases of either an harmonic local pinning potential [4] or, more recently [5], for a periodiclattice distribution of pinning centers. Our analysis has produced an explicit analytic result, and an estimate of the magnitude of the vortex pair-nucleation rate has shown that the e?ect may become experimentally accessible at low temperatures and high enough current densities. In this article we investigate, by means of our QFT approach, the original issue of tunneling of quantised vortices from the sample’s boundary as an alternative mechanism for the decay of an applied supercurrent in a thin-?lm geometry. The problem has already been introduced by Ao and Thouless [2] and by Stephen [3], with and without the inclusion of quantum dissipation, and discussions that give full weight to the inertial mass of the moving vortex can be found in the literature [2] alongside treatments [3,6] in which the inertial mass is taken to be negligible. The physics of the problem has been captured mainly through semi-classical evaluations of the tunneling rate and discussion has centered on the e?ects of dissipation and pinning in opposing the stabilising in?uence of the Magnus force on the classical orbits of a vortex [2,3]. Beside the issue of the residual superconductor’s resistence, the quantum tunneling of the magnetic ?ux lines into the bulk of the material has been investigated to understand theoretically [6,7] the observed [8] ?ux-creep phenomenon in the presence of a ?eld. In this work we will analyze further the problem of vortex tunneling from the point of view of the instability of the “vacuum” represented by a thin superconducting ?lm in the presence of an externally-driven supercurrent. Our aim is in fact that of obtaining an explicit formula (complete of prefactors) for the tunneling rate. From the theoretical point of view, our approach di?ers from other semi-classical evaluations of the path integral in that use is made of the Schwinger formalism for pair creation in QFT. Since this fully-relativistic formulation provides for a description of vortex-antivortex pair nucleation, we will be concerned here with the completely non-relativistic limit of this approach, formally corresponding to the case of a negligible vortex inertial mass and to contributions to the path integral relating to quantum particles moving “forward” in the time coordinate. This non-relativistic version of the Schwinger formalism indeed appears as very convenient for carrying out explicit and straightforward calculations, especially in the dissipation-dominated

2

regime. We in fact work out the tunneling rate in the case when dissipation dominates over inertia, and for the case of a potential barrier made up of the Coulomb-like attraction to the edge and the electric-like potential extracting the vortex into the bulk. Our central result shows that the tunneling rate Γ has a strong exponential dependence on the number current density J , much as in the case of vortex nucleation in the bulk of the sample [4]. The numerical value of the tunneling rate per unit length, as obtained from our approach with material parameters typical of the high-temperature cuprate superconductors, compares favourably with analogous results obtained for the bulk nucleation rate. This quali?es the novel vortex nucleation mechanism by us proposed in recent articles [4,5] as almost as likely as vortex tunneling from the sample’s edge for the decay of a supercurrent. In our estimates, edge tunneling appears to be more favourable than bulk nucleation for equal given extension of edge length and surface area. It is useful to describe the problem at hand by means of the classical equation of motion for a single vortex moving at relatively low velocity in a supercurrent having density J ¨ = ??U (q) + eE ? eq ˙ ×B ? η q ˙ mq Here m is the (negligible) inertial mass of the vortex carrying topological charge e = ±2π and treated as a single, point-like particle of 2-D coordinate q(t). Also, U (q) is the phenomenological potential acting on the vortex, in the present case the “Coulomb-like” interaction binding it to the sample’s edge. The supercurrent J gives rise to an electric-like ?eld E = ×J (a no(3) tation implying E · J = 0) superposed to a magnetic-like ?eld B = ? zdρs z and having thickness d with a sufor a thin ?lm orthogonal to the vector ? (3) per?uid component characterised by a 3-D number density ρs . Finally, η is a phenomenological friction coe?cient taking dissipation into account. A derivation of this electromagnetic analogy for the Magnus force was given by us in Ref. [4] (but see also [9–11]). Further contributions to B arising from other quantum e?ects are possible and have been recently debated [12], however in the dissipation-dominated regime the actual explicit dependence of B on the material’s parameters will be irrelevant for our study. The quantummechanical counterpart of Eq. (1.1) is constructed through the Feynman path-integral transposition in which the dissipation is treated quantistically through the formulation due to Caldeira and Leggett [13]. This approach views quantum dissipation as described by the linear coupling of the vortex coordinate to the coordinates of a bath of harmonic oscillators of prescribed dynamics. By means of this description of quantum dissipation, we formulate our own approach to the dissipative tunneling of vortices subject to the ?elds E and B (the magnetic-like ?eld being treated in an e?ective way). The organisation of our article is as follows. In Section II we sketch some basic facts about our Schwinger method for the relativistic vortex dynamics and show how the completely non-relativistic limit yields a convenient formulation of the vortex tunneling problem. This is expanded in Section III, where (1.1)

3

we treat in detail the situation in which dissipation and a Coulomb interaction are present and dominate over vortex inertia. In Section IV we estimate the tunneling rate for a range of reasonable values of the material parameters and compare the results with previous estimates for the bulk nucleation rate. This Section contains also our conclusions. We work in the units system for which h ? = 1.

II. VORTEX QUANTUM DYNAMICS: A UNIFIED APPROACH TO VACUUM DECAY THROUGH NUCLEATION AND TUNNELING The calculation that follows is the entirely non-relativistic version of our published QFT formulation [4] for the decay, via vortex-antivortex pair nucleation, of the “vacuum” represented by a supercurrent having number density J ?owing in a superconducting thin ?lm. Since the would-be particles should experience a force entirely analogous to the Lorentz force from a uniform electromagnetic ?eld, we expect this vacuum to be unstable and decay via vortex pair creation. We evaluate the probability amplitude for the vacuum decay in time T , Z = 0|e?iH T |0 ≡e?iT W0 where W0 = E (vac) ? i Γ 2 gives the energy of this vacuum, E (vac), and its decay rate, Γ. With a suitable normalization factor, the probability amplitude is given by a functional integral over ?eld con?gurations. In the presence of a gauge ?eld A? (r, t) and external potential V (r), we have Z=N D φ exp ?i

2 d2 r dt φ? ?D0 + ?

(2.1)

1 2 2 D ? E0 ? V (r) φ γ (2.2)

1 2 2 = exp ?T r ln ? D2 ? D3 + E0 + V (r) γ which can be evaluated, in the Euclidean metric, by means of the identity

2 + E 2 + V (r) ?DE 0 Λ2 ∞ dτ 2 2 2 = ? lim T r e?(?DE +E0 +V )τ ? e?Λ τ ?→0 ? τ

T r ln

(2.3)

2 = D 2 + 1 (D 2 + D 2 ) and where D = ? ? iA is where (with x3 ≡it) DE ? ? ? 2 1 3 γ the usual covariant derivative. Notice that γ = m/E0 , the ratio between the inertial mass m and the activation energy E0 , plays the role of the inverse square of the velocity of light. Thus the non-relativistic limit is implicitly taken in the dissipation-dominated case, where m→0. The evaluation of the trace is straightforward by means of the Feynman path-integral. We have, with a suitable normalization factor N (τ )

T r e?(?DE +E0 +V )τ

2

2

= N (τ )

dq0

q (0)=q (τ )=q0

D q (s)e?

τ 0

dsLE

(2.4)

4

where the Euclidean version of the relativistic Lagrangian reads 1 2 γ 2 2 ˙ ? iq LE = q ˙ + q ˙? A? (q ) + E0 + V (q) 4 t 4 Here q is taken to be a (d + 1)-dimensional coordinate describing the closed trajectories in the Euclidean space-time as a function of the Schwinger proper time s. We denote q = (qt (s), q(s)) in terms of its time- and space-like components, respectively, the dot representing the derivative q ˙ = dq ds . The part of the trajectory moving backward in time qt is therefore interpreted as describing the antivortex. The expression for the vacuum decay rate is therefore, quite generally Γ = Im 2

∞ 0

(2.5)

dτ τ

dq0 N (τ )

q (0)=q (τ )=q0

D q (s)e?

τ 0

dsLE

(2.6)

where the integral over q0 , the initial point in space of the closed paths, runs over the region of particle nucleation. In the case of vortex-antivortex pair creation in the bulk of the thin ?lm, this region is the surface of area L2 of the ?lm. In the problem considered in the present paper, vortices are already present in a narrow strip of width a≈ξ (with ξ the coherence length of the superconducting order parameter) along the edge of the sample where ? × J is di?erent from zero. Thus the “vacuum” in this case corresponds to a uniform distribution of vortices along the edge, whilst none of them is present in the bulk in the absence of an external magnetic ?eld. The “vacuum decay” thus corresponds to the possibility that some of the vortices are dragged away from the edge and enter the bulk. We can still take Eq. (2.6) as the starting point for determining the decay rate, provided we factor out the contributions coming from all paths describing particles propagating backwards in time qt and replace them with a suitable normalization factor. Also, since vortices are already present, a chemical potential ?? will be introduced to cancel out their nucleation energy. We will take the non-relativistic limit explicitely, since now the advantage of the relativistic formulation (which in our case provided for a simultaneous description of both particles and antiparticles) is no longer useful. We ?nd that this way of approaching the tunneling problem, based on Eq. (2.6), has in our opinion a number of advantages over the standard instantontype calculation [7,14]. The instanton (or WKB for point-like particles) calculation, although well established, nevertheless calls for the determination of the “bounce” solution of the path integral’s saddle point equation and the functional integration of the ?uctuations around it, a task often too hard to carry out explicitely. The present formulation based on an entirely nonrelativistic Schwinger approach leads to a promising viable alternative, as we now show. The integration over the time paths qt (s) in Eq. (2.6) is carried out by means of a sadde-point approximation which ?xes the correct nonrelativistic form of the Lagrangian. Our uniform-?eld situation corresponds

5

to A? (q ) = 1 2 F?ν qν , with the (2+1)-dimensional ?eld tensor given by, after the analytic continuation to Euclidean time 0 B iEx ? ? = ? ?B 0 iEy ? ?iEx ?iEy 0

? ?

F?ν

(2.7)

In the present treatment, the vortex chemical potential ?? is also added to the scalar potential, A0 → A0 + ?? , so that with a suitable choice the forwardmoving vortices can be selected from the backward-moving antivortices. In our previous work [4] we have shown that the main role of the magneticlike ?eld in the nucleation of vortex-antivortex pairs is to renormalise the friction coe?cient (mimicked by the bath of harmonic oscillators included in the potential V (q)), η → ηR = (η 2 + B 2 )/η (denoted simply by η in what follows), as well as the nucleation energy

2 2 2 2 E0 → E0 R = E0 + ?E (B )

(2.8)

(2.9)

where ?E (B ) is a B - and η -dependent renormalization. However, the friction coe?cient η is a rather uncertain normal-metal parameter and, moreover, we assume that there is an in?nite reservoir of vortices at the border of the ?lm, their ?? cancelling the value of the e?ective nucleation energy as well as all its renormalizations. Thus, we may safely neglect the e?ects of the magnetic-like component of the Magnus force and write the relativistic action in the form

τ

SE (B = 0) =

τ

ds

0

γ 2 1 2 2 ˙ + q q ˙ ?q ˙t (E · q + ?? ) + E0 R + V (q) 4 4 t

(2.10)

=

0

ds

γ 2 1 2 ˙ + (q q ˙t ? 2(E · q + ?? ))2 ? (E · q + ?? )2 + E0 R + V (q) 4 4

The saddle point approximation ?xes the function q˙t = dqt /ds so as to have an extremum for the action: namely, we have the saddle-point condition δSE (0) 1 = (q ˙t ? 2(E · q + ?? )) = 0 δq ˙t 2 from which we obtain dqt = 2(E · q + ?? ) → 2?? ds where we take ?? = ±E0R to be the dominant energy scale. Here, the positive sign applies to particles moving forward in the time qt and the negative one to backward-moving antiparticles. The resulting saddle-point action for those trajectories moving forward up to the standard Euclidean time T = 2E0R τ can therefore be taken as (indicating for short with t the time component qt ) (2.12) (2.11)

6

+ (0)≈ SE

T

dt

0

dq 1 m 2 dt

2

? E · q + U (q) ≡SN R

(2.13)

where the residual contribution in terms of the nucleation energy is cancelled by our choice for the chemical potential and where a term (E · q)2 /2E0R has been dropped. Also, we have denoted by U (q) = V (q)/2E0R the nonrelativistic potential energy. We now carry out the functional integral over the trajectories moving forward in time, replacing the contribution of the integral over the antivortex trajectories with some factor Φ. This factor will also include the contribution arising from the ?uctuations around the saddle point. The resulting expression for the tunneling rate through the barrier represented by the overall potential U (q) ? E · q becomes, from Eq. (2.6) Γ ≈Im 2

∞ 0

dT ΦNN R (T ) T

q(T )=q(0)

D q(t)e?SNR

(2.14)

This expression contains only closed paths forward-moving in time t and weighted by the non-relativistic action. The standard normalization factor NN R (T ) corresponds to the path integral for the non-relativistic free particle in D = 2 dimensions, in the absence of dissipation NN R (T ) D q(t)e?

T 0

˙2 mq dt 1 2

=

q(0)=q(T )

m 2πT

(2.15)

To ?x the overall factor Φ, we proceed in the following manner. The expression for the non-relativistic vacuum decay amplitude is as follows 0|e?HNR T |0 = NN R (T ) D qe?SNR Γ T = e?E (vac)T 2 cos ΓT ΓT + i sin 2 2 (2.16)

= exp ? E (vac) ? i

Since we are dealing with a highly-stable vacuum, we take the lowest order term in the expansion in powers of ΓT for Γ?E (vac) to obtain NN R (T ) D qe?SNR = i ΓT ?E (vac)T e + ··· 2 (2.17)

which determines, inserted in Eq. (2.14), the following expression for Φ

∞

Φ = E (vac) =

dT e?E (vac)T

?1

(2.18)

0

In deriving this expression we have taken into account the fact that the calculation for the rate Γ is based on a saddle-point approximation where Φ is a function of the material’s parameters only. The exponential e?E (vac)T represents the vacuum probability amplitude in the absence of the electric ?eld instability, thus we ?nally write

∞ dT ?SNR Im 0 Γ T NN R (T ) D qe = (0) ∞ ?SNR 2 0 dT NN R (T ) D qe

(2.19)

7

where the action SN R is the non-relativistic action in the absence of the electric ?eld. We have in this way reached a formulation for the tunneling rate which leads to an expression in agreement with the standard WKB method. This can be checked by examples insofar as the exponential term β is concerned, in the characteristic expression for the tunneling rate Γ = αe?β .

(0)

III. EVALUATION OF THE TUNNELING RATE THROUGH THE EDGE POTENTIAL. In this Section we will apply the above general formalism to the situation at hand, where the vortices tunnel from the edge strip under the combined e?ect of the “electric potential” ?E · q and the 2-D “Coulomb electrostatic potential”. The latter is due to the attraction of the vortex by the edge, which is equivalent to the attraction by a virtual antivortex, like in the familiar virtualcharge method in standard electrostatics. This Coulomb potential takes one 2 of the equivalent forms UC (y ) = K ln(1+ y/a) or UC (y ) = 1 2 K ln 1 + (y/a) , where y is the distance from the edge, both acceptable extrapolations of the known large-distance behavior. The coupling constant K depends on the (3) super?uid density ρs and carrier’s mass m0 through the relationship [2] (3) K = 2πρs d/m0 . However, its precise value in real materials is unknown and this parameter will be treated, like many others in this calculation, phenomenologically. When the Caldeira-Leggett quantum dissipation is taken into account [13,4], we end up with an overall non-relativistic potential U (q) = UD (q) =

k

1 V (q) = UC (y ) + UD (q) 2E0R

? ?1

with the oscillators’ masses mk and frequencies ωk constrained in such a way that the classical equation of motion for the resulting action reproduces the form (1.1). This requires the ck to satisfy the constraint (for the so-called ohmic case, which we consider) π 2 c2 k δ(ω ? ωk )≡J (ω ) = ηω mk ω k (3.2)

1 ck 2 ˙ 2 + mk ω k m x xk + 2 ?2 k k 2 2mk ωk

2?

? ?

(3.1)

k

η being the phenomenological friction coe?cient of Eq. (1.1) (renormalised by B ). Notice that this coe?cient is una?ected by the non-relativistic limit. After a Fourier transformation in which the xk modes can be integrated out [4], we are lead to an e?ective non-relativistic action which in the dissipationdominated (m → 0) regime reads, with q = (x, y ) SN R = 2πη

n>0

nqn · q? ?+K n ? ET y

T

dt ln 1 +

0

y (t) a

(3.3)

8

? + n=0 qn eiωn t , where ωn = 2πn/T , and furthermore y ? = Here q(t) = q T 0 dty (t)/T stands for the n = 0 mode in the y -direction. In order to render the problem tractable, at this point, we introduce a further approximation for the interaction term of the action in Eq. (3.3), by writing

T

dt ln 1 +

0

y y ? ≈T ln 1 + a a

(3.4)

which is justi?ed in the limit of large exit times T (see Appendix). We are therefore in a position to evaluate the numerator of the formula, Eq. (2.19), for the tunneling rate, which we write schematically as Γ/2 = N /D . We de?ne, for the numerator

∞

N ≡Im

0

dT NN R (T ) T

D qe?SNR = Im

∞ 0

dT NN R (T )Ix Iy T

(3.5)

where the path integral over q(t) factorises into Ix = L ∞ n=1 (1/2ηn), representing the contribution of the free dissipative motion along the x-axis of the edge having length L, and in Iy =

??KT ln(1+? y /a) dy ?IG (? y , T )eET y

(3.6)

the factor containing the dynamical e?ects, which we now evaluate. This we do most conveniently by integrating out ?rst the real and imaginary parts, Re yn = ψn and Im yn = ξn , of the modes yn for n > 0. Since the path integral is restricted to loops which have their origin at y (0) = y (T )≡y0 = 0, ? + 2 n>0 ψn = 0 has to be imposed, the constraint y0 = y ? + n=0 ψn = y corresponding to the vacuum wave-function |Ψ0 (y0 )|2 ?δ(y0 ). This leads to the constrained Gaussian integrals

∞ ∞

IG (? y, T ) =

n=1 ∞

dξn dψn δ y ?+2

n=1 ∞ n=2 1/2

ψn e?2πη

∞ n,m=2

∞ n=1

2 +ψ 2 ) n(ξn n

1 1 ?2 = e? 2 πηy × 2

∞

×

n=1

1 2ηn

dψn exp ?2πη

?

? ?

ψn (nδnm + 1)ψm ? 2πη y ?

ψn

n=2

? ? ?

(3.7)

where the δ-function constraint has been taken care of through, e.g., the ψ1 integration. Introducing the matrix (n, m = 2, 3, 4 . . .) Mnm = 2πη (nδnm + 1) the integrals can be evaluated, formally, to give 1 IG (? y, T ) = √ 2 π

∞ n=1

(3.8)

π 2ηn

1/2

det M ?1/2 e

?1 πηy ?2 +π 2 η2 y ?2 2

∞ n,m=2

?1 Mnm

(3.9)

Notice that the resulting expression involves in?nite products and sums that would lead to divergencies. However, since the dissipation is believed to be suppressed above a characteristic material-dependent frequency ωc , these

9

products and sums are to be cut at a characteristic integer n? = [ωc T /2π ]. ?1 Writing M = 2πηM0 (1 + M0 V ), with M0nm = nδnm and Vnm = 1, we get

n?

det M =

?1 (2πηn) exp T r ln(1 + M0 V) n=2

(3.10)

?1 The trace can be evaluated now by formally expanding ln(1 + M0 V ), to get ?1 det(1 + M0 V)=1+

1 ≡Q n n=2

1 n δnm

n?

(3.11) , yielding (3.12)

?1 = In a similar way, we evaluate Mnm n? ?1 Mnm = n,m=2

1 2πη

?

1 Qnm

1 1 (1 ? ) 2πη Q

from which we obtain IG (? y, T ) = 1 2 2η Q

1/2 n>0

1 ?2 ? πη y e 2Q 2ηn

(3.13)

The numerator of our formula for the tunneling rate Γ is therefore the double integral 1 2η N = Im L 2 Q

1/2 0 ∞

1 dT NN R (T ) T 2ηn n>0

2 0

∞

? dy ?e? 2Q y

πη

2 +ET y ??KT

ln(1+? y /a)

(3.14)

We remark that in the above formula the following formal expression appears: 2 1 ? ≡NN R (T ) N . We interpret this expression as an m → 0 limit:

n>0 2ηn

? = NN R (T ) N

n>0

πT ?1 2 + ηω mωn n

2

m = 2πT

n?

1+

n=1

ηT 1 2πm n

?2

η → exp ?T ?E 2π

η (1 + ln(mωc /η )) is a further renormalization of the activation where ?E = πm energy which is also cancelled by tuning the vortex chemical potential ?? , so ?→ η . that, e?ectively, N 2π The integral is now evaluated by means of the steepest descent approximation in both y ? and T ; the saddle-point equations are, with

(3.15)

S= as the reduced action,

πη 2 y ? y ? ? ET y ? + KT ln(1 + ) 2Q a

(3.16)

y ? ?S = ?E y ? + K ln(1 + ) = 0 ?T a πη 1 ?S ? + KT ? = y ? ? ET =0 ?y ? Q a+y ?

(3.17)

10

?, the mean exit distance and the ?rst of which yields y ? and the second T time, respectively (since n? is a large cuto? integer, Q is kept ?xed in the variation of the e?ective action). This leads to the expression S0 = πη y ?2 /2Q for the saddle-point e?ective action. One has, furthermore, to integrate over ?. One can easily verify that there the Gaussian ?uctuations around y ? and T is a negative mode, thus giving an imaginary contribution, coming from the integration over the ?uctuations in T . We in fact expand S = S0 + αδy 2 + βδyδT = S0 + α(δy +

T 1 πη ( Q ? (aK with α = 2 +? y )2 ) and β = ?E + evaluation of the numerator of Eq. (2.19) ∞ ?

β β2 2 δT )2 ? δT 2α 4α This leads to the following e?S0 η ? (E ? K/(a + y 2Q T ?))

(3.18)

K a+? y.

N = Im

0

dT NN R (T ) T

D qe?SNR = ηL

πη 2 y ? S0 = 2Q

(3.19)

?/2π ] for large enough T ?. where Q = ln[ωc T We now come to the evaluation of the normalization denominator in Eq. (2.19). We remark that, the numerator being already of the form αe?β , this denominator contributes only to the speci?cation of the prefactor α in the expression for the tunneling rate Γ. To begin with, we carry out the Gaussian integrals over ξn and ψn . Here we use the alternative form of the Coulomb 2 interaction, UC = 1 2 K ln 1 + (y (t)/a) , and we expand the logarithm around the minimum (now at y = 0) to lowest order (thus keeping all the integrals Gaussian). We de?ne

∞

D≡

0

dT NN R (T )

D qe?SNR =

(0)

∞ 0

(0) dT NN R (T )Ix Iy

(3.20)

where Ix is the same as for the numerator above, while

∞ (0) Iy ∞

=

n=1

dξn dψn

∞

dy ?δ(? y+2

n=1 2 2 n(ξn + ψn )?

ψn ) × (3.21)

× exp ?2πη

n=1

∞ KT 2 [2 (ξ 2 + ψn )+y ?2 ] 2a2 n=1 n

Finally, repeating many of the steps of the calculation done for the Gaussian integrals of the numerator N , we end up with a2 η 2 L D= K

∞

du (1 + 2uQ(u))

0

?1/2

n? n=1

?

(1 + u/n)?1 ≡

a2 η 2 ωc L ?( a2 η ) K K

(3.22)

where u = KT /2πa2 η , Q(u) = n n=1 1/(n + u) and ? is a dimensionless function of order unity. The evaluation of the tunneling rate is now formally

11

completed, and an explicit analytic formula ensues from our (albeit approximate) treatment leading, from Eq.s (3.19) and (3.22), to the tunneling rate per vortex: Γ K e?S0 √ = N /D = ? (E ? K/(a + y 2 ?)) ?a2 2Qη T This expression is readily evaluated, once the parameters a, η, ωc , K and J are ?xed, by evaluating numerically the solutions of Eq. (3.17) and the integral in Eq. (3.22) by means of n? = [ωc T /2π ] = [(ωc a2 η/K )u] ([] denoting the integer part). To obtain the tunneling rate per unit length, we must compute the density of vortices along the edge. Since ρv = 21 π ? × ?θ is the bulk density in terms of the condensate phase θ , and since the number current is J=

ρs d m0 (?θ

(3)

(3.23)

? A), we obtain dyρv = m0 2πρs d

(3)

dy?y Jx = J/K

(3.24)

The ?nal expression for R, the tunneling rate per unit length, is thus R= J Γ K (3.25)

We conclude this Section by noticing that, since E = 2πJ in terms of the supercurrent density, and since the saddle-point equation (3.17) has iterative solution y ?= K ?/a) = K E ln(1+ y E {ln(1+ K/Ea)+ · · ·}, we can cast the dominant exponential factor determining Γ in the form S0 = η {K ln(1 + K/2πJa) + · · ·}2 8πJ 2 Q (3.26)

This yields almost the same leading dependence (logarithmic corrections 2 apart) Γ≈e?(J0 /J ) on the external current that was also obtained for the homogeneous bulk-nucleation phenomenon [4] (in terms of the tunneling length 2 ?T ?K/E we could also rewrite Γ?e?η?T ). Indeed, the e?ective Coulomb energy K ln(1+ K/2πJa) can be interpreted as playing the role of the activation energy E0 renormalised by the vortex-antivortex interactions. IV. DISCUSSION AND ESTIMATE OF THE TUNNELING RATE. We have presented a new treatment for the quantum tunneling of vortices from the boundary of a thin superconducting ?lm. The formulation makes use of the idea that tunneling can be viewed as a pair-creation process in which only forward-moving time trajectories contribute to the Schwinger path integral. Backward-moving antivortices represent contributions to the path integral that are factorised out, and we have made use of the fact that an in?nite reservoir of vortices is present at the edge. This leads to a mathematically convenient formulation of the evaluation of the tunneling rate, which

12

we have worked out in detail by assuming that the vortices experience an attractive 2D Coulomb-like potential con?ning them to the edge. By means of some approximate treatment of the logarithmic Coulomb interaction, and of the saddle-point evaluation of the path integral, we have reduced all integrations to Gaussian integrals which a?ord a closed analytical expression for the tunneling rate. This can then be estimated by chosing material parameters suitable for typical cuprate superconducting systems, e.g. YBCO ?lms, that a?ord some of the highest critical current densities (Jc ?107 Acm?2 at 77 K [15]). To ?x the indicative value of the friction coe?cient η , we make use of the the Bardeen-Stephen formula [16] linking the upper critical ?eld Bc2 to the normal metal resistivity ρn : η = Φ0 Bc2 /ρn (Φ0 being the ?ux quantum). A?2 for YBCO. A rather uncertain parameter is the cuto? This yields η ≈10?2 ? frequency ωc , although the dependence on it of the tunneling rate is rather weak. We therefore take indicatively ωc ≈80 K, just under the expected value of the Kosterlitz-Thouless transition temperature in these ?lms [1]. As for the edge thickness, we take a≈ξ ≈10? A of the order of magnitude of the coherence length. Finally, there is the value of the Coulomb interaction strength, K . Since we must have K < ωc , we present in Fig. 1 the dependence of the tunneling rate on the current density for some indicative values of K . The e?ect of temperature is taken into account phenomenologically [4], by introducing thermal current ?uctuations in the exponential of the formula for the rate, J 2 → J 2 + ?J (T )2 . The ?uctuation term ?J (T ) is ?xed by requiring that for J = 0 an Arrhenius form Γ ∝ e?Umax /T is obtained for the rate, with the barrier’s height Umax = K ln(K/2πJa) ? K + 2πJa as the activation energy. Notice that the barrier is in?nite for J →0, so that there is no zero-current tunneling of vortices even for non-zero temperature. It can be seen from Fig. 1 that the e?ect of thermal ?uctuations, as estimated from the above interpolative assumption, is indicatively to raise the tunneling rate by some orders of magnitude. The results are summarised in Fig. 1, where it is shown that tunneling of vortices from the ?lm’s boundary can become a highly likely phenomenon for su?ciently large current densities J and relatively low Coulomb couplings K . The process is aided by the presence of thermal current ?uctuations. For even larger currents than those considered here, the process could be described by means of a classical treatment based on the time-dependent Ginzburg-Landau equation [17]. It is interesting to compare the results obtained in Fig. 1 with the estimates carried out for the bulk nucleation process [4]. There, the maximum nucleation rates observed (also taking temperature into account) were in the region of ≈ 1013 ?m?2 s?1 for a current J = 107 A cm?2 and assuming that pinning centers act as nucleation seeds. We therefore conclude that edge tunneling appears to be a more likely supercurrent decay mechanism for normal superconducting sample geometries. Yet, the bulk-nucleation of vortex-antivortex pairs is also a competitive parallel mechanism that could be singled-out by means of an appropriate choice of sample geometry. Although the measurements of the residual resistence in a superconductor

13

may not allow to distinguish between these two competing mechanisms, we stress that from the point of view of vortex-tunneling microscopy the two processes remain entirely separate observable phenomena. Bulk pair-nucleation remains, in particular, a completely open challenge for observation of a phenomenon that even in the context of standard QED has remained, to our knowledge, so far elusive for static ?elds. Acknowledgements One of us (G.J.) is grateful to the International School for Advanced Studies in Trieste, where part of his work was carried out, for hospitality and use of its facilities. APPENDIX In this Appendix, we justify the use of the approximation, Eq. (3.4), used in our calculation for handling the “Coulomb” interaction in the path integral of the numerator of our formula for the nucleation rate. For large y ?, and with ωn = 2πn/T :

T 0

y (t) dt ln 1 + a

∞ y ? (?)l+1 ≈ T ln 1 + + a l l=1

T 0

At the saddle point, yn = ?y ?/2nQ (as can be easily veri?ed by varying Eq. (3.3)) and therefore we have, for example

T 0

yn iωn t ? dt ? e y ? n=0

?

?l

(A.1)

ηK cT Since Q = ln[ ω 2π ] and T ? E 2 , we get the estimate T

dt ?

?

T yn iωn t ? e = y ? 4 Q2 n=0

?2

1 2 n n=1

∞

(A.2)

dt ln 1 +

0

y (t) a

≈ T ln 1 +

y ? ηωc K + O ln a E2

?2

(A.3)

Thus the approximation of taking the “Coulomb” potential of the form K ln(? y /a) is justi?ed for very low currents (E = 2πJ →0). We still keep this form, when the current is not so low, as we believe it captures the essential point of the dynamical picture.

14

REFERENCES

[1] For recent surveys of this ?eld, see: G. Blatter, M.V. Feigel’man, V.B. Geshkenbein, A.I. Larkin and V.M. Vinokur, Rev. Mod. Phys. 66, 1125 (1994); E.H. Brandt, Rep. Progr. Phys. 58, 1465 (1995); also, see the articles in: Phase Transitions and Relaxation in Systems with Competing Energy Scales, T. Riste and D. Sherrington (Ed.s) (Kluwer, Dordrecht 1993). [2] Ping Ao and D.J. Thouless, Phys. Rev. Lett. 72, 132 (1994). [3] M.J. Stephen, Phys. Rev. Lett. 72, 1534 (1994). [4] R. Iengo and G. Jug, Phys. Rev. B 52, 7536 (1995). [5] R. Iengo and G. Jug, submitted to Phys. Rev. Lett. (1996). [6] B.I. Ivlev, Yu.N. Ovchinnikov and R.S. Thompson, Phys. Rev. B44 7023 (1991). [7] G. Blatter, V.B. Geshkenbain and V.M. Vinokur, Phys. Rev. Lett. 66 3297 (1991); G. Blatter and V.B. Geshkenbein, Phys. Rev. B47 2725 (1993). [8] D. Fiorani, A. Garcia, X.X. Zhang, A.M. Testa and J. Tejada, in: Quantum Tunneling of the Magnetization, Proceedings of the 1994 NATO ASI in Grenoble (Kluwer, Dordrecht 1995). [9] D.-H. Lee and M.P.A. Fisher, Int. J. Mod. Phys. B 5, 2675 (1991). [10] P. Ao and D.J. Thouless, Phys. Rev. Lett. 70, 2158 (1993); D.J. Thouless, P. Ao and Q. Niu, Physica A 200, 42 (1993); Q. Niu, P. Ao and D.J. Thouless, Phys. Rev. Lett. 72, 1706 (1994). [11] F. Gaitan, Phys. Rev. B 51, 9061 (1995). [12] A. van Otterlo, M. Feigel’man, V. Geshkenbein and G. Blatter, Phys. Rev. Lett. 75, 3736 (1995); and references therein. [13] A.O. Caldeira and A.J. Leggett, Ann. Phys. 149, 374 (1983). [14] S. Coleman, Phys. Rev. D 15, 2929 (1977); U. Weiss, Dissipative Quantum Systems (World Scienti?c, Singapore 1993). [15] M. Cyrot and D. Pavuna, Introduction to Superconductivity and High-Tc Materials (World Scienti?c, Singapore 1992). [16] J. Bardeen and M.J. Stephen, Phys. Rev. 140, 1197 (1965). [17] I. Aranson, M. Gittermann and B.Ya. Shapiro, Phys. Rev. B51 3092 (1995). FIGURE CAPTIONS Figure 1. Plot of the tunneling rate per unit length, R, as function of the supercurrent density, J , for the values K = 30 K and K = 50 K. Dashed lines, T = 0 K; full lines, T = 2.5 K.

15

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