THEORY OF DISORDERED UNCONVENTIONAL SUPERCONDUCTORS
A. Kelesa, A. V. Andreeva, B. Z. Spivaka* S. A. Kivelsonb
Department of Physics, University of Washington, Seattle, WA 98195, USA hDepartment of Physics, Stanford University, CA 94305, USA
Received May 27, 2014
In contrast to conventional .s-wave superconductivity, unconventional (e.g., p- or d-wave) superconductivity is strongly suppressed even by relatively weak disorder. Upon approaching the superconductor-metal transition, the order parameter amplitude becomes increasingly inhomogeneous, leading to effective granularity and a phase ordering transition described by the Mattis model of spin glasses. One consequence of this is that at sufficiently low temperatures, between the clean unconventional superconducting and the diffusive metallic phases, there is necessarily an intermediate superconducting phase that exhibits .s-wave symmetry on macroscopic scales.
This article is dedicated to A. F. Andreev on the occasion of his 75th birthday
DOI: 10.7868/S0044451014120128
1. INTRODUCTION
Generally, the superconducting order parameter depends on two coordinates and two spin indices, -X,., (i\ r'). A classification of possible superconducting phases in crystalline materials was given in Rcfs. [1,2]. The majority of crystalline superconductors with low transition temperatures have a singlet order parameter with an .s-wave orbital symmetry that does not change under rotation of the coordinates. In the simplest case,
depends significantly only on a single coordinate, where <t2 is the second Pauli matrix in spin space, ¿l(s)(r) is a complex-valued function, and the superscript ,s indicates that it has s-wave symmetry. However, over the last decades, a number of superconductors have been discovered in which the order parameter transforms according to a nontrivial representation of the point group of the underlying crystal. Although such superconductors are quite common by now, following
E-mail: spivak'ffluw.edu
the terminology in Rcf. [3], we refer to them as "unconventional."
Important examples include the high-temperature cuprate superconductors that have a singlet d-wave symmetry [2, 4]: Afl.,(i\ r') = <(<т2 )aii AitP> (r - r'), where changes sign under coordinate ro-
tation by тг/2. The best-known example of a p-wave superfluid is superfluid 3He. One of the leading candidates for j)-wave pairing in electronic systems is Sr2Ru04 [5]. There are numerous pieces of experimental evidence that the superconducting state of Sr2Ru04 has odd parity, breaks time reversal symmetry, and is a spin triplet [5 lO]1^. An order parameter consistent with these experiments is given by the chiral j>-wave state [13], which has the form Л„.-,-(р) ~ pr±ipy, where -X,.,-(p) is the Fourier transform of — r'). An-
derson's theorem accounts for the fact that superconductivity in .s-wave superconductors is destroyed only when the disorder is so strong that ppl ~ 1, where pp is the Fermi momentum and / is the electronic elastic mean free path. However, in unconventional superconductors, A„.-,-(p) depends on the direction of the rela-
11 There are, however, some subset of experimental observations that are not easily reconciled with the existence of a chiral p-wave state in SroRuO.i. See, e.g., Refs. [11] and [12] for a discussion.
tivo nioniontuni p of oloctrons in the Cooper pair, and therefore they are much more sensitive to disorder; even at the temperature T = 0, unconventional superconductivity is destroyed when / is comparable to the zero temperature coherence length in the pure superconductor, / ~ Co 1 I'Pf- The fate of unconventional superconductivity subject to increasing disorder depends on the sign of the coupling constant in the ,s-wavc channel. It is straightforward to see that if the interaction in the ,s-wavc channel is attractive, but weaker than the attraction in an unconventional channel, then as a function of increasing disorder, there first occurs a transition from the unconventional to an ,s-wavc phase when / ~ £o, which is followed by a transition to a nonsupcrconducting phase when / ~ pj,1.
In this article, we consider the more interesting and realistic case where the interaction in the «-channel is repulsive. In this case, we show that there is necessarily a range of disorder strengths in which, although locally the pairing remains unconventional, the system has a global ,s-wavc symmetry with respect to any macroscopic superconducting interference experiments. Therefore, there must be at least two phase transitions as a function of increasing disorder: a d-wave (or p-wave) to .s-wave, followed by an .s-wave to normal metal transition. Qualitatively, the phase diagram of disordered unconventional superconductors is shown in Fig. 2 (see below). (An incomplete derivation of these results, only in the d-wave case, was obtained in Refs. [14,15].)
The existence of the intermediate .s-wave superconducting phase between the unconventional superconductor and the normal metal (and of the associated •s-wave to unconventional superconductor transition) can be understood at a mean-field level, which neglects both classical and quantum fluctuations of the order parameter. The electron mean free path is an average characteristic of disorder. We introduce a local value of the mean free path T(r) averaged over regions with a size of the order of £0- When the disorder is sufficiently strong such that, on average, I < £0. the superconducting order parameter can be large only in the rare regions where T(r) > £0- In this case, the system can be visualized as a matrix of superconducting islands that are coupled through Joscphson links in a nonsupcrconducting metal. (The superconductivity inside an island can also be enhanced if the pairing interaction is stronger than average, i.e., if the local value of is anomalously small.) At sufficiently large values of disorder, the distance between the islands is larger than both their size and the mean free path.
2. MATTIS MODEL DESCRIPTION OF DISORDERED UNCONVENTIONAL SUPERCONDUCTORS
Below, we show that in the vicinity of the superconductor normal-metal transition, the superconducting phase can be described by the Mattis model.
2.1. An isolated superconducting island
We first consider the mean-field description of an isolated superconducting island. The order parameter in an individual island is written as AQ(r,r'), where the hat indicates the two-by-two matrix structure in spin space and we label individual islands with Latin indices a, b, ... Generally, as a consequence of the random disorder, neither the shape of the island nor the texture of pairing tendencies within it have any particular symmetry, and hence the resulting gap function AQ(r,r') mixes the symmetries of different bulk phases. Since there is no translational symmetry, it is convenient to define AQ(r,p) as the Fourier transform of AQ(r,r') with respect to the relative coordinate r r' and to use r = (r + r')/2 as the ccntcr-of-mass coordinate. (Because all coordinates to appear in what follows are the ccntcr-of-mass coordinates, we henceforth drop the tilde.) In the absence of spin orbit coupling, a sharp distinction exists between spin-0 (singlet) and spin-1 (triplet) pairing, although even that distinction is entirely lost in the presence of spin orbit coupling. The most general form of the gap function (with a phase convention that we specify later) expressed as a second-rank spinor in terms of Pauli matrices is
AQ(r,p) = ei,p° ¿¿r2 (Aal + Aa ■ &) , (1)
where the r and p dependence of the scalar Aa and vector Aq quantities that represent the singlet and triplet components of the order parameter is implicit.
The energy of a single grain is independent of the overall phase of the order parameter <f>a. In the absence of spin orbit interaction, it is also independent of the direction Aa. An additional discrete degeneracy can be associated with time-reversal invariancc of the problem. It implies that the state described by a time-reversed order parameter
AQ(r,p) = -/<T2[Aa(r,-p)p<T2 (2)
leads to the same energy of the grain. In the absence of spontaneous breaking of time reversal symmetry, the time reversal operation leads to the same physical state Aq = Aq; otherwise, the time-reversed state is physically different.
1267
9*
It is important to note that generally (at the present mean-field level), time reversal symmetry is violated in droplets of unconventional superconductors of a random shape. This occurs even in the case where the bulk phase of the unconventional superconductor is time-reversal invariant, such as d-wave superconductors or tho/jy and/fy phases realized in strained Sr2Ru04 [16]. For example, d-wave superconducting droplets of a random shape embedded into a bulk metal can have, with a nonvanishing probability, a local geometry analogous to that of a corner SQUID experiment [4], in which two sides of a droplet with different signs of the order parameter are connected by a metallic Josephson link with an effective negative critical current. An equilibrium current then flows if the critical current of the "negative link" is sufficiently large.
We characterize the degeneracy with respect to time reversal by a pseudo-spin index £a = ±1. In this case, it is convenient to introduce a pseudospin £a in each grain to distinguish the two time-reversed states,
AHr.P)=i Aa(r'P)' ?Q = +L (3) Q \ A„(i\ p). £a = -1,
and write the general expression for the order parameter in each grain as
. " ' Aj;* (l*. p).
(4)
where we explicitly separate the U(l) phase of the order parameter.
2.2. Josephson coupling between islands
Electrons propagating in nonsuperconducting metals experience Andreev reflection [17] from the superconducting islands. This induces Josephson coupling between the islands. So long as the separation between islands is large, the spatial dependence of the order parameter within each grain, Aj>a(r,p), is not affected. Therefore, the lo
Для дальнейшего прочтения статьи необходимо приобрести полный текст. Статьи высылаются в формате PDF на указанную при оплате почту. Время доставки составляет менее 10 минут. Стоимость одной статьи — 150 рублей.