Cooper L.N., Feldman D. (Eds.) BCS: 50 Years
Подождите немного. Документ загружается.
September 8, 2010 10:53 World Scientific Review Volume - 9.75in x 6.5in ch9
SQUIDs: Then and Now 181
10. P. W. Anderson and J. M. Rowell, Phys. Rev. Lett. 10, 230 (1963).
11. J. M. Rowell, Phys. Rev. Lett. 11, 200 (1963).
12. R. C. Jaklevic, J. Lambe, A. H. Silver and J. E. Mercereau, Phys. Rev. Lett.
12, 159 (1964).
13. J. E. Zimmerman and A. H. Silver, Phys. Lett. 10, 47 (1964).
14. A. B. Pippard, J. G. Shepherd and D. A. Tindall, Proc. Roy, Soc. (London)
A 324, 17 (1971).
15. A. B. Pippard and G. T. Pullan, Proc. Camb. Phil. Soc. 48, 188 (1952).
16. J. Clarke, Phil. Mag. 13, 115 (1966).
17. J. Clarke, Proc. Roy. Soc. A 308, 447 (1969).
18. E. Rumbo, Phil. Mag. 13, 689 (1969).
19. J. Clarke, Phys. Rev. Lett. 21, 1566 (1968).
20. B. N. Taylor, W. H. Parker, D. N. Langenberg and A. Denenstein, Metrologia
3, 89 (1967).
21. J. S. Tsai, A. K. Jain and J. E. Lukens, Phys. Rev. Lett. 51, 316 (1983).
22. J. Clarke, Phys. Rev. Lett. 28, 1363 (1972).
23. M. Tinkham and J. Clarke, Phys. Rev. Lett. 28, 1366 (1972).
24. J. Clarke, in Nonequilibrium Superconductivity, eds. D. N. Langenberg and
A. I. Larkin (Elsevier Science Pub. B.V., Amsterdam, 1986), p. 1.
25. J. E. Zimmerman and A. H. Silver, Phys. Rev. 141, 367 (1966).
26. P. L. Richards, S. Shapiro and C. C. Grimes, Amer. J. Phys. 36, 690 (1968).
27. J. Clarke, Ph.D. thesis, University of Cambridge (1968).
28. R. L. Forgacs and A. Warnick, Rev. Sci. Instrum. 38, 214 (1967).
29. D. Drung, R. Cantor, M. Peters, H. J. Scheer and H. Koch, Appl. Phys. Lett.
57, 406 (1990).
30. A. H. Silver and J. E. Zimmerman, Phys. Rev. 157, 317 (1967).
31. J. E. Zimmerman, P. Thiene and J. T. Harding, J. Appl. Phys. 41, 1572
(1970).
32. J. E. Mercereau, Phys. Rev. Appl. 5, 13 (1970).
33. M. Nisenoff, Phys. Rev. Appl. 5, 21 (1970).
34. J. Clarke, Proc. IEEE 61, 8 (1973).
35. C. M. Falco, W. H. Parker, S. E. Trullinger and P. K. Hansma, Phys. Rev. B
10, 1865 (1974).
36. J. Clarke, W. M. Goubau and M. B. Ketchen, J. Low Temp. Phys. 25, 99
(1976).
37. T. D. Gamble, W. M. Goubau and J. Clarke, Geophys. 44, 53 (1979).
38. T. D. Gamble, W. M. Goubau and J. Clarke, Geophys. 44, 959 (1979).
39. M. B. Ketchen and J. M. Jaycox, Appl. Phys. Lett. 40, 736 (1982).
40. J. M. Rowell, M. Gurvitch and J. Geerk, Phys. Rev. B 24, 2278 (1981).
41. M. Gurvitch, M. A. Washington and H. A. Huggins, Appl. Phys. Lett. 42, 472
(1983).
42. D. E. McCumber, J. Appl. Phys. 39, 3113 (1968).
43. W. C. Stewart, Appl. Phys. Lett. 12, 277 (1968).
44. V. Ambegaokar and B. I. Halperin, Phys. Rev. Lett. 22, 1364 (1969).
45. K. K. Likharev and V. K. Semenov, JETP Lett. 15, 442 (1972).
46. C. D. Tesche and J. Clarke, J. Low Temp. Phys. 29, 301 (1977).
September 8, 2010 10:53 World Scientific Review Volume - 9.75in x 6.5in ch9
182 J. Clarke
47. C. D. Tesche and J. Clarke, J. Low Temp. Phys. 37, 397 (1979).
48. J. Kurkijarvi, J. Appl. Phys. 44, 3729 (1973).
49. L. D. Jackel and R. A. Buhrman, J. Low Temp. Phys. 19, 201 (1975).
50. G. J. Ehnhohn, J. Low Temp. Phys. 29, 1 (1977).
51. K. K. Likharev, Dynamics of Josephson Junctions and Circuits (Gordon
Breach, New York, 1986).
52. T. Ryh¨anen, H. Seppa, R. Ilmoniemi and J. Knuutila, J. Low Temp. Phys.
76, 287 (1989).
53. B. Chesca, J. Low Temp. Phys. 110, 963 (1998).
54. P. K. Hansma, J. Appl. Phys. 44, 4191 (1973).
55. L. S. Kuzmin, K. K. Likharev, V. V. Migulin, E. A. Polunin and N. A.
Simonov, in SQUID’85, Superconducting Quantum Interference Devices and
their Applications, eds. H. D. Hahlbohm and H. L¨ubbig (Walter de Gruyter,
Berlin, 1985), p. 1029.
56. D. Koelle, R. Kleiner, F. Ludwig, E. Dantsker and J. Clarke, Rev. Mod. Phys.
71, 631 (1999).
57. J. Vrba, J. Nenonen and L. Trahms, in The SQUID Handbook. Vol. II:
Applications of SQUIDs and SQUID Systems, eds. J. Clarke and A. I.
Braginski (Wiley-VCH GmbH & Co., Weinheim, 2006), Ch. 11.
58. R. C. Black and F. C. Wellstood, in The SQUID Handbook. Vol. II: Appli-
cations of SQUIDs and SQUID Systems, eds. J. Clarke and A. I. Braginski
(Wiley-VCH GmbH & Co., Weinheim, 2006), Ch. 12.
59. T. R. Clem, C. P. Foley and M. N. Keene, in The SQUID Handbook. Vol.
II: Applications of SQUIDs and SQUID Systems, eds. J. Clarke and A. I.
Braginski (Wiley-VCH GmbH & Co., Weinheim, 2006), Ch. 14.
60. H.-J. Krause and G. Donaldson, in The SQUID Handbook. Vol. II: Appli-
cations of SQUIDs and SQUID Systems, eds. J. Clarke and A. I. Braginski
(Wiley-VCH GmbH & Co., Weinheim, 2006), Ch. 13.
61. H. J. Paik, in The SQUID Handbook. Vol. II: Applications of SQUIDs and
SQUID Systems, eds. J. Clarke and A. I. Braginski (Wiley-VCH GmbH &
Co., Weinheim, 2006), Ch. 15.
62. J. Gallop and F. Piquermal, in The SQUID Handbook. Vol. II: Applications of
SQUIDs and SQUID Systems, eds. J. Clarke and A. I. Braginski (Wiley-VCH
GmbH & Co., Weinheim, 2006), Ch. 9.
63. J. Clarke and F. K. Wilhelm, Nature 453, 1031 (2008).
64. J. Clarke, A. T. Lee, M. M¨uck and P. L. Richards, in The SQUID Hand-
book. Vol. II: Applications of SQUIDs and SQUID Systems, eds. J. Clarke and
A. I. Braginski (Wiley-VCH GmbH & Co., Weinheim, 2006), Ch. 8.
65. P. James, E. Peebles, L. A. Page Jr. and R. B. Partridge, Finding the Big
Bang (Cambridge University Press, Cambridge, England, 2009).
66. R. A. Sunyaev and Y. B. Zel’Dovich, Comments on Astrophysics 4, 173 (1972).
67. K. D. Irwin, Appl. Phys. Lett. 66, 1998 (1995).
68. D. Schwan, P. A. R. Ade, K. Basu, A. N. Bender, B. A. Benson, F. Bertoldi,
H.-M. Cho, G. Chon, J. Clarke, M. Dobbs, D. Ferrusca, R. G¨usten, N. W.
Halverson, W. L. Holzapfel, B. Johnson, A. Kov´acs, J. Kennedy, Z. Ker-
mish, R. Kneissl, T. Lanting, A. T. Lee, M. Lueker, J. Mehl, K. M. Menten,
September 8, 2010 10:53 World Scientific Review Volume - 9.75in x 6.5in ch9
SQUIDs: Then and Now 183
D. Muders, M. Nord, F. Pacaud, T. Plagge, C. L. Reichardt, P. L. Richards,
R. Schaaf, P. Schilke, F. Schuller, H. Spieler, C. Tucker, A. Weiss, B. West-
brook, O. Zahn, Rev. Sci. Instrum., submitted.
69. T. M. Lanting, H.-M. Cho, J. Clarke, W. L. Holzapfel, A. T. Lee, M. Lueker,
P. L. Richards, M. A. Dobbs, H. Spieler and A. Smith, Appl. Phys. Lett. 86,
112511 (2005).
70. N. W. Halverson, T. Lanting, P. A. R. Ade, K. Basu, A. N. Bender, B. A.
Benson, F. Bertoldi, H.-M. Cho, G. Chon, J. Clarke, M. Dobbs, D. Ferrusca,
R. G¨usten, W. L. Holzapfel, A. Kov´acs, J. Kennedy, Z. Kermish, R. Kneissl,
A. T. Lee, M. Lueker, J. Mehl, K. M. Menten, D. Muders, M. Nord, F. Pacaud,
T. Plagge, C. Reichardt, P. L. Richards, R. Schaaf, P. Schilke, F. Schuller,
D. Schwan, H. Spieler, C. Tucker, A. Weiss and O. Zahn, Astrophys. J. 701,
42 (2009).
71. B. Sadoulet, Science 315, 61 (2007).
72. R. Peccei and H. Quinn, Phys. Rev. Lett. 38, 1440 (1977).
73. S. Weinberg, Phys. Rev. Lett. 40, 223 (1978).
74. F. Wilczek, Phys. Rev. Lett. 40, 279 (1978).
75. P. Sikivie, Phys. Rev. Lett. 51, 1415 (1983).
76. P. Sikivie, Phys. Rev. D 32, 2988 (1985).
77. R. Bradley, J. Clarke, D. Kinion, L. J. Rosenberg, K. van Bibber, S. Matsuki,
M. M¨uck and P. Sikivie, Rev. Mod. Phys. 75, 777 (2003).
78. J. E. Kim, Phys. Rev. Lett. 43, 103 (1979).
79. M. A. Shifman, A. I. Vainshtein and V. I. Zakharov, Nucl. Phys. B 166, 493
(1980).
80. M. Dine, W. Fischler and M. Srednicki, Phys. Lett. B 104, 199 (1981).
81. A. P. Zhitnitsky, Yad. Fiz. 31, 497 (1980) [Sov. J. Nucl. Phys. 31, 260].
82. M. M¨uck, M.-O. Andr´e, J. Clarke, J. Gail and Ch. Heiden, Appl. Phys. Lett.
72, 2885 (1998).
83. M. M¨uck and J. Clarke, J. Appl. Phys. 88, 6910 (2000).
84. M. M¨uck, C. Welzel and J. Clarke, Appl. Phys. Lett. 82, 3266 (2003).
85. M. M¨uck, M.-O. Andr´e, J. Clarke, J. Gail and Ch. Heiden, Appl. Phys. Lett.
75, 3545 (1999).
86. M. M¨uck, J. B. Kycia and J. Clarke, Appl. Phys. Lett. 78, 967 (2001).
87. F. C. Wellstood, C. Urbina and J. Clarke, Phys. Rev. B 49, 5942 (1994).
88. D. Kinion and J. Clarke, Appl. Phys. Lett., submitted.
89. S. J. Asztalos, G. Carosi, C. Hagmann, D. Kinion, K. van Bibber, M. Hotz,
L. Rosenberg, G. Rybka, J. Hoskins, J. Hwang, P. Sikivie, D. B. Tanner,
R. Bradley and J. Clarke, Phys. Rev. Lett. 104, 041301 (2010).
90. E. M. Haacke, R. W. Brown, M. R. Thompson and R. Venkatesan, Magnetic
Resonance Imaging: Physical Principles and Sequence Design (Wiley & Sons,
New York, 1999).
91. A. Abragam, The Principles of Nuclear Magnetism (Clarendon Press, Oxford,
1961).
92. C. P. Slichter, Principles of Magnetic Resonance, 3rd edition (Springer, New
York, 1989).
93. E. L. Hahn, Phys. Rev. 80, 580 (1950).
September 8, 2010 10:53 World Scientific Review Volume - 9.75in x 6.5in ch9
184 J. Clarke
94. M. Packard and R. Varian, Phys. Rev. 93, 941 (1954).
95. R. McDermott, A. H. Trabesinger, M. M¨uck, E. L. Hahn. A. Pines and
J. Clarke, Science 295, 2247 (2002).
96. R. McDermott, S.-K. Lee, B. ten Haken, A. H. Trabesinger, A. Pines and
J. Clarke, Proc. Natl. Acad. Sci. 101, 7857 (2004).
97. R. McDermott, N. Kelso, S.-K. Lee, M. M¨ossle, M. M¨uck, W. Myers, B. ten
Haken, H. C. Seton, A. H. Trabesinger, A. Pines and J. Clarke, J. Low Temp.
Phys. 135, 793 (2004).
98. J. Clarke, M. Hatridge and M. M¨ossle, Annu. Rev. Biomed. Eng. 9, 389 (2007).
99. S.-K. Lee, M. M¨ossle, W. Myers, N. Kelso, A. H. Trabesinger, A. Pines and
J. Clarke, Magn. Reson. Med. 53, 9 (2005).
100. J. Kurhanewicz, D. Vigneron, P. Carroll and F. Coakley, Curr. Opin. Urol.
18, 71 (2008).
101. L. Qiu, Y. Zhang, H.-J. Krause, A. I. Braginski, M. Burghoff and L. Trahms,
Appl. Phys. Lett. 91, 072505 (2007).
102. J. Bernarding, G. Buntkowsky, S. Macholl, S. Hartwig, M. Burghoff and
L. Trahms, J. Am. Chem. Soc. 128, 714 (2006).
103. P. Volegov, A. N. Matlachov, M. A. Espy, J. S. George and R. H. Krause, Jr.,
Magn. Reson. Med. 52, 467 (2004).
104. V. S. Zotev, A. N. Matlashov, P. L. Volegov, I. M. Savukov, M. A. Espy,
J. C. Mosher, J. J. Gomez and R. H. Kraus, Jr., J. Magn. Reson. 194, 115
(2008).
September 14, 2010 9:18 World Scientific Review Volume - 9.75in x 6.5in ch10
RESISTANCE IN SUPERCONDUCTORS
Bertrand I. Halperin
∗
,GilRefael
†
and Eugene Demler
∗
∗
Physics Dep artment, Harvard University, Cambridge, MA 02138, USA
†
Department of Physics, California Institute of Technology,
Pasadena, CA 91125, USA
In this pedagogical review, we discuss how electrical resistance can arise
in superconductors. Starting with the idea of the superconducting order
parameter as a condensate wave function, we introduce vortices as topologi-
cal excitations with quantized phase winding, and we show how phase slips
occur when vortices cross the sample. Superconductors exhibit non-zero
electrical resistance under circumstances where phase slips occur at a finite
rate. For one-dimensional superconductors or Josephson junctions, phase
slips can occur at isolated points in space-time. Phase slip rates may be
controlled by thermal activation over a free-energy barrier, or in some cir-
cumstances, at low temperatures, by quantum tunneling through a barrier.
We present an overview of several phenomena involving vortices that have
direct implications for the electrical resistance of superconductors, includ-
ing the Berezinskii–Kosterlitz–Thouless transition for vortex-proliferation
in thin films, and the effects of vortex pinning in bulk type II supercon-
ductors on the nonlinear resistivity of these materials in an applied mag-
netic field. We discuss how quantum fluctuations can cause phase slips and
review the non-trivial role of dissipation on such fluctuations. We present a
basic picture of the superconductor-to-insulator quantum phase transitions
in films, wires, and Josephson junctions. We point out related problems in
superfluid helium films and systems of ultra-cold trapped atoms. While our
emphasis is on theoretical concepts, we also briefly describe experimental
results, and we underline some of the open questions.
1. Introduction
The ability of a wire to carry an electrical current with no apparent dissi-
pation is surely the most dramatic property of the superconducting state.
Under favorable conditions, the electrical resistance of a superconducting
185
September 14, 2010 9:18 World Scientific Review Volume - 9.75in x 6.5in ch10
186 B. I. Halperin, G. Refael and E. Demler
wire can be very low indeed. Mathematical models predict lifetimes that far
exceed the age of the universe for sufficiently thick wires under appropriate
conditions. In one experiment, a superconducting ring was observed to carry
a persistent current for more than a year without measurable decay, with
an upper bound for the decay rate of a part in 10
5
inthecourseofayear.
1
However, in other circumstances, as for sufficiently thin wires or films, or in
the presence of penetrating strong magnetic fields, non-zero resistances are
observed. Over the past fifty years, a great deal of theoretical and experimen-
tal effort has been devoted to obtaining better qualitative and quantitative
understanding of how this resistivity arises, and how superconductivity
breaks down, in the variety of possible situations.
Let us first recall how superconductors can exhibit negligible resistance
in favorable situations. In a paper that appeared seven years before BCS,
Ginzburg and Landau (GL) proposed that a superconductor should be char-
acterized by a complex-valued function of position, Ψ(r), referred to as the
superconducting order parameter. They proposed, further, that one could
define a free-energy functional F , which depends on Ψ(r), and which would
be minimized, in the case of a homogeneous superconductor with no exter-
nal magnetic field, at temperatures below the critical temperature T
c
,byan
order parameter Ψ(r) whose magnitude was a constant Ψ
0
> 0, determined
by the temperature T . In order to minimize the free energy, the phase of the
complex order parameter should be independent of position, but the value of
the free energy would be independent of the choice of this constant’s overall
phase.
Following BCS, we now understand that the microscopic origin of the
GL order parameter is the condensate wave function for Cooper pairs.
Except for an arbitrary normalization constant, this is equal to the anoma-
lous expectation value ψ
↑
(r)ψ
↓
(r),whereψ
↑
(r)andψ
↓
(r) are the annihi-
lation operators for an electron at position r, with spin up and spin down,
respectively. The existence of this non-zero expectation value signifies that
the superconducting state has a broken symmetry, namely the U(1) symme-
try, commonly referred to as gauge symmetry, associated with the conserva-
tion of charge or electron number.
The GL assumption enables us to understand the phenomenon of per-
sistent currents. Consider a superconducting ring in a situation where the
magnitude of the order parameter is a constant, but where the phase varies
around the ring. Since the order parameter Ψ must be single-valued at any
given position, the net phase change around the ring must be an integral
multiple, n,of2π. A state with non-zero winding number n will have an
September 14, 2010 9:18 World Scientific Review Volume - 9.75in x 6.5in ch10
Resistance in Superconductors 187
excess free energy, proportional to n
2
for moderate values of n.Asweshall
see below, it will also carry an electrical current proportional to n. One finds
that in order for the initial state to decay to a state with a smaller winding
number, and hence, a smaller value of the free energy and a smaller value of
the current, it is necessary for the system to pass through an intermediate
configuration of Ψ(r) where the free energy is larger than the free energy of
the initial current carrying state, by an amount ∆F . Because Ψ(r)isthe
wave function for a large number of Cooper pairs, the free energy barrier ∆F
can be very large compared to the thermal energy k
B
T , and the probability
to be carried over the barrier by a thermal fluctuation can be exceedingly
small. On the other hand, by adjusting parameters, including the temper-
ature and the dimensions of the system, one may reduce ∆F to the point
where transitions occur at a small but measurable rate.
Quite generally, if we know the minimum value of the free energy barrier
∆F separating a state with winding number n from a state with n − 1,
we may estimate a thermally-activated rate of transitions between the two
states as
η =Ωe
−∆F/T
, (1)
where the prefactor Ω depends on details of the system, but is generally a
less-rapidly varying function of the parameters than the exponential factor,
in the region of interest. (We measure temperature in energy units, so that
k
B
= 1.) The most important step, therefore, in estimating the decay rate of
persistent current in a superconducting loop is to identify the process with
smallest barrier and to calculate ∆F as accurately as possible. After that,
one must estimate the associated prefactor Ω.
The most favorable path for a change in winding number, and the cor-
responding value of ∆F , can be quite different in different geometries, as
we shall discuss below. However, there are some general considerations.
The winding number n may be defined as (2π)
−1
times the integral of the
phase gradient ∇φ along a closed path around the ring, embedded in the
superconductor. If we assume that Ψ(r) varies continuously as a function of
position and time, however, the only way that the accumulated phase can
jump discontinuously at some time t, is if Ψ vanishes at some point on the
path. Since Ψ has both a real and imaginary part, the locus of points (r,t)
with Ψ = 0 will generically form a set of co-dimension 2. In the case of a
two-dimensional superconductor, at a fixed time t, zeroes of Ψ should occur
at isolated points, known as vortex points, which are described as positive
or negative, depending on whether the phase changes by ±2π as one winds
September 14, 2010 9:18 World Scientific Review Volume - 9.75in x 6.5in ch10
188 B. I. Halperin, G. Refael and E. Demler
Fig. 1. An illustration of a vortex in a superconducting strip. The height of
the function corresponds to the magnitude of the order parameter Ψ(r), and the
grayscale to its phase φ(r), mod 2π. The grayscale discontinuity along the line
y =0.5, for x>0, does not represent a discontinuity in Ψ, because e
iφ
is continu-
ous. The coherence length used here is ξ =0.1.
around the vortex in a counterclockwise sense. In three dimensions, the vor-
tex core, where Ψ = 0, becomes a line segment ending at the boundaries of
the superconductor, or possibly a closed loop, embedded within the material.
For a given location of the vortex core, the state of minimum free energy will
generally occur when the magnitude of Ψ returns to its equilibrium value
over a healing distance comparable to the BCS coherence length ξ(T ). The
structure of a vortex is depicted in Fig. 1.
Consider a superconducting annulus with a given initial phase-winding
number n>0. One way to change n with the smallest barrier ∆F would be
to introduce a single positive vortex at the outer edge of the annulus, and
move it across the ring until it disappears at the inner edge. Alternatively,
we could move a negative vortex from the inner edge to the outer. In either
case, any closed curve around the ring would have been cut once by the
vortex, and the winding number would have changed by −1. The free energy
cost for introducing a vortex into the film will depend on the details of the
material, but will clearly increase with increasing thickness of the film. The
free energy barrier can therefore be made arbitrarily large by going to a wire
of sufficient thickness.
In order to go further, it is helpful to make some assumptions about the
form of the free energy functional F (Ψ). The specific form proposed by
September 14, 2010 9:18 World Scientific Review Volume - 9.75in x 6.5in ch10
Resistance in Superconductors 189
Ginzburg and Landau may be written as
F =
d
3
r
−
α
2
|Ψ|
2
+
β
4
|Ψ|
4
+
γ
2
∇−
2ie
c
A
Ψ
2
, (2)
where A(r) is the electromagnetic vector potential (we use Gaussian cgs units
throughout this chapter). The parameters α, β, γ,areallpositivebelowT
c
,
but α is assumed to vanish linearly as T → T
c
. If there is no magnetic field
present, the free energy is minimized when |Ψ| =(α/β)
1/2
, and the resulting
free energy reduction per unit volume is f
0
=(α
2
/4β). The parameters of
(2) result in a coherence length ξ and a magnetic penetration depth λ given
by
ξ =(γ/α)
1/2
,λ= κξ , κ
2
≡
2
c
2
β
16πe
2
γ
. (3)
We shall be interested primarily in “type II” superconductors, where κ 1.
The free energy cost for creating a vortex is typically of order f
0
ξ
2
L
v
,where
L
v
is the length of the vortex.
The free energy (2) implies that in a state of stable or metastable equi-
librium, there will be an electrical current (“supercurrent”), whose density
is given by
j
s
(r)=−c
δF
δA(r)
=
2eγ
|Ψ|
2
∇φ −
2e
c
A
, (4)
where φ(r) is the phase of Ψ(r). Although the specific forms (2) and (4) are
quantitatively correct only close to T
c
, they are qualitatively correct more
generally, and they will serve well for our purposes. The BCS theory gives
a prediction for the Ginzburg–Landau parameters in terms of microscopic
parameters of the material, such as the electron density of states, the effective
mass, the mean-free-path, and the electron–phonon interaction strength.
The decay rate of a persistent current can be changed dramatically if
there is an applied magnetic field exceeding the threshold H
c1
=
hc/2e
4πλ
2
ln κ
for penetration into the superconductor. In this case one finds that there
are vortices present even in thermal equilibrium, with a vortex density n
v
proportional to the magnetic field:
n
v
= B/Φ
0
, (5)
where Φ
0
≡ hc/2e is the superconducting flux quantum. Although the vor-
tices will ideally form a triangular Abrikosov lattice, in practice the lattice is
usually distorted and the vortices tend to be pinned by various types of inho-
mogeneities in the material. In this case, since vortices are already present,
September 14, 2010 9:18 World Scientific Review Volume - 9.75in x 6.5in ch10
190 B. I. Halperin, G. Refael and E. Demler
the free energy barrier for relaxation of a supercurrent at low temperatures
will be the activation energy necessary for vortices to become unpinned so
they can move across the current-carrying sample.
An analysis of the decay rate for persistent current in a ring can be
extended directly to the onset of resistance in an open superconducting
wire carrying a current between two contacts. Because Ψ(r)representsa
wave function for Cooper pairs of charge 2e, there is a commutation relation
between the phase φ and the total electron number N,givenby
[φ, N]=2i. (6)
Then, because electrical charge is conserved, it can be shown that in a state
of local equilibrium, at any temperature T , the phase φ must evolve in time
according to the Josephson relation
dφ/dt =2eV/ , (7)
where V is the electrochemical potential (i.e. the voltage) at the point r.
(Equivalently, (7) may be understood as a consequence of gauge invariance.)
If the two ends of a superconducting wire were each in local equilibrium, with
their voltages differing by an amount ∆V = 0, and if no vortices or phase
slips were allowed to cross the intervening wire, the phase difference between
the two ends of the wire would increase linearly in time, and the resulting
supercurrent would increase without limit. In order to reach a steady state
with a constant current, there must be a net flow of vortices or phase slips
across the wire, at a rate η that just relaxes the phase build up due to the
voltage, which requires
∆V = πη/e . (8)
If, for a given current, there is a large free energy barrier, so that the rate η
given by Eq. (1) is very small, then the resulting voltage will be proportion-
ally small.
In the presence of a non-zero electric field, there can be an additional
contribution to the current from the “normal fluid”, which arises from ther-
mally excited quasiparticles. When the phase slip rate η is small, the normal
current will be negligible compared to the supercurrent in a dc measurement,
but it can be an important source of dissipation in ac applications, as dis-
cussed in Sec. 4.
Our general considerations may be applied to a variety of situations. In
wires that are thin compared to the coherence length ξ, one may neglect the
variation of Ψ across the diameter of the wire, and consider that the order
parameter is only a function of x, the distance along the wire. In this case,