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Muon spin rotation and the vortex lattice in superconductors Ernst Helmut Brandt Max Planck Institute for Metals Research, D-70506 Stuttgart, Germany

a r t i c l e in fo

abstract

Keywords: Muon spin rotation Superconductivity Vortex lattice

The magnetic ﬁeld probability PðBÞ is calculated from Ginzburg–Landau theory for various lattices of vortex lines in type-II superconductors: ideal triangular lattices, lattices with various shear strains and with a superlattice of vacancies, and lattices of short vortices in ﬁlms whose magnetic ﬁeld ‘‘mushrooms’’ near the surface. & 2008 Elsevier B.V. All rights reserved.

1. Introduction Type-II superconductors like niobium and many alloys allow magnetic ﬂux to penetrate in the form of magnetic ﬂux lines, i.e., vortices of the supercurrent, each vortex carrying one quantum of magnetic ﬂux. This effect was predicted in 1957 by Abrikosov [1], who got for this the Nobel Prize in Physics 2003. Abrikosov ﬂux lines arrange to a more or less perfect triangular vortex lattice that exhibits interesting structural defects that may be calculated from Ginzburg–Landau (GL) theory [2] or by treating the vortex lattice as a continuum with non-local elasticity [3]. The vortex lattice may even melt into a ‘‘vortex liquid’’ [4]. Pinning of vortices by material inhomogeneities [5] together with thermal ﬂuctuations of the vortices can cause a rich phase diagram in the magnetic ﬁeld–temperature plane [6], with a melting line and an order– disorder line [7,8] at which the weak elastic disorder (‘‘Bragg glass’’) suddenly changes to a plastically deformed or even amorphous vortex arrangement. The vortex lattice can be observed by decoration, magneto-optics, Hall probes, neutron scattering, magnetic force microscopy, and by muon spin rotation (mSR). mSR experiments can give valuable information about the vortex lattice, see, e.g., Ref. [9] and references therein. When pinning and thermal ﬂuctuations may be disregarded (e.g., in clean niobium with very weak pinning) the vortex lattice exists when the applied magnetic ﬁeld Ba lies between the lower critical ﬁeld Bc1 and the upper critical ﬁeld Bc2 . In this interval the internal average induction B¯ is smaller than Ba , i.e., the magnetization M ¼ B¯ Ba is negative (diamagnetic behavior), ranging from Bc1 to 0 while B¯ ranges from 0 to Bc2 . For Ba oBc1 the superconductor expels the magnetic ﬁeld (B¯ ¼ 0, ideal Meissner state) and for Ba 4Bc2 the superconductor is in the normal conducting state (B¯ ¼ Ba ). This applies to long superconductor cylinders or slabs in parallel Ba . For other geometries and for inhomogeneous materials, demagnetization

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effects modify this picture and the magnetization curve in general has to be computed numerically, e.g., for thick or thin strips, disks, and platelets in perpendicular Ba [10]. However, in the particular case when a homogeneous specimen with the shape of an ellipsoid is put into a uniform applied ﬁeld Ba , then the demagnetizing ﬁeld (caused by the magnetization) inside the ellipsoid is also uniform and superimposes to Ba , thus generating an effective applied ﬁeld Bi ¼ Ba NMðBi ; N ¼ 0Þ.

(1)

Solving Eq. (1) for the effective internal ﬁeld Bi , one obtains M ¼ MðBa ; NÞ ¼ MðBi ; N ¼ 0Þ. Here N is the demagnetization factor, and the ideal magnetization curve [MðBa ; 0Þ for N ¼ 0, e.g., from GL theory] should be inserted. In general, N is a tensor, but when Ba is along one of the three principal axes of the ellipsoid, then N and all ﬁelds in (1) are scalars. For long cylinders or slabs in parallel Ba one has N ¼ 0, for spheres N ¼ 13, for long cylinders in perpendicular ﬁeld N ¼ 12, and for thin plates and ﬁlms one has 1 N51. The sum of the N along the three axes is always N1 þ N2 þ N 3 ¼ 1. For the Meissner state one ﬁnds B¯ ¼ 0, MðBa ; 0Þ ¼ Ba , Bi ¼ Ba =ð1 NÞ, and MðBa ; NÞ ¼ Ba =ð1 NÞ, which means the vortex penetration starts at the effective penetration ﬁeld B0c1 ¼ ð1 NÞBc1 where Bi ¼ Bc1 . In the interval B0c1 oBa oBc2 such a pin-free ellipsoid contains a perfect vortex lattice. In this paper the probability PðBÞ that at a random point inside the superconductor a muon sees an induction value B is considered for ideal triangular vortex lattices and for various possible perturbations of it, namely, various types of shear deformation [11], a superlattice of vortex vacancies, the surfaces of a ﬁlm in a perpendicular ﬁeld, and random displacements. This probability (or ﬁeld density) is deﬁned as the spatial average PðB0 Þ ¼ hdðB0 BðrÞÞir . 0

(2)

In it B is the independent variable, BðrÞ the spatially varying magnetic ﬁeld, and dðxÞ is the 1D delta function, which for computation may be replaced by a narrow Gaussian whose width

ARTICLE IN PRESS E.H. Brandt / Physica B 404 (2009) 695–699

1

(3)

1

2. 2D ideal triangular vortex lattice The local magnetic ﬁeld Bðx; yÞ of the vortex lattice may be calculated for all values of the reduced average induction b ¼ ¯ c2 and GL parameter k by an elegant iteration method that B=B minimizes the GL free energy F with respect to the Fourier coefﬁcients of the periodic solutions Bðx; yÞ and order parameter oðx; yÞ ¼ f 2 ¼ jcj2 where cðx; yÞ ¼ f expðijÞ is the complex GL function [12,13]. In the usual reduced units (length l, induction pﬃﬃﬃ 2Bc , energy density B2c =m0, where l is the magnetic penetration pﬃﬃﬃ depth and Bc ¼ Bc2 = 2k is the thermodynamic critical ﬁeld with 2 Bc2 ¼ F0 =ð2px Þ, x ¼ l=k the coherence length and F0 ¼ h=2e ¼ 15 2:07 10 T m2 the quantum of magnetic ﬂux) the spatially averaged free energy density F of the GL theory referred to the Meissner state (c ¼ 1, B ¼ 0) in the superconductor reads * + 2 ð1 jcj2 Þ2 r þ F¼ A c þ B2 . (4) 2 ik Here BðrÞ ¼ r A, AðrÞ is the vector potential, and h. . .i ¼ R 3 ð1=VÞ V d r . . . means spatial averaging over the superconductor with volume V. Introducing the supervelocity Q ðrÞ ¼ A rj=k and the magnitude f ðrÞ ¼ jcj one may write F as a functional of these real and gauge-invariant functions, * + 2 ð1 f Þ2 ðrf Þ2 2 2 2 þ F¼ þ f Q þ ðr Q Þ . (5) 2 k2 In the presence of vortices Q ðrÞ has to be chosen such that r Q has the appropriate singularities along the vortex cores, where f vanishes. By minimizing this F with respect to c, A or f, Q , one obtains the two GL equations together with the appropriate boundary conditions. For periodic lattices with one ﬂux quantum per vortex, in the sense of a Ritz variational method one uses Fourier series for the periodic trial functions with a ﬁnite number of Fourier coefﬁcients aK and bK , X oðrÞ ¼ aK ð1 cos KrÞ, (6) K

BðrÞ ¼ B¯ þ

X

bK cos Kr,

(7)

where oA ðx; yÞ is the Abrikosov Bc2 solution given by a rapidly converging series (6) with coefﬁcients aAK ¼ ðÞmnþmþn expðK 2mn x1 y2 =8pÞ. The solutions oðrÞ and BðrÞ are then computed by iterating equations for the coefﬁcients aK and bK that are derived from the GL equations dF=do ¼ 0 and dF=dQ ¼ 0. This method and the obtained GL solutions are presented in detail in Refs. [12,13]. The ﬁeld density PðBÞ of the ideal triangular vortex lattice is pﬃﬃﬃ ¯ c2 values in Figs. 1 (k ¼ 1= 2) and 2 shown for several b ¼ B=B (k ¼ 1:4). For larger k the PðBÞ look similar to Fig. 2. For small b and large k the PðBÞ obtained from the London approximation are depicted in Ref. [11]. The maximum and the two equal minima per unit cell yield two steps in PðBÞ at B ¼ Bmax ð¼ 1Þ and B ¼ Bmin ð¼ 0Þ, and the three equal saddle points yield a logarithmic inﬁnity at B ¼ Bsad where PðBÞ / ln jB Bsad j. ¯ 2i ¼ The variance of the magnetic ﬁeld, s ¼ h½Bðx; yÞ B P 2 b , is plotted for the entire ranges of b and k in Fig. 3. In Ka0 K the low-ﬁeld range 0:13=k2 5b51 one has for the triangular

b = 0.25

9 8

0.4

7

triangular vortex 1 lattice from GL theory 0.8

κ = 0.707 0.6

6

P (B)

may depend on B0 . One easily shows that Z 1 Z 1 ¯ PðBÞ dB ¼ 1; PðBÞB dB ¼ B,

P (B)

696

5

0.95 0.4

0.95

4

0.6

b = 0.25

0.2

3 2

0 0.2

1

0.4

0.6 B

0.8

1

0 0

0.2

0.4

0.6

0.8

1

B Fig. 1. The ﬁeld density PðBÞ of the ideal triangular vortex lattice obtained from GL pﬃﬃﬃ theory for k ¼ 1= 2 ¼ 0:707 and b ¼ 0:25–0.95. Here and in the ﬁgures below the variable B is normalized such that B ¼ 0 means Bmin and B ¼ 1 means Bmax , and the inset enlarges the region near Bmax.

Ka0

X

bK

Ka0

z^ K K2

sin Kr,

(8)

12 b = 0.1

where K ¼ Kmn ¼ ðK x ; K y Þ are the reciprocal lattice vectors of the vortex lattice with positions Rmn , (9) (10)

(m; n ¼p 0;ﬃﬃﬃ1; 2; . . .; triangular lattice: x1 ¼ a, x2 ¼ x1 =2, y2 ¼ x1 3=2; square lattice: x1 ¼ y2 ¼ a, x2 ¼ 0). In (8) Q A ðx; yÞ is the supervelocity of the Abrikosov Bc2 solution, which satisﬁes " # X ¯ (11) r Q A ¼ B F0 d2 ðr RÞ z^ ,

0.2

0.8

κ = 1.4

8 0.25 P (B)

Rmn ¼ ðmx1 þ nx2 ; ny2 Þ, Kmn ¼ ð2p=x1 y2 Þðmy2 ; mx2 þ nx1 Þ,

0.15

10

triangular vortex 1 lattice from GL theory

P (B)

Q ðrÞ ¼ Q A ðrÞ þ

0.95

0.4

0.4 0.6

6

0.6

0.2

0.95 4

b = 0.1

0 0.2

0.4

0.6

R

where d2 ðrÞ ¼ dðxÞdðyÞ is the 2D delta function. This shows that Q A is the velocity ﬁeld of a lattice of ideal vortex lines but with zero average rotation. Close to each vortex center one has Q A ðrÞ r0 z^ =ð2kr 02 Þ and oðrÞ / r 02 with r0 ¼ r R. We take Q A from the exact relation Q A ðrÞ ¼

roA z^ , 2koA

0.8

1

B 2

0 0

0.05

0.1

0.15

B

(12)

Fig. 2. The PðBÞ as in Fig. 1 but for k ¼ 1:4.

0.2

ARTICLE IN PRESS E.H. Brandt / Physica B 404 (2009) 695–699

9

0.383 London limit

5

0.9

6

200

0.2

× (1 − b)−1

P (B)

0.95 7

P (B)

σ1/2 × (κ2 − 0.069) / Bc2

x2’/x2 = 1 0.4

0

0.3

1

8

σ1/2× (κ2 − 0.069) / Bc2

0.3

697

0.8

0.2 0.1

5 0.7 4

0.1

0

3

κ = 0.85

0.6 0.5

0.8 B 0.3

1

0 2

κ = 0.85, 1, 1.2, 1.5, 2, 3, 5, 7, 10, 20, 50, 100, 200

0 0

0.1

0.2

0.3

0.4

0.5 b1/2

0.6

0.7

0.8

0.9

Vortex lattice sheared along x x2’/x2 = 1, .95, .9, .8, .7, .5, .3, 0,

1

1

κ = 0.707, b = 0.5

0 0

0.1

0.15

0.2

0.25

B

¯ 2 i of the triangular FLL for Fig. 3. The magnetic ﬁeld variance s ¼ h½Bðx; yÞ B pﬃﬃﬃﬃ k ¼ 0:85–200 plotted in units of Bc2 as s ðk2 0:069Þ=Bc2 (solid lines) such that the curves for all k collapse near b ¼ 1. The dashed lines show the same functions divided by ð1 bÞ such that they tend to a ﬁnite constant 0.172 at b ¼ 1. All curves pﬃﬃﬃ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ¯ c2 to stretch them at small b values and show that are plotted versus b ¼ B=B

Fig. 4. Vortex lattice sheared along the x-axis.

9

they go to zero linearly. The upper frame 0.383 is the usual London approximation. The limit for very small b is shown as two dash-dotted straight lines for k ¼ 5 and 10. The upper frame 0.383 shows the usual London approximation.

y2’/y2 = x1/x1’ = 1

1.03

8 1.06

7

0.97

P (B)

Vortex lattice sheared along diagonal x = y y2’ / y2 = x1 / x1’ κ = 0.707, b = 0.5

0.94

6

4

lattice the London limit s ¼ 0:00371F20 =l (upper frame in Fig. 3), 4 at very small b50:13=k2 one has s ¼ ðbk2 =8p2 ÞF20 =l (dashdotted straight lines in Fig. 3), and near b ¼ 1 one has the 4 Abrikosov limit s ¼ 7:52 104 ðF20 =l Þ½k2 ð1 bÞ=ðk2 0:069Þ2 , approximately valid even at b]0:3, see Ref. [13]. Note that the usual London limit for s applies only in a narrow range of small (but not too small) b and for k450. The same is true for the London limit of the magnetization curve, where the often used ‘‘logarithmic law valid at Bc1 5Ba 5Bc2 ’’ for MðBa Þ ¼ B¯ Ba has a small range of validity [13].

0.05

0.91

5 4 3 2 1 0 0

3. Sheared vortex lattices

0.05

0.1

0.15

0.2

B The above Fourier method applies to any vortex lattice symmetry with vortex positions Rmn (9), also to sheared triangular lattices and to square and rectangular basic cells. In Fig. 4 the ﬁeld density PðBÞ is shown for a lattice sheared away from the ideal triangular lattice by decreasing in Eq. (9) the length x2 ¼ 0:5 by a factor c ¼ x02 =x2 ¼ p 1 ﬃﬃﬃ(triangular), 0:95; 0:9; . . . ; 0 (rectangular lattice). With y2 ¼ 3=2 the pﬃﬃﬃ shear strain in these cases is g ¼ ð1 cÞx2 =y2 ¼ ð1 cÞ= 3. Note that for this shear the saddle-point peak splits into two peaks, i.e., there are now two different types of saddle points in Bðx; yÞ, but still one maximum and two equal minima. Fig. 5 shows a different orientation of shearing the triangular lattice, namely, now the lengths y2 and x1 are changed by a factor c ¼ y02 =y2 ¼ x1 =x01 ¼ 1:06; 1:03; 1; 0:97; 0:94; 0:91 such that the unit cell area x1 y2 ¼ x01 y02 does not change (no compression). This corresponds to a shear strain of size g ¼ 2ð1 cÞ oriented along the diagonal x ¼ y. One can see that for c41 the saddle-point peak splits into three peaks (i.e., all three saddle points now occur at different B) while for co1 there occur two different saddle-point peaks. One notes that even very small shear of the vortex lattice causes pronounced change in the ﬁeld probability PðBÞ. Small shear costs very little energy since the shear modulus c66 of the vortex lattice is much smaller than its compressional modulus c11

Fig. 5. Vortex lattice sheared along the diagonal x ¼ y.

or its tilt modulus c44 . One has approximately [3] 2 1 B¯ qBa þc , m0 qB¯ ð1 þ k2 l02 Þ 66

(13)

¯ c2 ð1 bÞ2 ð2k2 1Þ2k2 BB , 8k2 m0 ð2k2 1 þ 1=bA Þ2

(14)

c11 ðkÞ ¼ c66

c44 ðkÞ ¼

2 B¯

1

m0 1 þ k2 l02

þ

¯ ¯ a BÞ BðB

m0

(15)

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 0 with l ¼ l= 1 b. In c66 , bA ¼ 1:1596 is the Abrikosov parameter of the triangular lattice (the square lattice is unstable and thus has negative c66 ), and the factor ð2k2 1Þ means the shear stiffness of the vortex lattice is zero in superconductors with k ¼ 0:707 (pure Nb). An interesting property is the dependence of c11 (13) and c44 (15) on the magnitude k ¼ jkj of the wave vector k ¼ ðkx ; ky ; kz Þ of spatially periodic strain, which means the elasticity of the vortex lattice is non-local. In the limit of uniform stress, k ! 0, these expressions reproduce the known values of the uniform compression and tilt moduli obtained by thermodynamics,

ARTICLE IN PRESS E.H. Brandt / Physica B 404 (2009) 695–699

2 ¯ a =m0 . However, when the ¯ c44 ¼ BB c11 c66 ¼ ðB¯ =m0 ÞqBa=qB, wavelength of the periodic compression or tilt decreases, i.e., k increases, these moduli decrease. This means, the vortex lattice is softer for short-wavelengths compression and tilt than it is for long wavelengths. In anisotropic superconductors these moduli at ﬁnite wavelengths are even smaller [14] and the vortex lattice is softer and can be distorted and melted more easily in high-T c superconductors.

a = no vacancies, ideal triangular FLL

4

b = few vacancies 3

4. Vortex lattice containing vacancies

c

2.5 2

d e = many vacancies

1.5

As an example for structural defects I consider a vortex lattice (spacing a) with a superlattice of vacancies with spacing Na, N ¼ 2; 3; 4; . . . : This problem was solved in Ref. [2] both within London theory (at b51) and GL theory near b ¼ 1 where BðrÞ ¼ B¯ þ ½hoi oðrÞBc2 =ð2k2 Þ

o ðrÞ Y oA ½ðr Rn sn Þ=N , oðrÞ ¼ c1 A oA ðr=NÞ n oA ½ðr Rn Þ=N

e

1 0.5

(16)

holds, i.e., the shapes of oðx; yÞ and Bðx; yÞ are the same. For this vacancy lattice near b ¼ 1 one has

Vortex lattice with a lattice of vacancies, solution near Bc2

3.5

P (B)

698

a 0

0.2

0.4

0.6

0.8

1

B Fig. 7. Field density PðBÞ of a vortex lattice with various vacancy concentrations 1=N 2 : (a) no vacancy, (b) N ¼ 9, (c) N ¼ 6, (d) N ¼ 4, (e) N ¼ 3. GL solution near Bc2.

(17)

10 9

1.2

triangular vortex lattice in bulk

vacuum 1

8 κ = 0.71

6

z/a

b = 0.6 7 P (B)

where oA ðx; yÞ is the Abrikosov Bc2 solution given below Eq. (12), c1 is a normalization constant, the product is over all vortex positions Rn ¼ Rmn within the supercell, and the vortex displacements sn are chosen such as to minimize the free energy and the Abrikosov parameter b ¼ ho2 i=hoi2 41. This relaxation of the vortex positions around the vacancy yields an oðx; yÞ with nearly constant spatial amplitude, i.e., the maximum oð0; 0Þ at the vacancy position has about the same height as all the maxima of o between the vortex positions. Fig. 6 shows the contours of o (and thus of B) for a vortex lattice with one vacancy (limit N ! 1) at the pﬃﬃﬃ origin and with central symmetric displacements sn ¼ Rn ½ 3a2 =ð4pR2n Þþ 0:068a4 =R4n . The ﬁeld density PðBÞ of vortex lattices with various vacancy concentrations 1=N 2 is shown in Fig. 7. The new peaks indicate that new saddle points and minima of Bðx; yÞ (i.e., maxima of o) appear near the vacancy, as seen also in Fig. 6.

0.8

0.6 triangular vortex lattice in film 0.4 d = 1.2a = 6λ 0.2

5 4

film

3 0 −0.5

2 1

0 x/a

0.5

0 −0.1

2

0

0.1 0.2 (B − 〈B〉) / Bc2

0.3

0.4

1.5 Fig. 8. The ﬁeld density PðBÞ of the triangular vortex lattice in a superconducting ﬁlm of thickness d ¼ 1:2a ¼ 6l in a perpendicular ﬁeld Ba ¼ B¯ for k ¼ 0:707, ¯ c2 ¼ 0:6, from GL theory. Shown are PðBÞ for bulk (2D, solid line) and for b ¼ B=B ﬁlm (3D, dashed line). The inset shows the magnetic ﬁeld lines in and near the ﬁlm; the dashed line marks the ﬁlm surface z ¼ d=2 ¼ 0:6a, a; vortex spacing.

1

y/a

0.5 0

5. 3D vortex lattice in ﬁlms

−0.5 −1 −1.5 −2 −1.5

−1

−0.5

0

0.5

1 x/a

1.5

2

2.5

3

Fig. 6. Contour lines of order parameter oðx; yÞ, Eq. (17), identical to the contours of induction Bðx; yÞ, Eq. (16), for a vortex lattice with one vacancy. The displacements sn of the relaxing vortices are shown as short bold lines connecting two dots. At all vortex positions oðx; yÞ has a minimum and is zero, but at the origin x ¼ y ¼ 0, o is maximum since a vortex was removed from there.

The Fourier method of Section 2 may be generalized to superconducting ﬁlms of arbitrary thickness d (jzjpd=2) containing a vortex lattice that is periodic in the ðx; yÞ plane. For this one has to add to F, Eq. (5), the energy F stray of the magnetic stray ﬁeld outside the ﬁlm and has to use z dependent trial functions, e.g., (6)–(8) with Fourier coefﬁcients depending on 3D vectors K ¼ ðK x ; K y ; K z Þ [15]. When the applied ﬁeld Ba is along z (perpendicular to the ﬁlm plane) then the short vortex lines are straight and along z. The inset of Fig. 8 shows how the magnetic ﬁeld lines of the vortices become less dense when they approach the ﬁlm surface, they ‘‘mushroom’’ and go over smoothly into the stray ﬁeld. Though in the depicted example (b ¼ 0:6, k ¼ 0:707, d ¼ 1:2a 6l 4x) this widening of the ﬁeld lines is only weak,

ARTICLE IN PRESS E.H. Brandt / Physica B 404 (2009) 695–699

it still has a strong effect on the ﬁeld density PðBÞ shown as dashed line, namely, the jumps of Bmin and Bmax and the saddle-point peak ¯ and PðBÞ is increased (has a are smeared and shifted towards B, ¯ These features are expected since hump) between B ¼ Bsad and B. for very thin ﬁlms with dol the ﬁeld amplitude is strongly reduced inside the ﬁlm and PðBÞ narrows to a line positioned at ¯ This PðBÞ should be observable by mSR in specimens B ¼ B. composed of many thin layers separated by a distance Xa=4 where the stray-ﬁeld modulation amplitude / exp½2pðjzj d=2Þ=a has almost vanished. In such inﬁnitely extended ﬁlms one has B¯ ¼ Ba since all ﬁeld lines have to pass the ﬁlm. The magnetization M of the ﬁlm, therefore, cannot be calculated as a difference of ﬁelds, but one has to take the derivative of the total free energy, ¯ A more elegant method calculates M by M ¼ qðF þ F stray =dÞ=qB. Doria’s virial theorem directly from the GL solution for the ﬁlm, with no need to take an energy derivative [16].

6. Random perturbations In real superconductors randomly positioned weak pinning centers, or random pinning forces, may lead to more or less random small displacements of the vortices from their ideal lattice positions. As shown in Ref. [11], in bulk superconductors this leads approximately to a convolution of the ideal-lattice PðBÞ with a Gaussian. Computer simulations of this problem based on London theory (i.e., pairwise interacting vortex lines and linear superposition of vortex ﬁelds) are presented in Ref. [11]. Interestingly, while disorder of a 2D vortex lattice broadens PðBÞ and its singularities, disorder in the 3D point-vortex (or pancakevortex) lattice occurring in layered high-T c superconductors [6,17] typically will lead to narrowing of PðBÞ [18] since the vortex lines (pancake stacks) become wider. A further effect that contributes to the broadening of PðBÞ is the (quantum) diffusion of muons after they have stopped, e.g., in ultrapure Nb [19–21].

699

Improved pinning simulations using GL theory and considering also thermal ﬂuctuations of vortices are desirable, as well as microscopic calculations going beyond the GL approach. From BCS-Gor’kov theory it is shown in Refs. [11,19] that for pure Nb near Bc2 the PðBÞ depends on temperature T, and at T5T c it looks quite different from the GL result valid near the critical temperature T c since then Bðx; yÞ has sharp conical maxima and minima, and two saddle points with three-fold symmetry yielding an inﬁnity of the form PðBÞ / jB Bsad j1=3 . References [1] A.A. Abrikosov, Zh. Exp. Teor. Fiz. 32 (1957) 1442 (Sov. Phys.-JETP 5 (1957) 1174). [2] E.H. Brandt, Phys. Stat. Sol. 35 (1969) 1027; E.H. Brandt, Phys. Stat. Sol. 36 (1969) 371; E.H. Brandt, Phys. Stat. Sol. 36 (1969) 381; E.H. Brandt, Phys. Stat. Sol. 36 (1969) 393. [3] E.H. Brandt, J. Low. Temp. Phys. 26 (1977) 709; E.H. Brandt, J. Low. Temp. Phys. 26 (1977) 735; E.H. Brandt, J. Low. Temp. Phys. 28 (1977) 263; E.H. Brandt, J. Low. Temp. Phys. 28 (1977) 291. [4] E. Zeldov, et al., Nature 375 (1995) 373. [5] A.M. Campbell, J.E. Evetts, Adv. Phys. 21 (1972) 199. [6] G. Blatter, M.V. Feigel’man, V.B. Geshkenbein, A.I. Larkin, V.M. Vinokur, Rev. Mod. Phys. 66 (1994) 1125. [7] K. Shibata, T. Nishizaki, T. Sasaki, N. Kobayashi, Phys. Rev. B 66 (2002) 214518. [8] G.P. Mikitik, E.H. Brandt, Phys. Rev. B 64 (2001) 184514; G.P. Mikitik, E.H. Brandt, Phys. Rev. B 68 (2003) 054509. [9] I.L. Landau, H. Keller, Phys. C 466 (2007) 131. [10] E.H. Brandt, Phys. Rev. B 64 (2001) 024505. [11] E.H. Brandt, J. Low Temp. Phys. 73 (1988) 355. [12] E.H. Brandt, Phys. Rev. Lett. 78 (1997) 2208. [13] E.H. Brandt, Phys. Rev. B 68 (2003) 054506. [14] E.H. Brandt, Rep. Prog. Phys. 58 (2003) 1465. [15] E.H. Brandt, Phys. Rev. B 71 (2005) 014521. [16] M.M. Doria, E.H. Brandt, F.M. Peeters, Phys. Rev. B 78 (2008) 054407. [17] J.R. Clem, Phys. Rev. B 43 (1991) 7837. [18] E.H. Brandt, Phys. Rev. Lett. 66 (1991) 3213. [19] E.H. Brandt, A. Seeger, Adv. Phys. 35 (1986) 189. [20] A. Schwarz, et al., Hyperﬁne Interactions 31 (1987) 247. [21] D. Herlach, et al., Hyperﬁne Interactions 63 (1990) 41.