Shear instabilities of a vortex lattice in layered superconductors

Shear instabilities of a vortex lattice in layered superconductors

Physica B 169 (1991) 619-620 North-Holland SHEAR INSTABILITIES OF A VORTEX B.I. IVLEV, N.B. KOPNIN: L.D. Landau Institute LATTICE IN LAYERED S...

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Physica B 169 (1991) 619-620 North-Holland

SHEAR

INSTABILITIES

OF A VORTEX

B.I. IVLEV, N.B. KOPNIN: L.D. Landau

Institute

LATTICE

IN LAYERED

SUPERCONDUCTORS

and V.L. POKROVSKY

for Theoretical

Physics,

USSR Academy

of Sciences.

117334 Moscow,

USSR

We study possible vortex-lattice structures in highly layered superconductors for the magnetic field parallel to the layers. We found instabilities of two different kinds. (1) When the maqnetic field is large the-shear modulus is exponentially small, and one can expect the vortexlattice to melt in-between the layers. (2) For low magnetic fields, on the other hand, several new types of lattices with almost same energies can be formed instead of an usual triangular array.

Most high-temperature superconductors have rather well-pronounced layered structures. The layered structure strongly affects the thermodynamics and motion of vortices, especially when they are parallel to the layers. The interaction between vortices and the layered structure of the superconductor produces an intrinsic interlayer pinning which attracts now considerable interest from both theoretical (l)-(3) and experimental (see, f.i., Ref.(4)) points of view. In the present paper, we study possible vortex confiqurations for the masnetic field parallel to the layers in case of a very strong interlayer pinning, when vortices are not able to move across the layers. We found that the vortex lattice can become unstable with respect to a melting in-between the layers or to a formation of several new types of lattices, depending on its prehistory and on the value of the applied magnetic field. At intermediate magnetic fields, Hc,<
H_~d!

_A

c ax2

2 8H ab

,22

k,e A

C

and xab

are the penetration

depths.

The

z-axis is along the cristal c-axis, and y is along H. We consider the simplest case of a vortex lattice with one of its unit-cell vectors parallel to the layers. The strong-pinning condition requires that the projection of the * Present address: Low Temperature SF-02150 Espoo, Finland.

Laboratory,

is fixed since the vor-

tices cannot cross the layers. When the magnetic field is varied, the unit cell area will vary due to vortex displacements along the layers only. With ZD fixed, q is the only variable parameter. can be found through The solution of Eq.(l) and the free-energy a Fourier transformation, density is F=%+

zLl + G(p,q)j, ;@O {Zln14~)

B'

3271 Xabhc

C

where .?D = {(~c/B)(hab/~c)l'

(2)

is of order of the

vortex lattice period without the pinning (i.e., a period of an usual Abrikosov lattice). The parameter p = 2nzo2/i

0

2 =

*~z,*($)(3,

and sinh(pa)

G(p,q) =a=, y gL ccosh(pe)-cos(2

nqa)

- 11

- ln(pj + g The function G(p,q) is periodic in q with the period 1, and it is invariant under the transfor!%i~~a~t~c'm,d~ius

= $0 ch(z - Zo”j61x - Xo(k + qk)), where

period ZD on the z-axis

CL;) for a shear defor-

mation of the lattice along the x-axis (changes in q) can be calculated from Eqs.(2),(3). One can easily find that a rectangular lattice (q = = 0) is always unstable (C66 < 0). For a rhombic lattice (q = s), one has

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