Peculiar magnetic properties of anisotropic London superconductors

Physica C 168 (1990) 421-425 North-Holland

PECULIAR MAGNETIC PROPERTIES OF ANISOTROPIC LONDON SUPERCONDUCTORS A.I. B U Z D I N and A.Yu. S I M O N O V Laboratory of High- Tc Superconductivity, Physics Department, Moscow State University, 117234 Moscow, USSR Received 22 February 1990 Revised manuscript received 7 May 1990

The interaction between inclined Abrikosov vortices in layered superconductors is investigated. The lower critical field corresponds to the penetration of chains of vortices. The equilibrium period of vortices in such chains for various field orientations and effective mass ratios is calculated.

In layered superconductors, screening currents o f an inclined vortex are preferably oriented in the conducting planes, which are C u - O planes in the copper-oxide superconductors. We consider the properties o f anisotropic type-II superconductors in the London approach. The density o f free energy can be written as (see, for instance refs. [ 1,21 ] ): F = (h2 + 2 2 ( c u r l h)/J(curl h) ) / 8 n ,

(1)

where h is the local field, 2 = 2 b and ft is the reduced effective mass tensor. We use the following notation for the principal values o f the mass tensor g a = M a / M b = e l + 1, ~b= 1, ~ c = M c / M b = e + 1. Straightforward minimization o f the free energy (eq. ( 1 ) ) yields the L o n d o n equation [ 1,2] h+22curl[/Tt(curlh) ] = ~ o l ~ g ( r - r ~ ) .

(2)

i

Here I is the unit vector along the vortex axis and @o is the flux quantum. In the isotropic case, eq. (2) coincides with the usual L o n d o n equation, and, as usual, we neglect the structure o f the vortex core. It is also assumed that the distance between vortices is ot >> ~, where ~ is the coherence length. To be more specific, we consider firstly the important case o f a layered material with Ma=Mb> a2/2, 2. Using eq. (11 ), one can find [5] that the magnetic induction in field H>Hcl is

q)o

l

B = a2x/1 + e ln[ q)oU2/2a2 ( 1 +{) ( H - H c I ) l

(12)

Note that for usual isotropic superconductors [ 4 ]

B ~ 1/ln2[H-Hcl] . In eqs. ( 10)- ( 12 ) it has been assumed for simplicity that the vortex lattice is rectangular, but if the condition L >> a is fulfilled, the difference in energy between triangular and rectangular vortex lattices is negligible. So, the attraction between vortices in an inclined field leads to extreme anisotropy of the vortex lattice at H~. Hc~. The question of the peculiarities of magnetic field penetration and vortex lattice formation in an inclined field will be discussed elsewhere. We also "consider briefly the interaction between vortices in biaxial superconductors. This case is important for strongly anisotropic organic superconductors. For example, in R-(TMTSF)2C104 the upper critical field ratio is T./a /4bc2'. HoE c __ "" c2". /""4 bc2"" -- 15 : 1 : 50 [ 5 ]. T a k i n g i n t o a c c o u n t t h a t e x p r e s s i o n ( 5 ) f o r t h e f r e e - e n e r g y d e n s i t y is suitable for any anisotropic superconductor, one can rewrite eqs. (3) and (6) for a biaxial superconductor [61: O0

(/}2

F = 8nS2 ~ l'hk = ~

1 +~,2 k2 ( l f i / )

~ (1 +;tZkZ(lfi/) ) (1 ..~)2 [kl]ft [kl] ) -,~.4k2([kl]~l) 2'

(13)

where (if the vortex lies in the (a, c)-plane of the biaxial superconductor)

t x x = ( 1 + E~) cos20+ ( 1 + {) sin20, tYy= 1 , t z z = ( 1 + el ) sin20+ ( 1 + {) cos20, t x z = ( ~ - { ) cos 0 sin 0,

tzx=txz,

]Jyz=]JZy.~-~yX~.J.Lxy=O

(14)

.

AS in the case of layered material, in the plane formed by the c-axis and the vortex axis, vortice chains may appear. The energy of a vortex lying in one of the main planes of a biaxial superconductor, can be rewritten in the form of eq. (8), where Qt and Q2 a r e the roots of the biquadratic polynomial Z ( Q ) = ].lrr]AzzQ4-F []2ZZ + J2yv + qE ( t z z t r y

F(q)=

1 + t z z ( q 2 - Q 2) tzz(Q2_Q2) )

+ JJatc ) ] Q 2 + [ l + q Z ( t a +tic) + q4 t a t c ] ,

( 15 )

The results of numerical calculations for the case of R-(TMTSF)2CIO4, for various vortice orientations, are shown in fig. 4. Note that the m a x i m u m of ( E v ( O ) - E ° (0)) is strongly dependent on the vortex orientation with respect to the a-axis in the (a, b)-plane and for el < 0 is shifted to the region of smaller 0. For the layered high-temperature superconductor (Re)Ba2Cu307, the effective mass ratio M±/Mll .~ 25 [7 ] and we can expect a decrease of the vortex energy at 0 ~ 60 °. Let us discuss briefly the case of an eUipsoidal sample with a rotation axis coinciding with the anisotropy axis (the demagnetization factor n along this direction being nonzero). In this case, the internal Maxwell mag-

425

A.I. Buzdin, A. Yu. Simonov I Vortices in anisotropicsuperconductors

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10

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