Attraction between tilted vortices in magnetic superconductors

MU

Physica C 185-189 (1991) 2621-2622 North-Holland

ATTRACTION BETWEEN TILTED VORTICES IN MAGNETIC SUPERCONDUCTORS A. I. BUZDIN, S. S. KROTOV, D. A. KUPTSOV Physics Department, Moscow State University, 117234 Moscow, USSR It is shown that in magnetic superconductors with easy axis magnetic anisotropy the flux lines attract each other at large distances if they are inclined with respec~ to the magnetic anisotropy axis. The lower critical field//el corresponds to the penetration of a vortex chain rather than a single vortex into the sample.

Superconductivity and magnetism coexist in a large number of compounds. Certain ternary rare-earth systems of the classes (RE)Mo6S 8 and (RE)Rh4B4 (RE is rare earth) reveal magnetic ordering in the superconducting state 1,2. Some of them, e.g. ErRh4B4,

HoMo6S 8 and HoMo6Ses, are ferromagnetic superconductors where the Cooper pairing gives rise to inhomogeneous magnetic structure in the coexistence phase just below the transition temperature Tm 1,2

between the vortices at sufficiently large distances (of order of the London penetration depth) in magnetic superconductors 5. Then at Hcl vortex chains rather than single vortices should penetrate into the sample. These effects could be observed in magnetic superconductors with relatively weak interaction between (RE) ions and the electrons as in this case magnetic subsystem susceptibility may be rather large.

At temperatures slightly higher than Tm magnetic superconductors also display unusual behaviour. The results of the measurements of magnetization and crltical fields indicate pronounced magnetic anisotropy of (RE) spin susceptibility in magnetic superconductors 1 Above Tm the anisotropy of (RE) spin susceptibility in the superconducting state must influence qualitatively the character of vortex-vortex interaction. Recently it was demo;~trated that in case of nonmagnetic superconductors the electronic subsystem anisotropy must lead to the attraction between inclined vortices along certain direction in the crystal 3,4 We calculate the field of a vortex in a magnetic superconductor with a uniaxial magnetic anisotropy but with isotropic electronic subsystem. Uniaxial-type ani-

x FIGURE 1 In the London approach (~ >> 1) the free energy of magnetic superconductor is given by: - 4,'rM)2 + ~,~(curl{B -- 4r, r¢~))2÷

sotropy is realized, for instance, in samples of HoMofS 8

F =/{

1. The axis I of the vortex is inclined with respect to

(1 - 4,'rXll)(gM)2 + (1 -- 4~rk±)( M

the magnetic anisotropy axis g (see fig.l). In this case,

2xii

a specific plane (lg) may be defined. We show that the magnetic field component

IB(r) (parallel to the axis of

the vortex) changes its sign on moving away from the vortex axis in this plane. This leads to the attraction

2x± BHex~ )d2r ' 47r

~(gM)) 2 -

(1)

where AL is the London penetration depth. In ca~c of

0921-4534/91/$03.50 © 1991 - Elsevier Science Publishers B.V. Anl rights reserved.

A.L Buzdin et at / Attraction between tilted vortices in magnetic superconductors

2622

Intervortex attraction leads to the formation of vor-

uuiaxial anisotropy magnetization can be written as: M = x.kB +

(Xll

tex chains which lie in the plane (IF) 5. Obviously, the - X.I.)~(~B)

(2)

chains disappear if the external field becomes parallel

The Maxwell equation for B which follows from (1) is readily solved by the Fourier transformation:

to the magnetic anisotropy axis. The results of vortex structure calculations presented in fig.3 aemonstrate that a crossover between the chain-like structure and

¢0 os 0 , ( 3 ) CD)], ~Bq = "~'c

IBq=dPO[c+eCO82d(~

field is rotated with respect to the anisotropy axis.

C = 1 + A2q2, D = 1 + A2q2 + eA2(sin20q 2 - q2), where e = 41(x~:x±) - x± characterizes the strength of the magnetic anisotropy, cosO = (~), A2 = A2(1 - 47rX.k) and coordinate axes are shown in fig.1. The most interesting case is when y = 0 :

IB(x, O) c~ - x -3/2 e x p ( - A )

usual honeycomb vortex lattice occurs as the external

40

30

(4)

for x > > A. This result holds for e > 0 and t9 # 0, 0 7r/2. It is easy to show that Hz = Bz - 47rM~ has the

lo

same behaviour. In terms of the Fourier components the free energy (1) can be rewritten as F = ~

¢0

i IHqd2q

(5)

As follows from (5), in contrast to nonmagnetic superconductors the vortex energy is determined by H = B -- 4rrM and not by B and the vortex self energy F 0 i~ nearly independent of 0 even in the case of strong

o J,,,............. 3'6 ............ e~ ............ ~, degrees FIGURE 3 The ratio of the lattice parameters L/d versus 0 for e = 0.9, X.I. = 0 and different cell areas S : S = A2(curve t), S = 10A2(curve 2), S = 100A2(curve 3). The unit cell of the flux line lattice is shown in the inset to the figure. T h e anomaly for S >> A 2 corresponds to 0 15 - 20 ° where the attraction between the vortices at large distances is most pronounced.

magnetic anisotropy. To demonstrate the existence of intervortex attraction we plot Hz distribution in fig2. The formation of vortex chains could be observed by

2 ly

magnetooptic and NMR measurements and in neutron diffraction experiments.

REFERENCES . Superconductivity in Ternary Compounds. II, ed. by M.B. Maple and O. Fisher, Springer-Verlag, 1982.

-3

,

-2

t

-1

0 X

J/

1

,

2

. L.N. Buiaevskii, A.I. Buzdin, M.L. Kulic, S.V. Punjukov, Adv. Phys. 34 (1985) 175. 3 -2

A

3. A. Buzdin, A. Simonov, J E T P Lett., 51 (1990) 191; Physica C 168 (1990) ~ 1 . 4. A.M. Grishin, A.Yu. Martynovich, S.V. Yampol'skii, Zh. Eksp. Tcor. Fiz. 97 (1990)1930.

FIGURE 2 Contours of constant Hz(x, y) (in units of ._9_0_ 2,A~) for e = 0.95, O = 20 °. The vortex axis is situated at x=0, y=O.

5. A.I. Buzdin, S.S. Krotov, D.A. Kuptsov, Physica C 175 (1991) 4m