Upper critical field of anisotropic superconductors

Solid State Communications, Vol. 42, No. 2, pp. 123. 1982. Printed in Great Britain.

0038-1098/82/140123 --01 $03.00/0 Pergamon Press Ltd.

UPPER CRITICAL FIELD OF ANISOTROPIC SUPERCONDUCTORS K. Takanaka Department of Engineering Science, Tohoku University, Sendai, Japan (Received 13 October 1981; in revised form 29 October 1981 by I4. Sasaki)

The upper critical field of anisotropic superconductors described by the effective mass model is calculated in the general direction of the applied magnetic field. RECENTLY, many anisotropic superconductors have been discovered. The upper critical field of some of them is analyzed by the effective mass model and it shows that the orientation dependence of anisotropy is well expressed by this simple model [ 1]. The expression of the upper critical field, however, is restricted to the case ml = m2 :~ m3 [2]. In this communication, we calculate the upper critical field of superconductors described by the effective mass model with three different masses in the general direction of the applied field. Choosing the x, y and z axes to be parallel to the principal axes of the effective mass (ml, m2, m3), we have the equation to determine the upper critical field He2;

where m is a constant with the dimension of mass, equation (1) reduces to 2m

-

A(~) = --aA(~),

where V = (a[0~, 0[a~, a/a~) and ,~ = (H x~/)/2. Thus, in the new system Ci, H, A), we have the isotropic equation. The upper critical field is, therefore, obtained as H¢2 - cmlc~l

eh

(7)

From equation (5),/t~ is rewritten as n

~ 1 (~. a /=l 2m---~. ax/

27/)2

A(r) = -- c~A(r),

(1)

where eF is the Fermi energy and r is the relaxation time of electrons, and in the pure limit at temperature near T e by -

(3)

7~(3)eF T~

1

( x / ~ l x , Vr~'2y, Vc~3z)

n2x +

nz2 ]-1/2

n~ + m3ml

mlm2/

cl~l (n 2x+ny2 -4- nz2/-1/2 He2 = ~ \ mxm3 m--~l/ "

(4)

,

(8)

(9)

"

For a uniaxial material ml = rn2 ~Vm3, He2 reduces to the well-known expression [2] (10)

REFERENCES

and the magnetic field FI = m ~ x / - ~ - ~ 2 m , x / - - ~ 3 m l , ~ ]

]"2.

To the author's knowledge, there are no materials whose upper critical field is described by the simple expression (9) with three different effective masses.

We assume that the magnetic field is applied in the direction of the unit vector n and use the gauge A = (H x r)12. If one introduces the coordinate system = ~

cirri(

= ~\m2m3

(2)

12n2Te2 T - - T e

n~

mlm2]

The upper critical field He2 is finally given by He2

~b(½) = 0

2 + + nY

/~¢2 = m//¢2 m2~n 3 m3m,

where A is the vector potential. The quantity t~ is given in the dirty limit by

(1 + log ~-~e+ ~b

(6)

(5)

2.

123

For example: J.C. Armici, M. Decroux, O. Fischer, M. Potel, R. Chevrel & M. Sergent, Solid State Commun. 33,607 (1980). R.C. Morris, R.V. Coleman & R. Bhandari, Phys. Rev. B5,895 (1972).