K. TAKANAKA: Magnetic Properties of Superconductors 623

phys. stat. sol. (b) 68, 623 (1975)

Subject classification: 14.2

Institut fur Physik am Max- Planck- Institut fur Metallforschumg, Stuttgart

Magnetic Properties of Superconductors with Uniaxia,l Symmetry BY

K. TAKANAKA~)

In type-I1 superconductors with uniaxial symmetry the three critical fields (Hc l , Hc2, Hca) are calculated near the transition temperature T, using the anisotropic Ginzburg-Landau equations. It is shown that in general an induced magnetic field A H 1 perpendicular to the applied magnetic field exists and that the anisotropy removes the degeneracy in the direc- tion of the flux-line lattice. The pattern and the orientat,ion of the flux-line lattice are in good agreement with the experimental results of technetium.

In Supraleitern 11. Art mit einzchsiger Symmetrie werden die drei kritischen Feldstarken (Hcl , Hc2, H,s) in der Nahe der Ubergangstemperatur T, mit den anisotropen Ginzburg- Landau-Gleichungen berechnet. Es wird gezeigt, daO im allgemeinen ein induziertcsMagnet- feld A H 1 senkrecht zum auI3eren Magnetfeld existiert und da13 die Anisotropie die Ent- artung in Richtung des FluBliniengitters aufhebt. Das Muster und die Richtung dcs Flus- liniengitters sind in guter Ubereinstimmung mit den experimentellen Ergebnissen an Tech- netium.

1. Introduction

Many experimental [ l to 41 and t'heoret>ical [5 to 91 works on the upper critical fields in the layered superconduct'ors have been published. The upper crit'ical fields of those materials are highly anisotropic and very large in the direct'ion parallel to the layers. For t'he snperconducting dichalcogenide NbSe, Muto et al. found that the ratio of H,,, to H,~II is dependent on the temperature where Hc21(Hc211) is the upper critical field parallel (perpendicular) to the layer planes [a]. A positive curvature of Hc2 versus T near 1', boundary was observed by Woollani et al. in all materials they studied [4]. Lawrence and Doniacli proposed a tJieoretica1 model for layered superconductors in which the Ginzburg-Landau order parameter in adjacent layers is coupled by Josephson tunnelling [GI. Bulaevskii calculat'ed the upper critical field of this model a t all t'einperatures and obtained a good agreement with the experimental result's for TaS2(Py)l/z [7]. Aoi et al. investigated t'he possible occurrence of part>ially depaired states in layered superconduct'ors and discussed those states as a possible explanation of the high critical field [9].

The upper critical fields of technetium have been experimentally measured [lo] and neutron diffraction measurements on flnx-line lattice (FLL) in Tc have been reported by Schelten et al. [ l l ] .

The author and Ehisawa studied t'he FLL in a superconductor with hexagonal synimetry near T, and showed t'hat other FLL struct'ures than the triangular occur [IZ]. Recently Dobrosavljevi6 and Ra,ffy considered the FLL by means of an effective mass model and determined te equilibrium FLL [13]. The effec- tive mass model without' gap anisotropy is, however, insufficient for layered

1) On leave from Tohoku University, Sendai, Japan.

624 K. TAKANAKA

superconductors as shown below. [12] and [13] did not notice the coniponent of the niagnetic induction perpendicular to the applied magnetic field.

In this paper we use the anisotropic Ginzburg-Landau equations to calculate the three critical fields of superconductors with uniaxial symmetry near T,. The BCS coupling, the electron energy spectrum and the Debye cut-off frequency are assumed to be anisotropic. This model may be applied to the layered superconductors if tlie superconduct ing order parameters of the adjacent layers vary slowly on the scale of the layer spacing. Taking account of the z-component of the current density, we derive tlie FLL structure and the magnetic induction perpendicular to the applied magnetic field.

In Section 2 we give the G-L equations including tlie higher order nonlocality. I n Section 3 the three critical fields are derived. Section 4 is devoted to the calculations of tlie FLL and the niagnetic induction. Concluding remarks are given in Section 5.

We use the unit system tz = Ex = c = 1 throughout.

2. a - L Equations

Near the transition temperature the G-L equations with anisotropy (aniso- tropies of the BCS coupling, the electron energy spectrum and the Debye cutoff frequency) were derived by Gorkov and Melik-Barkhudarov [ 141. An easy generalization to the case involving higher order terms in the non-locality gives us [15]

A-+(r) = K ( r ) i+(r) + T ( y i , 4 2 , 43, 4 4 ) ~ ' ( Y I ) J(r2) ~ + ( ~ ~ ) I T ~ = T ~ = T ~ = T , (2.1) 4

1 = 1 q7 = iv - 2e(- 1)7 A , 2 q1 = 0

and co n

- - rotrot A

J ( r ) = ~ ~ - 2 e N ( O ) 2 Anti 2 (y2(p) v(- v . q l ) + t - p n x 4n n = l m = O

where

and 2Gu,7

log-- n T for n = 0 ,

(2 .3)

l o for n odd .

Magnetic Properties of Superconductors with Uniaxial Symmetry 625

Here the angular brackets denotes the angular average over the Permi surface. N(O), V, p, and <(n + 1) are the average density of states a t Permi surface, the Ferini velocity, Euler's constant and Riemann's zeta function, respectively. We take the external magnetic field H, parallel to the z-axis and the vector potential as A = (0, H,, x, 0). j ( r ) and y ( p ) are the position-dependent and momentum-dependent parts of the order parameter A ( r , p ) , which is written as A ( r , p ) = y ( p ) a(,) + y l ( p ) . Here y l ( p ) is assumed to be a small correction term. Equation (2.1) is a condition for the existence of y l ( p ) . The factor y ( p ) characterizes the anisotropy of the gap a t zero temperature and is an eigen- function of the following equation with an eigenvalue A [ 161

Y ( P ) = AWO) ( U ( P , P') Y b')) ) (2.6) normalized as ( y 2 ( p ) ) = 1. The interaction between electrons is separated as gU(p, p ' ) , where g is the dimensionless coupling constant (g < 1) and U ( p , p ' ) is the matrix element of the interaction with order of unity. GjD in (2.5) is connected with the anisotropic Debye cutoff frequency &(p) according to [ 161

1% 6, = (Y2(P) 1% &(PI> - The transition temperature of this system is given by [16]

T, = (2ijDy/72) e-"/g. (2.8)

3. Critical Fields In order to calculate the three critical fields, we introduce the crystal coordi-

nates ( X , Y , Z) where the Z-axis is parallel to the symmetry axis. The symmetry axis means the c-axis for the hexagonal materials or the axis perpendicular to the layers for the layered structures. For simplicity we choose the x-axis in the X - Y plane. In this coordinate system

(VXV,), = (v&, = 0 holds by the symmetry, where (-..), = ( y 2 .-) .

3.1 He2

The author and Nagashima calculated the upper critical field and the order parameter for the cubic materials involving the higher order non-locality [ 151. Those results for Hc2 and d + ( r ) can be used if we notice the properties (3.1) and replace Hc2, v + , and (...) in (3.7) and (3.8) of [15] by t H c 2 , Zk(= 1/2(v, f ivu/c)) and (...),,,, respectively. Explicit expressions for Hc2 and d+(r) are as follows:

where 6 = log (TIT,) and = ( ( v ~ ) , / ( v ~ ) , ) ~ / ~ . 41 phgsiea (b) 6812

626 K. TAKANAKA

To see the dependence of Hc2 on the temperature and the direction of the magnetic field, we investigate the above expression by using a simple model in which the electron energy spectrum E ( p ) and the energy gap y ( p ) are given by

and Y ( P ) = C(1 + 5 2 c0s2 6 ) 7 (C = (1 + 2 t2 /3 + Ei/5)-1’2) 3 (3.5)

where 8 is the angle between the Z-axis and the momentum p . Since the final expression of Hc2 is very lengthy, we retain only the first two terms and first order of E2 in the second term of (3.2). Then (3.2) reduces to

3(2nT)2 mA(y) 6 Hc2 = - 7e.5,

07 08

Fig. 1. The upper critical field h:z(t) (= Hcn(t)/(-dHcz(t)/dtlt=l)), and the ratio (hFZil(t)/

h:211(t)) as functions of the reduced temperature (= T/T,) for the different values of Ez, y = l

-0.5 (A), -0.1 (B), 0 (C), 0.1 (D), and 0.5 (E) and for El = 5 (a) and = 10 (b)

Magnetic Properties of Superconductors with Uniaxial Symmetry 627

Fig. 2. The angular dependence of the upper critical field for different values of

and 0.5 (E) and for 5,=5 (a) andl,=lO (b) at t = 0.9

5 2 , - 0.5(A), - 0.1 (B), 0 (C), 0.1 (D),

where y is the direction cosine be- tween the Z-axis and the magnetic field. For some values of f1 andt,, we show the expression (3 .2) for Hc2 in the figures. When 5, = 0, the ratio HczJH,.21~ is independent of the temperature from (3.6) and the Fig. 1, where Hcz l , (Hc21) is the upper critical field parallel (perpen- dicular) to the symmetry axis. This fact holds even if the higher order terms of S are included as long as we use the energy spectrum (3.4).

A

B c D

1 E

b !I E

L- o" 30° 60" 90" O" 30" 60" 90°

e- 0-

-" - The experiments of the layered superconductor NbSe, by Muto et al. [ a ] show

that l ! I c z ~ / H p 2 ~ ~ depends on the temperature and increases as the temperature decreases. The same properties are also measured in the sample Sro,2MoS, by Woollam et al. who observed the positive curvature of the Hc2 versus T', boundary in all layered superconductors they studied. Further, the observed angular dependence of HCz in [ a ] is better fitted if we introduce a negative value of 6,. Then f z should be negative for the layered superconductors. This result is natural because the gap in the layered plane may be larger than that perpen- dicular to the layers. Thus the effective mass model without the anisotropic gap is insufficient for the layered superconductors.

3.2 IZcl and Hc3

Since it is difficult to obtain the lower critical field HC1 and the surface critical field H r S including the higher orders of non-locality, we restrict ourselves to temperatures near T,. While it is sufficient to take only the linear part of the order parameter in (2.1) to calculate He3, we must also take account of the current density ( 2 . 2 ) for Hc1. In an arbitrary direction of the external field there appears a z-component of the current density, which makes the calculations very complicated. Therefore we consider for Hcl only the special cases in which the external field is either parallel or perpendicular to the symmetry axis. I n these cases the z-component of the current density vanishes and HC1 can be easily obtained.

The substitutions y ( r ) = 0°C.) (- ZmN(0) A2VF/3)1/2 ,

a = - (3/2mA,ug) log (TIT,) <0, p = - 2N(O) A,(mvp)2 3(w4(pB,))_ > 0 (3.7)

with

41 *

628 K. TAKANAKA

I (3.8) Hi = (rot'A'), = -- H , A' = (A;, A;) = (a&, bA,) ,

/2 H,

and

(3.9)

transform (2.1) and (2.2) to

(3.10)

The surface critical field Hc3 and the lower critical field H,1 of the above equa- tions are well-known. The surface critical field is [17]

Hk3 = 1.6% = 1.GH;z (3.11)

and the ratio H13/HL2 remains the same as in the isotropic case within the linear approximation of (1 - t ) for arbitrary direction of the applied magnetic field. We obtain for the lower critical field [18]

(3.12) 1

2% = - (log x + 0.08) .

Since the angular dependence of log 3t is small in the case of T, [lo], the product HL1H;2 may be nearly constant for the magnetic fields parallel and perpendicular to the symmetry axis.

4. Induced Magnetic Field and Flux-Line Lattice

In contrast to cubic materials an anisotropy of the order of (v,vj) exists in uniaxial substances. In the previous paper [ la] we concluded that there remains a degeneracy of the FLL orientation with respect to the crystal axes. However, we neglected an important term of the current density, namely, its z-component, which is not always zero for arbitrary directions of the magnetic field. To show the nondegeneracy of the array it suffices to consider only the lowest anisotropy term retaining the z-component of the current density.

Making a linear combination of the degenerate solutions which give the same eigenvalue H,, we have the order parameter A+(r) in the external magnetic field H , just below the upper critical field Hc2 [18]

( p integer)

Magnetic Properties of Superconductors with Uniaxial Symmetry 629

and

where we assumed the primitive lattice vectors r, = (0, yl) and r2 = (x2, yz) which are restricted by tlie flux quantization condition 2n/ylx2 = 2eH. l h e coefficient C' determines the order of the magnitude of the order parameter.

In the preceding section the x-axis has been choosen in the X - Y plane, so ql = f and q2 = 0. For the determination of the stable configuration of the FLL with respect to the crystal axes, it is necessary to consider the freedom of the rotation about the magnetic field. This is done by introducing the Xuler angles (0, R v) ~ 9 1 .

The substitution of (4.1) into (2.2) gives for the current density

and d = 2eN(O) A,.

magnetization AH Equation (4.3) and Maxwell's equation rot AH = 4nJ give tlie induced

AH = 47cd o":(~) A,(r) (- A , - B, (v;),vl) . (4.5) P, P

The perpendicular components A H , and A H , are not always zero because ( V ~ I . ~ ) ~

and (vvvZ), do not vanish in general. Expressing (wu,vl), by the values of (0%. vj,) (i', j' = X , Y , Z ) fixed in the

crystal and the angle @) we get for A , B, and (v;),v1

where y = cos 8 (0

in the y-axis by the symmetry. Then 0 must be equal to zero.

0 n / 2 ) and C2 = (&>,/(vg), . If the x-axis lies in the X - Y plane, the magnetic induction must appear only

630 I<. TAKANAKA

By calculations similar to those of Abrikosov [18], the coefficient C' can be determined from

__ ~~

(H,, - H , ~ + A H ) . AH + 8ziv(o) ~,lA+l4 = 0 , (4.7) where the bar denotes the average in the primitive cell of FLL. Furt8her follow- ing Abrikosov, we obtain the free energy

where B is the magnetic induction (H, + AH).

The minimum of the free energy is given by the minimum of /I,, with respect to x2/yl and y2/yl. The conditions are as follows [ l l ] :

(4.10)

This means that the FLL is an isosceles triangular lattice and that the strongest deviation from the triangular PLL appears when y = 0, i.e.,when the mag- netic field is perpendicular to the symnietry axis. When y = 0, the y-axis is parallel to the symmetry axis and ql = 5. The y-component of the FLL be- comes larger than that of the equilateral triangle if ( > 1 and shorter if ( < 1. This fact can be understood by considering the relation for the components of the current density, J , N (v:), and the current conservation law around each flux- line.

5 can be determined by the experimental data of the upper critical fields [2,10] and

Substituting y = 0 and the above values for 5 into (4.10) and (4.11), we obtain 54" for Tc and 77" for NbSe, for the two angles of the isosceles triangle. The angle and the array of the FLL for Tc agree quite well with recent experi- mental observations [ l l ] . The observed angle is 52.5" when the magnetic field is perpendicular to the symmetry axis.

The perpendicular magnetic induction A Z is a rather large quantity and niay be observable. The maximum value of the ratio AH,/AH, is derived from (4.5) and (4.6)

_ _ _ _

__ 1 - p2 0.2 for Tc ={ - 1 for NbSe,

__ A H , - - aH, 25

at 0 = 39" for Tc and 69" for NbSe,, where cos 8 = y = (p/(l + 52))1/2.

Magnetic Properties of Superconductors with Uniaxial Symmetry

5. Concluding Remarks

Near Tc we have calculated the three critical fields, the stable FLL and the perpendicular magnetic induction by using the anisotropic Ginzburg-Landau equations. The importance of the anisotropic coupling has been shown in See- tion 3 by assuming the effective inass model to explain the experimental obser- vations of the upper critical fields. The shape of the BLL for Tc is in good agree- ment with the experimental results. To see the validity of this niodel for the layered materials, i t is desirable to measure the energy gap, the perpendicular magnetic induction and the shape of the FLL.

63 1

Acknowledgements

The author is very grateful for the hospitality of the Max-Planck Institut. He wishes to express his sincere thanks to Prof. A. Seeger for valuable discus- sions, and the Alexander von Humboldt Stiftung for financial support.

References

[l] R. C. MORRIS, R. V. COLEMAN, and RAJENURA BHANDRI, Phys. Rev. B 5, 895 (1972). R. C. MORRIS and R. V. COLEMAN, Phys. Rev. B 7, 991 (1973).

[2] Y. MUTO, N. TOYOTA, K. NOTO, and A. HOSHI, Phys. Letters A 45, 99 (1973). [3] S. FONER and E. J. MCNIFF, JR., Phys. Letters. A 45, 429 (1973). [4] J. A. WOOLLAM, R. B. SOMOANO, and P. O’CONNOR, Phys. Rev. Letters 32, 712 (1974). [5] E. I. KATS, Soviet Phys. - J. exper. theor. Phys. 29, 897 (1969). [GI W. E. LAWRENCE and S. DONIACH, Proc. 12th Internat. Conf. Low Temp. Phys.,

[7] L. N. BULAEVSKII, Soviet Phys. - J. exper. theor. Phys. 37, 1133 (1974); 38, 634

[SJ R. A. KLEMM, M. R. BEASLEY, and A. LUTHER, J. low-Temp. Phys. 16, 607 (1974). [9] K. Aor, W. DIETERICH, and P. FULDE, Z. Phys. 267, 223 (1974).

Kyoto, 1970, Ed. E. KANDA, Keigaku, Tokyo 1971 (p. 361).

(1974).

[lo] G. KOSTORTZ, L. L. ISAACS, R. L. PANOSH, and C. C. KOCII, Phys. Rev. Letters 27, 304

[ll] J. SCHELTEN, G. LIPPMANN, and H. ULLMAIER, J. low-Temp. Phys. 14, 213 (1974). [12] K. TAKANAKA and H. EBISAWA, Progr. theor. Phys. 47, 1781 (1972). [13] L. DOBROSAVLJEVI~! and H. RAFFY, phys. stat. sol. (b) 64, 229 (1974). [la] L. P. GORKOV and T. K. MELIK-BARKHUDA~V, Soviet Phys. - J. exper. theor. Phys.

[15] K. TAKANAKA and T. NACASHIMA, Progr. theor. Phys. 43, 18 (1970). [16] V. L. POKRQVSKII, Soviet Phys. - J. exper. theor. Phys. 13, 447 (1961). [17] D. SAINT-JAMES and P. G. GENNES, Phys. Letters 7, 306 (1963). [18] A. A. ABRIKOSOV, Soviet Phys. - J. exper. theor. Phys. 5, 1364 (1956). [19] P. M. MORSE and H. FESIIBACH, Methods of Theoretical Physics, McGraw-Hill Publ.

(1971).

18, 1031 (1964).

Go., New York 1953 (p. 62).

(Received January 20, 1975)