Vortex structure in a thin film of an unconventional superconductor

Physica B 206&207 (1995) 634-637

ELSEVIER

Vortex structure in a thin film of an unconventional superconductor N . O g a w a a'*, M. Sigrist b, K. U e d a c ~Department of Mechanical Engineering, Gifu National College of Technology, Shinsei-cho, Motosu-gun 501-04, Japan bDepartment of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA Clnstitute for Solid State Physics, University of Tokyo, Roppongi 7-22-1, Tokyo 106, Japan

Abstract

In bulk, an unconventional superconductor with two components of the order parameter can have the anisotropic internal vortex structure, since it is possible for the two components to have their own cores at different places. In a thin film there is a critical thickness d c for such vortices where the first critical magnetic field He1 has an anomaly. The vortex below the critical thickness is fractionally quantized where the core of one component of the order parameter is pushed out from the surface. The zero supercurrent at one point on the surface is a characteristic of the fractionally quantized vortex.

It is generally believed that at least some of the heavy fermion superconductors belong to the class of unconventional superconductors with non s-wave C o o p e r pairs [1]. Theoretically, this is due to the fact that it is very hard to form s-wave C o o p e r pairs b e t w e e n strongly correlated electrons. F r o m an experimental point of view, there is an accumulation of experimental results supporting unconventional pairings. Recently, many experimental investigations have been reported to indicate an unconventional d-wave pairing also for high-T c cuprates [2]. A conventional superconductor has a scaler order parameter. O n the other hand, an unconventional superconductor can have a multi-dimensional order p a r a m e t e r and many new phenomena, which are caused by the multi-components of the order parameter, are expected in the problem. In this paper, we study one simple example of the multi-dimensional order parameter, which corresponds to tetragonal crystal symmetry O4h F; representation. In this case, there are two components, r h and rh, of the order parameter. We consider this kind of superconductivity

* Corresponding author.

in a thin film in this paper. The G i n z b u r g - L a n d a u free energy constructed by group theoretical considerations in dimensionless units is f F = J d3r[(t - 1)(1~1 = + t~zl =)

÷ (1~11 = ÷ In=12) 2 ÷ & ( n ~ n 2 - ~ l n ; ) ~ ÷ ~3ln, lZln21~

+ k,(Io;7,12

+ IOyn212) +

k2(IOyn, 12 +

IOxn212)

+ k3{(D~ol)*(Dyn2) + (D~l~)(Dyn2)*} + k4{(D~12)*(Dy~7~) + (D~Th)(D/q~)* } + k~(ID~n,I ~ + I D ~ n f )

+ K~B - (s - 2nox)],

(1)

where Hex is the external magnetic field and D = V iA. To treat a thin film problem, additional energy F s which is finite only on the surface must be considered, since the existence of the surface makes the symmetry lower than in the bulk case. The constructed surface energy Fs with the surface parallel to the y - z plane is

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635

N. Ogawa et al. I Physica B 206 & 207 (1995) 634-637

t F~ = J,.,,.c~ d3r[+g,(l~/l[ 2 +

I

MI 2) + g~(l~,l ~ - I ~ f ) ] ,

I

I

I

I

I

I

I

(2) This surface energy gives different boundary conditions for each component of the order parameter. Therefore, in a thin film novel phenomena which does not happen in bulk, an unconventional superconductor is realized [3]. In bulk, the surface energy can be negligible. Furthermore, if the external magnetic field is zero, the order parameter is homogeneous. Under this situation, there are three superconducting phases depending on the fourth order coefficients /32 and /33 of the order parameter. A:

(n,,'rh)=(1, -i),

/32>0,4/32>/33,

B:

(7/1,T/2)=(1, +-1),

/3:<0,/33<0,

C:

(m , n:) ~ (1, 0) or (0,1) ,

/33>0,4/3:
These phases are called A, B, and C in the following. All of three phases have the twofold degeneracy. In A phase, time reversal symmetry in addition to the U(1) gauge symmetry is broken at the superconducting transition. In B and C phases, point group symmetry is broken additionally. Above the first critical external magnetic field, the vortices penetrate into the superconductor. In our case, three typical types of vortices occur in bulk. The first type is a vortex in the C phase. In the C phase, almost the same vortices are realized as the vortices of conventional superconductor, since only one component of the order parameter is finite. The other two types occur in the A and B phases where both components of the order parameter are finite. The second type is a vortex with split cores where each component of the order parameter has its vortex core at different places. These vortices will be called split vortices [4]. The third type of vortex is a vortex with sing!e core in the A or B phase. The vortices are called non-split vortices in this paper. In the A phase, since the point group symmetry is not broken, the non-split vortex is axially symmetric similar to the conventional Abrikosov type vortex. In the B phase, the vortex is not axially symmetric, because of the breakdown of the point group symmetry. When the absolute value of/32 is very small compared with the value of/33, split vortices are realized and for the other cases, non-split vortices are realized. In a thin film, the first magnetic field He1 is defined as the external magnetic field where the Gibbs free

0.02

-r"

0.01

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20

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|

40

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60

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80

d

Fig. 1. Dependence on the thickness d of the first critical magnetic field He1 for the non-split vortex. The following parameters are used. /32 = - 1 , /33 = - 0 . 5 , k I = 3k 2 = 3 k 3 = 3k 4 = 0.75, r = 5, gl = g2 = 100, t = 0.75. energy with one and no vortex are the same. We calculate the dependence on the thickness d of the first critical magnetic field He1 for split and non-split

i

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I 0.02

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o.ol

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20

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60 d

Fig. 2. Dependence on the thickness d dependence of the first critical magnetic field Hc~ for the split vortex./32 = -0.01 and the other parameters are the same as in Fig. 1.

636

N. Ogawa et al. / Physica B 206 & 207 (1995) 634-637

vortices. The dependence on the thickness d of H ~ for non-split vortex is shown in Fig. 1. As d decreases, H ~ increases, since in a thin film the influence of penetration of the external magnetic field from the film surface becomes stronger. There is no anomaly in this case. However, for the split vortex, H~ has an anomaly as shown in Fig. 2. The global behavior of H ~ for the split vortex is almost the same as the non-split vortex. There are two lines in Fig. 2. The lower part of the plotted two lines is the real / / ~ ,

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Fig. 4. Supercurrent distribution for (a) split vortex and (b) fractional vortex. The parameters are the same as in Fig. 2.

Fig. 3. Schematic figures of the domain wall for (a) split vortex and (b) fractional vortex.

since the first critical magnetic field is the lowest external magnetic field where a vortex penetrates into the superconductor. Therefore there is a kink at the critical thickness d~. Above the critical thickness, the vortex is almost the same as the bulk split vortex. On the other hand, below the critical thickness the vortex is essentially different from the split vortex. In a very thin film, a

N. Ogawa et al. / Physica B 206 & 207 (1995) 634-637

vortex core corresponding to one of the components of the order parameter is pushed out from the surface and a fractionally quantized vortex (fractional vortex) is realized. The magnetic flux at just below the critical thickness is smaller than at just above the critical thickness. Since the phase of each component of the order parameter changes its value 2~r once around its vortex core, a split vortex has a domain wall encircling through the cores as shown schematically in Fig. 3(a). If the inside of the domain wall (T/1,'02) O( (1, i), the outside of the domain wall (7/1,*12)~ (1, - i ) . Thus the domain wall is the boundary wall between the degenerate two states. On the other hand, for the case of a fractional vortex, the domain wall is cut into two and both ends of the domain wall are connected on one side of the surface as shown in Fig. 3(b). The reflection of the fractional vortex is seen in the pattern of the supercurrent. The supercurrent distribution for the fractional vortex, which is shown in Fig. 4(b), gives the zero current point on one side of the surface. On the other hand, in the supercurrent

637

pattern for the split vortex, which is shown in Fig. 4(a), there is no such point. In conclusion, we find an anomaly for the thickness dependence of He1 for a split vortex. Below the critical thickness de, a fractional vortex is realized in this case. There is a zero supercurrent point on one side of the surface, as a reflection of the fractional vortex.

References [1] M. Sigrist and K. Ueda, Rev. Mod. Phys. 63 (1991) 239. [2] D.A. Wollman, D.J. Van Harlingen, W.C. Lee, D.M. Ginsberg and A.J. Legget, Phys. Rev. Lett. 71 (1993) 2134; M. Sigrist and T.M. Rice, J. Phys. Soc. Japan 61 (1992) 4283. [3] N. Ogawa, M. Sigrist and K. Ueda, J. Phys. Soc. Japan 61 (1992) 1730. [4] T.A. Tokuyasu, D.W. Hess and J.A. Sauls, Phys. Rev. B 41 (1990) 8891. Yu.A. Izyumov and V.M. Laptev, Phase Transitions 20 (1990) 95.