Model of consecutive superconducting transitions

Physica B 165&166 (1990) 1051-1052 North-Holland

MODEL OF CONSECUTIVE SUPERCONDUCTING TRANSITIONS D.S. HIRASHIMA and T. MATSUURA Department of Physics, Nagoya University, Furo-cho, Chikusa-ku, Nagoya 464-01, Japan A model leading to consecutive superconducting transitions is investigated. Several thermodynamic quantities are calculated and the effects of the second transition on them are clarified. Possible relevance to high-T c superconductors is discussed. 1. INTRODUCTION Recently, superconductivity in highly correlated systems, such as heavy fermion- and high-T c superconductivity, has attracted much interest. In these systems the conventional BCS mechanism is unlikely to be effective. In heavy fermion superconductors (HFS), several properties in superconducting (SC) state are indeed deviating from the prediction of the BCS theory. One of the most intriguing phenomema in these unconventional superconductors is possible consecutive SC transitions, i.e., SC phase can undergo another transition with which additional component mixes into the equilibrium gap. Several authors (1",,3) have indeed suggested the consecutive SC transitions to account for the second phase transition observed in the SC phase in Ul-xThxBe13 (4). We investigate a model leading to the consecutive SC transitions (5), paying attention to the effects of the second transition on several quantities. Possible relevance to high-T c materials is discussed. 2. MODEL For simplicity, we consider a 2D square lattice system. As a model hamiltonian, we take }{ =

L ~pa~
P,
where ~p

= -2t

(cos Px

P,p'

+ cos p y) -J.!. and

with (p = 2 cos Px cos Py and ¢>p = 2 sin Px sin Py; our model hamiltonian represents a quasiparticle system where "8"-( vd and "d" -( v can be derived. Each of v( and v induces the SC transition at its characteristic transition temperature, T( and T. When T > T(, the normal phase is first condensed into the SC 0921-4526/90/$03.50

© 1990 -

state characterized by the orderparameter, A(p) = f'::,.¢>p, (the ¢>-phase) at T = T. Investigating the stability of the ¢>p ±iA((p (¢>±i(phase). To show that the consecutive SC transitions indeed take place, we have solved the gap equation with the above interaction; in the calculation, the chemical potential is determined so that the total electronic number is 1.15. First, we set v = 1.31 (t = 1), which leads to T = 0.1. Then, we find that, as far as 0.79 < v( < 1.34, the consecutive SC transitions necessarily occur, i.e., 0 < TZ < T. Solution to the gap equation with v = 1.31 and v( = 1.08 is shown in Fig. 1. It is seen that a second transition indeed occurs at TZ = 0.05, below which A( begins to grow.

T,

T~=O.

T~=O.

O. 1

0.05

1 05

/),., (T)

0.5

T/T~

1

FIGURE 1 Solution to the gap equation (v = 1.31 and v( = 1.08).

Elsevier Science Publishers B.V. (North-Holland)

D.S. Hirashima, T. Matsuura

1052

O. 2

T¢=O.l

0.6

T~=O.05

O. 15

0.4

~~

O. 1

0.2

0.05

o

O. 5

T/T¢

1

FIGURE 2 Electronic specific heat in the SC state. 3. PHYSICAL QUANTITIES 3.1. Thermodynamic Critical Field He Since the second transition at T = Tt brings about additional free energy gain, there is additional increase in He below T = Tt. Note that the derivative of He' dHe(T)/dT, is continuous at T = Tt, since the transition is of the second order. He is related to the lower critical field Hell if not directly. Hence, additional increase in He! below T = Tt is also expected. 3.2. Specific Heat Temperature dependence of specific heat below T = Tq, is shown in Fig. 2. A second jump appears accompanying the transition at T Tt. Furthermore, below T = Tt the exponential temperature dependence takes the place of power-law onej it is because the ¢>+i(-phase generally has no nodes in the excitation energy in contrast to the ¢>-phase, in which the excitation energy generally has nodes on those points where the Fermi surface (line) crosses the lines determined by ¢>p O. 3.3. Quasiparticle Density of States (DOS) The DOS in the tP-phase linearly vanishes as c -+ 0, and logarithmically diverges at the gap maximum. As shown in Fig. 3, in the ¢>+i(- phase, the DOS is zero around c = 0 and has another divergence, which results from a saddle point near the gap minimum. In the present case, the gap maximum and minimum are roughly given by t:. MAX ::: 2t:.q, and t:. MIN ::: 2t:.{, respectively. The DOS is directly probed by the tunneling experiment. Hence, below T = Tt, the tunneling conductance will have additional gap structure in between one observed at higher temperatures. 3.4. Penetration Depth We have calculated the so called Yosida function Y(T), which is related to the penetration depth )'(T)j in a clean, local-limit superconductor, ),(T) is given in terms

-0. 2

o

LJ~ 0.2 e

FIGURE 3 Quasiparticle DOS in the ¢>±i(-phase: t:.q, = 0.112, t:.{ = 0.051 and JL = 0.309. of Y(T) as ),(0) j)'(T) = 1 - Y(T). The result shows that the penetration depth is slightly shortened owing to the second transition. Furthermore, below T = Tt, it varies exponentially just like the specific heat. 4. DISCUSSION At present, the HFS is the most probable candidate for the consecutive SC transitions. In high-T e materials, however, there are also suggestive observations: Particularly, in YBa2Cu307, anomalous increase in H el (6) and appearance of additional gap-like structure in tunneling conductance (7""'9) have been observed at lower temperatures. Furthermore, the present model can simultaneously account for the absence of the coherence peak in liT! below Te and exponential temperature dependence of the penetration depth at lower temperatures. Careful measurement of specific heat at lower temperatures is highly desired. Even apart from high-T e superconductivity, occurrence of the consecutive SC transitions is a fascinating potential in non-BCS superconductors. Furthermore, the complex SC phase realized below T = Tt is also a matter of interest in itself. REFERENCES (1) P. Kumar and P. WolHe, Phys. Rev. Lett. 59(1987) 1954. (2) M. Sigrist and T.M. Rice, Phys. Rev. B39(1989) 2200. (3) D.S. Hirashima, Prog. TheoL Phys. 80(1988) 840. (4) H.R. Ott et al., Phys. Rev. B31(1985) 1651. (5) D.S. Hirashima and T. Matsuura, J. Phys. Soc. Jpn. 59(1990) 24. (6) H. Adrian et al., Physica C 162-164(1989) 329. (7) A. Fournel et al., Europhys. Lett. 6(1988) 653. (8) J. Geerk et al., Z. Phys. B 73(1988) 329. (9) M. Gurvitch et al., Phys. Rev. Lett. 63(1989) 1009.