PHYSICA ELSEVIER
Physica C 282-287 (1997) 1859-1860
Restoration of Superconductivity at High Magnetic Fields in Layered High-T~ and Organic Superconductors Andrei G. Lebed ~'b and Kunihiko Yamaji ~ ~Electrotechnical Laboratory, 1-1-4 Umezono, Tsukuba 305, Japan bL.D. Landau Institute for Theoretical Physics, 2 Kosygina St., Moscow, Russia An integral equation determining the upper critical field parallel to the conducting plane of a layered superconductor is derived from microscopic BCS theory. From this equation, it follows that the superconductivity-destruction mechanism due to the orbital effect becomes ineffective at magnetic fields higher than a critical field H~4 due to a quantum nature of an electron motion in a magnetic field. This leads to the appearance of reentrant superconducting phase with dT~/dH > 0 in the case of p-pairing and under certain conditions in the case of s(d)-pairing. We also show that the orbital effect cannot destroy superconductivity at T = 0 at arbitrary magnetic fields. Therefore qualitative deviations from the description of G L A G (WHH) theory have to appear for arbitrary symmetry of the order p a r a m e t e r at low enough temperatures. We argue that p(s)-wave superconductor Sr2RuO4 as well as s(d)-wave superconductors T1Ba2CaCu2OT, T12Ba2CaaCu40~, #-(ET)zAuI2, and ~-(ET)2Ia are good candidates for the observation of the above mentioned phenomena.
INTRODUCTION A search for nontrivial mechanisms of superconductivity has been a primary motivation for the intensive study of high-To and organic superconductors during past ten years. On the other hand, all these compounds possess a property of a strong quasi-twodimensional (Q2D) anisotropy of an electron spectrum that also distinguishes them from traditional superconducting materials. So far, the effects of strong anisotropy have been discussed within the framework of the Lawrence-Doniach (LD) model [1]. The aim of our presentation is to point out that in high-T¢ and organic compounds having a strong Q2D anisotropy feasibly high magnetic fields of 10-100 T can lead to the "quantum limit" when the "size" of semiclassical electron orbits in the direction perpendicular to the plane is smaller than the interplane distance, d (the boundary field He4 is shown in Fig.1 and Table 1). In this situation BCS-pairs are "twodimensionalized" and the Lorentz force (which is perpendicular to the plane) becomes unable to destroy these 2D BCS-pairs. The critical temperature of superconducting transition, To(H), is shown to increase with increasing magnetic field at H > He4 in the case of p-pairing. This demonstrates a principle possibility of using p-wave superconductors at arbitrary high magnetic fields. In the case of s(d)-pairing it is shown that the orbital effects of electron motion 0921-4534/97/$17.00 © Elsevier Science B.V. All rights reserved. PlI S0921-4534(97)01092-7
in a magnetic field start to improve superconducting pairing at H >__ He4 and reentrant superconducting phase with dTc/dH > 0 is expected to occur if He,, H~/(0) < Hp, where H ! / ( 0 ) i s the upper critical field in the G L A G (WHH) theory [2,3] and Hp is the paramagnetic-limited field. The existence of the reentrant superconductivity is beyond the descriptions of both G L A G (WHH) theory and LD model which are valid at H << H¢4 and are derived as the limiting cases from the obtained integral equation. In the case when He, > Hp > H~/(0) quafitative deviations from the G L A G (WHH) theory are found to appear at low enough temperatures, T < T*. We stress t h a t the phenomena suggested in this paper extend the effect of the reentrant superconductivity proposed by us for Q1D superconductors [4] (see also experimental work [5]) to a Q2D case. They are distinct from the reentrant superconductivity effects in an isotropic case [6] since electron orbits are not quantized in a Q2D conductor in a parallel magnetic field. RESULTS
Let us consider the following solutions: ~,~(x,y,ri) = eP~eiP*~ ¢,~(x,pu,p~) of the Schrodinger equation:
[-±( d~~ + ~d~ ~)1 2m dx ~y ) + 2tx c ° s ( - i d d ~ vF
(1)
A.G. Lebed, K. Yamaji/Physica C 282-287 (1997) 1859-1860
1860
¢~(=,p~,p±)=, ¢.~(=,p~,p.)
(2)
for a Q2D conductor with the electron spectrum: 1 , 2+
2,
e(p) = ~mmtP= pv/ + 2 t , c o s ( p , d )
t , << eF
Tc~
-A
,
(3)
in a parallel magnetic field, H = (0, H, 0). [Here w¢ = e v F d H / c , VF is the Fermi velocity, ¢F =
mv~/2].
Using the inequalities t . , w ¢ << eF, we get the solutions of Eq. (2) in the following form:
B T*~,......... 0
/"}~t'~"cP'"-~)-'~"c"'~)l/(,F
-,.)'/'
(4)
where 6e = e - e f , ey = p ~ / 2 m , ~ = 4t x/w¢. Green functions can be constructed by means of the standard procedure [7] and the linearized gap equation determining superconducting transition temperature can be derived using Gor'kov equations for nonuniform superconductivity [8]. As a result we obtain:
h(¢,~) =
d¢~
~ u(¢,¢0 _ x l l > a l s i n 4h [
27rT VF sin ¢1 _:_l.: ~utnl, 2~Tl=-xl vF sin ¢1 I)
c o s [ 2 / t B H ( x - xl)(1 - a) VF sin ¢a
]
, ~ II Hc2(O)
C
i Hc4
Hp
Hc2(T)
Figure 1. Possible temperature dependences of H~2(T) calculated from Eq. (5) are shown. Curve A stands for the case of p-pairing. Curves B and C correspond to s-pairing in the cases H~4, H~/(0) < Hp and
H~'~(0)< Up< He,,correspondingly. Table 1. Estimated values of He4 , H c2~ II :0"), H p, and T* Hc4
H Icl2(0)
Hp
T*
Sr2RuO4 TIBa2CaCu207
30T 100T
? 150T
? 200T
? 30K
Bi2Sr2CaCu20x TI2Ba2Ca3Cu40x 13-(ET)2Aul 2
10T 150T 80T
? 100T 6.5T
150T 190T 7.5T
.'? 15K 0.3K
ACKNOWLEDGMENTS 02cX
:0(sq~T[COS(TT) - cos(~/=~ )]) A(¢.,~,)
(5)
The superconducting order parameter, A ( ¢ , x ) , depends on the coordinate of the center of mass of the BCS-pair, x, as well as on the position on the Fermi surface, where ¢ is the polar angle between the magnetic field and the momentum of the pairing electrons, p = - p * . [Here a = 0 for s-pairing and ¢ = 1 for the so-called equal spin p-pairing [8], correspondingly; a is a cutoff distance; U(¢, Ca) is the matrix element of the interaction of two BCS-pairs]. The fact that the orbital effect cannot destroy superconductivity at arbitrary magnetic field is seen in a periodicity of the Bessel function J0(...) in Eq. (5) in variables x and xa. Indeed, the choice of the periodic solution A o ( $ , x + r v F / w c ) = A0(¢, x) in the case of a = 1 leads to a logarithmic divergence in Eq. (5) as T ---* 0 . At high enough magnetic fields, H > He4 (i.e., when A < 1 by definition), Jo(...) -~ 1 in Eq. (5) and the orbital effects start to improve superconducting pairing. Detailed analysis of the interplay of the orbital and paramagnetic effects will be published elsewhere [9].
One of us (A.G.L.) would like to thank to N.N. Bagmet and E.V. Brusse for useful discussions. REFERENCES
1. W.E. Lawrence and S. Doniach, Proc. of 12th Conf. Low Temp. Phys., Kyoto, 1970, p. 361. 2. A.A. Abrikosov, Soy. Phys. JETP, 5, 1174 (1957); L.P. Gor'kov, Soy. Phys. JETP, 10, 593 (1960). 3. N. Werthamer et al., Phys. Rev., 147, 295 (1966). 4. A.G. Lebed, J E T P Lett., 44, 114 (1986). 5. I.J. Lee et. al., Synth. Met., 70,747 (1995). 6. M. Rasolt and Z. Tesanovic, Rev. Mod. Phys., 64, 709 (1992). 7. A.A. Abrikosov, L.P. Gor'kov, and I.E. Dzyaloshinskii, Methods of Quantum Field Theory in Statistical Mechanics (Dover, NY, 1963). 8. M. Sigrist and K. Ueda, Rev. Mod. Phys., 63,239 (1991). 9. A.G. Lebed and K. Yamaji, in preparation.