Behera, S N

Ghatak, S K

Ray, D K

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Indian J . Phys. 70A (6), 721-727* (1996)IJP A— an international journalCoexistence of superconductivity and structural distortion in system s with degenerate bands1S. N. Behera and Hafanath Ghosh2 Institute of Physics, Sachivalaya Marg, Bhubaneswar-751005, INDIA.S. K. Ghatak.Department of Physics & Meteorology, Indian Institute of Technology,Kharagpur-721302, INDIA, and D. K. Ray.Laboratoire P.M.T.M, C.N.R.S., Universite Paris-Nord,93430, Villetaneuse, FranceA bstractThe effect of the band Jahn-Teller distortion on superconductivity is studied for systems with orbitally degenerate bands using a simple the­oretical model. The model assumes a two-fold degenerate band coupled to lattice strain and an attractive interaction mediated by some boson producing DCS pairing superconductivity. It therefore, incorporates two competing physical processes involving the removal of degeneracy by lat­tice distortion and the formation of the condensate state of the cooper pairs. It is found that as superconductivity sets in the growth of the lattice distortion is arrested. The magnitude of the distortion decreases with decreasing electron density, while the superconducting order para­meter increases. Our observations agree with the anamalous structural behaviour of the superconducting compound Lai &sBao i^CuO* as detec­ted experimentally.K eyw ords : Jahn-Teller Distortion, Doubly Degenerate band, Superconduct­ivity.PACS Nos. : 31.30.-1, 74.25.Dw, 74.20.Fg, 74.62.-c1 This work was supported by the “co-ordinated research programme of the Institute of Physics, Bhubaneswar.2e-mail: hng@iopb.ernet.in1  IntroductionCompounds which exhibit superconductivity are often prone to undergo struc­tural instability [1]. Recently, in high Tc cuprates [2, 3] and in many intermetal- lic compounds [4, 5, 6], the structural transition associated with the lowering of crystallographic symmetry has been observed. The existence of orbital degener­acy in the ground state with the Fermi level (FL) lying within a degenerate band is a common feature to all these systems. The distortion (which is a measure of the orthorhombic splitting in the tetragonal to orthorhombic transition) lifts the degeneracy of the band and the electron redistribution between the shifted bands lowers the electronic energy provided the lowering in electronic energy outweighs the cost in elastic energy due to distortion. With such transition there is an associated change in the density of states around the FL and there­fore, it is expected that the band Jahn-Teller (J-T) distortion would strongly influence the superconductivity in such a system. If the FL lies at the centre of the degenerate band then the density of states at FL is drastically reduced and the superconducting transition temperature Tc will go down. On the other hand, Tc can be higher as phonon softening due to J-T effect may increase the coupling constant which is inversely proportional to mean square phonon energy.It is well known that some high temperature superconductors like the C u -0  (e.g, La^CuO^) systems undergo structural transitions from the tetragonal to the orthorhombic phase. In the tetragonal phase, each Cu atom occupies a site of squared Bravais lattice and is surrounded by an octahedra of six neighbouring oxygen atoms, two of them lying above and below the plane and the four others occupying the sites inside the plane. The oxygen octahedra gives rise to a strong crystal field, the leading term of which is cubic and thus splits the five fold degenerate d-orbitals of Cu into a t2g triplet (orbitals dxy, dyz and dzr) which lies far below the Fermi energy and is thus completely occupied, and an eg doublet (orbitals dx2_y2 and d3z2_r2 ) which is partially occupied. In this context, experimental observation of high resolution neutron diffraction experiments on the superconducting Lai,ssBao.isCu0 4  compound was reported [7]. A brief summary of their essential findings are presented below.Their measurements [7] clearly exhibit a tetragonal to orthorhombic trans­ition at 180 K  together with further structural anomalies on approaching the onset of superconducting phase transition at 35 K. At temperatures greater than 185 K  the tetragonal symmetry (as in the parent LCL2CUO4 compound above 500 K) is observed. Below this temperature, a symrtietry breaking or­722 S N Behera, Haranath Ghosh, S K Ghatak and D K  Raythorhombic transition occurs with a splitting ratio that increases monotonicaJly with decreasing temperature, reaching a maximum value at 75 K. Below 75 K  a highly unusual and unexpected decrease in lattice distortion is observed which saturates at 35 K , coincident with the onset of superconductivity in this sample. Direct evidence of orthorhombic distortion is also detected from the splitting of the Bragg reflection peaks of the orthorhombic phase at 120 K  which col­lapse at lower temperatures [7]. Most interestingly, the anomalous decrease in the orthorhombic splitting correlates with the plateau in resistivity and further, that the onset of superconductivity at 35 K  corresponds to the saturation of the splitting at a very low label. One of the important experimental results, namely, the anamalous temperature dependence of the orthorhombic splitting which is defined as spontaneous strain is displayed in fig.l.C oexistence o f  superconductivity an d  structural distortion etc  723F ig .l The temperature variation of the spontaneous strain as a function of temperat­ure, obtained from the neutron diffraction experiment Ref. [7].Therefore, we emphasize that it is extremely important for high Tc super­conductors like La-iCuO* to have a detailed and acurate knowledge of structural instabilities which may throw light about the origin of superconductivity in such materials. Also, such observed anamalous structural behaviour will have strin­gent constraint to any realistic theoretical model for high Tc superconductors. Hence, our main aim in this paper is to present a detailed study of the coex­istence of superconductivity and band J-T distortion in the two-fold degenerate tg band as a function of the electron concentration.2 Model for coexistence of superconductivity and lat­tice distortion724 S N Behera, Haranath Ghosh, S K  Ghatak and D  K RayWe consider a model system in which the electrons are in a two fold degenerate band and interact with the lattice as well as between themselves, the later via a BCS pairing interaction. Therefore, the total Hamiltonian H consists of four terms,k ta«r k,a| ( C n - C 22)eJ -  E  V w ^ C l tC l^ C - w C w .  (1)k,k'The first term in the hamiltonian represents the non-interacting electrons in a two-fold degenerate (eg) band (a =  1,2 ; is the orbital index) with single­particle energy f*. The coupling between the electron density in the degenerate electron band and the static elastic strain in presence of tetragonal distortion is described by the 2nd term of equation (1). The third term is the elastic energy in absence of strain. The pairing hamiltonian described by the 4th term of(1) assumes that the BCS pairing exists only within the same orbitals and the strength of interaction is same for both the orbitals [8].Therefore, the interaction of the electrons in the degenerate conduction band with the lattice acts as external perturbation to the free electrons in the band which tries to creat a population difference between the two bands. As the population difference increases, the strain builds up [10] in the system resulting in a splitting of the single degenerate band into two with band energies £a (fc) = Ck -  (~ l)aGe, provided there is a net gain in the electronic energy due to the redistribution of electrons between the splitted subbands in comparison to the cost in the elastic energy.In order to study the actual nature of the superconducting condensate state m the distorted orthorhombic phase one has to understand the superconducting gap equation, the associated gap equation for the strain mode and the position of the chemical potential. The respective gap equations as well as an equation for the chemical potential can be obtained from equation (1) within the mean field approximation [8]. For example the equation for the SC gap is given as usual by, i = £k ,av2 Ea(k)tanh (/3Ea(k ) /  2) (2)Coexistence of superconductivity and structural distortion etc 725where the superconducting quasi particle energies Ea(k) are obtained as, Ea{k) =  ±\/(£«(fc) -  p )2+ I A |2 for a chemical potential p. ’Similarly, the gap equation corresponding to the strain order parameter given by,isk^ag S L ^ t M L2 Ea{k) 2 (3)and the total number of electrons determines the chemical potential.It is to be noticed that the above equations are coupled integral equations in the sense that the equation for the superconducting gap function A  is a func­tional of itself, the splitting in the band due to strain as well as the chemical potential p  i.e, A  =  A ( A ,e , /x ) .  Similarly we have the strain c = e(A,e,/x) and the number conservation equation n £  n (A ,e ,p ) . Therefore, in order to determine the temperature variation of any of these quantities which is im­portant to understand the nature of the system, we solve these three equations numerically self-consistently for different band fillings [8].3 Results and discussionsThe equations (2,3) are numerically solved for a given set of the values of input param eters (V, G, u>£>, C and B) with varying electron concentration n. For all the results presented here the values of these parameters are chosen as fol­lows : the dimensionless pairing potential JV(0)y =  0.11, the Debye frequency u>D — 0.038 <°y, the elastic stiffness constant C  =  30 eV and the half band­width B  =  0.1 eV are kept fixed. The peak density of state (DOS) used has an energy dependence given by, N( t )  =  ^ ( 0 ) ^ 1 -  | jj |In | -^r | ;B being the half-bandwidth [9]. In fig. 2 (a, b) superconducting gap and strain are plotted as a function of tem perature for different dopant concentrations (£ =  0.05 and 0.1 respectively) with G  =  1.15 eV. We find that close to half filling (6 small) the s tructu ra l transition  takes place at higher temperatures and it increases with decreasing tem perature till the onset of superconductivity where it attains the maximum magnitude, then decreases drastically and saturates to a lower magnitude (alm ost 50% of the maximum magnitude) at lower temperatures (cf. fig.2(b)). Such a behaviour is in complete agreement with the experimental observation (cf. fig .l). The structural transition temperature Ts as well as the maximum m agnitude of the strain  emax(T) reduces as the hole dopant concen­tration increases. To note, the emo*CO shifts towards higher temperatures with increasing dopant concentration. In contrast, experimentally the maximum in70A (61-10the strain occurs at a temperature much higher than Tc whereas that from our results coincide exactly at Tc. Such experimental data may be an indication of the fact that there is large fluctuation in the high-Tc superconductors close to Tc so that the superconducting pairing exists even at a higher temperature than Tc and hence the suppression of strain starts at T > Tc. On the otherhand, ours is an exact mean field result which does not take into account fluctuation effects, resulting in such qualitative differences with experiment.8 =  0.05 <5 =  0.1726 S N Behera, Haranath Ghosh, S K Ghatak and D K  RayT e m p e r a tu r e  T e m p e r a tu r eFig. 2 The temperature variation of the SC and the strain order parameters for dif­ferent hole concentrations (a) S — 0.05 and (b) 8 = 0.1, obtained from our modelIn contrast, the SC Tc as well as the A(0) increases with increasing dopant con­centration (J). At half-filling, the lower splitted sub-band is fully occupied whereas the upper one completely empty, resulting in a maximum population difference between the bands and hence maximum splitting. Consequently, a large amount of electronic states we taken away from the Fermi level resulting in strong suppression of Tc for the half-filling case. On the other hand, a small doping (n<l) reduces the effect of strain and Tc increases rapidly. Such strong influence of distortion is intimately related to energy dependence of the density of states near the Fermi level [8].However, the present model is too simplified because the strong electron correla­tion, which is known to play an important role in Cu -  O systems is not considered here. The effect of intra and inter-orbital correlation and exchange interaction has also not been incorporated in the present calculation partly for numerical simplifica­tion and partly for the well known difficulties in treating them. In systems like alkali doped fullerides (which shows no magnetism) such Jahn-Teller distortion is well es-tablished. The strong correlation treated within the Slave Boson approach has been under study in order to make the applicability of our present model more approbate for such systems [10].CoemUnce of superconductivity and structural distortion etc 7274 ConclusionInterplay between the band Jahn-Teller distortion and superconductivity in a narrow two-fold degenerate band has been found to be strongly dependent on band occu­pation and electron -lattice interaction strength. The transition temperature for the distorted phase goes down with hole doping and the coexistence phase appears at low temperatures. Observed temperature dependence of the lattice strain is in qualitative agreement with the experimental observation.References[1] Cohen. M. L. and Anderson. P. W., in Proceedings of the conference on su­perconductivity in d- and /-band metals - 1972, edited by Douglas. H. J., AIP conference Proceedings No. 4 (American Institute of Physics, New York, 1972).[2] Axe. J. D. et al., Phys. Rev. Lett. 62, 27551 (1989); Takagi. T. et, al„ Phys. Rev. Lett. 68, 3777 (1992).[3] Sato. M., in 'Physics of High Temperature superconductivity’, Springer-Verlag, Berlin Heidelberg (1992) p-239.[4] Testardi. L. R., Rev. Mod. Phys. 43,657 (1975).[5] Balster. H., Ihrig. H., Kockel. A. and Methfess. K. L. E., Z. Physik B 21[6] Suites. J. C., Sol. State Cominun., 18, 423 (1976)[7] Paul. McK. D., Balakrishnan. G., Bernhoeft. N. R., David. W. I. F. and Harrison. W. T. A:, Phys. Rev. Lett. 5 8 ,1976 (1987).[8] For details please see, Ghosh Haranath, Behera. S. N., Ghatak. S. K. and Ray.D. K, Preprint IP/BBSR/95-68 k  the references therein, also communicated to Phys. Rev. B (1995).[9] W. Nolting et al., Phys. Rev. B 43,1117 (1991).[10] Ghosh Haranath, Mitra Manidipa, Behera. S. N. et al., unpublished calculation.

Coexistence of Superconductivity and Structural Distortion in Systems with Degenerate Bands

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Coexistence of Superconductivity and Structural Distortion in Systems with Degenerate Bands

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