Solid State Communications, Printed in Great Britain.
Vol. 75, No. 2, pp. 105-108, 1990.
0038-1098/90 $3.00 + .OO Pergamon Press plc
ISOTOPE EFFECT AND HOLE SUPERCONDUCTIVITY J. Kasperczyk* and H. Biittner Physikalisches Institut, UniversitHt Bayreuth, D-8580 Bayreuth Federal Republic of Germany also *Physics Institute, Pedagogical University, Al. Zawadzkiego 13/ 15, PL-4220 1 Czestochowa, Poland (Received 22 February 1990 by B. Miihlschlegel)
We consider a hole superconductor described by a Hamiltonian proposed by Hirsch. For a discussion of the isotope effect within this model the influence of lattice vibrations on the hopping interaction is proposed. Both a harmonic interaction as well as an anharmonic approximation are used to calculate the superconducting critical temperature and the isotope effect parameter u in the weak coupling limit. While the harmonic hole-lattice coupling describes the usual isotope effect, increasing anharmonicity of the hole-lattice coupling decreases the isotope effect leading to a situation known in the high-temperature superconducting oxides. hopping integral and At is the hopping interaction, which is necessary for pairing of holes. The Hubbard on-site (inter-site) repulsion energies are denoted by U(V), respectively. All the parameters t, At, U and V are positive and U > V > At. As shown in  the effective interaction between oxygen holes with opposite momenta is attractive for an appropriate choice of parameters and high enough coordination number z. Therefore, the problem of superconducting transition can be solved within the usual weak coupling BCS scheme. Under the condition D/2 T p $- kBTc, where D is the bandwidth and p is the chemical potential and assuming a constant density of states g = l/D, the following critical temperature was obtained [4, 51
MANY THEORETICAL approaches were proposed since the discovery of high-temperature superconductivity to explain this phenomenon. However, Hirsch has recently developed a fundamental mechanism for superconductivity arising from the interaction of a hole with the outer electron in atoms with nearly filled shells [l-3]. The essential assumption of the model is that holes behave different from electrons, i.e. that there is a dynamical asymmetry between electrons and holes. The physical origin of this consists in the fact that a hole in the valence band causes a severe change in the bonding condition in contrast to a conduction electron which moves with no essential change in cation shells. As a result the effective mass of the hole is large giving rise to high resistivity in normal state. On the other hand, this local effect of the hole helps it to pair with another hole to lower the total energy which cannot be lowered effectively by delocalization of conduction electrons. The above pairing effect gives rise to superconductivity. The essential physics of the hole metal can be considered on the ground of the Hamiltonian proposed by Hirsch 
Ci Cj 1 !dCjn+
where a and b are functions of n, a(n)
1 + 2k(l - n) -w(l
+ (k* - wu) (1 - n)2 b(n)
for cr, are the
2k(l - n) - w( 1 - n)’ - u + (kz -
where cl(ci,,) are creation (annihilation) operators a hole at the i-th site with the spin component n, = c:,cia and ni = ni.O+ ni,_o. The sites i and j nearest neighbours. The parameter t is the value of
- n))“* exp (-a/b),
wu) (1 - n + 0.5n2),
with u = gU, w = zgV, k = 2zgAt and n is the concentration of holes per single site. It can be shown that the critical temperature strongly depends on this concentration having a maximum for a certain value of n. It is in qualitative agreement with experimental findings although a more quantitative agreement can be received by use of a more realistic density of states (i.e. non-constant). It was also shown in  that a parameter range of k’s and w’s necessary for occur-
ISOTOPE EFFECT AND HOLE SUPERCONDUCTIVITY
rence of superconductivity is strongly restricted for a given value of the parameter u. The high critical temperatures can be easily understood taking into account the fact that the interaction between a hole and the electronic cloud at the same site is purely electronic and that the excitation energies of the electron cloud are at least two orders of magnitude greater than typical phonon energies. However, Hirsch stated [l] that his model is quite general and can be applied to the high-temperature superconductors as well as to those classical superconductors (with low critical temperatures) which seemed to be well described by the usual electron-phonon interaction. A crucial point to elucidate such a statement is an analysis of the isotope effect which is rather weak in the high-temperature superconductors although some classical superconductors also show deviations from the value predicted by the BCS theory. There are no phonons and no electron-phonon interaction in the original Hamiltonian equation (l), therefore no isotope effect can be expected for the model based only on a purely electronic mechanism. It is quite physical to assume that lattice vibrations should influence the hopping interaction. Hirsch proposed  to modulate the size of At as follows At = AG(l - B(qi - qj))
where qj and qj are the nearest-neighbour ionic coordinates and fl is a harmonic coupling constant for the interaction between holes and phonons. This interaction is thus described by equation (5) within a harmonic phonon approximation. Later, Marsiglio  proposed also an electron-phonon coupling within the single-electron hopping t. But this term is only of interest in the strong coupling limit. In the weak coupling regime, however, the At-variation is most important. However, equation (5) seems to be insufficient, as it will be shown, for a common treatment of classical and high-temperature superconductors. Therefore, one postulates to replace equation (5) by the following one At = AtO(l + ,!?J - yd3),
where 1-3> 0, y > 0 is an anharmonic coupling constant and 62 =
qj)2) = LM-1’2
with a constant L and ( ) denoting a thermodynamical average. We argue that the harmonic hole-lattice constant coupling b is positive. Indeed, the hopping interaction At is a nonlinearly decreasing function of a distance between neighbouring sites, and Ato is its value for the equilibrium distance r. If one averages At
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over the region (r - 6, r + S), the averaged value is a bit higher than Ar, and, therefore, we have p > 0 in equation (6). On the other hand, the anharmonic hole-lattice coupling constant y > 0 enters equation (6) with negative sign because of the well-known fact that anharmonicity of a lattice causes its thermal expansion. Therefore, increasing anharmonicity should increase the equilibrium distance r between sites and decrease simultaneously the hopping interaction At. It is generally expected that the anharmonic effects should be especially important for oxide systems because of the large local polarizability [8, 91. As a consequence we have two competing mechanisms (a harmonic and an anharmonic) which modulate the hopping interaction At and superconductivity of a given system. It is, therefore, interesting to investigate an influence of the harmonic as well as the anharmonic hole-lattice couplings on the superconducting transition and the isntooe effect. Let us therefore consider the isotope effect parameter CI =
In T,/d In M
where M is the isotope mass. The only dependence on isotope mass in the expression for the critical temperature comes from the exponent equations (2)-(4). Therefore one obtains CI =
M d (a/b)/dM.
Combining equations (3), (4), (6) and (9), the following isotope effect parameter is calculated c1 =
(ab-‘(1 - a-‘(1
- n + k(1 - n + O.Sn’)) - n)(l
+ k - kn))
It is easily seen that parameter c1depends explicitly on the hole concentration 12and the hole-lattice coupling parameters /? and y. To analyze influence of the latter on a superconducting transition and an isotope effect we have chosen for calculations the following set of typical model parameters [2, 31: bandwidth D = 0.5eV, on-site repulsion U = 5eV, inter-site repulsion V = 2eV, hopping interaction Ato = 0.8 eV, coordination number z = 4. The value of 6 is taken for simplicity as a unit of length, therefore, we have 6 = 1.0 in all cases in this paper. Note that the lattice displacement is not calculated self-consistently but that it only changes the At-value. This is a first step in including the electronphonon coupling. We calculated at first a critical temperature as a function of hole concentration (equations (2), (3), (4)) for different harmonic hole-lattice couplings fi, assum-
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ISOTOPE EFFECT AND HOLE SUPERCONDUCTIVITY
Fig. 1. Superconducting critical temperature T, vs hole concentration n calculated for different values of the harmonic hole-lattice coupling parameter p (dotted: B = 0; full: 0.01; broken: 0.02). The model parameters are: D = 0.5eV, U = 5 eV, V = 2eV, At, = O.geV and z = 4.
Fig. 3. Critical temperature T, vs hole concentration n for different values of the anharmonic hole-lattice coupling parameter y (dotted: 0,001; full: 0.002; broken: 0.003). Same parameters as in Fig. 1. The harmonic coupling parameter /3 = 0.012 is constant.
ing no anharmonicity (y = 0.0). The results are presented in Fig. 1. Each curve shows a strong dependence on hole concentration n with a maximum critical temperature TcTaxfor n = nmax. Such a behaviour is actually observed in high-temperature oxide superconductors. The maximum critical temperature increases strongly with increasing harmonic coupling /I. It is, of course, caused by the change of hopping interaction due to the harmonic modulation described by equation (6). However, a relevant hole concentration nmax shows rather a weak dependence on jl although it also increases.
The isotope effect parameter c1as a function of hole concentration shows a relatively weak dependence for small values of n. The classical value c1 = 0.5, which is expected by the BCS theory on the ground of the usual electron-phonon coupling, can be reached within the present model for concentrations providing rather small critical temperatures (compare Fig. 1 and Fig. 2). This is to support the Hirsch’s statement [l] that low temperature (classical) superconductivity gives CI= 0.5 as a case, which is not related to the usual electron-phonon interaction. Many deviations from the value 0.5, observed in these materials, can be explained as an influence of hole concentration and other parameters, introduced within the present considerations. It should be pointed out that the isotope effect received within the above harmonic approximation (y = 0.0) is quite correct for low-temperature superconductors, however, it deviates significantly from small values observed in high-temperature materials. Therefore, we propose to include anharmonic approach (equation (6)) in order to receive more reasonable values of c(. Results are presented for the critical temperature and the isotope effect as functions of hole concentraton for different anharmonic hole-lattice coupling parameter y. Both quantities depend on changes of y, but to different extent. The changes of critical temperature are not too great in contrast to isotope effect parameter, which can be substantially lowered by an appropriate choice of y. These results are presented in Fig. 3 and Fig. 4, respectively. The case y = 0.003 seems to be very interesting because it
Fig. 2. Isotope effect parameter CL(equation (10)) as a function of hole concentration n for two values of the parameter fi (full: 0.01; broken: 0.02). Same parameter as in Fig. 1.
ISOTOPE EFFECT AND HOLE SUPERCONDUCTIVITY O.-e
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I I .:.
. . ..
. . .. /
: ’ 0.5
Fig. 4. Dependence of the isotope effect parameter a (equation (10)) on hole concentration n for some values of the parameter y as in Fig. 3. describes the case of relatively high maximum critical temperature and relatively small value of a, relevant to this temperature. Such a behaviour can be observed in many high-temperature superconductors. Finally, the isotope effect parameter a is considered as a function of the anharmonic coupling y. It is seen that a decreases with increasing anharmonicity. Moreover, the parameter a vanishes for critical value of anharmonicity y0 = /3(3#)‘.
Fig. 5. Isotope effect parameter a as a function of anharmonic hole-lattice parameter y for /I = 0.012 and 6 = 1.O. Other model parameters are the same as in Fig. 1. Isotope effect becomes weaker and weaker with increasing anharmonicity y and even disappears for y. = P(3#)-‘. An inverse isotope effect (a < 0) is predicted if y > yo. influence of the lattice is included in an anharmonic approximation proposed in the present paper. REFERENCES J.E. Hirsch, J.E. Hirsch (1989). J.E. Hirsch 122 (1989). R. Micnas,
An inverse isotope effect (a negative value of a) can be expected for y > yO. Such results were also reported for high-temperature superconductors. Taking the above results into account we can state that anharmonic effects are very important, especially for high-temperature materials. Of course, high critical temperatures and vanishing isotope effect can be explained within the purely “hole-like” model proposed by Hirsch [l-3, 5, 61. However, a common description of the low-temperature superconductors as well as the high-temperature ones is possible if
Phys. L&t. A134, 451 (1989). & S. Tang, Phys. Rev. B40, 2179 & F. Marsiglio, Phys. Lett. A140, J. Ranninger & S. Robaszkiewicz,
Phys. Rev. B39, 11653 (1989).
6. 7. 8. 9.
J.E. Hirsch & F. Marsiglio, presented during M*S-HTSC Conference, Stanford, CA, July 2328, 1989. J.E. Hirsch, Physica C 158, 326 (1989). F. Marsiglio, Physica C 160, 305 (1989). H. Bilz, H. Biittner, A. Bussmann-Holder & P. Vogl, Ferroelectrics 73, 493 (1987). A. Bussmann-Holder, A. Simon & H. Biittner, Phys. Rev. B39, 207 (1989).