SI9800021 179 Bipolaron theory of high-Tc oxides and some recent experiments A.S. ALEXANDROV IRC in Superconductivity, University of Cambridge, U.K. ABSTRACT The electron heat capacity, the London penetration depth, symmetry of the order parameter and gaps in YBa2CusO7-6 as well as c-axis transport and recent photoemission spectra of high-rc copper oxides are discussed within the bipolaron scenario. We suggest that the intersite O~ — O~ hole bipolarons in the CuOi plane are responsible for the anomalous kinetics and thermodynamics. Copper electrons also form intersite singlets which however are immobile because their Brillouin zone is half of the original one and completely filled at zero doping. 1. Introduction - basic model of CuO2 plane The crucial question in our understanding of high-Tc copper oxides is the origin of their normal state. According to the Fermi liquid (FL) scenario these superconductors are metals with a large Fermi surface of copper electrons. Within this scenario the pronounced deviations from the canonical FL behavior are due to the Fermi surface anisotropy and a large damping . The existence of the Mott parent insulators suggests on the opposite side that high-Tc superconductors are doped semiconductors with hole carriers. Studies of photoinduced carriers in the dielectric 'parent' compounds like La2Cu0^ YBa-iCu^Oe and others confirm the formation of self-localised polarons and provide direct evidence of the importance of the electron-phonon interaction in metal oxides. Mihailovic et al.1 described the spectral shape of the photoconductivity with the small polaron transport theory. They also argued that the similar spectral shape and systematic trends in both photoconductivity of optically doped dielectric samples and infrared conductivity of chemically doped high—Tc oxides indicate that carriers in the concentrated (metallic) regime retain much of the character of carriers in the dilute (photoexcited) regime implying that the charge carriers in the normal state of high-Tc cuprates are polarons or bipolarons. The direct evidence for polarons in n-type superconducting copper oxides was provided by Calvani and co-workers 2. Different types of possible polaron pairing in La2Cu0^ were investigated by Zhang and Catlow 3 by computer simulation techniques based on lattice energy minimisation. The binding energy of the pair was found to be strongly related to the pair distance and geometry of the pair. The nearest neighbor pairing is energetically favorable with the binding energy O.l\9eV and 0.059eV for different O~ — O~ bipolarons. Taking this into account Alexandrov and Mott (AM) 4>5 proposed a simple model for the copper-oxygen plane, which is a key structural element of all copper-based 180 high-Tc oxides. Our assumption is that all electrons are bound in small singlet or triplet intersite bipolarons stabilised by the lattice and spin in-plane and out-of- plane distortions. Because the undoped plane has a half-filled Cu&d? band there is no space for intersite bipolarons to move. Their Brillouin zone is half of the original electron one and completely filled with hard-core bosons. Hole pairs, which appear with doping, have enough space to move and they are responsible for the low-energy charge excitations of the CuOi plane. Above Tc a material such as YBCO contains a non-degenerate gas of these hole bipolarons in a singlet, or in a triplet state. Triplets are separated from singlets by the spin gap J and have a slightly lower mass due to the lower binding energy. The main part of the electron-electron correlation energy (Hubbard U) and the electron-phonon interaction is included in the binding energy of bipolarons and in their band-width renormalisation. To describe kinetic properties including the metal-insulator transition and the Hall effect one should also take into account the Anderson localisation of bipolarons by disorder 4 . We suggest that in the doped cuprates the low-energy band structure consists of two bosonic bands (singlet and triplet), separated by the singlet-triplet exchange energy J, estimated to be of the order of a few tens meV, which is responsible for the spin gap effects. The bipolaron binding energy is assumed to be large, A >> T and therefore unbound polarons are irrelevant in the temperature region under consideration except for overdoping. In this paper I argue that several recent observations are compatible with the AM model. 2. Normal state heat capacity The striking feature in which the superfluid Bose liquid differs significantly from the superfluid Fermi liquid is the specific heat near the transition temperature. Bose liquids (or more precisely He4) show the characteristic A like singularity in the specific heat. Superfluid Fermi liquids (He3 and the BCS superconductors), on the contrary, exhibit a sharp second order transition accompanied by a finite jump in the specific heat. It has been established beyond doubt that in high Tc superconductors the anomaly in the specific heat spreads to about \T — Tc\/Tc ~ 0.1 or larger, and the estimations with the canonical gaussian fluctuation theory yield an unusually small coherence volume, comparable or even less than the unit cell volume 7. That means that the overlap of pairs is small (if any). One can rescale the absolute value of the specific heat and the temperature to compare the experimentally determined specific heat of He4 with that of high-Tc oxides. The specific heat per boson in the high Tc oxides practically coincides with that of He4 (n^ = 1) in the entire region of the A singularity. In fact for the 2223 compound the A shape is experimentally better verified than in He4 itself because of the fifty times larger value of the critical temperature 8. In general it is difficult to analyse the specific heat above Tc because Tc is so high that the normal state temperature regime is dominated by phonons. However 181 Loram and co-workers 9 using the differential technique determined the electronic contribution in the normal state above Tc up to room temperature and found an electronic entropy rather large arid linear in temperature in optimally doped YBCO. In the underdoped samples they measured smaller entropy and reported unexpected deviations from linearity plausibly connected with the spin gap, Fig. 1 (insert). One can calculate the entropy of the extended bosons above Tc with the classical expression for the Bose gas >ext - xy)\ l-xy x(l-xy)J (1) where the density of states (l/2t) in the bipolaron band is assumed to be energy independent because of low (2 + e) dimensionality. The chemical potential is pinned to the mobility edge (y = exp(fi/T) ~ 1) as discussed in Ref.6. This expression predicts a practically linear entropy with the absolute value fitting well the experimental observations, Fig.l. S/kB 02 - 0.5 0.4 0.3 0.2 0.1 50 100 150 200 250 300 T(K) Fig.l. Theoretical entropy of extended bosons (2t = 1000/sT) and carrier entropy in (insert 9); different curves correspond to different values of x = 1 — 8. The localised bosons contribute also to the electronic entropy; however their con- tribution depends on the Coulomb repulsion and can be suppressed. The lowering 182 in the value of the entropy above Tc with the increase in oxygen deficiency (Fig.l) is due to the localisation by a random field, which is also responsible for the drop of Tc and of the heat capacity jump in Zn doped samples. The deviations of S from linearity in the underdoped samples are explained within our model by the existence of thermally excited triplet bipolarons, which are responsible for the spin gap effect in NMR, neutron scattering and resistivity ( for details see Ref. 5). A large value of the entropy above Tc and its lowering in underdoped samples are incompatible with the FL scenario. 3. London penetration depth A weak magnetic field penetrates into a charged Bose-gas to a depth 2 1 4ire* n0(T) (2) where for bipolarons e* = 2e and m** ~ 1/22. * * 20 lit I I I I | I I I I | I-H I | I II I | I II UJ--M M | lit II 700 - E c 500 * ti 300 •+++ I I | H M | IN M [ I I M | I I I I | I I I n 1.4 S U U o 0.6 0.2 ... I I . I ,. I .... I .... I .... I I I I , I ... 0 0.1 0.2 0.3 0.4 0.5 0.6 8 Fig.2. In-plane and out- of- plane effective masses of the bipolaron as measured from the London penetration depth and the density of extended bosons (T = 0) determined from the Hall effect 183 (Athanassopoulou N and Cooper J R 1994, unpublished). In a near 2d Coulomb Bose-gas with the Coulomb field parameter rs < 1 the condensate density is linear in temperature no ~ (1 — T/Tc) and so 1/A^ . At low temperature three dimensional corrections to the Bogoliubov spectrum becomes important and can be responsible for the deviation of A^2(T) from linearity. The general consensus is that the London penetration depth in copper based high-Tc oxides does not follow the BCS formula and in a wide temperature range has a power 2 u law behavior Xjf ~ 1 — (T/Tc) with the exponent v close to unity in YBCO in the intermediate temperature range. Extrapolating the Hall constant RH above Tc, Fig.2 to zero temperature one determines the density of extended bosons at T = 0, which is close to the condensate density for weakly interacting bosons. With n0 = rc//(0) = l/i?//2e and with the experimentally determined low temperature (T ~ AK) values of the in-plane and out-of- plane A#(0) one obtains the low-frequency mass enhancement both in the ab and c directions, Fig.2.