SI9800021 179

Bipolaron theory of high-Tc oxides and some recent experiments

A.S. ALEXANDROV IRC in , University of Cambridge, U.K.

ABSTRACT

The 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 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 and provide direct evidence of the importance of the electron- interaction in metal oxides. Mihailovic et al.1 described the spectral shape of the photoconductivity with the small 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 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 . 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 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 . 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 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 (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. The value of the in -plane effective mass depends on the composition and increases with the oxygen depletion being in all samples of YBCO larger than 10me. This is consistent with the earlier optical estimations *. The out-of plane mass is more than one order of magnitude larger. These observations are in line with the polaronic mass enhancement.

4. C-axis transport Transport perpendicular to the CuO2 layers is important for the understanding of the nature of carriers. In optimally doped YBCO c-axis resistivity shows metallic like temperature behavior with the mean-free path comparable or slightly less than the interplane distance. If the interplane hopping integral is very small the normal state transport along the c-axis should be thermally activated. This is usually observed in strongly anisotropic Bi and T7-based copper oxides at temperature just above Tc because the (bi)polaron band in the c-direction is very narrow in these materials, the bandwidth below a few tens K. In the superconducting state a very small temperature dependent plasma gap was observed with the c-axis optical reflectivity by Uchida 10 1 (ujps ~ 50cm" in Lai^^SroieCuO^. The value of the c-axis plasma gap is below the estimated BCS gap 2A which is quite anomalous situation and has never been observed in conventional superconducting materials. The gap disappears above Tc, Fig.3, which plausibly corresponds to a plasma frequency of the condensate. We propose a simple interpretation of u)ps based on the Bogoliubov spectrum of a charged condensed Bose-gas:

ek = y/El + 2EkV(k)no(T) (3) where E^ is the anisotropic single bipolaron band dispersion. In the long-wave limit &|| = 0, k± —+ 0 V(k) is the three-dimensional Coulomb interaction V^k) ~ l/k2 and 184

the excitation spectrum Eq.(3) has a plasma -like temperature dependent gap

2 167rno(r)e (4) \

2 with m*c* = l/2t±d the effective bipolaron mass in the c-direction. The gap Eq.(4) depends on temperature, disappearing above Tc because only condensed bosons con- tribute to it.

5. Gap and superconducting order parameter

There are two gaps for charge excitations in the bipolaronic superconductor. The largest one is the binding energy of a single bipolaron of order O.leV. This gap is temperature independent on the scale compared with Tc, exists above Tc and its n value does not depend directly on the value of Tc. . The large (2A/TC ~ 7 - 8 for Tc ~ 90K) temperature and oxygen independent gap existing above Tc was found by Schlesinger et a/.12 and by Demuth et a/.13 in YBCO using the infrared reflectivity and the electron energy loss spectroscopy. The second smaller gap is the plasma-like gap in the Bogoliubov spectrum, Eq.(3)

QT) (5)

This gap is closely related to the superconducting order parameter (n0), depends on temperature and disappears at T = Tc because the condensate disappears no(Tc) = 0. Its temperature dependence is influenced by the dimensionality of bosons. In an anisotropic crystal it depends also on the wave vector k according to the formula (3) and can be zero along particular directions if the anisotropy is large.

1 Lan Sr CuC> 2-x x 4 x=0.16 CO 0.8 \ two-fluid P 0.4 ^0.2 \ (b) 0 0 10 20 30 40 50 T(K) Fig.3. Plasma frequency of the condensate plotted as a function of temperature 10. 185

This anisotropy is important for the temperature dependence of the London pen- etration depth. In the bipolaronic superconductor the symmetry of the collective gap u>ps(k) should be distinguished from that of the 'internal' wave function of a single intersite bipolaron, which is determined by the relative orbital motion of two polarons. The latter is important for the Josephson tunneling but hardly affects the Bose-Einstein condensation. Therefore different experiments can measure two different symmetries.

6. Angle-resolved photoemission

It was argued that ARPES shows the well-defined Fermi-surface, which for 6 = 0.1 corresponds well with the plane-related Fermi surfaces calculated from band structure for YBa2Cu307-s- Aside from this, however, the dispersion is much more flat than that from the band structure as has been established in recent studies 14'15. I believe that the band narrowing can be explained by the small polaron theory of ARPES 16. The reduced intensity of the coherent (i.e. angle- dependent) contribution, the band narrowing and the broad incoherent background are due to the phonon (magnon) cloud which constitutes a small polaron. If bipolarons are formed the spectral weight is shifted down by the value A/2 from the chemical potential while the shape remains practically unchanged. That could be an explanation of the fact that a flat band observed with ARPES in YBa2Cu408 lies approximately 20meV below the chemical potential 15. That means that the bipolaron binding energy is about AOmeV in this material. In conclusion the AM model provides a simple explanation of the experimental observations discussed above as well as many others 5.

Acknowledgment

The author is thankful to Sir Nevill Mott and Yao Liang for their support. I appreciate the valuable discussions with A.Arko, P.Calvani, C.Castellani, T.Egami, H.Eschrig, J.Goodenough, D.Mihailovic, K.Muller, J.Ranninger, W.Reichardt and other participants of the International Workshop on high-Tc cuprates in Bled, Slovenia and the financial support from the Leverhulme Trust.

References

1. D. Mihailovic et a/, Phys.Rev. B42 (1990) 7989. 2. P. Calvani et al, Solid State Commun. 91 (1994) 113. 3. X. Zhang X and C.R.A. Catlow, J.Mater.Chem. 1 (1991) 233. 4. A.S. Alexandrov and N.F. Mott, J.Supercond.(US) 7 (1994) 599. 5. A.S. Alexandrov and N.F. Mott, 'High Temperature Superconductors and Other Superfluids' (1994) Taylor and Francis (London). 6. A.S. Alexandrov, A.M. Bratkovsky A M and N.F. Mott, Phys.Rev.Lett 72 (1994) 1734. 186

7. J.W. Loram et al, Phil.Mag. B65 (1992) 1405. 8. A.S. Alexandrov and J. Ranninger, Solid St.Commun.81 (1992) 403. 9. J.W. Loram et al, Phys.Rev.Lett. 71 (1993), 1740. 10. S. Uchida , J.Phys.Chem.Solids 53 (1992) 1603. 11. A.S. Alexandrov and D.K. Ray, Phil.Mag.Lett. 63 (1991) 295. 12. Z. Schlesinger et al, Physica C185 - 189 (1991) 57. 13 J.E. Demuth et al, Phys.Rev.Lett. 64 (1990) 603. 14.D.S. Dessau et al, Phys.Rev.Lett. 71 (1993) 2781. 15. A. Arko, Invited paper at the International Workshop on the Anharmonic Properties of High-Tc Cuprates (1994) Bled (Slovenia) unpublished. 16. A.S. Alexandrov and J. Ranninger, Phys.Rev. B45 (1992) 13109.