Bipolaron Anisotropic Flat Bands, Hall Mobility Edge, and Metal-Semiconductor Duality of Overdoped High-Tc Oxides

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PHYSICAL REVIEW B VOLUME 53, NUMBER 5 1 FEBRUARY 1996-I

Bipolaron anisotropic ﬂat bands, Hall mobility edge, and metal-semiconductor duality of overdoped high-Tc oxides

A. S. Alexandrov Department of Physics, Loughborough University of Technology, Loughborough LE11 3TU, United Kingdom and Interdisciplinary Research Centre in Superconductivity, University of Cambridge, Madingley Road, Cambridge CB3 0HE, United Kingdom ͑Received 30 March 1995; revised manuscript received 1 September 1995͒ Hole bipolaron band structure with two ﬂat anisotropic bands is derived for oxide superconductors. Strong anisotropy leads to one-dimensional localization in a random ﬁeld which explains the metal-like value of the Hall effect and the semiconductorlike doping dependence of resistivity of overdoped oxides. Doping dependence of Tc and ␭H(0) as well as the low-temperature dependence of resistivity, of the Hall effect, Hc2(T) and robust features of angle-resolved photoemission spectroscopy of several high-Tc copper oxides are explained.

I. INTRODUCTION doped oxides is inconsistent with the Fermi-liquid picture14,11 and with the spin-charge separation picture.9 In

The ‘‘parent’’ Mott insulators suggest that high-Tc super- particular, semiconductorlike scaling with x of dc conductiv- conductors are in fact doped semiconductors. There is now a ity of La2ϪxSrxCuO4 for a wide temperature and doping 10 growing consensus that the dopant-induced charge carriers in region and the linear low-temperature resistivity ␳ as well 1 as the linear Hall effect in overdoped Tl Ba CuO ,11 have high-Tc oxides exhibit a significant dressing due to spin, 2 2 6 charge,2 and lattice3 fluctuations. Studies of strongly corre- been observed. lated models like the Holstein t-J model show that the criti- Sometimes it is argued that unusual features of overdoped cal electron-phonon coupling strength for polaron formation high-Tc oxides can be understood as a result of a strong is considerably reduced by an antiferromagnetic exchange magnetic pair breaking if the spin-flip mean free path ls is shorter than the coherence length . However high-T ox- interaction compared to that in the uncorrelated model.4 ␰0 c ides are at a ‘‘clean’’ limit, the mean free path l is at least 20 There is also a growing experimental evidence that polarons times larger than ␰ .12,11 This makes the magnetic pair break- and bipolarons are carriers in high-temperature supercon- 0 ing irrelevant for high-T because the strong inequality l Ӷl ductors. In particular, studies of photoinduced carriers in di- c s is unrealistic; normally lsӷl. electric parent compounds like La2CuO4 and YBa2Cu3O6 as In this paper the oxygen hole energy dispersion is studied well as the infrared conductivity of metallic compounds con- 5,6 with the model electron-phonon interaction taking into ac- firm the formation of self-localized polarons. A direct evi- count the self-trapping effect and the attraction between in- dence for small polarons in doped copper oxides has been provided by Calvani et al.7 with infrared spectroscopy. An oxygen isotope effect on the Neél temperature has been found in La2CuO4 suggesting the oxygen-mass dependence of superexchange J due to the small polaron band narrowing.8 The crucial role of apex oxygen ions in the polaron formation has been verified for La-Sr-Cu-O ͑LSCO͒. Assuming that the carriers in a doped Mott insulator are spin-lattice bipolarons moving in a random potential Mott and the author explained several unusual features of the low- energy kinetics and thermodynamics of underdoped copper oxides.3 On the other hand, it has been suggested that optimally doped and overdoped oxides are metals with a large Fermi surface as follows from angle-resolved photoemission spectroscopy ͑ARPES͒, the T2 temperature dependence of resistivity, and from the small value of the Hall effect. A three- band model involving a strong oxygen-copper hybridization ͓Fig. 1͑a͔͒ has been studied by several authors as a relevant one. However, the recent progress in elucidating the normal 9,10 x state of the prototypical cuprate La2ϪxSrxCuO4 , and of FIG. 1. Counterplot of the x bipolaron dispersion Ek. Dark re- 11 y overdoped Tl2Ba2CuO6 , as well as the unusual Hc2(T)of gions correspond to the bottom of the band. Ek energy surfaces are Tl2Ba2CuO6 ͑Ref. 12͒ and Bi2Sr2CuOy ͑Ref. 13͒ led several obtained by ␲/2 rotation. Three-band (t-J) model ͑a͒ and two-band authors to the conclusion that low-energy kinetics of over- apex bipolaron model ͑b͒.

2 Ϫ1 Ϫ1 plane and apex holes. The role of copper in electronic trans- with the dimensionless coupling constant ␣ϭe (⑀ ϱ Ϫ⑀ 0 ) port is signiﬁcantly reduced by bipolaron formation. A ͱm/2␻, introduced by Fro¨hlich,17 and the momentum- simple two-band model is derived with a strong a-b anisot- independent optical phonon frequency ␻qϭ␻. The lattice ropy of two narrow bipolaron oxygen bands. The effective- polarization is coupled with the electron density, therefore mass anisotropy is 4 or larger. A random potential increases the interaction is diagonal in the site representation and the the anisotropy, so low-energy carriers are effectively local- coupling constant does not depend on the particular orbital. ized in one direction for a wide range of doping. Then there In doped oxides optical phonons are partially screened. Then is a ‘‘Hall mobility edge’’ EcH . The states below EcH con- molecular ␻qϭ␻0 and acoustical ␻qϭsq phonons contribute tribute to the longitudinal conductivity rather than to the also to the interaction with the coupling constants ␥ϵ␥0 and transverse one. A quantitative explanation for a high value of ␥ϳ1/ͱq, respectively. The canonical displacement transfor- the Hall density in La Sr CuO is proposed compatible 2Ϫx x 4 mation Sϭexp͕͚q, jn j͓u j͑q͒dqϪH.c.͔͖ eliminates an essential with the x scaling of dc conductivity. The doping dependence part of the electron-phonon interaction. The transformed 3 of Tc and of ␭H both in underdoped and overdoped samples, Hamiltonian is given by the low-temperature dependence of ␳, RH , and Hc2 of overdoped Tl2Ba2CuO6 and the robust features of ultrahigh en- 15,16 ˜ Ϫ1 ergy resolution angle-resolved photoemission spectra HϭSHS ϭ͑TpϪEp͚͒ ni͑p͒ϩ͑TdϪEd͚͒ ni͑d͒ ͑ARPES͒ are explained with bipolarons. i͑p͒ i͑d͒

† † ϩ ␴ˆ ijci cjϩ ␻qdqdq II. TWO-BAND BIPOLARON MODEL ͚iÞj ͚q The hole in a square oxygen-copper lattice hops directly 1 from one oxygen ion to its oxygen neighbor due to an over- Ϫ ͓2␻qui͑q͒u*j ͑q͒ϪVij͔ninj . ͑4͒ 2 q͚,i,j lap of p oxygen orbitals, Fig. 1͑b͒, or via a second-order indirect transition involving d orbitals of copper, Fig. 1͑a͒. The ﬁrst oxygen (p) and the second copper (d) diagonal While the direct tunneling is linear in the oxygen-oxygen terms include the polaronic level shift, which is the same for hopping integral T ppЈ, the indirect transition is of the second oxygen and copper ions order in the oxygen-copper hopping T pd . There is an as- sumption within the three-band model that the indirect hop- 2 ping is more important because of a shorter copper-oxygen E pϭEdϭ͚ ͉u j͑q͉͒ ␻q . ͑5͒ distance compared with an oxygen-oxygen one and of the q relatively small size of the charge-transfer gap EgӍ1–2 eV. The transformed hopping term involves phonon operators A strong Fro¨hlich-type interaction modiﬁes essentially the energy spectrum. To show this I consider the model Hamil- ˆ † tonian ␴ijϭTij exp ͚ ui*͑q͒dqϪH.c. exp ͚ u j͑q͒dqϪH.c. . ͩ q ͪ ͩ q ͪ ͑6͒ † † Hϭ Tijci cjϩ ␻qnj͓uj͑q͒dqϩH.c.͔ϩ ␻qdqdq ͚i, j ͚q,j ͚q There are two major effects of the electron-phonon interaction. One is the band narrowing due to a phonon cloud 1 around the hole. In case of Egӷ␻ the bandwidth reduction ϩ ͚ Vijninj , ͑1͒ 2 i, j factor is the same for the direct t ppЈ and the second-order via copper t(2) oxygen-oxygen transfer integrals see the Appen- ppЈ ͑ where Tij determines the bare band structure in the site rep- dix͒ resentation; ci , c j are hole annihilation operators for oxygen † 2 or copper sites i, j; n jϭc j c j is the number operator, Vij is Ϫg t ϵ͗0͉␴ˆ ͉0͘ϭT e ppЈ, ͑7͒ the direct Coulomb repulsion, which does not include the ppЈ ppЈ ppЈ on-site term iϭ j for parallel spins; ␻q ,dq are the phonon 2 0 ˆ ˜ 0 T 2 frequency and annihilation operator, respectively. The ͑2͒ ͗ ͉␴pd͉␯͗͘␯͉␴dpЈ͉ ͘ pd Ϫg tpp ϵ͚ Ӎ e ppЈ, ͑8͒ electron-phonon coupling in the site representation for elec- Ј ␯ E0ϪE␯ Eg trons is given by where ͉␯͘, E␯ are eigenstates and eigenvalues of the trans- 1 formed Hamiltonian, Eq. ͑4͒ without the hopping term, ͉0͘ iq•mj u j͑q͒ϭ ␥͑q͒e , ͑2͒ the phonon vacuum, and the reduction factor is ͱ2N 1 2 2 where N is the number of cells in the normalized volume g pp ϭ ͉␥͑q͉͒ ͕1Ϫcos͓q ͑mpϪmp ͔͖͒. ͑9͒ Ј 2N͚q • Ј N⍀, mj is the lattice vector. Oxides are strongly polarizable materials, so coupling with optical phonons dominates in the Because the nearest-neighbor oxygen-oxygen distance in electron-phonon interaction copper oxides is less than the lattice constant the calculation 2 yields a remarkably lower value of g Ӎ0.2E /␻ than one iͱ8␲␣ ppЈ p ␥͑q͒ϭϪ ͑3͒ can expect with a naive estimation (ӍE p/␻) ͑see the Appen- ͱ⍀͑2m␻͒1/4q dix͒. 53 BIPOLARON ANISOTROPIC FLAT BANDS, HALL MOBILITY... 2865

respect to the transfer integral responsible for the bipolaron tunneling to the nearest-neighbor cell mϩa is

1 apex 2 tϭ͗m͉H˜ ͉mϩa͘ϭ T eϪg , ͑12͒ 4 ppЈ

where Tapex is the single hole hopping between two apex ppЈ ions, and

1 FIG. 2. Apex bipolaron tunneling between two copper 2 2 g ϭ ͉␥͑q͉͒ ͓1Ϫcos͑qxa͔͒ ͑13͒ polyhedra. 2N͚q

The other effect of the electron-phonon coupling is the is the polaron narrowing factor. Here a is the in-plane lattice attraction between two polarons given by the last term in Eq. constant, which is also the nearest-neighbor apex-apex dis- ͑4͒. For the Fro¨hlich interaction the polaron level shift deter- tance. As a result the hole bipolaron energy spectrum in a mined by Eq. ͑5͒ is tight-binding approximation consists of two bands Ex,y formed by the overlap of px and py apex polaron orbitals, 2 respectively, Fig. 1͑b͒: qDe Ϫ1 Ϫ1 E pӍ ͑⑀ϱ Ϫ⑀0 ͒, ͑10͒ ␲ x EkϭϪt cos͑kx͒ϩtЈcos͑ky͒, ͑14͒ 2 1/3 where qdϭ(6␲ /⍀) is the Debye momentum. With the y static and high-frequency dielectric constants ⑀0ӷ⑀ϱӍ5 and EkϭtЈ cos͑kx͒Ϫt cos͑ky͒, ͑15͒ Ϫ1 qDӍ0.7 Å one estimates E pӍ0.64 eV and with ␻ϭ0.06 2 where the in-plane lattice constant is taken to be aϭ1, t is eV, g Ӎ2. As a result a large attraction between two po- ppЈ the renormalized hopping integral, Eq. ͑12͒ between p orbit- larons of the order of 2E 1 eV is possible accompanied by pӍ als of the same symmetry elongated in the direction of the the band mass enhancement less than one order of magni- hopping (pp␴), and tЈ is the renormalized hopping integral tude. This is in contrast with some assessments of the bipo- in the perpendicular direction (pp␲).19 Their ratio t/tЈϭ laronic mechanism of high-T superconductivity based on c Tapex/T apexϭ4 as follows from the tables of hopping inte- the incorrect estimation of the effective mass. ppЈ ЈppЈ grals in solids.20 Two different bands are not mixed because The polaron-polaron interaction is the sum of two large apex contributions of the opposite sign, last term in Eq. ͑4͒, which T p ,p ϭ0 for the nearest neighbors. The random potential x Јy generally are large compared with the reduced polaron band- does not mix them either if it varies smoothly on the lattice width. This is just the opposite regime to that of the BCS scale. Consequently, one can distinguish x and y bipolarons superconductor where the Fermi energy is the largest. In that with a lighter effective mass in the x or y direction, respec- case one can expect real-space bipolarons. Different types of tively. The apex z bipolaron, if formed is ca. four times less bipolarons in La2CuO4 were investigated by Zhang and mobile than the x and y bipolarons. The bipolaron bandwidth 18 Catlow with computer simulation techniques based on the is of the same order as the polaron one, which is a specific minimization of the ground-state energy E0 of the Hamil- feature of the intersite bipolarons. tonian Eq. ͑4͒ without the hopping term. The intersite pairing The polaronic features of the energy spectrum and bipo- of the in-plane oxygen hole polaron with the apex one was laron formation are in line with the extremely flat anisotropic found energetically favorable with the binding energy bands measured recently with ARPES ͑Refs. 16 and 15͒ in ⌬ϭ0.119 eV. Obviously this apex bipolaron can tunnel from several copper high-Tc oxides which display at least an order one cell to another via a direct single polaron hopping from of magnitude less dispersion than the first-principles band- one apex oxygen to its apex neighbor, Fig. 2. The bipolaron structure methodology can provide.16 I believe that this flat- hopping integral t is obtained by projecting the Hamiltonian, ness is due to the polaron narrowing of the band, Eq. ͑12͒ Eq. ͑4͒ onto the reduced Hilbert space containing only empty and the anisotropy is due to the remarkable difference of px or doubly occupied elementary cells. The wave function of overlaps in x and y direction, respectively. If bipolarons are the apex bipolaron localized, say in the cell m is written as formed the spectral weight is shifted down by half of the bipolaron binding energy with respect to the chemical poten- 4 † † tial. This could provide an explanation why the flat band ͉m͘ϭ Aici capex͉0͘, ͑11͒ i͚ϭ1 observed with ARPES in YBa2Cu4O8 does not cross the Fermi level. It lies approximately 20 meV below the chemi- 16 where i denotes the px,y orbitals and spins of the four plane cal potential which means that the bipolaron binding en- † oxygen ions in the cell m, Fig. 2 and c apex is the creation ergy is about ⌬Ӎ40 meV in this material. The ‘‘static’’ cal- operator for the hole on one of the three apex oxygen orbitals culations by Zhang and Catlow18 of the bipolaron binding with the spin, which is same or opposite to the spin of the energy depends on details of the perovskite crystal struc- plane hole, depending on the total spin of the bipolaron. The tures. In their modeling of small bipolarons in doped high-Tc 3ϩ Ϫ probability amplitudes Ai are normalized by the condition La2CuO4 they treated holes as Cu or O species placed in ͉Ai͉ϭ1/2 because only four plane orbitals px1, py2, px3, and the dielectric matrix with the CuO6 unit, Fig. 2. The energy py4 are relevant within the three-band model. The matrix of a region of the crystal surrounding the hole or the hole element of the Hamiltonian Eq. ͑4͒ of the first order with pair is then minimized with respect to the coordinates of the 2866 A. S. ALEXANDROV 53 ions within the region containing ϳ200–300 ions. The re- III. THE HALL CONSTANT sponse of the more distant regions of the crystal is calculated using approximate procedures based on continuum model It is well known that the effective-mass anisotropy of en- employing the relative permittivity of the material. With a ergy ellipsoids in a square ͑or cubic͒ lattice diminishes the proper choice of the interatomic potentials one can find the value of the Hall constant as in Si or Ge. In the presence of binding energy of the small bipolaron of different geometry disorder an x bipolaron can be localized in the y direction with the accuracy within 0.01 eV. The pairing was studied tunneling practically freely along x and a y bipolaron can be for a variety of separations in three types of possible bipo- localized in the x direction remaining free along y.21 That laron ͑Cu3ϩ-Cu3ϩ,Cu3ϩ-OϪ, and OϪ-OϪ pairs͒. Intercopper gives a very low metalliclike RH which presumably is due to and copper-oxygen intersite bipolarons are unstable both for bipolarons with the energy above the Hall mobility edge, interlayer and intralayer pairing, where the binding energy is EϾE . At the same time the dc conductivity remains pro- negative for all separations studied. However, three stable cH portional to the number of bipolarons above the mobility OϪ-OϪ configurations were found. For in-plane configura- edge, which lies below, E ϽE . To support this conclusion tion the bipolaron is bound by ϳ0.06 eV, whereas apex bi- c cH quantitatively one can adopt the effective-mass approxima- polaron configuration is bound by ϳ0.12 eV. These two bound oxygen pairs are situated at the nearest-neighbor sites tion for a large part of the Brillouin zone near ͑0,␲͒ for the x ͑dϭ2.66 Å͒ and next-nearest-neighbor sites ͑dϭ3.11 Å͒. and ͑␲,0͒ for the y bipolaron, Fig. 1, There is also a slightly bound bipolaron with a binding energy of 0.001 eV at dϭ3.58 Å. When d is larger than 3.81 Å k2 k2 all the configurations are energetically unfavorable. Conse- x,y x y Ek ϭ ϩ ͑16͒ quently, the binding energy of small bipolarons is strongly 2mx,y 2my,x related not only to the distance of the pair but also to the detailed geometry of the site where the polaron is situated and to the dielectric properties of the matrix. It is not surpris- with kx,y taken relative to the band bottom positions and ing that the bipolaron binding energy is not universal among mxϭ1/t, myϭ4mx . The Boltzmann equation in the relax- different copper oxides. ation time approximation yields

n 2 n 2 n 2 n n 2 n kx͔͒ץkyץ/Ek ץkx͒͑ץ/Ekץky͒͑ץ/Ekץky͒ Ϫ͑ץ/Ekץkx͒͑ץ/Ek ץ͚k,nϭx,y f Ј͑Ek͓͒͑ RHϳ n n 2 2 , ͑17͒ ͔ kx͒ץ/Ekץ͚k,nfЈ͑Ek͓͒͑

n where f Ј(Ek) is the derivative of the distribution function. Here ␥0 is the characteristic binding energy independent of Counting bipolarons (n0), with the energy above EcH in the the dopant density. That yields Bϭ2/␥ͱ␲ and numerator and above Ec in the denominator of Eq. ͑17͒ one obtains n0ϭx/2ϩerf͑␬/ͱx͒Ϫ1 ͑21͒

with ␬ϭtЈ/␥0 . The number of bipolarons, n0ϩn1 , above the 4mxmyn0 2eRHϭ 2 , ͑18͒ mobility edge Ec contributing to the longitudinal conductiv- ͓͑mxϩmy͒n0ϩmyn1͔ ity remains practically equal to the chemical density x/2 in a wide range of ␬ which can be verified with Eq. ͑19͒ replac- where n0ϭx/2ϪnL is the number of bipolarons with the en- ing tЈ for tϭ4tЈ. As a result, the Hall density n ϭ1/2eR to ergy above EcH , which are free in both directions; nL is the H H number of bipolarons localized at least in one direction, and the chemical density ratio is given by n1 is the number of bipolarons localized only in one direc- 2 nH ͓5xϩ2 erf͑␬/ͱx͒Ϫ2͔ tion. The number of bipolarons per cell localized at least in ϭ ͑22͒ one direction is proportional to the number of random poten- x/2 16x͓xϩ2 erf͑␬/ͱx͒Ϫ2͔ tial wells with the depth U larger than tЈ z 2 with erf(z)ϭ(2/ͱ␲)͐ 0exp͑Ϫ␰ ͒d␰. The agreement with ex- ϪtЈ periment is almost perfect for ␬ϭ0.57, Fig. 3. Due to the n ϭB exp ϪU2/ 2 dU. 19 L ͑ ␥ ͒ ͑ ͒ mass anisotropy the low-temperature physical density nH re- ͵Ϫ ϱ mains ϳ1.6 times larger than the chemical x/2 even for low The coefficient B is determined by the condition that all doping when nLӶx/2. The dc conductivity scales with x in 10 states of the Brillouin zone should be localized (nLϭ1) if overdoped samples as observed because n0ϩn1Ӎx/2 for the random potential is very large, ␥ӷtЈ. The average depth all x. On the contrary, the density n0 of carriers extended in ␥ of random wells is proportional to the relative fluctuation both directions falls rapidly in overdoped samples, Fig. 3 of the dopant density, which is the square root of the mean ͑inset͒, due to increasing random potential fluctuations, pro- density x ͑Ref. 22͒ portional to ͱx. The mass anisotropy of the order of 4 can be seen commonly in doped semiconductors. However the an- ␥ϭ␥0ͱx. ͑20͒ isotropy increases rapidly in overdoped samples. In fact, the 53 BIPOLARON ANISOTROPIC FLAT BANDS, HALL MOBILITY... 2867

The critical temperature of the superﬂuid phase transition in 2ϩ⑀ dimensions is proportional and the London penetra- tion depth squared is inversely proportional to the density n0 of delocalized bosons. Therefore

Tcϳxϩ2 erf͑␬/ͱx͒Ϫ2 ͑23͒

and

2 1 ␭H͑0͒ϳ . ͑24͒ xϩ2 erf͑␬/ͱx͒Ϫ2

With Eqs. ͑23͒ and ͑24͒ one can easily explain the doping dependence of Tc(x) in superconducting oxides as well as 2 the so-called ‘‘Uemura’’ plot Tcϳ1/␭ H veriﬁed experimen- FIG. 3. The ratio of the Hall nHϭ1/2eRH to chemical x/2 den- tally in underdoped and overdoped samples, Fig. 3 ͑inset͒. sities in La Sr CuO as a function of doping compared with ex- 2Ϫx x 4 As we have shown earlier24 the density of delocalized periment ͑Ref. 9͒ at 40 K. The size of the experimental circles bosons depends on temperature, increasing linearly for TӶ␥ includes an error bar due to temperature dependence of RH below 50 K and the uncertainty in the oxygen content. Theoretical depen- in a ‘‘single-well–single-particle’’ approximation, n0(T) ϭn0(0)ϩTnL ln 2/␥. That explains a linear increase of RH dence of the density of extended bosons n0 ,ofTc and of the pen- 11 etration depth ͑in relative units͒ on doping ͑inset͒. with increasing temperature observed by Mackenzie et al. in overdoped Tl2Ba2CuO6 starting from the mK scale up to ϳ30 K as follows from Eq. ͑18͒. The linear temperature Hall to chemical density ratio, Fig. 3 is a measure of this 11 anisotropy. At temperatures compared or higher than the bi- dependence of resistivity at low temperatures is explained by the fact that the number of unoccupied potential wells is polaron binding energy bipolarons coexist with unbound 24 thermally excited polarons, which contribute also to the proportional to temperature as the number of extended transport. In-plane polaronic bands are not so anisotropic as bosons. Less screened they provide a strong temperature- the apex bipolaron ones. That can explain why the Hall con- dependent elastic scattering of bipolarons. With increasing temperature the boson-boson scattering dominates and the stant in LSCO depends less on doping at high temperatures 2 compared with the low-temperature values. resistivity becomes proportional to T as observed in overdoped oxides. One can also describe the unusual temperature depen- IV. BIPOLARONS AND CRITICAL PARAMETERS dence of H of a ‘‘low T ’’ overdoped Tl Ba CuO ,12 as OF HIGH-T COPPER OXIDES c2 c 2 2 6ϩ␦ c the critical ﬁeld H* of the Bose-Einstein condensation of charged bosons. Starting with the linearized stationary equa- The coherence volume determined with the heat-capacity tion for the macroscopic condensate wave function ␺0͑r͒ measurements near Tc in many copper oxides is comparable or even less than the unit-cell volume.23 That favors a 1 charged 2e Bose liquid of small bipolarons as a plausible Ϫ ٌ͓Ϫ2ieA͑r͔͒2ϩU ͑r͒ ␺ ͑r͒ϭ␮␺ ͑r͒, ͑25͒ microscopic model of the superconducting state.3 ͩ 2m im ͪ 0 0

FIG. 4. Upper critical field of doped Tl2Ba2CuO6ϩ␦ ͑Ref. 12͒͑points͒ compared with the critical field of the Bose-Einstein condensation, Eq. ͑26͒ from 15 mK up to TcӍ20 K; inset represents the low-temperature part from 15 mK up to Ӎ2K. 2868 A. S. ALEXANDROV 53 where A͑r͒, Uim͑r͒, and ␮ are the vector, random, and envisage oxygen holes as heavy spin-lattice polarons sur- chemical potentials, respectively, one arrives at25 rounded by lattice and copper spin-polarized regions. Then both underdoped and overdoped high-Tc oxides are doped 3/2 semiconductors with oxygen ͑bi͒polarons as carriers partly 1Ϫ2nL͑t͒/x H*͑T͒ϭconst Ϫͱt . ͑26͒ localized by disorder. Another conclusion is that an estima- ͩ t͓1Ϫ2nL͑1͒/x͔ ͪ tion of the small ͑bi͒polaron mass with the dispersionless Holstein model leads to an erroneous conclusion that small Here tϭT/Tc , where Tc is the experimental critical tempera- bipolarons are immobile. Taking into account the dispersion ture, x is the chemical polaron density determined in of the electron-phonon coupling constant and the perovskite Tl2Ba2CuO6ϩ␦ by the excess oxygen content ␦, xϭ2␦, and crystal structure one obtains significantly less mass enhance- nL(t) is the number of localized bipolarons below EcH . ment compared with this estimation and at the same time the From Fig. 3 ͑inset͒ x/2ϪnL is very small in strongly over- polaron binding energy sufficient to overcome the intersite doped samples. In fact, at zero temperature the condition direct Coulomb repulsion. 27 2nL(0)/xϭ1 is satisfied because each bipolaron is localized This work is motivated by Mott’s idea that the localiza- on the excess oxygen ion. In the ‘‘single-well–single- tion of carriers in a random potential is crucial to our under- particle’’ approximation the number of localized bipolarons standing of low-energy kinetics of high-T oxides which is 24 c is determined by receiving now an overwhelming experimental support.3

0 NL͑⑀͒ ACKNOWLEDGMENTS nL͑t͒ϭ d⑀ , ͑27͒ ͵Ϫϱ exp͑⑀/T͒ϩ1 The author is grateful to Sir Nevill Mott, Y. Liang, J. Wheatley, A. Bratkovsky, N. Hussey, V. Kabanov, and P. where NL(⑀) is the density of localized states. The positive Kornilovich for valuable discussions. Victor Kabanov’s help curvature of H*(T) on the temperature scale of the order of in numerical calculations and enlightening comments by Tc does not depend on the particular shape of NL(⑀). How- Andy Mackenzie on the properties of overdoped materials ever, at mK temperatures shallow potential wells are impor- are highly appreciated. tant. Therefore the low-temperature behavior of H*(T)is sensitive to the shape of N (⑀) just below the mobility edge. L APPENDIX One can model Here the ﬁrst- ͓Eqs. ͑7͒ and ͑12͔͒ and the second-order e⑀/␥ ͓Eq. ͑8͔͒ phonon averages and the reduction factors are cal- NL͑⑀͒ϭ0.5nL͑0͒ ϩ␦͑⑀ϪE0͒ , ͑28͒ culated using the standard technique ͑see Ref. 3 and refer- ͫ ␥ ͬ ences therein͒. The direct hopping is given by to imitate both the discrete levels with the energy E0 and the exponential shallow tail due to the randomness of the impu- * † t ppЈϭTppЈ͗0͉exp ͚ u p ͑q͒dqϪH.c. rity potential. Then one can quantitatively describe the ex- ͩ q ͪ perimental Hc2(T) with Eq. ͑26͒ and ␥/Tcϭ0.13 and E0/Tcϭ0.3 for three decades of temperature, Fig. 4. This 13 ϫexp ͚ u pЈ͑q͒dqϪH.c. ͉0͘. ͑A1͒ equation was also applied by Osofsky et al. to describe the ͩ q ͪ Hc2(T)ofBi2Sr2CuOy with an excellent agreement for the 26 critical temperature. With the help of eAϩBϭeAeBeϪ[AB]/2 for any operators A, B with a c-number commutator one obtains V. CONCLUSIONS

2 In summary, the oxygen hole bipolaron bands of high-Tc Ϫg * † t ppЈϭTppЈe ppЈ͗0͉exp ͚ u p ͑q͒dq oxides are derived manifesting a remarkable flatness and ͩ q ͪ effective-mass anisotropy. The metallic value of the Hall effect and the semiconducting scaling of dc conductivity in † ϫexp Ϫ͚ u pЈ͑q͒dq ͉0͘, ͑A2͒ overdoped high-Tc oxides are explained taking into account ͩ q ͪ the Anderson localization of bipolarons. The doping dependence of the critical temperature and of the London penetra- where tion depth, the low-temperature dependences of resistivity, of the Hall ‘‘constant’’ and of the upper critical field as well as 1 the robust features of ARPES are described. 2 2 2 gijϭ ͚ ͓͉ui͑q͉͒ ϩ͉uj͑q͉͒ Ϫ2ui*͑q͒uj͑q͔͒. ͑A3͒ The proposed two-band oxygen bipolaron model favors 2 q the spin-polaron formation as suggested by Mott for high-Tc copper oxides.27 One conclusion is that copper electrons re- The bracket in Eq. ͑A2͒ is equal unity. Then Eq. ͑9͒ follows main localized even in overdoped oxides because of the large from Eqs. ͑A2͒ and ͑A3͒ using the definition of u j͑q͒, Hubbard U on copper and the local lattice deformation, Eq. ͑2͒. which prevents their hopping. Therefore the role of copper in Taking into account that E␯ϪE0ϭEgϩ͚q␻qnq , the electronic transport is significantly reduced. If so, one can second-order indirect hopping Eq. ͑8͒ is written as 53 BIPOLARON ANISOTROPIC FLAT BANDS, HALL MOBILITY... 2869

ϱ 2 T 2 2 ͑2͒ ϪiEgt ͑2͒ pd Ϫg Ϫg t ϭi dt e ͗0͉␴ˆ ͑t͒␴ˆ ͉0͘, ͑A4͒ t ϭ e pde dpЈ ppЈ ͵ pd dpЈ ppЈ 0 Eg where ϱ k * k ͑Ϫ1͒ ͚͑q͓up͑q͒Ϫud͑q͔͓͒ud*͑q͒Ϫup ͑q͔͒͒ ϫ͚ Ј . kϭ0 k!͑1ϩk␻/Eg͒ ˆ * † ␴ pd͑t͒ϭTpd exp ͚ u p ͑q,t͒dqϪH.c. ͩ q ͪ ͑A7͒

ϫexp ͚ ud͑q,t͒dqϪH.c. . ͑A5͒ Equation ͑8͒ is obtained from Eq. ͑A7͒ in the limit Egӷ␻. ͩ q ͪ Substitution of Eq. ͑3͒ into Eqs. ͑9͒ and ͑13͒ yields

Here u j͑q,t͒ϵu j͑q͒exp(i␻qt) and nqϭ0,1,2,... the phonon occupation numbers. Calculating the bracket in Eq. ͑A4͒ one Ep Si͑qdm͒ obtains g2 ,g2ϭ 1Ϫ , ͑A8͒ ppЈ ␻ ͩ q m ͪ d 2 Ϫg2 Ϫg ͗...͘ϭe pde dpЈ exp Ϫ͚ ͓u p͑q͒Ϫud͑q͔͒ ͩ q if the Debye approximation for the Brillouin zone is applied. x Here Si(x)ϭ͐ 0 sin(t)dt/t, mϭa/ͱ2, and mϭa for the in- 2 2 * Ϫi␻qt plane g and for the apex reduction factor g , respectively. ϫ͓ud*͑q͒Ϫu p ͑q͔͒e . ͑A6͒ ppЈ Ј ͪ Ϫ1 For LSCO with qdӍ0.7 Å and aӍ3.8 Å one obtains g2 0.2E /␻ and g2 0.3E /␻, where E is given by Eq. If ␻q is q independent the integral in Eq. ͑A4͒ is calculated ppЈӍ p Ӎ p p by the expansion of the exponent in Eq. ͑A6͒: ͑10͒.

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