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

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Bipolaron Anisotropic Flat Bands, Hall Mobility Edge, and Metal-Semiconductor Duality of Overdoped High-Tc Oxides View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Loughborough University Institutional Repository 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 1995! 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 depen- dence of Tc and lH(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 La22xSrxCuO4 for a wide temperature and doping 10 growing consensus that the dopant-induced charge carriers in region and the linear low-temperature resistivity r 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 j0 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 j .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 !l ductors. In particular, studies of photoinduced carriers in di- c s is unrealistic; normally ls@l. 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 Ne´el 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 po- laron formation has been verified for La-Sr-Cu-O ~LSCO!. 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 spec- troscopy ~ARPES!, the T2 temperature dependence of resis- tivity, and from the small value of the Hall effect. A three- band model involving a strong oxygen-copper hybridization @Fig. 1~a!# 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 La22xSrxCuO4 , 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. 12! and Bi2Sr2CuOy ~Ref. 13! led several obtained by p/2 rotation. Three-band (t-J) model ~a! and two-band authors to the conclusion that low-energy kinetics of over- apex bipolaron model ~b!. 0163-1829/96/53~5!/2863~7!/$06.0053 2863 © 1996 The American Physical Society 2864 A. S. ALEXANDROV 53 2 21 21 plane and apex holes. The role of copper in electronic trans- with the dimensionless coupling constant a5e (e ` 2e 0 ) port is significantly reduced by bipolaron formation. A Am/2v, introduced by Fro¨hlich,17 and the momentum- simple two-band model is derived with a strong a-b anisot- independent optical phonon frequency vq5v. The lattice ropy of two narrow bipolaron oxygen bands. The effective- polarization is coupled with the electron density, therefore mass anisotropy is 4 or larger. A random potential increases the interaction is diagonal in the site representation and the the anisotropy, so low-energy carriers are effectively local- coupling constant does not depend on the particular orbital. ized in one direction for a wide range of doping. Then there In doped oxides optical phonons are partially screened. Then is a ‘‘Hall mobility edge’’ EcH . The states below EcH con- molecular vq5v0 and acoustical vq5sq phonons contribute tribute to the longitudinal conductivity rather than to the also to the interaction with the coupling constants g[g0 and transverse one. A quantitative explanation for a high value of g;1/Aq, respectively. The canonical displacement transfor- the Hall density in La Sr CuO is proposed compatible 22x x 4 mation S5exp$(q, jn j@u j~q!dq2H.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 lH both in underdoped and overdoped samples, Hamiltonian is given by the low-temperature dependence of r, RH , and Hc2 of over- doped Tl2Ba2CuO6 and the robust features of ultrahigh en- 15,16 ˜ 21 ergy resolution angle-resolved photoemission spectra H5SHS 5~Tp2Ep!( ni~p!1~Td2Ed!( ni~d! ~ARPES! are explained with bipolarons. i~p! i~d! † † 1 sˆ ijci cj1 vqdqdq 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- 2 @2vqui~q!u*j ~q!2Vij#ninj . ~4! 2 q(,i,j lap of p oxygen orbitals, Fig. 1~b!, or via a second-order indirect transition involving d orbitals of copper, Fig. 1~a!. 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 pp8, 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 p5Ed5( uu j~q!u vq . ~5! distance compared with an oxygen-oxygen one and of the q relatively small size of the charge-transfer gap Eg.1–2 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 sij5Tij exp ( ui*~q!dq2H.c. exp ( u j~q!dq2H.c. S q D S q D ~6! † † H5 Tijci cj1 vqnj@uj~q!dq1H.c.#1 vqdqdq (i, j (q,j (q There are two major effects of the electron-phonon interac- tion. One is the band narrowing due to a phonon cloud 1 around the hole. In case of Eg@v the bandwidth reduction 1 ( Vijninj , ~1! 2 i, j factor is the same for the direct t pp8 and the second-order via copper t(2) oxygen-oxygen transfer integrals see the Appen- pp8 ~ where Tij determines the bare band structure in the site rep- dix! resentation; ci , c j are hole annihilation operators for oxygen † 2 or copper sites i, j; n j5c j c j is the number operator, Vij is 2g t [^0usˆ u0&5T e pp8, ~7! the direct Coulomb repulsion, which does not include the pp8 pp8 pp8 on-site term i5 j for parallel spins; vq ,dq are the phonon 2 0 ˆ ˜ 0 T 2 frequency and annihilation operator, respectively. The ~2! ^ uspdun&^nusdp8u & pd 2g tpp [( . e pp8, ~8! electron-phonon coupling in the site representation for elec- 8 n E02En Eg trons is given by where un&, En are eigenstates and eigenvalues of the trans- 1 formed Hamiltonian, Eq. ~4! without the hopping term, u0& iq•mj u j~q!5 g~q!e , ~2! the phonon vacuum, and the reduction factor is A2N 1 2 2 where N is the number of cells in the normalized volume g pp 5 ug~q!u $12cos@q ~mp2mp !#%. ~9! 8 2N(q • 8 NV, 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 .0.2E /v than one iA8pa pp8 p g~q!52 ~3! can expect with a naive estimation (.E p/v) ~see the Appen- AV~2mv!1/4q dix!.
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