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Series 3. High Technology - Vol. 77
Defects and Surface-Induced Effects in Advanced Perovskites editedby
Gunnar Borstel Department of Physics, University of Osnabruck, Germany
Andris Krumins Institute of Solid State Physics, University of latvia, Riga, latvia
and
Donats Millers Institute of Solid State Physics, University of latvia, Riga, latvia
Springer-Science+Business Media, BV.
Proceedings of the NATD Advanced Research Workshop on Defects and Surface-Induced Effects in Advanced Perovskites Jurmala, Latvia 23-25 August 1999
A C.I.P. Catalogue record for this book is available from the Library of Congress.
ISBN 978-0-7923-6217-3 ISBN 978-94-011-4030-0 (eBook) DOI 10.1007/978-94-011-4030-0
Printed on acid-free paper
AII Rights Reserved © 2000 Springer Science+Business Media Dordrecht Originally published by Kluwer Academic Publishers in 2000 Softcover reprint of the hardcover 1st edition 2000 No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic ar mechanical, including photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner.
CONTENTS
I Modelling of Defeets and Surfaee8 1
Quantum Meebanical Modelling of Pure and Defective KNbOJ Perovskites
N.E. Christensen, E.A.Kotomin, RLEglitis, A. V.Postni/cov, G.Borstel, D.'L.Novikov, S.Tinte, MG.Stachiotti, e.O.Rodriguez 3
Fint-Principles Simulation of Substitutional Defeets in Perovskites A. V.Postnikov, G.Borstel, A.LPoteryaev, RlEglitis 17
Point Defeets. Dielectric Relaxation and Conductivity in Ferroelectric Perovskites
MMaglione 27 Use and Limitations of the Sbell Model in Calculations on Perovskites
P. W.MJacobs 37 Wbat can We Learn about Perfect and Defeetive MgO(OOI) Surface Using Density Functional Theory!
L.N.Kantorovich, A.L.Shluger, MJ.Gillan 49 Defect Calculations for Yttrium Aluminum Perovskite and Gamet Crystals
MMKuklja 61 Theoretical Studies of Impurity Doped and Undoped BaTiOJand SrTiOJ Crystals
H.Pinto, A.Stashans, P.Sanchez 67
II Experimental Study of Structure and Basic Properties; HTSC 73
EPR Investigations of Small Electron and Hole Polarons in Oxide Perovskites
O.F.Schirmer 75 Nb4+ Polaron and TiJ+ Sballow Donor Jabn-Teller Centen in LiNbOJ Systems
G.CO"adi 89 Polarons and Bipolarons in Oxides
J.T.Devreese 101 Structural Distortions and Oxygen Local Dynamic Instabilities in Superconductiog Perovskites
N.Kristoffel 113 Subpicoseeond Laser Spectroscopy of Blue-Ligbt-Induced Absorbtion in KNbO:, and LiNbO:,
H.M Yochum, K.B. Ufer, R. T. Williams, L. Grigorjeva, D.Millers, G. Corradi 125
vi
A.Kuzmin, J.Purans, A.Sternberg 145 IR Spectroscopy of Monoclinic Tungsten Oxide
J.Gabrusenoks, A. v.Czamowski, K.-HMeiwes-Broer 151 Low Temperature Optical Absorption by Magnons in KNiF3 and NiO Single-Crystals
NMironova, V.Skvortsova, A.Kuzmin, I.Sildos, N.Zazubovich 155 The Metastable Superior Phases in Bi-2212 Perovskite-Like High-Temperature Superconductor
E.Shatkovskis, L.Dapkus, v.Pyragas 161 Surface-Induced Drift and Self organization Features of Microwave Losses in High-Tc Superconductor Perovskites
M.IShirokov 167
Chemisorption by Simple Oxide Surfaces A.B.Kunz, D.E.Zwitter ] 97
Theoretical Simulations of Surface Relaxation for Perovskite Titanates E.A.Kotomin, E.Heifets, W.A.Goddard, P. W.M.Jacobs, G.Borstel 209
Wetting of Domain WaDs in Perovskites S.Dorjman 221
Ferroelectric Soft-Mode Spectrocopy in Disordered Bulk and Thin-Film Perovskites
J.Petzeit, T. Ostapchuk, S.Kamba 233 Metals on Metal Oxides: Study of Adsorption Mechanisms with the Metastable Impact Electron Spectroscopy (MIES)
V.Kempter, M.Brause 249 Properties of Multiphase Interfaces on the Tungsten Trioxide Particles in the Thin Films
A.Lusis, J.Kleperis, E.Pentjuss 261 The Joining of LiNb03, Quartz, TIBr-Til and Other Optical Materials by the Use of Thin Metal Films as Bonding Agents
J.Maniks, A.Simanovskis 267 Experimental Study of Texture and Self-Polarization of Sol-Gel Derived PZT Thin Films
J.Frey, F.Schlenkrich, A.Sch6necker, P.Obenaus, J. Thomas, R.K6h/er 273
Experimental Study of Heat Properties of Bat_.Sr. Ti03 Thin Films on a Substrate
B.A.Strukov, S. T.Davitadze, S.NKravchun, v. V.Lemanov, 279 B.M.Goltzman, S.G.Shuiman
Synthesis of Lead Zirconate Antiferroelectric Thin Films by Sol-Gel Processing
vii
IV Defects and Phase Transitions 291
Infrared Spectroscopy of OH-induced Defects in Fluoroperovskites R.Capelletti, A.Baraldi, P.Bertoli, MCornelli, UMGrassano, A.Ruffini, A.Scacco 293
Hydrogen-Related Effects in Oxides R Gonzalez 305
Ion-Beam-Induced Defects and Defects Interactions in Perovskite- Structure Titanates
WJ. Weber, WJiang, S. Thevuthasan, RE. Williford, A.Meldrom, L.A.Boatner 317
Phase Transitions in Incipient Ferroelectrics of Perovskite Structure with Impurities
V. V.Lemanov 329 Influence of Structure Ordering, Defects and External Conditions on Properties of Ferroelectric Perovskites
A.Sternberg, L.Shebanovs, E.Birks, M Tyunina, V.Zauls 341 Phase Transition Anomalies in Crystals with Defects
A.S.Sigov 355 Defects in Perovskites Induced by llIumination
MD. Glinchuk, R o.Kuzian, V. V.Laguta, IP.Bykov 367 Defect Luminescence Study in Tetragonal GeOz Crystals
A.N Trukhin, H-J.Fitting, T.Barfels, A. Veispals 379 Thermally Stimulated Ionic and Electronic Processes and Radiation- Induced Defect Annealing in LiBaF 3 Crystals
V.Ziraps, P.Kulis, I Tale, A. Veispals 387 Radiation Defects in LiBaF3 Perovskites
P.Kulis, I Tale, MSpringis, URogulis, J. Trokss, A. Veispals, H-J.Fitting 393
Radiation Induced Defects in Yttrium Aluminium Perovskite V.Skvortsova, NMironova-U1mane, A.Matkovski, S. Ubizskii 399
Laser-Calorimetric Study of Fundamental Absorption Edge in Pb, La(ZrTi)03 (PLZT) Perovskite Ceramics
MKnite, A.Krumins, D.Millers 405 Mesoscopic Scale Polarization Inhomogeneities in Electrooptic Ceramics: the Chaotic Phase Screen Model
E.Klotins 411 New Ideas in Relaxor Theory
R.FMamin 419
V Advanced Technologies, New Perovskite Materials and Applications
Nature of the Defects Induced by Pbotoinjection of Hydrogen in Transition in Metal Oxides
A.1.Gavrilyuk Assessment of Surface Quality by SHG in Oxide Crystals
G.V.Vazquez, P.D.Townsend, J.Roms, P.Taylor, RWootf Perovskite Materials for Optical Filtering and tbe Generation of Coberent Radiation
T.Brudevoll, A. Vii/anger Preparation and Properties of Lal_sAsMn~ (A = Ca,Sr) Single Crystals
D.Shulyatev, S.Karabashev, A.Arsenov, Ya.Mukovskii, S.Zverkov
Oxygen Diffusion in Donor Doped SrTi~: Influence of Tbermal Pretreatment
J.Helmbold, G.Borehardt, RMeyer, R Waser, S. Weber, S.Seherrer
EtTect of Reducing and Oxidizing Atmospberes on the PTCR Properties of BaTi03
T.Kolod;azhnyi, A.Petrie New Approacb for Boundary Conditions: Space Cbarge Controlled Concentrations of Cation Vacancies in Donor Doped SiTi~ for Sbort DitTusion Lengtb
RMeyer, R Waser Defect Luminescence of LiBaF3 Perovskites
MSpringis, P.Kuiis, 1. Tale, A. Veispals, H.-J.Fitting
Author Index
Subject Index
425
427
439
449
455
461
467
473
479
485
487
PREFACE
In recent years, complex oxide materials have attracted growing scientific interest due to their technological importance. Among them are AB03-type perovslcite materials, revealing unique electro-optic properties necessary for producing photore­ fractive effects. These light-induced changes in the refractive index of the material make a solid basis for many important devices in optical technology, including holographic storage, optical data processing and phase conjugation. LiNb03 is widely used in a variety of integrated and active acoustic-optical devices, including optical waveguides. Another perovskite, KNb03 is very efficient in laser frequency doubling. Many of these perovskites reveal ferroelectric properties, which are the subject of numerous international conferences.
It is well known that optical and mechanical properties of non-metals are strongly affected by defects and impurities unavoidably present in any real material. Another problem is related to the miniaturization of relevant devices. When they become of nanoscopic size, the surface effects start to play an important role. This is the more important since multi-layered AB03 structures are good candidates for high capacity memory cells.
All this demonstrates the importance of the NATO workshop Defects and Surface­ Induced Effects in Advanced Perovslcites held in Jurmala (Latvia) on August 23-25, 1999 in which 67 participants from 21 Western and East European countries came together to discuss the main results in this field. Latest development and new results were reported in 25 key lectures and 30 poster presentations, as well as in many discussions throughout the meeting. We do believe that the exchange of experience and ideas developed in many research centers over the globe will greatly stimulate the progress in high-tech technologies using perovskite materials.
The choice of the workshop's place was well justified by the intensive research activity conducted in the Institute of Solid State Physics, University of Latvia, Riga. The Institute has a large Ferroelectric Department and Theoretical Laboratory whose activities are well known abroad. The papers in this volume are divided in five sections:
modelling of defects and surfaces (7 papers), experimental study of structure and basic properies, HTSC (11 papers), effects on surfaces, interfaces and thin films ( 11 papers), defects and phase transitions (14 papers), advanced technologies, new perovskite materials and applications (8 papers).
The contributed papers were peer-reviewed according to the procedure of NATO series. The organizers of the Workshop wants to express its gratitude for the financial
support provided by the NATO Scientific Affairs Division and Latvian Council of Science.
The Workshop could not have been a success without the outstanding contributions of all the participants. Special thanks to the invited lecturers and to the session chairmen. Finally, the organizers thank N.E. Christensen and E.A.Kotomin for their intense support.
Gunnar Borstel Andris Krumins Donats Millers
ix
BOIKOVY. loffe Physico-Technical Institute RAS, Polytechnicheskaya 26, 194021 StPetersburg, RUSSIA
BRUDEVOLL T. Norwegian Defence Research Establishment, P.O.Box 25, N-2027 Kjeller, NORWAY
BURSIAN V.E. A.F.Ioffe Physical-Technical Institute of Russian Academy of Science, Polytekchnicheskaya 26, 194021 St.Petersburg, RUSSIA
CAPELLE TTl R. Universita degli studi di Parma, Dipartamento di Fisica, Vialle delle Scienze, 43100 Parma ITALY
CATLOW C.R.A. Royal Institution ofGB, 21 Albermarle Str., London WIX 4BS, UNITED KINGDOM
CHAKAREL. losefStefan Institute, lamova 39, Ljubljana 1000, SLOVENIA
CHRISTENSEN N.E. Institute of Physics, Aarhus University, DK-8000, Aarhus - C, DENMARK
CIKMACSP. Institute of Solid State Physics, Kengaraga 8, Riga LV-I063, LATVIA
CORRADIG. Crystal Physics Laboratory, Research Institute for Solis State Physics and Optics, Hungarian Academy of Sciences, P.O.Box 49, H-1525 Budapest, HUNGARY
DAVITADZE S.T. Moscow State University, Moscow 119899, RUSSIA
Xl
xii
DORFMANS. Dept. of Physics, Technion, Haifa 32000, ISRAEL
FALINM. Kazan Physical-Technical Institute ofRAS, Sibirsky Trakt 1017, 420029 Kazan, RUSSIA
GABRUSENOKS J. Institute of Solid State Physics, Kengaraga8, Riga LV-I063, LATVIA
GA VRILYUK A.L A.F.Ioffe Physical-Technical Institute of Russian Academy of Science Polytekchnicheskaya 26, 194021 St.Petersburg, RUSSIA
GONZALEZR. Dep. de Fisica, Universidad Carlos ill, Avda. de la Universidad 30, 2891lLeganes, Madrid, SPAIN
GOTIEA. Inorganic Chemistry, The Angstrom Laboratory, P.O.Box 538, S-75121 Uppsala, SWEDEN
GRIGORJEVA L. Institute of Solid State Physics, Kengaraga 8, Riga L V-I 063, LATVIA
HERMANSSON K. Inorganic Chemistry, The Angstrom Laboratory, P.O.Box 538, S-75121 Uppsala, SWEDEN
JACOBS P.W.M. Dept. Chemistry, University of Western Ontario, London NGA 5B7, CANADA
KANTOROVICH L. Dept. of Physics, University College London, Gower Str., London WCIE 6BT UNITED KINGDOM·
KARABACHEV S. Moscow State Steel and Alloys Institute, Leninskii prosp. 4, 119936 Moscow, RUSSIA
KEMPTERV. Physikalishes Institut, Technische Universitet Claustahl, Leibnizstr. 4, D-38678 Clausthal- Zellerfeld, GERMANY
KLEPERISJ. Institute of Solid State Physics, Kengaraga 8, Riga LV-1063, LATVIA
KLOTINSE. Institute of Solid State Physics, Kengaraga 8, Riga LV-1063, LATVIA
KNlTEM. Riga Technical University, Technical Physics Institute, la Kalku Str., Riga LV-1658, LATVIA
KOEBERNIK G. Institute for Solid State and Materials, Research Dresden, , PF 270016, D-O 1171 Dresden, GERMANY
KOTOMIN E.A. Institute of Solid State Physics, Kengaraga 8, Riga L V-I 063, LATVIA
KRISTOFFEL N. Institute of Physics, University ofTartu, Riia 142, 51014 Tartu, ESTONIA
KRUMINSA. Institute of Solid State Physics, Kengaraga 8, Riga LV -1063, LATVIA
KUKLJAM.M. Dept.Electrical Eng., Michigan Technological University, Houghton MI 49931, USA
KULISP. Institute of Solid State Physics, Kengaraga 8, Riga LV-I063, LATVIA
KUZIANR. Institute for Problems of Material Science, National Academy of Science of Ukraine, Krijijanovskogo 3, 252180 Kiev, UKRAINE
KUZMINSA. Institute of Solid State Physics, Kengaraga 8, Riga LV-I063, LATVIA
MAGLIONEM. Laboratoire de Physique, Universite de Bourgogne, UPRESA CNRS 5027, BP 47870, F-21078 Dijon, FRANCE
MAMIN R.F. Kazan Physical-Technical Institute, Russian Academy of Sciences, 420029 Kazan, RUSSIA
MANIKSJ. Institute of Solid State Physics, Kengaraga 8, Riga L V-1063, LATVIA
xiii
xiv
MEYERR. Sommerfeldstrasse 24,52074 Aachen, GERMANY
MIRONOVAN. Institute of Solid State Physics, Kengaraga 8, Riga L V-I 063, LATVIA
LEMANOV V.V. A.F.IotTe Physico-Technical Institute, 194021 St.Petersburg, RUSSIA
ORLINSKII S.B. Kazan State University, Kremlevskaia Str.18, 420008 Kazan, RUSSIA
PETZELT J. Institute of Physics, Acad.ofSciences of the Czech Rep., Na Slovance 2, 18221 Praha 8 CZECH REPUBLIK
PINTOH. Departamento de Fisica, Facultad de Ciencias, Escuela Politecnica Nacional, Apto. 17-12-637 Quito, ECUADOR
POSTNIKOV A.V. Institute of Metal Physics, Russian Acad. Science, S.Kowalewskoj 18, Jekaterinburg 620219, RUSSIA
RUZAE. FB Physik, University Osnabrueck, D-49069 Osnabrueck, GERMANY
SAVITSKYD. State University "Livska Politechnika", 12 Bandera Str., Lviv 290646, UKRAINE
SCHIRMER O.E. Fachbereich Physik, Universitat Osnabruck, D-49069 Osnabruck, GERMANY
SHA TKOVSKIS E. Semiconductor Physics Institute, Vilnius 2600, LITHUANIA
SHIROKOVM. Institute of Solid State Physics, Kengaraga 8, Riga LV -1063, LATVIA
SHULYATEV D.A. Moscow State Steel and Alloys Institute, Leninskii prosp. 4, 119936 Moscow, RUSSIA
SIGOV A.S. Electronics Dept., Pr.Vemadskogo 78, 117454, RUSSIA
SKVORCOVA V. Institute of Solid State Physics, Kengaraga 8, Riga LV-1063, LATVIA
STERNBERG A. Institute of Solid State Physics, Kengaraga 8, Riga L V-I 063, LATVIA
STRUKOV B. Moscow State University, Moscow 119899, RUSSIA
TRUHIN A.N. Institute of Solid State Physics, Kengaraga 8, Riga L V-1063, LATVIA
VAZQUEZG. School of Engineering, University of Sussex, Brighton BNl 9QH, UNITED KINGDOM
ZIRAPS V. Institute of Solid State Physics, Kengaraga 8, Riga LV-1063, LATVIA
WEBERW.J. Pasific Northwest Nat. Lab., P.O.Box 999, Mail Stop K2-44, Richland, WA 99352, USA
WILLIAMS R. T. Department of Physics, Wake Forest University, Winston - Salem, NC 27109, USA
WOJCIKM. Inorganic Chemistry, The Angstrom Laboratory, P.O.Box 538, S-75121 Uppsala, SWEDEN
xv
I
QU ANTUM MECHANICAL MODELLING OF PURE AND DEFECTIVE KNb03 PEROVSKITES
N. E. CHRISTENSEN Institute of Physics and Astronomy, University of Aarhus, Aarhus C, DK-BOOO, Denmark
E. A. KOTOMIN, R. I. EGLITIS Institute of Solid State Physics, University of Latvia,
B Kengaraga, Riga LV-1063, Latvia
A. V. POSTNIKOV, G. BORSTEL
Universitiit Osnabrock - Fachbereich Physik, D-49069 Osnabriick, Germany
D. L. NOVIKOV
Arthur D. Little, Inc. Acorn Park, Cambridge, MA 02140-2390, USA
S. TINTE, M.G. STACHIOTTI
Instituto de Fisica Rosario, Universidad Nacional de Rosario, 2000 Rosario, Argentina
AND
IFLYSIB, Grupo de Fisica Solido, 1900 La Plata, Argentina
Abstract. Ab initio electronic structure calculations using the density­ functional theory (DFT) are performed for KNb03 with and without de­ fects. Ferroelectric distortive transitions involve very small changes in en­ ergies and are therefore sensitive to DFT -approximations. This is discussed by comparing results obtained with the local density approximation (LDA) to those where generalized gradient approximations (GGA) are used. The results of ab initio calculations for F-type centers and bound hole polarons are compared to those obtained by a semiempirical method of the Interme­ diate Neglect of the Differential Overlap (INDO), based on the Hartree­ Fock formalism. Supercells with 40 and 320 atoms were used in these two approaches, respectively. The relevant experimental data are discussed.
3
G. Borstel el al. (eds.), Defects and Surface-Induced Effects in Advanced Perovskites, 3-16. @ 2000 Klllwu Academic Publishers.
4
1. Introd uction
KNb03 is one of the oldest known room-temperature ferroelectric mate­ rials, that undergoes a sequence of ferroelectric transitions from the high­ temperature cubic perovskite structure [1]. The simplicity of crystal struc­ ture makes it appealing for ab initio theory simulations, whereas interesting photorefractive characteristics and low cost keep KNb03 in a list of seri­ ous candidates for practical applications in electrooptical devices even with the advent of new ferroelectric substances. The KNb03 material applied in practice always contains defects - either introduced intentionally (like su b­ stitutional impurities introduced in the search for particular optical char­ acteristics), or those immanently present even in nominally pure samples (vacancies on different sublattices). In the latter case, the identification of the defects which are responsible for (sometimes undesirable) absorption bands may be difficult.
Ab initio microscopic simulations of electronic structure, lattice distor­ tions and related optical excitations in the presence of defects may be a useful tool in connection with identification of defects. However, the the­ ory faces certain problems already in the description of ideal defect-freE' KNb03 . For one thing, a small magnitude of atomic displacements and correspondingly small (of the order of 1 mRy per formula unit) energy gains related to ferroelectric transition set the standards of accuracy for an underlying calculation quite high. The experience of precision total-energy calculations based on the local density approximation (LDA) within the density functional theory (DFT) have shown that the equilibrium volume, although only slightly (by ~5%) underestimated with respect to the exper­ imental one, is nevertheless almost too small for the ferroelectric instability to become qualitatively possible [2]. This problem may be solved on the path of constructing better approximations for the exchange-correlation energy within the DFT than the LDA. When turning to the simulations of electronic excitations in defect systems, the DFT-based methods are not always fully satisfactory. As a simple alternative, the Hartree-Fock schemes may prove to be useful.
In the present paper, we summarize essential results of our recent stud­ ies devoted to different aspects of ab initio electronic structure simulations in perovskites. We addressed the problem of tuning the generalized gradi­ ent approximation (GGA) for KNb03 , among other perovskites, in Ref. [3]. Two different vacancy-type defects, namely F-centers associated with an 0 vacancy, and hole polarons, bound assumedly on a K vacancy, have been studied respectively in Refs. [4] and [5]. In the following, we briefly discuss the technical side of the calculations and concentrate on three abovemen­ tioned problems, refering to related experimental information.
5
2. Methods
Our ab initio DFT calculations in all three cases used linear band struc­ ture methods [6], the full-potential linear augmented-plane-wave (FLAPW) method [7] for pure KNb03 and the linear muffin-tin orbital (LMTO) scheme, also in a full-potential implementation [8], for the study of de­ fects (in the supercell approach). The 2 x 2 x 2 supercells were chosen, Le. the distance between repeated point defects was ~8 A. As a consequence of the large number of eigenstates per k-point in a reduced Brillouin zone (BZ) of the supercell and of the metallicity of the doped system, it was essential to maintain a dense mesh for the k-integration by the tetrahedron method over the BZ. Specifically, clear trends in the total energy as func­ tion of atomic displacements were only established at 10 x 10 x 10 divisions of the BZ (Le., 186 irreducible k-points for a one-site polaron).
In the study on pure KNb03 , we concentrated on the accuracy of total energy evaluations obtained with different approximation schemes in the DFT. The LDA in general leads to overbinding when applied to solids, and this error is particularly serious for perovskites where predictions of ferroelectric properties are incorrect if calculated at too small volumes. A simple way to improve accuracy simply consists in performing the calcula­ tions for the experimental volume, but this still leaves the question open about additional "LDA errors". Some improvement over the LDA has been obtained by using gradient approximations for better descriptions of the inhomogeneous electron gas. In the Kohn-Sham density functional theory only the exchange-correlation energy Exe =Ex+Ee which is a functional of the electron spin densities must be approximated, and for slowly varying densities, n, it can be expressed as the volume integrals of n times €~~if in the LDA case and f( nt, n.j.,'V nt,V n.j.) for the GGA case. For practical calculations the exchange-correlation energy density of a uniform electron gas, €~~if (nt, n.j.), and f must be parametrized. The form of €~~if is now well established but which is the best choice of f is still under debate.
The GGA version suggested by Perdew, Burke and Ernzerhof (PBE) [9] has been very useful in several cases, but some uncertainty in its use is related to the choice of the parameter K, in the enhancement factor Fx{s) which is directly associated to the degree of localization of the exchange­ correlation hole. (Here the variable s is a measure of the relative density gradient, s = IVnl/2kFn, kF giving the Fermi wavenumber of an electron gas of density n). In their original work PBE proposed F x (s) = 1 + K -
K/{1 + J.LS2 / K) which satisfied the inequality F x ~ 1.804 with K = 0.804 and with the value of J.L ~ 0.21951. Zhang and Yang found [10] that the results for several atoms and molecules were improved by increasing K beyond the originally proposed value of 0.804. But, this does not hold for all types of
6
bonds [11], and it may well happen that applications to solids appear to be improved when smaller values of", are used. The parameter", might be a weak function of the reduced Laplacian, '" = g[\72n /{2kF )2n].
The motivation for using the DFT-based and Hartree-Fock (HF) -based calculation methods in parallel, when applied to defects, is to combine strong sides of both in a single study. The DFT is expected to be able to provide good description of the ground state, i.e. to deliver reasonable relaxation energies and ground-state geometry. In the HF approach, the relaxation energies are generally less accurate because of the omission of correlation effects. On the other hand, the HF formalism is well suited for the evaluation of excitation energies, because the total energies can be cal­ culated for any (ground-state or excited) electronic configuration on equal footing. This is generally not the case in the DFT. Practical experience shows that HF and DFT results often exhibit similar qualitative trends in the description of dielectric properties but quantitatively lie on opposite sides of experimental data, thus effectively setting error bars for a theo­ retical prediction [12]. For a HF calculation scheme in a present study, the semi-empirical Intermediate Neglect of the Differential Overlap (INDO) [13] method, modified for ionic and partly ionic solids [14, 15], has been used. The supercells of the same size as with the FP-LMTO method, i.e. 2x2x2, were used for the study of F and F+ centers, and a larger supercell, 4x4x4 (320 atoms), for the more recent calculation of a hole polaron. The effect of the supercell size can bee seen from the comparison of the present data (reproduced from Ref. [5]) with those of Ref. [16] where a small 40-at. su­ percell has been used for the hole polaron as well. In the supercell INDO calculations, the Brillouin zone summation was restricted to the zone center only (in the appropriately reduced zone). This introduced a certain error, especially large for small supercells. A discussion on possible magnitude of such error is given in Ref. [4]. The parametrization of the INDO method for the calculations on KNb03 has been done in Ref. [17].
3. Pure KNb03 : GGA vs. LDA
The accuracy of the LDA when applied to the perovskites, as well as the effects of introducing the GGA improvements, are illustrated in the follow­ ing by the calculations of: 1) the energy-volume curves from which lattice parameters and bulk moduli are derived, 2) f15 phonon modes for the cubic structure, and 3) the tendency to undergo a ferroelectric transition when the atoms are displaced according to the soft-mode displacement pattern.
In that connection the sensitivity to the choice of '" in the PBE GGA is examined. In Fig. 1 the values of V /Vo (Vo is the corresponding exper­ imental value) are plotted for four perovskites over the range of '" values
0 > :>
1.04
1.03
1.02
1.01
1.00
• KNb03
• SarlO:!
IC
7
Figure 1. Equilibrium volumes calculated for KNb03 , BaTi03 , SrTi03 and KTa03 as functions of the PBE-GGA parameter K. (Va are the experimental equilibrium volumes.
TABLE 1. Frequencies of the r l5 modes (in em-I), lattice parameter a (in A) and bulk modulus B (in GPa) in KNbOa as calculated in the LDA and GGA approximations. i in the frequency values indicate that these are imaginary, i.e. soft mode. Note that all phonon frequencies are calculated at the experimental equilibrium volume. In parantheses: bulk modulus at the experimental equilibrium volume.
LDA GGA (K=0.804) GGA (K=Keq) Expt
Frequencies 21li 197i 195i soft 166 182 179 198
466 478 478 521
a 3.96 4.04 4.016
B 206(155) 171(186) 138
in the PBE-GGA functional varying from 0.3 to 0.804. As can be seen, both BaTi03 and KNb03 would give perfect determination of the lattice parameter (V /Vo=l) for K. ~ 0.6. In the case of SrTi03 and KTa03 the value should be further reduced to K. ~ 0.4. The need of varying K. from one system to another reflects the fact that the localization of the exchange­ correlation hole is system-dependent.
In order to clarify the effect of the GGA functional on the phonon en-
8
0.1 0.2
Displacement (A)
Figure~. Total energy as a function of the ferroelectric displacement of the Nb atom relative to the center-of-mass of the O-octahedron in KNbOa. The calculations were performed for the experimental equilibrium volumes, and the straight LDA (full lines and crosses) are compared to PBE-GGA results for two choices of K. (0.804 and 0.600).
ergies, we have performed frozen phonon calculations for KNb03 at its experimental lattice constant, and examined the effects of choosing dif­ ferent exchange-correlation approximations, i.e. LOA and PBE-GGA with different K. values. The calculated frequencies of the r15 modes and the ex­ perimental values are shown in Table 1. It is seen that the GGA hardens the phonon frequencies (as compared to the LOA results). This hardening produces a slight reduction of the errors, since the LOA provides phonon frequencies which are understimated by 10'::::20% as compared with experi­ ments. The second conclusion concerns the parameter K.. It is evident from Table 1 that the effect introduced on the GGA phonon frequencies by the modification of K. is negligible. In addition, we found [3] that the eigenvec­ tors are practically unchanged.
Finally, to test the sensitivity of the energetics involved in the ferro­ electric instabilities when the different exchange-correlation functionals are used, we performed total energy calculations as a function of the off-center displacement of Nb atom. In Fig. 2, we show the energy as a function of such displacement along the (111). Both LOA and GGA (with K.=0.804 and 0.6) yield a clear ferroelectric instability with similar energetics and displacements, and with an energy gain of ~ 1.8 mRy/cell. The similar observation has been done by Singh [18] in his GGA study of KNb03 .
9
4. Simulation of defects
It is well understood now that point defects play an important role in the electro-optical and non-linear optical applications of KNb03 and related ferroelectric materials [1]. The prospects of the use of KNb03 for the light frequency doubling are seriously affected by the presence of unidentified defects responsible for induced infrared absorption [19]. The photorefractive effect, important in particular for holographic storage, is also well known to depend on the presence of impurities and defects. Most of as-grown AB03 perovskite crystals are non-stoichiometric and contain considerable amounts of vacancies. The electron F and F+ centers (an 0 vacancy, Vo, which traps two or one electron, respectively) [20, 21, 22] belong to the most common defects in oxide crystals. In electron-irradiated KNb03 , a broad absorption band observed around 2.7 eV at room temperature has been tentatively ascribed to the F-type centers [23]. These two defects were the subject of recent ab initio LOA and semiempirical calculations [24, 4]. A transient optical absorption band at 1.2 eV has been associated recently [25]- in analogy with other perovskites- with a hole polaron (a hole bound, probably, to a K vacancy). The ESR study of KNb03 doped with TiH gives a proof that holes could be trapped by such negatively charged defects [26]. For example, in BaTi03 , the hole polarons bound to Na and K alkali ions replacing Ba and thus forming a negatively charged site attracting a hole [27] have also been found. Cation vacancies are the most likely candidates for binding hole polarons. In irradiated MgO, they are known to trap one or two holes giving rise to the V- and VO centers [20, 21] which are in their nature bound hole polaron and bipolaron, respectively. The results of the experimental studies of hole polarons in alkali halides and ferroelectric perovskites reveal two different forms of atomic structure of polarons: atomic one (one-site), when a hole is localized on a single atom, and molecular-type (two-site), when a hole is shared by two atoms forming a quasi-molecule [26, 27, 28]. In the present study, we simulate both electron centers and hole polarons associated with a K vacancy in KNb03 .
4.1. F-TYPE CENTERS
In the cubic KNb03 all 0 atoms are equivalent and have the local sym­ metry C4v (due to which the excited state of the F-type centers could be split into a nondegenerate and a doubly-degenerate levels). The optimized atomic relaxation around the F center as done by the LMTO shows that the Nb neighbours to the 0 vacancy are displaced outwards by 3.5% a. The associated lattice relaxation energy is shown in Table 2.
The optimized Nb relaxation found in the INDO simulations was 3.9%,
10
TABLE 2. Absorption (Eob.) and lattice relaxation energies (Erel, for the electron centers and hole po­ larons (relatively to the perfect crystal with a K vacancy) (in eV), calculated by LMTO and INDO methods.
INDO Eob.
F-center 2.68; 2.93 F+-center 2.30; 2.63 one-site polaron 0.9 two-site polaron 0.95
1.6
> 1.2
INDO Erel
(a)
(b)
Figure 9. Local density of states of the F-center (left panel) and of the Nb atom nearest to it as calculated by the LMTO method.
11
i.e. very close to the ab initio calculations. The outward relaxation of nearest K atoms and inward displacements of 0 atoms are much smaller. They contribute ::::::20% of the net relaxation energy of 1..35 eV. The F center local energy level lies ::::::0.6 eV above the top of the valence band. Its molecular orbital contains primarily the contribution from the atomic orbitals of the two nearest Nb atoms. Only ::::::0.6 e resides at the orbitals centered at the vacancy site; hence the electron localization inside vacancy is much weaker than for F centers in ionic oxides where typically 80 % of the electron density is localized in the ground state [22]. The symmetry analysis of the ground-state wave function asseciated with the F center, done by the TB­ LMTO method with the use of the LDA+U formalism [29] and by INDO, revealed the same result, namely that the major contribution comes from the eg states centered at Nb neighbors (more specifically, it is essentially the 3z2 -r2 component, with z in the direction towards the F center). The partial densities of states from the LMTO calculation are shown in Fig. 3.
For the F+ center the relaxation energy of 2.23 eV and the Nb displace­ ments of 5.1 % of a are larger than those for the F center due to a stronger Coulomb repulsion between unscreened 0 vacancy and Nb atoms: a share of the electron density inside the 0 vacancy decreases to 0.3 e. The optical absorption energies calculated by means of the LlSCF method (the differ­ ence of total energies in relaxed ground state and excited state) for the F+ and F centers are given in Table 2. Both defects are predicted to have one of the bands around 2.6-2.7 eV, which was observed experimentally [23].
4.2. HOLE POLARONS
In the K vacancy-containing supercell, the relaxation of either one (for the one-site polaron) or two neighboring (for the two-site polaron) 0 atoms, amongst twelve closest to the K vacancy, has been allowed for, and the changes in the total energy (as compared to the unrelaxed perovskite struc­ ture with a K atom removed) have been analyzed. Also, we studied the fully symmetric relaxation pattern (breathing of twelve 0 atoms) around the va­ cancy.
The removal from the supercell of a K atom with its 7 electrons con­ tributing to the valence band (VB) produces slightly different effects on the electronic structure, as described within the DFT and in the HF for­ malism. Acording to the LMTO result, the Fermi energy lowers, and the system becomes metallic (remaining non-magnetic). Therefore, no specific occupied localized state is associated with the vacancy. The local density of states (DOS) at the sites of interest is shown in Fig. 4. As is typical for LDA calculations, the one-electron band gap in KNb03 comes out un­ derestimated (::::::2 eV) as compared to the experimental optical gap (::::::3.3
12
2
-5 o 5 10 Energy (eV)
Figure 4. Local DOS at the K vacancy site and at the adjacent oxygen atom (top panel) and at Nb and 0 sites in perfect KNb03 (bottom panel). as calculated by LMTO.
eV). The removal of a K4s electron amounts to adding a hole which forme. a localized state at ~10 eV above the Fermi level, i.e. above the unoccu­ pied Nb4d band. In the 2p-DOS of 0 atoms neighboring the vacancy, a quasi-local state (that effectively screens the hole) is visible just below the Fermi level. Apart from that, the 02p-DOS is largely unaffected by the presence of vacancy, and the changes in the DOS of more distant sites (K, Nb) are negligible as compared with those in the perfect crystal. As the cubic symmetry is lifted by allowing a non-uniform relaxation of 0 atoms, the "screening" quasi-local state is clearly localized at the atom closest to the vacancy. At the same time, the hole state becomes smeared out in en­ ergy. This amounts to the bonding being established between the hole and the screening charge on one of its neighbors.
In the case of one-site polaron, a single 0- ion is displaced towards the K vacancy by 1.5 % of the lattice constant (LMTO) or by 3% (INDO) - see Fig. 6. The INDO calculations show that simultaneously, 11 other nearest oxygens surrounding the vacancy tend to be slightly displaced outwards
13
Figure 5. Sketch of the polaron optical transition from the quasi-local state 1 near the top of the valence band to the hole state 2 below the conduction band bottom. 3 indicates the level of an Wlpaired electron.
the vacancy. In the two-site (molecular) configuration, a hole is shared by the two 0 atoms which approach each other - by 0.5% (LMTO) or 3.5% (INDO) - and both shift towards a vacancy - by 1.1% (LMTO) or 2.5% (INDO). The lattice relaxation energies (which could be associated with the experimentally measurable hole thermal ionization energies) are presented in Table 2. In both methods the two-site configuration of a polaron is lower in energy.
In the INDO treatment, the one-electron optical gap is overestimated, as is typical for the HF calculations (~6 eV [17]), but the ~SCF gap for the triplet state is 2.9 eV, close to the experiment. The quasi-local screening state is described by a wide band close to the VB top. This is consistent with the LDA description. The INDO calculation also suggests, and this differs from the LDA, that the removal of an electron leaves an unpaired electron state split-off at ~1eV above the VB band top. In case of asymmetrical o relaxation, the molecular orbital associated with this state is centered at the displaced 0 atom, on which about 80 % of unpaired spin density is localized. The same applies qualitatively to the two-site polaron, with the only difference that the localized state is formed from the 2p orbitals of both 0 atoms approaching the vacancy, with a corresponding symmetry lowering. The localized hole state is also present in the HF description but lies much lower than the corresponding state in the LDA, forming a 0.9 eV -wide band located ~ 0.2 eV below the conduction band bottom (see Fig. 5). In agreement with the general theory of small-radius polarons in ionic solids [28, 30], the optical absorption corresponds to a hole transfer to the state delocalized over nearest oxygens. The absorption energies due to the electron transition from the quasi-local states near the VB top (1, Fig. 5) into the vacant polaron band (2, Fig. 5) for one-site and two-site palarons are close (Table 2), and both are twice smaller than the experi­ mental value for a hole polaron trapped by the Ti impurity [26]. This shows that the optical absorption energy of small bound polarons can be strongly dependent on the defect involved.
14
!!. 3. • .. ! 3. • c:
INDO
'i 2. q o 2. '0 r 1. e ! 1. 0
'i o.
0 2 3 4 o displacement ("10 of a) 0·0 c.nl.r·of·m ... dl.pl. ('lI. 01.)
Figure 6. Relaxation energy for one-site polaron according to LMTO and INDO cal· culations vs the displacement of one oxygen atom towards the vacancy (a); relaxation energy for two-site polaron as function of the center·of-mass displacement of the 0-0 pair and of the 0-0 distance according to the INDO calculation (b).
In spite of generally observed considerable degree of covalency in KNb03
and contrary to a delocalized character of the F center state [24, 4], the one-site polaron state remains well localized at the displaced 0 atom, with only a small contribution from atomic orbitals of other 0 ions but none from K or Nb ions. Although there are some differences in the description of the (one-particle) electronic structure within the DFT- and HF-based methods, the trends in the total energy driving the structure optimization remain essentially the same. In both approaches, both one-site and two-site configurations of the hole polaron are much more energetically favorable than the fully symmetric (breathing mode) relaxation of twelve 0 atoms around the K vacancy. This is in line with what is known about small­ radius polarons in other ionic solids [28, 30] and is caused by the fact that the lattice polarization induced by a point charge is much larger than that due to a delocalized charge.
5. Conclusions
We emphasize that when dealing with tiny (on the energy scale) effects related to the off-center instabilities in ferroelectric perovskites, one should be particularly careful to the details of the method used, like e.g. the treat·· ment of exchange-correlation in a DFT-based ab initio scheme. In the study of charged defects where characteristic energies are much larger, the use of appropriately tuned semiempirical methods may provide reliable results.
15
Especially for the study of optical excitations, the use of a HF-type scheme in parallel with a DFT-based analysis turns out to be useful.
We demonstrated that the F-type centers could be responsible for the optical absorption observed around 2.7 eV [23]. The calculated polaron absorption (~1 eV) is close to the observed short-lived absorption band energy [25); hence this band could indeed arise due to a hole polaron bound to a cation vacancy.
6. Acknowledgments
This study was partly supported by the NATO Program for science se­ nior visitors (through the Danish Research Agency grant No 9800484 to E. A. K.), as well as by DFG to A. P. and G. B. (the SFB 225), and the Volkswagen Foundation (grant to R. E.). Authors are greatly indebted to L. Grigorjeva, D. Millers, and A. I. Popov for fruitful discussions.
References
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3. Tinte, S., Stachiotti, M.G., Rodriguez, C.O., Novikov, D.L. and Christensen, N.E. (1998) Applications of the generalized gradient approximation to ferroelectric per­ ovskites, Phys. Rev. 8, 58, pp. 11959-11963.
4. Eglitis, RI., Christensen, N.E., Kotomin, E.A., Postnikov, A.V. and Borstel, G. (1997) First-principles and semiempirical calculations for F centers in KNb03, Phys. Rev. 8, 56, pp. 8599-8604.
5. Kotomin, E.A., Eglitis, R.I., Postnikov, A.V., Borstel, G. and N. E. Christensen, N.E. (1999) First-principles and semiempirical calculations for bound hole polarons in KNb03, Phys. Rev. 8,60, pp. 1-5.
6. Andersen, O.K. (1975) Linear methods in band theory, Phys. Rev. 8, 12, pp. 3060- 3083.
7. P. Blaha, K. Schwarz, P. Dufek and R Augustyn, WIEN97, Technical University of Vienna 1997. Improved and updated Unix version of the original copyrighted WIEN­ code, which was published by P. Blaha, K. Schwarz, P. Sorantin and S.B. Trickey, Comput. Phys. Commun. 59,399 (1990).
8. Methfessel, M. (1988) Elastic constants and phonon frequencies of Si calculated by a fast full-potential linear-muffin-tin-orbital method, Phy,. Rev. H, 38, pp. 1537-1540; van Schilfgaarde, M. (unpublished).
9. Perdew, J.P., Burke, K. and Ernzerhof, M. (1996) Generalized Gradient Approxima­ tion Made Simple, Ph"s. Rev. Lett., 77, pp. 3865-3868.
10. Zhang, Y. and Yang, W. (1998) Comment on "Generalized Gradient Approximation Made Simple", Phy,. Rev. Lett., 80, p. 890.
11. Perdew, J.P., Burke, K. and Ernzerhof, M. (1998) Perdew, Burke, and Ernzerhof Reply, Ph",. Rev. Lett., 80, p. 891.
12. Fu, L., Yaschenko, E., Resca, L. and Resta, R (1998) Hartree-Fock approach to macroscopic polarization: Dielectric constant and dynamical charges of KNb03
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Phys. Rev. B, 57, pp. 6967-6971. 13. Pople, D.A. and Beveridge, D.L. (1970) Approximate Molecular Orbital Theory,
McGraw-Hill, New York. 14. Stefanovich, E., Shidlovskaya, E., Shluger, A.L. and Zakharov, M. (1990) Mod­
ification of the INDO calculation scheme and parametrization for ionic crystals, phys. stat. sol. (b), 160, pp. 529-540.
15. A. Shluger and E. Stefanovich (1990) Models ofthe self-trapped exciton and nearest­ neighbor defect pair in Si02 , Phys. Rev. B 42, pp. 9664-9673.
16. R. I. Eglitis, E. A. Kotomin, and G. Borstel (1998) Semi-Empirical Calculations of Hole Polarons in MgO and KNbOs Crystals, phys. status solidi (b), 208, pp. 15-20
17. Eglitis, R.I., Postnikov, A.Y. and Borstel, G. (1996) Semiempirical Hartree-Fock calculations for KNbOs, Phys. Rev. B, 54, pp. 2421-2427.
18. Singh, D.J. (1995) Local density and generalized gradient approximation studies of KNbOs and BaTiOs , Ferroelectrics, 164, pp. 143-152.
19. L. Shiv, J. L. S0rensen, E. S. Polzik, and G. MizeU (1995) Inhibited light-induced absorption in KNbOs, Optics Letters, 20, pp. 2270-2272.
20. Y. Chen and M. M. Abraham (1990) Trapped-hole centers in alkaline-earth oxides, J. Phys. Chem. Solids 51, pp. 747-764.
21. A. E. Hughes and B. Henderson (1972) In: Point Defects in Solids: ed. J.H. Crawford Jr and L.M. Slifkin, Plenum Press, New York.
22. E. A. Kotomin and A. I. Popov (1998) Radiation-induced point defects in simple oxides, Nucl. [nstr. Methods B 141, pp. 1-15.
23. E. R. Hodgson, C. Zaldo, and F. Agulla-Lopez (1990) Atomic displacement damage in electron irradiated KNb03, Solid State Commun. 75, pp. 351-353.
24. Kotomin, E.A., Eglitis, R.1. and Popov, A.1. (1997) Charge distribution and opti­ cal properties of F+ and F centers in KNbOs crystals, J. Phys.: Condo Matter, 9, pp. L315-L321.
25. L. Grigorjeva, D. Millers, E. A. Kotomin, and E. S. Polzik (1997) Transient opti­ cal absorption in KNbOs crystals irradiated with pulsed electron beam, Solid State Commun. 104, pp. 327-330.
26. E. Possenriede, B. HeUermann, and O. F. Schirmer (1988) 0- trapped holes in acceptor doped KNbOs, Solid State Commun. 65, pp. 31-33.
27. T. Varnhorst, O. F. Schirmer, H. Krase, R. Scharfschwerdt, and Th. W. Kool (1996) 0- holes associated with alkali acceptors in BaTi03, Phys. Rev. B 53, pp. 116-125.
28. A.L. Shluger and A.M. Stoneham (1993) Small polarons in real crystals: concepts and problems, J. Phys.: Condo Matter 5, pp. 3049-3086.
29. Eglitis, R.I., Kotomin, E.A., Postnikov, A.V., Christensen, N.E., Korotin, M.A. and Borstel, G. (1999) Computer simulations of defects in KNb03, to be published in Ferroelectrics; preprint http://xxx.lanl.gov/cond-mat/9812078.
30. O. F. Schirmer, P. Koidl, and H. G. Reik (1974) Bound Small Polaron Optical Absorption in V- - Type Centres in MgO, phys. status solidi (b) 62, pp. 385-391; O. F. Schirmer (1976) Optical Absorption of Small Polarons Bound in Octahedral Symmetry: Y- Type Centers in Alkaline Earth Oxides, Z. Phys. B 24, pp. 235-244.
FIRST-PRINCIPLES SIMULATION OF SUBSTITUTIONAL DEFECTS IN PEROVSKITES
A. V.POSTNIKOV,G.BORSTEL Universitiit Osnabriick - Fachbereich Physik D-49069 Osnabriick, Germany
A. I. POTERYAEV Institute of Metal Physics S. Kowalewskoj 18, GSP-170 Yekaterinburg, Russia
AND
R. I. EG LITIS Institute of Solid State Physics, University of Latvia 8 Kengaraga, Riga LV-1063, Latvia
Abstract. The results of supercell calculations of electronic structure and related properties of substitutional impurities in perovskite oxides KNb03 and KTa03 are discussed. For Fe impurities in KNb03, the results obtained in the local density approximation (LDA) and in the LDA+U approach (that allows an ad hoc treatment of nonlocality in exchange-correlation) are compared, and different impurity charge configurations are discussed. The study of off-centre Li defects in incipient ferroelectric KTa03 have been done by the appropriately parametrized Intermediate Neglect of Differen­ tial Overlap (INDO) method. The interaction energies of two off-centre impurities in different relative configurations are discussed.
1. Introduction
Mixed oxides of perovskite structure exhibit a variety of interesting and practically important properties. Among this family one can find supercon­ ductors, systems with colossal magnetoresistance, piezoelectrics and ferro­ electrics. A common feature of perovskite materials is the pliancy of their generic cubic structure to distortion, due to the coupling of a particular soft mode with electrostatic or magnetic interactions. The properties may
17
G. Borstel et al. (eds.), Defects and Surface-Induced Effects in Advanced Perovskites, 17-26. @ 2000 Kluwer Academic Publishers.
18
be further complicated, or intentionally tuned in a desirable range, by in­ troducing extrinsic defects. The quantitative theoretical study of doped systems is much more complicated than simulation of perfect crystals, and its success is based on the progress achieved in the study of the latter. Among ferroelectric systems, cubic perovskites, like BaTi03 and KNb03 ,
traditionally serve as benchmark systems for first-principle calculations of structure instability and lattice dynamics [1, 2]. In a sequence of works, we performed a detailed study of KNb03 (in comparison with incipient ferroelectric KTa03) regarding first-principle optimization of the ground­ state crystal structure [3], calculation of zone-centre phonons [4, 5, 6] and kinetic properties [7]. In the pI:esent contribution, we review our essential results obtained over the last years in modeling the electronic structure of perovskite-based impurity systems. We" do not discuss intrinsic point defects such as F-centres or bound hole polarons which were treated in Ref. [8, !)] within the same computational approach as applicable for substitutional impurities (see also the contribution by Christensen et ai. at this work­ shop). Among substitutional impurities, we consider two important specii. Light isovalent impurities (e.g., Li or Na substituting K in KTa03) do not affect the electronic structure in a noticeable way, but give rise to a strue­ ture distortion and long-range polarization in the crystal lattice. Impurities of chemically different type (e.g., Fe substituting Nb in KNb03 ) may de­ velop specific localized levels in the band gap of the matrix; they may exi8t in different charged states, so that different mechanisms of charge compen­ sation come into discussion. The treatment of these two different classes of impurity systems demands the use of different computational schemes.
2. Light isovalent impurities in KTa03
As it is experimentally known since the work by Yacoby and Just in 1974 [10], substitutional Li impurity in KTa03 gets spontaneously displaced along the [100] (or equivalent) direction from the K site. The magnitude of this displacement and (in some cases) the lattice relaxation related to it have been estimated by empirical models [11, 12, 13], the shell model [14, 15], full-potential (FP-) first-principles linear muffin-tin orbitals (LMTO) method [16] and by the Intermediate Neglect of the Differential Overlap (INDO) method [6]. The magnitudes of the Li displacement brought into discussion vary from 0.64 A[15] via 0.86 A [11], 1.1 A [12], and 1.35 A [13] to 1.44 A [14], based on the data extracted from (indirect) experimental measurements and on the results of numerical simulations. On the basis of our calculations [6], we advocate the Li off-centre displacement magnitude of ",,0.6 A and the energy gain associated with the displacement of single Li impurity of ",,60 meV. It was shown in Ref. [6] that the relaxation of
19
oxygen atoms nearest to the Li impurity almost triples the net energy gain due to the off-centre displacement of the latter. This tendency could be­ come even more pronounced if the lattice relaxation at farther distances from the impurity be properly taken into account.
The aspects of the Li-Li interaction in LixK1- x Ta03 have been ad­ dressed by Vugmeister and Glinchuk in 1990 in a review [17J that summa­ rized essential experimental information available by then and approached theoretical foundations of the interaction picture that could result in either dipole glass-type, or ferroelectric, long-range ordering of off-centre defects. Numerical simulation of the energetics of interacting Li dipoles of different distances and orientations in the lattice has been done by Stachiotti et al. [18J using the Green function method applied to the non-linear polarizable shell model. In Ref. [19], we estimated the Li-Li interaction energies as
Eint = E(2 Li) - 2E(Li) ,
where E(2Li) is the energy gain associated with the combined displacement of two atoms in the 6x3x3 supercell, and E(Li) the energy gain due to the off-centre displacement of a single Li impurity in the supercell of the same size. The relaxation of nearest oxygen atoms was allowed for for several geometries (see below).
We considered impurity pairs of different distances and different relative orientations within the supercell. This has been done in order to probe the electrostatic dipole field created by a Li impurity displaced along [100J by 0.62 A, i.e. the equilibrium displacement for the single off-centre Li ion. Positions of the other Li impurity labeled (a) through (h) with respect to the first impurity are shown in the left panel of Fig. 1, and the corresponding interaction energies - in the right panel. In this sequence of calculations, however, we did not yet allow for the lattice relaxation around impurities. It means that the interaction energies considered so far incorporate the effect of the polarization of the electron density, as it comes out from the self-consistent INDO calculation, in the background of all atomic cores (but those of two Li impurities) fixed. We compared the calculated interaction energies to the classical dipole-dipole interaction
E- = _~ [did2 _ 3 (rdi )(rd2)] mt C r3 r5·
In the right panel of Fig. 1, the (dimensionless) classical dipole-dipole interaction corresponding to the interaction of dipoles d= 1 positioned on the sites of the cubic lattice with a=l is shown on the right scale, and the calculated values - on the left scale. The periodicity of the translated dipoles (as in the supercell approach) was not accounted for when consid­ ering the dipole-dipole interaction. Because of this, a noticeable deviation
20
--~ 80 g Ci) 2.0 >.
1.5 f CD , C , + a. 0 , -c. n 40 , 0
~ , P 1.0 1D
III i 20 ,
0.5 S iii
a b c d e f 9 Impurity position
Figure 1. Left panel: Relative geometries of the Li impurity pairs considered. Impurity 1 is indicated by an open circle, impurity 2 occupies one of the positions a ... h. The supercell used in the calculations is shown, along with the translated Li atoms (impurity 1) in adjacent supercells. Right panel: Interaction energy of Li impurities in different geometries a ... g as obtained in the supercell calculation (crosses; left scale) and ,given by the dipole-dipole interaction for d=l and a=l (circles; right scale).
occurs, e.g., for the geometry (d) where the dipole-dipole interaction energy would be exactly zero in case of two interacting impurities; in the superceU calculation, however, the interaction energy of this impurity pair is sub­ stantial, because the impurity in (d) interacts with translated impurities in the adjacent supercells as well. Apart from this, the anisotropy of the dipole field (the difference between the interaction energy in (e) and (f) configurations) and the trend in the interaction strength falling down from (a) to (b) to (c) are in a reasonable agreement with what one could expect on the basis of the dipole-dipole interaction picture. The interaction energy in the (c) configuration is larger in the supercell calculation than follows from the simple dipole-dipole interaction, due to the fact that impurity 2 in (c) is already relatively close to the translated impurity 1. For geometries (a, b, e, f), where impurity 2 is the closest to impurity 1, the agreement with the dipole-dipole interaction is very good. From the constant factor by which the dipole-dipole interaction needs to be scaled to match the cal­ culated interaction energies, an estimation of the effective dipole value can be done. Specifically, the dimensionless dipole-dipole interaction (as shown
21
in Fig. 1) scales in a real system with (d*)2 j (c,a3), hence for a ",4 A and c ",400 (see, e.g., Ref. [20]) in the high-frequency region, i.e. without the contribution from the lattice relaxation, the enhancement of the nominal local dipole moment at the Li site d due to polarization of the electron system is d*jd "'13. With relaxation of oxygen atoms taken into account, the enhancement of the effective dipole moments is expected to be much larger than that, because both calculated Li-Li interaction energies and the static dielectric constant become larger (c "'4000) than those used in the above analysis.
As mentioned above, a model more sophisticated than the classical dipole-dipole coupling has been introduced by Vugmeister and Glinchuk (see Ref. [17] and references therein) to describe the interaction between individual Li dipoles in a highly polarizable medium like KTa03. The mean­ ingful comparison with the predictions of this model, however, needs the oxygen relaxation around impurity pairs to be taken into account in all geometries considered.
TABLE 1. The energy of Li-Li interaction with (Erel ax ) and without (Ebare) the 0 relaxation. Edd is the dipole-dipole inter­ action energy
Configuration Ebare (meV) Erelax (me V) Edd (d 2/r3)
(a) 62.0 105.0 2
(h) 2.7 19.1 -1
Displacements of neighboring oxygen atoms, however small (of the order of 1 % oflattice constant, on the largest - see ref. [19] for details), contribute more than 100% to the relaxation energy (as compared to that of a bare Li displacement in frozen oxygen cage) at each impurity site, and the inter­ action energy of two impurities in the presence of oxygen polarization may be scaled by a factor of 1 to 10, depending on the impurity configuration. In Table 2, interaction energies for three selected configurations of near­ est Li impurities are shown. The (dimensionless) energies of the nominal dipole-dipole interaction are listed in the last column. The enhancement of interaction due to the oxygen relaxation is well seen, as well as the fact that an isotropic correction to the nominal dipole-dipole interaction is present. The tendency towards long-range ferroelectric coupling appearing within the modified interaction model of Ref. [17] may explain the energy gain in the interaction between parallel dipoles at a distance normal to the dipole vector, like in the geometry (h) in our case and corresponding geometries
22
in Ref. [18]. For a more detailed analysis (Le., the extraction of the effective interaction parameters) calculated data for additional configurations would be essential.
3. Charged substitutional impurities: Fe2+, Fe3+ in KNb03
As an example of an impurity system which develops energy levels in the band gap and is important in photorefractive applications, one can mention substitutional Fe (in different charge configurations) in KNb03 , studied by us with the LMTO method in the supercell approach [21]. We summarize below the essential results of this study. Certain transition metal dopants in KNb03 are known to give rise to photorefractive effect, that is, the change of dielectric properties (and hence of the refraction index) under illumina­ tion. The explanation of the photorefractive effect typically assumes the existence of several charge states of an impurity, which can be switched by the drift of electrons or holes in the process of illumination. Regarding specifically Fe-doped KNb03 , one can address Ref. [22] for the description of the models available. There seems to be now a general agreement between the experimentalists that Fe preferentially enters the Nb site in KNb03 . In Ref. [21] we attempted to obtain reliable description of the local electronic structure of impurity, concentrating specifically on the treatment of dif­ ferent charge configurations within the localized d-shell. In particular, the treatment within the local density approximation (LDA) was compared with the so-called LDA+U approach.
The LDA+U method (see Ref. [23] for a review) was proposed as a convenient extension of the LDA scheme, aimed at overcoming the de­ ficiency of the exchange-correlation being strictly local in the latter. In terms of practical calculation for a particular compound, the nonlocality of exchange (which is automatically present in the Hartree-Fock formal­ ism, but the absence of correlation introduces much larger error there) may play an important role whenever strongly localized states are involved. The LDA+U method allows for nonlocality in a simplified way, by singling out "localized" states ad hoc and treating them in a different manner than conventional ones. The potential becomes dependent on the occupation of selected "localized" states in addition to its usual dependency on the to­ tal electron (spin) density. As has been shown in a number of studies (see Ref. [23]), the LDA+U method for certain systems corrects a qualitatively wrong description provided by the LDA. On the other side, selection of "localized" states may be ambiguous. In our study of the Fe impurity in KNb03 , both LDA and LDA+U schemes have been used in parallel, with the Fe3d orbitals as natural candidates for "localized" states selected for special treatment within the LDA+U formalism.
23
Since, contrary to the case of off-centre light impurities, precision total­ energy evaluation and structure relaxation around impurity were not (yet) primary objectives of our study, the calculations have been done with the LMTO method in the atomic sphere approximation [24], using the Stuttgart tight-binding LMTO code [25] as its practical implementation. In the choice of an appropriate supercell for simulation of (ideally) isolated 3d impurities, we compared the results obtained with the perovskite cell doubled in all three directions (i.e., including 40 atoms in total) with two times larger supercell. The latter was found to be better suited for the modeling of isolated impurities, with their localized energy levels well pronounced in the band gap (see Ref. [21] for details). However, the charge and the magnetic moment at the Fe site are almost identical in the calculation with both supercells. The results discussed below refer to the 40-at. supercell.
TABLE 2. Charges Q and magnetic moments M within atomic spheres of Fe impurity and its several neighboring atoms as calculated for different numbers of added electrons within the LDA and with the LDA+U method. The atomic sphere radii are 1.639 A for Fe and Nb, 1.964 A for K and 1.050 A for 0
Fe Oxy Oz Nbxy Nbz K extra e Q M Q M Q M Q M Q M Q M
0 8.78 1.69 5.88 0.24 5.90 0.27 4.96 0.01 5.04 -0.01 6.98 0.00 1 8.79 1.68 5.90 0.19 5.91 0.24 4.98 0.01 5.05 0.00 6.99 0.00 2 8.93 3.13 5.90 0.43 5.90 0.53 5.02 0.02 5.09 0.01 7.00 0.00
2 C+U) 8.85 3.31 5.92 0.41 5.91 0.51 5.01 0.02 5.09 0.01 7.01 0.00 3 8.92 3.13 5.93 0.33 5.93 0.42 5.03 0.04 5.10 0.03 7.02 0.00
3 C+U) 8.88 3.20 5.93 0.34 5.93 0.45 5.05 0.02 5.11 0.00 7.02 0.00
A special treatment of charge compensation is not necessary in a first­ principles simulation, but the nominal charge state Fe5+ in the absence of charge compensation is hardly realistic in the crystal in question. The Fe impurity tends to reduce this oxidation state by binding 2 or 3 extra elec­ trons available due to the presence of oxygen vacancies, or other defects. There are essentially only two ways to account for a charge compensation in the calculation - one either specifies explicitly the configuration of impuri­ ties/vacancies that provides such compensation, or adds extra electrons to the system, implying that their donors are in some distant parts of crystal and do not affect the local electronic structure. The second way is probably technically easier, because one needs only to search for the Fermi energy corresponding to the specified number of extra electrons in each iteration. In order to keep the supercell neutral, a compensating positive charge is
24
8
4 t2g eg
> t2g eg l
-4 t2g t2g
4 eg
~ I 0
-8 -8 -4 0 4 8 -8 -4 0 4
Energy (eV) Energy (eV)
Figure 2. Local 3d-DOS for charge configurations with 2 and 3 additional electrons per SO-at. supercell of KNb03 :Fe, as calculated in the LDA and LDA+U.
added in the background. This is the way we used for simulating Fe2+ and Fe3+ charge configurations.
The number of electrons and magnetic moments within atomic spheres of Fe and its nearest neighbors of different types are shown in Table 2 for several numbers of extra electrons. Since KNb03 is not so strongly ionic, and because of our choice of atomic sphere radii, the electron numbers in Table 2 are close to those of neutral atoms. Magnetic moment, induced by
25
that of the impurity, primarily resides on the first oxygen sphere.
Comparing the results of LDA and LDA+U calculations (Fig. 2), one should keep in mind that the overall effect of the latter is the lowering of en­ ergies of the occupied states and the upward shift of the vacant ones. This is exactly what happens in the configuration with 2 extra electrons. Since es­ sentially all majority-spin states are already occupied in this configuration and all minority-spin states empty, the inclusion of the U-correction has a negligible effect on all integral properties (both charges and moments). However, the exact position of the t2g-impurity state is changed, which may lead to the change in the optical absorption. The situation in the configuration with 3 extra electrons is completely different. Since here the minority-spin t2g state is partly occupied in the LDA, in the LDA+U treat­ ment the potentials acting on the xy, xz and yz components of it become different, thus lifting the orbital degeneracy of these states and distorting the local DOS considerably.
Summarizing, our study of the electronic structure of Fe impurity in the Nb site of KNb03 with the analysis of different charge compensation has shown, that only two considerably different configurations of impurity oc­ cur. The first one, with the magnetic moment rv 1. 7 I"B' corresponds to the substitutional impurity without compensation - the situation that prob­ ably is not very common in reality. This configuration, however, survives under addition of one extra electron per impurity. Two extra electrons (for instance, due to a distant oxygen vacancy) induce a transition into the high-spin state with the magnetic moment rv3.1I"B' with the minority-spin t2g level in the band gap. The charge and magnetic moment of this config­ uration remain intact with the addition of the third extra electron, but the exact position of the level in the gap (probably, its splitting as well) may be affected. The last two configurations with 2 or 3 extra electrons correspond to the most practically relevant impurity configurations, referred to as Fe3+ and Fe2+. A more precise study of their energetics is possible with the use of a full-potential calculation scheme and simultaneous analysis of lattice relaxation around impurity and, optionally, oxygen vacancy.
Acknowledgments This study was supported by the German Research Society (SFB 225) and by the Volkswagen Foundation (grant to R. E.).
References
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2. Singh, D. J. and Boyer, L. L. (1992) First principles analysis of vibrational modes in KNb03 , Feroelectrics 136, 95-103.
26
3. Postnikov, A. V., Neumann, T., Borstel, G. and Methfessel. M. (1993) Ferroelectric structure of KNbOs and KTaOa from first-principles calculations, Phys. Rev. B 48, 5920-5918.
4. Postnikov, A. V. and Borstel, G. (1994) r phonons and microscopic structure of or­ thorhombic KNbOa from first-principles calculations, Phys. Rev. B 50, 16403-16409.
5. Postnikov, A. V., Neumann, T. and Borstel, G. (1994) Phonon properties of KNbOa and KTaOa from first-principles calculations, Phys. Rev. B 50, 758-763.
6. Eglitis, R. I. and Postnikov, A. V. and Borstel, G. (1997) Semiempirical Hartree-Fock calculations for pure and Li-doped KTa03, Phys. Rev. B 55, 12976-12981.
7. Dorfman, S., Fuks, D., Gordon, A., Postnikov, A. V. and Borstel, G. (1995) Movement of the interphase boundary in KNb03 under pressure, Phys. Rev. B 52, 7135-7141.
8. Eglitis, R. I., Christensen, N. E., Kotomiri, E. A., Postnikov, A. V. and Borstel, G. (1997) First-principles and semiempirical calculations for F centers in KNbOa , Phys. Rev. B 55, pp. 8599-8604.
9. Kotomin, E. A., Eglitis, R. I., Postnikov, A. V., Borstel, G., and Christensen, N. E. (1999) First-principles and semiempirical calculations for bound-hole polarons in KNb03 , Phys. Rev. B 60, 1-5.
10. Yakoby, Y. and Just, S. (1974) Differential Raman scattering from impurity soft modes in mixed crystals of K1 - xNax Ta03 and K1-xLix TaOa, Solid State Commun. 15, 715-718.
11. Borsa, F., Hochli, U. T., van der Klink, J. J., and Rytz, D. (1980) Condensation of random-site electric dipoles: Li in KTa03, Phys. Rev. Lett. 45, 1884-1887.
12. Hochli, U. T., Knorr, K., and Loidl, A. (1990) Orientational glasses. Adv. Phys. 39, 405-615.
13. van der Klink, J. J. and Khanna, S. N. (1984) Off-center lithium ions in KTaOa, Phys. Rev. B 29, 2415-2422.
14. Stachiotti, M. G. and Migoni, R. L. (1990) Lattice polarization around off-centre Li in LixK1- x TaOs, J. Phys.: Condens. Matter 2, 4341-4350.
15. Exner, M., Catlow, C. R. A., Donerberg, H., and Schirmer, O. F. (1994) Atomistic simulation studies of LiK off-centre defects in KTaOs: I. Isolated defects, J. Phy.~.: Condens. Matter 6, 3379-3387.
16. Postnikov, A. V., Neumann, T., and Borstel, G. (1995) Equilibrium ground state structure and phonon properties of pure and doped KNbOs and KTaOs, Ferroelectrics 164, 101-112.
17. Vugmeister, B. E. and Glinchuk, M. D. (1990) Dipole glass and ferroelectricity Lll
random-site electric dipole systems, Rev. Mod. Phys. 62, 993-1026. 18. Stachiotti, M., Migoni, R., Christen, H.-M., Kohanoff, J., and Hochli, U. T. (1994)
Effective Li-Li interactions in K1-xLix TaOs, J. Phys.: Condens. Matter 6, 4297-4306. 19. Eglitis, R. I., Postnikov, A. V., and Borstel, G. (1998) Semi-empirical Hartree-Fock
simulations of lattice relaxation and effective interactions in Li-doped KTa03, Phys. Status Solidi B 209, 187-193.
20. Maglione, M., Rod, S., and Hiichli, U. T. (1987) Order and disorder in SrTiOa and in pure and doped KTaOs, Europhys. Lett. 4, 631-636.
21. Postnikov, A. V., Poteryaev, A. I., and Borstel, G. (1998) First-principles calcula­ tions for Fe impurities in KNb03, Ferroelectric8 206-207, 69-78.
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Phys. Rev. Lett. 53, pp. 2571-2574.
POINT DEFECTS, DIELECTRIC RELAXATION AND CONDUCTIVITY IN FERROELECTRIC PEROVSKITES
M.MAGLIONE LPUB Universite de Bourgogne CNRS BP 47870 F-21078 DIJONCedex France e-mail: maglione@u-bourgogneJr
Abstract. In all ferroelectric perovskites, intentionally introduced or "unwanted" point defects do playa role in the dielectric spectra and in the conductivity.
Special emphasis is brought on SrTi03 which has received renewed interest at the beginning of the nineties and for which reliable experimental data are available. Considering that a gradual freezing of polarized objects is occurring at low temperatures, one can reconcile most of these data within the same model. Even more interesting, one can show that point defects and quantum fluctuations do contribute to this freezing. This leads us to the new concept of quantum
polarons which are very specific to SrTi03.
l.Introduction
Ferroelectric AB03 perovskites are very sensitive to ionic substitution in the A or B site. It is well known that ferroelectricity may be induced or suppressed when the amount of such point defects reaches critical values [1, 2]. Such transition does not result from the interaction between individual defects but from the 3D percolation of extended polarization clouds surrounding each defect[3]. In some cases, it was possible to quantify the space extension of such polarized clusters by a systematic comparison between several experiments[4]. Such achievement is successful only in high quality single crystals in which the number of residual or "unwanted" point defects is really negligible. This requirement is by no way simple in lattices with high correlation length: non correlated defects should be very far away from each other which means that the
27 G. Borstel et al. (eds.), Defects and Surface-Induced Effects in Advanced Perovsldtes, 27-36. @ 2000 Kluwu Academic Publishers.
28
average distance between point defects is to be very large and their density very small. These concepts of correlation length, polarized clusters and percolation are applied by several groups to relaxors [5], dipolar glasses [6], and quantum ferroelectrics [7]. In the latter case, when no point defects are intentionally brought in the host lattice, some observations may be ascribed to unwanted
defects. In nominally pure KTa03 a slight but well defined maximum of dielectric losses was reported [8] and confirmed in doped crystals [9,10]. Such losses were shown to be related to unwanted point defects in a number of perovskites and a polaronic model was· drawn for this relaxation [9]. In
nominally pure SrTi03, a similar anomalous loss was observed [11] and we suggested that a similar polaronic model should apply to this compound. In this paper, it is shown that a continuous, thermally activated relaxation is able to reconcile most of the experimental reports available in SrTi03. Moreover, in the lowest temperature range, it is proposed that the "unwanted" polarons fall in the quantum regime of motion.
2.Three-Dimmensional percolation in mixed perovskites
2.1 POLARIZED CLUSTERS AROUND EACH POINT DEFECT
When inserting point defects in a polarizable lattice like AB03 ferroelectric perovskites, polarized clusters set in around each defect. As an example, in pure and doped KTa03, one could successfully compare the cluster radius or correlation length gain from several spectra: Inelastic Neutron Scattering, Hyper Raman, Nuclear Magnetic Resonance, Second Harmonic Generation [4,12]. In nominally pure samples, the cluster seed is a symmetry-breaking point defect which cannot be avoided even in the best single crystals. Among these are oxygen vacancies resulting from the equilibrium between the oxygen partial pressure and the crystal during the growth [8] and iron coming from the starting powders [13]. Using this concept of polarized cluster, one can draw a schematic map (Fig. I) of the cluster density versus the impurity content with the following assumptions :
I-the point defects -the cluster seed- are randomly distributed within the host lattice
2-the correlation length is a property of the polarizable host lattice and does not depend on the impurity content On this map, the density of clusters stands for the ratio of the total volume of the clusters to the crystal volume. Thus, full percolation occurs when the cluster density is 1. Both the above assumptions are very crude mainly in the 50/50
29
substitution rate where inhomogeneous impurity substitution is unavoidable. When the correlation length is equal to the lattice unit cell (Lc= 1), the full percolation between point defects occurs only when all the lattice sites have been replaced by a point defect. This extreme case is never observed in ferroelectric perovskites. In these polarizable compounds, the linear correlation length lies between 5 and 100 unit cells, the higher the lattice correlation length, the smaller the density of point defects where the percolation occurs (homogeneous range in Fig. 1). As recalled above, this picture obviously fails when the density of defects is
high. This is usually the case of relaxor compounds like PbMg1l3Nb2l303 in which a diffuse transition results from the occurrence of 111 chemically ordered regions on the B site. The size and shape of such regions is still a matter of debate but for our purposes, one can state that even for small correlation lengths (Lc<5 unit cells), percolation between polarized clusters always occurs (Inhomogeneous and correlated range in Fig. 1). As a consequence, a diffuse dielectric anomaly is observed at rather high temperatures (T>200K) with a strong dispersion in the dielectric spectrum [5].
0,8
Impurity content
Figure J : Schematical map of the density of clusters versus the impurity content in ferroelectric perovskites. The density of clusters is the ratio between the total volume of clusters to the crystal volume. Different correlation length (in units of the lattice parameter) leads to different behaviour as indicated.
30
2.2 PERCOLATION AND NON-LINEARITY
Reducing the density of point defects like 10 KTa03:Li (xu<5%) or in
SrTi03:Ca (xca<O.5%) implies a higher correlation length Lc>5 unit cells. In this concentration range, the random interaction between polarized clusters may result in a glass-like phase called dipolar glass [6]. Among the many interesting features of this familly of compounds, one can stress that the dielectric non­ linearity is extremely high at the percolation threshold. This threshold can be defined as the impurity content from which a well defined dielectric maximum is
observed at a given temperature. This occurs in KTal-xNhx03 for XNb=O.8%, in
Kl_xNaxTa03 for xNa=12% and in Sq_xCaxTi03 for XCa=O.4%. The lowest
order dielectric non-linearity ENL was probed under dc bias using the following expansion:
(1)
10 ... • • • ...l 10 -5 ~ .... • t!)z • ...... :t • 10 -6 " \
KTa03:Na .. 10 ·7 • xNa=20%
, .. " .. 10..a
0 10 20 30 40 50 60 T (K)
Figure 2 : dielectric non-linearity in Sq _xCax Ti03 for xCa=O to 0.58% and in K 1-
xNax Ta03 from xNa=O to 24%. In both cases, the maximum non-linearity is observed
near the percolation threshold xCa=O.4% and xNa=16%. Same observations in KTal_
xNbx03 for xNb=1.8%. From reference 14
31
Dielectric non-linearities ENL were shown to be of maximum value at the percolation threshold of the above compounds as plotted in Fig. 2 [14].
2.3 DIELECTRIC RELAXATION AND POLARONS
In the lowest concentration range of point defects, x<0.1 %, no well defined dielectric anomaly related to the defects are observed in the lowest temperature
range. This is the case for incipient ferroelectrics KTa03 and SrTi03 and also
for classical ferroelectrics BaTi03 and PbTi03. In the former case the linear
dielectric susceptibility eL increases when cooling to OK without any maximum,
in the latter EL is almost stable because the temperatures under interest here are far from the ferroelectric transition temperature. In such a case one is looking to
the frequency spectrum of EL in order to find possible dielectric dispersion resulting from the localised dynamics of the polar clusters. This was indeed observed in a large number of pure and substituted perovskites starting from
nominally pure KTa03 [8]. A maximum of dielectric losses (imaginary part of the linear dielectric susceptibility) was observed in the vicinity of 30K, the exact temperature increasing with the operating frequency. This observation closely fit in the Cole Cole model:
* ( ) &8 - &"" & \OJ = &"" + . I-a
1 + 1OJ" (2)
Where the L index will be omitted in the following, 1::00 is the high frequency
extrapolation of I:: and I::s the low frequency one. The Cole Cole exponent quantifies to what extend the dielectric dispersion is far from single Debye (a=O). The relaxation time was shown to follow an Arrhenius law with an activation energy of about 70me V. The same results were reported in a large number of perovskite single crystals and ceramics whatever their formula, texture, lattice symmetry and ferroelectric properties [9,10]. The Arrhenius activation energy was always 70-80meV as evidenced in Fig. 3. On this figure all the data points on the left hand side are the relaxation time for about 50 samples from 4 different research teams [8,9,10,15]. As to the model for this common observation, not everybody agrees but some features are found in all the above reports :
-the amplitude of the loss maximum is sensitive to the density of heterovalent point defects such as oxygen vacancies, Fe3+, La3+
-in nominally pure samples, heterovalent defects are present as unwanted impurities as recalled above
32
10-4 1.·+ += 0 x 0 Q)
10-6 a::
10~ ~~~~~~~~~~~~~~-L~~ o 0,1 0,2 0,3 0,4
l/T(K) Figure 3 : relaxation map of pure and substituted ferroelectrics. The Arrhenius straight
line on the left include independent observation on about 50 samples from 4 research
teams [8,9,10,15) (see sub-section 2.3 of the text). The specific behaviour of pure and Ca
substituted SrTi03 is clear on the right [9,11,18) (see section 3).
Taking into account these dielectric data and recalling that vibronic states usually result from heterovalent substitution[12,13], one can draw a model including polarized clusters as well as hopping conduction. We assumed that this coupling to be analogous to polaronic excitons[9] as proposed long ago [16] in heavily doped perovskites. Such coupling of electrons with polarized clusters was also used to explain the semi-conducting properties of ferroelectric
AB03 perovskites [17]. It is only during the last few years that the signature of such excitations could be found directly in the dielectric spectra of these compounds. In figure 3, we call these polarons classical since no bending of the Arrhenius law is observed, meaning that the activation energy is temperature
independent. This is not the case for SrTi03 which is discussed in part 3 below.
3. Polarized clusters in SrTi03•
As in all perovskites cited above, a dielectric loss anomaly was observed in pure
and Ca-substituted SrTi03 [9,11,18]. Since a lot of debate is going on about this compound in the recent literature, it is not the aim of this part to discuss all the available data. It is rather proposed that most of the spectroscopies agree with a dynamical slowing down of polarized clusters and that the polaronic picture
drawn for other perovskites can apply to SrTi03.
33
3.1 HIGH TEMPERATURE SLOWING DOWN
In figure 3, one can see that for temperatures higher than 10K, the mean relaxation time follows an Arrhenius law [9,11,18]. Even though the temperature range of this Arrhenius law is limited, its activation energy is less than O.leV which is not far from the one computed in other perovskites(see part 2.3 above). This limited temperature range is due to the operating frequency range of
dielectric spectroscopy (l0·3Hz to 106Hz), Thanks to the renewed interest in
SrTi03 which followed the paper by Muller et al [19], a number of spectroscopic data are available. Since these spectroscopies are operating at higher frequencies than dielectric spectroscopy, one can try to fit all the data on the same Arrhenius plot. To this aim, we will focus on EPR and Brillouin spectroscopies which provided reliable dynamical information. From Brillouin scattering, two main observations were recently reported:
-a doublet in the very low frequency part of the Brillouin spectra clearly appears at temperatures below 40K in single domain crystals [20,21]. This doublet was ascribed to second sound [20]. Since no temperature anomaly was reported for the width and frequency of this doublet, no definite point can be added to the Arrhenius plot
-the first T A mode was found to couple strongly with the lowest frequency TO soft mode through "gradient of polarization" [20]. As a
consequence, the full width at half maximum of this TA mode rTA displays a
minimum at about 30K. Since rTA includes the dynamical motion of polarization gradients, we can put a new point on the relaxation map whose
abscissa is the temperature where the minimum of r T A occurs and whose
ordinate is the inverse of 21trTA at this temperature (Fig. 4). In fact, The data
and error bars are taken from reference 20: rTA =1 5 OMHZ±5OMHz. We can gain another dynamical data point from EPR data [19]. Using the EPR
signal ofFe3+ impurities substituted to Ti4+, Muller et al. could find a dip in the
hyperfine splitting at 37K. It is not the absorption of the SrTi03:Fe crystals which was modified because this would only raise the intensity of EPR lines and
this was not observed. It is the Electric Field Gradient(EFG) at the Fe3+ site which is modulated by polarization fluctuations. When this modulation frequency is equal to the hyperfine frequency, the spatial averaging of the EFG by the polarization fluctuations can decrease the hyperfine splitting. This 37K
dip in the Fe3+ hyperfine splitting can thus be plotted on the relaxation map of
figure 4 with 1I37K as an abscissa and 1/21tf as an ordinate; f = 900MHz is the EPR hyperfine splitting of 340Gauss translated in units of frequencies[ 19]. The
34
10-4 ,....-------