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A.S.Alexandrov (Ed.)Department of PhysicsLoughborough UniversityLoughborough LE11 3TUUnited Kingdom

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Polarons in Advanced Materials

A. S. Alexandrov (ed.)

Department of Physics, Loughborough University, Loughborough LE11 3TU,United KingdomPhone: 01509 223303Fax: 01509 [email protected]

Dedicated to Sir Nevill Francis Mott (1905-1996), whose researchon metal-insulator transitions, polarons and amorphous

semiconductors has had tremendous impact on our currentunderstanding of strongly correlated quantum systems

Preface

Conducting electrons in inorganic and organic matter interact with vibratingions. If the interaction is sufficiently strong, a local deformation of ions, cre-ated by an electron, transforms the electron into a new quasiparticle calleda polaron as observed in a great number of conventional semiconductors andpolymers. The polaron problem has been actively researched for a long time.The electron Bloch states and bare lattice vibrations (phonons) are well de-fined in insulating parent compounds of semiconductors including the ad-vanced materials discussed in this book. However, microscopic separation ofelectrons and phonons might be rather complicated in doped insulators sincethe electron-phonon interaction (EPI) is strong and carriers are correlated.When EPI is strong, the electron Bloch states and phonons are affected.

If characteristic phonon frequencies are sufficiently low, local deforma-tions of ions, caused by the electron itself, create a potential well, which bindsthe electron even in a perfect crystal lattice. This self-trapping phenomenonwas predicted by Landau in 1933. It was studied in great detail by Pekar,Fröhlich, Feynman, Rashba, Devreese, Emin, Toyozawa and others in the ef-fective mass approximation for the electron placed in a continuous polariz-able (or deformable) medium, which leads to a so-called large or continuumpolaron. Large polaron wave functions and corresponding lattice distortionsspread over many lattice sites, which makes the lattice discreteness unimpor-tant. The self-trapping is never complete in a perfect lattice. Since phononfrequencies are finite, ion polarizations can follow polaron motion if the motionis sufficiently slow. Hence, large polarons with a low kinetic energy propagatethrough the lattice as free electrons with an enhanced effective mass.

When the characteristic polaron binding energy Ep becomes compara-ble with or larger than the electron half-bandwidth, D, of the rigid lattice,all states in the Bloch band become “dressed” by phonons. In this strong-coupling regime, λ = Ep/D > 1, the finite bandwidth and lattice discretenessare important and polaronic carriers are called small or lattice polarons. In thelast century many properties of small polarons were understood by Tyablikov,Yamashita and Kurosava, Sewell, Holstein and his school, Firsov, Lang and

VIII Preface

Kudinov, Reik, Klinger, Eagles, Böttger and Bryksin and others. The charac-teristic fingerprints of small polarons are a band narrowing and multi-phononfeatures in their spectral function (the so-called phonon side-bands).

Interest in the role of EPI and polaron dynamics in contemporary ma-terials has recently gone through a vigorous revival. There is overwhelmingevidence for polaronic carriers in novel high-temperature superconductors,colossal-magnetoresistance (CMR) oxides, conducting polymers and molecularnanowires. Here we encounter novel multi-polaron physics, which is qualita-tively different from conventional Fermi-liquids and conventional superconduc-tors. The recent interest in polarons extends, of course, well beyond physicaldescriptions of advanced materials. No general solution to the polaron problemexists for intermediate λ in finite dimensions. It is the enormous differencesbetween weak and strong coupling limits and adiabatic and nonadiabatic lim-its which make the polaron problem in the intermediate λ regime extremelydifficult to study analytically and numerically. The field is a testing ground formodern analytical techniques, including the path integral approach, unitarytransformations, diagrammatic expansions, and numerical techniques, suchas exact numerical diagonalisations, advanced variational methods, and novelQuantum-Monte-Carlo (QMC) algorithms reviewed in this book.

Polarons in Advanced Materials is written in the form of self-consistentpedagogical reviews authored by well-established researchers actively workingin the field. It will lead the reader from single-polaron problems to multi-polaron systems and finally to a description of many interesting phenomenain high-temperature superconductors, ferromagnetic oxides, and molecularnanowires.

The book naturally divides into four parts, following historical reminis-cences by Rashba on the early days of polarons. Part I opens with a com-prehensive overview by Devreese of the optical properties of continuum all-coupling polarons in all dimensions in the path-integral based theory. The sec-ond chapter by Firsov introduces small polarons, the Lang-Firsov canonicaltransformation and small polaron kinetics. Detailed analysis of magnetotrans-port and spin transport in the hopping regime of small polarons is presentedby Böttger, Bryksin and Damker in chapter 3. Chapter 4 (Cataudella, DeFilippis and Perroni) presents large and small polaron models from a unifiedvariational point of view. The fifth chapter by Kornilovitch offers a compre-hensive tutorial on the path-integral approach to all-coupling lattice polaronswith any-range EPI including Jahn-Teller polarons based on novel continuous-time QMC (CTQMC). Part I closes with the path integral description ofpolarons by Zoli in the Su-Schrieffer-Heeger model of the EPI important inlow-dimensional conjugated polymers and related systems.

Part II opens with the strong-coupling bipolaron theory of superconduc-tivity and a discussion of small mobile bipolarons in cuprate superconductorsby Alexandrov (chapter 7). Aubry analyses a small adiabatic polaron, an adia-batic bipolaron and multi-polaron adiabatic systems in chapter 8, where theo-rems for the adiabatic Holstein-Hubbard model are formulated and the role of

Preface IX

quantum fluctuations for bipolaronic superconductivity is emphasised. Nano-scale phase separation and different mesoscopic structures in multi-polaronsystems are described by Kabanov in chapter 9 with the realistic EPI and along-range Coulomb repulsion.

Part III starts with a complete numerical solution of the Holstein polaronproblem by exact diagonalization (ED) including bipolaron formation andrelates the results to strongly-correlated polarons in high-temperature super-conductors, CMR oxides and other materials (Fehske and Trugman). Chapter11 by Hohenadler and von der Linden describes the canonical transforma-tion based QMC and variational approaches to the Holstein-type models withany number of electrons. Strongly-correlated polarons in relation to high-temperature superconductors are further reviewed by Mishchenko and Na-gaosa in chapter 12, where basics of recently developed Diagrammatic MonteCarlo (DMC) method are discussed.

Part IV includes a comprehensive review by Mihailovic of photoinducedpolaron signatures in conducting polymers, cuprates, manganites, and otherrelated materials (chapter 13). Zhao reviews polaronic isotope effects and elec-tric transport in CMR oxides and high-temperature superconductors (chapter14), which are further reviewed by Bussmann-Holder and Keller in chapter15, where a two-component approach to cuprate superconductors with pola-ronic carriers is described. The final chapter by Bratkovsky presents a detaileddescription of electron transport in molecular scale devices including rectifica-tion, extrinsic switching, noise, and a theoretically proposed polaronic intrinsicswitching of molecular quantum dots.

This contemporary encyclopedia of polarons is easy to follow for seniorundergraduate and graduate students with a basic knowledge of quantummechanics. The combination of viewpoints presented within the book canprovide comprehensive understanding of strongly correlated electrons andphonons in solids. The book would be appropriate as supplementary read-ing for courses in Solid State Physics, Condensed Matter Theory, Theory ofSuperconductivity, Advanced Quantum Mechanics, and Many-Body Phenom-ena taught to final year undergraduate and postgraduate students in physicsand math departments. The subject of the book is of direct relevance tothe design of novel semiconducting, superconducting, and magnetic bulk andnano-materials. Their long term potential could be fully realised if an increasein fundamental understanding is achieved. The book will benefit researchersworking in condensed matter, theoretical and experimental physics, quantumchemistry and nanotechnology.

It is a great pleasure and honor for the Editor to present these collectedreviews. I thank our distinguished authors for sharing their insights and ex-pertise in polarons.

Loughborough University, August 2006 Sasha Alexandrov

Reminiscences of the Early Days of PolaronTheory

Emmanuel I. Rashba

Department of Physics, Harvard University, Cambridge, Massachusetts 02138,USA [email protected]

This volume, covering various aspects of modern polaron physics, its highlightsand challenges, appears at a time that is quite remarkable in the history ofpolarons. Exactly 60 years before the compilation of this volume, the seminalpaper by Solomon Pekar that initiated the theory of polarons was published [1,2]. In it, a model of the large polaron was developed and the term polaron wasproposed. The volume will appear early in 2007, close to the 90th anniversaryof Pekar’s birthday, March 16, 1917.

Referring to the famous Landau paper in which the possibility of electronself-trapping was first conceived [3], Pekar developed a macroscopic modelthat became a cornerstone of the theories that were to follow. The coupling ofan electron to a polar lattice was expressed in terms of a dielectric continuum.The inertial part of its polarization supported the electron self-consistently ina self-trapped state. The coupling constant for this mechanism is the Pekarfactor κ−1 = �−1∞ −�−10 , where �∞ and �0 are high and low frequency dielectricconstants, respectively. For the strong (adiabatic) coupling limit, Pekar cal-culated the ground state energy of a polaron, proved that the energies of itsoptical and thermal dissociation differ by a factor of 3, and established exactrelations between different contributions to the polaron energy. The title ofthe paper emphasizes local states in an ideal ionic crystal, but drift of thepolaron in an electric field was also envisioned. The concept of a polaron asa charge carrier in ionic crystals was developed in a following paper [4], andwas supported by calculating the polaron mass by Landau and Pekar [5]. Inthis way, the original concept of self-trapping as formation of crystal defectslike F-centers in alkali-halides [3] evolved into the concept of polarons as freecharge carriers in polar crystals.

The beginning of Pekar’s scientific career was quite remarkable. On theeve of the Nazi invasion of the USSR in the spring of 1941, he was awardedthe degree of a Doctor of Science (similar to Habilitation) for his PhD dis-sertation, which was an extraordinary event. Landau concluded Pekar’s talkat his seminar with the comment: “The self-conception of theoretical physicshas happened in Kiev.” During the war Pekar worked on defense projects,

XII Emmanuel I. Rashba

and after returning to Kiev he established there a Theoretical Division atthe Institute of Physics and a Chair in Theoretical Physics at the University.Pekar’s former colleagues and friends, who came back from their service in theArmy, became his graduate students and worked enthusiastically on differentaspects of the polaron theory. In the framework of the strong coupling limit, itwas shown that for electrons coupled to the lattice by a deformation potentialthere are no macroscopic self-trapped states in 3D [6], but they exist in 1Dwith 2D as a critical dimension [7], and the possibility of exciton self-trappingin polar crystals was also proven [8].

As an undergraduate student, I joined Pekar’s group in the late 1940s andremain the last witness of those developments in Kiev where polaron theorywas initiated, and of the loose contacts with the related work in the Westthat were possible only through published papers, under the conditions ofthe self-imposed political isolation of the USSR. Hence, the present brief noteis restricted to this subject of which I have primary knowledge, and is notintended to cover the different aspects of polaron theory that are reflected inthe vast review literature.

The next step was generalizing the semiclassical approach of [1] and [5] toa consistent quantum theory. Pekar drafted the first version of such a theory[9] by introducing a zero-mode related to the motion of the polaron center. Itsthree degrees of freedom come from the phonon system whose energy is auto-matically reduced by 3ω/2, ω being the phonon frequency. The scattering ofcarriers by phonons is reduced because the dominant part of electron–phononcoupling is included through the polaron energy. The field-theoretical aspectsof polaron theory attracted the attention of Bogoliubov who was working inKiev at that time, and in collaboration with Tyablikov he developed anotherversion of the quantum dynamics of adiabatic polarons [10, 11]. Because, afterdressing an electron by a phonon cloud, the effective Hamiltonian is quadraticin phonon amplitudes, it allows the finding of the renormalized phonon spec-trum and makes two-phonon processes a major scattering mechanism for adi-abatic polarons [12].

There are several factors to the theory of a large polaron that attractedthe close attention of theorists with a wide range of scientific interests. First,it presented a field theory model without divergences that enabled a consis-tent analysis at an arbitrary coupling constant and became a prototype for anumber of self-localized states in nonlinear field theories. Second, the electron-phonon interaction emerged as a prospective mechanism of superconductivitybecause of the discovery of the isotopic effect and different arguments [13].Third, electronic transport in polar conductors was of fundamental interestper se. Meanwhile, although the adiabatic limit is highly instructive in clari-fying the essential differences between free electrons and polarons, there existessential constraints on its applicability. Indeed, the Fröhlich electron-phononcoupling constant is α = (e2/�κ)

√m/2�ω, m being the electron effective

mass, and the ground state energy of an adiabatic polaron E0 and its effectivemass mp are E0 ≈ −0.1α2�ω [1] and mp ≈ 0.02α4m [5]. The small numerical

Reminiscences of the Early Days of Polaron Theory XIII

coefficients in both equations imply a strict criterion for the adiabatic limit,α � 10. With m ≈ m0 and κ ∼ 1 this inequality can be fulfilled because αis about (M/m0)1/4 and therefore large (here M is the ion mass). However,under these conditions macroscopic description fails because the polaron ra-dius becomes approximately a lattice constant. Fortunately, in many crystalsm � m0 and κ � 1, hence, the macroscopic description that is central to alarge polaron theory is justified.

At this point, polaron theory splits into two branches. The first one dealswith small (Holstein [14]) polarons where the detailed mechanism of a strongelectron-phonon coupling is not of primary importance, while the second dealswith large (Pekar–Fröhlich) polarons with polar electron-phonon interactionthat is not necessarily strong. The difference in their properties may be sig-nificant, e.g., the effective mass of small polarons increases with the couplingconstant exponentially, while for large adiabatic polarons by a power law.

Fröhlich et al. [15] developed a theory of weakly coupled large polarons,α � 1. The next step in bridging the gap between weakly and strongly cou-pled polarons was made by Lee, Low, and Pines whose variational approachworks for α � 5 [16]. Among the different variational schemes, the Feynmantechnique that allowed the finding of E0 and mp with high accuracy for allα values had the largest impact [17]. Pekar and his collaborators general-ized Feynman’s approach for calculating thermodynamical functions [18] andproposed an independent, simple, and efficient variational procedure [19].

Comparing the results of polaron theory with experimental data is a chal-lenging task. Historically, such a comparison began with multiphonon spec-tra of impurity centers that are conceptually closely related to the theory ofstrongly coupled polarons. The theory of such spectra was initiated indepen-dently by Huang and Rhys [20] and Pekar [21]. It is seen from the reviewpaper by Markham [22] that some of Pekar’s papers on this subject, pub-lished in Russian, were translated into English by different researchers andcirculated in the West long before the regular translation of Soviet journalsby the American Institute of Physics began.

Polaron effects manifest themselves even more spectacularly in the co-existence of free and self-trapped states of excitons that was observed byoptical techniques. While Landau envisioned a barrier for self-trapping [3],Pekar has shown [9] that such a barrier is absent for polarons that are formedby a gradual lowering of their energy, which is tantamount to a sequence ofsingle-phonon processes. However, for short range coupling to phonons thefree states may persist as metastable states even in the presence of deepself-trapped states [7]. Such free states are protected by a barrier and de-cay through a collective (instanton) tunneling of a coupled electron-phononsystem [23]. Conditions for the existence of the barrier for a Wannier–Mottexciton in a polar medium are nontrivial because the whole particle is neutralbut each component of it is coupled to the lattice by polar interaction. Theywere clarified in [24], which was the very last of Pekar’s papers related to po-larons. Since 1957, he directed his attention mostly to additional light waves

XIV Emmanuel I. Rashba

near exciton resonances that he predicted [25], and to some other problemsof the solid state theory.

I am grateful to M. I. Dykman and V. A. Kochelap for their help andadvice.

References

1. S. I. Pekar, Journal of Physics USSR 10, 341 (1946).2. S. Pekar, Zh. Eksp. Teor. Fiz. 16, 341 (1946).3. L. D. Landau, Sow. Phys. 3,664 (1933).4. S. I. Pekar, Zh. Eksp. Teor. Fiz. 18, 105 (1948).5. L. D. Landau and S. I. Pekar, Zh. Eksp. Teor. Fiz. 18, 419 (1948).6. M. F. Deigen and S. I. Pekar, Zh. Eksp. Teor. Fiz. 21, 803 (1951).7. E. I. Rashba, Opt. Spektrosk. 2, 75 and 88 (1957).8. I. M. Dykman and S. I. Pekar, Dokl. Acad. Nauk SSSR 83, 825 (1952).9. S. I. Pekar, Research in Electron Theory of Crystals, US AEC Transl. AEC-tr-555

(1963) [Russian edition 1951, German edition 1954].10. N. N. Bogoliubov, Ukr. Mat. Zh. 2, 3 (1950).11. S. V. Tyablikov, Zh. Eksp. Teor. Fiz. 21, 377 (1951).12. G. E. Volovik, V. I. Mel’nikov, and V. M. Edel’stein, JETP Lett. 18, 81 (1973).13. R. A. Ogg, Jr., Phys. Rev. 69, 243 (1946), where Bose condensation of trapped

electron pairs in metal-ammonia solutions was proposed.14. T. Holstein, Annals of Physics 8, 325 and 243 (1959).15. H. Frohlich, H. Pelzer, and S. Zienau, Phil. Mag. 41, 221 (1950).16. T. D. Lee, F. E. Low, and D. Pines, Phys. Rev. 90, 297 (1953).17. R. P. Feynman, Phys. Rev. 97, 660 (1955).18. M. A. Krivoglaz and S. I. Pekar, Izv. AN SSSR, ser. fiz., 21, 3 and 16 (1957).19. V. M. Buimistrov and S. I. Pekar, Zh. Eksp. Teor. Fiz. 33, 1193 and 1271 (1957).20. K. Huang and A. Rhys, Proc. Roy. Soc. (London) A204, 406 (1950).21. S. I. Pekar, Zh. Eksp. Teor. Fiz. 20, 510 (1950).22. J. J. Markham, Rev. Mod. Phys. 31, 956 (1959).23. A. S. Ioselevich and E. I. Rashba, in: Quantum Tunneling, ed. by Yu. Kagan

and A. J. Leggett (Elsevier) 1992, p. 347.24. S. I. Pekar, E. I. Rashba, and V. I. Sheka, Sov. Phys. JETP, 49, 129 (1979).25. S. I. Pekar, Zh. Eksp. Teor. Fiz. 33, 1022 (1957) [1958, Sov. Phys. JETP 6, 785].

Contents

Polarons in Advanced MaterialsA. S. Alexandrov (ed.) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . V

Reminiscences of the Early Days of Polaron TheoryEmmanuel I. Rashba . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . XI

Part I Large and Small Polarons

Optical Properties of Few and Many Fröhlich Polarons from3D to 0DJozef T. Devreese . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

Small Polarons: Transport PhenomenaYurii A. Firsov . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

Magnetic and Spin Effects in Small Polaron HoppingHarald Böttger, Valerij V. Bryksin, and Thomas Damker . . . . . . . . . . . . . 107

Single Polaron Properties in Different Electron PhononModelsV. Cataudella, G. De Filippis, C.A. Perroni . . . . . . . . . . . . . . . . . . . . . . . . . 149

Path Integrals in the Physics of Lattice PolaronsPavel Kornilovitch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191

Path Integral Methods in the Su–Schrieffer–Heeger PolaronProblemMarco Zoli . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231

Part II Bipolarons in Multi-Polaron Systems

XVI Contents

Superconducting Polarons and BipolaronsA. S. Alexandrov . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257

Small Adiabatic Polarons and BipolaronsSerge Aubry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311

From Single Polaron to Short Scale Phase SeparationV.V. Kabanov . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373

Part III Strongly Correlated Polarons

Numerical Solution of the Holstein Polaron ProblemH. Fehske, S. A. Trugman . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393

Lang-Firsov Approaches to Polaron Physics: From VariationalMethods to Unbiased Quantum Monte Carlo SimulationsMartin Hohenadler, Wolfgang von der Linden . . . . . . . . . . . . . . . . . . . . . . . 463

Spectroscopic Properties of Polarons in Strongly CorrelatedSystems by Exact Diagrammatic Monte Carlo MethodA. S. Mishchenko, N. Nagaosa . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503

Part IV Polarons in Contemporary Materials

Photoinduced Polaron Signatures in Infrared SpectroscopyDragan Mihailovic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 547

Polarons in Colossal Magnetoresistive and High-TemperatureSuperconducting MaterialsGuo-meng Zhao . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 569

Polaron Effects in High-Temperature CuprateSuperconductorsAnnette Bussmann-Holder, Hugo Keller . . . . . . . . . . . . . . . . . . . . . . . . . . . . 599

Current Rectification, Switching, Polarons, and Defects inMolecular Electronic DevicesA.M. Bratkovsky . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 623

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 665

List of Contributors

Serge AubryLaboratoire Léon Brillouin, CEASaclay (CEA-CNRS),91191 Gif-sur-Yvette (France)[email protected]

A. S. AlexandrovDepartment of Physics, Loughbor-ough University, LoughboroughLE11 3TU, United [email protected]

A. M. BratkovskyHewlett-Packard Laboratories, 1501Page Mill Road, PaloAlto, California [email protected]

Harald BöttgerInstitute for Theoretical Physics,Otto-von-Guericke-University PF4120, D-39016 Magdeburg, [email protected]

Valerij V. BryksinA. F. Ioffe Physico-TechnicalInstitute, Politekhnicheskaya 26,19526 St. Petersburg Russia

Annette Bussmann-HolderMax-Planck-Institut fürFestkörperforschung, Heisenbergstr.1, D-70569 Stuttgart, [email protected]

V. CataudellaCNR-INFM Coherentia andUniversity of Napoli,V. Cintia 80126 Napoli, [email protected]

Thomas DamkerInstitute for Theoretical PhysicsOtto-von-Guericke-University PF4120, D-39016 Magdeburg Germany

Jozef T. DevreeseUniversity of Antwerp,Groenenborgerlaan 171, B-2020Antwerpen, [email protected]

H. FehskeInstitut für Physik, Ernst-Moritz-Arndt-Universität Greifswald,D-17487 Greifswald, [email protected]

G. De FilippisCNR-INFM Coherentia andUniversity of Napoli,V. Cintia 80126 Napoli, [email protected]

XVIII List of Contributors

Yu. A. FirsovSolid State Physics Division, IoffeInstitute, 26 Polytekhnicheskaya,194021 St. Petersburg, [email protected]

Martin HohenadlerInstitute for Theoretical andComputational Physics, TU Graz,[email protected]

V. V. KabanovJ. Stefan Institute, Jamova 39, 1001,Ljubljana, [email protected]

Hugo KellerPhysik-Institut derUniversität Zürich, Winterthurerstr.190, CH-8057 Zürich,Switzerland

Pavel KornilovitchHewlett-Packard, Corvallis, Oregon,97330, [email protected]

Dragan MihailovicJozef Stefan Institute andInternational Postgraduate School,SI-1000 Ljubljana, [email protected]

A. S. MishchenkoCREST, Japan Science andTechnology Agency (JST),AIST, 1-1-1, Higashi, Tsukuba,Ibaraki 305-8562, [email protected]

N. NagaosaDepartment of Applied Physics, TheUniversity of Tokyo, 7-3-1 Hongo,Bunkyo-ku, Tokyo 113, [email protected]

C. A. PerroniInstitut für Festkörperforschung(IFF), Forschungszentrum Jülich,52425 Jülich, [email protected]

E. I. RashbaDepartment of Physics, HarvardUniversity, Cambridge, Mas-sachusetts 02138, U. S. [email protected]

S. A. TrugmanTheoretical Division, Los AlamosNational Laboratory, Los Alamos,New Mexico 87545, U. S. [email protected]

Wolfgang von der LindenInstitute for Theoretical andComputational Physics, TU Graz,[email protected]

Guo-meng ZhaoDepartment of Physics and Astron-omy, California State University,Los Angeles, CA [email protected]

Marco ZoliIstituto Nazionale Fisica dellaMateria - Dipartimento di Fisica,Universitá di Camerino, 62032Camerino,Italy [email protected]

Part I

Large and Small Polarons

Optical Properties of Few and Many FröhlichPolarons from 3D to 0D

Jozef T. Devreese1,2

1 Universiteit Antwerpen, T.F.V.S., Groenenborgerlaan 171, B-2020 Antwerpen,Belgium [email protected]

2 Technische Universiteit Eindhoven, P. O. Box 513, NL-5600 MB Eindhoven,The Netherlands

Summary. In this chapter I treat basic concepts and recent developments in thefield of optical properties of few and many Fröhlich polarons in systems of differ-ent dimensions and dimensionality. The key subjects are: comparison of the opti-cal conductivity spectra for a Fröhlich polaron calculated within the all-couplingpath-integral based theory with the results obtained using the numerical Diagram-matic Quantum Monte Carlo method and recently developed analytical approx-imations. The polaron excited state spectrum and the mechanism of the opticalabsorption by Fröhlich polarons are analysed in the light of early theoretical mod-els (from 1964 on) and of recent results. Further subjects are the scaling relationsfor Fröhlich polarons in different dimensions; the all-coupling path-integral basedtheory of the magneto-optical absorption of polarons; Fröhlich bipolarons and theirstability; the many-body problem (including the electron-electron interaction andFermi statistics) in the few- and many-polaron theory; the theory of the opticalabsorption spectra of many-polaron systems; the ground-state properties and theoptical response of interacting polarons in quantum dots; non-adiabaticity of pola-ronic excitons in semiconductor quantum dots. Numerous examples are shown ofcomparison between Fröhlich polaron theory and experiments in high-Tc materi-als, manganites, silver halides, semiconductors and semiconductor nanostructures,including GaAs/AlGaAs quantum wells, various quantum dots etc. Brief sectionsare devoted to the electronic polaron, to small polarons and to recent extensions ofLandau’s concept, including ripplopolarons.

1 Introduction

As is generally known, the polaron concept was introduced by Landau in 1933[1]. Initial theoretical [2–8] and experimental [9] works laid the foundation ofpolaron physics. Among the comprehensive review papers and books coveringthe subject, I refer to [10–17].

Significant extensions and recent developments of the polaron concept havebeen realised (see, for example, [17–21] and references therein). Polarons have

4 Jozef T. Devreese

been invoked, e.g., to study the properties of conjugated polymers, colossalmagnetoresistance perovskites, high-Tc superconductors, layered MgB2 su-perconductors, fullerenes, quasi-1D conductors and semiconductor nanostruc-tures.

A distinction was made between polarons in the continuum approximationwhere long-range electron-lattice interaction prevails (“Fröhlich”polarons)and polarons for which the short-range interaction is essential (Holstein,Holstein-Hubbard, Su-Schrieffer-Heeger models).

The chapter starts with a review of basic concepts and recent developmentsin the study of the optical absorption of Fröhlich polarons in three dimensions.Scaling relations are discussed for Fröhlich polarons in different dimensions.The scaling relation for the polaron free energy is checked for the path integralMonte Carlo results. The next section is devoted to the all-coupling path-integral based theory of the magneto-optical absorption of polarons, whichallows for an interpretation – with high spectroscopic precision – of cyclotronresonance experiments in various solid structures of different dimensionality.In particular, the analysis of the cyclotron resonance spectra of silver halidesprovided one of the most convincing and clearest demonstrations of polaronfeatures in solids. Fröhlich bipolarons, small bipolarons and their extensionsare represented in the context of applications of bipolaron theory to high-Tcsuperconductivity.

Furthermore, recent results on the many-body problem (“the N -polaronproblem”) are discussed. The related theory of the optical absorption spectraof many-polaron systems has been applied to explain the experimental peaksin the mid-infrared optical absorption spectra of cuprates and manganites.The ground-state properties, the optical response of interacting polarons andthe non-adiabaticity of the polaronic excitons in quantum dots are discussedin the concluding sections.

1.1 Fröhlich Polarons

A conduction electron (or hole) in an ionic crystal or a polar semiconductoris the prototype of a polaron. Fröhlich proposed a model Hamiltonian forthis polaron through which its dynamics is treated quantum mechanically(the “Fröhlich Hamiltonian”[6]). The polarization, carried by the longitudinaloptical (LO) phonons, is represented by a set of quantum oscillators withfrequency ωLO, the long-wavelength LO-phonon frequency, and the interactionbetween the charge and the polarization field is linear in the field. The strengthof the electron–phonon interaction is expressed by a dimensionless couplingconstant α. Polaron coupling constants for selected materials are given inTable 1.

This model has been the subject of extensive investigations. The first stud-ies on polarons were devoted to the calculation of the self-energy and theeffective mass of polarons in the limit of large α, or “strong coupling”[2–4].

Optical Properties of Fröhlich Polarons 5

Table 1. Electron-phonon coupling constants (Reprinted with permission after [22].c©2003 by the American Institute of Physics.)

Material α Material αInSb 0.023 AgCl 1.84InAs 0.052 KI 2.5GaAs 0.068 TlBr 2.55GaP 0.20 KBr 3.05CdTe 0.29 Bi12SiO20 3.18ZnSe 0.43 CdF2 3.2CdS 0.53 KCl 3.44α-Al2O3 1.25 CsI 3.67AgBr 1.53 SrTiO3 3.77α-SiO2 1.59 RbCl 3.81

The “weak-coupling” limit, first explored by H. Fröhlich [6], is obtainedfrom the leading terms of the perturbation theory for α → 0. Inspired bythe work of Tomonaga, Lee et al. [7] derived the self-energy and the effectivemass of polarons from a canonical-transformation formulation; the range ofvalidity of their approximation is in principle not larger than that of the weak-coupling approximation. The main significance of the approximation of [7] isin the elegance of the used canonical transformation, together with the factthat it puts the Fröhlich result [6] on a variational basis.

An all-coupling polaron approximation was developed by Feynman usinghis path-integral formalism [8]. In a trial action he simulated the interactionbetween the electron and the polarization modes by a harmonic interactionbetween a hypothetical particle and the electron and introduced a variationalprinciple for path integrals. Feynman derived first the self-energy E0 and theeffective mass m∗ of the polaron [8]. The analysis of an exactly solvable (“sym-metrical”) 1D-polaron model [23, 24] demonstrated the accuracy of Feynman’spath-integral approach to the polaron ground-state energy. Later Feynman etal. formulated a response theory for path integrals, derived a formal expressionfor the impedance and studied the mobility of all-coupling polarons [25, 26].

Subsequently the path-integral approach to the polaron problem was gen-eralised and developed to become a tool to study optical absorption, magne-tophonon resonance, cyclotron resonance etc.

1.2 Optical Absorption of Fröhlich Polarons at ArbitraryCoupling. Analytical Theory

The study of the internal excitations of Fröhlich polarons and their opticalabsorption started in 1964 [23, 27] with the analysis of the spectrum of anexactly solvable “symmetrical” 1D-polaron model. It was also shown that two

6 Jozef T. Devreese

types of excitations exist for this polaron model: a) scattering states (called“diffusion” states in [23], and b) “relaxed excited states” (RES). It was arguedin [23, 27] that the RES is only stable for a sufficiently large electron-LO-phonon coupling constant.

In 1969, starting from [23, 27, 28], a mechanism for the optical absorptionof strong-coupling Fröhlich polarons was proposed in [29]. This mechanismconsists of transitions to a RES and to its LO-phonon sidebands that consti-tute a Franck-Condon band. In 1972, Devreese et al. (DSG; [30]) publishedall-coupling results for the optical absorption of the Fröhlich polaron. Refer-ence [30] uses the Feynman ground-state polaron model and the path-integralresponse formalism [25] as its starting point. For α � 1 the DSG-spectrumconsists of a one-LO-phonon sideband (along with, for T = 0, a δ-peak atzero frequency). This result confirms the one-polaron limit of the perturba-tive treatment in [31]. For intermediate coupling (3 � α � 6) DSG predict atransition to a RES and its LO-phonon sidebands (FC band). For α � 6, DSGidentify a narrow RES-transition with a narrow sideband (as already statedin [30, 32] the resulting RES-peak is too narrow and – at sufficiently large α– inconsistent with the Heisenberg uncertainty principle).

Recent numerical [33] and analytical [34] studies have allowed for a morecomplete understanding of the optical absorption of the Fröhlich polaron atall α, as discussed in [35]. Later in this chapter I will analyse to what extentrecent calculations confirm the mechanisms for the polaron optical absorptionproposed in [29] and by DSG [30]. I refer also to the chapters by Mishchenkoand Nagaosa [36] and Cataudella et al [37] in the present volume.

A path-integral Monte Carlo scheme was presented [38] to study theFröhlich polaron model in three and two dimensions. The ground state featuresof the Fröhlich polaron model were revisited numerically using a Diagram-matic Quantum Monte-Carlo (DQMC) method [39] and analytically usingan “all-coupling” variational Hamiltonian approach [34]. The three aforemen-tioned schemes confirm the remarkable accuracy of the Feynman path-integralmodel [8] to calculate the polaron ground-state energy. The dependence of thecalculated polaron ground state properties on the electron-phonon couplingstrength supports the earlier conclusion [40–42] that the crossover betweenthe two asymptotic regimes characterizing a polaron occurs smoothly and donot suggest any sharp “self-trapping” transition.

1.3 Small Polarons. Recent Extensions of the Polaron Concept

Holstein, using a 1D model, has pioneered the study of what are often called“small polarons”, for which the lattice polarisation, induced by a charge car-rier, is essentially confined to a unit cell [43, 44]. Hopping of electrons fromone lattice site to another in the presence of the electron-phonon interactionis the key process determining the dynamical properties of small polarons (seee.g. [45–48]) and spin polarons (cf. [49]). A crossover regime of the Holstein


polaron has been studied using a variational analysis based on a superposi-tion of Bloch states that describe large polarons and small polarons by V.Cataudella et al. [50] and within a numerical variational approach [51]. Dy-namical polaron solutions, which are characterised by very long lifetime atlow temperatures, have been proposed for the Holstein model on a latticewith anharmonic local potential [52].

The first identification of small polarons in solids was made for non-stoichiometric uranium dioxide by the present author [53, 54]. The mecha-nisms of self-trapping, static and dynamic properties of small polarons inalkali halides and in several other ionic crystals were analysed e. g. in [55, 56].Quantitative evidence for critical quantum fluctuations and superlocalisationof the small polarons in one, two and three dimensions was presented on thebasis of the Quantum Monte Carlo approach in [57, 58]. It was demonstratedthat for all lattice dimensionalities there exists a critical value of the electron-lattice coupling constant, below which self-trapping of Holstein polarons doesnot occur [59]. Several recent experimental and theoretical investigations haveprovided convincing evidence for the occurrence of small (bi)polarons in “con-temporary” materials. In the case of short-range electron-phonon interaction,when a small (bi)polaron hops between lattice sites, the total lattice deforma-tion vanishes at one site and then re-appears at a new one. The effective massof a one-site bipolaron is then very large, and the predicted critical tempera-ture Tc is very low (see [60]). A two-site small bipolaron model by A. Alexan-drov and N. Mott [14] provides a parameter-free estimate of Tc for high-Tcsuperconducting cuprates [61]. A long-range Fröhlich-type, rather than short-range, electron-phonon interaction on a discrete ionic lattice [62] is assumedwithin the “Fröhlich-Coulomb” model of the high-Tc superconductivity pro-posed by A. Alexandrov [63]. For a long-range interaction, only a fraction ofthe total deformation changes as a (bi)polaron hops between the lattice sites.This leads to a dramatic mass reduction as compared to that of the Holsteinsmall (bi)polaron. It was then proposed that in the superconducting phasethe carriers are “superlight mobile bipolarons”. As distinct from the conven-tional continuum Fröhlich polaron, a multipolaron lattice model is used withelectrostatic forces taking into account the discreteness of the lattice, finiteelectron bandwidth and the quantum nature of phonons. This model is appliedin an attempt to explain the physical properties of superconducting cupratessuch as their Tc-values, the isotope effects, the normal-state diamagnetism,the pseudogap and spectral functions measured in tunnelling and photoemis-sion (see [17, 64] for an extensive review). Experiments have been interpretedas due to small polarons in the paramagnetic (see e.g. [65]), ferromagnetic[66] and antiferromagnetic [67] states of manganites. The magnetization andresistivity of manganites near the ferromagnetic transition were interpretedin terms of pairing of oxygen holes into heavy bipolarons in the paramagneticphase and their magnetic pair breaking in the ferromagnetic phase [68]. Thesestudies do not preclude the occurrence of Fröhlich polarons in manganites, asevidenced in the work of Hartinger et al. [69]

8 Jozef T. Devreese

1.4 Electronic Polarons

An extension of the polaron concept arises by considering the interactionbetween a carrier and the exciton field. One of the early formulations of thismodel was developed by Toyozawa [70]. The resulting quasi-particle is calledthe electronic polaron.

The self-energy of the electronic polaron (which is almost independent ofwave number) must be taken into account when the bandgap of an insulatoror semiconductor is calculated using pseudopotentials. For example if onecalculates, with Hartree-Fock theory, the bandgap of an alkali halide, oneis typically off by a factor of two. This was the original problem which wassolved conceptually with the introduction of the electronic polaron [70]. Also inthe soft X-ray spectra of alkali halides exciton sidebands have been observedwhich we attributed to the electronic polaron coupling [71] (see also [72]).For a review of the current experimental status of the “electronic polaroncomplexes”, as predicted in [71, 73], I refer to [74].

Using the all-coupling theory of the polaron optical absorption [30, 32], wefound [73] that the electronic polaron produces peaks in the optical absorptionspectra beginning about an exciton energy above the absorption edge, allowingfor the interpretation of the experiments on LiF, LiCl, and LiBr. This theoryhas been invoked recently e. g. for the interpretation of the experimental dataon inelastic soft X-ray scattering in solid LiCl, resonantly enhanced at stateswith two Li 1s vacancies [75].

2 Optical Absorption of Fröhlich Polarons in 3D

2.1 Optical Absorption at Weak Coupling. The Role ofMany-Polarons

At zero temperature and in the weak-coupling limit, the optical absorptionof a Fröhlich polaron is due to the elementary polaron scattering process,schematically shown in Fig. 1.

In the weak-coupling limit (α � 1) the polaron absorption coefficient fora many-polaron gas was first obtained by V. Gurevich, I. Lang and Yu. Firsov[31]. Their optical-absorption coefficient is equivalent to a particular case ofthe result of J. Tempere and J. T. Devreese [76], with the dynamic structurefactor S(q, Ω) corresponding to the Hartree-Fock approximation. In [76] theoptical absorption coefficient of a many-polaron gas was shown to be given,to order α, by

Re[σ(Ω)] = n0e223α

12πΩ3

∞∫0

dqq2S(q, Ω − ωLO), (1)

where n0 is the density of charge carriers.


Fig. 1. Elementary polaron scattering process describing the absorption of an incom-ing photon and the generation of an outgoing phonon. (Reprinted with permissionfrom [22]. c©2003, American Institute of Physics.)

In the zero-temperature limit, starting from the Kubo formula ([77], p.165), the optical conductivity of a single Fröhlich polaron can be representedin the form

σ(Ω) = i e2

mb(Ω+iε)+ e

2

m2b�1

(Ω+iε)3∫∞0 e

−εt (eiΩt − 1)∑q,q′ qxq′x×〈Ψ0

∣∣∣∣∣[ (

Vqbq (t) + V ∗−qb+−q (t)

)eiq·r(t),(

V−q′b−q′ + V ∗q′b+q′

)e−iq

′·r

]∣∣∣∣∣Ψ0〉

dt, (2)

where ε = +0 and |Ψ0〉 is the ground-state wave function of the electron-phonon system. Within the weak coupling approximation, the following an-alytic expression for the real part of the polaron optical conductivity resultsfrom (2):

Reσ (Ω) =πe2

2m∗δ (Ω) +

2e2

3mbωLOα

Ω3

√Ω − ωLOΘ (Ω − ωLO) , (3)

where

Θ(Ω − ωLO) ={

1 if Ω > ωLO,0 if Ω < ωLO.

The spectrum of the real part of the polaron optical conductivity (3) is rep-resented in Fig. 2.

According to (3), the absorption coefficient for absorption of light withfrequency Ω by free polarons for α −→ 0 takes the form

Γp(Ω) =1

�0cn

2n0e2αω2LO3mbΩ3

√Ω

ωLO− 1 Θ (Ω − ωLO) , (4)

where �0 is the dielectric permittivity of the vacuum, n is the refractive indexof the medium, n0 is the concentration of polarons. A simple derivation in

10 Jozef T. Devreese

Fig. 2. Polaron optical conductivity for α = 1 in the weak-coupling approximation,according to [32], p. 92. A δ-like central peak (at Ω = 0) is schematically shownby a vertical line. (Reprinted with permission from [78]. c©2006, Società Italiana diFisica.)

[79] using a canonical transformation method gives the absorption coefficientof free polarons, which coincides with the result (4). The step function in (4)reflects the fact that at zero temperature the absorption of light accompaniedby the emission of a phonon can occur only if the energy of the incidentphoton is larger than that of a phonon (Ω > ωLO). In the weak-coupling limit,according to (4), the absorption spectrum consists of a “one-phonon line”. Atnonzero temperature, the absorption of a photon can be accompanied notonly by emission, but also by absorption of one or more phonons. Similaritybetween the temperature dependence of several features of the experimentalinfrared absorption spectra in high-Tc superconductors and the temperaturedependence predicted for the optical absorption of a single Fröhlich polaron[30, 32] has been revealed in [80].

Experimentally, this one-phonon line has been observed for free polaronsin the infrared absorption spectra of CdO-films, see Fig. 3. In CdO, which is aweakly polar material with α ≈ 0.74, the polaron absorption band is observedin the spectral region between 6 and 20 µm (above the LO phonon frequency).The difference between theory and experiment in the wavelength region wherepolaron absorption dominates the spectrum is due to many-polaron effects,see [76].


Fig. 3. Optical absorption spectrum of a CdO-film with carrier concentration n0 =5.9 × 1019 cm−3 at T = 300 K. The experimental data (solid dots) of [81] arecompared to different theoretical results: with (solid curve) and without (dashedline) the single-polaron contribution of [31, 79] and for many polarons (dash-dottedcurve) of [76]. The following values of material parameters of CdO were used forthe calculations: α = 0.74 [81], ωLO = 490 cm−1 (from the experimental opticalabsorption spectrum, Fig. 2 of [81]), mb = 0.11me [82], ε0 = 21.9, ε∞ = 5.3 [82].(Reprinted with permission from [22]. c©2003, American Institute of Physics.)

2.2 Optical Absorption at Strong Coupling

The problem of the structure of the Fröhlich polaron excitation spectrumconstituted a central question in the early stages of the development of polarontheory. The exactly solvable polaron model of [27] was used to demonstratethe existence of the so-called “relaxed excited states”of Fröhlich polarons [23].In [28], and after earlier intuitive analysis, this problem was studied using theclassical equations of motion and Poisson-brackets. The insight gained as aresult of those investigations concerning the structure of the excited polaronstates, was subsequently used to develop a theory of the optical absorptionspectra of polarons. The first work was limited to the strong coupling limit[29]. Reference [29] is the first work that reveals the impact of the internaldegrees of freedom of polarons on their optical properties.

The optical absorption of light by free Fröhlich polarons was treated in[29] using the polaron states obtained within the adiabatic strong-couplingapproximation. It was argued in [29], that for sufficiently large α (α � 3),the (first) relaxed excited state (RES) of a polaron is a relatively stable state,which gives rise to a “resonance” in the polaron optical absorption spectrum.


This idea was necessary to understand the polaron optical absorption spec-trum in the strong-coupling regime. The following scenario of a transition,which leads to a “zero-phonon” peak in the absorption by a strong-couplingpolaron, was then suggested. If the frequency of the incoming photon is equalto ΩRES = 0.065α2ωLO, the electron jumps from the ground state (which, atlarge coupling, is well-characterised by “s”-symmetry for the electron) to anexcited state (“2p”), while the lattice polarization in the final state is adaptedto the “2p” electronic state of the polaron. In [29], considering the decay of theRES with emission of one real phonon, it is argued that the “zero-phonon”peak can be described using the Wigner-Weisskopf formula valid when thelinewidth of that peak is much smaller than �ωLO.

For photon energies larger than ΩRES + ωLO, a transition of the polarontowards the first scattering state, belonging to the RES, becomes possible.The final state of the optical absorption process then consists of a polaron inits lowest RES plus a free phonon. A “one-phonon sideband” then appears inthe polaron absorption spectrum. This process is called one-phonon sidebandabsorption. The one-, two-, ... K-, ... phonon sidebands of the zero-phononpeak give rise to a broad structure in the absorption spectrum. It turns outthat the first moment of the phonon sidebands corresponds to the Franck-Condon (FC) frequency ΩFC = 0.141α2ωLO.

To summarise, following [29], the polaron optical absorption spectrum atstrong coupling is characterised by the following features (at T = 0):

a) An absorption peak (“zero-phonon line”) appears, which corresponds toa transition from the ground state to the first RES at ΩRES.

b) For Ω > ΩRES + ωLO, a phonon sideband structure arises. This sidebandstructure peaks around ΩFC. Even when the zero-phonon line becomesweak, and most oscillator strength is in the LO-phonon sidebands, thezero-phonon line continues to determine the onset of the phonon sidebandstructure.

The basic qualitative strong coupling behaviour predicted in [29], namely,zero-phonon (RES) line with a broader sideband at the high-frequency side,was confirmed by later studies, as will be discussed below.

2.3 Optical Absorption of Fröhlich Polarons at Arbitrary Coupling(DSG, [30])

In 1972 the optical absorption of the Fröhlich polaron was calculated by thepresent author et al. ([30, 32] (“DSG”)) for the Feynman polaron model (andusing path integrals). Until recently DSG (combined with [29]) constituted thebasic picture for the optical absorption of the Fröhlich polaron. In 1983 [83] theDSG-result was rederived using the memory function formalism (MFF). TheDSG-approach is successful at small electron-phonon coupling and is able toidentify the excitations at intermediate electron-phonon coupling (3 � α � 6).


In the strong coupling limit DSG still gives an accurate first moment for thepolaron optical absorption but does not reproduce the broad phonon sidebandstructure (cf. [29] and [84]). A comparison of the DSG results with the OCspectra given by recently developed “approximation-free‘” numerical [33] andapproximate analytical [34, 35] approaches was carried out recently in [35], seealso the chapters by V. Cataudella et al. and A. Mishchenko and N. Nagaosain the present volume.

The polaron absorption coefficient Γ (Ω) of light with frequency Ω at ar-bitrary coupling was first derived in [30]. It was represented in the form

Γ (Ω) = − 1n�0c

e2

mb

ImΣ(Ω)[Ω − ReΣ(Ω)]2 + [ImΣ(Ω)]2

. (5)

This general expression was the starting point for a derivation of the theoret-ical optical absorption spectrum of a single Fröhlich polaron at all electron-phonon coupling strengths by DSG in [30]. Σ(Ω) is the so-called “memoryfunction”, which contains the dynamics of the polaron and depends on Ω, α,temperature and applied external fields. The key contribution of the work in[30] was to introduce Γ (Ω) in the form (5) and to calculate ReΣ(Ω), which isessentially a (technically not trivial) Kramers–Kronig transform of the moresimple function ImΣ(Ω). Only function ImΣ(Ω) had been derived for theFeynman polaron [25] to study the polaron mobility µ from the impedancefunction, i. e. the static limit

µ−1 = limΩ→0

(ImΣ(Ω)

Ω

).

The basic nature of the Fröhlich polaron excitations was clearly revealedthrough this polaron optical absorption obtained in [30]. It was demonstratedin [30] that the Franck-Condon states for Fröhlich polarons are nothing elsebut a superposition of phonon sidebands. It was also established in [30] thata relatively large value of the electron-phonon coupling strength (α > 5.9) isneeded to stabilise the relaxed excited state of the polaron. It was, further,revealed that at weaker coupling only “scattering states”of the polaron playa significant role in the optical absorption [30, 85].

2.4 The Structure of the Polaron Excitation Spectrum

In the weak coupling limit, the optical absorption spectrum (5) of the po-laron is determined by the absorption of radiation energy, which is re-emittedin the form of LO phonons. As α increases between approximately 3 and6, a resonance with increasing stability appears in the optical absorption ofthe Fröhlich polaron of [30] (see Fig. 4). The RES peak in the optical absorp-tion spectrum also has a phonon sideband-structure, whose average transitionfrequency can be related to an FC-type transition. Furthermore, at zero tem-perature, the optical absorption spectrum of one polaron also exhibits a zero-frequency “central peak” [∝ δ(Ω)]. For nonzero temperature, this “central


peak” smears out and gives rise to an “anomalous” Drude-type low-frequencycomponent of the optical absorption spectrum.

For α > 6.5 the polaron optical absorption gradually develops the structurequalitatively proposed in [29]: a broad LO-phonon sideband structure appearswith the zero-phonon (“RES”) transition as onset. Reference [30] does notpredict the broad LO-phonon sidebands at large coupling constant, althoughit still gives an accurate first Stieltjes moment of the optical absorption spec-trum. Reference [35], discussed further in this chapter, sheds new light on thepolaron optical absorption.

In Fig. 4 (from [30]), the main peak of the polaron optical absorption forα = 5.25 at Ω = 3.71ωLO is interpreted as due to transitions to a RES. The“shoulder” at the low-frequency side of the main peak is attributed as mainlydue to one-phonon transitions to polaron“scattering states”. The broad struc-ture centred at about Ω = 6.6ωLO is interpreted as an FC band (composedof LO-phonon sidebands). As seen from Fig. 4, when increasing the electron-phonon coupling constant to α=6, the RES peak at Ω = 4.14ωLO stabilises.It is in [30] that an all-coupling optical absorption spectrum of a Fröhlich po-laron, together with the role of RES-states, FC-states and scattering states,was first presented. Up to α = 6, the DQMC results of [33] reproduce the mainfeatures of the optical absorption spectrum of a Fröhlich polaron as found in[30].

Based on [30], it was argued that it is Holstein polarons that determine theoptical properties of the charge carriers in oxides like SrTiO3, BaTiO3 [86],while Fröhlich weak-coupling polarons could be identified e.g. in CdO [79].

Fig. 4. Optical absorption spectrum of a Fröhlich polaron for α = 4.5, α = 5.25 andα = 6 after [30] (DSG). The RES peak is very intense compared with the FC peak.The δ-like central peaks (at Ω = 0) are schematically shown by vertical lines. TheDQMC results of [33] are shown with open circles.


The Fröhlich coupling constants of polar semiconductors and ionic crystalsare generally too small to allow for a static “RES”. In [87] the RES-peaks of[30] were involved to explain the optical absorption spectrum of Pr2NiO4.22.Further study of the spectra of [87] is called for. The RES-like resonances inΓ (Ω), (5), due to the zero’s of Ω − ReΣ(Ω), can effectively be displaced tosmaller polaron coupling by applying an external magnetic field B, in whichcase the contribution for what is formally a “RES-type resonance” arises atΩ−ωc−ReΣ(Ω) = 0 (ωc = eB/mbc is the cyclotron frequency). Resonancesin the magnetoabsorption governed by this contribution have been clearlyobserved and analysed in many solids and structures, see Sect. 4.

2.5 Optical Absorption at Arbitrary Coupling. DQMC and DSG

Accurate numerical methods have been developed for the calculation of spec-tral characteristics and correlation functions of the Holstein polaron (see e.g.[51, 57–59, 88, 89]), of the Fröhlich polaron [39], and of the long-range discreteFröhlich model [90]. The numerical calculations of the optical conductivity forthe Fröhlich polaron performed within the DQMC method by Mishchenko etal. [33], see [36], confirm the analytical results derived in [30] for α � 3. In theintermediate coupling regime 3 < α < 6, the low-energy behaviour and theposition of the RES-peak in the optical conductivity spectrum of [33] followclosely the prediction of [30]. There are some minor quantitative differencesbetween the two approaches in the intermediate coupling regime: in [33], thedominant (“RES”) peak is less intense in the Monte-Carlo numerical simu-lations and the second (“FC”) peak develops less prominently. The followingqualitative differences exist between the two approaches: in [33], the dominantpeak broadens for α � 6 and the second peak does not develop, but gives riseto a flat shoulder in the optical conductivity spectrum at α ≈ 6. As α increasesbeyond α ≈ 6, the DSG results for the OC do not produce the broad phononsideband spectrum of the RES-transition that was qualitatively predicted in[29] and obtained with DQMC.

Figure 5 shows that already for α = 1 noticeable differences arise betweenReσ(Ω) calculated with perturbation theory to O(α), resp. O(α2), and DSG orDQMC. Remarkably, the DQMC results for α = 1 seem to show a somewhatmore pronounced two-phonon-scattering contribution than the perturbationtheory result to O(α2). This point deserves further analysis.

An instructive comparison between the positions of the main peak in theoptical absorption spectra of Fröhlich polarons obtained within the DSG andDQMC approaches has been performed recently [92]. In Fig. 6 the frequencyof the main peak in the OC spectra calculated within the DSG approach [30]is plotted together with that given by DQMC [33, 35]. As seen from the figure,the main-peak positions, obtained within DSG, are in good agreement withthe results of DQMC for all considered values of α. At large α the positionsof the main peak in the DSG spectra are remarkably close to those givenby DQMC. The difference between the DSG and DQMC results is relatively


Fig. 5. One-polaron optical conductivity Reσ (Ω) for α = 1 calculated within theDQMC approach [33] (open circles), derived using the expansion in powers of α upto α [79] (solid line), up to α2 [91] (dashed line) and within the DSG approach [30](dotted line). A δ-like central peak (at Ω = 0) is schematically shown by a verticalline.

Fig. 6. Main peak positions from DQMC optical conductivity spectra of Fröhlichpolarons [35] compared to those of the analytical DSG approach [30]. (From [92].)

larger at α = 8 and for α = 9.5, but even for those values of the couplingconstant the agreement is quite good.

I suggest that the RES-peak at α ≈ 6 in the DSG-treatment, as α increases,gradually transforms into an FC-peak. As stated above and in [30], DSG


predicts a much too narrow FC-peak in the strong coupling limit, but still atthe “correct” frequency.

The DSG spectrum also satisfies the zero and first moment sum rules atall α as will be discussed further in the present chapter.

2.6 Extended Memory Function Formalism

In order to describe the OC main peak line width at intermediate electron-phonon coupling, the DSG approach was modified [35] to include additionaldissipation processes, the strength of which is fixed by an exact sum rule, seethe chapter by Cataudella et al [37].

To include dissipation [35], a finite lifetime for the states of the relativemotion, which can be considered as the result of the residual e-ph interactionnot included in the Feynman variational model was introduced. If broadeningof the oscillator levels is neglected, the DSG results [30, 83] are recovered.

2.7 The Extended Strong-Coupling Expansion (SCE) of thePolaron Optical Conductivity [92]

Using the Kubo formula (at T = 0) the strong coupling OC of the polaroncan be expressed in terms of the dipole-dipole two-point correlation functionfzz (t):

Reσ (Ω) =Ω

2

∞∫−∞

eiωtfzz (t) dt, (6)

fzz (t) = 〈z (t) z (0)〉 . (7)

The polaron optical conductivity within the strong-coupling approach can nowbe calculated beyond the Landau-Pekar approximation [2] in order to obtainrigorous results in the strong-coupling limit. The electron-phonon system isdescribed by the Hamiltonian

H =p2

2+ Hph +

1√V

∑k

√2√

2παk

(bk + b+−k

)eik·r, (8)

Hph =∑k

(b+k bk +

12

), (9)

where mb = 1, � = 1, ωLO = 1. In the representation where the phononcoordinates and momenta

Qk =bk + b+−k√

2, Pk =

b−k − b+k√2i

,

Q+k = Q−k, P+k = P−k


are used, this Hamiltonian is

H =p2

2+

12

∑k

(PkP−k + QkQ−k) +1√V

∑k

2√√

2παk

Qkeik·r. (10)

In order to develop a strong-coupling approach for the polaron OC, a scalingtransformation of the coordinates and momenta of the electron-phonon systemis made following Allcock [93] (p. 48):

r = α−1x, p = −iα ∂∂x

,

k = ακ,Qk = αqκ, Pk = α−1pκ, (11)

∑κ

. . . =∑k

. . . =V

(2π)3

∫. . . dk =

V α3

(2π)3

∫. . . dκ =

V(2π)3

∫. . . dκ(

V ≡ V α3)

This transformation is necessary in order to see explicitly the order of mag-nitude of the different terms in the Hamiltonian. Expressed in terms of thenew variables, the Hamiltonian (10) is

H = −α2

2∂2

∂x2+

12

∑κ

(α−2pκp−κ + α2qκq−κ

)+ α2

2√√

2π√V

∑κ

1κqκe

iκ·x.

(12)This Hamiltonian can be written as a sum of two terms, which are of differentorder in powers of α:

H = H1 + H2,

where H1 ∼ α2 is the leading term,

H1 = α2(−1

2∂2

∂x2+

12

∑κ

qκq−κ +2√√

2π√V

∑κ

1κqκe

iκ·x)

, (13)

and H2 ∼ α−2 is the kinetic energy of the phonons,

H2 = α−212

∑κ

pκp−κ. (14)

The total ground-state wave function of the electron-phonon system in theadiabatic approximation is given by the strong-coupling Ansatz

|Ψ0〉 = |Φ0〉 |ψ0〉 , (15)

where |Φ0〉 and |ψ0〉 are, respectively, the phonon and electron wave functions.The phonon wave function is related to the phonon vacuum |0ph〉 by


|Φ0〉 = U |0ph〉 , (16)

where U is the unitary transformation:

U = e∑

k(fkbk−f∗kb+k ). (17)

The optimal values of the variational parameters fk are

fk =

√2√

2παk√V

ρk, (18)

where ρk is the averageρk =

〈ψ0∣∣eik·r∣∣ψ0〉 . (19)

Using the fact that ρ−k = ρk (due to the inversion symmetry of the groundstate), we express the unitary operator (17) in the new variables:

U = exp

(i∑κ

g−κpκ

)

with

gκ =2√√

2π√Vκ

ρκ, ρκ =〈ψ0∣∣eiκ·x∣∣ψ0〉 .

H̃ = U−1HU is the transformed Hamiltonian:

H̃ = −α2

2∂2

∂x2+Ua (x)+∆E+

12

∑κ

(1α2

pκp−κ + α2qκq−κ

)+∑κ

Wκ (x) qκ

(20)with the notations

∆E ≡ α2

2

∑κ

|gκ|2 , (21)

Ua (x) = −α22√√

2π√V

∑κ

1κg−κeiκ·x, (22)

Wκ (x) ≡ α22√√

2π√V

1κ

(eiκ·x − ρκ

). (23)

Here, Wκ (x) are the amplitudes of the renormalised electron-phonon interac-tion and Ua (x) is the self-consistent adiabatic potential energy for the elec-tron.

As a result, the correlation function fzz (t) takes the form

fzz (t) =〈0ph

∣∣∣〈ψ0 ∣∣∣eitH̃ze−itH̃z∣∣∣ψ0〉∣∣∣ 0ph〉 . (24)The transformed Hamiltonian H̃ is the sum of two terms:


H̃ = H̃0 + W (25)

with

H̃0 = −α2

2∂2

∂x2+ Ua (x) + ∆E +

12

∑κ

(1α2

pκp−κ + α2qκq−κ

), (26)

W =∑κ

Wκ (x) qκ. (27)

The unperturbed Hamiltonian H̃0 and the renormalised electron-phonon in-teraction are, respectively,

H̃0 =p2

2+∑k

|fk|2 + Va (r) +∑k

(b+k bk +

12

), (28)

W =∑k

(Wkbk + W ∗kb

+k

). (29)

Here, the Wk are the amplitudes of the renormalised electron-phonon inter-action

Wk =

√2√

2παk√V

(eik·r − ρk

)(30)

and Va (r) is the self-consistent adiabatic potential energy for the electron

Va (r) = −∑k

4√

2παk2V

ρ−keik·r. (31)

Further on, a complete orthogonal basis consisting of the Franck-Condon(FC) states |ψn,l,m〉 is used, with l the quantum number of the angular momen-tum, m the z-projection of the angular momentum, n the radial quantum num-ber. (In this classification, the ground-state wave function is |ψ0,0,0〉 ≡ |ψ0〉.)The FC wave functions |ψn,l,m〉 are the exact eigenstates of the HamiltonianH̃0.

Up to this point, the only approximation made in fzz (t) was the strong-coupling Ansatz for the polaron ground-state wave function. The next step isto apply the Born-Oppenheimer (BO) approximation [93], which neglects thenon-adiabatic transitions between different polaron levels for the renormalisedoperator of the electron-phonon interaction W . The dipole-dipole correlationfunction fzz (t) in the BO approximation is [35, 92]

fzz (t) =∑n,l,m

|〈ψ0 |z|ψn,1,0〉|2 eit(E0−En,1)

×〈

0ph

∣∣∣∣∣∣〈ψn,1,0

∣∣∣∣∣∣T exp⎛⎝−i t∫

0

dsW (s)

⎞⎠∣∣∣∣∣∣ψn,1,0〉∣∣∣∣∣∣ 0ph

〉(32)


Fig. 7. The polaron OC calculated within the extended SCE taking into accountcorrections of order α0 (solid curve), the OC calculated within the leading-termstrong-coupling approximation (dashed curve), with the leading term of the Landau-Pekar (LP) adiabatic approximation (dash-dotted curve), and the numerical DQMCdata (open circles) for α = 7, 9, 13 and 15. (From [92].)

with the time-dependent interaction Hamiltonian

W (s) ≡ eisH̃0W (s) e−isH̃0 . (33)

The polaron energies E0, En,1 and the wave functions ψ0, ψn,1,0 are calculatedtaking into account the corrections of order of α0.

Figure 7 shows the polaron OC spectra for different values of α calcu-lated numerically using (32) within different approximations. The OC spectracalculated within the extended SCE approach taking into account both theJahn-Teller effect – related to the degeneracy with respect to the quantumnumber m – and the corrections of order α0 are shown by the solid curves.The OC obtained with the leading-term strong-coupling approximation takinginto account the Jahn-Teller effect and with the leading term of the Landau-Pekar adiabatic approximation are plotted as dashed and dash-dotted curves,respectively. The open circles show the DQMC data [33, 35].


The polaron OC band of Fig. 7 obtained within the extended SCE gener-alises the Gaussian-like polaron OC band (as given e.g. by (3) of [35]) thanksto (i) the use of the numerically exact strong-coupling polaron wave functions[94] and (ii) the incorporation of both static and dynamic Jahn-Teller effects.The polaron OC broad structure calculated within the present extended SCEconsists of a series of LO-phonon sidebands and provides a realisation – withall LO-phonons involved for a given α – of the scheme proposed by KED [29].

As seen from Fig. 7, the polaron OC spectra calculated within the asymp-totically exact strong-coupling approach are shifted towards lower frequen-cies as compared with the OC spectra calculated within the LP approxima-tion. This shift is due to the use of the numerically exact (in the strong-coupling limit) energy levels and wave functions of the internal excited po-laron states, as well as the numerically exact self-consistent adiabatic polaronpotential. Furthermore, the inclusion of the corrections of order α0 leads toa shift of the OC spectra to lower frequencies with respect to the OC spec-tra calculated within the leading-term approximation. The value of this shift∆Ωn,0/ωLO ≈ −1.8 obtained within the extended SCE, is close to the LPvalue ∆Ω(LP )n,0 /ωLO = −(4 ln 2 − 1) ≈ −1.7726 (cf. [95, 96]). The distinctionbetween the OC spectra calculated with and without the Jahn-Teller effect isvery small.

Starting from α ≈ 9 towards larger values of α, the agreement betweenthe extended SCE polaron OC spectra and the numerical DQMC data be-comes gradually better, consistent with the fact that the extended SCE forthe polaron OC is asymptotically exact in the strong-coupling limit.

The results of the extended SCE as treated in the present section arequalitatively consistent with the interpretation advanced in [29]. In [29] onlythe 1-LO-phonon sideband was taken into account, while in [84] 2-LO-phononemission was included.

The extended SCE carries on the program started in [29]. The spectra inFig. 7, in the strong coupling approximation, consist of LO-phonon sidebandsto the RES (which itself has negligible oscillator strength in this limit, similarto the optical absorption for some colour centres in alkali halides). TheseLO-phonon sidebands form a broad FC-structure.

2.8 Comparison Between the Optical Conductivity SpectraObtained Within Different Approaches

A comparison between the optical conductivity spectra obtained with theDQMC method, extended MFF, SCE and DSG for different values of α isshown in Figs. 8 and 9, taken from [35]. The key results of the comparison arethe following.

First, as expected, in the weak-coupling regime, both the extended MFFwith phonon broadening and DSG [30] are in very good agreement with theDQMC data [33], showing significant improvement with respect to the weak-coupling perturbation approach [31, 79] which provides a good description of


the OC spectra only for very small values of α. For 3 ≤ α ≤ 6, DSG predictsthe essential structure of the OA, with a RES-transition gradually building upfor increasing α, but underestimates the peak width. The damping, introducedin the extended MFF approach, becomes crucial in this coupling regime.

Second, comparing the peak and shoulder energies, obtained by DQMC,with the peak energies, given by MFF, and the FC transition energies fromthe SCE, it is concluded [35] that as α increases from 6 to 10 the spectralweights rapidly switch from the dynamic regime, where the lattice follows theelectron motion, to the adiabatic regime dominated by FC transitions. In theintermediate electron-phonon coupling regime, 6 < α < 10, both adiabaticFC and non-adiabatic dynamical excitations coexist.

For still larger coupling (α � 10), the polaron OA spectrum consists of abroad FC-structure, built of LO-phonon sidebands.

Fig. 8. Comparison of the optical conductivity calculated with the DQMC method(circles), extended MFF (solid line) and DSG [30, 83] (dotted line), for four differentvalues of α. (Reprinted with permission from [35]. c©2006 by the American PhysicalSociety.)

Fig. 9. Comparison of the optical conductivity calculated with the DQMC method(circles), the extended MFF (solid line) and SCE (dashed line) for three differentvalues of α. (Reprinted with permission from [35]. c©2006 by the American PhysicalSociety.)


In summary, the accurate numerical results obtained from DQMC – mod-ulo the linewidths for α > 6 – and from the recent analytical approximations[34, 35] confirm the essence of the mechanism for the optical absorption ofFröhlich polarons, proposed in [30, 32] combined with [29] and do add impor-tant new extensions and new insights (see the chapters by V. Cataudella etal. and by A. Mishchenko and N. Nagaosa in the present volume).

2.9 Sum Rules for the Optical Conductivity Spectra of FröhlichPolarons

In this section several sum rules for the optical conductivity spectra of Fröhlichpolarons are applied to test the DSG approach [30] and the DQMC results[33]. The values of the polaron effective mass for the DQMC approach aretaken from [39]. In Tables 2 and 3, we show the polaron ground-state E0 andthe following parameters calculated using the optical conductivity spectra:

M0 ≡∫ Ωmax1 Reσ (Ω) dΩ, (34)

M1 ≡∫ Ωmax1 ΩReσ (Ω) dΩ, (35)

where Ωmax is the upper value of the frequency available from [33],

M̃0 ≡π

2m∗+

Ωmax∫1

Reσ (Ω) dΩ. (36)

Here m∗ is the polaron mass, the optical conductivity is calculated in unitsn0e

2/(mbωLO), m∗ is measured in units of the band mass mb, and the fre-quency is measured in units of ωLO. The values of Ωmax are: Ωmax = 10 forα = 0.01, 1 and 3, Ωmax = 12 for α = 4.5, 5.25 and 6, Ωmax = 18 for α = 6.5,7 and 8.

The parameters corresponding to the DQMC calculation are obtained us-ing the numerical data kindly provided to the author by A. Mishchenko [97].The optical conductivity derived by DSG [30] exactly satisfies the sum rule[98]

π

2m∗+

∞∫1

Reσ (Ω) dΩ =π

2. (37)

Since the optical conductivity obtained from the DQMC results [33] is knownonly within a limited interval of frequencies 1 < Ω < Ωmax, the integral in(36) for the DSG-approach [30] is calculated over the same frequency intervalas for the Monte Carlo results [33].

The comparison of the resulting zero frequency moments M̃ (DQMC)0 andM̃

(DSG)0 with each other and with the value π/2 = 1.5707963... correspond-

ing to the right-hand-side of the sum rule (37) shows that the difference


Table 2. Polaron parameters M0,M1, M̃0 obtained from the diagrammatic MonteCarlo results (Reprinted with permission from [78]. c©2006, Società Italiana diFisica.)

α M(DQMC)0 m

∗(DQMC) M̃ (DQMC)0 M(DQMC)1 /α E

(DQMC)0

0.01 0.00249 1.0017 1.5706 0.634 −0.0101 0.24179 1.1865 1.5657 0.65789 −1.0133 0.67743 1.8467 1.5280 0.73123 −3.184.5 0.97540 2.8742 1.5219 0.862 −4.975.25 1.0904 3.8148 1.5022 0.90181 −5.686 1.1994 5.3708 1.4919 0.98248 −6.796.5 1.30 6.4989 1.5417 1.1356 −7.447 1.3558 9.7158 1.5175 1.2163 −8.318 1.4195 19.991 1.4981 1.3774 −9.85

Table 3. Polaron parameters M0,M1, M̃0 obtained within the path-integral ap-proach (Reprinted with permission from [78]. c©2006, Società Italiana di Fisica.)

α M(DSG)0 m

∗(Feynman) M̃ (DSG)0 M(DSG)1 /α E

(Feynman)0

0.01 0.00248 1.0017 1.5706 0.633 −0.0101 0.24318 1.1957 1.5569 0.65468 −1.01303 0.69696 1.8912 1.5275 0.71572 −3.13334.5 1.0162 3.1202 1.5196 0.83184 −4.83945.25 1.1504 4.3969 1.5077 0.88595 −5.74826 1.2608 6.8367 1.4906 0.95384 −6.71086.5 1.3657 9.7449 1.5269 1.1192 −7.39207 1.4278 14.395 1.5369 1.2170 −8.11278 1.4741 31.569 1.5239 1.4340 −9.6953

∣∣∣M̃ (DQMC)0 − M̃ (DSG)0 ∣∣∣ on the interval α ≤ 8 is smaller than the absolute valueof the contribution of the “tail” of the optical conductivity for Ω > Ωmax tothe integral in the sum rule (37):

∞∫Ωmax

Reσ(DSG) (Ω) dΩ ≡ π2− M̃ (DSG)0 . (38)

Within the accuracy determined by the neglect of the “tail” of the part ofthe spectrum for Ω > Ωmax, the contribution to the integral in the sum rule(37) for the optical conductivity obtained from the DQMC results [33] agreeswell with that for the optical conductivity found within the path-integralapproach in [30]. Hence, the conclusion follows that the optical conductivityobtained from the DQMC results [33] satisfies the sum rule (37) within theaforementioned accuracy.


We analyse the fulfilment of the “LSD” polaron ground-state theoremintroduced in [99]:

E0 (α)− E0 (0) = −3π

α∫0

dα′

α′

∞∫0

ΩReσ (Ω,α′) dΩ (39)

using the first-frequency moments M (DQMC)1 and M(DSG)1 . The results of this

comparison are presented in Fig. 10. The solid dots indicate the polaronground-st

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