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Polaron concept and its applications

Theory of Quantum and Complex systems. Polaron concept and its applications. J . T. Devreese Theorie van Kwantumsystemen en Complexe Systemen (TQC ), Universiteit Antwerpen , Belgium. Outline. Polaron concept Polaron optical absorption: analytic approximations

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Polaron concept and its applications

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  1. Theory of Quantum and Complex systems Polaron concept and its applications J. T. Devreese Theorievan Kwantumsystemen en ComplexeSystemen (TQC),UniversiteitAntwerpen, Belgium

  2. Outline • Polaron concept • Polaron optical absorption: analytic approximations • Many-polaron optical absorption of doped strontium titanate • Mechanisms of the Fermi-liquid response of Nb-doped strontium titanate • Superconductivity in the LaAlO3-SrTiO3heterostructure

  3. Polaronconcept

  4. Polaron concept A conduction electron (or hole) together with its self-induced polarization in a polar crystal forms a quasiparticle, which is called a polaron1-3 Properties of polarons have attracted increasing attention due to their possible relevance to physics of high-Tc superconductors 4 A conduction electron repels the negative ions and attracts the positive ions A self-induced potential arises, which acts back on the electron and modifies its physical properties An artist’s view of a polaron 5 1 L. D. Landau, Phys. Z. Sowjetunion 3, 664 (1933) 2 S. I. Pekar, Untersuchungen über die Elektronentheorie der Kristalle, Berlin, Akademie, 1954 3 H. Fröhlich, Adv. Phys. 3, 325 (1954) 4 A. S. Alexandrov and N. Mott, Polarons and bipolarons, World Scientific, Singapore, 1996 5 J. T. L. Devreese, Moles agitat mentem. Ontwikkelingen in de fysika van de vaste stof. Rede uitgesproken bij de aanvaarding van het ambt van buitengewoon hoogleraar in de fysica van de vaste stof, in het bijzonder de theorie van de vaste stof, bij de afdeling der technische natuurkunde aan de Technische Hogeschool Eindhoven, March 9, 19796 J. T. Devreese and A. S. Alexandrov, Advances in Polaron Physics, Springer Series in Solid-State Sciences, Vol. 159 (Springer, 2009)

  5. Electron-phonon coupling The large-polaron coupling constant was introduced by Fröhlich 1 wLO is the long-wavelength frequency of a LO phonon e∞ and e0 are, respectively, the electronic and the static dielectric constant of the polar crystal mb is the electron (hole) band mass 1 H. Fröhlich, Adv. Phys. 3, 325 (1954)

  6. Strong- and weak-coupling polaron The first studies on polarons were devoted to the calculation of the self-energy and the effective mass of polarons in the limit of large a, or “strong coupling” 1-3 The ground-state energy and the effective mass of a strong-coupling polaron 4 are: The “weak-coupling” limit 5 is obtained from the leading terms for a → 0. Weak-coupling results for the polaron parameters are The 3D case 6,7: The 2D case 8: where ws is the surface-optical-phonon frequency 1 L. D. Landau and S. I. Pekar, Zh. Eksper. Teor. Fiz. 18, 419 (1948) 2 S. I. Pekar, Untersuchungen über die Elektronentheorie der Kristalle, Akademie Verlag, Berlin, 1951 3 N. N. Bogolubov, Ukr. Matem. Zh. 2, 3 (1950) 4 S. J. Miyake, J. Phys. Soc. Jpn. 38, 181 (1975) 5 H. Fröhlich, Adv. Phys. 3, 325 (1954) 6 M. A. Smondyrev, Teor. Math. Fiz. 68, 29 (1986) [English translation: Theor. Math. Phys. 68, 653 (1986)] 7 J. Röseler, Phys. Stat. Sol. (b) 25, 311 (1968) 8 Wu Xiaoguang, F. M. Peeters, and J. T. Devreese, Phys. Rev. B 31, 3420 (1985)

  7. Feynman’s path-integral treatment The propagator resulting from the elimination of the phonon field b = 1/(kBT) All-coupling theory The polaron problem is formulated1 as an equivalent one-particle problem in which the interaction, non-local in time or "retarded", is between the electron and itself. 1 R. P. Feynman, Phys. Rev. 97, 660 (1955) 2 R. P. Feynman, R. W. Hellwarth, C. K. Iddings, and P. M. Platzman, Phys. Rev. 127, 1004 (1962) 3 K. K. Thornber and R. P. Feynman, Phys. Rev. B 1, 4099 (1970)

  8. Polaron optical absorption:analytic approximations

  9. Elementary polaron scattering process At T = 0, the optical ab-sorption coefficientcan be expressed in terms of elementary functions in two limiting cases Optical properties of polarons at weak coupling1,2 High densities, , where z is the Fermi energy: Low densities, : 1 V. L. Gurevich, I. G. Lang, and Yu. A. Firsov, Sov. Phys. Solid State 4, 918 (1962) 2 J. Devreese, W. Huybrechts, and L. Lemmens, Phys. Stat. Sol. (b) 48, 77 (1971)

  10. Internal excitations of a polaron at strong coupling

  11. Dynamic response of continuum polarons:the path-integral treatment The path-integral treatment 1-3 of the polaron response is based on the Feynman polaron model 4 Absorption coefficient is related to impedance as FHIP approximation 1 The expansion of the impedance leads to the resonant structure 1 R. P. Feynman, R. W. Hellwarth, C. K. Iddings, and P. M. Platzman, Phys. Rev. 127, 1004 (1962)2 K. K. Thornber and R. P. Feynman, Phys. Rev. B 1, 4099 (1970)3 J. Devreese, J. De Sitter, and M. Goovaerts, Phys. Rev. B 5, 2367 (1972)4 R. P. Feynman, Phys. Rev. 97, 660 (1955)

  12. All-coupling path-integral Feynman approach Memory-function formalism 1,2 The polaron optical conductivity within the memory-function formalism Memory function The two-point density-density Green’s function vandware the variational frequency parameters of the Feynman polaron model. 1 J. Devreese, J. De Sitter, and M. Goovaerts, Phys. Rev. B 5, 2367 (1972)2 F. M. Peeters and J. T. Devreese, Phys. Rev. B 28, 6051 (1983)

  13. Optical absorption of polarons at arbitrary coupling Optical-absorption spectrum of a single large polaron 1 • The memory function c(w) contains the dynamics of the polaron • At T = 0, a d-like “central peak” is at the origin • For larger a, peaks attributed to transitions to RES are more pronounced 1 J. Devreese, J. De Sitter, and M. Goovaerts, Phys. Rev. B 5, 2367 (1972)

  14. Polaron optical conductivity: comparison between DSG and DQMC Second-order perturbation theory DSG approach1 Diagrammatic Quantum Monte-Carlo method 2 1 J. Devreese, J. De Sitter, and M. Goovaerts, Phys. Rev. B 5, 2367 (1972)2A.S. Mishchenko, N. Nagaosa, N. V. Prokof’ev, A. Sakamoto, B. V. Svistunov, Phys. Rev. Lett. 91, 236401 (2003)3 J. T. Devreese and A. S. Alexandrov, Advances in Polaron Physics, Springer Series in Solid-State Sciences, Vol. 159 (Springer, 2009)

  15. DQMC Comparison of DQMC with the DSG model of the polaron optical conductivity The positions of the main peak of the polaron optical-conductivity band1, obtained within DSG2, are in a remarkable agreement with the results of DQMC 3. The peak width within DSG is smaller than that in DQMC, especially at strong coupling. The origin of the peak width at strong coupling is not yet understood. 1 J. T. Devreese and A. S. Alexandrov, Advances in Polaron Physics, Springer Series in Solid-State Sciences, Vol. 159 (Springer, 2009) 2 J. Devreese, J. De Sitter, and M. Goovaerts, Phys. Rev. B 5, 2367 (1972) 3 A. S. Mishchenko, N. Nagaosa, N. V. Prokofev, A. Sakamoto, and B. V. Svistunov, Phys. Rev. Lett. 91, 236401 (2003)

  16. Arbitrary-coupling polaron optical conductivity: new analytical approximations • Extended memory function formalism with phonon broadened levels • Strong coupling expansion based on the Franck-Condon principle

  17. (A) Extended memory-function formalism 1,2 Like in the memory-function formalism, where in the function a finite lifetime t for the states of the relative motion is introduced. This damping can be considered as the result of of the residual electron-phonon interaction, not included in the Feynman polaron model. and Dt(t) turns to D(t) of DSG. The lifetime is determined from the requirement that the sum rule is fulfilled: 1 G. De Filippis, V. Cataudella, A. S. Mishchenko, C. A. Perroni, and J. T. Devreese, Phys. Rev. Lett. 96, 136405 (2006)2 V. Cataudella, “Variational approaches to the polaron problem” (Invited lecture at the International School of Physics Enrico Fermi, CLXI Course "Polarons in Bulk Mateials and Systems with Reduced Dimensionality", Varenna, Italy, 21.6. - 1.7.2005 )

  18. (B) Strong-coupling expansion: the Franck-Condon picture 1 Taking into account all multiphonon processes and neglecting recoil as well as correlation between the emission and absorption of successive phonons, one arrives at the real part of the optical conductivity with the parameters For large enough coupling strengths, the envelope of the optical conductivity is well described by the Gaussian 1 G. De Filippis, V. Cataudella, A. S. Mishchenko, C. A. Perroni, and J. T. Devreese, Phys. Rev. Lett. 96, 136405 (2006)2 S. I. Pekar, Untersuchungen über die Elektronentheorie der Kristalle (Akademie-Verlag, Berlin, 1954).

  19. Polaron optical conductivity for various a: Comparison of methods Extended MFF1 Strong-coupling expansion1 DSG model 2 DQMC 3 1 G. De Filippis, V. Cataudella, A. S. Mishchenko, C. A. Perroni, and J. T. Devreese, Phys. Rev. Lett. 96, 136405 (2006)2 J. Devreese, J. De Sitter, and M. Goovaerts, Phys. Rev. B 5, 2367 (1972)3 A. S. Mishchenko, N. Nagaosa, N. V. Prokofev, A. Sakamoto, and B. V. Svistunov, Phys. Rev. Lett. 91, 236401 (2003)

  20. (C) Strong-coupling expansionbeyond the adiabatic approximation1 The Kubo formula with the dipole-dipole two-point correlation function A strong-coupling adiabatic Ansatz where 1 S. N. Klimin and J. T. Devreese, to be published (2010)

  21. (C) Strong-coupling expansionbeyond the adiabatic approximation1 The transformed Hamiltonian contains The self-consistent adiabatic potential energy for the electron A complete orthonormal basis of the Franck-Condon states , which are eigenstates of . The non-adiabatic transitions between different polaron levels for the renormalized operator of the electron-phonon interaction W are neglected: 1 S. N. Klimin and J. T. Devreese, to be published (2010)

  22. Polaron optical conductivity: extended strong-coupling expansion 1 • The asymptotically exact strong-coupling expansion 1,2 takes into account • the internal non-adiabaticity of the excited state • the corrections of order a0 • The bandwidth is determined by the multiphonon processes 1 S. N. Klimin and J. T. Devreese, to be published (2010)2 J. T. Devreese and A. S. Alexandrov, Advances in Polaron Physics, Springer Series in Solid-State Sciences, Vol. 159 (Springer, 2009)3 G. De Filippis, V. Cataudella, A. S. Mishchenko, C. A. Perroni, and J. T. Devreese, Phys. Rev. Lett. 96, 136405 (2006)4 A.S. Mishchenko, N. Nagaosa, N. V. Prokof’ev, A. Sakamoto, and B. V. Svistunov, Phys. Rev. Lett. 91, 236401 (2003)

  23. Recent applications of the polaron theory • Ripplonicpolarons in multielectron bubbles • Polaron physics in ultracold Fermi and Bose gases • Optical absorption of many-polaron systems • Optical conductivity of the doped strontium titanate • Superconductivity at the interface of two dielectrics

  24. Conclusions • It is remarkable how the Fröhlichpolaron, one of the simplest examples of a Quantum Field Theoretical problem, as it basically consists of a single fermion interacting with a scalar Bose-field, has resisted full analytical solution at all coupling since ~1950, when its Hamiltonian was first written. • Although a basic mechanism for the optical absorption of Fröhlichpolarons was already proposed between 1969and 1972, some subtle characteristics were only clarified in 2006 by combining numerical studies (DQMC) and improved variational approximate analytical methods. • The richness and profundity of Landau's polaron concept is further illustrated by its extensions, e.g., to the electronic polaron, to the Holstein polaron, to bipolarons, to excitons, to ripplopolarons...

  25. Many-polaron optical absorption of doped strontium titanate

  26. Experimental optical conductivity spectra of SrTi1-xNbxO3 1 The optical conductivity spectra of a doped SrTiO3 reveal the following features: • Drude peaks • Peaks due to the LO-phonon absorption ( ) • Midinfrared band ( ) Doping levels: The intensity of the midinfrared band increases with doping 1 J. L. van Mechelen, D. van der Marel, C. Grimaldi, A. B. Kuzmenko, N. P. Armitage, N. Reyren, H. Hagemann, and I. I. Mazin, Phys. Rev. Lett. 100, 226403 (2008) 2 P. Calvani, M. Capizzi, F. Donato, S. Lupi, P. Maselli, and D. Peschiaroli, Phys. Rev. B 47, 8917 (1993)

  27. Polaron effective mass and coupling strength in SrTi1-xNbxO3 The calculated LDA band mass1 in SrTiO3 is , which provides the effective mass determined from the spectral weight using the sum rule4for the optical conductivity Spectral weight for the optical conductivity The effective mass in SrTi1-xNbxO3 yields an intermediate electron-phonon coupling strength The electronic contribution Large-polaron mechanism is suggested 1 Within our approach for the polaron optical conductivity in a polar medium with multiple LO-phonon branches 2,3, the effective electron-phonon coupling strength in SrTi1-xNbxO3 is aeff ~ 2 where e and me are the free electron charge and mass, and n is the electron density. 1 J. L. van Mechelen, D. van der Marel, C. Grimaldi, A. B. Kuzmenko, N. P. Armitage, N. Reyren, H. Hagemann, and I. I. Mazin, Phys. Rev. Lett. 100, 226403 (2008); arXiv:0712.1607v1 2 S. N. Klimin, V. M. Fomin, and J. T. Devreese, “Optical conductivity of an interacting polaron gas in a polar crystal with multiple LO-phonon branches” (to be published) 3 J. T. Devreese, S. N. Klimin, J. L. M. van Mechelen, and D. van der Marel, Phys. Rev. B 81, 125119 (2010) 4 J. T. Devreese, L. F. Lemmens, and J. Van Royen, Phys. Rev. B 15, 1212 (1977).

  28. Sum rule leading to a relationbetween the effective mass and the optical absorption of polarons In 1977, we found how the effective mass of the polaron can be calculated given the optical absorption spectrum 1. We use the well-known f-sum rule: the zeroth moment of the real part of the freguency dependent conductivity for a particle is a constant We start from the expression for the optical conductivity 2 and find for the zero-frequency limits: with the function Here, v and w are the Feynman variational parameters 3. 1 J. T. Devreese, L. F. Lemmens, and J. Van Royen, Phys. Rev. B 15, 1212 (1977). 2 J. Devreese, J. De Sitter, and M. Goovaerts, Phys. Rev. B 5, 2367 (1972). 3 R. P. Feynman, Phys. Rev. 97, 660 (1955).

  29. Sum rule leading to a relationbetween the effective mass and the optical absorption of polarons It follows that for , The denominator m[1 —R(a)] coincides with the polaron effective mass calculated by Feynman. The latter equation then suggests the solution to the apparent inconsistency concerning the a dependence of the zerothmoment. This inconsistency is due to the neglect of the dfunction, multiplied by pe2/m*f(where m*fis the effective mass of the polaron). Thus the increase of the absorption, integrated for w > wLOis compensated by a decrease of the strength of the dfunction. Therefore we introduce the following sum rule for polarons: where m* is the effective mass of the polaron. 1 J. T. Devreese, L. F. Lemmens, and J. Van Royen, Phys. Rev. B 15, 1212 (1977). 2 J. Devreese, J. De Sitter, and M. Goovaerts, Phys. Rev. B 5, 2367 (1972). 3 R. P. Feynman, Phys. Rev. 97, 660 (1955).

  30. Optical conductivity of SrTi1-xNbxO3:the many-large-polaron theory and experiment Large-polaron optical conductivity spectra 1 in comparison with the experimental data 2 Many-body effects are taken into account Calculated spectra are in a fair agreement with the experiment for the position and the width of the optical conductivity band 1 J. T. Devreese, S. N. Klimin, J. L. M. van Mechelen, and D. van der Marel, Phys. Rev. B 81, 125119 (2010) 2 J. L. van Mechelen, D. van der Marel, C. Grimaldi, A. B. Kuzmenko, N. P. Armitage, N. Reyren, H. Hagemann, and I. I. Mazin, Phys. Rev. Lett. 100, 226403 (2008)

  31. Mechanisms of the Fermi-liquid response of Nb-doped strontium titanate

  32. Landau Fermi-liquid It can be shown2that the scattering rate At low temperatures, the average excitation energy is ~kBT, and hence http://landau100.itp.ac.ru http://www.bayarea.net In 1956, Landau developed a theory of interacting spin-1/2 fermions1 The Landau-Fermi liquid theory successfully describes metals, nuclear matter, liquid He-3 … 1 L. D. Landau, Zh. Eskp. Teor. Fiz. 30, 1058 (1956) [Sov. Phys. JETP 3, 920 (1957)]; [ 32, 59 (1957) [ 5, 101 (1957)]. 2 P. Phillips, Advanced solid state physics (Frontiers in Physics Series, Westview Press, 2003)

  33. SrTiO3: band structure Fermi surface of the cubic phase at 2% doping, showing the large anisotropy of the lowest band • Band dispersion of the lowest unoccupied bands of SrTiO3 D. van der Marel, J. L. M. van Mechelen, and I. I. Mazin, Phys. Rev. B 84, 205111 (2011)

  34. Fermi-liquid behavior of resistivity in n-doped SrTiO3 D. van der Marel, J. L. M. van Mechelen, and I. I. Mazin, Phys. Rev. B 84, 205111 (2011)

  35. Superconductivity in SrTiO3 1 J. G. Bednorz and K. A. M¨ueller, Rev. Mod. Phys. 60, 585 (1988) 1 S. Koonce, M. L. Cohen, J. F. Schooley,W. R. Hosler, and E. R. Pfeiffer, Phys. Rev. 163, 380 (1967) 2 G. Binnig, A. Baratoff, H. E. Hoenig, and J. G. Bednorz, Phys. Rev. Lett. 45, 1352 (1980) 3N. Reyren et al., Science 317, 1196 (2007) 4 N. Reyren, S. Gariglio, A. D. Caviglia, D. Jaccard, T. Schneider, and J.-M. Triscone, Appl. Phys. Lett. 94, 112506 (2009)

  36. Fermi-liquid and superconductivity in SrTiO3: a common origin? Relaxation rate The parameter u < 1 describes the fraction of the momentum changes that is transferred to the ionic lattice. The dimensionless parameter ltis the effective interaction strength. C. S. Koonce, M. L. Cohen, J. F. Schooley,W. R. Hosler, and E. R. Pfeiffer, Phys. Rev. 163, 380 (1967). G. Binnig, A. Baratoff, H. E. Hoenig, and J. G. Bednorz, Phys. Rev. Lett. 45, 1352 (1980). D. van der Marel, J. L. M. van Mechelen, and I. I. Mazin, Phys. Rev. B 84, 205111 (2011)

  37. Fermi-liquid and superconductivity in SrTiO3: Effective interaction The effective electron-electron interaction is provided both by the Coulomb repulsion and the phonon-mediated attraction • Picture from: D. van der Marel, Ginzburg Conference on Physics, Moscow May 28 - June 2, 2012 • S. N. Klimin, J. Tempere, D. van der Marel, and J. T. Devreese, Phys. Rev. B 86, 045113 (2012) • C. S. Koonce, M. L. Cohen, J. F. Schooley, W. R. Hosler, and E. R. Pfeiffer, Phys. Rev. 163, 380 (1967).

  38. Scattering mechanisms The relaxation rate in SrTi1-xNbxO3 can be accounted for by various scattering mechanisms: We calculate the effective relaxation rate treating all these mechanisms The calculation is performed using the kinetic equation and accounting for the band anisotropy and the splitting of the conduction band • Baber scattering • Umklapp electron-electron scattering • Scattering of the electrons by the donors • Direct electron – LO-phonon scattering S. N. Klimin, J. Tempere, D. van derMarel, and J. T. Devreese, Phys. Rev. B 86, 045113 (2012) D. van der Marel, J. L. M. van Mechelen, and I. I. Mazin, Phys. Rev. B 84, 205111 (2011)

  39. Method: kinetic equation Scattering probability Effective interaction between the electrons is the phonon Green’s function When both optical and acoustic phonons are taken into account, the effective electron-electron interaction can become attractive. • S. N. Klimin, J. Tempere, D. van derMarel, and J. T. Devreese, Phys. Rev. B 86, 045113 (2012)

  40. Contributions to the relaxation rate Baber scattering For a parabolic simple conductivity band, the electron-electron interaction due to the normal scattering processes does not contribute to the carrier electric response owing to momentum conservation. However, for a non-parabolic and/or complex conductivity band, the carrier response can be non-zero. • S. N. Klimin, J. Tempere, D. van derMarel, and J. T. Devreese, Phys. Rev. B 86, 045113 (2012) • W. G. Baber, Proc. R. Soc. A 158, 383 (1937)

  41. Contributions to the relaxation rate Scattering by donors The scattering on the potential landscape caused by impurities provides a non-negligible contribution to the total relaxation rate at low temperatures. • S. N. Klimin, J. Tempere, D. van derMarel, and J. T. Devreese, Phys. Rev. B 86, 045113 (2012)

  42. Contributions to the relaxation rate Electron – LO-phonon scattering The scattering of the electrons by the LO phonons due to the Frohlich interaction can bring a non-vanishing (but small) contribution to the resistivity only at sufficiently high temperatures. • S. N. Klimin, J. Tempere, D. van derMarel, and J. T. Devreese, Phys. Rev. B 86, 045113 (2012)

  43. Contributions to the relaxation rate Umklappelectron-electron scattering The umklapp processes contribute to the Fermi-liquid response of the electron gas in strontium titanate only at sufficiently high electron densities. This is not the case of the experiment 1 • 1 D. van derMarel, J. L. M. van Mechelen, and I. I. Mazin, Phys. Rev. B 84, 205111 (2011) • 2 S. N. Klimin, J. Tempere, D. van derMarel, and J. T. Devreese, Phys. Rev. B 86, 045113 (2012)

  44. Conclusions • The total relaxation rate in SrTi1−xNbxO3 is provided mainly by two mechanisms: • The Baber electron-electron scattering with participation of both Coulomb and phonon-mediated electron-electron interactions leads to the T2 – dependence of the relaxation rate. • The scattering on the potential landscape caused by impurities is responsible for the residual relaxation rate at low temperatures. • A good agreement with experiment is achieved accounting for all phonon branches in strontium titanate. • The effective electron-electron interaction can be attractive in strontium titanate. • This interaction provides superconductivity at low temperatures and Fermi-liquid response in a wide range of temperatures. • The used microscopic model supports the notion that superconductivity and Fermi-liquid properties of n-type SrTiO3 have a common origin.

  45. Superconductivity in the LaAlO3-SrTiO3heterostructure

  46. LaAlO3-SrTiO3heterostructure High-mobility quasi 2D electron gas Picture from: S. Thiel et al., Science 313, 1942 (2006) 1 A. Ohtomo, H. Y. Hwang, Nature 427, 423 (2004) 2 N. Nakagawa, H. Y. Hwang, D. A. Muller, Nat. Mater. 5, 204 (2006) 3 S. Thiel, G. Hammerl, A. Schmehl, C. W. Schneider, and J. Mannhart, Science 313, 1942 (2006)

  47. Experimentally detected superconductivityin the LAO-STO structure N. Reyren, S. Thiel, A. D. Caviglia, L. F. Kourkoutis, G. Hammerl, C. Richter, C. W. Schneider, T. Kopp, A.-S. Ruetschi, D. Jaccard, M. Gabay, D. A. Muller, J.-M. Triscone, and J. Mannhart, Science 317, 1196 (2007).

  48. Theoretical model: multilayer structure Scheme of a multilayer structure The coordinate z for the n-th layer lies in the range zn-1 < z < zn z LaAlO3-SrTiO3: the asymmetric three-layer structure zK K zK-1 zn ln n zn-1 z2 2 z1 1 z0 L

  49. Coulomb interaction in the LaAlO3-SrTiO3 structure The Poisson equation The Fourier transformation Boundary contitions The resulting interaction

  50. Optical phonons in the LaAlO3-SrTiO3 structure Spectra of the interface phonons The potential induced by the interfaceoptical vibrations obeys the Laplace equation The relation between the electric field and the displacement with the dielectric function and the electrostatic boundary conditionslead to the set of equations with the matrix elements The equation for eigenfrequencies: G.-Q Hai, F. M. Peeters, and J. T. Devreese, Phys. Rev. B 42, 11063 (1990)

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