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Energy spectrum of an electron moving in a periodical potential of rigid crystal lattice consists of allowed and forbidden bands and is known as the Bloch spectrum. An electron with energy inside an allowed band moves as a free electron but with effective mass (solid-state physics) that differs from the electron mass in vacuum. However, crystal lattice is deformable and displacements of atoms (ions) from their equilibrium positions are described in terms of phonons. Electrons interact with these displacements, and this interaction is known as electron-phonon coupling. One of possible scenarios was proposed in the seminal 1933 paper by Lev Landau, it includes production of a lattice defect such as an F-center and trapping the electron by this defect. A different scenario was proposed by Solomon Pekar that envisions dressing the electron with lattice deformation (a cloud of virtual phonons). Such an electron with the accompanying deformation moves freely across the crystal, but with increased effective mass.〔 L. D. Landau and S. I. Pekar, Effective mass of a polaron, Zh. Eksp. Teor. Fiz. 18, 419–423 (1948) (Russian ), English translation: Ukr. J. Phys., Special Issue, 53, p.71-74 (2008), http://ujp.bitp.kiev.ua/files/journals/53/si/53SI15p.pdf〕 Pekar coined for this charge carrier the term polaron. The general concept of a polaron has been extended to describe other interactions between the electrons and ions in metals that result in a bound state, or a lowering of energy compared to the non-interacting system. Major theoretical work has focused on solving Fröhlich and Holstein Hamiltonians. This is still an active field of research to find exact numerical solutions to the case of one or two electrons in a large crystal lattice, and to study the case of many interacting electrons. Experimentally, polarons are important to the understanding of a wide variety of materials. The electron mobility in semiconductors can be greatly decreased by the formation of polarons. Organic semiconductors are also sensitive to polaronic effects, which is particularly relevant in the design of organic solar cells that effectively transport charge. The electron phonon interaction that forms Cooper pairs in low-Tc superconductors (type-I superconductors) can also be modeled as a polaron, and two opposite spin electrons may form a bipolaron sharing a phonon cloud. This has been suggested as a mechanism for Cooper pair formation in high-Tc superconductors (type-II superconductors). Polarons are also important for interpreting the optical conductivity of these types of materials. The polaron, a fermionic quasiparticle, should not be confused with the polariton, a bosonic quasiparticle analogous to a hybridized state between a photon and an optical phonon. == Polaron theory== L. D. Landau 〔 〕 and S. I. Pekar 〔 . English translation: Research in Electron Theory of Crystals, AEC-tr-555, US Atomic Energy Commission (1963) 〕 formed the basis of polaron theory. A charge placed in a polarizable medium will be screened. Dielectric theory describes the phenomenon by the induction of a polarization around the charge carrier. The induced polarization will follow the charge carrier when it is moving through the medium. The carrier together with the induced polarization is considered as one entity, which is called a polaron (see Fig. 1). A conduction electron in an ionic crystal or a polar semiconductor is the prototype of a polaron. Herbert Fröhlich proposed a model Hamiltonian for this polaron through which its dynamics are treated quantum mechanically (Fröhlich Hamiltonian).〔 〕〔 〕 This model assumes that electron wavefunction is spread out over many ions which are all somewhat displaced from their equilibrium positions, or the continuum approximation. The strength of the electron-phonon interaction is expressed by a dimensionless coupling constant α introduced by Fröhlich.〔 In Table 1 the Fröhlich coupling constant is given for a few solids. The Fröhlich Hamiltonian for a single electron in a crystal using second quantization notation is: The exact form of gamma depends on the material and the type of phonon being used in the model. A detailed advanced discussion of the variations of the Fröhlich Hamiltonian can be found in J. T. Devreese and A. S. Alexandrov 〔 〕 The terms Fröhlich polaron and large polaron are sometimes used synonymously, since the Fröhlich Hamiltonian includes the continuum approximation and long range forces. There is no known exact solution for the Fröhlich Hamiltonian with longitudinal optical (LO) phonons and linear (the most commonly considered variant of the Fröhlich polaron) despite extensive investigations.〔〔〔〔〔 〕〔 〕〔 〕〔 〕〔 〕〔 〕 Despite the lack of an exact solution, some approximations of the polaron properties are known. The physical properties of a polaron differ from those of a band-carrier. A polaron is characterized by its ''self-energy'' , an ''effective mass'' and by its characteristic ''response'' to external electric and magnetic fields (e. g. dc mobility and optical absorption coefficient). When the coupling is weak ( small), the self-energy of the polaron can be approximated as:〔 〕 \approx -\alpha -0.015919622\alpha^2, | |} and the polaron mass , which can be measured by cyclotron resonance experiments, is larger than the band mass m of the charge carrier without self-induced polarization:〔 〕 \approx 1+\frac+0.0236\alpha^2. | |} When the coupling is strong (α large), a variational approach due to Landau and Pekar indicates that the self-energy is proportional to α² and the polaron mass scales as α⁴. The Landau-Pekar variational calculation 〔 yields an upper bound to the polaron self-energy , valid for ''all'' α, where is a constant determined by solving an integro-differential equation. It was an open question for many years whether this expression was asymptotically exact as α tends to infinity. Finally, Donsker and Varadhan,〔M. Donsker and R.Varadhan(1983). "Asymptotics for the Polaron", ''Commun. Pure Appl. Math.'' 36, 505–528.〕 applying large deviation theory to Feynman's path integral formulation for the self-energy, showed the large α exactitude of this Landau-Pekar formula. Later, Lieb and Thomas gave a shorter proof using more conventional methods, and with explicit bounds on the lower order corrections to the Landau-Pekar formula. Feynman 〔 〕 introduced a variational principle for path integrals to study the polaron. He simulated the interaction between the electron and the polarization modes by a harmonic interaction between a hypothetical particle and the electron. The analysis of an exactly solvable ("symmetrical") 1D-polaron model,〔 〕〔 〕 Monte Carlo schemes 〔 〕〔 〕 and other numerical schemes 〔 〕 demonstrate the remarkable accuracy of Feynman's path-integral approach to the polaron ground-state energy. Experimentally more directly accessible properties of the polaron, such as its mobility and optical absorption, have been investigated subsequently. In the strong coupling limit, , the spectrum of excited states of a polaron begins with polaron-phonon bound states with energies less than , where is the frequency of optical phonons.〔V. I. Mel'nikov and E. I. Rashba. ZhETF Pis Red., 10 1969, 95, 359 (1959), JETP Lett 10, 60 (1969). http://www.jetpletters.ac.ru/ps/1687/article_25692.pdf〕 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「polaron」の詳細全文を読む スポンサード リンク
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