Polaron


A polaron is a quasiparticle used in condensed matter physics to understand the interactions between electrons and atoms in a solid material. The polaron concept was proposed by Lev Landau in 1933 and Solomon Pekar in 1946 to describe an electron moving in a dielectric crystal where the atoms displace from their equilibrium positions to effectively screen the charge of an electron, known as a phonon cloud. This lowers the electron mobility and increases the electron's effective mass.
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. 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.

Theory

The energy spectrum of an electron moving in a periodical potential of a rigid crystal lattice is called the Bloch spectrum, which consists of allowed bands and forbidden bands. An electron with energy inside an allowed band moves as a free electron but has an effective mass that differs from the electron mass in vacuum. However, a crystal lattice is deformable and displacements of atoms 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 possible scenario was proposed in the seminal 1933 paper by Lev Landau, which includes the production of a lattice defect such as an F-center and a trapping of the electron by this defect. A different scenario was proposed by Solomon Pekar that envisions dressing the electron with lattice polarization. Such an electron with the accompanying deformation moves freely across the crystal, but with increased effective mass. Pekar coined for this charge carrier the term polaron.
Landau and Pekar constructed 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.
While polaron theory was originally developed for electrons, there is no fundamental reason why it could not be any other charged particle interacting with phonons. Indeed, other charged particles such as holes and ions generally follow the polaron theory. For example, the proton polaron was identified experimentally in 2017 and on ceramic electrolytes after its existence was hypothesized.
MaterialαMaterialα
InSb0.023KI2.5
InAs0.052TlBr2.55
GaAs0.068KBr3.05
GaP0.20RbI3.16
CdTe0.29Bi12SiO203.18
ZnSe0.43CdF23.2
CdS0.53KCl3.44
AgBr1.53CsI3.67
AgCl1.84SrTiO33.77
α-Al2O32.40RbCl3.81

Usually, in covalent semiconductors the coupling of electrons with lattice deformation is weak and polarons do not form. In polar semiconductors the electrostatic interaction with induced polarization is strong and polarons are formed at low temperature, provided that their concentration is not large and the screening is not efficient. Another class of materials in which polarons are observed is molecular crystals, where the interaction with molecular vibrations may be strong. In the case of polar semiconductors, the interaction with polar phonons is described by the Fröhlich Hamiltonian. On the other hand, the interaction of electrons with molecular phonons is described by the Holstein Hamiltonian. Usually, the models describing polarons may be divided into two classes. The first class represents continuum models where the discreteness of the crystal lattice is neglected. In that case, polarons are weakly coupled or strongly coupled depending on whether the polaron binding energy is small or large compared to the phonon frequency. The second class of systems commonly considered are lattice models of polarons. In this case, there may be small or large polarons, depending on the relative size of the polaron radius to the lattice constant.

Fröhlich Hamiltonian

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.
The strength of electron-phonon interaction is determined by the dimensionless coupling constant. Here is electron mass, is the phonon frequency and,, are static and high frequency dielectric constants. 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 γ depends on the material and the type of phonon being used in the model. In the case of a single polar mode
,
here is the volume of the unit cell. In the case of molecular crystal γ is usually momentum independent constant. 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 phonons and linear 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.
When the coupling is weak, the self-energy of the polaron can be approximated as:
and the polaron mass, which can be measured by cyclotron resonance experiments, is larger than the band mass of the charge carrier without self-induced polarization:
When the coupling is strong, 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, Monroe D. Donsker and S. R. Srinivasa Varadhan, applying large deviation theory to the path integral formulation for the self-energy, showed the large α exactitude of this Landau–Pekar formula. Later, Elliot H. Lieb and Lawrence E. Thomas gave a shorter proof using more conventional methods, and with explicit bounds on the lower order corrections to the Landau–Pekar formula.
Richard Feynman introduced the 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 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.
In the lattice models the main parameter is the polaron binding energy:, here summation is taken over the Brillouin zone. Note that this binding energy is purely adiabatic, i.e. does not depend on the ionic masses. For polar crystals the value of the polaron binding energy is strictly determined by the dielectric constants,, and is of the order of 0.3-0.8 eV. If polaron binding energy is smaller than the hopping integral the large polaron is formed for some type of electron-phonon interactions. In the case when the small polaron is formed. There are two limiting cases in the lattice polaron theory. In the physically important adiabatic limit all terms which involve ionic masses are cancelled and formation of polaron is described by nonlinear Schrödinger equation with nonadiabatic correction describing phonon frequency renormalization and polaron tunneling.
In the opposite limit the theory represents the expansion in.