Polaron Energy and Effective Mass

One of the celebrated applications of the Feynman path integral approach in statistical mechanics or solid-state physics is the motion of electrons in an ionic crystal resulting in a polaron where the Feynman path integral approach gives results that are visibly superior over those obtained via conventional approaches. Interest in the polaron concept is not only due to its description of particular physical properties of charge carriers in polarizable (or deformable) solids but because it constitutes also an interesting field theoretical model consisting of a fermion interacting with a scalar boson field [2,21].

The polaron concept was introduced by Landau [35] as an auto localization of an electron in a polar crystal and, subsequently, Landau and Pekar [27,36,37] investigated the self-energy and the effective mass of the polaron and was shown by Frohlich [38] to correspond to the adiabatic or strong-coupling regime [39]. From the Frohlich polaron Hamiltonian which is a sum of three operators (kinetic energy of the electron, energy of the phonon modes-treated as harmonic oscillators and the electron-phonon interaction energy) considers the electron to interact only with the longitudinal optical modes and the interaction being Coulombic. Further, in 1950 Frohlich and collaborators considered his 1937 polaron model two modifications where they treated the crystal as a dielectric continuum, requiring the spatial extension of the polaron to be much larger than the lattice spacing - the so-called large polaron model [40]. In this model they introduced a dimensionless constant, in front of the interaction term of the Hamiltonian with determining the strength of the electron-phonon coupling. Due to this extended interest, the dimensionless Frohlich electron-phonon coupling constant, a_p is regarded as an independent parameter. For weak coupling polaron, the dimensionless Frohlich electron-phonon coupling constant is a_t < 1, strong coupling a_F > 7, then the intermediate coupling is between these ranges. The strong coupling condition, for bulk crystals is not even achieved for strong ionic crystals such as alkali halides. One of the goals of theoretical research is to develop suitable mathematical techniques that are accurate for all coupling regimes that can also be appropriate in dealing with similar many-body problems like a nucleon interacting with its meson field.

It is possible to reduce the lower bound of the electron-phonon coupling constant's threshold value in nanocrystals to within weak or intermediate-coupling range when the electron and the phonon confinements potentials are considered and tailors the polaron characteristics [2,21]. The properties (binding or (self-)energy, effective mass, response to external electric or magnetic fields such as mobility and impedance) characterizing this quasi-particle differ from those of the band-electron. The polaron effective mass has been obtained experimentally by cyclotron resonance measurements made on different materials where the dimensionless polaron coupling constant, a_h, lies within the intermediate coupling range. So, it will be necessary, to develop suitable accurate formulae for the polaron characteristics and, in particular, the effective mass and the energy expressed through the dimensionless Frohlich electron- phonon coupling constant, a_t.

The above path integral method permits the evaluation of not only the partition function but also the ground state energy of the system as well as the dependence of the energy on the total momentum p of the system for an absolute zero temperature, f) -* oo. Consequently, the interaction of the electron with lattice vibrations should not change the form of the energy spectrum of the system. In the absence of interaction, the total momentum p of the system should be a unique continuous quantum number. The total momentum p is the full integral of motion and as a result of interaction the dependence of the energy on the momentum p in the case of an isotropic polarized crystal, should have the form [2]:

The quantities £₀(v), £,M , E₄(u), are understood as expansion coefficients for the given totality of

discrete quantum numbers v of the system. We now evaluate the partition function of the system with the energy spectrum 18.1. We limit ourselves to the first two terms at low temperatures, /? -> oo, since other terms are infinitesimally small for a given momentum.

We calculate the polaron effective mass from the expression of the polaron energy in 18.1 via

Considering only the first two terms of 18.1 for low temperatures, we find the partition function of the system:

or

From here, considering 18.2 then

and

where,

is the thermal de Broglie wavelength; m is the electron band-mass and V is a normalization volume which is to be taken infinitely large. From 18.6, the coefficient of /? gives the polaron ground state energy E„(i>) while the term independent of /i in the argument of the logarithmic function, the polaron dimensionless effective mass — and is useful in relation with the experimentally determined polaron

m

effective mass to compute the electron band-mass for the sake of comparison with independent theoretical calculations.