An explicit formula is presented for a (conditional) Wiener integral, the integrand of which is an exponential of a general quadratic functional of the path. The functional integrals arising in non‐Markovian Gaussian approximations to various problems of statistical physics (e.g., theory of the large polaron, theory of disordered systems) are easily recovered as special cases.

Explicit evaluation of certain Gaussian functional integrals arising in problems of statistical physics

We compute the average partition function for an electron moving in a Gaussian random potential. A path integral formulation is used, with a trial action like that in Feynman's polaron theory. We compute the variational bound as well as the first correction in a systematic cumulant expansion. The results are checked against exact formulas for the onedimensional white noise problem. The density of states in the low-energy tail has the correct exponential energy dependence, and energy-dependent prefactor to within a few percent. In addition, the partition function goes over smoothly to the perturbation theory result at high temperatures.

Partition function for an electron in a random potential

Path integration of a general two-time quadratic action characterising memory effects is performed within the framework of Feynman's polygonal path approach. Explicit evaluation of the propagator in exact analytical form is further carried out for the specific kernel used by Feynman in the polaron problem.

http://iopscience.iop.org/article/10.1088/0305-4470/16/18/022/pdf

Path integration of a two-time quadratic action

Applications of white noise calculus to the computation of Feynman integrals

Publisher Summary This chapter focuses on the properties of moderately and heavily doped silicon (Si) and germanium (Ge); lightly doped crystals are only briefly considered in it. It discusses the work (experimental as well as theoretical) that helps in understanding the physics of bandgap narrowing (BGN) and the effect of BGN on device performance. A historical survey of the work done on BGN is outlined in the chapter, followed by the theoretical work on BGN. The effect of concentration fluctuation and the theory of band tails are also discussed in the chapter, followed by the effect of BGN on optical properties of Si and Ge. In calculating the donor and acceptor impurity states in semiconductors, several approximations have to be made. In many calculations, the band is assumed to be isotropic and parabolic, the electrons and holes are described by effective masses, and the screened Coulomb potentials are used in the Hamiltonian.

Bandgap Narrowing and Its Effects on the Properties of Moderately and Heavily Doped Germanium and Silicon

A method for calculating the density of states and the effective mass of a model of an electron in a completely random system containing dense and weak scatterers, is presented. The scattering potential employed in the model is assumed to be a gaussian function. The technique used in the calculation is the path-integral method of Feynman applied to the polaron problem (1955). Special emphasis is devoted to the consideration of the asymptotic behaviours of the density of states m(E) and the effective mass m*, for two limiting values of the correlation length L. Some improvements upon the method as well as an extension to a real physical system are discussed.

https://iopscience.iop.org/article/10.1088/0022-3719/7/16/015/pdf

V Samathiyakanit

Papers

Path-integral theory of a model disordered system