Generation-recombination noise in solids

This paper is concerned with some of the mathematical aspects of noise in solids. Attention is given to generation-recombination noise. This noise is due to current fluctuations arising from the generation, recombination, or trapping of carriers. We treat only homogeneous systems, and therefore, our descriptive random processes are independent of spatial coordinates. In our model, we associate a counting process
\upsilon_{j}(t), j = 1, 2, \cdots , N
with each energy level
E_{j}
of the material, where
\upsilon_{j}(t)
represents the number of carriers in
E_{j}
at time
t
. Thus the statistics of the vector counting process
\bar{V}(t) = (\upsilon_{1}(t), \upsilon_{2}(t), \cdots , \upsilon_{N}(t))
completely determine the noise behavior of the material. Clearly the component processes of
\bar{V}(t)
are not statistically independent, and
\Sigma \upsilon_{j}
must be constrained since the total number of carriers is assumed to be fixed. Here we restrict ourselves to the study of the second-order statistics of
\bar{V}(t)
. Making several physically reasonable assumptions and approximations, we derive the mean, covariance, and spectral density of
\bar{V}(t)
, which agree with previously published results. While our basic results are not new, the underlying assumptions made in the derivations and the basic formulation of the problem are novel. Our approach is to expand time derivatives of the characteristic function for the vector random process
\bar{V}(t)
from which we extract systems of differential equations in the desired statistics. This method exhibits the results as solutions of linear state equations.