Computation of the Spectral Density of Current Fluctuations in Bulk Silicon Based on the Solution of the Boltzmann Transport Equation

Numerical simulation results for the spectral density of noise due to current fluctuations are presented. The mathematical framework is based on the interpretation of the equations describing electron transport in the semiclassical transport model as stochastic differential equations (SDE). Within this framework, it was previously shown that the autocovariance function of current fluctuations can be obtained from the transient solution of the Boltzmann transport equation (BTE) with special initial conditions. The key aspect which differentiates this approach from other noise models is that this approach directly connects noise characteristics with the physics of scattering in the semiclassical transport model and makes no additional assumptions regarding the nature of noise. The solution of the BTE is based on the Legendre polynomial method. A numerical algorithm is presented for the solution of the transient BTE. Numerical results are in good agreement with Monte Carlo noise simulations for the spectral density of current fluctuations in bulk silicon.


I. INTRODUCTION
Noise is generally characterized by the spectral density of current or voltage fluctuations. In this paper, we present numerical simulation results for the autocovariance function and spectral density of current fluctuations based on a new noise model [1 ]. Employing the machinery of SDE, this model shows that key computations for the noise autocovariance function are reduced to the solution of the BTE with special initial conditions. Based on this novel approach, we study the influence of temperature and electric field on noise spectral density in bulk silicon. We also compare our results with those obtained using the Corresponding author. 185 Monte Carlo technique [2]. The paper is organized as follows. Section II summarizes the noise model employed for the calculations. Section III describes the algorithm for the solution the transient BTE based on the Legendre polynomial method [3] and presents numerical results. The last section is devoted to the conclusions.

II. NOISE MODEL BASED ON SDE
According to semiclassical transport theory, an electron in a semiconductor drifts under the influence of electric field and experiences occasional random changes in its momentum due to the scattering mechanisms in the crystal. This process is described in terms of the following SDE" and r Zhffi)(t ti), (1) where x, v, p and k are the electron position, drit velocity, momentum and wave vector, respectively, E is the electric field, (k) is the energy-way>e vector relationship in the given energy band and Fr is the random impulse force on the electron due to scattering. The random force is characterized by the transition rate W(k, k'). Accordingly, the probability of scattering is given by (2) where )(k)is the scattering, rate. Therefore, given the electron wave vector k, )(k)At is the probability that a jump in momentum will occur in a small time interval At. Assuming that a scattering event has occurred at some time ti, the probability density function for the amplitude of the jump is given by, where k(t) ki and k(t[) ki+ui. Note that these are the same equations which are used to simulate the electron motion in Monte Carlo simulations. The stochastic differential equations (1) together with Eqs. (2) and (3) define a copound Poisson process which is discontinuous in k-space and is a Markov process. In stochastic differential equation theory, such a process is usually characterized by a transition probability density function, which satisfies the Kolmogorov-Feller forward equation. Employing the machinery of SDE theory it was previously shown that this function satisfies the BTE (see [4]).
Semiconductor noise is usually characterized by the spectral density of current fluctuations. The spectral density of current fluctuations is by definition the time Fourier transform of its autocovariance function.
Since the instantaneous electron current is proportional to the electron momentum, the autocovariance function of current fluctuations can be computed directly from the autocovariance function of the electron momentum. In order to compute the autocovariance function of a stochastic process k t, one needs to know the joint probability density function of the stochastic process at times + " and for all ,. Since the stochastic process is stationary, the reference time is arbitrary and can be chosen to be zero for convenience. Consequently, it is sufficient to compute the joint probability density function p(k,, k0). Employing conditional probabilities, the joint probability density function can be expressed as follows" 9(k,, k) 9(kzlk)Po(k). This conditional probability density function is nothing more than the transitional probability density function which characterizes the Markov process and it satisfies the Kolmogorov-Feller forward equation with the initial condition, ktlz=0 k. On the other hand, the probability density function P0(k) satisfies the stationary Kolmogorov-Feller equation since the process is stationary and the initial reference time is arbitrary. Since the Kolmogorov-Feller equation is identical to the BTE, we conclude that the key computations for the noise spectral density are reduced to special initial value problems for the BTE. The computation of the noise spectral density is based on this formalism and is described in [1]. According to this formalism, it is shown that the autocovariance matrix K],(,) of the electron wave-vector can be computed as follows. For bulk noise computations, the space-independent BTE, is ) solved subject to the )initial condition: a(k, ")[--0 kf(k), where f(k) denotes the stationary solution. Here, equation (4) should be understood in a component-wise sense. The above solution is substituted into the following equation in order to evaluate the autocovariance matrix: Since, the autocovariance function is a symmetric function of , the solution of Eq. for x > 0 is sufficient.

III. SOLUTION OF THE BTE EMPLOYING LEGENDRE POLYNOMIALS
It is assumed that a single band distribution function accurately represents the state of the momentum space. The Herring-Vogt transformation is employed to map the coordinate system k into k* which results in spherical equal energy surfaces: y(e) e + [3e 2 t2k*2/2mo In k* momentum space the direction of the average electron wave vector (]*) defines a symmetry axis () and the density function dependence on momentum can be expressed in terms of only two independent variables: E and 0. We expand f(k*, t) in Legendre polynomials according to: f(-*,t) fo(e, t) + k* g(13, t)cosO + k*Zh(e,t)(3cos20 1). (6) In terms of this representation for f(k*,t), the BTE can be formulated as follows  Figure (1