To investigate the full range of phenomena which occur in open systems, one needs a model of the dissipative processes (such as scattering of electrons by phonons in semiconductors) which occur within the system. However, the question of the correct description of such processes is at present far from resolved (see Jauho, 1989). Therefore, in the inductive spirit of the present work, we will assume a priori that the classical Boltzmann collision operator acting upon the Wigner distribution is an adequate approximation at some level. The form which the Boltzmann operator takes within the present one-dimensional model is developed below.
In solid-state physics the name ``Boltzmann equation'' is applied to any transport equation which combines the Liouville description of ballistic motion with a local Markovian model of the stochastic processes. This can include such processes as the scattering of electrons by phonons or impurities. These will be considered to be one-body processes because the phonon and impurity degrees of freedom are not explicitly included in the model, and thus (neglecting Fermi degeneracy) such processes lead to terms linear in the distribution function. The Boltzmann equation can also include a master-operator description of two-body interactions such as electron-electron scattering (and in statistical physics the name ``Boltzmann equation'' usually refers more specifically to this kinetic equation), and such a term will be nonlinear in the single-particle distribution function (assuming the Stosszahlansatz). For the present purposes we will only consider one-body interactions so that the collision operator is linear.
The Boltzmann collision term is usually written in the form (Ferry, 1980):
where is the rate of scattering from plane-wave state to state k. (To maintain consistency with the literature we will use the wavevector to label these states, rather than the momentum.) Equation (14.157) can be rewritten to emphasize the linear, homogeneous nature of the collision term:
Note that the collision term is local, so that in the complete kernel of there is a -function in q, which is suppressed from the above definition. Note that the potential superoperator has a similar dependence on q (4.39), and as a result and have the same sparsity structure in the discrete approximation [see (4.55)]. Thus, the addition of to the calculation requires no modification to the superoperator data structures or solution procedures.
The scattering rates are taken to be the Fermi golden rule rates. For electron-phonon scattering:
where is the Hamiltonian for the electron-phonon interaction and is the phonon frequency. In (14.159) and the following, the upper sign refers to phonon absorption and the lower sign refers to phonon emission. The transition rates depend upon the full three-dimensional of each state, whereas the numerical calculations at present consider only the longitudinal momentum . Thus, the scattering rates must be ``projected'' onto the one-dimensional model. To do so, we first assume that the distribution of electrons with respect to the transverse momenta of the initial state is a normalized Maxwellian distribution at a fixed temperature:
with defined in (3.18). The resulting scattering rates are then integrated over the transverse momenta of the final states:
where is the volume of the crystal. Henceforth we will drop the subscript from the .
For polar optical phonon scattering the absolute square of the matrix element is (in SI notation and from Fawcett, Boardman, and Swain, 1970):
where is the longitudinal-optical phonon frequency, and and are the low and high frequency permittivities of the semiconductor, respectively. The phonon occupation number is given by the Bose-Einstein distribution. The one-dimensional scattering rates are obtained by inserting (14.163) into (14.162). After some manipulation, one can write an expression for the scattering rate. First, define dimensionless quantities a and b as:
Then the scattering rate is
The collision operator for polar optical phonon scattering in the one-dimensional model is then obtained by inserting (14.164), for both phonon emission and absorption, into a discretized version of (14.158).
The collision operator for acoustic deformation-potential scattering may be similarly constructed. Assuming equipartition of energy in the acoustic modes, the matrix element is (Fawcett, Boardman, and Swain, 1970):
where is the acoustic deformation potential, is the mass density of the material, and s is the velocity of sound. The second expression is obtained by expanding the Bose distribution for low energies using . Inserting (14.165) into (14.162) and multiplying by 2 to include the equal emission and absorption rates, we obtain:
Given the expressions such as (14.164) and (14.166) we can readily construct the collision operator using (14.158). For the purposes of numerical evaluation, it is most convenient to accumulate the values of (in the discrete approximation) by performing the assignments
for all values of k and . One can implement this procedure in a single subprogram to which a function which evaluates is passed as an argument, and then invoke this subprogram for each of the processes of interest. A convenient test of the resulting is provided by the principle of detailed balance. It is , where is an equilibrium (Maxwellian) distribution. The collision operators obtained from (14.164) and (14.166) pass this test.
The effects of the Boltzmann collision operators for these phonon scattering processes on the steady-state characteristics of the RTD are illustrated in Fig. 25. In this calculation the matrix elements for GaAs using the parameters of Fawcett, Boardmann, and Swain (1970) were assumed to hold throughout the structure. The acoustic phonon scattering has a very small effect on the curve. The LO phonon scattering processes significantly decrease the peak current and increase the valley current. The initial report of this calculation (Frensley, 1988b) employed a scattering operator for the LO phonons which was one-half of the correct value, due to an algebraic error. Similar calculations have been done by Mains and Haddad (1988b). Kluksdahl et al. (1989) and Jensen and Buot (1990) have used a relaxation term to model the inelastic processes.
Figure 25. Effect of phonon scattering processes on the characteristic of the resonant-tunneling diode, using the Boltzmann collision operator. Scattering by LO phonons significantly reduces the peak current and increases the valley current. The effect of acoustic phonons is nearly negligible. The temperature was 300 K.