SIMILARITY REDUCTION OF ENERGY-TRANSPORT MODELS FOR SEMICONDUCTORS

- The main purpose of this work is to perform symmetry classification of a system of partial differential equations for energy-transport in semiconductors. In the case where there are symmetries, they are used to reduce the number of independent variables which in turn enables one to construct invariant solutions. Invariant solutions of a given equation satisfy an equation with fewer independent variables. Thus, the search of invariant solutions can be viewed as a sort of dimensional reduction. From a computational standpoint, the reduced system is easier to analyse both numerically and analytically than the original system.

When the original system contains arbitrary parameters or functions, the consistency conditions of the determining equations provides a means to specify their forms. This is the essence of the group classification method [8]. The importance of group classification stems from the fact that many models in application contain parameters or functions which cannot be determined from any known physical law. However, by analysing the consistency conditions of the determining equations we are able to identify maximally symmetric submodels.
In some problems of practical interest, generating the determining equations turns out to be tedious. Fortunately, Lie's method for calculating symmetries is algorithmical and can be implemented using packages for symbolic computation (Mathematica, Maple, Reduce, ...). In this work we will use the program, Yalie [4], written in Mathematica for the generation and manipulation of the determining equations for symmetries.
The outline of this paper is as follows. In section 2, we present the model to be investigated. In section 3, we perform the symmetry classification of the model. In section 4, we employ the symmetries to optimally reduce the number of independent variables in the model. Finally, we summarize our findings and hints on further work.

ENERGY-TRANSPORT MODEL FOR SEMICONDUCTORS
The detailed derivation of the energy-transport (ET) model from the Boltzmann equation is presented in [1]. On coupling the Poisson equation for the electric potential to the diffusion equations for the electron density and temperature, we have the following equations [10] 0, c(x) n φ λ 0, nC φ t  (1) where n is the electron density, J the electron momentum density, W the electron energy, S the energy flux density, W nC the energy production, 2 λ the dielectric constant, φ the electric potential, c(x) the doping profile and . z , where T is the electron temperature, T L is the lattice temperature (taken as constant), µ (i) are the electron mobilities and τ W is the energy relaxation time. In general the mobilities are temperature-dependent. The system (1) must be solved subject to appropriate initial and boundary conditions. Some special cases recently considered in the literature are: where µ 0 and τ 0 are positive constants, • the Lyumkis et al [6] model in which

THE SYMMETRY CLASSIFICATION OF THE ET MODEL FOR SEMICONDUCTORS
Recently Romano and Valenti [10] performed the symmetry analysis of the one-dimensional ET model for semiconductors. They also calculated invariant solutions. In this section we will continue their work by examining multidimensional models, namely, we perform the group classification of the spherically symmetric and the two-dimensional ET models for semiconductors. In order to generate and manipulate the determining equations for symmetries, we use the Mathematica software Yalie developed by Díaz [4].

The spherically symmetric ET model for semiconductors
Here we focus on the symmetry analysis of the spherically symmetric ET model for semiconductors. By spherically symmetric we mean that all spatial dependence of the dependent variables is through the radial coordinate , where d is the spatial dimension. Using the chain rule, we arrive at the following spherically symmetric equations where the subscript denotes partial differentiation, The cases of interest are 1,2,3 d = , i.e. 0,1,2 k = .
From now on we will use τ(T) to mean τ W (T). According to Lie's algorithm, the vector field whenever (2) When expanded and separated, the determining equations (3) span many pages. Using the program Yalie [4] written in Mathematica, we obtain 38 equations. After simplifying the 38 determining equations we get the following equations (we only consider the case 0 k ≠ , the case 0 k = was investigated in [10]).
[ ] are constants and F(t) is a constant function.
If we assume that µ (1) (T), µ (2) (T), τ(T) and c(r) are arbitrary functions of their arguments, we end up with the following symmetries where F(t) is a smooth function of t. The symmetries F 1 X and X span the so-called principal symmetry Lie algebra of (2). Now our goal is to find specifications of the arbitrary elements that extend the principal Lie algebra. Consider equation (6b). If b 1 +a 1 q = 0, then b 1 +a 1 (q−1) = 0. Thus, a 1 = b 1 = 0. This implies that c(r) and τ(T) are arbitrary. We end up with the principal Lie algebra. As a result, this case does not lead to an extension. Therefore, we assume from now on that b 1 +a 1 q ≠ 0. Solving equation in (6b), we obtain where b 1 = a 1 p. If a 1 = 0, then equation (6a) implies that b 1 = 0. i.e. we get the case a 1 = b 1 = 0 which has already been dealt with. The solution to equation (6a) is τ(T) = τ 0 (T−T L )T p , 0 τ constant. Thus, the extension of the principal symmetry Lie algebra is given by the operator Remark: For q = 0 and 2 1 q = we obtain the forms of J and S for Chen et al [3] model and Lyumkis et al [6] model respectively.

The two-dimensional ET model for semiconductors
Using the definition of the divergence and the Laplacian operators, the ET model system in two-dimensions becomes , 0 J J n Employing the Yalie program [4] to generate the determining equations, we obtain 91 determining equations. After simplifications we get the following equations [ ] Where a 0 , a 1 , a 2 , b 0 , m, (1) 0 µ and µ are constants and G(t) is a constant function.
If we assume that µ (i) (T), τ(T) and c(x,y) are arbitrary functions of their arguments, it can be shown that the principal symmetry Lie algebra of (7) is spanned by the operators where G(t) is a smooth function of t.
We now seek specifications of the arbitrary elements that extend the principal symmetry Lie algebra. The solution to equation (10) where H is an arbitrary function of its argument. Therefore, the extension of the principal Lie algebra is given by Note that the case 0 m = does not yield an extension of the principal Lie algebra.

SIMILARITY REDUCTIONS OF THE ET MODEL FOR SEMICONDUCTORS
In many applications it is desirable to reduce partial differential equations (PDEs) to ordinary differential equations (ODEs), or at least reduce the number of independent variables. One of the procedures commonly used is dimensional analysis. Dimensional analysis is reminiscent of scaling symmetries and dimensionless variables are simply invariants of the scaling symmetry group [8]. By using symmetries more general than scaling symmetries it is possible to reduce the number of independent variables by introducing the invariants of these symmetries as new independent variables. Thus, the existence of symmetries for PDEs allows a sort of dimensional reduction. In this section we shall use symmetry methods to simplify our submodels whenever possible.
From section 3, we note that the symmetry structure of the submodels is a combination of a two-dimensional Lie algebra and an infinite-dimensional Lie algebra. In the following subsections we shall exploit mainly the finite-dimensional part of the symmetry Lie algebra. Precisely, using the notations of section 3, we shall look for solutions invariant under the operator

The spherically symmetric case
(15) In the above system and the subsequent systems a prime denotes differentiation with respect to the similarity variable γ.
The characteristic equation for the invariants of where ν, ω and χ are arbitrary functions of γ and β. The equations (20) are substituted into system (7) to give the following reduced system Here and thereafter the subscripts on ν, ω and χ denote partial differentiation.

CONCLUSION
In this paper, we have performed a complete Lie symmetry classification of the spherically symmetric and two-dimensional ET models for semiconductor. i.e we obtained all the forms of the arbitrary elements (energy relaxation time, carrier mobility and the doping profile) that maximize the symmetry Lie algebra.
Following the usual modus operandi in symmetry analysis [2,5,7,8], we exploited the symmetries of the submodels to perform similarity reductions. The reduced submodels are still highly nonlinear and hence, difficult to solve analytically. The next logical step of this work will be a numerical investigation of the reduced submodels. This investigation might be of great importance in the simulation and design of semiconductors.
Finally it might be important to investigate the full 1+3 model without a priori symmetry assumptions (spherical symmetry for instance). It might not be a simple task as the classifying relations can be very difficult to analyse. If this happens, the so-called method of preliminary group classification may be used.