Bi-Dimensional Simulation of the Simplified Hydrodynamic and Energy-Transport Models Heter ] unction Semiconductors Devices Using Mixed Finite Elements

We study the application of the mixed finite elements method (MFE) for the bidimensional simulation of the simplified hydrodynamic and energy-transport models. The two main points are the use of entropic variables which gives a symmetric positive definite problem and a coupled computation of the equations for electrons which requires a generalization of the MFE method for vector valued problems. We give numerical results on JFET and HEMT devices.


INTRODUCTION
The Drift-Diffusion (DD) model, which is cur- rently the most usual model in industrial simula- tion of electronic devices, is not sufficiently accurate for sub micrometer device modeling.The two main contenders are the Hydrodynamic (HD) [2, 6, 10] and the Energy Transport (ET) [11,1] models which provides a better description of such effects as velocity overshoot and carrier heating.Both models can be derived from the Boltzmann Transport Equation (BTE) by the moments method.If we remove the convective term from the HD model, we obtain the Simplified Hydrodynamic (SHD) model.Indeed, this con- vective term can be neglected for some applica- tions, otherwise numerical difficulties may occurs as in fluid dynamics occurs.
The Mixed Finite Elements (MFE) method, which was first developed in the study of structural mechanics, is well adapted to these equations because it gives a good conservation of the currents.For the DD equations this method was used by Brezzi et al. [3, 8] with the carrier concentration variables and by Marrocco et al. [5, 7] with the quasi-Fermi level variables.This last *Corresponding author.
A. MARROCCO AND Pn.MONTARNAL approach seems well adapted when considering heterojunctions.Indeed, the quasi-Fermi levels are continuous at the heterojunctions when densities are not.
We consider the bi-dimensional stationary case and take a simplified hydrodynamic model or an energy transport model for the electrons coupled with a drift-diffusion model for the holes through the Poisson equation.We use a Boltzmann statistic.We express the problem in a conservative form.In order to obtain a symmetric positive definite system of elliptic equations, we introduce appropriate variables.We first use the electrostatic potential 4, the hole quasi-Fermi level p and for the electrons we introduce entropic variables vl and v2 (functions of the electron quasi-Fermi levels n and the electron temperature Tn, see (9)) which lead to a symmetric formulation.We consider the usual "flow" variables like the electric displacement D, the hole current density Jp, the electron current density Jn and a "flow" function Jw which depends on Jn and on the thermal energy flow density S (see (10)).The boundary conditions are non-homogeneous Dirichlet conditions on the contacts and homogeneous Neumann conditions on the insulated boundaries.
We extend the numerical methods developed for the DD model in [5, 7] and for the uni-dimensional ET model in [9].The solution of the problem is obtained as the limit, when goes to infinity, of an artificial transient problem discretized with local time steps.We use block relaxation techniques as we consider successively the Poisson equation, the equation for the holes and the two equations for the electrons.The two first equations are solved like in [5, 7].On the other hand, in order to treat the electronic conservation laws, we extend the MFE method for a vector valued problem.For each case "implicit type" discretization for time and mixed finite elements for space are used.The non-linear problems are solved by Newton-Raphson algorithms.
An outline of the paper is as follows.In section 2 the physical model is presented in his symmetric form.We present the general method of solution in section 3. The extension of the MFE method for vector valued problems is developed in section 4.And finally we present numerical results on JFET and HEMT in section 5.
Let us remark that the matrix A is symmetric positive definite.This proves the interest of the variables g)n v--n and vz--n. (9)e link between Jw and Jn, S, is the following: Jw --Sn X)Jn.
The boundary conditions are non-homogeneous Dirichlet conditions on the ohmic contacts and homogeneous Neumann conditions on the insu- lated boundaries: q5 go, P gp, N-gN, Tn gTn on FD, Odp OP ON OTn 0 on F N -F D.

On On On On
The difference between the models comes from the definition of a,/3 and c a 1,/3 5/2 and 0 _< c _< 5/2 for the SHD model, a 1,/3 3/2 and c 0 for the Stratton ET model and a 1/2, /3 -2 and c-0 for the Degond et al.ET model.

NUMERICAL METHODS
The solution of the problem (1)-( 8) is obtained as the limit, when goes to infinity, of the following artificial transient problem sOOtdp div D + q(N P dop) O, (11) S qop OtqOp div Jp q U 0, ( SOt( vl )v2 -div( t J n t J w ) qU ( --( (Wn q(q X.))U__ nWO_Wn ) --0, "rw Our choice of functions s , sep and Si,j is motivated by the following considerations: we must keep the homogeneity of the equations, S must be a positive definite matrix and functions s0, sp must be strictly positive and we want to optimize the convergence.In our computations, we take as in [5, 7]   S o E S p (Ax)2 (Ax)2 q#PP' and by natural extension S:A. (ZXx) We use block relaxation techniques as we consider successively the Poisson Eq. ( 11), the equation for the holes (12) and the system of equations for the electrons (13).So we have to solve scalar and vector valued non linear parabolic equations.We use an implicit type discretization for time and mixed finite elements for space.The non-linear problems are solved by Newton-Raphson algorithms.
The two first problems (11,12) are solved like in [5,7].On the other hand, for the computation of the system (13), we extend the MFE method for a vector valued problem.We develop this in the next section.
For the linear systems, direct solvers (Cholesky or LU factorization) are used.The non-symmetric linear system associated with the coupled Eq. (13) can efficiently (memory/time) be solved via GMRES algorithms with block-diagonal pre-con- ditioning.

SOLUTION OF THE TWO ELECTRON EQUATIONS: EXTENSION OF THE MFE METHOD FOR A VECTOR VALUED PROBLEM
where D, Jp, Jn, Jw are respectively given by Eqs. (5-8).
At each time step k, after solving the Poisson equation and the hole conservation equation, we -----""'-"'-t-t' dimensional devices such that JFET or HEMT with 0.2 lam gates length.
We give here the charateristic Is (Vz)s)curves and the electron concentration and temperature distribution for the HEMT device for applied potential Vs 0V, Va + IV, VG =-2V.We use a mesh with 20 000 elements which gives 30 000 degrees of freedom for each flow variable.