Analysis and Simulation of Extended Hydrodynamic Models: The Multi-Valley Gunn Oscillator and MESFET Symmetries

We introduce a novel two carrier hydrodynamic model, which incorporates higher dimen- sional geometric effects into a one dimensional model. We study (1) the GaAs device in the notched oscillator circuit, and, (2) a MESFET channel, and its symmetries. We present new mathematical results for a reduced model.


INTRODUCTION
In previous work, we have demonstrated the robustness of an algorithm (ENO: Essentially Non-Oscillatory) designed for the simulation of the hydrodynamic model for semiconductors over a wide range of parameters. In [8] and [9], n+-n-n / diodes in one dimension and MESFETS in two dimensions were simulated. Here we allow multi-species and geometric source terms.
where Vo(t) is the voltage at the device terminal, V B is the bias voltage, l(t) is the current flowing through the battery, and C is the total capacitance, which includes the so-called cold capacitance, ld(t) is the particle current. In 10], a Monte-Carlo simulation of the Boltzmann equation was used to update la(t). Earlier, a single valley hydrodynamic model was used by [6], whereas here we employ a two-valley hydrodynamic model. The coupling terms and the system have the structure of ].

Description of the Gunn Oscillator
The equations describing an RLC tank circuit, connected to a Gunn oscillator, are:  [12]). The lattice temperature is taken as T o 300 K. In Fig. 1, right, we show the contour of the concentration n when vbias 2V and vgate -0.8V. We can see an approximate spherical symmetry around the upper middle point. This serves as a basis for our reduced 1D model with spherically symmetric forcing terms in the next two sections. The velocity v also shows a similar spherical symmetry. However, temperature T and potential do not reveal spherical symmetry at this vbias. , Ix= ,l (t) GL , * 1-:2-)2(t) e L , (2.4) where the field term x is nonlocal (self-consistent) and a(x) is a C function that can be represented by The Euler-Possion equations for two carriers with a(x) 0 have been studied for some special couplings: The case R H 0 in 11 by the Godunov scheme with fractional step techniques and the case Ri= (1 PlP2)Q(Pl,P2), Hi 0, 0 < Q(Pl,P2) < Q0 in [7] by the viscosity method. The system +Pl +P2 for one carrier with general a(x) C is solved in [5].
We develop a new shock capturing numerical scheme and apply this scheme to construct global entropy solutions to the system (2.3-2.4) with nonzero a(x) and general R and H i. More precisely, we consider the following coupling terms Ri(Pl, P2) and Hi(Pl, 132, El, E2)" mi(x,t)), of (2.3-2.4) such that 0 < Pi(x,t) < C(T) < oo, Imi(x,t)/Pi(X,t)l < C(T) < 0% for 0 < < T < oo, x R,  [3]. Thus, we use the piecewise steady-state solutions, which incorporate such source terms, to replace the piecewise constants from the Riemann solutions as the building blocks. Due to the nonlocal source term, we also incorporate the fractional step procedure into our construction of approximate solutions, with the steady-state solutions as their fundamental building blocks. To obtain a uniform bound for the approximate solutions, we estimate the Riemann invariants involving the nonlocal term with the aid of the conservation of mass and the estimates of the approximate steady-state solutions and Riemann solutions. The H -l compactness of the weak entropy dissipation measures can be achieved as in [2], [3]. These requirements enable us to deduce the strong convergence of the approximate solutions with the aid of a compactness framework (see [2]), proved by DiPerna 2 (1983) for ,---+ 2s + s > 2 an integer, and by Ding-Chen-Luo (1987) and Chen (1988) for the general case < T < 5/3.
In [4] we consider the one-dimensional Euler-Poisson equations. It is proved that the relaxation terms prevent the development of shock waves for the smooth initial data with small oscillation, which is not true for large initial data. The nonlinear singular limit of the smooth solution to the drift-diffusion equation is shown when the relaxation times tend to zero.

SIMULATION RESULTS
We first present simulation results for the Gunn oscillator defined in Section 1.1. Notice that this involves solving a time dependent equation system with strong hyperbolic components, hence upwinding and high order accuracy in space and time are important, justifying the usage of ENO schemes [9]. In Fig. 2, left, we show the time history of the applied voltage Vo(t).
We can see sustained oscillations of a slow frequency on top of a fast frequency. In Fig. 2  "snaps" over one period of the oscillation. We can see the movement of the structure clearly in such a period. Next, we show the result of attempting to use the 1D model with a spherical symmetry assumption, to approximate the 2D MESFET described in Section 1.2. We take our 1D domain to be from r 0.025 to r=0.1, measured from the top middle point at (x,y) (0.3, 0.2) downwards. The boundary conditions for the concentration n, the temperature T and the potential are prescribed, using the values of the 2D simulations; the boundary condition for the velocity is floating (Neumann). In Fig. 3, left, we show the comparison, for the concentration n, of the 2D MES-FET result with the 1D model assuming spherical symmetry, at vbias 2.0V and vgate -0.8V. We can see a qualitatively correct agreement (the agreement is even better for lower bias). This is good since it means that other quantities (such as T and ) which are not spherically symmetric have minimal effect on the concentration through the nonlinear coupling of the equations. In Fig. 3, right, we show the same comparison for the temperature T. We can see that now the 1D model disagrees with the 2D results to a much greater extent, manifesting the fact that T is not spherically symmetric. If n is the only quantity of interest, then the 1D model can be used, saving substantial computing time in the simulation. Otherwise