Time-Dependent Solution of a Full Hydrodynamic Model Including Convective Terms and Viscous Effect

Sub-picosecond turn-on transient behavior of ballistic diodes (N+N N+ structures) is studied by solving a system of time-dependent hydrodynamic (HD) equations. Convective terms as well as viscous effect are included in the study. The simulation result indicates that the diode undergoes approximately one quarter of a plasma oscillation before it relaxes to the steady-state value through collisions.


INTRODUCTION
Recently the hydrodynamic model has been widely used in submicrometer semiconductor device simula- tions in order to more accurately describe the non-equilibrium and non-local phenomena occurring in these devices.However, most of these simulations are limited to the steady-state operation.Although some work has been performed for transient simula- tion [1 ], [2], the transport models used are often simpli- fied.In this study, we used a unique set of HD model which has been calibrated by Monte Carlo (MC) data [3].

Convective and Viscous Terms
The macroscopic quantity A and the inverse effective mass M-lappearing in Eq.( 3) are modeled as [3]" dV A VV + 2.85(2W)VV + a(W)-rl'CvVV(--;-), (6)   where c 0.5eV !denotes the non-parabolicity factor and a(W) and b(W) represent the empirical formulas fitted from the MC simulation data [3].The first two terms on the r.h.s, of (5) and the second and third terms on the r.h.s, of (6) represent the convective terms.These terms cannot be easily handled by the conventional Scharfetter-Gummel (S-G) [4] type dis- cretization scheme and therefore are often neglected in the device simulation.The last term on the r.h.s, of (5) represents the viscous term.The viscous effect is a second-order effect which becomes important when the velocity-gradient is large.The dimensionless parameter q is determined from the MC simulation and is approximately 2-3 [5].

NUMERICAL SCHEMES Spatial Discretization Scheme
The well known S-G scheme [4] for spatial discretiza- tion is very effective in dealing with the drift-diffu- sion(DD) equations as well as the simplified energy-balance equation.However, all exponentially weighted schemes have one common disadvantage: the velocity V is not treated as one of the state varia- bles.Thus neither the convective terms nor the vis- cous effect can be included in the HD model in a natural way.
In this work we use the second upwind as the spa- tial discretization scheme which has been success- fully used by Gardner [6] to solve the simplified HD system under the steady-state condition.where I]leq, neqand Weq are the equilibrium values of the electrostatic potential, the electron concentration and the average energy, respectively.

Time Advancing Scheme
In this study we use Bank's trapezoidal rule/back- ward-differentiation-formula (TR-BDF2) composite method as the time-stepping scheme.The detailed discussion of this method can be found in [2].An automatic time-step adjustment mechanism is built in with this scheme through monitoring the local trunca- tion error at each time step.The relative error during the course of time advancement is chosen to be 10-3.

SIMULATION RESULTS
We use an N + N-N + structure (not shown) as our testing device.A step bias of 2 Volts is applied at 0+. Figs. 1, 2, and 3 show the time evolution of the electron velocity, the average energy and the electron current density, respectively.The average energy increases smoothly from Weq to its final distribution (Fig. 2).However, the velocity has a rapid overshoot (Fig. 1) at around 0. lps and then gradually relaxes to a much lower steady-state value.During approximately the same time period (t. 0.1ps).the electron current density also shows a similar peak (Fig. 3).The time duration from 0 to O.lps represents approxi- mately one quarter of the plasma oscillation period.
After O.lps, the free plasma oscillation is quickly taken over by the collision processes and the velocity, as well as the current density, relax towards their steady-state values.
As we mentioned before, the viscous term, which has a dissipating effect, is included in our HD system.
The effect of including the viscous term on the final steady-state result is shown in Fig. 4. It is clear that the velocity overshoot obtained with the viscous term near the N + N junction region is substantially lower than that without the viscous term, and is in a better agreement with the MC data.This difference is expected to be more significant in the simulation of the narrow-base BJT's.

CONCLUSIONS
We have tested our new HD transport model and the existing numerical schemes with the transient simula- tion of an N + N-N + diode.The HD model is con- sidered as "full" in the sense that the conservation of "momentum" is solved explicitly with time.Although the problem considered is somewhat artificial (a step voltage is directly applied to the diode), it serves to demonstrate the utility of the HD model and the numerical solution schemes being used.We have also shown that the inclusion of the convective and vis- cous terms in the HD model vields a result in better agreement with the MC simulation data.We are now extending this work to the simulation of narrow-base BJT's in which the viscous effect is expected to play a more significant role.

FIGURE 4 AFIGURE 3
FIGURE Time evolution of the electron velocity