Hydrodynamic Model For Semiconductor Devices Research Articles

A thermal-fully hydrodynamic model for semiconductor devices and applications to III-V HBT simulation

Because of the interaction between self-heating and hot carriers effects, neither isothermal nor conventional macrothermal models are adequate for the simulation of state-of-the-art power devices; instead, a detailed electro-thermal model accounting for nonstationary transport, such as the Thermal-Fully Hydrodynamic (T-FH) model, is required. We apply a one-dimensional (1-D) implementation of such a model to the simulation of AlGaAs/GaAs and InP/InGaAs Heterojunction Bipolar Transistors (HBTs), comparing the results with those provided by simplified models, and highlighting how deeply both nonlocal transport and self-heating affect the predicted device performance. The importance of the convective terms is assessed, and a new nonthermal mechanism for the output Negative Differential Resistance (NDR) is proposed.

Well-posedness of the Hydrodynamic Model for Semiconductors

This paper concerns the well-posedness of the hydrodynamic model for semiconductor devices, a quasilinear elliptic-parbolic-hyperbolic system. Boundary conditions for elliptic and parabolic equations are Dirichlet conditions while boundary conditions for the hyperbolic equations are assumed to be well-posed in L2 sense. Maximally strictly dissipative boundary conditions for the hyperbolic equations satisfy the assumption of well-posedness in L2 sense. The well-posedness of the model under the boundary conditions is demonstrated. This paper addresses the well-posedness of the hydrodynamic model for semiconductors. The model is derived from moments of the Boltzmann’s equation, taken over group velocity space. When coupled with the charge conservation equation, it describes the behaviour of small semiconductor devices and accounts for special features such as hot electrons and velocity overshoots. The model consists of a set of non-linear conservation laws for particle number, momentum, and energy, coupled to Poisson’s equation for the electric potential. It is a perturbation of the drift diffusion model [7]. We consider a ballistic diode problem which models the channel of a MOSFET, so the effect of holes in the model can be neglected. The model is [4,11]

Simulation of the hydrodynamic model of semiconductor devices by a finite element method

Proposes a finite element method for numerical simulation of the hydrodynamic model of semiconductor devices. Presents some scaling factors and a variational formulation. Uses the P1 ‐ isoP2 element to discretize this formulation and the GMRES (Generalized Minimum RESidual) algorithm to solve the associated non‐linear system. Proposes an artificial viscous term to stabilize the non‐linear system. Gives a choice for an initial solution. Presents the numerical solutions for n+ ‐ n ‐ n+ diodes and 0.25μm gate length Si MESFETs. Calculates a shock wave at 300K. Observes velocity overshoot phenomenon and the effect of heat conduction term.

A finite element method for the quantum hydrodynamic model for semiconductor devices

A finite element method for numerical simulation of the transient quantum hydrodynamic model for semiconductor devices is presented. This model treats electron flow in a semiconductor device in the same manner as the classical hydrodynamic model, but the energy density and stress tensor include additional quantum terms, which allow particles to tunnel through potential barriers and to build up in potential wells. The finite element method under consideration is based on use of a mixed method for the approximation of the electric field and a shock-capturing Runge-Kutta discontinuous Galerkin method for the quantum hydrodynamic conservation laws. Numerical simulations of a resonant tunneling diode are presented, which show charge buildup in the quantum well and negative differential resistance and a hysteresis in the current-voltage curve. A comparison between the present method and a finite difference method is given.

On a steady state euler-poisson model for semiconduc

The Euler-Poisson model, a simplified version of the hydrodynamic model for semiconductor devices, consists of a first order nonlinear system coupled to Poisson`s equation. The nonlinear system is just the steady state Euler equations of gas dynamics for a gas of charged particles in a electric field that includes a velocity relaxation term. In the one-dimensional steady state case, the differential equations are elliptic in the subsonic regions, and hyperbolic/elliptic in the transonic regions solution in the one-dimensional case, we refer researchers. For a transonic solution in the one-dimensional case are discussed. They prescribed boundary conditions for electron density and are discussed. They prescribed boundary conditions for electron density and electrostatic potential, and showed that if the current density is small enough, a unique subsonic solution exists. In it was shown that the existence of a subsonic solution in the two-dimensional case, though uniqueness of solution is unknown. Furthermore, in existence and uniqueness of a subsonic solution were proved for potential flow in the three-dimensional case. Without the potential flow assumption, we study in this paper the existence of a subsonic solution in the three-dimensional case. By prescribing a vorticity on inflow boundary, a small normal component of velocity onmore » the whole boundary, and assuming a small variation of the velocity relaxation time in the entire semiconductor domain, we prove the existence of a smooth subsonic solution by the Schauder fixed point theorem. Then, a way of prescribing admissible vorticities on the inflow boundary so that subsonic solutions exist is discussed.« less

Simulation of the hydrodynamic device model on distributed memory parallel computers

Stable and robust finite element methods for the convective hydrodynamic model of semiconductor devices are developed and implemented on distributed memory parallel computers. Specifically, a stable and accurate space-time and Galerkin/least-squares finite element formulation is developed for the hydrodynamic transport equations for the conservation laws. In addition, Galerkin finite element methods are employed for the Poisson and lattice thermal diffusion equations. The inclusion of the lattice thermal diffusion equation in the numerical solution of the convective hydrodynamic model is presented for the first time. Numerical results for a bipolar transistor are included to illustrate the effectiveness of the numerical methods. Numerical simulations of semiconductor devices using the convective hydrodynamic model require a significant amount of computations. A single-program-multiple-data (SPMD) programming model is proposed for the implementation of the hydrodynamic model on distributed memory parallel computers. Employing the SPMD programming model, a serial program developed on a workstation can be converted into a parallel program with minimal changes. The parallel program has been ported to a wide range of distributed memory parallel computers including the iPSC/860 hypercube, the Touchstone Delta machine, and the IBM SP-1. Parallel performance results are reported for a bipolar transistor, silicon MESFET and diodes. The results indicate that the parallel hydrodynamic device simulator exhibits excellent speedups and scalability on distributed memory parallel computers with minimum communication overhead.

Global existence and asymptotic stability to the full 1D hydrodynamic model for semiconductor devices

Global existence and asymptotic stability to the full 1D hydrodynamic model for semiconductor devices

Resonant Tunneling in the Quantum Hydrodynamic Model

The phenomenon of resonant tunneling is simulated and analyzed in the quantum hydrodynamic (QHD) model for semiconductor devices. Simulations of a parabolic well resonant tunneling diode at 77 K are presented which show multiple regions of negative differential resistance (NDR) in the current-voltage curve. These are the first simulations of the QHD equations to show multiple regions of NDR.Resonant tunneling (and NDR) depend on the quantum interference of electron wavefunctions and therefore on the phases of the wavefunctions. An analysis of the QHD equations using a moment expansion of the Wigner-Boltzmann equation indicates how phase information is retained in the hydrodynamic equations.

On a Local Existence Theorem for a Simplified One-Dimensional Hydrodynamic Model for Semiconductor Devices

A simplified hydrodynamic model for semiconductor devices, where the energy equation is replaced by a pressure-density relationship, is studied. The system of Euler–Poisson equations is changed to a quasilinear wave equation in Lagrangian mass coordinates. The local existence of a smooth solution of the Euler–Poisson equations is then obtained by using a known result for the quasilinear wave equation.

The Quantum Hydrodynamic Model for Semiconductor Devices

The classical hydrodynamic equations can be extended to include quantum effects by incorporating the first quantum corrections. These quantum corrections are $O( {\hbar ^2 } )$. The full three-dimensional quantum hydrodynamic (QHD) model is derived for the first time by a moment expansion of the Wigner–Boltzmann equation. The QHD conservation laws have the same form as the classical hydrodynamic equations, but the energy density and stress tensor have additional quantum terms. These quantum terms allow particles to tunnel through potential barriers and to build up in potential wells.The three-dimensional QHD transport equations are mathematically classified as having two Schrodinger modes, two hyperbolic modes, and one parabolic mode. The one-dimensional steady-state QHD equations are discretized in conservation form using the second upwind method.Simulations of a resonant tunneling diode are presented that show charge buildup in the quantum well and negative differential resistance (NDR) in the current-v...

Convergence of the Godunov scheme for a simplified one-dimensional hydrodynamic model for semiconductor devices

Convergence of approximate solutions derived by the Godunov scheme for a simplified one-dimensional hydrodynamic model for semiconductor devices is established by using the compensated compactness method. A global existence theorem is shown; and a numerical method for the computation of the physical global solution of this model is provided by this approach.

A finite element formulation for the hydrodynamic semiconductor device equations

A new formulation employing the Galerkin/least-squares finite element method is presented for the simulation of the hydrodynamic model of semiconductor devices. Numerical simulations are performed on the coupled Poisson and hydrodynamic equations for one carrier devices. The hydrodynamic equations for a single carrier, i.e. for the electrons or holes, resemble the compressible Navier-Stokes equations with the addition of highly nonlinear source terms and without the viscous terms. The governing equations are nondimensionalized to improve the conditioning on the resulting system of equations and the efficiency of the numerical algorithms. Furthermore, to establish the stability of the discrete solution, the system of hydrodynamic equations is symmetrized by considering generalized entropy functions. A staggered solution strategy is employed to treat the coupled hydrodynamic and Poisson equations. Numerical results are presented for one-dimensional and two-dimensional one-carrier n +- n- n + devices. The presence of velocity overshoot has been observed and it is recognized that the heat flux term plays an important role in the simulation of semiconductor devices employing the hydrodynamic model.

Hydrodynamic simulation of impact-ionization effects in p-n junctions

Using the hydrodynamic model of semiconductor devices, the authors simulate an abrupt p-n junction under strong reverse bias, accounting for impact ionization by means of electron and hole generation rates dependent on the carrier temperature. They demonstrate the influence of impact ionization as a cooling mechanism on the mean energy and velocity of the carriers, and show that the ionization coefficients alpha /sub n/ and alpha /sub p/ are spatially retarded with respect to the normally used electric-field dependent ones; this results in larger multiplication factors M/sub n/ and M/sub p/. The reason for this effect is traced back to the increased impact-ionization probability of secondary generated carriers having the chance to travel through a larger portion of the high-field region. This is shown qualitatively by discussing the impact ionization integrals, and is estimated quantitatively through the simulation.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

ON THE WELL‐POSEDNESS OF THE TWO‐DIMENSIONAL HYDRODYNAMIC MODEL FOR SEMICONDUCTOR DEVICES

Some results concerning the well‐posedness of the hydrodynamic model of semiconductor devices in two dimensions are given. We show the non‐ellipticity of the stationary model; give representations which exhibit its elliptic and hyperbolic components, and obtain some appropriate boundary conditions from an examination of the time‐dependent problem.

Numerical methods for the hydrodynamic device model: subsonic flow

An introduction to the hydrodynamic model for semiconductor devices is presented. Special attention is paid to classifying the hydrodynamic PDEs (partial differential equations) and analyzing their nonlinear wave structure. Numerical simulations of the ballistic diode using the hydrodynamical device model are presented, as an illustrative elliptic problem. The importance of nonlinear block iterative methods is emphasized. Arguments for existence of solutions and convergence of numerical methods are given for the case of subsonic electron flow. >

MULTI‐DIMENSIONAL DISCRETIZATION SCHEME FOR THE HYDRODYNAMIC MODEL OF SEMICONDUCTOR DEVICES

A discretization technique is proposed for the multi‐dimensional, steady‐state hydrodynamic model of semiconductor devices, and a derivation of the model's appropriate boundary conditions is given. The model includes the complete balance equations for charge, momentum and energy, coupled with Poisson's equation, thus accounting for both diffusion and convection phenomena. The technique, like the Scharfetter—Gummel scheme for the simpler drift‐diffusion model, provides an efficient method for solving the hydrodynamic equations, allowing for a more detailed investigation of carrier dynamics in semiconductor devices.

Plastic encapsulated transistors : C. F. Maguire, F. J. Koons and Q. T. Jarrett. Solid State Technol., August (1971), p. 37

A new formulation employing the Galerkin/least-squares finite element method is presented for the simulation of the hydrodynamic model of semiconductor devices. Numerical simulations are performed on the coupled Poisson and hydrodynamic equations for one carrier devices. The hydrodynamic equations for a single carrier, i.e. for the electrons or holes, resemble the compressible Navier-Stokes equations with the addition of highly nonlinear source terms and without the viscous terms. The governing equations are nondimensionalized to improve the conditioning on the resulting system of equations and the efficiency of the numerical algorithms. Furthermore, to establish the stability of the discrete solution, the system of hydrodynamic equations is symmetrized by considering generalized entropy functions. A staggered solution strategy is employed to treat the coupled hydrodynamic and Poisson equations. Numerical results are presented for one-dimensional and two-dimensional one-carrier n+-n-n+ devices. The presence of velocity overshoot has been observed and it is recognized that the heat flux term plays an important role in the simulation of semiconductor devices employing the hydrodynamic model.