Numerical Coupling of Electric Circuit Equations and Energy-Transport Models for Semiconductors

Abstract

A coupled semiconductor-circuit model including thermal effects is proposed. The charged particle flow in the semiconductor devices is described by the energy-transport equations for the electrons and the drift-diffusion equations for the holes. The electric circuit is modeled by the network equations from modified nodal analysis. The coupling is realized by the node potentials providing the voltages applied to the semiconductor devices and the output device currents for the network model. The resulting partial differential-algebraic system is discretized in time by the 2-stage backward difference formula and in space by a mixed-hybrid finite-element method using Marini-Pietra elements. A static condensation procedure is applied to the coupled system reducing the number of unknowns. Numerical simulations of a one-dimensional pn-junction diode with timedependent voltage and of a rectifier circuit show the heating of the electrons which allows one to identify hot spots in the devices. Moreover, the choice of the boundary conditions for the electron density and energy density is numerically discussed. Introduction.In industrial applications, complex semiconductor device models are usually substituted by circuits of basic network elements (resistors, capacitors, inductors, voltage, and current sources) resulting in simpler so-called compact models. Such a strategy was advantageous up to now since integrated circuit simulations were possible without computationally expensive device simulations. Parasitic effects and high frequencies in the circuits, however, require one to take into account a very large number of basic elements and to adjust carefully a large number of parameters in order to achieve the required accuracy. Moreover, device heating and hot spots cannot be easily modeled by this approach.Therefore, it is preferable to model those semiconductor devices which are critical for the parasitic effects by semiconductor transport equations. Since structural information about the resulting device-circuit equations was missing for a long time, the first approaches to couple circuits and devices were based on an extension of existing device simulators by more complex boundary conditions [36,40] or the combination of device simulators with circuit simulators as a "black box" solver [15]. Both approaches, however, are not suitable for complex circuits in the high-frequency domain. In this work, we follow an approach that includes the device model into the network equations directly. As the device model is described by partial differential equations and the network equations are given by differential-algebraic equations

Self-heating in a coupled thermo-electric circuit-device model

“…In fact, Dirichlet conditions for T n and T p may lead to artificial boundary layers [11]. Moreover, in [5] it was argued that the use of homogeneous Neumann conditions for T n and T p can be justified in highly doped regions close to the contacts.…”

“…More recently, electric network models were coupled to semiconductor transport equations, such as driftdiffusion [34] or energy-transport models [11], leading to a coupled system of partial differential-algebraic equations. The work [11] includes temperature models for the charge carriers, but the lattice and circuit element temperatures were assumed to be constant.…”

“…The equations for the energy-transport model [22], the electric circuit model [38], and the thermal network model [7] are well established. The thermo-electric coupling is described in [3,7] and the electric circuit-device coupling is detailed in [11,34]. In the following, we explain our modeling of the lattice heating and the coupling between the thermal network and semiconductor devices, which seems to be new in this context.…”

Numerical Simulation of Thermal Effects in Coupled Optoelectronic Device-circuit Systems

“…For the temperature, homogenous Neumann boundary conditions are assumed as in [1]. We have shown in [5] that boundary layers for the particle densities can be avoided if Robin-type boundary conditions similar as in [14] are employed on the remaining boundary parts,…”

“…As a test example we consider a rectifying circuit containing four silicon pn diodes as in [5] (Fig. 1).…”