Simulation of Thermal Effects in Optoelectronic Devices Using Coupled Energy-Transport and Circuit Models

Abstract

A coupled model with optoelectronic semiconductor devices in electric circuits is proposed. The circuit is modeled by differential-algebraic equations derived from modified nodal analysis. The transport of charge carriers in the semiconductor devices (laser diode and photo diode) is described by the energy-transport equations for the electron density and temperature, the drift-diffusion equations for the hole density, and the Poisson equation for the electric potential. The generation of photons in the laser diode is modeled by spontaneous and stimulated recombination terms appearing in the transport equations. The devices are coupled to the circuit by the semiconductor current entering the circuit and by the applied voltage at the device contacts, coming from the circuit. The resulting time-dependent model is a system of nonlinear partial differential-algebraic equations. The one-dimensional transient transport equations are numerically discretized in time by the backward Euler method and in space by a hybridized mixed finite-element method. Numerical results for a circuit consisting of a single-mode heterostructure laser diode, a silicon photo diode, and a high-pass filter are presented.

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

“…We employ the finite elements of [12] since they guarantee the positivity of the discrete variables if positive initial and Dirichlet boundary data are prescribed and if σ j ≥ 0, f j ≥ 0 for j = 1, 2. This property also holds for the Robin conditions (8) [6]. Finally, the nonlinear discrete system is solved by Newton's method.…”

“…where the circuit equations (1)- (2) have to be appropriately scaled [6]. The complete coupled system consists of equations (1)- (10) forming an initial boundary-value problem of partial differential-algebraic equations.…”

“…The interaction between optical and electrical effects is modeled by recombination-generation terms appearing in (7). In the following, we present the model used in the numerical simulations and we refer to [6] for a discussion about the model simplifications.…”

“…respectively, where B is the spontaneous recombination parameter, c the speed of light µ opt the refractive index of the material, g(n) the optical gain depending on the electron density, |Ξ| 2 the intensity distribution of the optical field, which is a solution of the waveguide equation [6], and S = S(x, t) is the number of photons in the device.…”

“…In the lasing mode we can employ the quasineutral assumption n ≈ p in the active region such that the gain becomes approximately g(p) = g 0 (p − n th ) in the recombination term occurring in the hole equation (3). This allows for a discretization that guarantees positivity of the discretized hole density [6]. The number of photons S is balanced by the rate equation…”