Numerical Simulation of Thermal Effects in Coupled Optoelectronic Device-circuit Systems
The control of thermal effects becomes more and more important in modern semiconductor circuits like in the simplified CMOS transceiver representation described by U. Feldmann in the above article Numerical simulation of multiscale models for radio freque
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nstitut f¨ ur Mathematik, Universit¨ at Mainz, Staudingerweg 9, 55099 Mainz, Germany, [email protected] Institut f¨ ur Analysis und Scientific Computing, Technische Universit¨ at Wien, Wiedner Hauptstr. 8–10, 1040 Wien, Austria [email protected]
1 Introduction The control of thermal effects becomes more and more important in modern semiconductor circuits like in the simplified CMOS transceiver representation described by U. Feldmann in the above article Numerical simulation of multiscale models for radio frequency circuits in the time domain. The standard approach for modeling integrated circuits is to replace the semiconductor devices by equivalent circuits consisting of basic elements and resulting in so-called compact models. Parasitic thermal effects, however, require a very large number of basic elements and a careful adjustment of the resulting large number of parameters in order to achieve the needed accuracy. Therefore, it is preferable to model those semiconductor devices which are critical for the parasitic effects by semiconductor transport equations. The transport of electrons in the devices is modeled here by the one-dimensional energy-transport model allowing for the simulation of the electron temperature. The electric circuits are described by modified nodal analysis. Thus, the devices are modeled by (nonlinear) partial differential equations, whereas the circuit is described by differential-algebraic equations. The coupled model, which becomes a system of (nonlinear) partial differential-algebraic equations, is numerically discretized in time by the 2-stage backward difference formula (BDF2), since this scheme allows to maintain the M-matrix property, and the semi-discrete equations are approximated by a mixed finite-element method. The objective is the simulation of a benchmark high-frequency transceiver circuit, using a laser diode as transmitter and a photo diode as receiver. The optical field in the laser diode is modeled by recombination terms and a rate equation for the number of photons in the device. The optical effects in the photo diode are described by generation terms. The numerical results show that the thermal effects can modify significantly the behavior of the transmitter circuit.
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M. Brunk, A. J¨ ungel
2 Modeling Circuit Modeling A well-established mathematical description of electric circuits, consisting of resistors, capacitors, and inductors (RCL circuit) is the modified nodal analysis (MNA) which can be easily extended to circuits containing semiconductor devices. In the following, the circuit model is described. The circuit is replaced by a directed graph. The RLC branches are characterized by the incidence matrix A, and the semiconductor branches are characterized by the semiconductor incidence matrix AS . The basic tools for the MNA are the Kirchhoff laws and the current-voltage characteristics for the basic elements, Ai + AS jS = 0,
v = A e,
iR = g(vR ),
iC =
dq (vC ), dt
vL =
dΦ (iL ), dt
where i, v, and e are the vectors of branch currents, branch voltages,
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