Mathematical description of the DGL system solved by AFORS-HET

In the following, the differential equations and corresponding boundary conditions, which are solved by AFORS-HET under the various conditions, are stated.

An arbitrary stack of semiconductor layers can be modeled. Within each semiconductor layer the Poisson equation and the transport and continuity equations for electrons and holes have to be solved. At each semiconductor/semiconductor interface and at the front and back side boundary of the stack the current transport through these interfaces/boundaries can be described by different physical models. It results a highly non­linear coupled system of three differential equations with respect to time and space derivatives. The electron density n(x, t), the hole density p(x, t), and the electric potential p(x, t) are the independent variables, for which this system of differential equations is solved. It is solved according to the numerical discretisation scheme as outlined by Selberherr (Selberherr, 1984) in order to linearize the problem and using the linear SparLin solver which is available in the internet (Kundert et. al., 1988).

It can be solved for different calculation modes: (1) EQ calculation mode, describing thermodynamic equilibrium at a given temperature, (2) DC calculation mode, describing steady-state conditions under an external applied voltage or current and/or illumination, (3) AC calculation mode, describing small additional sinusoidal modulations of the external applied voltage/illumination, and (4) TR calculation mode, describing transient changes of the system, due to general time dependent changes of the external applied voltage or current and/or illumination.

In case of using the EQ or the DC calculation mode, all time derivatives vanish, resulting in a simplified system of differential equations. The system of differential equations is then solved for the time independent, but position dependent functions, nEQI DC (x), pEQI DC (x), pEQ1DC (x).

n(x, t) = nEQ (x), n(x, t) = nDC (x)

p(x, t) = pEQ (x) p(x, t) = pDC (x)

px, t)=pEQ(x) p(x, t) = pDC(x)

In case of using the AC calculation mode, it is assumed that all time dependencies can be described by small additional sinusoidal modulations of the steady-state solutions. All time dependent quantities are then modelled with complex numbers (marked by a dash ~), which allows to determine the amplitudes and the phase shifts between them. I. e., for the independent variables of the system of differential equations, one gets:

n(x, t) = nDC(x) + nAC(x) eiat

p(x, t) = pDC (x) + pAC (x) eiat

(p(x, t) = pDC (x) + p AC (x) eiat

In case of using the TR calculation mode, the description of the system starts with a steady – state (DC-mode) simulation, specifying an external applied voltage or current and/or illumination. An arbitrary evolution in time of the external applied voltage or current and/or illumination can then be specified by loading an appropriate file. Then, the time evolution of the system, i. e. the functions n(x, t), p(x, t), p(x, t) during and after the externally applied changes are calculated.

Updated: August 18, 2015 — 2:35 pm