An advanced equation assembly module

Abstract

We present an advanced equation assembly module which has been developed for the simulation of semiconductor devices based on the Finite Boxes discretization scheme and is currently used in the general purpose device and circuit simulator Minimos-NT. Such simulations require the solution of a specific set of nonlinear partial differential equations which are discretized on a grid. The resulting nonlinear problem is solved by a damped Newton algorithm that demands the solution of a linear equation system at each step. The presented module is responsible for assembling these systems and takes into account several requirements of the simulation process. The underlying concepts, namely the representation of boundary conditions, physically motivated variable transformation, preelimination and numerical conditioning, are presented together with some examples.

Appendix

As an example consider a simple ohmic boundary condition for the electron concentration n _i = N ₀ with N ₀ being a constant value depending only on the doping of the semiconductor. To further simplify the example we only consider the static Poisson equation and the static continuity equation for electrons. Furthermore, the separate variable for the electron contact current $I_{C_{n}}$ is omitted. For a one-dimensional device (index 0 refers to the incomplete point at the contact, 1 to the first complete point), the segment constitutive relations read

$$f_{\psi_{0}}^{\rm S} = \Psi_{01}+\hbox{q}\cdot n_{0}\cdot V_{0}\ne 0,$$

(41)

$$f_{\psi_{1}}^{\rm S} = \Psi_{10} + \Psi_{12} + \hbox{q}\cdot n_{1}\cdot V_{1} = 0,$$

(42)

$$f_{n_{0}}^{\rm S} = I_{01} \ne 0,$$

(43)

$$f_{n_{1}}^{\rm S} = I_{01} + I_{12} = 0,$$

(44)

with

$$\Psi_{i, j} = \Psi(\psi_{i},\psi_{j}) = -\Psi_{ji}$$

$$I_{i, j} = I(n_{i}, n_{j}, \psi_{i}, \psi_{j}) = -I_{ji}.$$

At the boundary, the constitutive relations are

$$f_{\psi_{0}} = \psi_{0} - \psi_{\rm C} = 0,$$

(45)

$$f_{n_{0}} = n_{0} - N_{0} = 0,$$

(46)

$$f_{I_{\rm C}} = I_{\rm C} + f_{n_{0}}^{\rm S} = 0,$$

(47)

$$f_{Q_{\rm C}} = Q_{\rm C} + f_{\psi_{0}}^{\rm S} = 0.$$

(48)

The boundary constitutive relations will be used to determine the quantity values at the boundary while the segment constitutive relations will be used to build up an expression for the boundary charge Q _C and for the boundary current I _C. This is done by the boundary models which set the appropriate entries in the transformation matrix T _B . The solution vector x contains the following quantities

$$\mathbf{x} = \left( \psi_0, \psi_1, \ldots n_0, n_1, \ldots \psi_{\rm C}, I_{\rm C}, Q_{\rm C}, \ldots \right)^{\rm T}.$$

(49)

For voltage controlled contacts with V ₀ applied to the contact one gets (50), when applying the current I ₀ to the contact $f_{\psi_{\rm C}}$ changes to (51).

$$f_{\psi_{\rm C}} = \psi_{\rm C} - V_{\rm 0} = 0,$$

(50)

$$f_{\psi_{\rm C}} = I_{\rm C} - I_{\rm 0} = 0.$$

(51)

The contact related entries in the three matrices (A _b + T _b· A _s) are assembled as follows (the complete equations such as Eqs. 42 or 44 are assembled in A _s only and have T _b diagonal entries of 1):

A b	ψ0	n 0	I C	Q C	ψC	see	T b	ψ0	n 0	I C	Q C	ψC
ψ0	-1	0	0	0	0	(45)	ψ0	0	0	0	0	0
n 0	0	-1	0	0	0	(46)	n 0	0	0	0	0	0
I C	0	1	-1	0	0	(47)	I C	0	1	0	0	0
Q C	0	0	0	-1	0	(48)	Q C	1	0	0	0	0
ψC	0	0	-1	0	0	(51)	ψC	0	0	0	0	0

A b	ψ0	n 0	I C	Q C	ψC	see	transferred to
ψ0	−∂Ψ01/∂ψ0	-q·V0	0	0	0	(41)	Q C
n 0	− ∂I 01/∂ψ0	−∂I 01/∂n 0	0	0	0	(43)	I C
I C	0	0	0	0	0
Q C	0	0	0	0	0
ψC	0	0	0	0	0

The compiled linear equation system for iteration step k is

A	ψ0	n 0	ψC	I C	Q C	RHS
ψ0	−1	0	1	0	0	$f^{k}_{\psi_{0}}$
n 0	0	−1	0	0	0	$f^{k}_{n_{0}}$
ψC	0	0	0	−1	0	$f^{k}_{\psi_{\rm C}}$
I C	−∂I 01/∂ψ0	−∂I 01/∂n 0	0	−1	0	$f^{k}_{I_{\rm C}}$
Q C	−∂Ψ01/∂ψ0	−q·V0	0	0	−1	$f^{k}_{Q_{\rm C}}$

As the constitutive relations for the quantities ψ₀, n ₀, and I _C are preeliminated one ends up with the following equation for ψ_C:

j x, y	ψ	RHS
ψC	−∂I 01/∂ψ0	$f_{\psi_{\rm c}}^{k}-f_{I_{\rm C}}^{k}-f_{\psi_{0}}^{k}\cdot \partial I_{01}/\partial\psi_{0} + f_{n_{0}}^{k}\cdot \partial I_{01}/\partial n_{0}$