5.7 Example

As an example a one-dimensional semiconductor structure with a simple ohmic boundary condition is considered. The index of the quantities is the number of the grid point they belong to, with 0 being the left contact. For the electron concentration n₀ = N₀ with N₀ being a constant value depending only on the doping of the semiconductor. To further simplify the example only the static Poisson equation and the static continuity equation for electrons are considered. Furthermore, the separate variable for the electron contact current I_Cn is omitted. For a one-dimensional device the segment constitutive relations read

fS$\psi_{0}$	=			$\Psi_{01}^{}$	+	q . n0 . V0	$\neq$	0
fS$\psi_{1}$	=	$\Psi_{10}^{}$	+	$\Psi_{12}^{}$	+	q . n1 . V1	=	0
$\vdots$
fSn0	=			I01			$\neq$	0
fSn1	=	I10	+	I12			=	0
$\vdots$

with

$\Psi_{ij}^{}$	=	$\Psi$($\psi_{i}^{}$,$\psi_{j}^{}$)	=	- $\Psi_{ji}^{}$
Iij	=	I(ni, nj,$\psi_{i}^{}$,$\psi_{j}^{}$)	=	- Iji

At the boundary, the constitutive relations are

f$\psi_{0}$	=	$\psi_{0}^{}$	-	$\psi_{C}^{}$	=	0
fn0	=	n0	-	N0	=	0
fIC	=	IC	+	fSn0	=	0
fQC	=	QC	+	fS$\psi_{0}$	=	0

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 achieved by the boundary models which set the appropriate entries in the transformation matrix $\mathbb {T}$_B which reads

tx, y	$\psi_{0}^{}$	$\psi_{1}^{}$	n0	n1	IC	QC
$\psi_{0}^{}$
$\psi_{1}^{}$		1
n0
n1				1
IC			1
QC	1

The solution vector x contains the following quantities

x

=

$\left(\vphantom{ \psi_{0}, \ \psi_{1}, \ \ldots \ n_{0}, \ n_{1}, \ \ldots \ \psi_{C}, \ I_{C}, \ Q_{C}, \ \ldots }\right.$$\psi_{0}^{}$, $\psi_{1}^{}$, ... n0, n1, ... $\psi_{C}^{}$, IC, QC, ...$\left.\vphantom{ \psi_{0}, \ \psi_{1}, \ \ldots \ n_{0}, \ n_{1}, \ \ldots \ \psi_{C}, \ I_{C}, \ Q_{C}, \ \ldots }\right)^{T}_{}$

For voltage controlled contacts with V₀ applied to the contact one gets

f$\psi_{C}$

=

$\psi_{C}^{}$

-

V0

=

0

When applying the current I₀ to the contact f_{$\psi_{C}$} changes to

f$\psi_{C}$

=

IC

-

I0

=

0

The system matrix for iteration step k is

jx, y	$\psi_{0}^{}$	n0	$\psi_{C}^{}$	IC	QC	r
$\psi_{0}^{}$	-1		1			f$\psi_{0}$k
n0		-1				fn0k
$\psi_{C}^{}$				-1		f$\psi_{C}$k
IC	- ${\frac{\partial \displaystyle I_{01}}{\partial \displaystyle \psi_{0}}}$	- ${\frac{\partial \displaystyle I_{01}}{\partial \displaystyle n_{0}}}$		-1		fICk
QC	- ${\frac{\partial \displaystyle \Psi_{01}}{\partial \displaystyle \psi_{0}}}$	-q . V0			-1	fQCk

As the constitutive relations for the quantities $\psi_{0}^{}$, n₀, and I_C are eliminated first, one ends up with the following matrix

jx, y	$\psi_{C}^{}$	r
$\psi_{C}^{}$	- ${\frac{\partial \displaystyle I_{01}}{\partial \displaystyle \psi_{0}}}$	f$\psi_{C}$k - fICk + f$\psi_{0}$k . ${\frac{\partial \displaystyle I_{01}}{\partial \displaystyle \psi_{0}}}$ + fn0k . ${\frac{\partial \displaystyle I_{01}}{\partial \displaystyle n_{0}}}$