3.5.2 The GUNDLACH Method
The GUNDLACH method [137] provides an analytical solution of
SCHRÖDINGER's equation for a linear energy barrier. The onedimensional
timeindependent SCHRÖDINGER equation in this case reads

(3.61) 
with the linear potential energy between the points and ,
, and
,

(3.62) 
for
. Using the abbreviations

(3.63) 
and
, expression (3.61) turns into

(3.64) 
With

(3.65) 
SCHRÖDINGER's equation evolves into the AIRY^{3.7} differential equation

(3.66) 
The solutions of this differential equation are the AIRY functions
and
[138], which are depicted in
Fig. 3.10 together with their derivatives. The wave functions
consist of linear superpositions of these AIRY functions

(3.67) 
where the function is given as

(3.68) 
Figure 3.10:
The AIRY functions Ai and Bi and their
derivatives.

Assuming a constant electron mass in the dielectric, GUNDLACH derives an
expression for the transmission coefficient

(3.69) 
where the abbreviations

(3.70) 
have been used and the symbols , , and are given by

(3.71) 
and

(3.72) 
The symbols
and
denote the two edges of the energy barrier
as shown in Fig. 3.9. The GUNDLACH method is frequently used in the
literature [121,139] and implemented in device
simulators. Numerical problems may occur for flat barriers (
) due to the exponential increase of the AIRY functions
and
for positive arguments. In practical implementations the values of
and have been bounded to values below
to avoid
floating point overflow.
A. Gehring: Simulation of Tunneling in Semiconductor Devices