3.6.1 Eigenvalues of a Triangular Energy Well
To first order the conduction band edge in a MOSFET inversion layer can be
approximated by a linear potential (this is actually done by various authors,
see [161,162,163,164]). The solution of
SCHRÖDINGER's equation for a linear potential has been derived in Section 3.5.2
and consists of a linear superposition of AIRY's functions. If the triangular
energy well is defined as
|
(3.94) |
and no wave function penetration for is taken into account, the
wave function for can be written as [156]
Therefore, must equal one of the zeros of the AIRY function :
|
(3.97) |
With from expression (3.68) the energy eigenvalues are found as
|
(3.98) |
The first five zeros of the AIRY function are , , , ,
and . These values are often used to approximate the quantized carrier
concentration in the channel of MOS devices. The value of the normalizing
constant becomes (the derivation is shown in Appendix C)
|
(3.99) |
where is the constant electric field in the energy well, and the value of
depends on the energy eigenvalue
via
|
(3.100) |
This method can be used to get an estimate of the first few eigenvalues of the
system, or to find initial values for the calculation of the eigenvalues
described in the next section.
A. Gehring: Simulation of Tunneling in Semiconductor Devices