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4.2.1 Semiconductor Hetero Interfaces
Two adjacent semiconductor segments are connected in the simulation
by calculating the current J_{n1} leaving segment 1 and
J_{n2} entering segment 2 as shown in Figure
4.1. The current is described by the thermionicfield emission model
with the equations [6]

(27)

and
. 
(28)

The electron energy fluxes S_{n1} and S_{n2}
across the heterojunction are given by

(29)

and

(30)

where

(31)

is the emission velocity. The height of the energy barrier for electrons
between two semiconductors is
. 
(32)

Figure 4.1 Schematic conduction band diagram of the
heterojunction between an InGaAs channel and an AlGaAs barrier. The effective
barrier height DE_{C} is lowered
by dE_{C} due to tunneling of
electrons through the energy barrier.
The function f(dE_{C})
which accounts for barrier lowering due to tunneling will be explained
in the following. The profile of the energy barrier near the interface
can be approximated by a triangular shape as shown in Figure
4.1. The lowering of the effective barrier height dE_{C}
due to tunneling through the energy barrier is modeled by
, 
(33)

where
is the electric field perpendicular to the interface in segment 2 and x_{eff}
is an effective tunnel length which is assumed to be 7 nm. The tunneling
current through a potential barrier in first order is .
Thus, the exponential function in (28)
and (30) would read .
To describe deviations from the idealized tunneling model the exponential
function is expanded into a Tailor series. Therefore f(dE_{C})
in (28) and (30)
becomes

(34)

where B_{i} have to be considered as fitting parameters
for the simulation. Figure
4.2 illustrates a physical reason for the inadequacy of the idealized
tunnel characteristics. It schematically shows the electron distribution
in the channel of a delta doped DHHEMT. The electrons in the whole
simulation area are treated as classical particles, i. e. with zero spatial
extension. This means that a simulated electron in the channel close to
the interface experiences only the properties of the channel material.
It is well known from quantum mechanics that the electron wave extends
several nanometers which leads to non local effects such as quantization
in a potential well as discussed in Section 2.2
[46].
Figure 4.2 Schematic conduction band diagram and
electron distribution in the channel of a delta doped DHHEMT. The
wave of the electron distribution obtained from quantum mechanical considerations
extends several nano meters into the barrier layers.
The interface model given here describes a non local effect by a local
relation. For instance the reduction in the energy barrier due to tunneling
is determined by the electric field between the grid point right on the
interface and the next grid point in the barrier perpendicular to the interface.
Their separation is usually below 1 nm but the electron wave extends several
nanometers into that layer as shown in Figure
4.2. Therefore the actual energy barrier experienced by the electron
cannot be predicted exactly. Based on the typical conduction band profile
of HEMTs one expects that tunneling would be overestimated significantly
if an exponential dependence on the electric field were used.
Next: 4.2.2 Semiconductor Passivation
Interface Up: 4.2 Interface Models
Previous: 4.2 Interface Models
Helmut Brech
19980311