The semiconductor heterostructure used in the analysis in Sec.
1.2 consisted of an undoped 3.39 nm (12 unit
cells) layer of Al
Ga
As embedded in GaAs crystal doped such
that the mobile electron density was 
cm
, and the
temperature was 300 K. (This particular structure was chosen to provide a
clear demonstration of the failure of the standard tunneling theory.)
The calculations were done for a bias of 0.2 V (
) applied
to the structure.
The initial approximation for the self-consistent potential was obtained from a generalized (to finite temperature) Thomas-Fermi screening approximation. At its most fundamental level, the Thomas-Fermi approximation can be viewed as an expression for the Wigner distribution function:
where 
is obtained by taking the kinetic-energy term
of the Hamiltonian in the neighborhood of 
, extending this form
over all space, and taking the expectation value of the resulting
operator on the plane-wave state 
. This typically
gives 
, where the
effective mass 
can vary with position, as discussed in Appendix
13. Integrating over all momenta gives the more familiar
expression (Blakemore, 1982):
where 
is the ``effective
density of states,'' and 
is the Fermi-Dirac integral of
order 
.
The potential v can be separated into a Hartree potential 
and a
``heterostructure'' potential 
which describes the heterostructure
band offsets:
The Hartree potential satisfies Poisson's equation:
where 
is the background positive charge density (ionized donor
density). Inserting (9.128) and (9.129) into
(9.130) produces a Poisson equation with a nonlinear source
term, which is readily solved in a finite-difference approximation by a
multidimensional Newton iteration technique (Selberherr, 1984, ch. 7).
The boundary conditions
for (9.130) are obtained from the requirement that the system
asymptotically approach charge-neutrality:
with all quantities evaluated in the appropriate asymptotic region. In
practice, these boundary conditions are applied at fixed locations
which are sufficiently distant that charge neutrality is well satisfied
(see Fig. 1). Note that the reference energy for
may be chosen arbitrarily; this reference and the externally
imposed 
then uniquely determine 
. Strictly speaking, the
Thomas-Fermi approximation is only an equilibrium approximation.
However, in some structures, such as the present single-barrier device,
one can identify regions in which a local quasi-equilibrium ought to
hold. In such cases one can obtain useful results for the nonequilibrium
case by assuming that the chemical potentials differ from one region to
another, as illustrated in Fig. 1.
To evaluate the self-consistent potential within the conventional independent-electron tunneling theory, we need to precisely define the (mixed) quantum state of the system. The fundamental assumption of tunneling theory is that the electrons will be found in the eigenstates of the Hamiltonian (generally unnormalizable scattering states), and the probability of occupation of the left- and right-incident states is given by the different Fermi distributions of the respective contacts. We may summarize these assumptions by writing a density operator for the system:
where 
are the asymptotic potentials to the left and right,
is the velocity of an electron of energy E at the
respective boundary. Here 
is the Fermi-Dirac distribution function
integrated over the transverse momenta:
The 
are the solutions of Schrödinger's equation in an
effective-mass approximation,
with unit incident amplitude from the left or right, respectively.
Using (9.132) we can evaluate any physical observable of the
tunneling system, although in the literature, the content of
(9.132) is usually expressed only in an equation for the current
density. However, to evaluate the self-consistent potential we
need to evaluate the electron density, which is simply 
.
Inserting this into Poisson's equation (9.130) and again
applying the condition (9.131) at each boundary, we obtain
the potential shown by the dashed line in Fig. 1.
This potential is clearly unphysical, as discussed in the text,
because inelastic processes are neglected. A proper description of such
processes requires a kinetic theory.
The quantum-kinetic calculations shown in Fig. 3
were done by solving the steady-state kinetic equation (4.56) and
Poisson's equation (9.130) self-consistently,
again by a multidimensional Newton iteration scheme. The electron
density n in Poisson's equation was obtained from the Wigner function
using (7.113). Phonon scattering was included by adding the
Boltzmann collision operator described in Appendix 14, for
both longitudinal-optic and acoustic phonons, to the Liouville operator
used in (4.56). The calculation of Fig. 3(a)
assumed fixed boundary distributions (5.57). The calculation
of Fig. 3(b) assumed displaced equilibrium
boundary distributions, to take into account the transport processes in
the contacting layers (Mains and Haddad, 1988c).
These distributions were just (5.57)
with argument 
, where 
,
is the electron mobility (taken to be 5000
cm
V
s
), and the electric field was evaluated
at the respective boundary. This shifts the distribution function so
that a greater density of electrons enters on the upstream side and a
lesser density enters on the downstream side, which makes the screening
of the electric field more effective.
Other self-consistent calculations of far-from-equilibrium tunneling structures have focused upon the double-barrier resonant-tunneling diode because of its greater technological significance. Cahay et al. (1987) performed a self-consistent Schrödinger calculation, as described above. However, they assumed a device structure with undoped spacer layers on either side of the double barrier. The contact potentials of the doped-undoped junctions create an additional energy barrier which, by confining the electrons, helps to enforce charge neutrality, and thus the unphysical effects described above are avoided. If the undoped spacer layers had not been present, an unphysical potential would have been obtained.
Pötz (1989) also performed a self-consistent Schrödinger calculation. In this case the unphysical results were avoided by modifying the definition of the electron ensemble from (9.132) to one in which the notch states were weighted with the Fermi distribution of the upstream electrode, in effect assuming a high rate of inelastic processes to fill these states. A displaced distribution function as described above was also used in this calculation, but the drift momentum was chosen so as to satisfy charge neutrality, rather than to approximate Ohmic conduction.
Kluksdahl et al. (1989) performed a self-consistent kinetic (Wigner function) calculation of the type described above, with a relaxation-time approximation for the collision operator. The results showed an unphysically large electric field at the upstream boundary. Similar results were obtained by the present author (Frensley, 1989a, 1989b) from a kinetic model lacking the collision term. As in the single-barrier case, the inclusion of phonon collisions and displaced boundary distributions led to more credible results (that is, more complete screening of the field) for the self-consistent potential (Mains and Haddad, 1988c; Frensley 1989a, 1989b).
