The semiconductor heterostructure used in the analysis in Sec. 1.2 consisted of an undoped 3.39 nm (12 unit cells) layer of AlGaAs 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 cmVs), 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).