Since the existing theoretical work on open systems consists primarily of the definition of boundary conditions on transport equations, it is appropriate to examine various approaches to transport theory to see how they have dealt with this issue. This examination will center upon electron transport theory, because we wish to include quantum-coherence effects in the theory and these are much more prominent in systems of electrons than in systems of more massive particles.
By far the most common approach to defining the boundary conditions on a transport problem is to circumvent the issue entirely. This is most easily done by restricting one's attention to the special case of spatially uniform systems, so that (at the kinetic level) all spatial derivatives disappear, and with them the need to specify the boundary conditions. Applications of the Boltzmann equation (as expressed in terms of the usual Euler variables) have most often been restricted to the case of uniform driving fields (Dresden, 1961; Conwell, 1967). When the Boltzmann equation has been applied to nonuniform systems (see, for example, Castagné, 1985; Reggiani, 1985; Constant, 1985; Baranger and Wilkins, 1987), techniques which require that the equation be recast in terms of the Lagrange variables have generally been employed. Boundary conditions for such formulations are discussed in Appendix 11. Much of the work on quantum transport has also assumed uniform fields (see, for example, Levinson 1969; Mahan, 1987).
The other popular approach is to assume periodic boundary conditions (Kohn and Luttinger, 1967), which assure the Hermiticity of all relevant operators (Yennie, 1987). This in effect closes the system, forestalling the possibility of studying any open-system aspects of the problem. It also prevents one from studying any situation in which the change in chemical potential across the system is of finite magnitude (because the potential must also be periodic). Periodic boundary conditions are thus adapted to the requirements of linear response theory (Kubo, 1957), but not to those of far-from-equilibrium problems.
A fundamental approach which does take cognizance of the open nature of transporting systems is that advocated by Landauer (1957, 1970; also Büttiker et al., 1985). This approach envisions a system within which dissipative processes do not occur, but which is coupled to two or more ideal particle reservoirs. The conductance of such a structure is then expressed in terms of the quantum-mechanical transmission coefficients of the system. The ideal reservoirs have properties analogous to those of a blackbody: They absorb without reflection any electrons leaving the system and emit an equilibrium thermal distribution into the system. We shall see that such a picture is indeed the key to constructing a useful open-system model. However, let us note that this approach does not specify the boundary conditions on a boundary-value problem. The boundary conditions are actually applied to Schrödinger's equation and are the asymptotic conditions upon which the formal theory of scattering is based (see Appendix 12). The concept of thermal reservoirs is invoked to specify how the various wavefunctions are to be incorporated into a density operator for the system, from which observables may be evaluated.
The Landauer approach has successfully described a number of quantum conductance phenomena (Stone and Szafer, 1988): Aharonov-Bohm oscillations, universal conductance fluctuations, and quantized conductance through constrictions (Szafer and Stone, 1989). (Many recent results in this area can be found in Heinrich, Bauer, and Kuchar, 1988, and in Reed and Kirk, 1989.) However, it is important to recognize that these phenomena occur only under a very restricted range of circumstances (Webb, 1989): cryogenic temperatures (typically 1 K) and low voltages (typically 1 meV). The reason for this is not so much the fragility of quantum interference effects in themselves, but rather the constraints placed upon the phenomena by the requirement that they be observable in the linear-response regime (which is to say, very near to thermal equilibrium). Near equilibrium, only the states near the Fermi level contribute to the conductance, but all such states participate. As the temperature or bias voltage is raised, more states participate in the conduction, with slightly different energies or wavevectors, and the observable effects are ``washed out.''
In a far-from-equilibrium situation one has the opportunity to selectively populate a narrow set of quantum states, leaving nearby states unpopulated. This can lead to quantum-interference phenomena which are quantitatively dominant at or above room temperature. The prototypical example of such a situation is provided by the quantum-well resonant-tunneling diode (Chang, Esaki, and Tsu, 1974; Sollner et al., 1983), which is discussed more extensively in Sec. 5. Such devices have demonstrated peak-to-valley current ratios as high as 30 at 300 K (Broekaert, Lee, and Fonstad, 1988; for a tabulation of device results see Mehdi and Haddad, 1989).
Given that far-from-equilibrium quantum-interference effects can be large and are thus important to study, one must ask whether such effects can be adequately described by elementary quantum theory. For the case of tunneling structures the standard elementary theory models the electron states as stationary scattering-state solutions of Schrödinger's equation (Duke, 1969; Tsu and Esaki, 1973; Wolf, 1985). Does this provide an adequate description of nonequilibrium phenomena? The answer is, in general, no, and we will explore this issue below. The elementary tunneling theory does seem to give good results for the current density, but for other physical observables, such as the charge distribution, a more sophisticated approach is required.
To demonstrate the problems one encounters with elementary quantum-mechanical models in a far-from-equilibrium situation, let us consider the apparently simple problem of finding the self-consistent electrostatic potential in a single-barrier tunneling structure. A semiconductor heterostructure is assumed and the details of the structure and analysis are given in Appendix 9. The approach that we will use is to first approximate the self-consistent potential using the Thomas-Fermi screening theory. The resulting potential and electron distribution are shown in Fig. 1.
Figure 1. Potential (upper) and charge density (lower) of a semiconductor tunneling heterostructure biased far from equilibrium. The solid lines show the results of a Thomas-Fermi screening model, and the dashed lines show the charge density and first iteration of the potential obtained by solving Schrödinger's equation in a conventional tunneling calculation. The tunneling result fails to display an accumulation of electrons on the upstream side of the barrier because inelastic processes are not included, and as a result the self-consistent potential is quite unphysical. The dotted line shows the distribution of positive charges, and the dash-dot line shows the chemical potentials.The Thomas-Fermi potential shows the smooth bending that one would expect in a system in which the charge densities are several orders of magnitude less than those in metallic systems. Now we use the Thomas-Fermi potential in Schrödinger's equation and start an iterative procedure to find the ``true'' self-consistent potential. The results of the first iteration are also shown in Fig. 1, and it is quite clear that we will not obtain a physically credible result. The charge density obtained from Schrödinger's equation differs markedly from the Thomas-Fermi solution on the left-hand (upstream) side of the barrier. Where Thomas-Fermi indicates an accumulation of electrons, Schrödinger's equation gives a depletion of electrons. The reason for this is that the tunneling theory assumes that the electron states in the potential ``notch'' on the left-hand side of the barrier are in equilibrium with the right-hand side reservoir, because that is the side from which these wavefunctions are incident. The depletion of electron density may be traced to the requirement of current continuity in the propagating states: As an electron propagates into a region of decreasing potential, its velocity increases; but to maintain the current density constant, its amplitude must then decrease. Because the tunneling-theory charge density does not produce overall charge-neutrality in the structure, the solution of Poisson's equation has large electric fields at the boundaries, which in turn exacerbates the problem of charge neutrality. (The final self-consistent result would show the energy barrier lying near the bottom of a parabolic potential well considerably deeper than that of the first iteration.)
The physical processes which work to enforce charge neutrality are those which work to restore thermal equilibrium, which is to say inelastic processes. In the present case, these are the inelastic scattering events (primarily phonon scattering) which dissipate the electrons' energy and cause electrons in the propagating states entering from the left-hand reservoir to fall into the lower-energy notch states. The resulting population in the notch states produces the accumulation of negative charge required to screen the electric field. Thus the true self-consistent potential will depend upon the number of electrons in the notch states, which in turn will depend upon the relative rates at which electrons are scattered into the notch states and subsequently tunnel out (Wingreen and Wilkins, 1987). Therefore, a physically reasonable self-consistent potential will not be obtained unless the inelastic processes are included in the analysis.
The usual way to incorporate inelastic processes, to the first order, is to use the Fermi golden rule to evaluate the transition rates between states. In a more complete description these transition rates actually appear as terms in a Pauli master equation (see Kreuzer, 1981, ch. 10). The Pauli master equation assumes that the electrons occupy only eigenstates of the Hamiltonian, not superpositions of those eigenstates. In other words, the density operator of the system is and remains diagonal in the eigenbasis of the Hamiltonian. In the present case, this assumption violates continuity. An example of this is shown in Fig. 2, which shows two eigenstates of Schrödinger's equation, one incoming from the left and one confined in the notch (though it is coupled by tunneling to a propagating state on the right-hand side of the barrier).
Figure 2. Typical eigenstates in a tunneling structure. The solid line shows a propagating state, and the dashed line shows a state which is confined in the potential ``notch.'' The spatial distributions of these states are quite different, as shown in the lower plot ofAn inelastic process described by the Pauli master equation will cause probability density to disappear from one state and reappear in the other. Because the spatial distributions of the two states are different, this means that the probability distribution must change with time. But because the states are both eigenstates, their current densities are uniform. Thus the Pauli master equation violates the continuity equation. This is explored more formally in Appendix 10. Presumably, inelastic transitions are more localized processes, involving superpositions of eigenstates which describe such localization. However, this implies that the off-diagonal elements of the density operator are non-negligible, and theories which comprehend off-diagonal density operators are kinetic theories.. Thus, the Pauli master-equation description of an inelastic process which couples these states must violate the continuity equation.
To demonstrate that a plausible solution to the self-consistent potential problem can be obtained using kinetic theory, the results of such a calculation are shown in Fig. 3.
Figure 3. Calculations of the self-consistent potential of the tunneling heterostructure using a kinetic theory which includes inelastic processes. In (a) the longitudinal-optic and acoustic phonon scattering processes are included, but the incoming distribution of electrons is fixed. An accumulation layer is formed on the upstream side of the barrier, but the screening of the electric field is far from complete. In (b) the incoming distribution of electrons is allowed to shift in response to the electric field at the boundary, to simulate an ohmic conductor outside the boundary. The screening is more complete, and the resulting potential is more physically credible.The approach described in Secs. 4 and 5 was used, and inelastic processes (phonon scattering) were included using the Boltzmann collision operator described in Appendix 14. When the phonon scattering processes are included, Fig. 3(a), an accumulation layer is formed in the potential notch. However, the accumulation is not sufficient to effectively screen the electric field as it approaches the boundary. Evidently there are other effects which need to be included. One such effect is the resistivity of the contacting layers (outside of the calculation domain). If these layers are ohmic conductors, the distribution of electrons in them must shift away from its equilibrium value when a current is conducted. When this effect is incorporated into the boundary conditions on the kinetic model, the self-consistent potential shown in Fig. 3(b) is obtained. This is a much more credible result, as the potential varies smoothly through the structure and the electric field approaches a small value at the boundaries. The screening length from the kinetic model is significantly longer than the value indicated by the Thomas-Fermi calculation of Fig.\ 1. This might have been expected from the effects of size quantization in the notch (Ando, Fowler, and Stern, 1982), and also from the finite rate of inelastic transitions which fill the notch.
Thus, the problem of calculating the self-consistent potential in a tunneling structure is about as complicated as it could possibly be, in the sense that the qualitative result depends upon all the processes occurring within the system. It thus provides a vivid example of the problems one encounters in attempting to apply elementary quantum-mechanical concepts to a far-from-equilibrium situation. A satisfactory treatment of far-from-equilibrium phenomena requires an approach at a level of sophistication at least equal to that of kinetic theory.
