To evaluate any physical observables, such as the current density, we must specify how the scattering solutions are statistically weighted in the final result. For the case of a continuous spectrum of states, with -function normalization, the derivation of the correct expressions are rather tricky, because we seek expressions for densities of charge, current, energy, etc., rather than total quantities (which are of course infinite in an unbounded system). To illustrate the procedure, let us follow through the derivation of the electron density in a spatially uniform three-dimensional semiconductor in equilibrium. We approximate the conduction band structure by a simple parabolic dispersion relation:
where is the wavevector. The probability that each state will be occupied by an electron is given by the Fermi-Dirac distribution function:
where is the Fermi level or chemical potential and , T being the absolute temperature. (To avoid confusion with the transmission probability which is also denoted by T, the absolute temperature will always be shown multiplied by Boltzmann's constant .) Let us now make an ad hoc assumption that the semiconductor crystal is a cube with each side of length L, and apply periodic boundary conditions. Then the stationary quantum states are plane waves (normalized to unit amplitude) of the form
Due to the periodic boundary conditions, must assume discrete values:
where , , and are integers. The total number of electrons in the crystal N is just the sum over all of the states of the probability that each state is occupied
where the factor of 2 comes from the two spin states. Now, because L is large, the allowed values of are very closely spaced, and the sum over can be well approximated by an integral:
We can now write an expression for the density of electrons n:
Notice that the arbitrary crystal dimension L has dropped out of the final expression.
In order to evaluate densities using expressions such as (23) it is usually more convenient to transform the integration variable to E. By expressing in spherical coordinates and manipulating the dispersion relation (18) one finds :
We will also have occasion to use the corresponding expressions for integrals over two- and one-dimensional vectors. For the two-dimensional case (still assuming a parabolic dispersion relation):
For integration over a one-dimensional k, the definition of the group velocity (8) may be used to obtain an expression valid for any dispersion relation:
Inserting (24) into (23) leads to the usual expression for the electron density in a semiconductor (as discussed in Chapter 1 of the present volume) .
The procedure for evaluating a physical observable in an equilibrium system of infinite extent may thus be generalized from the above discussion. The expectation value of the observable quantity is calculated for each state, taking the scattering states to be normalized to unit amplitude. The density of the observable is then determined by inserting this expectation value into the sum in (23) and evaluating the resulting integral, usually using the relations (2). The two most important observables are the electron density and the current density j (which is independent of position in one dimension and steady-state). The expectation value of the density for a state is simply
The expectation value of j is simple, though the operator itself often is not. (Link to manuscript with full details.) If the dispersion relation is not parabolic and independent of position, the form of the operator j is not given by the simple textbook expression . The current density operator is instead whatever remains of the kinetic energy term of the Hamiltonian after the application of Green's identity as in the derivation of (1), and this obviously depends upon the form of the Hamiltonian itself. For unit-incident-amplitude scattering states, however, the result is invariably
Of course, in equilibrium, these two currents cancel each other (by the principle of detailed balance) and there is no net current flow.
To investigate the transport properties of a quantum system one must generally evaluate the current flow through the system, and this requires that one examine systems that are out of thermal equilibrium. A common situation, in both experimental apparatus and technological systems, is that one has two (or more) physically large regions densely populated with electrons in which the current density is low, coupled by a smaller region through which the current density is much larger. It is convenient to regard the large regions as ``electron reservoirs'' within which the electrons are all in equilibrium with a constant temperature and Fermi level, and which are so large that the current flow into or out of the smaller ``device'' represents a negligible perturbation. The reservoirs represent the metallic contacting leads to discrete devices or experimental samples, or the power-supply busses at the system level. Consequently the electrons flowing from from a reservoir into the device occupy that equilibrium distribution which characterizes the reservoir. In a simple one-dimensional system with two reservoirs, the electrons flowing in from the left-hand reservoir have k > 0 and those flowing from the right-hand reservoir have k < 0.
Within this picture, the current that is injected from the left-hand reservoir is
and the current injected from the right-hand reservoir is
In order to simplify the calculation of J further, we must invoke some special properties of the system. The most useful such property is that symmetry which permits the separation of the spatial variables. The separation of variables is possible if the Hamiltonian can be separated into two parts:
(Here the notation and is defined with respect to the direction of current transport.) Then the wavefunction separates into a product of two factors:
and the energy can be separated into a product of two terms:
The expression for the total current density J can now be simplified to
where is the larger of the two asymptotic potentials (minimum energy for a propagating state) and F is the Fermi-Dirac distribution function summed over the transverse states:
The form of the sum over depends upon the spatial configuration of the tunneling system. Note that the velocity factor does not appear in (35) because it was canceled by the density of states.
If the system in question is macroscopically large in its transverse dimensions, the form a two-dimensional continuum, and . Then using the two-dimensional analog of (22) and (25) F can be analytically evaluated:
The current density can now be written in the form usually given for the tunneling current :
Note that this expression is valid in general with respect to the dispersion relation in the x direction, but requires a parabolic dispersion relation in the transverse directions. The separation of variables leading to (38) is never rigorously valid in a semiconductor heterostructure. The reason for this is that the transverse effective mass will vary with semiconductor composition, which varies in the x direction. In principle, one must do at least a two-dimensional integral (if axial symmetry holds, otherwise a three-dimensional integral) as implied by (2). Nevertheless, (38) is widely used to model the current density in heterostructure devices. The error introduced by assuming separation of variables is probably less severe than that due to the assumption of an infinite coherence length.
If the transverse dimensions are constrained, but separation of variables is still possible, the transverse motion of the electrons consists of a discrete set of standing waves or normal modes. Such systems are referred to as ``one-dimensional'' systems, quantum wires, or electron waveguides. The symbol is now interpreted as an index for the discrete transverse modes, and the expression for the current density now becomes