In a heterostructure, the band structure necessarily varies with position. This variation requires that the drift-diffusion equation for the current density be modified. This is most easily demonstrated by considering the case of thermal equilibrium, where the total current density must be zero. If the electron density is non-degenerate it may be approximated by the Boltzmann distribution:
If we insert this into the ordinary expression for the diffusion current, we obtain an expression which must equal the negative of the drift current:
The effective density of states depends upon position through the effective mass , which is a function of the semiconductor composition. Thus, from eq. (13) for parabolic bands,
Adding the drift and diffusion currents together, and making use of the Einstein relationship, we find that the electron current must be given by an expression of the form
By a similar argument one obtains an expression for the hole current:
The first and third terms of eqs. (1) are the usual drift and diffusion, respectively. The second and fourth terms are due to the spatial variability of the band structure. The second term resembles the drift term, but describes the carriers' response to changes in the band-edge energy, rather than to changes in the electrostatic potential. This effect is called a ``quasi-electric field'' , and is the origin of much of the usefulness of heterostructures. This term is readily understood on the basis that the carriers respond to gradients in the total band-edge energies and . The fourth term is more closely related to the diffusion term, and it describes the dynamical effects of a variable . To visualize this effect, consider two materials, having different effective masses but equal potentials and equal temperatures, in intimate contact. The thermal energies in each material are equal, but the average thermal velocity will be larger in the material with the smaller . Those carriers will diffuse across the interface between the materials faster than the heavier carriers, leading to a net flux of particles out of the region of smaller . The heterostructure drift-diffusion equations (1) may also be derived microscopically, starting from the Bolzmann equation . Equations (1) may also be written more compactly as
which is a more convenient form for subsequent manipulations.
Equations (1) may be solved analytically for the case of steady-state transport in one dimension, provided that recombination and generation may be neglected. The current density will then be independent of x. The carrier densities may be rewritten in terms of the quasi-Fermi levels, or, equivalently, one multiplies the drift-diffusion equation by an appropriate integrating factor. Let us consider the electron current first. Recognizing that both and (and thus ) will be functions of the position x, the integrating factor is . Multiplying both sides of (28) by this factor and integrating between points x = a and x = c, where the electron density is presumed to be fixed, we find
The drift-diffusion equation for holes may be similarly solved to yield
This solution is mathematically valid even when there are discontinuities in the parameters such as . It thus provides a convenient way to deal with abrupt heterojunctions. If one takes a and c to bound a differential element centered upon an abrupt heterojunction, one finds (not surprisingly) that the quasi-Fermi level should be continuous through the heterojunction. Equations (1) may also be used in numerical simulations, to evaluate the current density between discrete mesh points.
The heterostructure drift-diffusion equations (1) and their solutions (1) can be incorporated into the conventional pn junction theory to obtain expressions for the characteristics of a heterojunction. The variety of band alignment topologies makes it difficult to write generally valid expressions. However, the general behavior of heterojunctions is easy to understand intuitively and to describe (neglecting the broken-gap or extremely staggered cases). The barrier for carriers in the wider-gap semiconductor to pass into the narrower-gap one is lowered as compared to the barrier for carriers to pass from the narrower-gap material to the wider-gap one. Thus the great majority of the forward current in a heterojunction consists of one type of carrier, or in the language of bipolar transistors, the injection efficiency is quite large. This effect is exploited in the heterojunction bipolar transistor (HBT) [4,3].
Equations (1) also provide a model for the rather common case of current transport over an energy barrier. Suppose that has a maximum in the interval at x= b. Then, because of the exponential dependence upon , most of the contribution to the integral will come from the vicinity of the barrier at b. One may define an effective width for the barrier as that value such that The current density then becomes
This demonstrates the exponential dependence upon applied voltage (through ) expected for barrier-limited current flow. If one considers very narrow barriers, the factor of in the denominator leads to a very large pre-exponential factor. In such a case the energy band profile resembles that of a Schottky barrier, and the drift-diffusion equation is not the most appropriate model for current flow.