|Fig. 1: The electronic energy levels of shallow donor and acceptor impurities. Left: a DA pair exciton consisting of a bound neutral donor and acceptor; right: upon radiative decay of the exciton, the donor and acceptor are now charged, strengthening their interaction.|
It is common for semiconductors to be simultaneously doped by both donor and acceptor impurities—intentionally or unintentionally. In IC processing, this is typical and n-type doping is usually achieved by overriding the p-type acceptor impurities with a higher concentration of donor impurities. In this case, most of the acceptors will be ionized not by valence band electrons, but by donor electrons so that neither impurity will contribute to the free carrier population. These donor and acceptor impurities have effectively compensated each other. However, if the semiconductor is optically excited or if carriers are injected at very low temperatures, electrons can be trapped forming neutral donors (D0) and holes can be trapped forming neutral acceptors (A0). With a well compensated (roughly equal D and A concentrations) sample, most of these sites will form donor-acceptor pair (DAP) excitons with their neighboring complementary impurity, and these excitons can efficiently luminesce.
Excited electrons and holes in semiconductors exist in many forms, ranging from essentially free in high quality high temperature crystals, to tightly bound (0.1 – 1 eV) excitons (electron-hole pairs) in disordered or molecular systems (termed Frenkel excitons or charge transfer excitons). In between, are weakly bound excitons (Wannier excitons), occurring, for example, in crystals at very low temperature . A Wannier exciton can be further categorized as a free exciton, excitons bound to one another (exciton complex), an exciton bound to a single impurity, or an exciton bound to both a D and A impurity in close proximity. This paper discusses luminescence from the last type. Studies on this effect were done mostly in the 1960-1970’s. It is interesting mostly because, unlike a free exciton, the electron-hole separation in a DA pair is determined by the D-A ion core separation, which is discretized in a crystal. The emitted photon energy depends on this separation and this makes for some spectacular spectra that can yield very precise information on properties such as the D-A energy levels and positions.
A photon emitted from a DAP recombination event has the energy
where Rm is the D-A separation, m is an integer and e is the dielectric constant . The last term is the interaction of the ion cores in the final state—the ionized donor and acceptor levels are perturbed toward the band edges as a consequence of their stabilizing Coulombic interaction (figure 1). This somewhat counterintuitive effect increases the energy of the photons and it varies as 1/Rm . To begin to predict what the spectra of an excited sample will look like, we can calculate the allowed values of Rm and the number of lattice sights (N(Rm)) at each Rm. If we take the FCC lattice as an example and we assume that the D and A impurities are randomly distributed (not always a good assumption, especially in light of the D+A- interaction) and that the impurities only sit at substitutional sites, then there is a simple formula for Rm 
where ao is the lattice constant. And N(Rm) can be counted by finding all the permutations of the conventional unit cell vectors ni that satisfy
|Fig. 2: Number of lattice sites as a function of distance from (000) for an FCC lattice.|
Figure 2 shows N(Rm) up to m=20. Note that it is not a steadily increasing function of R and that there are some values of m (such as 14) where N(Rm) even equals 0. This behavior has been experimental observed closely enough to assign luminescence peaks to particular D-A separations. The most extensively studied material is the semiconductor GaP doped with S and Mg. Upwards of 300 resolved lines from 2.32 to 2.22 eV have been attributed to DA transitions for radii ranging from the 11th to the 62nd nearest neighbor . At a temperature of 1.6K, it was even possible to distinguish between the Ga versus P substitutional sites and interstitial sites. The range of Rm over which DA pair luminescence peaks are observed is, of course, limited. For very close pairs (m < 11 in the GaP case above) where Rm is less than the sum of the Bohr radii of the electron and hole, the energetics are dominated by the wave function overlap effects and the spectrum becomes broad, and for large separations, the intensity of DA luminescence dies off because the probability of radiative decay decreases as the exponential of the pair separation.
The last statement above raises the question of how the luminescence varies with time. In normal band to band photoluminescence, the signal will decrease exponentially in time as the carriers diffuse and find each other. However, in our case, if the donors and acceptors are saturated at t=0, the electrons and holes are fixed in location and the decay rate of a pair W(R) goes as exp(-2R/ax), where ax is the sum of electron and hole Bohr radii . This means that the signal will shift to longer wavelengths with time as the close pairs decay rapidly and the distant pairs decay slowly.
|Fig. 3: Experimental setup similar to that used in . The two chopper blades are twisted with respect to each other and have different openings. Light first passes through blade 1 and is reflected off of blade 2 onto the sample, Blade 1 then closes and after a delay, blade 2 opens and allows luminescence to be detected.|
In a particular experiment with silicon, Enck and Honig were able to calculate Wo by monitoring the intensity with time . Figure 3 shows an example of an experimental setup similar to that used by Enck and Honig, which is convenient for collecting time resolved spectra. In this experiment, samples were doped ~1016cm-3 of both donors and acceptors. Thick samples (4x5x3 mm) were illuminated with a white light source while immersed in a 1.4-4 K liquid He dewar. With the dual chopper setup shown, they could easily vary both the time between end of illumination and beginning of measurement and the time duration of measurement. As a final note, for silicon due to the significant indirect nature of the transition, a phonon production term must be included in the equation for photon energy. The energies of the wave-vector-conserving TO and TA phonons were observed to be 58.3 and 18.9 meV, respectively. For impurities with binding energies greater than 39 meV, a significant no-phonon transition was also observed as the more strongly bound carriers have a greater spread in momentum space about the band extremum .
© 2007 Michael W. Rowell. The author grants permission to copy, distribute and display this work in unaltered form, with attribution to the author, for noncommercial purposes only. All other rights, including commercial rights, are reserved to the author.
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