March 20, 2007

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 (D^{0}) and
holes can be trapped forming neutral acceptors (A^{0}). 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 [1]. 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 R_{m} is the D-A separation, m is an
integer and e is the dielectric constant [2]. 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/R_{m} [3]. To begin to predict what the spectra of
an excited sample will look like, we can calculate the allowed values of
R_{m} and the number of lattice sights (N(R_{m})) at
each R_{m}. 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 R_{m} [4]

where a_{o} is the lattice constant. And
N(R_{m}) can be counted by finding all the permutations of the
conventional unit cell vectors n_{i} that satisfy

Fig. 2: Number of lattice sites as a function
of distance from (000) for an FCC lattice. |

Figure 2 shows N(R_{m}) 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(R_{m}) 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 11^{th} to the 62^{nd} nearest neighbor [5]. 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 R_{m} over which DA pair luminescence peaks are observed is, of course, limited. For very close pairs (m < 11 in the GaP case above) where R_{m} 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/a_{x}), where a_{x} is the sum of electron and
hole Bohr radii [6]. This means that the signal will shift to longer
wavelengths with time as the close pairs decay rapidly and the distant
pairs decay slowly.

In a particular experiment with silicon, Enck and
Honig were able to calculate W_{o} by monitoring the intensity
with time [6]. 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 ~10^{16}cm^{-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 [6].

© 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.

[1] E. I. Rashba and M.D. Sturge, eds.
*Excitons*, Modern Problems in Condensed Matter Sciences, ed. V.M.
Agranovich and A.A. Maradudin. Vol. 2. (North-Holland, Amsterdam,
1982).

[2] F. Williams, "Donor-acceptor Pairs in
Semiconductors," Phys. Stat. Solidi **25**, 493 (1968).

[3] P. Y. Yu and M. Cardona, *Fundamentals
of Semiconductors--Physics and Materials Properties, 2 ed.*
(Springer, 1999).

[4] D. G. Thomas, M. Gershenzon and F.A.
Trumbore, "Pair Spectra and 'Edge' Emission in Gallium Phosphide,"
Phys. Rev. **133**, A269 (1964).

[5] P. J. Dean, E.G. Schönherr and R. B.
Zetterstrom, "Pair spectra involving the shallow acceptor Mg in GaP,"
J. Appl. Phys. **41**, 3475 (1970).

[6] R. C. Enck and A. Honig, "Radiative
Spectra from Shallow Donor-Acceptor Electron Transfer in Silicon,"
Phys. Rev. **177**, 1182 (1969).