June 18, 2007

Fig. 1: Energy band diagram showing simplified
schematic of recombination processes: a) radiative, b)
defect mediated, anc c) band-to-band Auger
recombination. |

Excess electrons and holes in a semiconductor will recombine back to their equilibrium distribution releasing energy equal to the band gap. The recombination event is usually categorized in one of three ways: radiative emission, defect mediated, or non-radiative band-to-band Auger recombination. In most materials, one of those processes will be much faster and will largely determine the effective lifetime, t, of the excited carriers.

where t_{r}, t_{d}, t_{Aug},
are the radiative, defect, and Auger lifetimes, respectively.

In high quality direct band gap semiconductors,
radiative emission, occurring within a 1-10 ns time scale, can be the
dominant mechanism, and the transition rate is described by Fermi’s
Golden rule. In heavily doped semiconductors, doped more than
10^{17} cm^{-3}, the Auger rate, which is proportional
to the doping density squared, can be the dominant mechanism. For most
semiconductors, however, especially those with indirect band gaps, short
lifetimes are due to impurities and defects. Empirically, the worst
defects are usually those with a mid-gap ground state, as depicted in
figure 1. Such deep level recombination is often loosely referred to as
Shockley-Read-Hall[1,2] recombination after those who first successfully
modeled recombination with Fermi statistics and empirical carrier
capture cross-sections. Shockley-Read-Hall statistics, however, say
little about the underlying mechanism of the event itself.

The recombination process for deep level traps is usually non-radiative, so the energy must be dissipated as phonons. The energetic jump between states is large, however, so the fact that phonons have energies of only 10-50 meV makes the prospect for a single multi-particle reaction (i.e. an electron and tens of phonons) unlikely. How then do deep level defects act as such efficient recombination centers? The answer varies from case to case and is often not so clear[3]. The lack of clarity and difficulty in studying deep level recombination is largely due to the immense chemistry and difficulty in controlling defects. The presence of deep levels usually means the bonds with the host are weak and/or highly strained, which make solubility low and defect precipitation likely. The two most accepted mechanisms are an Auger process or a significant lattice distortion with multiple phonon release. This paper will discus qualitatively these two deep level processes as well as the initial free carrier capture and localization about a defect site.

The complete recombination process can be broken into
two steps: an initial, fast capture or trapping of a free carrier into
an energy level near the band edge followed by two large energy jumps to
the deep level and then to the other band. The initial event is well
understood using the "effective mass approximation" or, equivalently,
"hydrogenic theory" used to describe donor and acceptor dopant
behavior[4, 5]. The term effective mass implies that theory for the
perfect crystal is still applicable: namely that wave functions
describing the localized electron can be described by Bloch functions
modified with a spherical "hydrogen" function and that the screening of
the Coulombic attraction between the carrier and the defect atom is
still described by the long range static dielectric constant. In
analogy with the hydrogen atom, there are then a series of closely
spaced energy states that the carrier can step through rapidly, emitting
a single phonon between each level. The energy levels, E_{n},
and Bohr radii, a_{n}, are just modifed by the effective mass
and dielectric constant.

This cascade model describes very well the behavior of shallow defects and dopant impurities. However, once the electron is stuck at point A in figure 2, the next energy drop is too large to be achieved by a simple phonon scattering event and either de-trapping will occur or one of the following mechanisms will take place.

At high enough temperatures there can be large enough lattice vibrations such that the energy states for a free electron in the conduction band (or hole in the valence band) can cross with the energy state of the localized defect. Fig. 3 is a diagram of the electronic and elastic energy of the conduction band, defect level and valence band versus a single lattice coordinate, Q, based on the Hamiltonian used by Henry and Lang to model this process[6]:

where H_{E} is the normal electronic
Hamiltonian independent of Q, H_{EL} is the change in the
potential well of the defect with displacement in the lattice and is the
perturbation enabling the transition between the delocalized conduction
band state and the localized defect states (b is a measured constant),
and H_{L} is a harmonic oscillator of the defect. For a carrier
weakly trapped at point A, thermal vibrations cause a displacement in
the lattice from Q_{o} to Q_{B} where the energy states
cross and it is possible for a transition between free carrier wave
function (or weakly trapped wave function, as described above) and the
localized defect wave function. The transition from the delocalized to
localized wave function (B to C) is nearly adiabatic (little phonon
production); however, relaxation of the surrounding lattice to
accommodate the occupation of the defect state will result in a rapid
release of multiple phonons in transitioning from C to D. This energy
release is often enough to cause damage to the lattice or physical
movement of the defect. If a hole is then weakly trapped in the valence
band, the process can occur again, completing the recombination event.

The Auger process (the inverse of impact ionization) is the transfer of energy to a second nearby carrier. The particle receiving the energy is then in a position to rapidly thermalize back to the band edge or even back into the trap. The process is effective for heavily doped materials where, for example, the probability of an electron finding both a hole to recombine with and a second hole to impart energy to is likely. But it is also effective for some defects that can attract two carriers, which is often the case for impurities that exist in the host lattice as a doubly charged site[7].

© 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] W. Shockley and W.T. Read, "Statistics of the
Recombinations of Holes and Electrons," Phys. Rev. **87**, 835
(1952).

[2] R. N. Hall, "Electron-hole Recombination in
Germanium," Phys. Rev. **87**, 387 (1952).

[3] H. J. Queisser, "Recombination at deep
traps," Solid-State Electr. **21**, 1495 (1978).

[4] M. Lax, "Cascade capture of electrons in
solids," Phys. Rev. **119**, 1502 (1960).

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

[6] C. H. Henry and D.V. Lang, "Nonradiative
capture and recombination by multiphonon emission in GaAs and GaP,"
Phys. Rev. B **15**, 989 (1977).

[7] M. Jaros, "A case for large Auger
recombination cross sections associated with deep centers in
semiconductors," Solid State Comm. **25**, 1071 (1978).