A computerized Model Calculation of Emission Rate Coefficients involving a
Transient Bonding State in Semiconductors
A. ZEHE, A. RAMÍREZ and E. REYNOSO
Facultad de Ciencias de la Electrónica
Universidad Autónoma de Puebla,
Ciudad Universitaria, 72000 Puebla
MÉXICO
Keywords: - Bound donor-acceptor pair state, semiconductor, AUGER-molecule, emission quenching
Abstract: - The effective Bohr radius of the donor electron in semiconductors extends over several interatomic distances,
providing for a potential bonding with a neighboring donor state at both high doping and external excitation. Such a
bound state has actually been identified in GaAlAs, which is formed of a close donor-acceptor molecule and a
neighboring second donor and/or acceptor. The recombination behavior of such a bound state is best described by
AUGER-type transitions, reason for which this state is called by us an AUGER molecule. The donor-acceptor pair
radiation intensity is affected in a characteristic manner by the existence of the AUGER molecule state at elevated
excitation levels and not too low temperatures. A computerized model calculation of corresponding rate coefficients turns
out to be in good agreement with reported experimental results.
1 Introduction
The existence of radiative luminescence transitions
which carry crystal site information is well documented
as Donor-Acceptor-Pair (DAP) radiation in bulk
semiconductors [1]. Due to a dependence on R, the
separation distance between pair forming donor and
acceptor ions in real space, DAP transitions must reflect
their mutual position and are still a widely studied
phenomenon [2]. Quenching of the DAP luminescence,
and at the same time an enhancement of the free-tobound (FB) luminescence was observed, when an
electric field was applied to the sample during
luminescence measurements [3]. The involvement of
AUGER transitions is discussed, when emission bands
are quenched [4]. Under the particular situation of highly
doped and almost compensated semiconductors, a new
kind of bound state may happen at high excitation levels
which is formed of a close da-molecule and a
neighboring second donor or acceptor. The de-excitation
behavior of such a bound state resembles characteristics
known from AUGER transitions, for this constellation is
called an AUGER molecule. A three-body or AUGER
type recombination is not unlikely to happen in direct
energy-gap semiconductors, that is, quenching of the
radiative recombination efficiency of the first transition,
-often not simple to detect and to relate to this kind of
non-radiative process. The DAP recombination, on the
other hand, provides a means to dismantle the existence
of AUGER molecules by virtue of the common intensity
saturation and peak energy behavior of the main
emission band. We present a study of the excitation of
dda-AUGER molecule states in a direct-gap
semiconductor host, and its effect on the external
quantum efficiency of photon emission.
2 Transient bonding state
An AUGER molecule in the here discussed context is a
bound state between three interacting impurity atoms
[d1d2a2] or [d1a1a2] within a crystalline semiconductor
host, as shown in Fig. 1. The real-space arrangement of
donor-acceptor molecules on substitution sites of the
host lattice is also demonstrated. In the present case the
donor of a close pair, formed of donor d1 and acceptor
a1, separated by a distance R1 from each other, is situated
near the donor of a distant pair (R2), formed of donor d2
and acceptor a2. The spatial separation distance R3
between the two donors is an important measure for the
formation of the AUGER molecule. It is clear, that only
highly doped (n-type in the here discussed case) and
almost compensated semiconductors provide for such a
situation. The expected relaxation processes after an
external excitation are seen in Fig. 1(b). Given a relative
closeness of d1 and d2, the otherwise radiative transition
PDAM between d1 and a1 is quenched due to a nonradiative AUGER transition d1a1d2. As a consequence,
an energetic electron is pushed into the CB, which after
thermalization allows a transition PFB between the free
electron and acceptor a2.
acceptor molecule (DAM), the AUGER molecule (AM),
and the conduction band-to-acceptor transition (FB).
Additionally, a characteristic photon energy distribution
of the recombination intensities can be expected.
The recombination intensity I(R) of the DAP-radiation
involves the pair distribution function, G(R), the
probability of pair transitions, P(R), and the probability
of pair occupation, F(R), respectively, being
G(R) = 4R2N exp(-R3N4/3)
(1)
P(R) = P0 exp(2R/aB)
(2)
Figure 1: (a) The donor-acceptor molecules (DAM´s) d1a1 and
d2a2 are separated by a distance R3 in the lattice on
substitution sites of the crystal. R1 and R2 are corresponding
donor-acceptor pair separation distances. The capture cross
sections  of a single DA molecule is proportional to the
squared separation distance, R. (b) Radiative recombination
transition PDAM and PFB, and non-radiative AUGER transition
PAM.
F(R) = vAgR2/[ vAgR2+ P0 exp(2R/aB)+X0f(g)] (3)
Donor-acceptor molecules (DAM´s) with large pair
separation R are occupied first at external excitation due
to their large capture cross section R2. At a low
external excitation rate, only common DAM-spectra of
distant pairs are observed.
There exist particular conditions of the pair distribution,
and a situation can be build, where a distant and a close
pair are situated at a separation distance R3 such, that
once occupied, both pairs interact with each other.
Indeed, at high excitation intensities, ever closer pairs
are occupied, providing for the formation of a molecule
consisting of the three components as shown in Fig. 1.
Such a dda-excitation represents an AUGER molecule
with a strongly increased probability for a three-body
collision, involving two electrons and one hole. No net
photon emission occurs, as the energy released by the
recombination of the short living donor-acceptor
molecule of close separation is immediately absorbed by
the electron of the longer living distant pair, which is
excited deep into the conduction band and then
dissipates this energy by emitting phonons. This effect
would be difficult to be proven by its own, though.
Eq. (3) contains the action of the AUGER-transition,
incorporated in F(R) as competing with the probability
of pair occupation; X0 is the collision probability; gt
D
threshold excitation density, E12
is the energetic
distance between the ground and the first excited donor
state:
3 Recombination channels
The [dda]-process leaves an occupied acceptor of the
distant pair molecule behind, as well as an additional
electron in the conduction band. A conduction band-toacceptor transition is likely to happen, with the
consequence of a competing intensity between this now
three recombination channels, that is the common donor-
with N-majority impurity concentration, v-thermal
carrier velocity, AR2-capture cross section, g-external
excitation density, aB-BOHR's radius, T-lattice
temperature. It is proportional to
I(R) = R2 G(R) P(R) F(R)
(4)
X 0 ( R, T )  X '  exp(  R / a B )  exp( E12D / kT ) (5)
f ( g )  (1  exp( g / g t ) /(1  exp( g / g t ).
(6)
The latter exponential term in Eq. (5) is related to the
bonding mechanism of the AUGER molecule. In
analogy to the hydrogen molecule bonding provides the
closeness of charged donors, once occupied, for their
mutual bonding. At high impurity concentrations, the
electron wave functions at the impurity level overlap.
This interaction changes slightly the potential of each
level, resulting in the formation of a band of states in the
region of overlap. Its length E0 on an energy scale
depends on the impurity concentration N, and is for
ND=NA in GaAs approximately given by E0 = 7.7 meV
(ND· 10-18 cm3). The otherwise discrete donor energy
levels, taken in the picture of the hydrogen model as En
= 13.6 eV · m*/mε2n2 broaden the more the closer the
impurity atoms are packed in the crystal (m*-effective
electron mass, ε-crystal lattice permittivity, n=1,2,3...).
The latter aspect is important, when the temperature
dependence is concerned. The thermal excitation of an
electron from the ground state E1 to the first excited state
E2 is more effective at higher temperatures at the one
hand, but might even be operative under conditions,
where the thermal energy kT is smaller than the distance
(E2-E1) of the energy levels of the isolated donor atom.
The requirement of spatial closeness of donors in order
to interact can be relieved, if high external excitation
densities come into play. Then both the ground energy
state and the second energy level can be occupied. The
by a factor of 4 larger BOHR radius of the E2-level
allows more distant donors to interact. This makes E2 the
bonding state of AUGER molecules and implies a
characteristic temperature behavior. The probability of
AUGER molecules to exist, say at liquid nitrogen
temperature, is reduced considerably at helium
temperature for the occupation probability of E2 levels
are proportional to exp[-(E2-E1)/kT], which should be
observed experimentally.
The collision probability X0 is introduced as an
analytical function. Their determination is an involved
problem, as X0 depends in the first place on R3, but this
in turn on the occupation function and the recombination
probability of DAM´s, and thus finally on the excitation
density g. Supposedly, the formation of AUGER
molecules does not happen before the excitation density
g runs over a threshold value gt, where close DAM´s in
the vicinity of distant DAM´s are beginning to be
occupied.
4 Recombination spectra
Calculated transition spectra of DAP-radiation are
shown in Fig. 2 (a). It is clearly seen, that for increasing
excitation density both the line shape and peak position
are modified corresponding to the changed pair
occupation. Under the action of the AUGER-molecule at
higher excitation levels, the integral intensity is reduced
as consequence of the competing effect between
radiative and non radiative transitions. Both spectral
intensities are compared in Fig. 2(b).
Donor-to-acceptor transitions are the dominant radiative
recombination process and occur already at low
excitation levels. The described intensity behavior of
measured DAP-transition has previously been reported
[5,6]. A single crystalline Ga1-xAlxAs host (x=0.8) was
doped almost to compensation with silicon. Due to its
amphoteric behavior, silicon forms both donors with Ed=
54 meV, and deep acceptors with EA200 meV.
Cathodoluminescence of the samples was excited and
the concentration of excited electrons and holes in the
host was controlled by the current of a 30 keV primary
electron beam. In the [dda]-AUGER molecule region,
for any two donor-acceptor pairs, bound into the
AUGER molecule and taken out of the radiative DAM-
transition channel, one additional excess electron
appears in the conduction band, and one additional
neutral acceptor (see Fig. 1b) participates in a FBrecombination channel instead of the DAM-channel.
Figure 2 (a): Donor-acceptor pair luminescence spectra in
dependence on the excitation density. Excitation density
g[cm3] is (1)- 1018; (2)- 1017; (3)- 1016; (4)- 1015; (5)- 1014; (6)1013. (b) Calculated integral intensities without (upper curve)
and under participation of the AUGER molecule state (X' =
108 s-1), in dependence on the external excitation density g.
The emission intensity in the latter case is strongly reduced.
Both radiative transition channels have been reported to
show a blue shift with increasing excitation density,
which in any case confirms the participation of excited
donor states in the radiative transition process.
References
[1] B. Guo, Z.R. Qui, K.S. Wong, Appl. Phys Letters 82,
2003, 2290
[2] A. Aydinli, M.N. Gasanly, A. Serpengüzel, Semicond.
Sci. Technol. 14, 1999, 599
[3]A. Cesna, J. Kundrotas, A. Dargys, J. Luminescence, 78,
1998, 157
[4]A. Zehe, A. Ramirez, Theoret. Chemistry Acc.104, 2000,
331
[5]C. Wang, D. As, B. Schöttker, D. Schikora, K. Lischka,
Semicond. Sci. Technol.14, 1999, 161
[6]T. Haensel, A. Zehe, "Competing Donor-Acceptor Pair
Recombination in GaAlAs under Electron-Beam Excitation,
Crystal Research and Technology 12, 1974, K 95