ANNUAL
REVIEWS
Ann. Rev. Mater. Sci. 1977. 7: 341-76
Copyright © 1977 by Annual Reviews Inc. All riuhts reserved
DEEP LEVEL IMPURITIES
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IN SEMICONDUCTORS
=-=8607
H. G. Grimmeiss
Department of Solid State Physics, Lund Institute of Technology, Lund 7, Sweden
INTRODUCTION
Solids are usually divided into metals and nonmetals. In a simplified scheme we may
say that nonmetals consist of insulators and semiconductors. Both can be described
by energy bands separated by a forbidden energy gap Eg• Whether a nonmetal is
considered to be an insulator or a semiconductor often depends on the temperature
at which its properties are to be investigated or used. In this chapter we use "semi­
conductor" to mean a nonmetal with a band gap from at least a few tenths of an
eV up to a maximum of 10 eV.
The considerable interest in semiconductors since the 1930s has been stimulated
by their technical significance. Unlikl� metals, the physical properties of semi­
conductors can be considerably modified by introducing small amounts of foreign
atoms. In this manner the resistivity of a silicon crystal can be changed by about
seven orders of magnitude when one-millionth of the atoms in the crystal are
replaced by suitable foreign atoms. Note that impurity substitution of this magni­
tude does not affect the band gap or other basic characteristics of the host material.
Moreover, depending on the kind of foreign atom, the electrical current through
the crystal is carried by electrons or holes. Hence multilayer structures with different
types of conductivity can be constructed by doping a semiconductor crystal in­
homogeneously, resulting in devices such as diodes, transistors, lasers, etc.
Experimentally it is found that replacing an atom of the host lattice by a foreign
atom results in lattice defects with physical properties depending considerably on
the particular atom introduced. Good understanding of such impurity centers has
been achieved at least for Si and Ge when the foreign atom belongs to the
groups of the periodic table closest to that of the semiconductor. They introduce
localized donor and acceptor levels in the otherwise forbidden energy gap and are
often described by the effective mass theory of Kahn & Luttinger (KL-EMT) (1-3),
giving a hydrogen-like spectrum of levels with binding energies En' which can be
written as
En
=�. »1*
n2 2h2
(q2)2
s
Further
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341
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342
GRIMMEISS
Here m* is an effective mass for the electron or hole, 6 is the static dielectric constant
of the semiconductor, and n is an integer. Equation 1 gives excellent agreement
with experimental results obtained for excited states but has limited success for the
ground state. Better agreement between calculated and measured values for ground­
state energies is obtained using a pseudo-impurity theory developed by Pantelides
(4, 5) which combines the general theory of pseudopotentials with effective-mass
ideas. Nevertheless, adapting equation 1 for an estimation of ground-state energies
in silicon gives values of the order of 50 meV for singly charged impurity levels.
Such centers are therefore called "shallow" impurity levels because they lie close to
one of the energy band edges. Shallow impurities are widely used in semiconductor
technology for modifying the amount and type of electrical conductivity.
Impurity levels which lie at a considerable distance from a band edge are called
"deep" impurity levels. They are easily obtained in silicon or germanium by doping
these materials with group II or group VI impurities. In general, all elements which
replace an atom of the host lattice and do not belong to the periodic table groups
closest to that (or those) of the semiconductor create deep impurity levels. Although
deep level impurities have been a subject of study for about 20 years, the field is
still characterized by a lack of unambiguous experimental data and insufficient
theoretical understanding. Previous reviews of deep centers (6, 7) have already
underlined that no general formalism is presently available to predict the physical
properties of these centers.
This review summarizes some aspects of present-day thinking about phenomena
particular to deep level impurities. Emphasis is placed on measuring techniques
and new experimental results rather than on theoretical speculations. To show the
consistency of experimental data and to give the reader the possibility of comparing
different experimental results, one of the most investigated systems (gold-doped
silicon) has been selected as a typical material to illustrate the essential features of
deep centers. The results are then compared with data obtained for other deep
level impurities in silicon as well as in III-V and II-VI compounds.
We have chosen not to outline all the different features proposed for deep
centers but rather to concentrate on their electrical and optical properties and,
hence, on a discussion of thermal and optical capture and emission rates and
energy positions. Many other important topics are therefore not adequately covered
in this review. Among these are Auger-effects, lahn-Teller distortions, and 'effects
arising from so-called isovalent impurities which are in the same column .of the
periodic table as the host crystal.
INFLUENCE ON FREE-CARRIER LIFETIME
Deep level centers seem to be present in all known semiconducting materials. One
of their most important properties is the abi lity to control carrier lifetime significantly
even in small concentrations. This is readily explained by Shockley-Read-Hall
statistics (8, 9). According to these statistics, the lifetime T of excess charge carriers
in a semiconductor with a single impurity level is given by
t=
cp(Po+ pIl + cn(no + nil
cncpN TT(nO + Po)
•
2
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DEEP LEVEL IMPURITIES
343
where Cn and cp are the average valu(:s of the capture constants of electrons and
holes over the states in the bands, no and Po are the free-carrier concentrations in
thermal equilibrium, and nl and PI are the concentrations of conduction band
electrons and valence band holes for the case in which the Fermi level EF falls at
ET, the energy position of the impurity level. N TT is the number of deep level
centers per unit volume. Equation 2 is valid as long as N TT is small compared to
any one of the four quantities no, Po, nl, and PI ( 1 0).
Let us now consider an n-type semiconductor (EF > EiF' where EiF is the intrinsic
Fermi level) with an impurity level above the Fermi level (ET > EF)' We then have
nl P no p Po p PI, a condition which, for example, is given in the space-charge
region of a pin junction or a Schottky barrier. Under these circumstances, equation
2 reduces to
nl
'=---.
NTTCpnO
3.
Since nl depends on the energy position of the impurity level, the free-carrier
lifetime changes exponentially with E,.. How significant the energy position of the
impurity center is for the free-carrier lifetime is easily demonstrated for constant
background doping by considering two impurity levels having different energy
positions. Denoting the lifetimes resulting from each of the centers by ,(1 ) and ,(2),
the ratio of the lifetimes is then given by
�
,(2)
=
(ET(l)-ET(2)).
cp(2)NTT(2)
exp
cp(l)NTT(I)
kT
4.
Taking the same value for the capture constants of both levels and ET(l) > ET(2),
the ratio is always much larger than unity as long as nt(l)NTT(2) p nt(2)NTT(1).
This means that at room temperature an energy difference of 0.4 eV and at 77 K
of only 0. 1 eV is sufficient for the lifetime of excess charge carriers to be determined
by the deeper level impurity even if the concentration of the deeper impurity is 6
orders of magnitude smaller than the wncentration of the shallow impurity level.
If an impurity has a sufficiently large binding energy, it will affect the carrier lifetime
even when present in a concentration so small that separate experiments for
impurity identification have little chance of success. Thus one negative (but
important) reason for the current interest in deep impurities is the need to be
aware of and hence able to a void undesired conseq uences of impurities inadvertently
present. Other sometimes harmful effects of deep impurities include undesired
trapping effects which may change switching times in devices such as photo­
conductors and nonradiative recombination processes which can affect the efficiency
of light-emitting diodes.
However, deep level impurities may also create desirable effects in devices. The
use of gold to give fast recombination and, hence, an increase of switching
frequency in silicon junctions is well known. The high quantum gain of photo­
conductors is usually obtained by just these deep centers. The high resistance of,
and many interesting effects in, semi-insulating GaAs are due to the presence of deep
level impurities. However, a better understanding of the physical properties of deep
centers is needed not only for semiconductor technology; from several points of
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344
GRIMMEISS
view concerning fundamental questions of solid state physics, an improvement of
our knowledge about deep centers is highly desirable. For deep levels, the bound
electrons or holes have orbits of small radius. Their ground staters) and excited
states each have a characteristic wave function. Depending on how sharply localized
the wave function is, wave packets may arise which extend far out in the Brillouin
zone. The magnitude and spectrum of the photo-ionization cross section, for example,
are both direct consequences of the bound-charge distribution. Hence, for all
theoretical models a knowledge of the impurity potentials is of decisive importance.
Apart from some impurities in silicon and germanium, however, the perturbation
potential is usually not known. Experimental techniques are thus of vital importance
and are likely to remain so in the near future.
CHARACTERIZATION OF DEEP LEVEL IMPURITIES
For a discussion of deep level impurities, it is necessary to know which parameters
characterize the physical properties of these centers. It is quite obvious for the
reasons just mentioned that the energy position of the impurity level within the
forbidden energy gap is an important parameter. Furthermore, to describe re­
combination and excitation kinetics, the probabilities for capture and emission
processes have to be known. However, the rate of recombination and excitation
is proportional to the number of impurity levels. Although the density N TT of the
centers is not directly correlated to the physical properties of impurity levels, a
knowledge of this quantity is nevertheless desirable. Hence, to characterize the
electrical and optical properties of deep level impurities, at least ten quantities have
to be known: thermal and optical capture constants of elcctrons and holcs; thermal
and optical emission constants of electrons and holes, the energy position, and the
density.
Several attempts have been made to calculate these parameters using different
theoretical models. According to Pantelides (11), the effective mass approach (KL­
EMT) can still be valid for deep levels when the impurity atom is substitutional
and is from the same row of the periodic table as the host atom. Such atoms have
the same number of core electrons and are therefore called isocoric. Without
employing adjustable parameters, good agreement with experiment is found for
sulfur impurities in silicon. By developing a new pseudo-impurity theory, the same
author also calculated the binding energy of non-isocoric donors in silicon (4) and
oxygen at a phosphorus site in GaP (12). Baldereschi & Lipari, using the effective­
mass point-charge model, calculated the binding energies for some divalent and
trivalent acceptor levels in germanium in fair agreement with experiment (13). Several
techniques have been employed to estimate the energy positions of transition-metal
impurities (14-16). Fleurov & Kikoin consider the scattering of band electrons
accompanied by the excitation of an unfilled d shel l of an impurity atom to be
the origin of the deep levels associated with transition-metal impurities ( 1 7). To
describe the energy position of the impurity levels, they used the Anderson
model (18).
The situation for capture processes is less satisfactory (Figure I). Experimental
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DEEP LEVEL IMPURITIES
345
values for capture cross sections vary between 10- 22 cm 2 and 1 0 - 12 cm 2, and in
most cases are very different from the geometrical size of an impurity center (about
1 0 -16 cm 2). This is quite reasonable considering that capture of a free-charge
carrier not only results in a bound-charge carrier but that simultaneously an energy
of the order of 0.5 eV must be carried away. In radiative processes, this is
accounted for by photons, whereas in nonradiative processes the energy is carried
by particles such as phonons [cascade (19) or multiphonon processes (20, 21)],
electrons [Auger processes (22-24)], or plasmons (22). Lax proposed a semiclassical
model in which large capture cross sections may arise due to phonon cascade
processes in which the carriers recombine through a ladder of excited states
emitting a phonon at every step. In most cases, there exists considerable dis­
agreement between calculated values and experiment [see for example (25)J.
Attempts have been made (26-29) to refine phonon cascade theory, although further
work remains to be done before nonradiative recombination at deep centers can be
regarded as being fully understood. A similar unsatisfactory situation exists for
thermal emission rates e' (Figure 1 ) . They are often calculated from thermal
capture rates using the detailed balance relationship. For electron excitation
,
, exp (ET-EC) ,
processes this gives
en = gA 'cnNe
5.
�
where gA is the degeneracy factor of the center, Ne is the effective density of states
in the conduction band, and Ec is the bottom of the conduction band. Due to the
uncertainty of the thermal capture rate c� and sometimes also in the energy position
ET• published data of calculated tht�rmal emission rates often disagree with
•
jen
•
Figure J
Ev
Capture and emission pro ce sses. Ee- ET and NTT denote the binding energy of
electrons and the concentration of energy levels, respectively. c.[cm3/s] is -the capture rate
and en[l/sJ the emission rate of electrons. The total net rate of electron capture is dnT/d t
c.n(N TT- nT) - ennT, where nT is the concentration of occupied energy levels. Similar nota­
tions are used for hole capture processes.
=
346
GRIMMEISS
experiment. So far the best agreement between theory and experiment has been
obtained for optical emission rates eO, which are correlated to the photo-ionization
cross sections (f0 by the relation
eO
=
(f0<p,
6.
where <p is the photon flux used for measuring eO. This cross section is customarily
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calculated (30, 31) in time-dependent perturbation theory. The probability for a
transition from a bound state with wave function t/Ji to a state t/Jf in the band by
application of the optical perturbation il' can be expressed as
7.
where p(Ef) is the density of final states at the cnergy in the band to which the
transition takes place. It is quite clear from equation 7 that the magnitude and
spectral dependence of the optical transition probability greatly depend on the
choice of t/Jf' t/Ji' and P(Ef). Many calculations of optical cross sections a"(E) using
different impurity models are found in the literature (32-41).
Although remarkable progress has been made in the theoretical study of deep
impurities, it becomes more and more apparent that the derivation of a general
theory of deep level impurities is probably impossible because of the great variety
of features inherent for these centers. This would place experimental techniques
in a very important position for further investigation of deep level impurities. A few
years ago there was very little information about deep level parameters mostly
because of the lack of proper measuring techniques. Recently, however, several
new methods have been developed which make it possible to measure emission
rates, capture rates, energy positions, and impurity concentrations very accurately.
The various quantities of interest just listed can in principle be derived from
measurements with bulk materials, with Schottky barrier contacts to single crystals
and with p-n junctions. The next section concentrates on results derivable from
p-n junction measurements since the junction technique has the important ability
to yield information on several of these quantities from a single sample with
various forms of stimulus. However, junction methods are not the complete answer
and therefore some bulk material measurements are discussed in a later section.
THERMAL EMISSION RATES
Statistically, the thermal emission rate of electrons e� is defined as (8)
e. = r 00 e�(E)N(E)(l-!c) dE,
JE,
where e� is the number of electrons emitted per unit time and center from the
impurity level at ET to an empty level in the conduction band, and !c is the Fermi
function of the conduction band states. A similar definition applies for the thermal
emission rates of holes. Thermal emission rates are commonly investigated by
techniques which make use of extrinsic excitation processes in reverse-biased p-n
347
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DEEP LEVEL IMPURITIES
junctions. The external applied voltage acts almost only on the space-charge region
and not on the neutral region. The concentrations of free-charge carriers are
therefore reduced well below their equilibrium concentrations only in the transition
region. The reduction in charge-carrier concentrations implies that only emission
processes contribute to the dark current and that capture processes can be neglected.
Dark currents of reverse-biased p-n junctions are therefore especially suitable for
studies of thermal emission rates.
A proper analysis of these investigations requires a good knowledge of excitation
and recombination processes in the junction region. In the past, it has generally
been believed that the dark reverse current is generated in the total space-charge
region W. Braun & Grimmeiss (42), however, showed that due to the tails of free­
charge carriers extending from the neutral regions into the space-charge region, only
part of the transition region contributes to the thermally generated reverse current.
Hence, the dark-current density of a reverse-biased p-n junction can be calculated
according to
+
tl� Un (1- �)UpJdX,
X2-XI
is the effective generation region width (Figure 2), Un and
Upwhere
are the total net rates of electron and hole emission, respectively, and is the
ratio between the excited charge within the effective generation region - and
the measured charge in the external circuit (43). Later it is shown that the factor
plays an important role in the determination of emission rates from transient
measurements. The boundaries and X2 are defined by the conditions (42)
J� = q
=
8.
W- Wo
X2
D
XI
D
XI
9.
I
1
I
1-.
_____
:
I
1
Figure 2
"
"
"'E---
Xn
W --��
Band diagram of a p-n junction wit h deep energy levels at ET•
Ev
348
GRIMMEISS
10.
respectively, where (8)
'nO = (C�NTT)�1 = n de�NTT
1 1.
'pO = (C�NTT)�1 = p de �NTT
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is the lifetime of electrons in highly p-type samples and
1 2.
is the lifetime of holes in highly n-type materials. Hence, for an impurity level in the
upper halfof the hand gap (nt � pdand TnO::::; 'pO, Xl and X2 correspond to positions
in the space-charge region at which the distance from the quasi-Fermi level for holes
to the valence band edge is Ec- ET and the quasi-Fermi level of electrons coincides
with the deep impurity level, respectively (42). X2 - Xl therefore decreases with
increasing depth of the impurity level and is less than half the space-charge region
for impurity levels close to the middle of the forbiddcn band gap (such as, for
instance, the gold-acceptor level in silicon).
Because capture processes can be neglected in reverse-biased p-n junctions, the
net rate of electron and hole emission is given by
13.
and
14.
respectively, where nT and PT are the concentrations of occupied and empty impurity
levels, respectively, and IT is the Fermi function or occupancy of the impurity level (8).
Substituting equations 13 and 14 into equation 8 yields for a reverse voltage
VR � kT/q
15.
The factor D depends among other things on ET, VR, and the concentration of the
shallow background doping due to the free-charge carrier tails extending from the
neutral region into the space-charge region. Commonly these tails are neglected and
D then takes a value of 2. However, depending on the experimental conditions, D
can take any value greater than 1 (43). At steady state Un Up and hence
=
nT(oo )
en+ep
e'
=
�NTT'
16.
Equation 15 then reduces to the simple expression
J�(oo)
=
e n + ep
et et
q(x2-xdNTT �,
17.
showing that e� and e� cannot be investigated separately from steady-state reverse
currents. The most widely used technique for investigating thermal emission rates
is therefore the measurement of dark-current transients in reverse-biased p-n
DEEP LEVEL IMPURITIES
349
junctions as suggested by Sah et al (44). The technique involves the measurements
of the decay time constant T and the initial and final dark current J�(O) and
J�(oo), respectively, when the junction voltage is switched from zero to a large
reverse bias VR (see Figure 3). The current transient contains a fast and a slow
component. The fast component is due to the sweep-out of majority carriers from
the increased space-charge region and has normally a time constant of the order of
10-10 sec. The slow current transient is generated by the change in occupancy of
deep energy levels. Using the fact that d nT/dt Up- Un' one obtains for the
time dependence of the slow dark-current transient in a p + n junction [nT(O) = N TT]
by integration and substitution into eq uat ion 15
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=
,
JR(t)
=
q(x2-xllNTT'
{(
e�
(t)]
1 )
[e�
--I + -'-- exp
'
1 - - e� + -T
en+ ep en+ep
D
18.
From the transient wave form given by equation 18, e� and e� of an impurity level
at ET can b e calculated according to the following relations
Jk(O)
e�
� = D -- -I
e�
19.
Jk(oo)
and
20.
2 �------�------�
r-,
.-.
8
-..+-11::
J
+-
+-11::
J
L--...J
CI
0
r
0
0
Figure 3
2
6
4
--�
t(S)
8
Typical dark-current transient in (a) semilogarithmic and (b) linear scale. (e) is
the actual recorder trace.
10
350
GRlMMEISS
Recalling Ihal D can vary considerably depending on the experimental conditions,
it is quite evident that an exact knowledge of D is important for an accurate deter­
mination of emission rates. In the mQst general case, the time dependence of Xl - X,
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2,5
/"
2,0
3:
N
><
1,5
;(
r
1,0
..."
W
.
../'.
�
.//'
./,.
/--�
./
xz.o
�:
/{
Ii
/ /}
;J
P
/
I
W
/I
A
.
0,5
5
10
--->,.
Vft (volt)
15
Figure <1 Boundaries X, and x, (see also Figure 2) within the transition region of a
linearly graded p+n juncti'''' as It function of the reverse bias VR in darkness (XI ; -- ;
x�',�-) and during illumination (x,: ... ; x,:···-' ). For comparison the width Wof the
total transition region is also plotted (_ .. ').
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DEEP LEVEL IMPURITIES
351
has to be taken into account. However, for NTT values small compared with the
density of shallow impurity levels, this time dependence can be neglected.
One of the drawbacks of this technique is that all excitation processes occur
within the space-charge region of a reverse-biased p-n junction, in which the electric
field is 1 04-105 V/cm. This high field should not be ignored, since it is certainly
conceivable that emission rates could be field sensitive. Several authors have
published data on emission rates in silicon doped with Au, Ag, and S (45-47),
which have been interpreted as showing a field dependence. Those analyses were
based on the assumption that the total space-charge thickness W contributed to
the dark-current density, which was noted earlier in this review as being invalid for
deep impurities.
Replacing X2-XI by W - Wo in equation 1 7 shows that information about a
possible field dependence of thermal emission rates is readily obtained by plotting
Jk(w) against W. Wo comprises those parts of the transition region where the net
thermal excitation process can be neglected. It has been demonstrated previously (42)
that in one-sided step junctions, for example, Wo becomes rather voltage insensitive
with increasing reverse bias VR and can therefore be considered as constant (see
Figure 4). Hence, if the thermal emission rates are field independent, a plot of
Jk( CX)) against W should give a straight line with slope qNTTe�e�/(e� + e�). This has
�
:IE
or:(
00,52 5
15'10-8
----+
10
VR (VOL T)
20
30
IZ
IU
It:
!5 10'10.8
0
IU
tJ)
It:
IU
>
IU
It:
� 5'168
It:
or:(
0
f
0
1,0
1
2,0
3,0
_W(�J
4,0
Figure 5 Dark reverse current of a gold-doped silicon p + n junction as a func;tion of the
width W of the total transition region at room temperature. The intercept of the extra­
polated straight line with the p ositive abscissa shows Wo the width of that part of the
transition region where the net emission process can be neglected Uunction area I mm2).
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352
GRIMMEISS
been demonstrated for e� of gold-acceptor levels in silicon (48) for which e� 3> e�
(Figure 5). No field dependence was observed for average electric fields up to 105
Vfcm. Furthermore, these measurements allow an experimental determination of
Wo (and hence X2-XI ) from the intercept on the positive abscissa obtained by
extrapolating the straight line at larger VR values.
Thermal emission rates can also be determined from the decay-time constant r
and the initial and final dark capacitance C(O) and C( CX) if the concentration ratio
between the deep and shallow impurity levels is known. This makes capacity
measurements less attractive in comparison with the dark-current technique. How­
ever, for investigations ofthe temperature dependence ofthermal emission rates, both
methods are often used.
OPTICAL EMISSION RATES
By replacing thermal emission rates e' with optical emission rates eO in the
preceding equations, all expressions obtained for thermal excitation processes can be
used in a similar way for the description of optical excitation processes below the
freeze-out temperature (i.e. eO 3> e'l. According to equation 17, the steady-state
photocurrent density J� through a p-n junction is then given by 21.
The boundaries XI and X2 (see Figure 2) of the effective generation region are now
defined by the conditions (42)
c�p(xd
=
and
c�n(X2)
=
e� + e�
22.
e� + e�.
23.
The effective generation region is thus independent of ET• but depends on intensity.
However, due to the exponential variation of n and p. the boundaries x I and X2 are
rather insensitive to illumination (42) (see also Figure 4).
Absolute values of optical emission rates cannot be determined from steady-state
photo currents with one light source. They are therefore usually measured by
transient techniques (44, 49), which involve measurements of the decay time constant
and the initial and final photocurrent J�(O) and J�(CX). respectively, when the sample
is illuminated with monochromatic photons of energy hv. From the time dependence
of the photocurrent, e� and e� can bc calculated according to equations 19 and 20.
The initial conditions for these measurements may be established either electrically
or optically. As pointed out by Bjorklund & Grimmeiss (49), this transient technique
reduces to a very simple measurement when photons of energy less than half the
band gap are used. One of the optical emission rates is then zero and r is equal
to the inverse of the other emission rate. Knowing the photon flux, absolute
values of photo-ionization cross sections are obtained from equation 6 without any
further information. Repeating these measurements with different photon energies,
the spectral distribution of (J0 is readily obtained.
DEEP LEVEL IMPURITIES
353
Another simple method of measuring the spectral distribution of photo-ionization
cross sections is to plot the initial photocurrent of the transient JR(D) versus hv
and calibrate the spectrum by one transient measurement. According to equation 1 8,
J�(O) is proportional to e� if "T( O)
N TT' This can, for example, be achieved in a
p + n junction when the initial condition is established by reverse biasing the
junction. In this case, the photon energy hv is not limited to energies smaller than
half the band gap.
Transient measurements are sometimes difficult to analyze. The exact value of the
D factor and the time dependence of X2 - XI in more heavily doped junctions are
not always known. Steady-state measurements are not affected by these drawbacks
and are therefore in most cases sup<:rior, as is demonstrated later for a single
impurity level in the upper half of the band gap in a p +" junction. For more compli­
cated systems the reader is referred to (50). The reason that spectra of photo­
ionization cross sections cannot be measured from steady-state photocurrents with
one light source is the change in occupancy of the centers when the photon
energy is varied. A constant occupancy is, however, easily achieved by using a second
light source with properly chosen constant p hot on energy hvs and high photon
flux 1s (Figure 6). The simultaneous illumination of a junction with two light
sources is taken care of in the relations describing the photocurrent by replacing
e� with e� + e� and e; with e; + e;s, where e�s and e;s are the optical emission rates
due to the second light source. Choosing the photon energy hvs and photon flux
1s of the second light source such that E,- E T < hvs < ET- Ev (i.e. e�s 0) and
e�s � e� + e� (i.e. ¢s � ¢), the steady-state photocurrent density is given by (see
equation 17)
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=
=
24.
,
--------+-
-----
Ev
Generation of photocurrent Jh due to simultaneous ilIumination with two light
sources of photon energies E, - ET < hv, < E1" - Ev and ET - Ev < hv < Eg. respectively.
Figure 6
354
GRJMMEISS
where Cl can be considered as constant because the occupancy and hence X2-Xl
does not change during the measurement. Equation 24 is valid for all photon
energies hv smallcr than thc band gap Eg• The spectrum of e� in the whole
extrinsic energy region is obtained by choosing hvs and cPs such that
ET- Ev < hvs < Eg and both e�s and e�s are much larger than e� and e� for variable
photon energies (Figure 7). In the most general case, the increase in the steady­
state photocurrent density IlJ�1 J�l-J� can then be expressed (50) as
t'l.J'Rl = (1-b)2J'Rh+q(x2-xIlNTTb2e�
=
=
(1-b)2J'Rh+clb2e�.
25.
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where
26.
is the electron occupancy of the impurity level when thc junction is illuminated
with photons of energy hvs alone (see equation 16); b can be considered as
constant during the measurements. Plotting t'l.J1ll- (1- b)2JRh versus hv gives the
spectral distribution of e�. For hv < Er- Ev the optical emission rate of holes e�
vanishes and J�h = O. In this region, the absolute value of e� is readily determined
by a transient measurement, and Cl can threfore be calculated from equation 25.
Hence, from measurements of the steady-state currents J'R. J'Rh and t'l.JR1 together
with one transient measurement, the spectral distributions of e� and e; are obtained
in absolute values for all energies smaller than the band gap. This technique is also
easily used in samples with more than one deep level impurity (50).
/14
J�
0
Ec
0
ensnT
en nT
� I'
/I,
0
epSPT
Ev
Figure 7 Generation of photocurrent AJR1 due to simultaneous illumination with two
light sources of photon energies ET- Ev < hv, < Egand Ec- ET < hv < ET- Ev. respectively.
DEEP LEVEL IMPURITIES
355
Another technique for the determination of optical emission rates involves the
measurement of photocapacitance (44,,51). The reverse-biased capacitance of a p+n
junction depends on the depletion width W which exists in the lightly doped
side of the junction and can therefore be expressed as
Ae
C - --
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_
_
W
1/2
A2 e
N1
q
2(VD+ VR) J
[
27.
where A is the area of the junction, Vn is the diffusion voltage of the diode and
N1 is the total concentration of ionized impurity levels in the transition region (52).
For the purpose of demonstrating different kinds of procedures involved, the
impurity level is assumed to be an acceptor state in the upper half of the band gap.
N1 is then given by N D - nT(t) where ND is the shallow background doping. Recalling
that dnT/dt Up- Un. we obtain for the time dependence of the junction capacitance
C2(t) =
A2qe
=
2(VD+ VR)
{ND- nT(oo) - [nT(O) nT(oo )]e-tl'}.
-
28.
where, (e�+e�) -I. Depending on whether nT((0) � nT(O) the final capacitance
C(00) will be smaller, equal to, or larger than the initial capacitance C(O). Plotting
In [C2(OO)-C2(t)] versus t. the sum qf the optical emission rates is found from
the slope of the straight line. For hv < ET-Ev. e; 0 and, reveals directly e�. The
optical emission rate of holes e� is obtained by choosing the initial conditions such
that nT(O) = O. This can, for example, be achieved by shining light of photon energy
smaller than ET- Evon the diode. Illuminating the sample with photons of energy
hv > ET- Ev then results in a capacitance change, which according to equations
16 and 28, can be expressed as
=
=
�C = C2(OO)-C2(O) =
Plotting �C·,
A 2 qe
2( Vn+ VR)
eO
NTT -p-.
e�+e�
29.
1 versus hv gives the spectrum of e� apart from a factor which is
constant if NTT � ND• and can be calculated if NTT is known.
The photo capacitance technique is a differential method and thus limited by
resolution. This makes measurements close to the threshold energy unfeasible (51).
In diodes with high deep impurity concentrations, complications arise due to the
change in W. These drawbacks can often be avoided by using steady-state photo­
current measurements.
Photocurrent and photocapacitanc:e techniques using p-n junctions are only
applicable in semiconductors which can be made n-type and p-type. Furthermore,
information about deep level impurities is obtained only in the vicinity of the
junction, and the impurities may originate from or be affected by the junction
fabrication process. Some of these disadvantages can be avoided by using metal­
semiconductor barriers. Under appropriate conditions, at least part of the space­
charge region generated by the metal contact is not contaminated by the fabrication
process. Some of the measuring methods using metal-semiconductor barriers have
been described in a review paper (53). No matter whether p-n junctions or Schottky
barriers are used, the measurements involve fields of the order of 104-105 V/cm.
-
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356
GRIMMEISS
So far no field dependence of optical emission rates has been observed far away from
threshold energies. However, for many investigations such as those of excited
states a low-field method would be advantageous. This makes photoconductivity
measurements very attractive considering that, because of the photoconductivity
gain, the sensitivity of a photoconductor can be much larger than that of a diode.
However, photoconductivity measurements may be vitiated by long decay times,
which in certain cases can amount to hours (54-56), and which make the use of this
technique not only rather unreliable but lengthy. Furthermore, in most materials
the photoconductivity current is a complicated function of the optical emission (and
capture) rates, which makes a straightforward analysis rather difficult (54, 57). A
technique for measuring the spectral distribution of optical emission rates in photo­
conductors was developed by Grimmeiss & Ledebo (57). The method is based on the
fact that the occupancy of an impurity level is not changed during illumination with
photons of dilTerent energy if the photocurrent is kept constant by adjusting the
light intensity. This technique does not suffer from long decay times and is not
affected by the fact that the photocurrent sometimes has different intensity
dependences at different photon energies. The spectral distribution of eO is obtained
as the inverse of the photon flux as a function of photon energy. Excellent agreement
between photocurrent measurements using p-n junctions and those using bulk
materials was obtained.
For im pu rity centers with radiative recombination, optical emission rates can be
determined from photoluminescence excitation (PLE) and quenching (PLQ) spectra
(58-61). The high sensitivity of these techniques can provide accurate spectral
measurements of photo-ionization cross sections and they can be carried out even
at very low temperatures. They are therefore useful for a study of the details of
phonon interactions for such transitions (60, 61), in particular since measurements
are performed on bulk material where electric field-induced broadening is absent.
In cases where such phonon interactions (relaxation effects) are appreciable,
accurate experimental data to deduce the pure electronic spectrum for the optical
cross section (j0 [which can be done via a deconvolution procedure (60, 6 1 )J are
necessary. Such experiments have so far been carried out only for a few centers in
III-V compounds.
THERMAL AND OPTICAL CAPTURE RATES
Previously capture rates were usually obtained from lifetime measurements, many
of which are rather complicated and useful only under certain conditions. A list of
methods is given by Bullis in a survey of lifetime measuring techniques (62). One of
the most widely used methods is the measurement of photoconductivity decay.
These measurements often encounter difficulties such as (a) the identification of the
recombination process with a particular impurity, (b) the problem of providing
sufficiently strong light pulses with a decay time that is considerably less than the
expected lifetime decay, or (c) sufficiently small RC time constants. An excellent
investigation of copper-doped germanium has been performed by Norton &
Levinstein (25) using the decay of the photoexcited-carrier population.
Another method of evaluating capture rates is the analysis of luminescence data.
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DEEP LEVEL IMPURITIES
357
Jayson et ai, for example, used this technique for investigations of the oxygen
donor in GaP (63).
A different group of techniques is concerned with the study of capacitance
transients under reverse-bias voltage step conditions. This method was originally
suggested by Williams (64), who studied deep impurity levels in n-GaAs. A
comprehensive treatment of capacitance transients is given by Sah et al (44). Recently
Lang (65) extended this technique to a rapid (and therefore sensitive) method which
simplifies the initial characterization of many traps in an unknown sample. This high­
frequency capacitance transient thermal scanning method (DLTS) has several advan­
tages compared to other techniques such as admittance spectroscopy (66) because
DLTS is not limited to majority carri,:r traps and has a greater range of observable
trap depths. For very accurate measurement, however, the photocapacitance method
still seems to be the best (67).
The basic idea of the capacitance transient technique is that the reverse-bias
voltage VR of a Schottky barrier or one-sided junction is momentarily reduced for a
short time tsc. The depletion layer then contracts, thereby making free-charge carriers
available for recombination processes. The time available for these processes is just
equal to tsc, while the spatial location in which recombination can take place is the
region through which the depletion layer is moved. The measurements therefore
provide capture cross sections in a neutral material. The amount of charge captured
can then be determined by any of the capacitance methods. If the impurity levels
in a p+n junction are empty before the reverse bias is reduced, the concentration of
centers occupied after the reduction of the reverse bias is given by
Plotting In (1- nT/NTT) vs tsc gives cnn. Majority carrier capture rates are therefore
straightforward to measure. Minority capture is more complex since the rate depends
on the concentration of injected minority charge carriers. In certain materials,
absolute values of minority capture cross sections are therefore hard to obtain
because of the difficulty in relating the injected minority carrier concentration to the
measured injection pulse current under forward bias. In most cases, the occupancy
of impurity levels is determined from the time dependence of the capacitance value
when the cen ters in the depletion region release captured charge carriers either by
thermal or optical excitation. In more heavily compensated samples, the analysis
becomes easier when the capacitance value is kept constant during the experiment
and the reverse-bias voltage is measured instead (68, 69).
Recently capture rates of the gold-acceptor level in silicon have been measured
by single-injection SCLC (70) and current-controlled negative resistance (71)
measurements. Regardless of the methods of measurement, the capture rate obtained
is the sum of thermal and optical capture rates. If the capture process is non­
radiative, the data can be directly correlated to thermal capture rates. The situation
is somewhat more complicated for radiative recombination processes. Careful
efficiency measurements have to be performed to separate thermal capture rates
from optical capture ratcs. At prescnt there is no general techniquc available for such
detailed investigations.
30.
358
GRIMMEISS
CONCENTRATION OF DEEP LEVEL CENTERS
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In the past, many errors in published data of deep level impurities resulted from
difficulties in the determination of deep level concentrations. Recently, several new
techniques have been developed which make possible the determination of deep
center densities under almost any conditions. One of the simplest methods employs
the initial current JR(O) of a dark-current transient. In a homogeneously doped
p + n junction or a Schottky barrier on n-type material, one obtains from equation 1 8
Jk(O)
=
�
31.
q(x2-xdNTTe�.
Recalling that X2-X, = W -W o and, hence, plotting Jk(O) v s W, the product
NTTe� can be calculated from the slope of the straight line. W is varied by applying
a bias voltage and is calculated from the diode capacitance. e� is readily determined
from separate measurements according to equations 19 and 20. This method can be
used only in perfect samples with no leakage currents. Sometimes it may therefore
be advantageous to use optical investigations for the determination of NTT (53, 72).
Initially extrinsic light of photon energy somewhat smaller than the band gap is
directed upon the diode, resulting in a change of the occupancy irrespective of
whether the energy level is in the upper or the lower half of the band gap. Below
the freeze-out temperature, the electron occupancy b of the centers is given by
equation 26 and is not changed when the light is removed. Light of photon energy
less than half the band gap can refill or empty the centers completely depending
on their energy position. For an energy level in the lower half of the band gap
(e� = 0), the resulting time dependence of the transient photocurrent density is given
by
,
J'k(t) = Q(X2-X,)NTT(1-b) 1 -
( �) exp
e�
( e�t)
-
32.
(see equation 15). Integration of the photocurrent transient with respect to time
gives the charge Q flowing in the external circuit when the centers are refilled
Q
=
too J'k(t) dt ( �)
=
q 1
-
(1-b)NTT(W-Wo).
33.
From a plot of Q vs W, the density NT T is readily calculated if band D flre known.
In an n + p junction, the initial condition can be established by reverse biasing the
diode giving b O. If e� is already known, it may be advantageous to plot JR(O) vs W
in order to determine NTT. It should be noted that, for energy levels close to the
middle of the band gap, W-Wo becomes rather small and that substantial errors
arise by neglecting Wo (53). It is evident that similar techniques can be used for
concentration profiling of deep level centers by variation of the reverse bias. The
capacitance of a barrier junction is proportional to the total concentration of
ionized impurity levels according to equation 27. N TT is therefore often determined
from capacitance measurements. With the photocapacitance method, the levels are
=
359
emptied optically (44, 67). By reverse biasing a pTn junction [nr(O) = Nrr] and
illuminating with photons of energy somewhat smaller than half the band gap
(e; 0), it follows, e.g. from equation 28 for a level in the upper half of the band
gap, that
DEEP LEVEL IMPURITIES
=
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C 2 (<XJ) -C 2 (0)
A2qB
=
2(VD+ VR)
NTT·
In the capacitance transient methods, the deep centers are repetitively filled or
emptied by voltage pulses and the capacitance transients due to the thermal emission
of the trapped carriers are measured, giving NTT' Such transients may be studied
at a fixed temperature (44, 64) or by a spectrum of peaks corresponding to the various
deep levels as a function of temperature (65, 73).
ENERGY POSITION
The energy position of an impurity kvel can be characterized by the distances
between the level and the bands, Ee - Er, and Er- Ev, respectively. These energies
may be determined either optically or thermally. Previously, thermal activation
energies of deep centers were determined from the temperature dependence of Hall
constant and conductivity. Such data are often uncertain by several tens of milli­
volts. Most reliable recent data on thermal activation energies have been obtained
from the temperature dependence of thermal emission rates. These emission rates
are thermally activated and, by the principle of detailed balance, can be given by
equation 5. In this relation, c� o"�<VII> is the thermal capture rate of electrons,
(T� is their thermal capture cross section, and <vn> is the mean thermal velocity of
electrons. A similar equation holds for holes. The capture cross section (T� will in
general have some temperature dependence, characteristic of the nature of the
capture process which is dominant, its I�nergy dependence, and the range of energy
from which thermal electrons are captured. For convenience, suppose that (T� oc T -N.
Then, since Ne ex T3IZ, while <v> ex TIIZ,
=
e� = AnT2 -N exp (-AE�o/kT).
34.
The constant An contains all temperature-independent factors. A standard means of
characterizing the depth of an energy level is often to construct a plot of In e'IT2 -N
vs liT and to report the slope of the resulting straight line as the activation energy
of the energy level. However, because no higher order temperature dependence of
Ee and ET is assumed, E�o corresponds to the linearly extrapolated thermal
ionization energy at T O°K. Again, an exactly similar expression holds for holes,
=
e�
=
Ap T2 -p exp (-AE�o/kT).
35.
The sum of the two thermal ionization energies AE�o + AE�o is expected to be
equal to the linearly extrapolated band-gap energy Ego at T = OK (74). I n silicon,
for example, the temperature dependence of Eg is, however, not linear for tem­
peratures below 300K. Thus the value of Ego depends on the temperature region
in which the measurements are performed (15) (Figure 8). For temperatures above
360
GRIMMEISS
absolute zero, the energy position Ec - ET can therefore be determined only from
measurements of thermal emission rates when the temperature dependence of the
energy position is known.
l 22 � --- -- - - - -- - -- - - - - - - - - - -- - - - --- - - - - --
�------�
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,
,
,
"\
I
--- -- - - - ,- - - - - - - - - - - - - - - - - - - - - - - -- £
l20
,
�18
l16
Ego
"
"\
,
"
"\.
"\
"\.
�
"\
"\
"\
"\
,-- - - - - - - ,- - - - ,, - - - - - - - - - -- E
-_
9
"\.
"\
"\.
"\
"\
(0)
,
'�
"\
� "\
"\
"\
"\
�
�12
T.08
o
---�--�
�-
100
----
200
T [K]
300
400
Figure 8 Temperature dependence of the band gap E. in silicon (95). E�o and E.o are the
linearly extrapolated band-gap energies at T OK.
=
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DEEP LEVEL IMPURITIES
361
Different techniques have been employed to measure the temperature dependence
of thermal emission rates. Among these can he mentioned thermally stimulated
current measurements in bulk materials and p-n junctions (54, 76-81 ), thermally
stimulated capacitance measurements (73), investigations of the frequency depen­
dence of junction capacitance (82, 83), and dark-current transients (44).
If phpnon broadening is small, the most reliable information about optical
ionization energies is obtained by comparing measured spectral distributions of
photo-ionization cross sections with theory. Recently, several attempts have been
made to calculate photo-ionization cross sections aU trying different kinds of impurity
potentials. Lucovsky (34) calculated aO(hv) by studying the transition from an
impurity level to a parabolic band with the assumption of a delta-function potential
for the impurity. He also assumed all masses entering the calculation to be equal
to the effective mass in the band. In certain cases, the measured spectrum of
cross. sections for deep impurity levels corresponds fairly well with Lucovsky's
model (6, 58, 84-86). In other cases, the agreement with experimental data is worse
(50, 57, 87). A number of theoretical models have therefore been developed with
the aim of improving the agreement between calculated and experimentally observed
cross sections (36, 37, 4 1 , 88). A common feature of these models is that the impurity
atom is described with more accuracy than is offered by the use of a delta-function
potential. Moreover, the band structure is still assumed to be parabolic. The success
of these models has been rather limited (40). In a recent paper, Grimmeiss & Ledebo
showed (40) that the spectral distribution of photo-ionization cross sections is more
strongly affected by the band structure than by the impurity potential. Using a delta­
function for the impurity potential, they extended the Lucovsky model by per­
forming calculations without the assumption that all masses entering the model are
equal to the effective mass in the band and by including the influence of both
nonparabolic and nonspherical bands. The strong influence on the spectral dis­
tribution of (JO(hv) arising from the assumption that all masses entering the model
are not equal to the effective mass is readily demonstrated for the case of excitations
to a single parabolic band. The photo-ionization cross section is then given by (40)
a (hv) = RgJ(EJ
°
where
R=
(hv-Ej)3/2
.j(mTm*)
h v [h v + 1'�i(mT/m* - I)J 2 '
mH2
36.
q2h (rffeff)2
16n
.
3 £OCII
37.
rff o
Here, Ej is the optical ionization energy of the impurity level, q is a correction
if a many-electron configuration is excited, and mT is the mass of the bound particle.
mH is defined by the perturbation operator for the radiation fl
qfi ' A/mH' The
validity of this equation has been tested for transitions from an impurity level to
the conduction band in oxygen-doped GaAs (57). Because in GaAs the conduction
band is spherical and parabolic up to fairly high energies, equation 36 should
describe the photo-ionization cross section for this transition if the impurity poten­
tial can be approximated by a delta-function. Good agreement between measured
=
-
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362
GRIMMEISS
and calculated cross sections was observed (Figure 9). However, equation 36 gives
very poor agreement with experimental data in silicon for excitation processes
from the valence band into the impurity levcl even if propcr masses are inserted
(40, 50) (see Figure 1 0). This is due to the rather complicated band structure of the
valence band close to the band edge. In addition, for larger energies, transitions
to states deep in the bands where the assumption of a constant effective mass
is no longer adequate are involved. No theory for these more complicated transi­
tions which starts from first principles is presently available. However, using experi­
mental data for the construction of a spherically symmetrical "average band,"
Grimmeiss & Ledebo showed that good agreement between measured and calculated
photo-ionization cross sections of holes can be obtained for several different
impurities in silicon by taking into account not only heavy and light holes but
also the spin-orbit splitting and, at higher energies, the nonparabolicity of the
band (40).
10 0
,.
...
�'
en
+-
.,,
.ri
,
I !
, I
,
I
GaAs : O
•
I !I
c
1
10
��
--: .�..:..
..
o;o
f
,·'i1l
�
01:)
.,
I II
c:
�
A
. ..."....-:::: . -- - - B
C
•
I.
1 '
. '
,
I I
j I
i I
0,8
.
Ei
f
�
I
1,0
-�) h v (eV)
Ec
_
Ev
I
1,2
Figure 9 The photo-ionization cross section IT� of electrons for an unidentified impurity
level in oxygen-doped GaAs. Curve A corresponds to equation 36 with mT mo. Curve B
has been calculated using equation 36 assuming mH m T = m * [Lucovsky model (34)].
Curve C is the photo-ionization cross section of a Coulomb potential with mT "" m* . The
experimental points (.) were obtained by a transient photocurrent method using a p+n
junction (40).
=
=
363
As already mentioned, it has recently been argued that KL-EMT is valid for both
shallow and deep levels if the impurity is substitutional and isocoric (4, 5). For
silicon this means that phosphorus and sulfur should be describable by KL-EMT.
The potentials before dielectric screening of these two impurities were found to be
nearly equal to the bare Coulomb potentials of one or two point charges, respec­
tively. For sulfur in silicon, the spectral distribution of the photo-ionization cross
section should therefore be expected to be different from those of other deep
impurities which have been described by narrower potentials such as a delta­
function potential. Following the intentions of Anderson (39), i.e. assuming Bloch
waves for the final states and a one-valley localized impurity wave function, O'� for
the excitation of electrons from the localized level to the conduction band can be
expressed as (89)
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DEEP LEVEL IMPURITIES
o
O'n =
(To)
tffeff 2
q
3- eoncm*
128 n q 2 hEfl 2 (hv - Ei)3/ 2
�r
38.
E
N
� 10- 17
(.)
Si : Au
0,4
0,6
Ec
�� 1 O"p�1 !
0,8
>
hv (eV)
1,0
Ev
1,2
Figure 10 Measured photo-ionization cross sections of holes for the gold-donor and gold­
acceptor levels in silicon ( ) at 90K (50). The broken curves are calculated values according
to the Lucovsky model (34). The full curves are calculated cross sections using a delta­
function potential and a spherically symmetrical "average valence band" (40). By fitting the
experimental data, values of 0.35 eV and 0.61 eV are obtained for the optical threshold
energies.
0
364
GRIMMEISS
In the particular case of sulfur in silicon, it may be shown (90) that equation
38 is consistent with the correct many-valley wave function to a good approxima­
tion. Hence, plotting [(j�(hv)5y/3 as a function of photon energy hv, a straight line
is obtained and by extrapolating this line the intercept on the energy axis gives
Ei (Figure 1 1 ).
4 ------
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•
X
T = lOOK
T = 1 63 K
•
./
. )'
xl
I·
-
•
1'1.
.':
�x
/
x �x
·f
I
• •• lX .
....
I
•
I I-
•
x
.
..
:
.,. /
I �/
O L-A��·�_�
x�
x
,?
/
x
.
,
x
•
i
x.
�__�
I ____�__�I____�
__
0, 5
Figure J J
0,7
---+)
0, 9
h v ( eV)
1,1
Photo-ionization cross section O'� for electrons in the deep sulfur-donor level
linearized with respect to a Coulomb impurity potential (89).
DEEP LEVEL IMPURITIES
365
Turning to excitations of electrons from the valence band to a localized sulfur
level, calculations of 6; become more complicated, e.g. as in the case of impurities
the perturbation potentials of which are described by a delta-function potential.
Since the extremity of the band is at k 0, and the localized wave function is
constructed from regions of k space differcnt from k 0, i.e., where the conduction
band minima are, a single formula like equation 38 is no longer valid (89). At
present, no general theory to calculate cr� is available.
=
=
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RELAXATION PROCESSES
It has been believed that deep level impurities are likely to have substantial relaxa­
tion of the lattice equilibrium position near the impurity after the electronic state
is changed due to the capture or release of a charge carrier [see for example (9 1 )J .
Knowledge about such relaxation processes is of vital importance for, among other
things, the study of nonradiative recombination processes. Two main difficulties
prevent this domain from being better understood. Even in cases where phonon
interactions are negligible, theoretical models are still not available for an accurate
and unambiguous determination of binding energies. The analysis is therefore
commonly performed on the basis of experimental interpretations. In the presence of
relaxation effects, the situation is even worse. Depending on the strength of the
electron-lattice interaction in the optical transition, more or less of the oscillator
strength is transferred from the zero phonon transition to transitions into excited
vibrational states. This complicates the analysis, since the total experimental cross
section is composed of many overlapping displaced cross sections each with a
different weight. It has been shown (60, 6 1 ) that in cases where details about the
relaxation can be extracted, e.g. from low temperature luminescence, a deconvolution
of the experiment can bc performed, thus giving both accurate data about the
relaxation and the cross section expect,�d if there was no phonon interaction active.
An important feature of the cross sections thus obtained is that their threshold
energies should always add up to the band gap, independent of the relaxation.
One of the few systems that has been analyzed thoroughly is gold-doped silicon.
To illustrate the present state-of-the-art, this system is used here as a basis for
further discussion. Comparison is then made with other deep level impurities in
silicon as well as in IIl-V and II-VI compounds.
Gold causes two deep centers in silicon, one energy level at 0.55 eV below the
conduction band edge and another one at 0.35 eV above the valence band. The
latter is usually considered to be the gold-donor level and the former to be the gold­
acceptor level. To avoid confusing the reader, this notation is used throughout
this review, although van Vechten & Thurmond (92) proposed that these defects
are not simple substitutional defects but complexes of the impurity atom with
vacancies. Relaxation of the gold impurity levels in silicon can he assumed to be
small if the thermally and optically determined experimental energies E, - ET and
ET - Ev are equal, and the sum of both energies adds up to the band gap EM "
Previously published data of thermal activation energies for the gold-acceptor level
suggested that the sum of E, - ET and ET- Ev may deviate from Eg by as much as
366
GRIMMEISS
60 meV (see Table 1 ) (45, 93, 94). In silicon, however, one expects that the deviation
should be small. Parillo & Johnson presented therefore a new interpretation (74)
by assuming that the gold-acceptor level at T OK is 0. 1 eV higher than previously
reported and that the level shifts proportionally with E9 when the temperature is
changed. This resulted in thermal emission rates which disagreed with earlier data
(45, 94). Engstrom & Grimmeiss remeasured the thermal emission rates of the gold­
acceptor level in silicon as a function of temperature (75) and obtained thermal
activation energies of dE�o 0.553 eV and dE�o 0.641 eV by taking into account
the temperature dependence of the capture cross sections. The sum of these two
energies differs by about 1 meV from the linearly extrapolated band-gap energy
E�o at T = OK, which is 1 . 195 eV in the temperature range 200-300K (95), where
the measurements were performed (see Figure 8). Similar small deviations from E.
have been obtained by Wong & Penchina (96) for the gold-donor level by a com­
bined analysis of photoconductivity and thermal emission rate measurements.
These results can be confirmed by a similar analysis of optical activation energies.
By comparing measured and calculated spectral distributions of photo-ionization
cross sections, rather accurate values of the optical activation energies dE� and
dE� are obtained for both the gold-donor and gold-acceptor levels (40, 50, 96).
The sum 1: dE� + dE� has in this case to be compared with the band-gap energy
at the temperature at which the measurements were performed (see Table 2).
=
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=
=
=
Again � differs very little from Eg• For comparison, results obtained for other
impurities are also included in Table 2. The deviation of 1: from Ey is within
the experimental error for sulfur (89, 97, 98), platinum (99), and cobalt (100- 1 02)
in silicon, the first electron of the isolated oxygen donor in GaP (67, 1 03-1 06),
and two unidentified impurity levels in GaAs (57). For these impurities, one might
therefore expect that relaxation effects are small. It is rather surprising that 1:
for the manganese center in ZnSe differs so little from E9 considering the well­
known ionicity of the II-VI compounds and the results obtained with GaN where
1: is 1 1 0- 1 50 meV larger than Eg.
Table 1
Published data on thermal activation energies for the gold acceptor in silicon.
Ego
Temperature (E,- ET)
range (K) + (ET - Ev) (meV)
E, - ET
ET - E v
(meV)
(meV)
540 ± 20
620 ± 20
200-330
547 ± 2
589 ± 2
200-330
545 ± 1 .4
588 ± 1.3
200-300
490 ± 20
720 ± 10
2 10-370
553
641
200-300
Sou rce
Method
1 195 Coll ins et al (93) Hall efftct and
resistivity
1 1 36 ± 4
1 195 Sah et al (94)
Dark-current
transients
1 195 Tasch & Sah
Photocurrent
1 1 33 ± 3
transient
(45)
1210 ± 30 � 1 206 Parillo &
Dark-current
Johnson (74)
transients
1 194
1 195 Engstrom &
Dark-current
Grimmeiss (75) transients
1 164 ± 40
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DEEP LEVEL IMPURITIES
367
A further confirmation for the absence of larger relaxation effects in silicon and
in other semiconductors is obtained by comparing the values of the energy position
obtained from optical measurements with those obtained from thermal measure­
ments. As already mentioned, thermal activation energies for temperatures above
OK usually cannot be obtained directly. However, they can be calculated from
experimental data if the temperature dependence of the energy position is known.
For donors described by KL-EMT, the wave functions of captured electrons can be
expanded in terms of Bloch functions for electrons in the conduction band. This
implies that donor energy levels should be pinned to the conduction band and
correspondingly that acceptor energy levels should be pinned to the valence band
when the temperature is changed. This picture is in agreement with experimental
results obtained for substitutional sulfur in silicon (89), which introduces deep donor
states and can be described by EMT. Although there has been much theoretical
effort recently devoted to computing the energy levels of deep centers (4, 5, 1 1, 41,
107, 108), there is no detailed theory to describe the temperature dependence of
these centers. In the past, it has often been assumed [see for example (74, 92, 102,
1 09)] that deep impurity states vary their binding energies in proportion to the
variation of the band gap with temperature. However, for many deep level im­
purities the situation is less straightforward.
Information about the temperature dependence of the energy position is obtained
from measurements of the spectral distribution of the photo-ionization cross section
at different temperatures. In a detailed series of experiments it has been shown
( 1 10) that in the temperature range b(:tween 90 and 200K there is no systematic
shift in the spectrum of the photo-excitation of electrons from the gold-acceptor
level to the conduction band. This is despite the fact that Ey in silicon varies by
Table 2
Optical threshold energies deduced from the spectral distributions of photoionization cross sections
Impurity
Si : Au
Si : S
Si : Pt
Si : Co
GaP : 0,
GaAs : O
GaN : Zn
GaN : Li
ZnSe : Mn
T
(K)
Eg
(eV)
M�
(eV)
90 1 . 1 66 0.555
90 1 . 1 66 0.75-0.83
223 1.144 0.61 2
100 1 . 1 65 0.36
80 1 . 1 67 0.250
80 1 . 1 67 0.82
295 1 . 126 0.535
100 1 .1 65 0.781
295 2.267 0.870
100 1 .508 0.46
100 1 .508 1 .03
78 3.50 3.1 7
78 3.50 2.86
293 2.71 2.065
M�
(eV)
L=
M� + M�
0.610 1 . 1 65
0.345 1 .095-1 . 1 75
0.536 1 . 148
0.81 1.17
0.908 1. 158
0.355 1.175
0.600 1 . 1 35
0.384 1. 165
1 .400 2.270
1 .04 1 .50
0.48 1.5 1
0.48 3.65
0.75 3.61
0.68 2.745
Eg - L
(eV)
Source
0.001
50
- 0.009- + 0.07 50
89, 97
- 0.004
- 0.005
89, 97, 98
99
0.009
99
- 0.008
101, 1 02
- 0.009
100
0 000
103, 104, 105
- 0.003
57
0.008
57
- 0.002
59
- 0. 150
59
- 0. 1 10
- 0.035
72
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368
GRIMMEISS
about 1 4 meV in this range, which is far beyond thc cxperimental uncertainty.
Moreover, the spectrum of (J'� shifts to smaller energies with increasing temperature
in exactly the same manner as Eg• This shows that the gold-acceptor level in
silicon is fixed in energy relative to the conduction band in the temperature region
between 90 and 242K. Unlikc previous assumptions and indirect arguments (74,
92, 1 0 1 ), this dircctly dcmonstrated for the first time the existence of a deep center
fixed in energy relative to one of the energy bands. Later it was also shown that
the gold-donor level in silicon is pinned to the conduction band (96) (see Table 3).
The absence of any fixed rule for the temperature dependence of dccp levels is
further demonstrated by investigations in cobalt-doped silicon. Here the cobalt
acceptor is fixed in energy relative to the conduction band edge ( 1 0 1 ) and the
cobalt donor is pinned to the valence band ( 1 00). Experimental results obtained
with GaP indicate that the copper-acceptor level is probably fixed to the conduction
band (58) and the first electron of the isolated oxygen-donor level is pinned to the
valence band ( 103, 1 1 1 ). On the other hand, the zinc- and lithium-acceptor levels in
GaN are probably fixed in energy relative to the valence band (59, 1 12). In none
of these cases is the relative position of the energy level unchanged, as was
previously assumed.
Knowing that the gold-acceptor level in silicon is pinned to the conduction
band, the thermal activation energies can be calculated for all temperatures using
results obtained from the measurements of thermal emission rates. Considering
that the temperature dependence of the band-gap energy in silicon (95) is described
by the semi-empirical expression proposed by Varshni ( 1 1 3) to within about 1 meV,
the temperature dependence of ET - Ev can be expressed as (75)
AE�(T) = AE�(O) - o a2/(1 1 08 + T),
39.
where CI. = 7 .02 X 1 0 - 4 eV/K. The (not extrapolated) thermal activation energy
AE�(O) is calculated at T 0 by inserting equation 39 in equation 35 and taking
P
=
=
2 (75). One then obtains
AE�(O)
Table 3
=
kT ln (A�/e�),
40.
Temperature dependence of impurity levels
Temperature
Type of
Impurity
level
Pinned to
range (K)
Source
Si : S
Si : Au
Donor
Acceptor
Donor
Acceptor
Donor
Acceptor
Donor
Acceptor
Acceptor
Conduction band
Conduction band
Conduction band
Conduction band
Valence band
Conduction band
Valence band
Valence band
Valence band
Valence band
77-223
90-242
20-195
89
1 10
96
101
101
58
1 03, 1 1 1
59, 1 12
59
72
Si : Co
GaP : Cu
GaP : O,
GaN : Zn
GaN : Li
ZnSe : Mn
100-247
10- 1 70
4-410
4-300
78-300
90-293
DEEP LEVEL IMPURITIES
/l,E" and /l,E', respectively, for the gold-acceptor level in silicon (75)
Table 4
T
(K)
o
90
369
Comparison between optical and (non-extrapolated) thermal activation energies
M�
M;�
M� + M�
M: (42)
M� (42)
/l,E: + M�
(meY)
(meY)
(meY)
(meY)
(meY)
(meY)
553
553
613
609
1 1 66
1 1 62
555
610
1 165
Eg (76)
(meY)
1 170
1 166
where A� Apexp [7.02 x 1 0 - 4 T2 /(1 108 + T)J. Inserting experimental data in cqua­
tion 40 gives �E�(O), and hence Ey E" for all temperatures according to equation
39. Because the gold-acceptor level is pinned to the conduction band, it is evident
that �E�(T) �E�(O) = �E�o 0.553 eY. These thermal activation energies can now
be compared with optical values. For convenience, the corresponding values of the
band-gap energy are also included in Table 4. The good agreement between the
thermal and optical ionization energies on the one hand and the sum of �En + �Ep
and the band-gap energy on the other confirms that no pronounced relaxation
effects might be expected due to excitation processes from the gold-acceptor level
in silicon.
From what has been said it should, however, not be concluded that there are no
relaxation effects connected with deep level impurities. The situation might be
entirely different in other systems. After capturing an electron, lattice relaxation
lowcrs, for example, the F center in NaCl ( 1 14) by about 3.5 eY. Messmer &
Wat kins showed ( 1 1 5) that after the nitrogen donor in diamond captures an
electron, the optical ionization energy of the electron increases by about 3.5 eV
due to a change in lattice configuration. Another example is oxygen in GaP ;
Kukimoto, Henry & Merritt ( 1 05) claim that substantial lattice relaxation takes
place after the capture of the second electron. They measured a large and tem­
perature independent cross section for the trapped electron to recombine non­
radiatively with a hole (ap2 � 2 x 1 O - 1 4 cm2 ) and consider the large relaxation as
being responsible for this large cross section due to mUlti-phonon emission (9 1 ).
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=
-
=
=
GENERAL CONSIDERATIONS
So far the photo-ionization cross section has been considered as useful only for the
determination of activation energies. However, measurements of aD might also be
valuable for other purposes. It has already been pointed out that different approxi­
mations of the impurity potential for the calculation of aD have been used. By
comparison with experimental data, information may be obtained about the actual
impurity potential which can be employed for calculations of binding energies and
capture cross sections. In addition, it is readily seen from equations 36 and 38
[see also for example (1 1 6)] that apart from the threshold energy E;, the effective
mass m*, and the dielectric constant B, no further material constants are involved.
This may be why photo-ionization cross sections in different materials differ very
little in contrast to capture cross sections. In Table 5 published values of aD for
several impurities in silicon, GaP, and ZnSe are summarized. Apart from the elec-
370
GRIMMEISS
Table 5 Absolute values of photo-ionization cross sections
Impurity
Si : Au
Si : S
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Si : Zn
S i : Pt
Si : In
GaP : O,
ZnSe : Mn
Energy
position
(eV)
K - 0.55
Ev + 0.35
Ee - 0.61
Ee - 0.36
Ev + 0.59
Ev + 0.32
Ee - 0.25
Ev + 0.36
Ev +0.16
Ee - 0.87
Ev + 0.68
(eV)
u: x 1 0 1 7
(cm 2)
0.85
0.95
1.00
0.5
2
0.7
30
30
hv
0.40
0.95
2
0.3
1 . 10
2.5
6
10
hv
(eV)
0.85
0.55
1 .00
0.90
0.59
0.60
1 . 10
0.30
1.80
0.85
u� X 1 0 1 7
(cm 2)
6
11
10
5
10
8
7
3
10
9
Source
45, 50
50
89
89
57, 87, 1 1 8
57, 87, 1 18
99
99
120, 1 2 3
104
72
tron transition oftheAu and Pt donors in silicon, all experimental values lie within one
decade.
As already mentioned, the magnitude and spectrum of the photo-ionization cross
section aO are both direct consequences of the bound-charge distribution for the
ground state of a deep level impurity. For substitutional impurities which are from
the same row of the periodic table as the host atom, the impurity potentials may
be constructed from basic crystal and atomic properties ( 1 1 ) in agreement with
KL-EMT. It has been suggested that they are very nearly equal to the bare Coulomb
potential of point charges. For non-isocoric impurities, a pseudo-impurity theory
(4) can be used to construct impurity pseudopotentials from first-principles proper­
ties. In certain cases, the deviation from a Coulomb potential might be so sub­
stantial that the impurity potential can be approximated by a strongly localized
potential such as a delta-function potential. Calculations using pseudo potentials are
less straightforward than those employing a delta-function potential, which is why the
latter often are used for simpler calculations. On the other hand, the use of delta­
function potentials (or, in general, model potentials) yields very little information
apart from numbers for binding energies and cross sections. In particular, no
information is obtained on the mechanisms operative in dctcrmining the photo­
ionization cross sections. The measured spectrum of a� for the isocoric sulfur
impurity in silicon is well understood on the basis of a Coulomb potential (Figure
1 1). Similar good agreement is observed for a� in zinc-doped silicon ( 1 1 7) when
the experimental results published by Zavadskii & Kornilov ( 1 1 8) and Herman &
Sah (87) are employed [see also ( 1 19)J and a spherically symmetrical "average
band" is used as in (40) (see Figure 1 2). There are reasons to believe that the
data published by Messenger & Blakemore ( 1 20) on indium in silicon also agree
with a Coulomb potential. However, the data presented in Figure 9 for an un­
identified deep impurity in GaAs can hardly be explained by a bare Coulomb
potential although the conduction band minimum in GaAs is spherical and no
371
DEEP LEVEL IMPURITIES
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uncertainty arises as to the value of effective mass to be used. The excellent
agreement between experiment and calculations using a delta-function indicates
that the perturbation potential is probably screened substantially. Experimental
data obtained in platinum-doped silicon (121) are also found to be in good agree-
-
NE
0
C
0 0.
1
10- 16
.,..,
/��
�
I�
0
......
r--.
0
p
�o
� . . .......
(
10- 17
f
U l
p
0
· S i : Zn
4
•
•
•
10- 19
0.2
0.5 9 eV
t
*
0.4
0.6
I
0. 8
0.32 eV
+
iii
I
1.0
1.2
hv ( e V )
Photo-ionization cross sections of Si : Zn. Full circles are data as measured by
Herman & Sah (87) and the open circles are points from Zavadskii & Kornilov (1 18).
The full curves are calculated cross sections (1 1 7) using a Coulomb potential and a
spherically symmetrical "average valence band" (4Q). By fitting the experimental data, values
of 0.32 eV and 0.59 eV for the optical threshold energies are obtained.
Figure 1 2
372
GRIMMEISS
ment with calculated values using a delta-function potential. Fair agreement is also
found for the copper acceptor in GaP (58) and the manganese impurity in ZnSe
(72). It should, however, be kept in mind that the good agreement is caused by
the theoretical model and might therefore be accidental. In other cases (59, 103),
such a comparison cannot be performed successfully because of poor knowledge
about details in the band structure.
Alt hough no coherent picture has yet emerged, one is tempted to raise the
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Table 6
Electron and hole capture rates c� and c:" respectively
c�(cm3 /sec)
System
Si : Au
Acceptor
2 x 10- 8
1 X 1 0- 7
( 1 . 3 x 1 0 - 9 - 5 x 1 O � 8)
6 x 10-7
2.4 X 1 0 - 7
Donor
Si : S
Shallow donor
Deep donor
Si : Zn
Shallow acceptor
Deep acceptor
Si : Fe
Si : W
Shallow acceptor
Middle m:ceptor
Deep acceptor
Si : Cu
Donor
Si : Pt
Acceptor
Donor
Si : ln
Si : Pd
Acceptor
Donor
Si : Rh
Shallow acceptor
Deep acceptor
GaP : O
First el.-state
Sec. el.-state
GaAs : O
ZnSe : Mn
c�(cm3 /sec)
� 1O - 8
� 1O-6
10-7
T (K)
300
Activation
energy
(eV)
2 x 10- 15
(2 x 1 0 - 1 5 _ 2 x 1 0 - 9)
1 x 10-8
10-6
3.2 x 1 0 - 8
2.4 x 1 0 - 9
3.6 x 1 0 - 8
4 x 10-8
2 x 10- 1 1
3 x 10-12
5 x 10-9
4 x l O - 14
2 X 10- 7
10-7
125
0.29
0. 5 5
47
47
=
0. 3 5
0.62
126
127
=
0.40
1 28
=
0.22
0.30
0.37
1 29
1 29
1 29
=
0.24
1 30
0.25
0.35
69
0.22
030
69
69
0.3 1
0.53
69
69
0.9
0.S9
0.75
0.66
67
67
132
72
E T - E"
=
80
SO
E, - ET
E,- ET
=
SO-200 ET - E"
90-200 E T - E"
E, - ET
E, - ET
E, - ET
5 x 10-8
0. 3 5
77
0.6 - 1 x 1 0 - 9
6 x 10- 10
2.5 x 10- 1 1
2 x 10-7
75, 1 24
=
3 X 1 0 - 9 250-300 ET - E"
> 10-6
0.55
E, - ET
5 x 10-9
10-9
( 1 0 - 1 3 - 1 0 - 6)
1.5 x 10-8
78- 1 20 ET - E"
=
=
=
=
90
Ec - ET
=
90
ET - E"
=
77
95
1 35
E, - ET
ET - E"
1 20
220
E,- ET
E, - ET
300
300
300
250
E, - ET
E, - ET
E, - ET
ET - E"
Reference
=
=
=
=
=
=
=
=
69
70, 1 3 1
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DEEP LEVEL IMPURITIES
373
question as to whether proper impurity potentials such as are used for calculations
of photo-ionization cross sections might give further information on other proper­
ties of the impurity. It seems plausible that the capture probability of free carriers
should be smaller for an impurity with a strongly localized impurity potential than
for an impurity with a delocalized potential. This would imply that isocoric im­
purities have larger capture rates than other impurities. Of course, the capture rate
is determined not only by the impurity potential but certainly also by other
features such as lattice relaxation, etc. However, the agreement found might never­
theless give some insight. Table 6 summarizes capture rates in Si, GaP, GaAs,
and ZnSe. The values cover more than 9 orders of magnitude. It is commonly
assumed that the capture probability increases with decreasing binding energy.
Moreover, it is known that the capture rate of free carriers is larger for charged
impurities than for neutral impurities All this can be understood within the frame
of the above simple picture. Looking now for those impurity levels which have
the largest capture rate (see Table 6), it is readily seen that Cn of the deep sulfur­
donor level and cp of the indium- and deep zinc-acceptor levels, respectively, are
the only impurity levels with values larger than 10-6 cm 3/sec. Only in these three
cases is the spectrum of the photo-ionization cross section best understood on
the basis of a Coulomb potential. That doubly ionized impurity levels have larger
capture rates than singly charged impurity states can bc understood by virtue of
the different charges. It should, however, be kept in mind that some of the values
presented in Table 6 might be erroneous due to uncertainties in determining
impurity concentrations and should therefore be remeasured using one of the new
techniques described earlier.
.
CONCLUDING REMARKS
At thc present time, there are still only a few relevant theoretical results and
particular experimental data which can be correlated to these calculations. In
many aspects we have therefore had to limit ourselves to qualitative considera­
tions. Many experiments fail to distinguish between mechanisms of quite different
physical origin due to the lack of proper theoretical models. A significant con­
tribution has been made by Pant elides (4, 5, 1 1, 1 2) by presenting a comprehensive
theory for impurity states and by analyzing donor states in silicon in detail. Excita­
tions from these states to the conduction band are now better understood. What
is needed next are detailed calculations of the more complicated transitions between
localized states and the valence band. In many indirect semiconductors, there is not
sufficient information about the energy dependence of the density of states. This
makes calculations of photo-ionization cross sections unreliablc. Rccently, an intcr­
esting attempt has been made (122) to deduce impurity wave functions from
empirical data on photo-ionization pro perties. If this approach proves to be more
generally valid, there is hope that theoretical models can be provided. In addition,
very little is known about the excited states of deep impurity centers. This makes
the analysis of the few available experimental data still more difficult. At present
nonradiative capture processes are in general very inadequately understood although
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374
GRIMMElSS
interesting progress in explaining some of their features (21 ) has been made. Quanti­
tative information about capture cross sections is therefore in many cases available
only from experimental data. Another feature of deep level impurities without any
general theory is the temperature dependence of non-isocoric impurity states. Such a
theory would contribute to a better understanding of both thermal emission and
recombination processes.
In the present situation, there is no sense in continuing general considerations
and expressing belief or disbelief in certain features of deep level impurities. How­
ever, it seems to be quite clear that a better understanding of these centers will
need to be established within the next few years. The significance in both basic
research and technical applications is too large for the field to remain without
a more comprehensive framework of understanding.
ACKNOWLEDGMENTS
The author would like to thank the Swedish Natural Science Research Council
and the Swedish Board for Technical Development for their financial support. He
gratefully acknowledges many stimulating discussions with J. W. Allen, J. B. Blake­
more, L-A . Ledebo, C. Ovren, and S. Pantelides. The critical reading' of the manu­
script by L. Hedin and B. Monemar is much appreciated.
Literature Cited
1. Luttinger, J. M., Kohn, W. 1955. Phys.
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ANNUAL
REVIEWS
Further
Quick links to online content
CONTENTS
PREFATORY CHAPTER
POINT DEFECTS AND THEIR INTERACTION, Carl Wagner
Annu. Rev. Mater. Sci. 1977.7:341-376. Downloaded from www.annualreviews.org
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STRUCTURE
DEFECT CHEMISTRY IN CRYSTALLINE SOLIDS, F. A. Kroger
449
DEEP LEVEL IMPURITIES IN SEMICONDUCTORS, H. G. Grimmeiss
341
STRucTuRAL ASPECTS OF ONE-DIMENSIONAL CONDUCTORS, Galen D. Stucky,
301
Arthur J. Schultz, and Jack M. Williams
PREPARATION, PROCESSING, AND STRUCTURAL CHANGES
STRUCTURAL TRANSFORMATIONS DURING AGING OF METAL ALLOYS, Yu. D.
���
�
HIGH RATE THICK FILM GROWTH, John A. Thornton
239
METAL FORMING: THE ApPLICATION OF LIMIT ANALYSIS, Betzalel Avitzur
261
PROPERTIES AND PHENOMENA
KINETICS AND MECHANISMS OF GAs-METAL INTERACTIONS, H. J. Grabke and
155
G. Harz
EROSION, Carolyn M. Preece and Norman H. Macmillan
95
REVERSIBLE TEMPER EMBRITTLEMENT, D. F. Stein
123
ACOUSTIC EMISSION IN BRITTLE MATERIALS, A. G. Evans and M. Linzer
179
CAPACITANCE TRANSIENT SPECTROSCOPY, G. L. Miller, D.
V. Lang, and
L. C. Kimerling
HOT CORROSION OF HIGH-TEMPERATURE ALLOYS, John Stringer
377
477
FUNDAMENTAL OPTICAL PHENOMENA IN INFRARED WINDOW MATERIALS,
Bernard Bendow
23
SPECIAL MATERIALS
DENTAL AMALGAM, Svein Espevik
55
TRANSPARENT CONDUCTING COATINGS, G. Haacke
73
INDEXES
AUTHOR INDEX
511
SUBJECT INDEX
524
CUMULATIVE INDEX OF CONTRIBUTING AUTHORS, VOLUMES 2-6
534
CUMULATIVE INDEX OF CHAPTER TITLES, VOLUMES 2-6
535