A new model for the absorption coefficient of narrow-gap (Hg,Cd)Te
that simultaneously considers band tails and band filling
K.
H.
Herrrnann,
M.
Happ,
H.
Kissel, K.-P. MOllrnann, and J.
W:.
Tornrn.
Fachbereich Physik der Humboldt-Universitiit
zu
Berlin, Institut
for
Festkiirperphyslk.
Invalidenstrasse
JlO,
O-Jl40
Berlin, Germany
C.
R.
Becker,
M.
M. Kraus,
S.
Yuan,a) and G. Landwehr
Physikalisches Institut der Universitiit Wiirzburg. MBE-Labor,
Am
Hubland,
W-8700
Wiirzburg. Germany
(Received
13
August
1992;
accepted for publication
14
December 1992)
A semiempirical model
is
presented that correlates the broadening
of
the absorption
~ge
with
both transitions below the energy gap and with transitions by the
Kane
band model. This model
correctly fits both the absorption
and
luminescence spectra
of
narrow-gap
(Hg,Cd)~e
samples
that
have been grown by the traveling heater method as well as by molecular-beam epltaxy. The
accuracy
of
the band-gap determination
is
enhanced by this model.
I.
INTRODUCTION
Narrow-gap (Hg,Cd)Te is
an
important material used
in infrared detectors
that
has a number
of
physical prop-
erties
of
considerable interest: The small effective mass
of
the electrons
[mn'
mlh~O.OO8mo<O
.
44mo~mhh
for x=O.2
at
T =
77
K
(Ref
. 1) results in rather large quantum-size
effects in structures with reduced dimensionality and with
pronounced Landau quantization in a magnetic field. Fur-
thermore, the mixed crystal system is seen as a model sub-
stance for studying alloy disorder effects as a function
of
the energy gap.
The absorption coefficient
of
(Hg,Cd)Te
can be em-
pirically described by different relationships for different
spectral regions. Above the energy gap
Eg
where the Kane
band structure is valid, the absorption coefficient follows a
modified square-root law,2 and
at
energies near
Eg
it ex-
hibits an exponential Urbach tail. Both regions have been
investigated in numerous experimental studies, and a large
amount
of
empirical data already exists (see, e.g., Ref. 3).
The energy dependence
of
the absorption coefficient
of
the Urbach tail may be described by
a(w)
=a(wo)
exp[
(w-wo}/WoJ.
(1)
Here
Wo
is
a parameter that
is
an indication
of
the width
of
the tail and
Wo
a parameter that defines the transition
from the modified square-root law to an exponential de-
pendence. Both
Wo
and
Wo
are experimental parameters
that
can be determined from absorption measurements ei-
ther directly from transmission spectra
or
indirectly from
photoconductivity
or
the spectral 'response
of
a photodi-
ode;
see,
e.g., Ref. 4.
The mechanisms
that
contribute to broadening have
been analyzed in Refs. 4 and 5. An analysis
of
the temper-
ature dependence
of
Wo
resulted in the following relation-
ship:
(2)
a)
On
leave from Shanghai Institute
of
Technical Physics, Academia Sin-
ica, 420
Zhong Shan
Bei
Yi
Road, Shanghai, People's Republic
of
China.
E which will be referred
to
as "permanent broadening"
r{maining
at
zero temperature,
is
independent
of
temper-
ature. Since the same broadening parameter
Wo
is obtained
in experiments with bipolar photogeneration, we proposed
earlier the hypothesis
6
that the exponential tail is caused by
states
that
have been shifted from the valence
and
conduc-
tion bands to energies within the gap. The composition
dependence
of
the broadening Wo(x)
of
high-quality crys-
tals grown by the traveling heater method
(THM)
near
thermodynamic equilibrium correlates well with the disor-
der
function W
o
-x(1-x).7
This leads us to
the
hypoth-
esis
that
E is due
to
alloy disorder. In contrast to Fuchs
and
Koidl:S charge-carrier localization due to fluctuating
band edges will not be considered here.
The temperature-dependent contribution to
Wo
is
probably due to more than one mechanism, judging by the
individual sample results.
For
example, experimental
evi-
dence has been given by Herrmann and co-workers
7
that
acceptor levels contribute to the temperature dependence
of
Wo.
In
this article a quantitative model for optical tran-
sitions in narrow-gap
(Hg,Cd)Te
is presented, which de-
scribes in a uniform manner normal optical absorption
as
well as nonequilibrium phenomena such as luminescence
and
pumped absorption. This required a description of
band filling in the spectral region where the Urbach tail
is
present.
In
addition, this model accurately describes the
corresponding spectra
of
samples with different disorder
structure
or
other parameters as judged by their exponen-
tial Urbach tail. Thus, an investigation
of
the physical na-
ture
of
the energy states which cause
the
Urbach tail
is
possible.
11.
EMPIRICAL DATA AND SEMIEMPIRICAL MODEL
In
the following analysis, optical absorption near the
interband edge is attributed to band-to-band transitions
that
conserve k.
2
Excitonic contributions can be com-
pletely neglected according to arguments concerning mag-
netophotoluminescence
9
and temperature-dependent
d
.
7
broa enmg.
3486
J. Appl. Phys. 73 (7), 1 April 1993
0021-8979/93/073486-07$06.00
@)
1993 American Institute
of
Physics
3486
Up
to now the separation
of
the exponential Urbach
component, Eq.
(1),
from the square-root dependence, has
been carried out by requiring that the absorption coefficient
is
continuous
at
the transition but without considering
band filling.
8
This
is
reflected in the value
of
the absorption
coefficient at the transition between exponential and
square-root dependenceIO compared to the value usually
assumed:
a = 1000 cm
-I.
This implies, apart from correc-
tions for k-dependent matrix elements,
that
the tail can be
described as due to a modified density
of
states at E <
Eg
in
the disordered crystal.
However, in order to establish a model
of
the spectral
region below
Eg
for doped samples
or
for a nonequilibrium
situation
we
have to consider how band filling will influ-
ence interband transitions that involve states in the tail.
If
we
assume that these tail states originated in one
of
the
bands, then
we
should use a continuous extraPolation
of
the real absorption coefficient [absorption coefficient mul-
tiplied by a band-filling factor
(BFF)]
for all kinds
oftran-
sitions. Because this band-filling factor depends on the en-
ergy at the k value where these transitions take place, a
suitable average over light and heavy holes must
be
used.
As a first approximation the band-filling factor for heavy-
hole to conduction-band transitions will be used:
(3)
Equation (3) implies that the tail states above the valence-
band edge are mainly due to the heavy hole. This
is
in
accordance with the large ratio
of
the density-of-state
masses and with the observation
that
for HgTe/CdTe het-
erostructures the valence-band offset
is
only about 20%
of
the gap difference.
The assumption that the
U rbach region can be de-
scribed by Eq. (1) multiplied by the band-filling factor
given by Eq. (3)
is
confirmed
by
pumped absorption
ex-
periments when the excitation
is
resonant with the tail
states, as described in
Sec.
IV
C.
The spectral dependence
of
the absorption coefficient
well above
Eg
is
described by the Kane band structure as
outlined by Anderson.
2
,1I
We applied his formula to the
nonequilibrium case (existence
of
quasi-Fermi levels), i.e.,
to luminescence and pumped absorption experiments, in a
previous publication.
7
Here we shall use similar notation as
in Refs. 7 and
11.
The main assumption
of
these considerations
is
that
permanent broadening
is
caused by real-space fluctuations
of the band edges due to alloy disorder. The resulting
po-
tential valleys are considered to be macroscopic crystalline
regions when compared to the spatial expansion
of
the
trapped carrier wave functions.
The
combined density
of
states
of
an inhomogeneous crystalline region, as defined
above, should be constant for various degrees
of
alloy dis-
order resulting from different growth methods
of
the sam-
ples.
The procedure used to
fit
the optical spectra
is
very
simple. The following input parameters are used: the
en-
ergy gap
Ego
effective charge-carrier temperature T
eo
tail
parameter
Wo>
and minority- and majority-carrier concen-
trations which define the positions
of
the quasi-Fermi lev-
3487
J. Appl. Phys., Vol. 73, No.
7,
1 April 1993
a(1lw.)CII
.....
..
...........
.
....
...
.
...
..
...
.
FIG
.
1.
Schematic drawing
of
the absorption coefficient vs photon energy
Iim.
Wo=O (absorption edge without broadening) (solid line);
Wf,1)
(J¥61) >
J¥62)
>
0)
(dotted line);
J¥62)
(dashed line).
e1s.
Both
liw
o
(the transition point between the exponential
and square-root regions) and
a(liw
o
)
were determined by
fulfilling the demand for a continuous transition between
the exponential and square-root regions when in
equilib-
rium or, in other words, without exitation.
Employing this model, one finds that larger tail
ener-
gies
Wo
are accompanied by larger spectral distances be-
tween
Eg
and
liw
o
and higher values
of
a(lico
o
),
as
is
al-
ready empirically known (see Fig.
1).
Figure 2 displays a
calculation applying our model with relevant sample pa-
rameters. Experimental data including
Wo
and
a(lim
o
) for
a large variety
of
samples published by several authors are
2000
15
1800
"-"~
,
12
..
~
.
..
,"
,
,>
,/'
. '
.
/.
,
>
':"
1200
•••
;;.I'.
E
..
B
•
~
.....
/
/
,
,
-+
E
/
/
--
/
.'
.
la
0
,/
W
3
800
.,.
8
I
.s::.
/
0
"-
/
3
~
/.
.s::.
400
3
/
o -------
.L
_______
-'
______
--'--
____
-'-
______
0
o 5
10
15
20 25
Tail
energy
Wo /meV
FIG
.
2.
Value
of
the absorption coefficient
at
the
energy
Iimo
where the
Urbach tail meets the modified square-root law according to
our
fit
pro-
cedure (left-hand-side scale) vs tailing energy
Wo
according to Eq.
(I)
.
The
right-hand-side axis indicates the difference between
Iimo
and
the
energy gap
Elf"
The
values
of
the parameters used are:
x=0.27.
T=20
K.
E
g
= 181.35 meV. Q
p
=20
meV.
Q.=
- (EG+Qp) (notations according to
Refs. 7
and
ll)
.
I'
Herrmann
et
al.
3487
TABLE
I.
Comparision
of
the measured and the calculated absorption
coefficients
at
the transition point between the exponential and the square-
root regions.
w o
-E
g (meV)
a(w
o)
(cm
-
I)
T
Eg Wo
x
(K)
(meV) (meV)
Meas. Calc. Meas.
ealc.
Ref.
0.140 300 70
24.0
10.6 12.0 2000 698
12
0.222 80
119
2.9 2.0 1.4 480 508
13
0.230 80 132 4.4
1.5
2.2 1040 656 4
0.290
85
228 4.2
2.3
2.1
840
843
3
0.290 245 264 4.0 10.4
2.0
1120
678 6
0.320
300
317 19.0
10
.6 9.8 2030 1577
9
0.344 300 350 8.3 9.7 4.2 1060 1078
14
0.480 80
531
9.4
6.0 4.8
1000 1994 6
compared with values calculated from
our
model in Table
1.
Note the good agreement between the two sets
of
data.
Ill.
EXPERIMENT
The samples investigated were grown by the traveling
heater method
(THM)
and molecular-beam epitaxy
(MBE), as described in Refs.
15
and
16.
Both kinds
of
samples were characterized by a number
of
standard meth-
ods which include transport measurements, photoconduc-
tive decay, and chemical analysis. These experiments pro-
vide the data necessary
to
determine the input parameters
such as the Fermi level and quasi-Fermi levels in the ab-
sence
of
excitation (absorption and photoconductivity) as
well as with excitation (luminescence). The main tech-
niques employed· in this study are as follows.
(i)
Steady-state photoconductivity
(PC):
The form
or
structure
of
the absorption edge when
ad
< 1 can be ob-
tained from the spectral response
of
the interband genera-
tion
of
mobile charge carriers. Here d is the sample thick-
ness. The spectral response measurements were conducted
with a globar, double-beam grating monochromator and
an optical cryostat. Because
of
the very small monochro-
matic photon
flux
of
10
13
_10
14
cm-
2
S-
I,
a very small de-
viation from equilibrium results.
(ii) Transmission: Transmission measurements were
carried out with a Fourier transform spectrometer, Bruker
IFS
88,
equipped with a liquid-helium cryostat. The ab-
sorption coefficient
a
(1luJ)
was determined without consid-
eration of the spectral dependence
of
the refractive index.
(iii) Infrared photoluminescence
(PL):
The Fourier
transform spectrometer was also employed for steady-state
luminescence experiments. The excitation radiation was
provided by a cw Nd:YAG laser
(1luJ= 1.17 eV) and signal
processing was carried out with a lock-in technique. When
high excitation levels were required, a Q-switched
Nd:YAG laser
('TFWHM=80 ns, where
FWHM
denotes the
full width
at
half-maximum) was employed as an excita-
tion source, spectral discrimination and detection were ac-
complished with a conventional double monochromator
and a fast (Hg,Cd)Te photodiode; for details, see Ref.
17.
3488 J. Appl. Phys., Vol. 73, No.
7,
1 April 1993
(iv) Pumped absorption
(PA):
PA
measurements
were done with a pump and probe method. A Q-switched
CO
2
laser ('TFWHM=220 ns) was employed both as an
ex-
citation source as well as a source for the transmission
measurement.
The
116.9 meV emission line was selected
by
means
of
a grating
and
the intensity
of
this laser line was
attenuated with a set of filters, whose transmittances are
independent
of
wavelength in the
COrlaser
frequency
range. The samples were cooled in an optical
cryostat, and
a
(Hg,Cd)Te
photodiode was used
as
a detector.
IV. RESULTS AND DISCUSSION
A.
Application
to
THM-grown samples
A
PC
spectrum
of
a THM-grown sample is displayed
in Fig. 3
(a)
and a
PL
spectrum as well as a transmission
spectrum near the energy gap
of
the same sample
at
T=4.2
K are shown in Fig.
3(b).
Two values for
Wo
are given in Fig.
3(a).
The value of
3.8 meV (dashed line)
fits
the
PC
spectrum for the entire
region from
130
to
150 me V whereas the lower value
of
2.1
meV (solid line) gives a better
fit
but only for the region
between
145 and 150 meV. Using both
of
these values
along with the corresponding values
of
lluJ
o
and a(lluJ
o
),
which are given in the figure caption, the spectral depen-
dence
of
the absorption coefficient was calculated as de-
scribed in
Sec.
11.
This absorption coefficient was used
to
fit
the lumines-
cence spectrum obtained
at
higher excitation levels which
is shown on the right-hand side in Fig. 3
(b).
The dashed
line corresponds to the larger value
of
Wo
and the solid line
to the smaller value. Note the very good agreement be-
tween the measured curve and calculated curves for the
spectral regions mentioned above, i.e., the dashed line
fits
better overall and especially the low-energy region whereas
the solid line
fits
better
at
higher energies. The effective
carrier temperature
T e used in the
fit
of
the high-energy
tail
of
the
PL
spectrum was determined from a semiloga-
rithmic plot
of
the luminescence. Obviously the
cw-
luminescence spectrum with a weaker excitation [dotted
line in Fig. 3
(b)]
is due to transitions involving levels in
the energy gap and does not include interband contribu-
tions.
Figure 3
(c)
shows the absorption coefficient as deter-
mined from a transmission spectrum (squares). The
dashed line
that
falls near the experimental absorption co-
efficient
is
the result
of
Eq. (1) with
lluJ
o
and a (lluJ
o
)
as
determined from the luminescence line shape
fit
using a
value for the tail parameter
Wo
of
3.8 meV. The solid line
results from Eq.
(1)
when W
o
=2
.1 meV, which
fits
the
luminescence line shape
at
higher energies. We cannot
compare the transmission data with this line
at
higher en-
ergies because its thickness
(d=500
jlm)
does not permit
an accurate determination
of
larger values
of
the absorp-
tion coefficient. Both lines represent the absorption
coeffi-
cient without a dynamic BM shift. This is correct for trans-
mission measurements because
of
the low excitation level
compared to levels used in luminescence experiments.
. Herrmann
et
al.
3488
10000
1000
:i
100
•
.....
S!
10
C
<I
1
0,1L---~~--~----~-----L--
--
~----~
100
120
140
180
180
200
220
Photon energy t'lw/meV
,-------------------------,
3000
:i
ai
.....
':'
ii
E c
01
U
ii
.....
~
..J
~
100
Photon energy
t'I
w/meV
If
one takes into account a realistic excitation level for
luminescence, but no degeneracy, then a dynamic
Burstein-Moss
(BM)
shift must be taken into consider-
ation which shifts the dashed line
(W
o
=3.8
meV) to the
dotted one. This spectral dependence
of
the absorption co-
efficient including a dynamic BM shift describes the ab-
sorption
of
the excited material and
is
also plotted in Fig.
3 (b)
as
a solid line.
In
order to further test the consistency
of
our model
the
PC
spectrum was simulated. Assuming a diffusion
length
of
10
/Lm
in order to describe the recombination, the
absorption coefficient has a value
of
1000 cm - \
at
e - \
times the plateau value which occurs
at
about
150
meV.
An extrapolation
of
the lines in Fig. 3 (
c)
also reaches 1000
cm
-\
at
about this energy.
Therefore, our model consistently describes all three
sets
of
experimental data (PL, transmittance, and
PC).
The smaller value
of
the broadening parameter describes
the band tails due to alloy disorder, whereas the larger one
describes an envelope which contains two acceptor levels in
the
PC
spectrum
at
136
meV (probably a hydrogenlike
acceptor) and
at
143
meV (probably caused by the first
3489
J. Appl. Phys., Vol. 73, No. 7, 1 April 1993
10000
/
,
,
,
,
,
,
/
1000
,
,'.
"
.'
""
.,
':'e
./
./
. ,
100
. ,
.g
. ,
. /
~
./ '
.
,'
.,
~
'
10
•
~
,,"
.,.
"
,
..
/
...
.
1
110 115
120
125
130
135
140 145 150
Photon energy
t'I
w/meV
FIG.
3.
Optical spectra and empirical results using the fitting procedure
for a THM-grown sample
at
T=4
.2 K with
x=0.25l5
and E
g
=
146
.5
meV.
(a)
PC spectrum: W
o
=2.l
meV (solid line); W
o
=3.8
meV
(dashed line).
(b)
PL
spectra
(PL
signal in arbitrary units)
at
low (sev-
eral W cm-
2
)
and high (50 kW cm-
2
)
excitation densities are shown as·
a
dott~
line on the left-hand side and a solid line on the right-hand side,
respectively. The solid line represents a line-shape
fit
using W
o
=2.1
meV
[see Fig.
2(a))
and the dashed line a
fit
for W
o
=3.8
meV. Using these two
values
in
Eq.
(I)
results in w o=
147
.5 and 148.3
meV
as well as in
a(wo)
=503
.1 and 649.2 cm- I, respectively.
An
effective carrier temper-
ature
of
T. = 130 K was determined from the high-excitation density
spectrum on the right-hand side. A diffusion length
of
10
/Lm
(from
transport and lifetime measurement) was used
in
calculation
of
the reab-
sorption. The position
of
the quasi-Fermi levels, Q. and
Qp>
was
-10
meV. The absorption coefficient behavior includes the dynamic BM shift.
The photon energy
is
shown on the right-hand side. (c) Absorption co-
efficients
vs
photon energy. Squares denote data calculated from the trans-
mission measurement. Both straight lines represent
a(w)
according
to
Eq.
(I)
for both values
of
the parameters without dynamic BM shift.
Solid line: W
o
=2
.1 meV, a(l!mo)
=511
cm
-I
, w o
=147
.6
meV
; dashed
line: W
o
=3.8
meV,
a(wo)
=680
cm-
I,
wo=
148.4 meV. The dotted
line is a modification
of
the dashed line when one takes into account the
dynamic BM shift as is usually done for
PL measurements.
ionization level
of
the Hg vacancy). Additional levels are
seen
at
130
meV in the
PL
spectrum and
at
120
meV in
both the
PC
and
PL
spectrum.
Obviously the position
of
the quasi-Fermi levels is not
correctly defined because it was determined indirectly from
lifetime measurements and the excitation intensity versus
the steady-state rate equation. Nevertheless this
"uncer-
tainity"
cannot be misused as an additional parameter to
fit
the luminescence line shape, because the demand for a
continuous transition between both spectral regions and
the uniform application
of
the BM factor conserves the
linear character of the spontaneous emission in our model.
B.
Application
to
MBE-grown samples
The procedure as described above was also applied to
the optical spectra obtained from MBE-grown samples.
The
PL
intensities
of
these samples are comparable to
those for THM-grown samples with the exception
that
the
full width
at
half-maximum
(FWHM)
is significantly
larger. In addition the application
of
pulsed, high-intensity
..
Herrmann
et
al.
3489
1~r--------------------------------.
-
O
100
-
c
<I
10L--L--~~--~~--~~--~~~~~
210
220
230
240
250
280
270 280
HO
300 310
320
Photon energy t'I ",/meV
,7
Photon energy
1'1
",/meV
FIG.
4.
Optical spectra and empirical results using the fitting procedure
for a MBE-grown sample
at
T=
18
K with
x=0
.
328
and
E,=276.6
meV.
(a)
PC
spectrum: The dotted line indicates the tail energy Wo= 14.5
meV. (b)
PL
spectrum with an excitation density
of
several W
cm-
2
•
The
line-shape
fit
takes into account the tail energy as determined in
(a).
This
results in
1iwo=283.8
meV
and a(liwo) = 1814 cm
-I
. The solid line indi-
cates values for
a(
Iiw)
which include the dynamic BM shift.
The
effective
carrier temperature
is
48
K and a diffusion length
of
3.0
I'm
describes the
reabsorption in the comparable thin
(d=9.2I'm)
sample.
For
the quasi-
Fermi levels
we
used Q.=O
meV
and Q
p
= - 7
meV
.
excitation does not shift the broadened luminescence as
expected from charge-carrier heating. These are the main
differences to
the
behavior
of
the
THM
samples described
above. Nevertheless, we note
that
THM
samples often ex-
hibit interband luminescence for both low-
and
high-
excitation
levels
l7
•
18
and, therefore,
do
not normally exhibit
a shift as large as shown in Fig. 3
(b).
Figure
4(a)
displays the
PC
spectrum
of
an
MBE
sam-
ple
that
exhibits a tail parameter
of
14.5 meV
at
T=
18
K.
The
PL
line-shape
fit
using our model [see Fig.
4(b)]
3490 J. Appl. Phys., Vol. 73, No.
7,
1 April 1993
shows reasonable agreement
at
higher energies, where the
tail parameter was determined.
The
use
of
the tail energy
Wo= 14.5 meV in
our
model results in the dotted line in
Fig.
4(a).
For
energies below 240 meV, a residual conductivity
is
present [Fig.
4(a)]
and the
PL
[Fig.
4(b)]
cannot be de-
scribed by this value (14.5 meV) for the tail parameter.
Therefore, there is good agreement between
PL
and PC.
In
order to test the limits
of
our
model, the room-temperature
optical spectra were also evaluated (see Fig. 5).
The
energies
of
16.3
and
26.2 meV for the tail param-
eter were determined from the transmittance and
PC
[Fig.
5(a)],
respectively. These data were used in the determi-
nation
of
Wo
and a
(wo).
The
resulting curves for a
(w)
are shown in Fig.
5(a).
The
fit
of
the luminescence line
shape in Fig.
5(b)
demonstrates good agreement between
the
data
obtained by various types
of
experiments. Even
though the temperature dependence
of
Ei
T)
according to
Hansen, Schmit,
and
Casselman
19
was used, the stronger
temperature dependence
of
the
PL
peak position [com-
pared to
EiT)]
is well described by
our
model.
At
room temperature the interband character
of
the
high-energy
part
of
the luminescence is not questionable,
whereas the low-energy tail
is
influenced by optical transi-
tions which involve localized states and
do
no contribute to
PC.
The
majority
of
the low-temperature
PL
[T=
18
K,
Fig.
4(b)]
obviously occurs at energies smaller than
Er
Nevertheless, bandlike states (both electrons and holes are
mobile) also contribute to the low-temperature lumines-
cence.
c.
Pumped absorption
Figure 6 displays the square
of
the
absorption
coeffi-
cient in order to distinguish between
the
Urbach tail and
the square-root region.
a(w)
was determined from a
transmission measurement using
an
IFS
88
Fourier trans-
form spectrometer. We chose the relevant parameters for
the sample
(x=0.212,
T=77
K,
d=24
/Lm)
in such a way
that
the absorption coefficient
at
the
COr
laser photon en-
ergy
(w=
116.9 meV, indicated by
an
arrow) falls in the
low-energy tail
of
the absorption edge.
Figure 7 displays the dependence
of
the absorption
coefficient
on
quantum flux density for excitation resonant
with
the
tail using a Q-switched CO
2
laser for the same
sample parameters. The rectangles are the measured val-
ues.
For
quantum flux densities Q above
2X
102
1
cm-
2
s
-I
we obtained a decrease in the absorption coefficient,
whereas it remained constant
(a::::: 1700 cm
-I)
for Q < 2
X
10
21
cm-
2
S-I.
The
curve in Fig. 7
is
a
fit
with the
following assumptions.
(a)
Spatial distribution
of
excitation:
The
sample un-
der consideration has
an
acceptor concentration
of
9.1
X
10
15
cm-
3
•
Thus, it follows
that
the minority-carrier dif-
fusion length
is
about Ln=25
/Lm.
Because the sample
thickness
is
d = 24
/Lm
we can assume bulk excitation.
(b)
Steady-state nonequilibrium carrier concentration:
The
excitation was provided by a Q-switched CO
2
laser
with a relatively large
TFWHM
of
220 ns compared with the
. Herrmann
et
al.
3490