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The rate of radiative recombination in the nitride semiconductors
and alloys
Alexey Dmitriev a) and Alexander Oruzheinikov
Department of Low Temperature Physics, Faculty of Physics, M. V. Lomonosov Moscow State University,
Moscow, 119899, Russia
#Received 7 January 1999; accepted for publication 27 May 1999#
The radiative recombination rates of free carriers and lifetimes of free excitons have been calculated
in the wide band gap semiconductors GaN, InN, and AlN of the hexagonal wurtzite structure, and
in their solid solutions Ga x Al 1#x N, In x Al 1#x N and Ga x In 1#x N on the base of existing data on the
energy band structure and optical absorption in these materials. We determined the interband matrix
elements for the direct optical transitions between the conduction and valence bands, using the
experimental photon energy dependence of absorption coefficient near the band edge. In our
calculations we assumed that the material parameters of the solid solutions #the interband matrix
element, carrier effective masses, and so on# could be obtained by a linear interpolation between
their values in the alloy components. The temperature dependence of the energy gap was taken in
the form proposed by Varshni #Physica 34, 149 #1967##. The calculations of the radiative
recombination rates were performed in a wide range of temperature and alloy compositions.
© 1999 American Institute of Physics. #S0021­8979#99#06917­0#
I. INTRODUCTION
Nowadays, the nitride semiconductors such as GaN,
AlN, and InN attract a considerable attention due to their
outstanding physical, chemical, and mechanical properties
and also because of the recent progress in technology that
allowed to produce high quality nitride films with the help of
metal­organic chemical vapor deposition #MOCVD# and
molecular­beam epitaxy #MBE# #for a recent review, see Ref.
1#. The attractive properties of the nitrides include high heat
conductivity, hardness, chemical stability, and high lumines­
cence intensity. These wide gap semiconductors are very
promising materials for light emitting diodes and semicon­
ductor lasers operating in a wide spectral interval from ultra­
violet to green and even orange 2 because their solid solutions
may have energy gap varying from 2 eV in InN to 6.2 eV in
AlN. Important material characteristics for luminescence de­
vices are the rates of different electron­hole recombination
processes. However, not much is known at the moment about
the intensities of these processes in the nitrides. In this work,
we concentrate on the calculation of the radiative band­band
recombination rate in GaN, InN, AlN, and their binary alloys
Ga x Al 1#x N, In x Al 1#x N, and Ga x In 1#x N on the basis of the
experimental absorption data that are present in the
literature, 3 -- 10 calculated electron energy band dispersion
laws, 11 -- 14 and temperature dependence of the band gaps in
GaN, InN, and AlN. 15 -- 17 In addition we calculated lifetimes
of free excitons in the main nitride semiconductors.
II. ENERGY SPECTRA OF THE NITRIDE
SEMICONDUCTORS
We consider more common and popular hexagonal
phase of the nitrides. All of them belong to the crystal class
C 6v . Their conduction bands are nondegenerate, and their
electron states originate from atomic s functions. If one first
neglects the relatively small spin­orbit interaction, then in the
# point of the Brillouin zone the electron wave functions
transform according to # 1 , the unit representation of C 6v .
The valence band is complicated and consists of two
branches. One of them transforms according to # 1 whereas
the other is degenerate and forms the two­dimensional rep­
resentation # 6 . If spin­orbit interaction is taken into account,
# 6 further splits into two bands, # 7
v and # 9
v , and # 1 in both
conduction and valence bands turn into # 7 : # 1
c , v ”# 7
c , v #see
Refs. 18 and 19#. However, the spin­orbit splitting manifests
itself along k x and k y strongly, but faint at the # point and
along k z , z being the direction of the hexagonal axis c which
usually coincides with the normal to the film.
According to the results of Refs. 11--13, 18, 20--22, the
order of levels in the valence band of the nitrides is different:
in GaN and InN, # 9
v (# 6 ) branch lies above # 7
v (# 1 ) , whereas
in AlN it lies below # 7
v (# 1 ) #for more details, see Ref. 18#.
The symmetry of the electron wave functions in different
bands and band branches leads to the following selection
rules for the radiative transitions: for the transition from # 7
c
to # 7
v , only z component of the transition matrix element
differs from zero, 4,22 and correspondingly, the emitted pho­
ton is polarized along z axis. On the contrary, for the transi­
tion from # 7
c to # 9
v , the z component of the transition matrix
element equals zero, 22 and the polarization direction of the
emitted photon is perpendicular to z axis.
III. RADIATIVE RECOMBINATION RATE:
THE METHOD OF CALCULATION
Usually, the Shockley--van Roosbroeck formula 23 is
used for the calculation of the radiative recombination inten­
sity, which allows one to calculate the transition rate if the
a# Electronic mail: Dmitriev@lt.phys.msu.su
JOURNAL OF APPLIED PHYSICS VOLUME 86, NUMBER 6 15 SEPTEMBER 1999
3241
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spectral dependence of the absorption coefficient is known.
However, this is not the case in the nitrides where optical
absorption has been measured only in a small vicinity of the
band edge, so another method is needed that would allow to
express the recombination rate through material parameters.
By definition, the spontaneous radiative recombination
rate R s is the number of spontaneous recombinations per
second in unit volume, and is expressed by an integral:
R s ##
0
#
r s # ## #d### ## # E g
#
r s # ## #d### #, #1#
where r s is a spectral function of spontaneous recombination.
It can be obtained on the basis of quantum mechanics.
Standard quantum­mechanical calculations #similar to
those given in Refs. 24 and 25# lead to the following expres­
sion for spectral density of the transition probability with
emission of the photon with energy E k and arbitrary polar­
ization and arbitrary direction of the wave vector k:
W cv #
2e 2 ####
m 0
2 Vc 3 # 2
M 2 #N#E k ##1# ##E c #E v #E k #, #2#
where ## is the refractive index, m 0 is free electron mass,
and
M 2 #
1
4# #
##1
2
# d# k #e k# --P cv # 2 ,
where e k# is the polarization vector of the photon with the
momentum k. The sum is over two polarizations, the inte­
gration is over all photon momentum directions, and P cv is
the interband transition matrix element at the # point of the
Brillouin zone.
For spontaneous emission of a photon one has to put the
number of photons N(E k ) in the expression #2# equal to zero.
Multiplying the probability by the carrier statistical factor
f (E c )#1# f (E v )# , f being the Fermi--Dirac function f ##1
#exp#(E#F)/k B T## #1 and E c and E v the electron and hole
energies, and integrating over all states of the particles in the
conductivity and valence bands, one can obtain an expres­
sion for the spectral function r s :
r s # ## ##
2e 2 ####
m 0
2 Vc 3 # 2
M 2 # 2V
# 2# # 3
d 3 k c d 3 k v f # E c #
##1# f # E v ####k c #k v ###E c #E v ### #. #3#
For nondegenerate semiconductors the Fermi--Dirac func­
tions are reduced to
f # E c #” f c # E c ##exp # F#E c
k B T
# ,
#1# f # E v ##” f v # E v ##exp # E v #F
k B T
# .
Thus, we obtain
r s # ## ##
e 2 ####
m 0
2 # 3 c 3 # 2
M 2 # e
E v #E c
k B T ##k c #k v #
###E c #E v ### # d 3 k c d 3 k v .
First we perform the summation over one of the wave vec­
tors #namely, over k v ) and obtain
r s # ## ##
e 2 ####
m 0
2 # 3 c 3 # 2
M 2 # e
E v #E c
k B T ##E c #E v ### #d 3 k.
#4#
Then, denoting by L the coefficient in Eq. #4#
L#
e 2 ##
m 0
2
# 3 c 3 # 2
M 2
and substituting it in Eq. #1# one can obtain
R s #L#
E g
#
##d### # # e
E v #E c
k B T ##E c # k##E v # k#####d 3 k,
where the integral is taken over photon energies ###E g .
The carrier energy dispersion laws in the nitrides are in
general nonparabolic. The degree of the nonparabolicity,
however, differs in different bands. According to recent
spectrum calculations, 18,19 this effect is more prominent for
# 7
v (# 1 ) and # 7
v (# 6 ) bands, whereas # 7
c and # 9
v can be con­
sidered as parabolic at typical carrier energies of few k B T .
We will see in the Sec. V that the most important recombi­
nation channel corresponds exactly to # 7
c ”# 9
v transitions
due to high density of states in # 9
v band. So we can use an
anisotropic band energy dispersion law in the parabolic ap­
proximation for these bands:
E c #E c
0 #
# 2
2
# k x
2
m ex
#
k y
2
m ey
#
k z
2
m ez
# ,
E v #E v
0 #
# 2
2
# k x
2
m hx
#
k y
2
m hy
#
k z
2
m hz
# ,
E c #E v #E g #
# 2
2
# k x
2
# x
#
k y
2
# y
#
k z
2
# z
# ,
E g #E c
0 #E v
0 ,
where # x #(m e , x
#1 #m h x
#1 ) #1 is the reduced carrier mass in x
direction, and similarly for y and z.
After a scale transformation k a ”k a / #2# a with the
Jacobian #8# x # y # z and integration in spherical coordinates,
we find:
R s #
2#
# 3
L#8# x # y # z
# E g
#
e #
##
k B T######E g # 1/2 d### #.
This integral is reduced by elementary transformations to
two # functions: #(3/2)###/2; #(5/2)#3##/4, and thus
we obtain, finally, the following expression for the spontane­
ous radiative recombination rate:
R#
##e 2
m 0
2 c 3 # 2
M 2 # 2k B T
## 2
# 3/2
## x # y # z E g# 1#
3k B T
2E g
# e #E g /k B T .
#5#
Defining the radiative recombination coefficient B ac­
cording to the equality
R#Bnp ,
3242 J. Appl. Phys., Vol. 86, No. 6, 15 September 1999 A. Dmitriev and A. Oruzheinikov
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where n and p are carrier concentrations and using the well­
known expression for concentration of electrons and holes in
a nondegenerate semiconductor:
np#4#m e m h # 3/2 # k B T
2## 2
# 3
exp # #
E g
k B T
# ,
one comes to the following expression:
B#
##e 2
m 0
2 c 3 # 2
M 2 # 2## 2
k B T
# 3/2 1
# m • x m • y m • z # 1/2
E g # T #
# # 1#
3k B T
2E g # T #
# , #6#
where m • x #m e , x #m h x and so on. One can see from this
equation that for the calculation of the radiative recombina­
tion coefficient B it is necessary to know the matrix element
M 2 and the temperature dependence of E g .
IV. TEMPERATURE AND ALLOY COMPOSITION
DEPENDENCE OF THE ENERGY GAP
In many semiconductors, including the nitrides, the em­
pirical Varshni formula 26 approximates well the observed
temperature dependence of the gap:
E g # T ##E g # 0 ##
#T 2
T##
,
where # and # are parameters. Their values for nitride thin
films were found in Refs. 15--17:
AlN: E g # 300 K##6.026 eV,
##1.799#10 #3 eV/K, ##1462 K,
GaN: E g # 0 ##3.427 eV,
##0.939#10 #3 eV/K, ##772 K,
InN: E g # 300 K##1.970 eV,
##0.245#10 #3 eV/K, ##624 K.
There are indications #see, for example, Ref. 17 for GaN#
showing that the Varshni formula parameters depend on the
fabrication method of the nitride films of the same wurtzite
structure. In our calculation we have taken the values of
these parameters found in single­crystal films produced by
MBE. For the gap width in the binary alloys of the compo­
nents A and B, we used the expression 27
E g
(AB) # x , T ###xE g
(A) # T ###1#x #E g
(B) # T ##dx#1#x ##
with d#1.3 for Ga x Al 1#x N, d#2.6 for In x Al 1#x N and d
#0.6 for Ga x In 1#x N #the data from works 3,10,28,29 #.
V. CALCULATION OF THE INTERBAND MATRIX
ELEMENT
To find the interband transition matrix elements, we
made use of the formula for the absorption coefficient that
also contains P cv #see, for example, Ref. 30#:
#### ##b####E g ,
b#
2e 2 #8# x # y # z
# 2 m 0
2 ##cE g
1
2# #
##1
2
# d##e k# .P cv # 2 , #7#
where the integration is over all polarization directions in the
plane perpendicular to the incident photon beam, as distinct
from the integration over all photon wave vector directions
in the calculations of the spontaneous radiative recombina­
tion rate R. Extracting the coefficient b from the measured
frequency dependence of # , one can find P cv components in
the xy plane assuming that the light beam was perpendicular
to the film surface. However, this method does not allow us
to find P cv,z . This is not important for the transitions from
# 7
c to # 9
v as in InN and GaN #see selection rules above#, but
in AlN where crystal split­off # 7
v band lies higher than # 9
v ,
this may cause problems. We still calculated B for AlN, as­
suming that # 7
v -- # 9
v splitting in AlN is rather small ##E split
#20 meV #Ref. 31##, and the hole population of # 9
v , and
hence the transition probability to this band, may be higher
than that in # 7
v due to much higher density of states in # 9
v .
The values of P cv,x #P cv,y found this way together with
other material parameters used in our calculations are listed
below:
AlN: P cv,x #9.5#10 #20 g cm/s, ###2.15,
m e #0.27, m h , z #3.68, m h , xy #6.33 # Ref. 18#;
GaN: P cv,x #9.2#10 #20 g cm/s, ###2.67,
m e #0.18, m h , z #1.76, m h xy #1.69 #Ref. 19#;
InN: P cv,x #11.9#10 #20 g cm/s, ###2.1,
m e #0.11, m h , z #1.56, m h xy #1.68 #Ref. 19#.
The photon energy dependence of the absorption coeffi­
cient was taken from Ref. 10 for AlN, Ref. 6 for GaN, and
Ref. 9 for InN. The method we used to obtain the matrix
element of the radiative transition includes the assumption
that all contributions of defects in the nitride thin films #such
as nitrogen vacancies, oxygen impurities, etc.# to the absorp­
tion spectrum are noticeable only below and around the fun­
damental absorption edge. Deeper in the region of the band--
band transitions where the fundamental absorption greatly
increases, these contributions can be neglected. This propo­
sition is consistent with results of experimental works #for
example, see Refs. 5 and 10#. So we extracted the matrix
element values from the experimental data using just this
area of maximum absorption.
The values of the matrix element found this way are in a
good agreement with those obtained in other works. 32,33 In
the next section we will show that the calculated exciton
lifetimes based on these values are also in agreement with
available experimental data.
Now let us turn to the calculations of the interband ra­
diative recombination rate of free carriers in the nitride semi­
conductor alloys. The values of the matrix elements and ef­
fective masses in the alloys were found using a linear
interpolation between the values in the alloy components.
The calculated temperature dependence of the radiative re­
combination coefficient B is shown in Figs. 1--3. One can see
3243
J. Appl. Phys., Vol. 86, No. 6, 15 September 1999 A. Dmitriev and A. Oruzheinikov
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that B is the highest in InN and the compositions close to it,
and the lowest in AlN and the alloys close to this material.
Figure 1 depicts that the radiative recombination rate of
Ga x In 1#x N is almost the alloy composition independent. The
effective masses and refractive indices have close values in
these materials, and the difference in the band gap values is
just compensated by the change in the matrix element #see
Eq. #6##.
The radiative recombination coefficient and the radiative
free carrier lifetime defined as # r
#1 #Bn are presented in
Table I for different materials and alloys. One can see that in
similar conditions the carrier lifetime due to free carrier ra­
diative recombination even in AlN is higher than in GaAs.
Please note that our B coefficient of GaN is shown to be
nearly twice as large as one from Ref. 34. This fact can be
explained by using the recent effective masses values we
took from new energy spectrum calculations. 18,19
VI. LIFETIMES OF EXCITONS IN NITRIDES
With the above matrix elements we can also calculate
the radiative lifetimes of free excitons in the nitride semicon­
ductors and compare them with the data of experimental
works. 22,35
It is known that exciton annihilation dominate the carrier
radiative recombination in the nitrides. This fact was con­
firmed by observations of photoluminescence in this
material. 22 The predominance of exciton radiative recombi­
nation is the characteristic property of wide­gap direct band
semiconductors #such as, for example, CdS and GaP 36 #. The
creation time of free exciton in them is an order of magni­
tude less than the radiative lifetime of free carriers. 37
There are three valence band subbands #see Sec. II# in
the electron spectra of the nitrides AlN, GaN, InN, which
give rise to three different free excitons. They are clearly
observable in GaN #see Refs. 4, 22, and 38#. We calculated
the temperature dependence of the free exciton lifetime for
the brightest line in photoluminescence spectrum of the ni­
tride semiconductors. This line corresponds to the free exci­
ton with a heavy hole from # 9
v subband.
In direct semiconductors the free exciton radiative re­
combination rate can be derived from the probability of the
photon emission under exciton recombination 24
W ph #
2
3
e 2 ###E g #E x #
m 0
2 c 3 # 2 #P cv # 2 # 2## 2
k B T
# 3/2 1
# m • x m • y m • z # 1/2
##F#0 ## 2 # k , #8#
where E x is the exciton binding energy:
E x #
m 0 e 4
2# 2
#
# 2
,
F(r) , an s­like envelope function, is a solution of the effec­
tive mass equation for the free exciton ground state
FIG. 1. The radiative recombination coefficient B vs temperature T and
alloy composition x for Ga x In 1#x N.
FIG. 2. The radiative recombination coefficient B vs temperature T and
alloy composition x for Ga x Al 1#x N.
FIG. 3. The radiative recombination coefficient B vs temperature T and
alloy composition x for In x Al 1#x N.
TABLE I. The radiative recombination coefficient B and an average lifetime
# r of carriers #at T#300 K and n#10 18 cm #3 ) .
B(10 #10 cm 3 /s) # r #ns#
AlN 0.18 55
InN 0.52 19
Ga 0.6 In 0.4 N 0.49 20
GaN 0.47 21
Ga 0.8 Al 0.2 N 0.31 32
In 0.6 Al 0.4 N 0.26 38
GaN a 0.24
GaAs b 7.2 1.3
a The data were taken from Ref. 34.
b The data were taken from Ref. 30.
3244 J. Appl. Phys., Vol. 86, No. 6, 15 September 1999 A. Dmitriev and A. Oruzheinikov
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F#r ###
V
#a x
3 exp # #
r
a x
# ,
a x #a B #/# being the exciton Bohr radius, and V is the sys­
tem volume. To take the anisotropy of the electron and hole
spectra into account, we take an averaged value for the ex­
citon reduced mass: 22 ###(2/3)# xy
#1 #(1/3)# z
#1 # #1 , where
the perpendicular components are # xy #(m e xy
#1 #m h , xy
#1 ) #1 ,
and the parallel one is # z #(m e , z
#1 #m h z
#1 ) #1 . We used the
electron mass values presented above in Sec. V.
Equation #8#, multiplied by the Boltzmann distribution
of excitons #valid in our case# and then integrated over all
exciton wave vectors k, gives an expression for the radiative
recombination probability of one free exciton per unit time:
B x #
2
3
##e 2
m 0
2 c 3 # 2
#P cv # 2 # 2## 2
k B T
# 3/2 1
# m • x m • y m • z # 1/2 # E g #E x #
1
#a x
3 .
#9#
Finally, # x #1/B x gives the lifetime of excitons
# x #KT 3/2 ,
where a coefficient K derived from Eq. #9# is
K#s/K 3/2 ##0.78#10 #16
# m • x m • y m • z # 1/2 a x
3
###P cv # 2 # E g #E x #
. #10#
Here all fundamental constants were substituted, and a x
is measured in å , energies are in eV, and the matrix element
#P cv # 2 is in #10 #38 g erg#. The exciton parameters can be
calculated now on the basis of the data presented above and
in Table II.
Our results for the coefficient K and exciton lifetimes at
T#60 and 300 K are tabulated in Table III. When this life­
times are compared with those of free carriers #see Table I#,
it is apparent that the lifetimes of free excitons are nearly one
order of magnitude less than the radiative lifetimes of free
carriers in the nitrides at the same temperature. We point out
a good agreement of our values for the exciton lifetimes in
GaN with experimental data in Refs. 22, 33--35. We could
not find experimental data on the exciton lifetimes in other
nitrides.
VII. CONCLUSION
We calculated radiative recombination coefficients for
three nitride semiconductors and their binary alloys. To do it,
we extracted values of the interband matrix elements from
the absorption data. The matrix elements do not differ con­
siderably in these semiconductors, and the prominent differ­
ence between the recombination coefficients is connected
also with the difference in the band gap values and carrier
masses.
We estimated radiative lifetimes of excitons in the ni­
tride semiconductors. Our results are in a good agreement
with the data presented in other works. 22,33--35
ACKNOWLEDGMENT
The work was supported in part by Russian Foundation
for Fundamental Research.
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#1993#.
TABLE II. Parameters of free excitons in the nitrides.
# E x #meV# a x #å #
InN 9.3 16 49.2
GaN 9.5 20 38.7
AlN 8.5 49 17.3
TABLE III. The lifetimes of free excitons in the nitrides and K factor for the
temperature dependence of radiative lifetime of free excitons.
# x #ns# # x #ns#
K #ps/K 3/2 ) T#60 K T#300 K
InN 1.4 0.65 7.3
GaN 0.73 0.34 3.8
AlN 0.46 0.21 2.4
GaN 0.71 a 0.35 b 3.5 c
a The data were taken from Ref. 34.
b The data were taken from Refs. 22 and 35.
c The data were taken from Ref. 33.
3245
J. Appl. Phys., Vol. 86, No. 6, 15 September 1999 A. Dmitriev and A. Oruzheinikov
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