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Transcript
Light-absorption effect on Bragg interference in multilayer semiconductor
heterostructures
Alexey V. Kavokina) and Mikhail A. Kaliteevski
A. F. Ioffe Physico-Technical Institute, Russian Academy of Sciences, 26, Polytechnicheskaya,
194021 St-Petersburg, Russia, CIS
~Received 22 August 1995; accepted for publication 9 October 1995!
The transfer matrix method has been employed to study the effect of light absorption on optical
spectra of GaAs/AlAs Bragg reflectors. Substantial distortion of the Bragg plateau due to the
absorption in GaAs layers is found. The saturation value of the reflection coefficient of the infinite
Bragg reflector is shown to change nonmonotonically with the absorption coefficient showing a
minimum. The particular case of low-temperature light reflection from the Bragg mirror in the
vicinity of the exciton resonance frequency in GaAs layer is considered. The dramatic enhancement
~by a factor of 50! of the exciton resonance amplitude in comparison with the reflectivity spectrum
of GaAs is found in case of GaAs/AlAs Bragg structure covered by l/2-thick GaAs
caplayer. © 1996 American Institute of Physics. @S0021-8979~96!01302-X#
INTRODUCTION
Bragg reflectors ~BRs! are widely used in various types
of optical devices such as vertical cavity surface-emitting
lasers and light-emitting diodes, waveguides, reflectivity
modulators, and solar cells.1–5 Usually the transparent materials are used to construct BRs in order to exclude any effects
connected with light absorption in Bragg mirrors. Evidently,
the condition of the absence of absorption in the mirror substantially limits the useful spectral range of the Bragg reflector. In this article we present the theoretical study of the
effect of light absorption on optical properties of Bragg reflectors constructed from multilayer semiconductor heterostructures such as GaAs/AlAs. Both cases of intersubband
and excitonic absorption are considered.
Let us consider a periodic structure containing N double
layers consisted of layers of semiconductor materials A and
B ~see Fig. 1!. In general, the refractive index is complex in
both the materials A and B,
n B5ñ B1ik B ,
~1!
where ñ A,B and k A,B are real and imaginary parts of the corresponding refractive index. In the following we will use the
transfer-matrix method.6,7 The natural boundary conditions
for the light wave at each heteroboundary can be written as a
condition of continuity of the vector
S D
t 115cos~ q Aa ! cos~ q B b ! 2
t 2252
nA
sin~ q Aa ! sin~ q B b ! ,
nB
nB
sin~ q Aa ! sin~ q B b ! 1cos~ q Aa ! cos~ q B b ! ,
nA
nA
cos~ q Aa ! sin~ q B b ! ,
t 125cos~ q B b ! sin~ q Aa ! 1
nB
t 2152
BASIC EQUATIONS
n A5ñ A1ik A ,
l is the wavelength of light in vacuum. The factor q 21
A is
introduced to provide the same dimensionality of the components of the vector F.
The transfer of the vector F from the interface between
B and A layers by the period d is a result of the action of the
transfer matrix t̂, with components
~3!
nB
cos~ q Aa ! sin~ q B b ! 2cos~ q B b ! sin~ q Aa ! ,
nA
where a,b are the thickness of layers A, B, respectively.
The eigenvalues of the matrix t̂ are
F
~6!
t 111t 22
6
5
2
FS
D G
t 111t 22 2
21
2
1/2
[exp~ 6iQd ! , ~4!
where Q plays the role of an effective wave vector of light in
a multilayer structure. The corresponding eigenvectors can
be written as
Ex
F5
1
qA
]Ex ,
]z
~2!
where E x is the x component of the electric field ~for definiteness we consider a linearly polarized light!, the z direction is normal to layer planes, and q A~B!5n A~B!~2p/l!, where
a!
Electronic mail: [email protected]
J. Appl. Phys. 79 (2), 15 January 1996
FIG. 1. The schema of the structure considered: N double layers consisted
of A and B layers each one. r and t are the amplitude reflection and transmission coefficients.
0021-8979/96/79(2)/595/4/$6.00
© 1996 American Institute of Physics
595
SD SD
1
,
x
1
,
y
~5!
F G
F G
11r
i ~ 12r !
with x5~F (6) 2t 11!/t 12, y5~F (2) 2t 11!/t 12 .
To find out the amplitude reflection and transmission coefficients r and t of the structure under consideration one has
to solve a system of two vector equations given by boundary
conditions at left- and right-hand-side boundaries of the periodic structure,
SD SD
1
1
n 0 5A
1B
,
x
y
nA
SD
~6!
SD
t
1
1
n 1 5A exp~ iQNd !
1B exp~ 2iQNd !
,
x
y
i t
n1
~68!
with n 0 and n 1 being refractive indices in the cap layer and
the substrate, respectively ~see in inset in Fig. 1!.
The solution of Eqs. ~6! and ~68! yields
r5
~ in 0 2n Ax !~ n Ay2in 1 ! 1 ~ in 0 2n Ay !~ in 1 2n Ax ! exp~ 2iQNd !
,
~ in 0 1n Ax !~ n Ay2in 1 ! 1 ~ in 0 1n Ay !~ in 1 2n Ax ! exp~ 2iQNd !
~7!
t5
in 0 1n Ax
in 0 2n Ax
2r
.
in 1 2n Ax
in 1 2n Ax
~8!
The general formulas ~7! and ~8! can be simplified in a few
important particular cases.
For N→`,
r5
in 0 2n Ax
,
in 0 1n Ax
y50.
~9!
At the Bragg interference condition,
2p
p
2p
ñ a5
ñ b5 ,
l A
l B
2
~10!
this picture is dramatically changed. The magnitude of the
reflectivity at the plateau gets lower and the oscillations disappear so that all the spectrum is substantially smoothed and
only one central maximum is kept. Correspondingly, f
changes its sign only one.
Figure 3 shows a comparison between the experimental
spectrum of an absorbing Bragg reflector8 and our theoretical
calculation. The studied structure contains 10 periods of
GaAs/AlAs superlattice with layer thickness a546 nm,
if k A5k B50, ñ A,ñ B , well-known expressions can be obtained,
r5
12 ~ n A /n B! 2N
,
11 ~ n A /n B! 2N
~11!
t5
2
.
11 ~ n A /n B! 2N
~12!
We note, however, that the formulas ~11! and ~12! are not
valid in case of nonzero absorption, and one should use Eqs.
~7! and ~8!.
RESULTS AND DISCUSSION
To have an idea about the magnitude of the effect of the
absorption on the spectra of Bragg mirrors we have considered a structure schematically shown in Fig. 1. This is a
periodic multilayer GaAs/AlAs structure surrounded by
semiinfinite layers of GaAs. All GaAs layers in the structure
are supposed to be absorbing unlike AlAs layers. Figures
2~a! and 2~b! shows the spectra of the squared modulus and
the phase of the reflection coefficient from such a structure
for different levels of absorption. The used parameters are:
thickness of layers in the superlattice a~GaAs!546 nm,
b~AlAs!561 nm; the number of periods N510. At the wavelength l5715 nm the Bragg interference condition is satisfied. One can see for low absorption levels the plateau and
oscillating tails in uru2 spectrum. At high absorption levels
596
J. Appl. Phys., Vol. 79, No. 2, 15 January 1996
FIG. 2. Calculated ~a! squared module and ~b! phase of the amplitude reflection coefficient for GaAs/AlAs periodic structure surrounded by semiinfinite media of GaAs, N510, a546 nm, b561 nm, for different levels of
absorption in GaAs: ~1! k50; ~2! k50.01; ~3! k50.05; ~4! k50.1; ~5!
k50.5.
A. V. Kavokin and M. A. Kaliteevski
FIG. 3. Experimentally measured Ref. 8 ~solid! and calculated ~dashed and
dotted! spectra of GaAs/AlAs absorbing Bragg reflector covered by GaAs
cap layer. Dashed and dotted curves represent calculations taking into account and ignoring absorption in GaAs layers, respectively.
b561 nm grown on GaAs substrate and covered by 88-nmthick GaAs cap layer. All GaAs layers were doped with the
donor ~Si! concentration n51018. There are no fitting parameters in the calculation, while the spectral dependence of k A
is taken from Ref. 9. Note that this is a case of the intersubband absorption, and excitonic effects are removed due to the
high screening. Peculiarities of the systems with lowtemperature excitonic absorption are discussed below. One
can see that quite a good agreement is achieved between
theory ~dashed line! and experiment ~solid line!. On the other
hand, the spectrum calculated neglecting the absorption in
GaAs layers ~dotted line! is strongly different from the experimental curve.
Figure 4 shows the reflection coefficient R5 u r u 2, when
the Bragg condition ~10! is satisfied, as a function of the
number of periods of the superlattice N for different values
FIG. 5. The saturative value of the reflection coefficient of the infinite
GaAs/AlAs Bragg reflector as a function of the level of absorption in GaAs
layers. Light is reflected to GaAs media.
of k A . One can see that this dependence always has a saturating character while the saturation value of the reflection
coefficient R ` drastically depends on absorption. It is worth
noting from Fig. 4 that the Bragg mirror containing 10 periods of a superlattice has roughly the same peak value of the
reflection coefficient as an infinite Bragg mirror. The saturation value of the reflection coefficient R ` as a function of k A
is shown in Fig. 5. This dependence is nonmonotonic. The
initial decrease of R ` with k A manifests the negative effect
of light absorption on the quality of a Bragg mirror. The
minimum of R ` ~k A! corresponds to k A}l /a. For higher k A
the interference effects get negligible due to strong absorption in the first GaAs layer. This first layer acts as a metallic
film, which results in the increase of the reflection coefficient.
An important particular case of the absorbing Bragg mirror is realized if the condition ~10! is satisfied at the wavelength close to the wavelength of the exciton resonance in
one of the materials of the mirror. ~The temperature is supposed to be low enough for appearance of excitonic effects.!
The reflection coefficient of such structure in the vicinity of
the exciton resonance can be found by using the same formalism as earlier, bearing in mind that the dielectric constant
e5n 2 in an absorbing layer can be written in the form
S
e ~ v ! 5 e B 11
D
v LT
,
v 0 2 v 2iG
~13!
where eB is the background dielectric constant, v0 and vLT
are the exciton resonance frequency and longitudinal transverse splitting, respectively, and G is the exciton damping.
Figure 6 shows the calculated reflectivity spectrum of
the GaAs/AlAs BR ~N58! with layer thickness a,b such as
2p
p
2p
ñ Aa5
ñ B b5 ,
l0
l0
2
FIG. 4. Reflection coefficient of N-period-thick GaAs/AlAs Bragg reflector
surrounded by the semi-infinite media of GaAs as a function of N, for
different levels of absorption in GaAs: ~1! k50; ~2! k50.01; ~3! k50.05; ~4!
k50.1; ~5! k50.5.
J. Appl. Phys., Vol. 79, No. 2, 15 January 1996
with l05v 0c ~curve 2!. One can see that the amplitude of
the exciton resonance in reflection is an order higher in
Bragg structure in comparison with bulk GaAs ~curve 1!. In
accordance with Ref. 10 we have taken \vLT50.1 meV,
\G51 meV. The reflectivity modulation of almost 100% is
A. V. Kavokin and M. A. Kaliteevski
597
Bragg plateau is only about 1 meV. On the other hand, the
effect discussed in the present work takes place for all values
of l0 belonging to the Bragg plateau, which is of width about
50 meV in the considered structures.
CONCLUSION
FIG. 6. Calculated reflectivity spectrum of ~1! bulk GaAs, ~2! Bragg reflector consisting of eight periods of GaAs/AlAs structure, and ~3! the same
Bragg reflector covered by GaAs caplayer of the thickness equal to a halfwavelength of light at the exciton resonance frequency in GaAs. The Bragg
interference condition is satisfied at the exciton resonance frequency in
GaAs.
achieved in case of a Bragg structure covered by GaAs cap
layer of thickness l5l 0 /2ñ A ~curve 3!. This dramatic enhancement of the modulation is caused by destructive interference of light waves reflected by the surface and the interface cap layer/superlattice. High sharpness of the excitonic
modulation of the reflection spectra of Bragg structures provides extremely high precision in detection of exciton parameters such as oscillator strength and radiative lifetime.
Note that Bragg reflection from multiple-quantum-well
structure near the exciton resonance frequency has been analyzed in a recent work.11 In that case, however, the enhancement of the excitonic modulation of the reflection spectrum
is substantial only in the close vicinity of the wavelength
corresponding to Bragg interference condition. The reason is
that in a multiple-quantum-well structure the width of the
598
J. Appl. Phys., Vol. 79, No. 2, 15 January 1996
In conclusion, the light-absorption effect on optical
properties of Bragg mirrors has been considered in the
framework of transfer matrix method. Dramatic distortion of
the reflectivity spectra of Bragg mirrors due to absorption is
revealed. We have established that the peak value of the reflection coefficient from Bragg mirror depends nonmonotonically on the absorption with a minimum corresponding to
the threshold between interference regime and metallic reflectance regime. The interference between exciton resonances in Bragg structures is shown to result in the giant
increase of the reflectivity modulation in the vicinity of the
exciton resonance, so that in particular cases the amplitude
of the excitonic peak in reflection is almost 100%.
1
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V. M. Andreev, V. V. Komin, I. V. Kochnev, V. M. Lantratov, and M. Z.
Shvarts, in Proceedings of the First World Conference on Photovoltaic
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4
G. Parry, M. Whitehead, P. Stevens, A. Rivers, P. Barnes, D. Atkinson, J.
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5
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Oxford, 1986!.
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Disordered Systems ~Cambridge University Press, Cambridge, 1979!.
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V. V. Evstropov, M. A. Kaliteevski, A. L. Lipco, M. A. Sinitsyn, B. V.
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9
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E. L. Ivchenko, P. S. Kop’ev, V. P. Kochereshko, I. N. Uraltsev, D. R.
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A. V. Kavokin and M. A. Kaliteevski