Light absorption by inhomogeneous semiconductor film
Processes of light absorption by thin semiconductor film in the framework of local-field method are studied. The film is inhomogeneously implanted with O⁺ ions. A distribution of implanted layer is characterized by different profiles. The effective susceptibility (response to the external field) and...
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Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України
2005
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| Цитувати: | Light absorption by inhomogeneous semiconductor film / L. Baraban, V. Lozovski // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2005. — Т. 8, № 3. — С. 66-73. — Бібліогр.: 24 назв. — англ. |
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irk-123456789-1209712017-06-14T03:04:50Z Light absorption by inhomogeneous semiconductor film Baraban, L. Lozovski, V. Processes of light absorption by thin semiconductor film in the framework of local-field method are studied. The film is inhomogeneously implanted with O⁺ ions. A distribution of implanted layer is characterized by different profiles. The effective susceptibility (response to the external field) and dissipative function of inhomogeneous in thickness semiconductor film were calculated. The absorption spectra are numerically calculated as a function of the frequency and angle of incidence. It was obtain that light absorption spectra are strongly dependent on profile distributions of implanted impurities along the film thickness. 2005 Article Light absorption by inhomogeneous semiconductor film / L. Baraban, V. Lozovski // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2005. — Т. 8, № 3. — С. 66-73. — Бібліогр.: 24 назв. — англ. 1560-8034 PACS: 78.20.Bh, 78.40.Fy http://dspace.nbuv.gov.ua/handle/123456789/120971 en Semiconductor Physics Quantum Electronics & Optoelectronics Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України |
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Processes of light absorption by thin semiconductor film in the framework of local-field method are studied. The film is inhomogeneously implanted with O⁺ ions. A distribution of implanted layer is characterized by different profiles. The effective susceptibility (response to the external field) and dissipative function of inhomogeneous in thickness semiconductor film were calculated. The absorption spectra are numerically calculated as a function of the frequency and angle of incidence. It was obtain that light absorption spectra are strongly dependent on profile distributions of implanted impurities along the film thickness. |
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Baraban, L. Lozovski, V. |
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Baraban, L. Lozovski, V. Light absorption by inhomogeneous semiconductor film Semiconductor Physics Quantum Electronics & Optoelectronics |
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Baraban, L. Lozovski, V. |
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Baraban, L. |
| title |
Light absorption by inhomogeneous semiconductor film |
| title_short |
Light absorption by inhomogeneous semiconductor film |
| title_full |
Light absorption by inhomogeneous semiconductor film |
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Light absorption by inhomogeneous semiconductor film |
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Light absorption by inhomogeneous semiconductor film |
| title_sort |
light absorption by inhomogeneous semiconductor film |
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Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України |
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2005 |
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http://dspace.nbuv.gov.ua/handle/123456789/120971 |
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Light absorption by inhomogeneous semiconductor film / L. Baraban, V. Lozovski // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2005. — Т. 8, № 3. — С. 66-73. — Бібліогр.: 24 назв. — англ. |
| series |
Semiconductor Physics Quantum Electronics & Optoelectronics |
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AT barabanl lightabsorptionbyinhomogeneoussemiconductorfilm AT lozovskiv lightabsorptionbyinhomogeneoussemiconductorfilm |
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2025-07-08T18:57:00Z |
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2025-07-08T18:57:00Z |
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1837106224300032000 |
| fulltext |
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2005. V. 8, N 3. P. 66-73.
© 2005, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
66
PACS: 78.20.Bh, 78.40.Fy
Light absorption by an inhomogeneous semiconductor film
L. Baraban1, V. Lozovski1,2
1Taras Shevchenko Kyiv National University, Radiophysics Department,
2/5, prospect Academician Glushkov, 03022 Kyiv, Ukraine, e-mail: laryssy@univ.kiev.ua
2V. Lashkaryov Institute of Semiconductor Physics, NAS of Ukraine,
45, prospect Nauky, 03028 Kyiv, Ukraine
Corresponding author: V. Lozovski, e-mail: lozovski@univ.kiev.ua
Abstract. Processes of light absorption by thin semiconductor film in the framework of
local-field method are studied. The film is inhomogeneously implanted with O+ ions. A
distribution of implanted layer is characterized by different profiles. The effective
susceptibility (response to the external field) and dissipative function of inhomogeneous
in thickness semiconductor film were calculated. The absorption spectra are numerically
calculated as a function of the frequency and angle of incidence. It was obtain that light
absorption spectra are strongly dependent on profile distributions of implanted impurities
along the film thickness.
Keywords: thin film, local field, effective susceptibility, absorption, semiconductor,
implanted ion.
Manuscript received 23.06.05; accepted for publication 25.10.05.
1. Introduction
Thin and ultra-thin films are in interest for fundamental
and applied sciences. This is associated with the
investigations of new materials for different areas of
human activities. Nowadays the technologies allow to
create the devices based on thin films. The letter can be
both solid films (for example, quantum wells and
superlattices [1-4]) and molecular coatings [5-6]. The
devices based on ultra-thin films, for example
superlattices, characterized by quantum-dimension
effects, are widely used in modern electronics [7-9]. It
should be also pointed on the example of using the
classical properties of thin films. One of the branches of
thin film science is associated with developing and
producing the sensors based on the surface plasmon-
polariton resonance effect [10, 11].
The macroscopic Maxwell equations can be applied
only to the systems, where the wavelength λ is much
longer than the characteristic dimensions of the
microscopic field variations l , and characteristic linear
dimension of the particle L is much longer than l.
Otherwise, other methods should be developed for
describing the electrodynamical properties. The case of
electrodynamics of mesoparticles where the thickness of
the transition layer near the surfaces of the particles is
about l and L ~ l can be mentioned as an example.
Transition layers are characterized by extremely
inhomogeneous local fields. In the cases of thin films,
the transition layers at both interfaces are about the
thickness of the film. It means that the condition lL >>
is not carried out for the thin film, and the standard
methods of macroscopical electrodynamics for these
films cannot be used. The development of near-field
physics gives the powerful method for calculation of
electrodynamical properties of nanoobjects based on
solution of the Lippmann-Schwinger equation for self-
consistent fields [12, 13]. The main idea of the local-
field method is to ascertain the relation between the local
field in an arbitrary point inside the system ),( ωRE
rr
and
external long-range field ),()0( ωRE
rr
that is written as
),(),(),( )0( ωωω RERLRE jiji
rrr
= (1)
with the local-field factor ),( ωRLij
r
[12]. The method of
effective susceptibility allows to calculate the local-field
factor via the linear response (to the external field)
function.
The theory of electrodynamical properties of ultra-
thin films based on the method analogous to the mean-
field theory [14, 15] was developed earlier. The main
idea of this theory was to exclude uniform field
characterizing the film from all equations for the Green
function. As a result, the equation for the electrody-
namical Green function of ultra-thin film was obtained.
The tensor of the linear response to the external field
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2005. V. 8, N 3. P. 66-73.
© 2005, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
67
Fig. 1. Dependence of electron concentration on coordinate
along the thickness of the film. The dark area corresponds to
the high concentration of charge carriers.
was also calculated. It should be emphasized that the
elctrodynamics of ultra-thin films could be built using
the specific boundary conditions that require the local
field abrupt across the upper and bottom sides of the film
(see, for example Refs [16] and [17]).
In this work, we consider the absorption of the
external electromagnetic field by a thin inhomogeneous
semiconductor film irradiated by an electromagnetic
wave. The consideration was performed using an
approach based on the solution of the Lippmann-
Schwinger equation with regard to the local-field effects
proposed in Refs [18, 19].
2. Model
Let us consider a thin semiconductor n+-GaAs film with
some distribution of compensating impurity. Electrons
are compensated by positive ions of oxygen in the
certain film area, defined by the distribution of
compensating impurity. The dielectric permeability of
the material is determined by both phonon and plasma of
free carriers excitations. In the simplest model of the
dielectric function for semiconductor, the dielectric
permeability can be written in the form
( ) ⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
+
−
−−
−
+= ∞
pl
2
2
ph
22
22
1,
ωγω
ω
ωγωω
ωω
εωε
ii
z P
T
TL , (2)
where ∞ε is the high frequency dielectric permeability;
Lω , Tω , Pω are the longitudinal and transversal fre-
quencies of phonons as well as the plasma frequency,
respectively; phγ , plγ are the phonons and plasma exci-
tation decay constants. The density of electrons along
the thickness of the film depends on the profile of com-
pensating impurities O+ (see. Fig. 1). It leads to depen-
dence of the plasmon frequency Pω on the z-coordinate
inside the film. The shape of impurity distribution can be
determined by the theory of ionic implantation by
Lindhard-Scharf-Shiott [20]. According to the basic
principles of this theory, implanted ions are distributed
along the thickness of the film with the Gauss function
( )
⎪⎭
⎪
⎬
⎫
⎪⎩
⎪
⎨
⎧
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
Δ
−
−
Δ
=
2
2
1exp
2 p
p
p
i R
Rz
R
Nzn
π
, (3)
with the doze of implanted ions N, the depth of
distribution Rp, and the dispersion of distribution ΔRp.
The space-dependent linear response to the total (local)
field is the characteristic feature of the model under
consideration. In agreement with Eq. (3), the plasmon
frequency is dependent on the z-coordinate
2/1
*
2
2 )(4
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
=
∞ε
πω
m
zne
P . (4)
Consequently, the dielectric function must be the
function of the z-coordinate. Then, the problem is to
calculate the local self-consistent field inside the film
with regard to both inhomogeneity of initial
susceptibility of the material of the film and local-field
effects caused by inhomogeneities. It is necessary to
point out that study of the same problem, where
consideration of the optical properties of thin
semiconductor film doped with the compensating
impurity was provided by using the model of the step-
like distribution of compensating impurity [21].
Nevertheless this extremely simplified model of ionic
implantation (see, Fig. 2a) gave suitable results for
optical properties of the implanted thin film. It is clear,
that in the real situation, the profile of impurity
distribution has a more complicated dependence on a z-
coordinate. In this case, the multistep-like model
(Fig. 2b) could be used analogously to the one-step-like
model and, of course, could give a more adequate
description of the film optical properties. Consequently,
the plasmon oscillations excite only on the single
frequency for each step, and to describe properties of the
whole film the multiplayer model should be used. This
model is rather complicated. Moreover, such assumption
does not correspond to a real physical situation
completely as far as the so-called “tails of distribution”
exist and can contribute to the general properties of a
film. Then, here we shall use the recently developed
approach [22] where the characteristics of the film
material can continuously change along its thickness.
The concentration of electrons along the film thickness
is ever-changing, its plasmon frequency is ever-
changing, too. It leads to the dependence of the initial
susceptibility (the response to a local field) both on the
frequency and z-coordinate (Fig. 3).
3. The linear response to the external field
The effective susceptibility of a thin film is the
important characteristic showing how the film responds
to an external field. To understand a physical sense of
the effective susceptibility, one need to consider the
general solution of the Lippmann-Schwinger equation,
that describes the self-consistent (local) field, formed
inside a particle [12, 13]. The equation for a thin film
can be written in the form
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2005. V. 8, N 3. P. 66-73.
© 2005, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
68
Fig. 2. Single step-like (a), multistep-like (b) and smooth Gauss-like (b) distribution of the compensating impurity.
Fig. 3. The real part of the initial susceptibility as a function
of the frequency and z-coordinate (along the thickness of the
film) ( )],Re[ 0 zωχ . The depth and dispersion of the dist-
ribution are hR p 5.0= and hR p 11.0=Δ , respectively.
,),(),(),,,(
4
1
),,(),,(
2/
2/0
)0(
∫
−
′′′′−
−=
h
h
ljlij
ii
zkEzzzkGzd
zkEzkE
rr
rr
ωχω
πε
ωω
(5)
with ),,()0( ωzkEi
r
as the external longwave field,
),( zjl ′ωχ as the linear response to the local field, and k
r
as the projection of the external field wavevector on the
film plane. The solution of the Lippmann-Schwinger
equation (5) can be written via an effective susceptibility
(the linear response to an external film) in the form:
,),(),,(),,,(
4
1
),,(),,(
2/
2/
)0(
0
)0(
∫
−
′′Χ′′−
−=
h
h
ljlij
ii
zkEzkzzkGzd
zkEzkE
rrr
rr
ωω
πε
ωω
(6)
where the effective susceptibility is [22]
1),,(),(),,( −=Χ ωωχω zkPzzk lmjljm
rr
, (7)
with
1
)(
2/
2/0
),(),,(
4
1
),,(
−
′−
−
⎥
⎥
⎦
⎤
′′′′′+
⎢
⎣
⎡
+=
∫ zziq
jlmj
h
h
mllm
ezzzkGzd
zkP
ωχ
πε
δω
r
r
(8)
local-field correction function. In this expression, q is
the z-component of the external field wavevector. Then,
the effective susceptibility (7) connects the longwave
external field with the local field ),,( ωzkEi
r
that is
quickly changed at short-range distances. As can be seen
from Eq. (8), the proposed approach can be easily used
for calculation of the effective susceptibility of an
inhomogeneous film, for example, for the film with
inhomogeneous distribution of compensating impurities
along its thickness. One should note that different widths
of the implanted ion distribution profile mean the
different relationship between the contributions of the
polariton and plasmon subsystems into the formation of
electrical response to a local field. As can be easily seen
from Eq. (7), the same behavior can be also observed for
the effective susceptibility.
4. Light absorption by thin semiconductor film
The self-consistent approach allows to describe the
specific features of the absorption of an electromagnetic
wave energy by a thin film. The absorbed energy of
electromagnetic wave in a unit film area per unit time
can be written:
))(( ** EEJJQ
rrrr
++= (9)
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2005. V. 8, N 3. P. 66-73.
© 2005, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
69
where )(RJ
rr
is the local current in the film caused by the
local field )(RE
rr
; (...) and ... are the time averaging
over the period (much larger than the period of the
electromagnetic field oscillation), and averaging over the
film volume, respectively. Local currents are related to
the effective susceptibility via equation
)()( )0()( REXiRJ E
rrtrr
⋅−= ω . (10)
On the other side, the local field is related to the
external field as
( ) )0()(1)()( EXRE EE
rttrr −
= χ . (11)
Using the substitution of Eqs (10) and (11) into
Eq. (9) and performing the averaging-out over time, one
obtains
( )
( ) ( ) .)(
)(
)0()0()(1)(*)(
*)(1)()(
EEXX
XXiQ
EEE
EEE
rrttt
ttt
〉−
−〈−=
−
−
χ
χω
(12)
Using Eq. (7) into Eq. (12), one can write for the
case of isotropic initial susceptibility that energy
absorbed by the inhomogeneous film can be written in
the form
eezkPzdz
h
IQ
h
h
rrrt
⋅⋅= ∫
−
22/
2/
0 ),,()),(Im(12 ωωχω (13)
with the unit vector of light polarization e
r
, the intensity
of incident light
2)0(
0 EI = . Thus, the absorbed energy
is controlled by the imaginary part of the initial
susceptibility of the film material ),(0 zωχ , and defined
by the film thickness and experiment geometry.
The main contribution of local field effects is
displayed by the pole part of the effective susceptibility,
or as seen from Eq. (7)
.),(),,(
4
1
det
)(
2/
2/0 ⎥
⎥
⎦
⎤
′′′′′+
⎢
⎣
⎡
+
′−
−
∫ zziq
jlmj
h
h
ml
ezzzkGzd ωχ
πε
δ
r (14a)
In other words, the specific absorption occurs in the
resonant case, when the real part of the pole part of the
function ),,( ωzkP
rt
is equal to zero
.0),(),,(
4
1
detRe
)(
2/
2/0
=
⎪⎭
⎪
⎬
⎫
⎥
⎥
⎦
⎤
′′′′′+
⎪⎩
⎪
⎨
⎧
⎢
⎣
⎡
+
′−
−
∫ zziq
jlmj
h
h
ml
ezzzkGzd ωχ
πε
δ
r
(14)
In this work, we numerically analyze the dependence
of the normalized dissipative function 0/ IQI = on the
frequency and angle of incidence for the probing
radiation. Eq. (13) is the main expression that was used
in this work for numerical calculations.
5. Numerical results and discussions
Using the analytical approach developed in Sec. 4, here
we analyze numerically the characteristic features of
light absorption by thin inhomogeneous semiconductor
film. To analyze numerically the features of the process
of light absorption by the inhomogeneous semiconductor
film, we have chosen the following parameters of the
film: thickness of the film h = 40 nm, concentration of
electrons in the undoped film n0 = 2·1018 cm−3, dose of
implanted ions Ni = 1.2·1014 cm−2, and dielectric
permeability of GaAs 1.13=ε . Study of light
absorption by the thin semiconductor film with
inhomogeneous distribution of compensating impurities
shows that the absorption profile has interesting
characteristic properties. First of all, it is necessary to
note that, in contrast to the well-known electric dipole
approximation where absorption at the normal light
incidence is absent [1, 24], the present calculations show
nonzero absorption at the same condition. This fact is
connected with the use of more general model that does
not base on the electric dipole approximation and local-
field effects. As it follows from the numerical results
(see, Figs 4-7), the main characteristic properties of
absorption processes is the presence of one or two peaks
in absorption profile determined by plasma oscillations.
Indeed, the continuous distribution of the carriers along
the thickness of the film should lead to rather wide
absorption spectrum caused by interaction between the
light and electron plasmas in semiconductor material of
the film. Moreover, the broadening of the impurity
distribution leads to a shift of the peaks, which
corresponds to the plasma oscillations, to a low-
frequency region. Another interesting feature of the
absorption processes is the growth of an additional
plasma peak at a lower frequency relatively to the main
plasmon peak. This low-frequency peak exhibits
tendency to the shift to the low-frequency region, when
the impurity distribution becomes broader. It should be
noted that the low-frequency peak shifts much faster
than the main one. The results of the numerical
calculations are shown in Figs 4-7. The calculations
demonstrate that continuous distribution of the
impurities, which mean that plasma frequency is
distributed continuously along the thickness of the film,
leads to the increase of absorbing profile with one or
several peaks. The characteristic feature of absorption
processes is the dependence of absorption spectra on the
width of the layer doped with impurities and on the
depth of their occurrence. The dependence of the
absorption intensity on the incident light frequency and
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2005. V. 8, N 3. P. 66-73.
© 2005, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
70
Fig. 4. Absorption profiles for the thin film (h = 40 nm) of
p-polarized light in the cases of narrow, intermediate, and
wide impurity distributions, respectively. The distribution
located in the middle of the film hRp 5.0= . a) hRp 01.0=Δ ,
b) hRp 06.0=Δ , c) hRp 3.0=Δ .
Fig. 5. Dependences of the absorption energy on the
frequency of external radiation for the cases of narrow,
intermediate, and wide distributions. The angle of incidence
6411 ′= oϑ . a) The distribution centered in the middle of the
film hR p 5.0= . b) The center of the distribution is shifted
from the middle of the film hR p 2.0= .
angle of light incidence is shown in Fig. 4. This figure
demonstrates the behavior of the absorption spectrum in
the case of different widths of impurity distribution:
narrow (a), intermediate (b), and wide (c). The
broadening of distribution leads to diminution of the
absorption intensity and to splitting the plasmon peak
(Fig. 4b). The plasmon peak disappears in the case of a
very wide distribution of impurities (see Fig. 4c) that
confirm total compensation of the electrons in the film
by ion impurities. Fig. 5 exhibits the profiles of the
absorption intensity dependence on the frequency with
the fixed angle of incidence ϑ = 11°46΄, which helps to
understand the behavior of 3D-figures in Fig. 4. The
comparison of absorption spectra for different positions
of the distribution of impurities is demonstrated in Fig. 5
for the case of maximum of impurity distribution located
in the middle of the film, namely, Rp = 0.5h (a), and for
the case of shifted distribution Rp = 0.2h (b).
The absorption spectra with sufficiently wide
distribution of implanted ions are shown in Fig. 6. The
evolution of splitting and shifting the plasmon peak of
absorption profile is also demonstrated in Fig. 6. First of
all, the splitting of the plasmon peak, when the
distribution becomes wider, arises. The splitting
increases, when broadening of impurity profile
continues, and both low-frequency and high-frequency
peaks shift to the longwave region. Moreover, the
shifting of the low-frequency peak is appreciably faster
than shifting of the high-frequency plasmon peak.
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2005. V. 8, N 3. P. 66-73.
© 2005, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
71
Fig. 6. The absorption profiles of the p-polarized wave in the
case of wide distributions with regard to “distribution tails”.
The angle of incidence 6411 ′= oϑ . a) The distribution is
centered in the middle of the film hR p 5.0= . b) The center
of distribution is shifted from the middle of the film
hR p 2.0= .
These characteristic features can be explained by the
following speculations. The film was characterized by
inhomogeneous distribution of plasma oscillations with
different frequencies. Different plasmons are
characterized by different phase volumes. It means that
contribution into interaction “plasma oscillations-light”
at different frequencies will be different. To clarify this
thought, let us consider Fig. 8 where the different widths
of impurity distributions along the film thickness are
shown. According to general properties of solid state
plasma, the response function that was connected to the
total induced charge with the total electric potential is
proportional to the correlation function of the “charge
density – charge density” type (see, for example
Ref. [23]). Then the interactions between light and
plasma subsystems of the film is weaker, the electron
concentration is lower. Let us consider the narrow
distribution of compensating impurity (Fig. 7a). It is
convenient to divide the distribution profile into three
regions, where the points “a” and “b” are the inflections
of the Gauss curve. As a result, one can contend that the
compensation is practically absent in the regions 1 and 3.
It can be supposed that the plasma oscillations in these
regions will be characterized by the frequency 3,1
pω
(defined by the electron concentration corresponding to
points in the surface region of the film) that is
approximately equal to the plasma frequency of undoped
semiconductor max
pω that is connected with the electron
concentration n0 of uncompensated material via the
expression
0
0
2
2max )(
εε
ω
m
nq
p = . (15)
In the region 2, distribution of impurities is evident.
The reason is that in the narrow region the impurity
concentration is quickly changed, the region 2 of
impurity distribution can be characterized by any
frequency 2
pω that can be selected approximately as a
frequency corresponding to the most strong interaction
between light and plasma subsystems. It is clear that 2
pω
will be not essentially differ from 3,1
pω . Moreover, the
contribution into the absorption processes of the region 2
is much less than those of regions 1 and 3, or
)()( 3,12
pp II ωω << . It means that absorption profile
corresponding to light-plasma interaction is determined
only by the oscillations with the frequency 3,1
pω . Then,
the total absorption profile should have two peaks – one
of them (lower frequency) corresponds to light-lattice
subsystem interaction, and the second (higher frequency)
corresponds to light-plasma interaction (Fig. 4a and
Fig. 5, curve 1).
If the profile of impurity distribution has an
intermediate width (Fig. 7b), the concentration profile
varies more smoothly. The inflections (pointed as a′
and b′ ) and boundaries of the regions (1΄, 2΄, 3΄) vary
their positions. The 1΄, 3΄ regions become narrower. It
means that the region where plasmons with the
frequency max
pω play the main role become narrower.
The electron concentration in these regions slightly
decreases because the “distribution tails” are involved
outside the film. This circumstance leads to weak
decreasing interaction between light and plasma
oscillations at frequencies about 3,1 ′′
pω . It means that the
peak of absorption profile characterized by frequency
3,1 ′′
pω becomes lower. The central frequency of the peak
remains equal to max3,1
pp ωω ≈′′ because the thickness of
regions 1΄, 3΄ keeps rather wide. A more interesting
behavior of plasma oscillations is displayed in the region
2′ . This region is characterized by lower frequencies as
in 1΄, 3΄ regions. Then, 22
pp ωω <′ , and the possibility of
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2005. V. 8, N 3. P. 66-73.
© 2005, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
72
Fig. 7. Three cases of location of “distribution tails” inside the film.
the additional peak growth appears. This situation is
shown in Figs 4b and 5 (curve 2) where the plasmon
peak splits. Further broadening of the distribution profile
(Fig. 7c) leads to enhancement of effect of plasmon peak
splitting in the absorption profile. Moreover, the
characteristic plasmon frequency for the regions 1′′ and
3 ′′ shifts to the low-frequency region, because
compensation processes become to play role in the
regions near film boundaries diminishing the electron
concentration in the regions. Then, the inequality
becomes valid max3,1
pp ωω <′′′′ . This fact leads to
decreasing the “light-electron plasma” interactions. Then
the intensity of the main plasmon peak in absorption
profile decreases. One should note that the shift velocity
of the peaks in the regions 1 and 3 is smaller than that of
the peaks caused by plasmons in the region 2. This is
because of distribution function profile after inflection
(in the tail regions 1 and 3) has more slow behavior than
in the middle of the region 2.
6. Conclusions
In this work, we used the local field method based on the
formally exact solution of the Lipmann-Schwinger
equation to analyze the light absorption processes by a
thin semiconductor film with an inhomogeneous
distribution of the compensating impurity. Then, the
electrodynamical properties of the thin film were
calculated in the frames of the self-consistent approach.
As a result, the effective susceptibility of thin film was
calculated analytically. To calculate the absorption
spectra for a strongly inhomogeneous system, we used
the dissipation function approach. This calculation
showed that the dissipative processes are not directly
determined by the imaginary part of the effective
susceptibility of the system (linear response to the
external field).
The absorption spectra were calculated numerically.
The spectra exhibited some unexpected features: first, a
splitting of the plasmon peak under the conditions of
sufficiently wide impurity distribution; second, different
shift velocities for each plasmon peak when the peaks
shift into the low-frequency range with the impurity
distribution widening. The characteristic behavior of the
absorption profile consisting of the absorption peaks
connected with plasmon oscillations, when the electron
concentration is distributed continuously, was discussed.
It should be mentioned that results, obtained in this
work, are qualitatively confirmed by experimental data
demonstrated in Ref. [21].
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