Stress-induced effects in light scattering by plasmons in p-type germanium
Infrared light scattering by plasmons in p-Ge has been studied under uniaxial stress along the [110] axis with polarization of incident light parallel to the stress direction. It is found that the deformation of the crystal results in an increase of the plasma frequency and lowering the asymmetry of...
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Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України
2003
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| Cite this: | Stress-induced effects in light scattering by plasmons in p-type germanium / V.N. Poroshin, A.V. Gaydar, A.A. Abramov, V.N. Tulupenko // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2003. — Т. 6, № 4. — С. 425-430. — Бібліогр.: 19 назв. — англ. |
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| author | Poroshin, V.N. Gaydar, A.V. Abramov, A.A. Tulupenko, V.N. |
| author_facet | Poroshin, V.N. Gaydar, A.V. Abramov, A.A. Tulupenko, V.N. |
| citation_txt | Stress-induced effects in light scattering by plasmons in p-type germanium / V.N. Poroshin, A.V. Gaydar, A.A. Abramov, V.N. Tulupenko // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2003. — Т. 6, № 4. — С. 425-430. — Бібліогр.: 19 назв. — англ. |
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| description | Infrared light scattering by plasmons in p-Ge has been studied under uniaxial stress along the [110] axis with polarization of incident light parallel to the stress direction. It is found that the deformation of the crystal results in an increase of the plasma frequency and lowering the asymmetry of the line by plasma scattering. These effects are explained by taking into account the change of contribution to the dielectric constant caused by the intra- and intersubband transitions as a consequence of variation of the energy band of p-Ge related with deformation.
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425© 2003, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
Semiconductor Physics, Quantum Electronics & Optoelectronics. 2003. V. 6, N 4. P. 425-430.
PACS: 78.40. Fy, 73.20. Mf
Stress-induced effects in light scattering
by plasmons in p-type germanium
V.N. Poroshin, A.V. Gaydar, A.A. Abramov*, V.N. Tulupenko*
Institute of Physics, NAS of Ukraine, 46 prospect Nauki, 03028 Kyiv, Ukraine
Fax: +380 (44) 2651589, E-mail: poroshin@iop.kiev.ua
*Donbass State Machine-building Academy, 72 Shkadinov Str., 84313 Kramatorsk, Ukraine
E-mail: tvn@laser.donetsk.ua
Abstract. Infrared light scattering by plasmons in p-Ge has been studied under uniaxial stress
along the [110] axis with polarization of incident light parallel to the stress direction. It is
found that the deformation of the crystal results in an increase of the plasma frequency and
lowering the asymmetry of the line by plasma scattering. These effects are explained by taking
into account the change of contribution to the dielectric constant caused by the intra- and
intersubband transitions as a consequence of variation of the energy band of p-Ge related
with deformation.
Keywords: light scattering, plasmon, plasma frequency, stress, dielectric constant, germa-
nium.
Paper received 13.10.03; accepted for publication 11.12.03.
1. Introduction
Uniaxial deformation of cubic semiconductors causes
substantial changes of valence band spectrum of the holes
[1]. Degeneracy of the light and heavy hole subbands in
the point of maximum is removed and their shape changes
due to reduction of crystal symmetry. This effect results
in changes of various crystal properties related to free
carriers.
Effects of uniaxial elastic deformation on electric con-
duction, galvanomagnetic and thermomagnetic phenom-
ena have been well investigated for various p-type semi-
conductors (see, for example [2]). A variation of optical
properties for semiconductors with a degenerate valence
band in the IR spectral range under uniaxial stress has
been observed. In particular, anisotropy of such optical
crystal characteristics as the refraction index and ab-
sorption coefficient, variation of magnitude and spectral
dependencies of linear and non-linear light absorption
coefficients due to hole intersubband transitions were
found [3�5]. Besides, a number of phenomena related to
intersubband hole transitions near the subband maximum
appeared in the crystals under deformation has been con-
sidered theoretically. In particular, resonance light ab-
sorption and corresponding spectral dependencies of the
photoconductivity and photon drag effect of holes, the
Raman scattering of light with frequency shift by the ener-
gy of subband maximum splitting were considered [6�8].
The present paper reports on observation and investi-
gation of new features in light scattering by collective
oscillations of free carriers, referred to as plasmons, in p-
Ge crystals under uniaxial elastic compressive stress.
2. Experimental procedure
Light scattering by plasmons was investigated in germa-
nium samples with acceptor (Ga) concentrations of
2.3⋅1017 cm�3 and 4.5⋅1017 cm�3. The concentration of
free carriers in the samples was determined using the Hall-
effect.
The samples were cut as rectangular parallelepipeds
with all faces parallel to the crystallographic plane (110).
The uniaxial compression within the range of 0 to 2.5 kbar
was applied to a sample along the crystallographic di-
rection [110].
We used the infrared (IR) light of the high-stable CO2 �
laser with λ = 10.6 µm as exciting radiation. The laser
pulse duration was 300 ns and the repetition frequency
20 Hz. The incident IR light was polarized along the
direction of the applied stress.
The light scattering measurements were performed at
80 K in the backscattering geometry. The scattered light
426
SQO, 6(4), 2003
V.N. Poroshin et al.: Stress-induced effects in light scattering by plasmons...
was analyzed by an IR monochromator using a cooled
CdHgTe photoconductor cell and computer-controlled
system of pulse accumulation. In addition, SF6 gas filter
was mounted in front of the input slit of the mono-
chromator. It enabled to avoid transmission of the light
scattered elastically in the crystal and elements of the
optical facilities into the recording system (the ratio of
the intensities of the incident and scattered light was of
the order of 1012).
The incident laser pulse intensity at the crystal sur-
face was 300 kW/cm2. Inelastically scattered light was of
the order 10�9 to 10�8 W. At the temperature under study
all the impurity centers in germanium were ionized and
carriers were in the valence band.
3. Experimental results
Fig.1 presents typical spectra of inelastic light scattering
(Raman spectra) for the p-Ge samples measured in the
range from �150 to +150 cm�1 under uniaxial stress X.
In the absence of stress, Raman spectra coincide with
those measured earlier [9]. The line due to light scatter-
ing by collective excitations of carriers (plasmons) was
observed on the background of the Stokes side of the wide
(from 0 to ±150 cm�1) asymmetric band caused by quasi-
elastic scattering connected with single-particle unscree-
ned excitations of free holes. The latter are fluctuations
of the quadrupole momentum of holes originating from
intrasubband transitions. The plasmon line was identi-
fied in [9] by the dependence of the maximum frequency
on the carrier concentration and by the dependence of
the scattering intensity on the mutual polarization direc-
tions of the incident ei and scattered es light (the line was
observed only in the case of ei // es ).
Fig. 2 shows the reduced Raman spectrum for the non-
stressed p-Ge obtained by subtraction of the intrasubband
scattering from the complete spectrum. The intrasubband
scattering band contour was fitted by the following ex-
pression
22
)(~)(
A
A
FI
+ω
ωω ,
(1)
)(ωF = �ω/{1 � exp (� �ω / kT)}
which well describes spectra of light scattered by single-
particle excitations in the case of collisions of carriers
with crystal lattice defects, as shown in [10]. The magni-
tude of fitting parameter A was selected with regard to a
best agreement between the calculated curve and experi-
mental Raman spectrum in the frequency range from �
150 cm�1 to about 40 cm�1, where scattering by plasmons
is in fact absent. It is seen from Fig. 2 that the plasmon
line for the non-stressed crystal is wide and asymmetric.
The uniaxial compression applied to the p-Ge crystal
led to changes of the Stokes part in the observed Raman
spectrum, in particular, to an increase of scattered light
In
te
n
si
ty
,
a
rb
.
u
n
it
s
20
40
60
80
�150 �100 �50 0 50 100 150
0
0
20
40
60
80
0
20
40
60
80
3
2
1
X = 2.5 kbar
1
2
3
X = 1 kbar
w, cm �1
1
X = 0 kbar
Fig. 1. Typical electronic Raman spectra of p-Ge at 80 K for
different values of uniaxial stress along [110]. The solid lines
show the calculated spectra for intrasubband (1), intersubband
(2) and total (3) electronic scattering.
Fig. 2. Plasmon line for different stress values: 1 � 0, 2 � 1, 3 �
2.5 kbar. The dots represent experimental data while the solid
line is a fit using (6). N = 4.5×1017 cm�3.
30 60 90 120 150
0
10
20
30
In
te
n
si
ty
,
a
rb
.
u
n
it
s
1
2
3
w, cm �1
V.N. Poroshin et al.: Stress-induced effects in light scattering by plasmons...
427SQO, 6(4), 2003
intensity and to a change of the plasmon line shape. Be-
sides, at pressure X > 1 kbar maximum frequency of the
line varies. At the same time, the anti-Stokes part of the
spectrum, which is determined basically by light scatter-
ing on the single-particle excitations of free carriers, did
not change under uniaxial stress up to its maximum mag-
nitude used in experiment X = 2.5 kbar.
On the one hand, the changes observed in Raman spec-
tra are related to changes in light scattering by plasmons
under the crystal stress, and, on the other hand, to an
appearance of light scattering due to hole quadrupole
momentum fluctuations during transitions between the
valence subbands splitted by the stress near their maxima.
The splitting subband energy at the maximum point, E0,
is determined by the applied uniaxial stress: E0 = χX and
changes from 0 to 90 cm�1 with increase of X from 0 to
2.5 kbar (according to [11], χ = 39 cm�1/ kbar for X||
[110]).
We observed intersubband scattering of IR light in
uniaxially stressed samples of p-Ge with the carrier con-
centration from 5⋅1015 cm � 3 to 3⋅1016 cm�3 when the plas-
mon line did not fall in the recorded spectral range [12].
The Raman line was observed at the frequency ωmax =
= E0/�. In the range of the applied pressures under study,
the line had the Lorentz shape, and its width was propor-
tional to the pressure X and varied from about 60 cm�1 to
78 cm�1 with X varying from 1 kbar to 2.5 kbar. At the
same time, the line contour area that determines the inte-
gral cross section remains constant.
The calculated lines of intersubband light scattering
at X = 1 and 2.5 kbar are shown in Fig. 1. The sum of
both intra and intersubband components of light scatter-
ing in the observed Raman spectrum is shown here, too.
The intensity and shape of the intrasubband light scat-
tering band were taken to be the same in the calculations
for both stressed and non-stressed crystals. For the
intersubband scattering, the intensity of the line with the
known width and maximum frequency was fitted to ob-
tain a good agreement between the calculated total sin-
gle-particle spectrum and the Stokes Raman spectrum
observed experimentally for stressed samples at low fre-
quencies (below approximately 50 cm�1) where light scat-
tering by plasmons is absent.
After subtraction of the single-particle scattering com-
ponent from the observed Raman spectrum, we obtain
the plasmon line. This line is shown in Fig. 2 for different
magnitudes of applied stress. As seen from Fig. 2, under
uniaxial stress of the p-Ge crystal, the line asymmetry
for plasma light scattering decreases. The line maximum
frequency that defines the plasma frequency of carriers ωp
does not vary indeed under the applied stress X up to
X = 1 kbar. At X > 1 kbar ωp increases with growing stress
X. Fig. 3 illustrates the dependence of the plasma frequency
versus stress for the p-Ge samples. At some given magni-
tude of the stress, the magnitude of the plasma frequency
change differs for samples with different concentrations of
free carriers. For example, at X = 2.5 kbar, it is about 20
and 36 cm�1 for N = 2.3⋅1017 and N = 4.5⋅1017 cm� 3, re-
spectively.
4. Discussion
The plasma frequency of free carriers and the line shape
for light scattered by plasmon are determined by the fre-
quency dependence of the dielectric constant of the crys-
tal. For cubic p-type semiconductors, the dielectric con-
stant may be written as
ε = ε0 + εintra + εinter, (2)
where ε0 is the lattice dielectric constant, εintra describes
the dielectric constant components due to virtual intra-
subband electron transitions, the last term εinter is the
intersubband term .
Therefore, the observed behavior of plasma frequency
and the plasmon line shape in the stressed p-Ge has to be
related to the change of the dielectric constant caused by
changes in the energy band spectrum of carriers on ac-
count of deformation. The energy gap arising by the de-
formation increases with growing stress X applied to a
crystal. The subbands change their shape so that with
growing stress the warped surface of a constant energy
becomes an ellipse compressed or elongated along the
deformation axis. As a consequence, the dielectric con-
stant and, hence, plasma frequency as well as the line
shape for light scattered by plasmons become dependent
on the magnitude of applied stress.
Note that a uniaxial crystal deformation violates the
cubic symmetry in distribution of free carriers in the k-
space. Hence, the dielectric constant becomes anisotropic.
For the direction of uniaxial compressive stress of the Ge
crystals along [110] one of the main axes of the dielectric
constant tensor coincides with this direction, while two
other are directed along [110] and [001]. All three main
values of the tensor (longitudinal ε | | and transverse ε⊥1,
ε⊥2 in regard to deformation direction) are non-zero.
Since the polarization direction of incident IR light in
our experiment coincides with the deformation direction
�
1
0.0 0.5 1.0 1.5 2.0 2.5
0
10
20
30
40
2
1
X, kbar
p
p
Fig. 3. Plasma frequency vs stress in p-Ge with different hole
concentration: 1 � N = 2.3×1017, 2 � N = 4.5×1017 cm�3.
428
SQO, 6(4), 2003
V.N. Poroshin et al.: Stress-induced effects in light scattering by plasmons...
the measured plasma frequency and spectral plasmon line
dependence are defined by the longitudinal component
of the dielectric constant tensor.
The dielectric constant of uniaxially stressed p-type
semiconductors was considered in [6�7]. Its real part is
given by:
( )=ωε '
|| ( ) ∑
∂
∂
+
∂
∂
Γ+
− −
−
+
+
p
2
||
2
2
||
2
22
2
0
24
p
E
f
p
E
f
V
e
ω
πε
(3)
( )( ) ( )
( ) ( )22
2
||2
2 24
ω
ω
ω
π
h
h
−−−
−Μ−
+−+−
+−∑
EEEE
ff
V
e
p
p
where ű is the energy of carriers in split subbands, f± is
their Fermi function, Ì| | (ð) is the matrix element of
intersubband direct transitions, Ã is the effective frequency
of carrier collision with lattice defects, which is assumed
to be the same for different subbands. The second and
third terms in (3) describe the components of virtual in-
tra- and intersubband transitions, respectively.
The matrix element of intersubband transitions in
stressed crystals is given in [6] without taking into ac-
count valence band warping and anisotropy of the elas-
tic properties of crystals
( )
( )
.
4
)(
2
/26
)(
2
0220
22
2
2
2
||
++−
×
×
−
=Μ ⊥
−+
E
pp
m
E
m
p
p
EE
m
z
γγ
γ
p
(4)
The dependence of carrier energy on their momentum
p in the subbands is defined as
42
2
0
2
0
222
1 E
p
p
m
p
E
m
p
m
p
E z
z +
Ρ+
±=±
γγγ
(5)
where γ1 and γ are the Luttinger parameters of the iso-
tropic approximation, pz is the projection of carrier mo-
mentum on the direction of stress, P2 is the second
Legendre polynomial, Å0 is the splitting energy of the
subband maxima, which depends on the applied stress.
Solution of the equation ε |'| (ωð) = 0 gives the depend-
ence of the plasma frequency ωð on the magnitude of
uniaxial stress. The variation of ωð is negligible up to 1
kbar. An increase of ωð occurs at X > 1 kbar. Fig. 5
illustrates the dependence of ωð on X calculated for p-Ge
with the hole concentration N = 4.5⋅1017 cm�3. These cal-
culations do not take into account any collisions of carri-
ers (Ã = 0). The following parameters of Ge were used:
ε0 = 16, γ1 =13.35 and γ = 5.25 [11].
Such a dependence of ωð on Õ is explained by the fact
that changes of the dielectric constant components for
intra-εintra and intersubband εinter transitions caused by
the crystal deformation have different impact on the
plasma frequency (curves 1 and 2 in Fig. 5). Thus, change
of the intrasubband component εintra results in an increase
of the plasma frequency δωintra = ωintra (X) � ωintra (0) > 0,
the increase being stronger at higher X. It is basically
connected with a decrease of the effective mass of holes in
the lower subband under deformation and the transitions
of carriers to it from the upper split-off subband. The
plasma frequency change related to intersubband transi-
tions is negative (δωinter< 0). It is maximal at the pressure
about 1 kbar. That is connected with the contribution to
the dielectric constant of the carrier transitions between
0.0 0.5 1.0 1.5 2.0 2.5 3.0
�5
0
5
10
15
20
3
2
1
X, kbar
�
1
p
p
Fig. 4. Plasma frequency change in p-Ge (N = 4.5∗1017 cm� 3) vs
stress. Contributions to the dielectric constant of intrasubband
(1), intersubband (2) and both types (3) of electron transitions.
Fig. 5. Intersubband contribution to the imaginary part of the
dielectric constant of -Ge with the hole concentration N =
= 4.5×1017 cm �3 for various uniaxial stress values: 0 � 0 kbar, 1 � 1,
2 � 2, 3 � 3 kbar.
in
te
r
w, cm �1
0 30 60 90 120 150 180
0
1
2
3
4
5
6
3
2
1
0
e
''
V.N. Poroshin et al.: Stress-induced effects in light scattering by plasmons...
429SQO, 6(4), 2003
the split subbands near the maxima. The magnitude of
this contribution reaches a maximum in the range of fre-
quencies corresponding to the sum of splitting energy at
the subband maxima and average energy of carriers, as
shown in [6,7]. At X<1 kbar the plasma frequency
changes related to the intrasubband transitions δωintra
are compensated by its variation due to intersubband
component δωinter. At X>1 kbar δωintra > δωinter. There-
fore, the plasma frequency increases with the growing
pressure applied to the crystal.
The calculated dependence of the plasma frequency
on the applied pressure is similar to that obtained experi-
mentally for p-Ge. Discrepancy between calculated and
experimental values δωp is caused by the fact that the
calculations did not take into account the warping of the
valence band and the anisotropy of elastic properties of
germanium. These factors taken into account have an
impact on the magnitude of dielectric constant compo-
nents due to intra and intersubband transitions and, con-
sequently, on the plasma frequency, as shown in [7].
However, it does not change its dependence on the ap-
plied pressure.
Let us consider now the influence of a uniaxial stress
on the plasmon line shape. The spectral dependence of
the cross section for light scattering by plasmons, which
defines the line shape, is given by:
σ(ω) ~ Im [�1/ ε(ω) ] (6)
In the non-stressed p-Ge crystals the asymmetry of
the plasmon line is caused by the component of inter-
subband transitions in the imaginary part of the dielec-
tric constant ε′′inter [9]. At the same time, the line width is
determined also by collisions of carriers, which occur
mainly with impurities in the crystals under study. There-
fore, changes in the line shape under uniaxial stress must
be related to changes in the collision frequency à and the
frequency dependence of ε′′inter.
The imaginary part of the dielectric constant in the
stressed p-type crystals may be written as [6]
( )ωε "
|| ( ) +
∂
∂
+
∂
∂
Γ+
Γ= ∑ −
−
+
+
p
2
||
2
2
||
2
22
2 24
p
E
f
p
E
f
V
e
ωω
π
(7)
+ ( ) ( ) ( )∑ −+−+ −−Μ
p
p EEff
V
e δ
ω
π
||2
224
Changes in the frequency dependence of ε′′inter under
stress calculated according to (7) are shown in Fig. 4. As
can be seen, with an increasing uniaxial pressure, the
curve ε′′inter (ω) shifts to a higher frequencies and the di-
electric constant component due to intersubband transi-
tions actually decreases in the spectral range containing
the plasmon line.
The dashed line in Fig. 2 shows the spectral depend-
ence of the plasma light scattering in p-Ge calculated for
various magnitudes of the applied uniaxial stress and
taking into account the changes of ε′′inter(ω). The colli-
sion frequency of holes was chosen to obtain the best agree-
ment between the theoretical curve and experimental data.
In the pressure range 0 < X ≤ 1.5 kbar, the collision fre-
quency of holes occurred to be close to that obtained from
the carrier mobility measured under dc electric field in
non-stressed crystals Ã(0). However, at X = 2.5 kbar the
collision frequency of carriers turned out to be almost 4
times as large as Ã(0). It gives evidence of the enhanced
scattering of carriers by impurities in the pressure range
from 1.5 to 2.5 kbar. From our point of view, the effect is
caused by the transformation of the localized acceptor
states into resonance (quasi-local) states. Existence of
these states in uniaxially stressed Ge was experimentally
evidenced, for example, in [14, 15]. Such states arise due
to stresses in the crystal leading to the splitting of impu-
rity levels. One of the two split impurity levels at some
magnitude of the deformation shifts into continuous spec-
trum of energy of the lower valence subband and becomes,
therefore, quasi-local. At the same time, hole scattering
by impurities increases sharply when the quasi-local state
energy reaches the average energy of holes (resonance
scattering) [16,17].
The resonance states in uniaxially stressed germa-
nium were calculated in many investigations basing on
various approximations for the impurity potential (for
example, [16-19]). The obtained values of valence
subband splitting energy due to deformation, in the case
when the main acceptor state becomes resonance, con-
siderably differ from each other. For Ga impurity they
lie within the range from 8 to 16 meV that corresponds to
pressures between 1.8 to 3.6 kbar for stresses along [110].
The magnitude of uniaxial stress, for which an increase
of hole scattering by impurities occurs, lies in this pres-
sure range.
The authors are grateful to F.T. Vasko and O.G. Sar-
bey for the discussion of results and to V.M. Vasetski for
his assistance in carrying out the experiments.
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| id | nasplib_isofts_kiev_ua-123456789-118080 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1560-8034 |
| language | English |
| last_indexed | 2025-12-07T16:23:51Z |
| publishDate | 2003 |
| publisher | Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України |
| record_format | dspace |
| spelling | Poroshin, V.N. Gaydar, A.V. Abramov, A.A. Tulupenko, V.N. 2017-05-28T16:38:46Z 2017-05-28T16:38:46Z 2003 Stress-induced effects in light scattering by plasmons in p-type germanium / V.N. Poroshin, A.V. Gaydar, A.A. Abramov, V.N. Tulupenko // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2003. — Т. 6, № 4. — С. 425-430. — Бібліогр.: 19 назв. — англ. 1560-8034 PACS: 78.40. Fy, 73.20. Mf https://nasplib.isofts.kiev.ua/handle/123456789/118080 Infrared light scattering by plasmons in p-Ge has been studied under uniaxial stress along the [110] axis with polarization of incident light parallel to the stress direction. It is found that the deformation of the crystal results in an increase of the plasma frequency and lowering the asymmetry of the line by plasma scattering. These effects are explained by taking into account the change of contribution to the dielectric constant caused by the intra- and intersubband transitions as a consequence of variation of the energy band of p-Ge related with deformation. The authors are grateful to F.T. Vasko and O.G. Sarbey for the discussion of results and to V.M. Vasetski for his assistance in carrying out the experiments. en Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України Semiconductor Physics Quantum Electronics & Optoelectronics Stress-induced effects in light scattering by plasmons in p-type germanium Article published earlier |
| spellingShingle | Stress-induced effects in light scattering by plasmons in p-type germanium Poroshin, V.N. Gaydar, A.V. Abramov, A.A. Tulupenko, V.N. |
| title | Stress-induced effects in light scattering by plasmons in p-type germanium |
| title_full | Stress-induced effects in light scattering by plasmons in p-type germanium |
| title_fullStr | Stress-induced effects in light scattering by plasmons in p-type germanium |
| title_full_unstemmed | Stress-induced effects in light scattering by plasmons in p-type germanium |
| title_short | Stress-induced effects in light scattering by plasmons in p-type germanium |
| title_sort | stress-induced effects in light scattering by plasmons in p-type germanium |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/118080 |
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