Anisotropy and non-linearity of absorption of an intensive IR light by free electrons in germanium
The anisotropy and non-linearity of absorption of the intensive CO₂ laser radiation by free electrons in germanium has been found. The effect is caused by redistribution of electrons among equivalent valleys occurring due to non-equal heating electrons in the valleys which have different orientation...
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Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України
2005
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irk-123456789-1209622017-06-14T03:03:45Z Anisotropy and non-linearity of absorption of an intensive IR light by free electrons in germanium Vasetskii, V.M. Ignatenko, V.A. Poroshin, V.N. The anisotropy and non-linearity of absorption of the intensive CO₂ laser radiation by free electrons in germanium has been found. The effect is caused by redistribution of electrons among equivalent valleys occurring due to non-equal heating electrons in the valleys which have different orientation in regard to the electric field of the light wave. 2005 Article Anisotropy and non-linearity of absorption of an intensive IR light by free electrons in germanium / V.M. Vasetskii, V.A. Ignatenko, V.N. Poroshin // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2005. — Т. 8, № 3. — С. 12-15. — Бібліогр.: 11 назв. — англ. 1560-8034 PACS: 78.20.-e, 78.20.Fy http://dspace.nbuv.gov.ua/handle/123456789/120962 en Semiconductor Physics Quantum Electronics & Optoelectronics Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України |
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The anisotropy and non-linearity of absorption of the intensive CO₂ laser radiation by free electrons in germanium has been found. The effect is caused by redistribution of electrons among equivalent valleys occurring due to non-equal heating electrons in the valleys which have different orientation in regard to the electric field of the light wave. |
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Vasetskii, V.M. Ignatenko, V.A. Poroshin, V.N. |
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Vasetskii, V.M. Ignatenko, V.A. Poroshin, V.N. Anisotropy and non-linearity of absorption of an intensive IR light by free electrons in germanium Semiconductor Physics Quantum Electronics & Optoelectronics |
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Vasetskii, V.M. Ignatenko, V.A. Poroshin, V.N. |
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Vasetskii, V.M. |
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Anisotropy and non-linearity of absorption of an intensive IR light by free electrons in germanium |
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Anisotropy and non-linearity of absorption of an intensive IR light by free electrons in germanium |
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Anisotropy and non-linearity of absorption of an intensive IR light by free electrons in germanium |
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Anisotropy and non-linearity of absorption of an intensive IR light by free electrons in germanium |
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Anisotropy and non-linearity of absorption of an intensive IR light by free electrons in germanium |
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anisotropy and non-linearity of absorption of an intensive ir light by free electrons in germanium |
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Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України |
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Anisotropy and non-linearity of absorption of an intensive IR light by free electrons in germanium / V.M. Vasetskii, V.A. Ignatenko, V.N. Poroshin // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2005. — Т. 8, № 3. — С. 12-15. — Бібліогр.: 11 назв. — англ. |
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Semiconductor Physics Quantum Electronics & Optoelectronics |
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AT vasetskiivm anisotropyandnonlinearityofabsorptionofanintensiveirlightbyfreeelectronsingermanium AT ignatenkova anisotropyandnonlinearityofabsorptionofanintensiveirlightbyfreeelectronsingermanium AT poroshinvn anisotropyandnonlinearityofabsorptionofanintensiveirlightbyfreeelectronsingermanium |
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2025-07-08T18:56:05Z |
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Semiconductor Physics, Quantum Electronics & Optoelectronics, 2005. V. 8, N 3. P. 12-15.
© 2005, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
12
PACS: 78.20.-e, 78.20.Fy
Anisotropy and non-linearity of absorption of intensive IR light
by free electrons in germanium
V.M. Vasetskii, V.A. Ignatenko, V.N. Poroshin
Institute of Physics, NAS of Ukraine, 46, prospect Nauky, 03028 Kyiv, Ukraine
Fax: (38 044) 525-15-89, e-mail: poroshin@iop.kiev.ua
Abstract. The anisotropy and non-linearity of absorption of the intensive CO2 laser
radiation by free electrons in germanium has been found. The effect is caused by
redistribution of electrons among equivalent valleys occurring due to non-equal heating
electrons in the valleys which have different orientation in regard to the electric field of
the light wave.
Keywords: multivalley semiconductors, light absorption by free carriers, anisotropy,
non-linearity, heating carriers, intervalley redistribution.
Manuscript received 05.07.05; accepted for publication 25.10.05.
1. Introduction
Light absorption by free carriers in the cubic
multivalley semiconductors (Ge, Si) is isotropic. The
anisotropy of absorption arises with violation of a
uniform distribution of electrons over the equivalent
valleys because of lowering the symmetry of their
distribution in the wavevector space. This phenomenon
was observed in n-Ge under elastic directional defor-
mation of a crystal as well as in the course of heating
carriers by electric field [1, 2].
Redistribution of carriers among the valleys may be
caused by a light wave itself in the case when its
intensity is sufficiently high [3]. Earlier we have found
experimentally the birefringence [4] and the four-wave
interaction (FWI) [5] for the CO2 laser radiation in Ge
crystals with the carrier concentration N = 5·1016 cm−3 at
the temperatures 300 and 77 K. These effects were rela-
ted with the refraction index variation caused by inter-
valley redistribution of electrons in a light wave field.
The main cause of redistribution consists in different
heating electrons during light absorption by them in
different valleys. Therefore, this redistribution exists
only in the case when the light wave electric field E is
directed asymmetrically in regard to the long valley axes
in a crystal. At the same time, the extent of the
intervalley redistribution depends on the light intensity.
Therefore, the light absorption by electrons must be both
anisotropic and non-linear. A theoretical analysis of the
anisotropy and non-linearity of light absorption by free
electrons caused by the intervalley redistribution was
carried out in [6]. In this paper, we report on
experimental demonstration and study of this
phenomenon in Ge crystals under irradiation by intense
IR light with the wavelength 10.6 μm.
We studied the reciprocal transmission Λ−1= Iin / Iout
(Iin and Iout are the intensities of the input and output light)
in the n-type germanium samples as a dependence on the
Iin for two orientations of the light wave electric field in
the crystal Е||(001) and Е||(111). The samples were cut
from the germanium crystals in which the birefringence
and FWI induced by the CO2 laser radiation were
observed earlier. They have a shape of plane-parallel
plates, two plates of which coincided with the (110)
crystallographic plane of germanium. These surfaces were
provided with antireflecting coatings. The linearly polari-
zed light propagated along the [110] axis of the sample.
The sample thickness in this direction was 0.5 cm.
The source of IR radiation was either the pulsed
periodic CO2 laser with the longitudinal electric
discharge and Q-switching or the TEA-CO2 laser
operating in the single pulse regime. The output pulse
power was 10 kW for the first laser and 1 MW for the
second one. The radiation pulse durations was about 300
and 100 ns, respectively. The pyroelectric detectors with
the time resolution better than 3 ns measured an intensity
of input and output light. A laser beam was focused on
the sample surface by the BaF2 lens. The light intensity
was varied from 1 to about 30 MW/cm2 by the calibrated
CaF2 filters and/or by varying the distance between the
lens and the sample. The measurements were carried out
at the temperatures 300 and 77 K. Since the light
intensity after passing through a crystal has changed
slightly, we calculated the absorption coefficient from
the measured transmittance using the Bouguer formula.
An error did not exceed 2 %.
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2005. V. 8, N 3. P. 12-15.
© 2005, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
13
Fig. 1. The coefficient of light absorption by free electrons in Ge
as a function of the IR light intensity for the light polarization
vector oriented along the (100) and (111) crystallographic axes.
The coefficient is normalized to the value corresponding to a
low intensity I = 10 kW/cm2. Circles – experiment, dashed
lines – calculations. The carrier concentration: N = 5 10 16 cm−3.
The light wavelength λ = 10.6 μm.
Fig. 1 shows the dependence of the absorption
coefficient K of the studied n-Ge crystals on the
intensity of the incident IR radiation of the CO2 laser Iin
with 10.6-μm wavelength. The value of K is normalized
to its magnitude at the low intensity K0 (I ≈ 10 kW/cm2).
It is seen that, at the crystal temperature 300 K and the
low intensities of light (I ≤ 12 MW/cm2), the absorption
coefficients are essentially equal for both field
orientations of a light wave under study (E||(111) and
E||(001)). However, with the high intensities of light
I ≥ 12 MW/cm2, the value of K for E||(001) becomes
larger than that for E||(111). At the crystal temperature
77 K, this difference is observed for all the light
intensities under study. Along with the dependence of
the absorption coefficient on the electric field orientation
of a light wave one observes also its increase with the
light intensity growth. This is indicative of the fact that
the absorption of light by free carriers in germanium is
anisotropic and non-linear at the IR light intensities
under consideration.
An increase of the absorption coefficient when
E||(111) and E||(001) is different. With increasing the
light intensity from 2 to 29 MW/cm2 at 300 K it equals
12 and 15 %, respectively, it equals 4 and 11 % while
77 K. As a consequence, the anisotropy of absorption
coefficients ( ( ) ( )111001 KK − ) increases with a growth of
the light intensity. At the maximal intensity
Iin ≈ 29 MW/cm2, it equals to about 3 % at 300 K. With
lowering the crystal temperature the anisotropy increa-
ses, and at 77 K for the same light intensity it is already
about 7 %, almost 2.3 times higher than at 300 K.
Let us consider the mechanism of these phenomena.
In germanium for E||(001), an intervalley redistribution
of electrons is absent because the valleys are oriented
symmetrically in regard to E and carriers in them are
heated up equally [7]. The heating rate increases with a
growth of the light intensity. Since with growing the
average electron energy (the electron temperature) the
electron scattering changes, the light absorption depends
on its intensity. For the light with E||(111), the heating
electrons leads to their transition from three valleys of
the )111( type into the valley of (111) type. For this
reason, the effective mass of carriers along the E
direction changes. Along with the changes in their
scattering, this causes a non-linearity of light absorption.
Because of the intervalley redistribution, the light
absorption by electrons becomes also anisotropic.
Let us calculate the absorption coefficient of the CO2
laser IR radiation by free electrons in Ge crystals, taking
into account the heating electrons in the light wave field
and the intervalley redistribution. We showed earlier [8]
that, at the temperatures 300 and 77 K, and under
conditions of equilibrium between free carriers and a
crystal lattice, in Ge crystals the light absorption by
these carriers at the wavelength 10.6 μm is mainly
connected with scattering the carriers by the acoustic and
optical phonons. The part of light absorption coefficient
related with the impurity scattering equals only 1.5 % of
the total absorption coefficient at 300 K and 6.5 % at
77 K. In [8, 9], obtained was the expression for the
coefficient of light absorption by free electrons in the
multivalley semiconductors through these scattering
mechanisms for the case when the electron and lattice
temperatures differ from each other. For scattering by
the acoustic phonons it is written as
( ) ( )
( )
( )
( ) ( ),cossin1
1
3
16
ac
212
32/1
2
ac
ii
l
t
i
tt
Z
i
i
ei
i
Z
T
T
Tm
e
kTN
c
eK i
Ψ⋅⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
Θ+Θ×
×−⋅=
−
−∑
τ
τ
α
τ
ωγε
π
h
(1)
( ) ( )
( ) ,2
2
2/1
0
2/3
2/1
0
2/1
ac
dxeZxx
dxeZxxZZ
x
i
x
iii
−
∞
−
∞
+++
+⋅=Ψ
∫
∫
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2005. V. 8, N 3. P. 12-15.
© 2005, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
14
where ω is the light frequency; iθ is the angle between
the axis of rotation of the i-th valley and the light
polarization vector E; Ni and ( )i
eT are the electron
concentration and temperature in the valleys;
( )i
ei TT=γ , ( )i
ei kTZ /ωh= , ( )Tlτ and ( )Ttτ are,
respectively, the longitudinal and transverse components
of the relaxation time tensor for electrons with the
energy kT when scattering by the acoustic phonons takes
place [10].
The light absorption coefficient when one deals with
the scattering electrons by non-polar optical phonons
equals to [8]
( ) ( )[ ]
( ) ( ) ( )[ ]
+×
×
⎪
⎪
⎭
⎪⎪
⎬
⎫
⎪
⎪
⎩
⎪⎪
⎨
⎧
−++−++
++++++
×
×=
−
∞
∫
∑
dxe
ZSxxZSxxN
ZSxxZSxxN
BK
x
i
i
iγ
0 2/32/12/12/3
0
2/32/12/12/3
0
opt
1
(2)
( ) ( )[ ]
( ) ( ) ( )[ ]
,
10 2/32/12/12/3
0
2/32/12/12/3
0
dxe
ZSxxZSxxN
ZSxxZSxxN
eB
x
S
i
i
i
i
γ
γ
−
∞
×
×
⎪
⎪
⎭
⎪⎪
⎬
⎫
⎪
⎪
⎩
⎪⎪
⎨
⎧
−−+−−+
++−++−
×
×+
∫
∑
where we use the following designation
( )
ii
l
t
i
il
i
N
m
m
kT
cn
mDe
B
⋅⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
+×
×=
θθ
ω
γ
ωρπ
22
3
2/32/3
0
4
2
opt
2
2/1
cossin
3
22
h (3)
Here 2
optD is the constant of interaction with the
optical phonons, N0 and 0ω are the number and
frequency of phonons, n is the refraction index,
kTS /0ωh= .
The optical absorption coefficient equals to the sum
of the coefficients of light absorption by electrons in
separate valleys, which depend on the light wave field
orientation in regard to the valley axes, the carrier
concentration in this valley and their temperature Ti. We
find the electron temperature in valleys, as usual, from
the balance of the powers added to the electron
subsystem at the light absorption and that lost due to
interaction of electrons with phonons and due to
interelectron collisions.
Multiplying the coefficient of light absorption caused
by electrons in one valley by the electromagnetic field
energy flux incident on the semiconductor, we obtain the
power transferred to the electron subsystem from a light
wave. In numerous works, reported are the expressions
for the average rate of energy losses due to the electrons
scattering by the acoustic and optical phonons. We used
such expressions in the form given in [11]. The energy
exchange between electrons of different valleys at their
collisions was calculated by the formulae presented in
[10]. The constants, that characterize the mechanisms of
electron scattering, the anisotropy parameter of
scattering by the acoustic phonons, the optical phonon
frequencies, and germanium band parameters were taken
the same as from [7].
The electron concentration in the valleys was
calculated from the equation of balance of the particle
number. The redistribution of the “hot” electrons among
valleys is assumed to occur in consequence of their
scattering by the intervalley phonons. In this case, the
equation of particle balance is written as follows [7]:
( )[ ] ( )
( ) ( ) ,1
1
2/1
0
2/112/3
2/1
0
2/112/3
dxeZxxeN
dxeZxxeN
x
M
Z
jj
x
M
Z
ii
jjM
iiM
γγ
γγ
γ
γ
−
∞
−
−
∞
−
+⎥⎦
⎤
⎢⎣
⎡ +
=++
∫
∫
(4)
where kTZ MM /ωh= ( Mω is the intervalley phonon
frequency).
Fig. 1 presents the results of calculations for the
absorption coefficient of the CO2 laser radiation by free
electrons in the n-Ge crystals studied as dependent on
the light intensity. It is seen from this figure that the
calculated values of the absorption coefficient, and its
behaviour with increasing the IR light intensity shows
good agreement with the experimental data for both
orientations of the light polarization in the crystal. This
is indicative of the fact that, in the case of n-Ge, the
anisotropy and non-linearity of the light absorption by
free electrons are caused by heating the carriers in the
light wave field and their consequent redistribution
among equivalent valleys.
Let us consider the influence only of heating
electrons on the light absorption. It corresponds to the
field orientation of a light wave along the (001) axis in
germanium, when, as already mentioned above, the
electron temperatures in all valleys and, consequently,
the valley populations are equal. As following from (1)
and (2), the dependence of the absorption coefficient on
the electron temperature is weaker in the quantum case
( ekT>>ωh ) as compared to the classic one
( ekT<<ωh )1. The calculated dependences of the
absorption coefficient on the electron temperature in n-
Ge at the lattice temperatures 300 and 77 K for the CO2
laser radiation with the polarization orientation
coinciding with the (001) axis of the crystal are shown in
1 For absorption with participation of the optical phonons one should
take into account also relationship between 0ωh and ekT
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2005. V. 8, N 3. P. 12-15.
© 2005, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
15
Fig. 2. Calculated dependence of the normalized absorption
coefficient of the IR light with the wavelength λ = 10.6 μm by
free electrons in n-Ge on the reduced electron temperature.
N = 5⋅1016 cm−3.
Fig. 2. It is seen that, at both temperatures, the light
absorption coefficient increases with growth of heating
the electrons. At equal variations of the electron
temperature, this increase is larger at 300 K as compared
to 77 K, in accordance with experimental results.
The quantum energy of the CO2 laser used in our
experiments is about 1400 K while for the light
intensities under study the electron temperature does not
exceed 400 and 120 K at the lattice temperatures of 300
and 77 K, respectively. This demonstrates that, in the
experiment, realized is an intermediate situation that
corresponds rather to the classic case at 300 K than to
the quantum one at 77 K. It seems to be connected with
different influence of heating electron on the light
absorption by electrons at these crystal temperatures.
The authors are greatly indebted to Prof. O.G. Sarbey
for fruitful discussion of the results.
References
1. A.K. Walton, Infrared modulation and energy band
parameters in multivalley semiconductors through
uniaxial stress dependence of free carrier
contribution to optical constants // Phys. status solidi
(b) 43, N1, p. 379-386 (1971).
2. K. Seeger, H. Vana, Hot-carrier infrared absorption in
n-type germanium // Ibid. 96, N 2, p. 605-610 (1979).
3. P.M. Tomchuk, A.A. Chumak, Nonlinear propa-
gation of infrared radiation in many-valley semicon-
ductors // Sov. Phys. Semicond. 19, p. 46-52 (1985).
4. V.N. Poroshin, O.G. Sarbey, V.M. Vasetskii,
Nonlinear optical phenomena related to “hot”
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redistribution of hot electrons // Semiconductor
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bey, Hot electrons in many-valley semiconductors,
Naukova Dumka, Kiev (1977) (in Russian).
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phonons in electrons heating by IR radiation in Ge //
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(1969).
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