Influence of polarization of free-electron system in semiconductor on the position of minimum in plasma light reflection
The dielectric permeability of the free-electron system in semiconductor is usually considered using the Drude-Lorentz model without taking into account this system polarization. But it seems reasonable to include polarization phenomena into consideration of the free-electron system behavior. In...
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Severin, V.S. 2017-05-26T12:50:11Z 2017-05-26T12:50:11Z 2011 Influence of polarization of free-electron system in semiconductor on the position of minimum in plasma light reflection / V.S. Severin // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2011. — Т. 14, № 2. — С. 175-178. — Бібліогр.: 15 назв. — англ. 1560-8034 PACS 77.22.Ch, Ej, 78.20.Bh, Ci, -e https://nasplib.isofts.kiev.ua/handle/123456789/117710 The dielectric permeability of the free-electron system in semiconductor is usually considered using the Drude-Lorentz model without taking into account this system polarization. But it seems reasonable to include polarization phenomena into consideration of the free-electron system behavior. In this paper, the position of minimum in plasma optical reflection by the system of free electrons is analyzed with allowance for this system polarization. This position can substantially differ from that given via calculation of it within the framework of the traditional Drude-Lorentz model. This difference is significant when analyzing the available experimental results. en Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України Semiconductor Physics Quantum Electronics & Optoelectronics Influence of polarization of free-electron system in semiconductor on the position of minimum in plasma light reflection Article published earlier |
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Influence of polarization of free-electron system in semiconductor on the position of minimum in plasma light reflection |
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Influence of polarization of free-electron system in semiconductor on the position of minimum in plasma light reflection Severin, V.S. |
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Influence of polarization of free-electron system in semiconductor on the position of minimum in plasma light reflection |
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Influence of polarization of free-electron system in semiconductor on the position of minimum in plasma light reflection |
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Influence of polarization of free-electron system in semiconductor on the position of minimum in plasma light reflection |
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Influence of polarization of free-electron system in semiconductor on the position of minimum in plasma light reflection |
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influence of polarization of free-electron system in semiconductor on the position of minimum in plasma light reflection |
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Severin, V.S. |
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Severin, V.S. |
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2011 |
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English |
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Semiconductor Physics Quantum Electronics & Optoelectronics |
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Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України |
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Article |
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The dielectric permeability of the free-electron system in semiconductor is
usually considered using the Drude-Lorentz model without taking into account this
system polarization. But it seems reasonable to include polarization phenomena into
consideration of the free-electron system behavior. In this paper, the position of
minimum in plasma optical reflection by the system of free electrons is analyzed with
allowance for this system polarization. This position can substantially differ from that
given via calculation of it within the framework of the traditional Drude-Lorentz model.
This difference is significant when analyzing the available experimental results.
|
| issn |
1560-8034 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/117710 |
| citation_txt |
Influence of polarization of free-electron system in semiconductor on the position of minimum in plasma light reflection / V.S. Severin // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2011. — Т. 14, № 2. — С. 175-178. — Бібліогр.: 15 назв. — англ. |
| work_keys_str_mv |
AT severinvs influenceofpolarizationoffreeelectronsysteminsemiconductoronthepositionofminimuminplasmalightreflection |
| first_indexed |
2025-11-24T16:25:17Z |
| last_indexed |
2025-11-24T16:25:17Z |
| _version_ |
1850482358843604992 |
| fulltext |
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2011. V. 14, N 2. P. 175-178.
PACS 77.22.Ch, Ej, 78.20.Bh, Ci, -e
Influence of polarization of free-electron system in semiconductor
on the position of minimum in plasma light reflection
V.S. Severin
National Aviation University, Chair of theoretical physics
1, Cosmonaut Komarov prospect, 03058 Kyiv, Ukraine
E-mail: severinvs@ukr.net
Abstract. The dielectric permeability of the free-electron system in semiconductor is
usually considered using the Drude-Lorentz model without taking into account this
system polarization. But it seems reasonable to include polarization phenomena into
consideration of the free-electron system behavior. In this paper, the position of
minimum in plasma optical reflection by the system of free electrons is analyzed with
allowance for this system polarization. This position can substantially differ from that
given via calculation of it within the framework of the traditional Drude-Lorentz model.
This difference is significant when analyzing the available experimental results.
Keywords: dielectric permeability, polarization, optical properties, reflection of light.
Manuscript received 06.05.10; accepted for publication 16.03.11; published online 30.06.11.
1. Introduction
Researching the plasma reflection of light by free
electrons in semiconductor is one of the methods to
determine electron effective mass m, concentration n and
the high-frequency dielectric permeability of crystal
lattice [1-6]. 0ε
The light frequency value to produce the luminous
reflectance minimum is a matter of principle in this
method. The traditional Drude-Lorentz model is now
utilized to find the above mentioned frequency [1-6].
Optical properties of substance are given by means
of its dielectric permeability expressed in terms of this
substance specific internal conductivity. However, the
substance specific external conductivity is traditionally
used in the dielectric permeability. It is the
approximation not taking this substance polarization into
account [7, 8].
The internal conductivity of a system of free
electrons can considerably differ from the conductivity
of the Drude-Lorentz model subject to the light
frequency [7, 8]. The Drude-Lorentz model does not
consider the above polarization and can be applied only
in the approximation of the large light frequency ω,
when the condition is satisfied, where ω22
pω>>ω p is
the plasma frequency [7, 8].
The dielectric permeability of the free-electron
system in semiconductor is considered taking the
polarization of such a system into account and in the
Drude-Lorentz model that does not consider this system
polarization in the second section of this paper.
The location of the plasma minimum in luminous
reflectance of the free-electron system is considered
taking the polarization of this system into account and
without this account in the third section. This location
can substantially differ from the position given via
calculation of it within the framework of the traditional
Drude-Lorentz model. This difference is substantial
when analyzing experimental results.
2. Dielectric permeability of the Drude-Lorentz
model. Inclusion of polarization into the dielectric
permeability
The Drude-Lorentz model examines free within an
energy band electrons as an electronic gas neutralized by
a positive crystal lattice background having the dielectric
permeability 0ε not depending on the light frequency.
The motion equation of an electron being under action of
the external electric field applied to a substance
has the following look in this model:
)(tD
)(1)()(
0
tt
m
e
dt
td vDv
τ
−
ε
= . (1)
Here, t is time; e is the electron charge; is its
velocity;
)(tv
γ=τ 1 is the electron momentum relaxation
© 2011, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
175
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2011. V. 14, N 2. P. 175-178.
time taking into account this electron scattering by
crystal lattice.
The electric current of these electrons having the
density arises under action of this field
. Here, n is the electron concentration.
)()( tent vj =
)(tD
The external electric field )exp()( tit ω= ωDD
having the amplitude creates the alternating current
of these electrons with the amplitude of electric current
density . The equation (1) implies the following form
of the linear response of this current to the external
field D:
ωD
ωj
ωω ε
ω= Dj
0
1)(s ,
where
)1(
)(
2
ωτ+
τ
=ω
im
nes (2)
is the specific conductivity of free electrons within the
Drude-Lorentz model. This conductivity is the external
one in compliance with its calculation.
The following dielectric permeability is
traditionally used to consider optical properties of free
electrons [1-6]:
( )⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
γ−ωω
ω
−ε=ω
ω
π
+ε=ωε
i
s
i
p
S
2
0 1)(4)( 0 , (3)
where
m
ne
p
0
2
2 4
ε
π
=ω
is the plasma frequency squared. This dielectric
permeability is based on the Drude-Lorentz model.
However, the Drude-Lorentz model does not
consider screening the external field D by mobile
electrons of a substance. This screening begins to reveal
itself at the time of the order of [9]. Therefore,
at the time of (that is at the light frequency of
) this substance conductivity electrons will be
under action of not the external in relation to them field
1−ω= ppt
ptt >
pω<ω
0εD , but they will experience action of the field
0ε≠ DE , which is the internal field in the system of
these electrons. The difference between the fields and E
0εD is caused by the vector of polarization P for the
system of electrons [9-14]:
( ) (
© 2011, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
)πε−= 40EDP . (4)
This difference is not considered when deriving the
expression (3) for the dielectric permeability )(ωεS .
Substance optical properties on the light frequency
ω are defined by its dielectric permeability )(ωε .
Derivation of the expression (3) for the substance
dielectric permeability does not consider screening the
external field by this substance charges that are optically
active at the light frequency ω under study.
Indeed, the current density j can be expressed both
through the internal electric field E and the field 0εD
[9, 10, 12-14]
0)()( εω=ωσ= ωωω DEj s , (5)
where is the amplitude of internal electric field. The
conductivity
ωE
)(ωs determines the current response to the
external field 0εD (with respect to substance charges
being optically active at the light frequency ω under
study), but the conductivity gives the current
response to the internal field . Consequently, the
physical quantity
)(ωσ
E
)(ωs is named as the substance
external conductivity and the physical quantity )(ωσ is
named as the internal one [12-14]. The distinction of the
conductivity )(ωσ from the conductivity )(ωs in the
expression (5) is caused by the difference of the
polarization P (4) from zero.
Substance charges not partaking in optical
transitions in the studied frequency interval create the
frequency-independent background contribution 0ε to
this substance dielectric permeability [1, 11]. Therefore,
the expression for this substance dielectric permeability
)(ωε has the following form:
)(4)( 0 ωσ
ω
π
+ε=ωε
i
, (6)
The expression (6) is derived from the relationship:
ωω ωε= ED )( , (7)
which is the definition of substance dielectric
permeability )(ωε , and from the relations (4), (5) [9-14].
The expression
)()( t
dt
dt Pj = (8)
is used under this derivation [9-14].
The definition of substance dielectric permeability
requires the presence of this substance internal
conductivity )(ωσ in it, but not the external one )(ωs
[9-14]. For simplicity, the quantities σ and s are here
supposed as scalar quantities, but not as tensor ones. The
formulas (5), (7) give the next relationship:
0
1)()()(
ε
ωεω=ωσ s . (9)
In addition, the formulas (6), (9) give the following
relation:
)(
)(41
1)(
0
ω
ω
ωε
π
−
=ωσ s
s
i
. (10)
After substituting the relation (10) into the formula
(6) and taking the formula (2) into account, the next
relationship is resulted:
176
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2011. V. 14, N 2. P. 175-178.
( )γ−ωω
ω
+
ε
=
ω
ωε
π
−
ε
=ωε
i
s
i
p
2
0
0
0
1)(41
)( . (11)
The expression (11) for the dielectric permeability
taking the above-stated polarization into account
has a considerable difference from the approximate
expression for the dielectric permeability
)(ωε
)(ωεS (3), in
which this polarization is not taken into account.
Having expanded the right side of the expression
(11) into series by powers of 2)( ωω p , we ascertain the
validity of the following relationship:
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎝
⎛
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
ω
ω
+ωε=ωε
4
)()( p
S O . (12)
Therefore, the dielectric permeability )(ωεS (3)
not taking the polarization influence into account can be
used only at the approximation of large frequencies, as
soon as the condition is satisfied. 22
pω>>ω
The formula (3) gives the following expressions for
real and imaginary parts of : )(ωεS
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
ω+γ
ω
−ε=ωε=ωε
22
2
01 1))(Re()( p
SS , (13)
22
2
02 ))(Im()(
ω+γ
γ
ω
ω
ε=ωε−=ωε p
SS . (14)
However, the formula (11) gives the real part and
the imaginary part of as follows: )(ωε
( )
( ) ,
)(1
)(1
1
))(Re()(
2222
22
0
1
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
ωω+ω+γ
ωω+ω
−ε=
=ωε=ωε
p
pp (15)
( ) .
)(1
))(Im()(
2222
2
0
2
ωω+ω+γ
γ
ω
ω
ε=
=ωε−=ωε
p
p (16)
3. Minimum of plasma optical reflection
The reflectivity of a body for the normal light incidence
is given by the next expression [1, 3-6]: )(ωR
22
22
)()1)((
)()1)(()(
ω++ω
ω+−ω
=ω
KN
KNR . (17)
Here,
⎟
⎠
⎞⎜
⎝
⎛ ωε+ωε+ωε−=ω )()()(
2
1)( 2
2
2
11K (18)
is the absorption index of body substance;
⎟
⎠
⎞⎜
⎝
⎛ ωε+ωε+ωε=ω )()()(
2
1)( 2
2
2
11N (19)
is the refraction index of it.
Under the assumption of weak light absorption,
when the condition (or
) is satisfied, the formula (17) gives
)()( 22 ω>>ω KN
)()( 2
2
2
1 ωε>>ωε
2
1)(
1)()( ⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
+ω
−ω
=ω
N
NR . (20)
It is already here )()( 1 ωε=ωN . In this case, the
minimum value of reflectivity occurs under
the following condition:
0)( =ωR
1)(1 =ωε . (21)
The weak light absorption by a system of free
electrons takes place at the fulfillment of the condition
. The substitution of dielectric permeability 22 γ>>ω
)(1 ωεS derived from the formula (13) at the
approximation 0=γ into the equation (21) gives the
reflectivity minimum at the following frequency [1,
5, 6]:
10
0
1 −ε
ε
ω=ω p . (22)
This frequency value that is derived using the
Drude-Lorentz model to obtain the dielectric
permeability )(ωεS does not consider the influence of
polarization.
The substitution of the dielectric permeability (15)
)(1 ωε into the equation (21) (at the approximation
0=γ ), under which derivation the influence of
polarization is taken into account, gives the reflectivity
minimum at the following frequency:
1
1
0
2
−ε
ω=ω p . (23)
It follows from the formulas (22), (23) that there is
0
12
1
ε
ω=ω . (24)
4. Discussion
The electric field in a substance influencing on charge
carriers differs from the external one because of this
substance polarization.
The available methods to account the substance
polarization: Clausius-Mossotti, Lorentz-Lorenz, Ewald,
Onsager [2, 3, 10, 15] examine the difference between
the electric field in a substance, which influences on
atoms (molecules) of this substance, and an external
electric field as if this difference is caused by the
substance polarization at the location of atoms
(molecules).
© 2011, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
177
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2011. V. 14, N 2. P. 175-178.
These methods, in essence, utilize models of the
spatial localization of electric charges and are unsuitable
for mobile charges delocalized in space. Electrons being
free within the energy band of a solid are exactly such
charges. The Drude-Lorentz model that is traditionally
used to study optical properties of these electrons does
not take the influence of substance polarization into
account and can be applied in the approximation of only
high frequencies . 22
pω>>ω
The examination method of the influence of
substance polarization used in this paper is applicable to
free electrons and allows using the results of the
traditional Drude-Lorentz model. The position of
minimum of plasma optical reflection obtained within
the framework of this method substantially differs from
the position of this minimum obtained within the
framework of the traditional Drude-Lorentz model. For
example, it is for germanium [3, 6] and the
formula (24) gives in that case.
160 =ε
21 4ω=ω
This disagreement must be taken into account when
determining the free electron effective mass,
concentration and the high-frequency dielectric
permeability of crystal lattice by means of reflection
spectra for free electrons.
Experiment gives the minimum in optical reflection
of free electrons at the wavelength λ1 = 5.7 μm for the
case of n-Ge having the electron concentration
(p. 249 [6]). 319 cm1009.8 −⋅
The formula (22) no considering the influence of
substance polarization gives the plasma frequency
(which is designated in this case as ) psω
0
0
10
0
1
121
ε
−ε
λ
π
=
ε
−ε
ω=ω
c
ps , (25)
where c is the velocity of light. The expression (25)
gives under ε115 s102.3 −⋅=ω ps 0 = 16.
The formula (23) considering the influence of
substance polarization gives the plasma frequency
(which is designated in this case as ) pω
121 0
1
02 −ε
λ
π
=−εω=ω
c
p . (26)
The expression (26) gives under
ε
116 s1028.1 −⋅=ω p
0 = 16. Evidently that there is the substantial difference
between values and . pω psω
The electron effective mass determines the plasma
frequency value. Having designated
s
ps m
ne
0
2
2 4
ε
π
=ω ,
m
ne
p
0
2
2 4
ε
π
=ω ,
we obtain the next relationship between effective masses
0
2
¦Е
1
¦Ш
¦Ш
=⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
=
p
ps
sm
m . (27)
Here ms is the electron effective mass compared
with the plasma frequency ωps; m is the electron
effective mass compared with the plasma frequency ωp.
The formula (27) gives the value m/ms = 0.06 for
the case of ε0 = 16. Results having a difference up to
60% were obtained under the experimental
determination of the electron effective mass for Ge by
means of reflection spectra (p. 250-251 [6]). These
results differ from values obtained by means of electric
measurements by 25% (p. 251 [6]). Moreover, the
comparison of them with results obtained from the
cyclotron resonance gives a difference more than twice
as much (p. 253 [6]). Possibly, such distinctions are the
display of the polarization influence not taken into
account when treating the experimental results.
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© 2011, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
178
V.S. Severin
2. Dielectric permeability of the Drude-Lorentz model. Inclusion of polarization into the dielectric permeability
3. Minimum of plasma optical reflection
4. Discussion
|