Electric field and carrier concentration distributions in the semiconductor under photorefractive Gunn effect
In the framework of the one-dimension field model of semiconductor simultaneosly subjected to the action of carrier-warming electric field and two quasi-monochromatic light waves the authors have numerically calculated the spatial-temporal distributions of inner electric field Е(x,τ) and conductivit...
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Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України
2006
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| Zitieren: | Electric field and carrier concentration distributions in the semiconductor under photorefractive Gunn effect / P.M. Gorley, P.P. Horley, S.M. Chupyra // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2006. — Т. 9, № 1. — С. 93-96. — Бібліогр.: 8 назв. — англ. |
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irk-123456789-1215872017-06-15T03:05:20Z Electric field and carrier concentration distributions in the semiconductor under photorefractive Gunn effect Gorley, P.M. Horley, P.P. Chupyra, S.M. In the framework of the one-dimension field model of semiconductor simultaneosly subjected to the action of carrier-warming electric field and two quasi-monochromatic light waves the authors have numerically calculated the spatial-temporal distributions of inner electric field Е(x,τ) and conductivity band electrons n(x,τ) in dependence on external control parameters (intensity of the incident light waves, their wave vector, external electric field and doping impurity concentration). It was found that the device operating on the base of photorefractive Gunn effect may be controllably switched between three following operation modes: low- and high-light wave intensity as well as a transition mode. The influence of the external control parameters on the Е(x,τ) distribution was determined for each mode in question. It was shown that one could efficiently control the refractive index increment nΔ by means of proper change of the control parameters. 2006 Article Electric field and carrier concentration distributions in the semiconductor under photorefractive Gunn effect / P.M. Gorley, P.P. Horley, S.M. Chupyra // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2006. — Т. 9, № 1. — С. 93-96. — Бібліогр.: 8 назв. — англ. 1560-8034 PACS 85.30.Fg, 42.70.Nq, 42.65.Sf http://dspace.nbuv.gov.ua/handle/123456789/121587 en Semiconductor Physics Quantum Electronics & Optoelectronics Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України |
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In the framework of the one-dimension field model of semiconductor simultaneosly subjected to the action of carrier-warming electric field and two quasi-monochromatic light waves the authors have numerically calculated the spatial-temporal distributions of inner electric field Е(x,τ) and conductivity band electrons n(x,τ) in dependence on external control parameters (intensity of the incident light waves, their wave vector, external electric field and doping impurity concentration). It was found that the device operating on the base of photorefractive Gunn effect may be controllably switched between three following operation modes: low- and high-light wave intensity as well as a transition mode. The influence of the external control parameters on the Е(x,τ) distribution was determined for each mode in question. It was shown that one could efficiently control the refractive index increment nΔ by means of proper change of the control parameters. |
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Gorley, P.M. Horley, P.P. Chupyra, S.M. |
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Gorley, P.M. Horley, P.P. Chupyra, S.M. Electric field and carrier concentration distributions in the semiconductor under photorefractive Gunn effect Semiconductor Physics Quantum Electronics & Optoelectronics |
| author_facet |
Gorley, P.M. Horley, P.P. Chupyra, S.M. |
| author_sort |
Gorley, P.M. |
| title |
Electric field and carrier concentration distributions in the semiconductor under photorefractive Gunn effect |
| title_short |
Electric field and carrier concentration distributions in the semiconductor under photorefractive Gunn effect |
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Electric field and carrier concentration distributions in the semiconductor under photorefractive Gunn effect |
| title_fullStr |
Electric field and carrier concentration distributions in the semiconductor under photorefractive Gunn effect |
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Electric field and carrier concentration distributions in the semiconductor under photorefractive Gunn effect |
| title_sort |
electric field and carrier concentration distributions in the semiconductor under photorefractive gunn effect |
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Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України |
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2006 |
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http://dspace.nbuv.gov.ua/handle/123456789/121587 |
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Electric field and carrier concentration distributions in the semiconductor under photorefractive Gunn effect / P.M. Gorley, P.P. Horley, S.M. Chupyra // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2006. — Т. 9, № 1. — С. 93-96. — Бібліогр.: 8 назв. — англ. |
| series |
Semiconductor Physics Quantum Electronics & Optoelectronics |
| work_keys_str_mv |
AT gorleypm electricfieldandcarrierconcentrationdistributionsinthesemiconductorunderphotorefractivegunneffect AT horleypp electricfieldandcarrierconcentrationdistributionsinthesemiconductorunderphotorefractivegunneffect AT chupyrasm electricfieldandcarrierconcentrationdistributionsinthesemiconductorunderphotorefractivegunneffect |
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2025-07-08T20:10:30Z |
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2025-07-08T20:10:30Z |
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Semiconductor Physics, Quantum Electronics & Optoelectronics, 2006. V. 9, N 1. P. 93-96.
© 2006, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
93
PACS 85.30.Fg, 42.70.Nq, 42.65.Sf
Electric field and carrier concentration distributions
in the semiconductor under photorefractive Gunn effect
P.M. Gorley, P.P. Horley, S.M. Chupyra
Yu. Fed’kovich Chernivtsi National University
2, Kotsyubynsky str., 58012 Chernivtsi, Ukraine; e-mail: semicon@chnu.cv.ua
Abstract. In the framework of the one-dimension field model of semiconductor
simultaneosly subjected to the action of carrier-warming electric field and two quasi-
monochromatic light waves the authors have numerically calculated the spatial-temporal
distributions of inner electric field Е(x,τ) and conductivity band electrons n(x,τ) in
dependence on external control parameters (intensity of the incident light waves, their
wave vector, external electric field and doping impurity concentration). It was found that
the device operating on the base of photorefractive Gunn effect may be controllably
switched between three following operation modes: low- and high-light wave intensity as
well as a transition mode. The influence of the external control parameters on the Е(x,τ)
distribution was determined for each mode in question. It was shown that one could
efficiently control the refractive index increment nΔ by means of proper change of the
control parameters.
Keywords: compensated semiconductor, drift instability, photorefractive Gunn effect,
refraction index.
Manuscript received 15.11.05; accepted for publication 15.12.05.
1. Introduction
The photorefractive effect [1] and Gunn effect [2] are
both well-known and widely used in the different fields
of electronics and sensorics. In 1996, the authors of the
paper [3] predicted a new non-linear optical effect –
photorefractive analog of the Gunn effect (photo-Gunn
effect, PhGE), representing a superposition of
photorefractive and Gunn effects. In comparison with
them, PhGE offered additional ways to control the
domain propagation process through the semiconductor
media by means of incident light wave intensity, and to
control the photorefractive coefficient by the warming
electric field. In the meantime, the investigation of the
photo-Gunn effect passes its initial stage. In particular,
the paper [4] shows that the presence of in-phase
interference structures generated by two quasi-
monochromatic waves incident on the crystal leads to
the formation of high-field domains. Papers [5] and [6]
reported investigation results regarding the self-
organization phenomena taking place under the PhGE in
GaAs semiconductor. To solve the model describing
PhGE, the authors of the original paper [3] used the
expansion of the spatial-temporal electron and field
distributions in the sample over trigonometric functions.
They derived the analytical formulas for the phase
variable increments regarding their stationary values,
considering only the first harmonics of the expansion. At
the same time, the influence of the control parameters
(such as the intensity and frequency of the incident light
wave, doping impurity concentration, carrier-warming
electric field) on the time- and coordinate-dependent
carrier concentrations and field distributions were not
investigated as well as the convergence problem for the
expansion used was considered. This paper is aimed at
the clarification of the both points mentioned, which will
contribute to the fundamental knowledge on the physics
of non-equilibrium phenomena in semiconductors and,
from the applied point of view, will favour the
development of recommendations regarding the
improvement of the technical characteristics of ultra-
high frequency electronic devices (Gunn diodes),
sensors for the visible optical range, etc.
2. Theoretical model
In this study, the authors used the model proposed in the
paper [3] to describe the carrier system under the
existence of PhGE. Let us consider partially-
compensated semiconductor with the structure of n-
GaAs, subjected simultaneously to carrier-warming
electric field and two quasi-monochromatic light waves
with slightly different frequencies ω0 and ω0+ω(ω<<ω0),
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2006. V. 9, N 1. P. 93-96.
© 2006, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
94
forming a travelling interference structure on the surface
of the sample with the intensity I(z, t) =
I0[1+mcos(Kz+ωt)], where K = 2π/Λ is the wave
number, Λ is the distance between the peaks of I(z,t), m
is the modulation depth, and I0 is the time-averaged wave
intensity. Changes of the conductivity band electrons
concentration n, ionized donors ND
i and electric field Е
with time t and space coordinate z are described in the
framework of the one-dimensional field model by the set
of partial differential equations (including continuity and
Poisson equations), written in dimensionless variables as
[5]:
211
1 )())cos(1( yyybkxma
τ
y
−−⋅+⋅+⋅=
∂
∂
ωτ ,
⎥
⎦
⎤
⎢
⎣
⎡
∂
∂
+
∂
∂
+
∂
∂
=
∂
∂
x
y
βυy
xτ
y
τ
y 2
2
12 α , (1)
)1(
1
12
3 +−−=
∂
∂
yy
αβx
y
,
with the parameters y1 = ND
i /NA, y2 = n/NA, y3 = E/Es,
τ = γ NA t, x = ε ε0 Es γ z /(eD), a = s I0 /(γ NA),
b = ND /NA, α = ε ε0 Es υs /(eD NA), β = ε ε0 Es γ/(eυs),
k = eDK/(εε0Esγ), and dimensionless electron drift velo-
city υ ≡ υ(y3)/υs =y3 (1+Ay3
3)/(1+Ay3
4). Here s is the
transversal ionization cross-section, γ and D – recom-
bination and diffusion coefficients; e is the elementary
charge, ε0 and ε – dielectric constants for the vacuum
and the semiconductor, respectively; υs is the saturation
value of electron drift velocity, Es is the saturation field;
and A is the parameter depending on material properties.
It is assumed that the coefficients D, s and γ do not
depend on both electric field and coordinate.
As it was shown in [5], the stationary space-
homogeneous (x → ∞, τ → ∞) solutions of the sys-
tem (1) y10 and y20 are
( )
( ) )).1(41(5.0
)),1(41(5.0
2
20
2
10
+−+−=
−−+−=
aabay
aabay
(2)
Stationary value of the electric field y30 in the sample
under the given electric current density j0 can be found
solving the equation j0 = y20υ(y30).
Let us look for the solution of the partial differential
equations set (1) in the form of the expansion [7]
( ) ( ) ( ) ( )( )
)3,2,1(
sincos
1
0
=
++= ∑
=
n
jkxyjkxyyy
p
j
im
nj
re
njnn τδτδ
(3)
with the weight coefficients imre
njy ,δ depending on time
but not the coordinate.
Considering three harmonics (p = 3) in the formula
(3) and substituting it into (1), and further collecting the
corresponding items by the sine and cosine of the same
arguments, one will obtain the system of 12 equations
regarding the coefficients imre
njy ,δ of the general form
( )
,
))sin(),cos(,,,,,,(),,(
,
)sin(),cos(,)(
11
1111
1
11
6
,54,3
,
2
21
2,
1
1
,
110
,
1
⎪⎭
⎪
⎬
⎫
+
++++
⎩
⎨
⎧
+Θ=
∂
∂
++
+Ψ−=
∂
∂
∏
∏∏∏
∏∏
jiji
im
pj
re
pi
im
pj
re
pijiji
ijj
imre
pj
im
pj
re
piijj
ij
im
pj
re
piij
j
imre
pjj
imre
j
ij
im
j
re
iij
j
imre
jj
imre
j
imre
j
yyyyh
yyyhyyhyh
bmakkf
y
yyhyh
yybam
y
ωτωτβαβα
τ
δ
ωτωτ
τ
δ
(4)
with the complex functions Ψ and Θ , and the
coefficients ,1
jh ,2
ijh ,3
jh ,4
ijh ,5
1ijjh 6
11 jijih , representing
a certain combinations of y10, y20, y30, α, β, k, a, m,
cos(ωτ), and sin(ωτ). The variables imre
jy ,
3δ depend on
imre
jjy ,
2,1δ according to the following expressions:
k
yy
j
y
k
yy
j
y
re
j
re
jim
j
im
j
im
jre
j αβ
δδ
δ
αβ
δδ
δ 21
3
12
3
1,1 −
=
−
= . (5)
Let us emphasize that the formulas (5) satisfy the
condition of electric neutrality of the crystal. In the
addition to (4) and (5), one will obtain also three
complementary equations for the non-trigonometric
components regarding the variables imre
njy ,δ , which were
used to verify the correctness of the numerical
calculations.
3. Calculation results and discussions
The system of equations (4) was solved numerically with
the 4th order Runge-Kutta method [8] using the
following material parameters for n-GaAs [1] (at
T = 300 K): ε = 13.2, μ = 0.5 m2V-1s-1, υs = 8.5⋅104 m/s,
Es = 1.7⋅105 V/m, γ = 10−12s−1, D = 0.0129 m2/s,
Na = 1022 m−3, A = 0.04.
It is necessary to emphasize that we have performed
the calculations for three cases taking into account one,
two and three harmonics of the expansion (3).
Comparison between the obtained data sets has proven
good convergence of the series expansion used. In
particular, consideration of the next higher-order
harmonic resulted only in the negligible (less than 2%)
changes to the dynamical variable values.
The obtained formulas (4) and (5) made it possible to
calculate time and space distributions of the band
electron concentration n and electric field E in the
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2006. V. 9, N 1. P. 93-96.
© 2006, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
95
Fig. 1. Surface plot of the field phase variable y3(x, τ).
Fig. 2. Surface plot of y3(k, lg a), calculated for x = 4.25 and
τ = 500.
sample as a function of the following control parameters:
intensity of the incident light wave а, frequency
mismatch of the quasi-monochromatic waves ω, their
wavenumber k, modulation depth m, donor doping
impurity concentration b and external electric field y30.
In Fig. 1, we have presented time-space surface plot
of the electric field y3(x, τ) for the fixed set of the control
parameters (k = 34, m = 0.1, ω = 10−4, y30 = 3, a = 10−4,
b = 25). As expected, it has a well-pronounced plane-
wave shape. As it turned out, the increase of the
frequency mismatch value ω does not influence the
oscillation amplitude, but leads to the increase of their
frequency. At the same time, the amplitude grows
linearly with the increase of y30 (within the negative
differential conductivity ranges 2.5 < y30 < 3.5) and m
(for 0 < m < 0.5). The spatial-temporal distribution of
y2(x, τ) also has a plane-wave character, but oscillations
of the carrier concentration advance those of the electric
field by π/2.
Fig. 2 presents the surface plot y3(k,a) calculated for
the sample with the normalized thickness x = 4.25 for
the fixed time τ = 500. As one can see, this surface
features three different operation modes depending on
the value of a: low intensity mode (a < 3⋅10−6), high
intensity mode (a > 1.3⋅10−5), and the transition one
(3⋅10−6 < a < 1.3⋅10−5). For these two former modes,
y3(k) curves represent two travelling waves with almost
the same amplitude, but phase-shifted by π/2. In the
transition region, the oscillation character changes
smoothly from that of low-intensity to high-intensity
mode. As it was found, both position and width of the
transition region regarding the incident light intensity
axis depend on the concentration of doping impurity and
the external electric field, namely: for the increasing b
and constant y30, the beginning of the transition region
becomes shifted towards the lower intensities. Increase
of y30 within the ranges of the negative differential
conductivity for the constant b results in decreasing the
transition region width.
It is worth noting that the electron concentration
distribution y2(k,a) features the same three regions
corresponding to different intensities plus a transition
region, with the same phase shift of π/2 between the
y2(k,a) oscillations in low- and high-intensity modes and
their smooth mutual transformation in the transition
region. The difference between y2(k,a) and y3(k,a)
surfaces can be seen in low- and high-illumination
intensity modes, where the mean field value remains
practically constant, while carrier concentration grows
with the increase of the parameter а. At the same time,
the growing speed of y2(a) curve in the high-intensity
mode is significantly larger than that for the low-light
intensities.
It is important that in both low- and high-intensity
modes one can obtain almost equal maximum values of
electric field y3 by choosing the proper value of k. This
result is of the significant application interest. As we
know, y3(x, τ, k, a) distribution determines the time- and
space-periodical changes of the refraction index
Δn(x,τ) = –0.5n3r41Esy3(x,τ) [1], where n and r41 are
refractive and electrooptical coefficients, respectively.
On the other hand, the same distribution influences the
value of Δn by means of the parameters k and a. As it
follows from the results of our numerical calculations,
the minimal values of Δn for n-GaAs presented in the
paper [3] appear to be slightly underestimated, i.e. they
are 1.3 to 1.5 and 3 to 4 times smaller for the high- and
low-intensity modes, respectively. Such a difference can
be attributed to the approximate character of the
formulas (18) and (22) in the paper [3].
4. Conclusions
This paper presents the investigation results regarding
the spatial-temporal distributions of the internal electric
field and conductivity electron concentration as a
function of the system control parameters (i.e., the
incident light wave intensity, wave vector, external
electric field and doping impurity concentration). It was
shown that both low- and high-intensity modes are
characterized with a travelling-wave form of the curve
y3(k) with almost the same amplitude but phase-shifted
by π/2. The starting point of the transition region
between these modes is determined by the concentration
of donor doping impurity, while its width can be
controlled using the applied electric field. As it follows
.
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2006. V. 9, N 1. P. 93-96.
© 2006, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
96
from our investigations, the character of y3(x, τ, k, a, b,
y30) dependence, and hence the values of integral
characteristics of the resulting device, operating on
photorefractive Gunn effect (in particular, the refractive
index increment nΔ ), can be efficiently controlled by
the proper choice of the control parameter values k, a, b
and y30. Our estimations have shown that the influence
of the latter on the dynamics of the system can be
significant and therefore experimentally detectable,
which agrees well with the conclusions of the paper [3].
This investigation was performed in the framework
of the research project GP/F11/0036 (Grant of the
President of Ukraine for the support of research work of
young scientists, 2006).
References
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