Diffusion properties of electrons in GaN crystals subjected to electric and magnetic fields
We have studied the diffusion coefficient of hot electrons in GaN crystals under moderate electric (1...10 kV/cm) and magnetic (1…4 T) fields. Two configurations, parallel and crossed fields, have been analyzed. The study was carried out for compensated bulk-like GaN samples for various lattice temp...
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| Опубліковано в: : | Semiconductor Physics Quantum Electronics & Optoelectronics |
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| Дата: | 2018 |
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Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України
2018
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| Цитувати: | Diffusion properties of electrons in GaN crystals subjected to electric and magnetic fields / G.I. Syngaivska, V.V. Koroteev, V.A. Kochelap // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2018. — Т. 21, № 4. — С. 325-335. — Бібліогр.: 42 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1860479660399263744 |
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| author | Syngaivska, G.I. Koroteev, V.V. Kochelap, V.A. |
| author_facet | Syngaivska, G.I. Koroteev, V.V. Kochelap, V.A. |
| citation_txt | Diffusion properties of electrons in GaN crystals subjected to electric and magnetic fields / G.I. Syngaivska, V.V. Koroteev, V.A. Kochelap // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2018. — Т. 21, № 4. — С. 325-335. — Бібліогр.: 42 назв. — англ. |
| collection | DSpace DC |
| container_title | Semiconductor Physics Quantum Electronics & Optoelectronics |
| description | We have studied the diffusion coefficient of hot electrons in GaN crystals under moderate electric (1...10 kV/cm) and magnetic (1…4 T) fields. Two configurations, parallel and crossed fields, have been analyzed. The study was carried out for compensated bulk-like GaN samples for various lattice temperatures (30…300 K) and impurity concentrations (10¹⁶…10¹⁷ cm⁻³). We found that at low lattice temperatures and low impurity concentrations, electric-field dependences of the transversal-to-current components of the diffusion tensor are non-monotonic for both configurations, while diffusion processes are mainly controlled by the magnetic field. With increasing the lattice temperature or impurity concentration, the behaviour of the diffusion tensor becomes more monotonous and less affected by the magnetic field. We showed that this behaviour of the diffusion processes is caused by the distinct kinetics of hot electrons in polar semiconductors with strong electron – optical phonon coupling. We have suggested that measurements of the diffusion coefficient of electrons subjected to electric and magnetic fields facilitate the identification of features of different electron transport regimes and the development of more efficient devices and practical applications.
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ISSN 1560-8034, 1605-6582 (On-line), SPQEO, 2018. V. 21, N 4. P. 325-335.
© 2018, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
325
Semiconductor physics
Diffusion properties of electrons in GaN crystals subjected to electric
and magnetic fields
G.I. Syngayivska
*
, V.V. Korotyeyev
**
, V.A. Kochelap
V. Lashkaryov Institute of Semiconductor Physics, NAS of Ukraine, 41, prospect Nauky, 03680 Kyiv, Ukraine
*
E-mail: singg@ukr.net,
**
E-mail: koroteev@ukr.net
Abstract. We have studied the diffusion coefficient of hot electrons in GaN crystals under
moderate electric (1...10 kV/cm) and magnetic (1…4 T) fields. Two configurations, parallel
and crossed fields, have been analyzed. The study was carried out for compensated bulk-
like GaN samples for various lattice temperatures (30…300 K) and impurity concentrations
(1016…1017 cm-3). We found that at low lattice temperatures and low impurity
concentrations, electric-field dependences of the transversal-to-current components of the
diffusion tensor are non-monotonic for both configurations, while diffusion processes are
mainly controlled by the magnetic field. With increasing the lattice temperature or impurity
concentration, behaviour of the diffusion tensor becomes more monotonous and less
affected by the magnetic field. We showed that this behaviour of the diffusion processes is
caused by the distinct kinetics of hot electrons in polar semiconductors with strong electron
– optical phonon coupling. We have suggested that measurements of the diffusion
coefficient of electrons subjected to electric and magnetic fields facilitate identification of
features of different electron transport regimes and development of more efficient devices
and practical applications.
Keywords: diffusion coefficient, magneto-transport properties, hot electrons, GaN, Monte
Carlo method.
doi: https://doi.org/10.15407/spqeo21.04.325
PACS 72.20.Ht, 72.20.Dp, 73.23.-b, 85.35.-p
Manuscript received 09.11.18; revised version received 27.11.18; accepted for publication
29.11.18; published online 03.12.18.
1. Introduction
The results of intensive investigations of wide-bandgap
semiconductor compounds, in particular, group-III
nitrides, such as GaN, InN, AlN and related quantum
heterostructures, can find various applications in modern
high-power and high-frequency microelectronics and
optoelectronics [1, 2]. The nitride compounds are
discussed as perspective materials for new devices like to
light-emitting diodes [3, 4], optical switches [5],
biosensors [6] and THz-active devices. The latter include
electrically pumping THz sources [7-9], detectors [10]
and modulators [5, 11]. A lot of attention is also paid to
the electron transport properties of the nitrides in
magnetic fields to develop novel devices working as
sensors and switches controlled by a magnetic field
[12-14].
In contrast to conventional AIIIBV materials,
namely: GaAs, InSb or InP, the wide-bandgap nitrides
(GaN has the bandgap 3.2 eV) are described by large
separation between the lower Г-valley and upper valleys
(~1.2…1.5 eV for GaN), the high optical phonon energy,
ћω0 (for GaN, ћω0 ≈ 92 meV), the strong electron –
optical phonon coupling (the Fröhlich constant is ~0.4 for
GaN) and the high low-field mobility (at room to liquid
nitrogen temperatures the mobility is
~1500…5000 cm2/V·s for GaN, [15]). These material
properties of the nitrides are favorable for realization of a
specific streaming-like electron transport regime, which
is characterized by a quasi-periodic electron motion in
the momentum space [16-20] due to the threshold
character of the electron – optical phonon emission. The
streaming transport regime is possible at low lattice
temperatures, T0, (kBT0 < ћω0, where kB is the Boltzmann
constant) and small electron concentrations, Ne. The
latter means that electron-electron scattering does not
control the electron kinetics.
In the papers [19, 20], it was shown that conditions
of the streaming transport regime can be realized in the
compensated high-quality bulk GaN samples. Under
streaming regime, both steady-state and high-frequency
characteristics have specific behaviour. In particular,
SPQEO, 2018. V. 21, N 4. P. 325-335.
Syngayivska G.I., Korotyeyev V.V., Kochelap V.A. Diffusion properties of electrons in GaN crystals subjected to …
326
current-voltage characteristics show saturation, while
field dependences of the diffusion coefficient
demonstrate strongly non-monotonic behaviour. In
addition, a high-frequency conductivity of electrons is
essentially anisotropic. Spectra of the high-frequency
conductivity along the steady-state field have the
oscillating behavior with appearance of the frequency
“windows” with negative values of the real part of the
conductivity. This effect is known as the optical phonon
transit-time resonance (OPTTR). The OPTTR is
considered as a perspective mechanism for electrically-
pumped amplifiers and generators in the THz frequency
range [8, 9]. The streaming regime and OPTTR can be
identified in the optical experiments by the methods of
THz-Fourier- or time-domain spectroscopies. The
peculiarities of the THz transmission/absorption spectra
for the GaN sample under conditions of OPTTR were
analyzed in [21].
Additional useful information about the streaming-
like transport regime can be obtained investigating the
galvano-magnetic characteristics at different
configurations of the electric, E, and magnetic, H, fields.
For compensated GaN, in crossed configuration of E and
H, it was found the strong effect of the magnetic field on
the streaming-like electron distribution function. In
particular, it was shown that in the range of the moderate
electric (3…10 kV/cm) and magnetic (1.5…5 T) fields,
the electron transport occurs in the form of a vortex-like
motion in the momentum space. At higher magnetic
fields, the effect of a collapse of the dissipative current
occurs due to the strong suppression of electron-optical
phonon emission [22]. The field dependences (vs E and
H) of the dissipative current, the Hall current, and the
Hall electric field were studied in details for the
compensated GaN [23].
This work continues investigations of the hot
electron transport in compensated GaN with a focus on
the diffusion processes of electrons in the real space.
These processes are actual for systems with non-uniform
electron concentrations and are described by the Fick
law:
j
e
iji
x
N
eDJ
∂
∂
−= , (1)
where Ji is i-projection of the diffusion flux density, Dij is
the diffusion coefficient tensor. Below we will study the
E- and H-field dependences of Dij.
It is well-known that under equilibrium conditions
(E = 0, H = 0) in cubic crystals, the diffusion processes
are characterized by a scalar diffusion coefficient, D,
which obeys the Einstein relationship:
e
Tk
D 0Bµ
= , (2)
where µ is the low-field electron mobility, and e is the
elementary charge. This relationship is valid for non-
degenerate electrons. For the case of the weak applied
electric fields, when the electron distribution function
remains quasi-isotropic, the diffusion coefficient can be
obtained by solving the Boltzmann transport equation
and can be expressed through the integrals of a steady-
state distribution function in the momentum space [24,
25]. Under strongly non-equilibrium conditions, two
approaches for calculations of the diffusion coefficient
are used. Both are based on the Monte Carlo simulation
of the electron transport. The first approach is based on
calculations of the variance of random distances of
travels of individual electron per unit time [26, 27]. The
obtained diffusion coefficient describes the spreading of
initially localized in space electron packet, i.e., it directly
corresponds to the Fick law. The second approach uses
computation of the velocity autocorrelation function in
the time domain [27, 28]. Both approaches give the same
results in the case of the absence of the electron-electron
correlation [29].
The experimental techniques of the diffusion
coefficient measurements are based on the electro-
gradient measurements of the thermo-electric force [30],
time-of-flight measurements [31], noise spectroscopy
[32], light-induced grating technique [33, 34], etc.
The mentioned above theoretical and experimental
studies of diffusion properties of the non-equilibrium
electron gas are dated by 1970–1980-th and concentrated
to conventional semiconductors such as Si, Ge, GaAs,
etc. In the past decades, the theoretical studies of the
diffusion properties of hot electrons were oriented on the
nitride compounds, GaN [35] and InN [36]. Particularly,
in these papers, the electric field dependences of the
diffusion coefficient were obtained for the wide range of
electric fields: E = 0...100 kV/cm. The non-monotonic
behaviour of the diffusion coefficient was identified for
both longitudinal and transversal diffusion with respect
to the electric field direction.
The present interest to the study of the diffusion
processes in a non-equilibrium electron gas is inspired, in
the particular, by the development of recent pure all-
optical pump-probe techniques, known as the light-
induced transient grating (LITG) technique [37]. This
contactless technique allows determining the diffusion
coefficient as well as the carrier recombination times by
investigating temporal changes of the diffraction
efficiency of the grating induced optically near the
surface of material.
The aim of this paper is to theoretically study the
diffusion processes of hot electrons in compensated GaN
subjected to the electric and magnetic fields of moderate
magnitudes. The paper is organized as follows. In
Section II, we briefly describe the model of the electron
transport and the Monte Carlo method used for
determination of the diffusion tensor. In Section III, we
discuss the steady-state electric characteristics for
different samples of GaN at zero magnetic field. Then,
the effect of the magnetic field on the diffusion
coefficient is considered for the parallel and crossed
configurations of electric and magnetic fields in Sections
IV and V, respectively. In Section VI, we summarize the
main results of these investigations.
SPQEO, 2018. V. 21, N 4. P. 325-335.
Syngayivska G.I., Korotyeyev V.V., Kochelap V.A. Diffusion properties of electrons in GaN crystals subjected to …
327
2. The model of the electron transport in electric and
magnetic fields
We consider the electron transport in the samples of GaN
of cubic modification under the parallel and crossed
configurations of electric, E, and magnetic, H, fields. At
E || H, we assume that E and H are directed along z-axis
(Fig. 1a). In the case HE ⊥ , it is supposed that E and H
are applied along z- and y-axis, respectively (Fig. 1b).
Here, the GaN-samples are assumed to be with short-
circuited Hall contacts.
To calculate electron transport characteristics, we
used the single-particle algorithm of the Monte Carlo
procedure [20, 22, 23, 38]. Three main scattering
mechanisms were taken into account: electron scattering
by ionized impurities, acoustic phonons and polar optical
phonons. We considered the range of the electric fields
for which the intervalley transitions to the upper valleys
are absent and the electron dispersion law can be set the
parabolic one. The electron-electron scattering was not
included into these calculations, because only small
electron concentrations were considered.
The standard single-particle Monte Carlo algorithm
is based on the stochastic simulation of the electron
trajectories in the momentum and coordinate spaces
including the scattering processes (for details, see [39,
40]). The instantaneous values of the electron velocity
(momentum and/or energy) and coordinate are recorded
at the specific moments of time and accumulated for the
further statistical processing. Motion of electrons in the
electric and magnetic fields were treated as semi-
classical. These fields were taken into account when
modelling the electron trajectory between sequential
collisions; however, the fields were not included to the
scattering probabilities.
In the case of parallel configuration for E and H, the
equations of the electron motion in momentum space are
as follows:
=
ω−=
ω=
eEp
pp
pp
z
xcy
ycx
&
&
&
(3a)
For the crossed configuration, they are read as:
ω+=
=
ω−=
xcz
y
zcx
peEp
p
pp
&
&
&
0 (3b)
Here, px, py, pz are the components of the electron
momentum, ωc is the cyclotron frequency, cmeHc
∗=ω ,
m
* and c are the effective electron mass and light
velocity, respectively.
Fig. 1. The scheme of parallel (a) and crossed (b)
configurations of E and H.
At the end of a free flight, the electron velocity is
calculated by the following equations for parallel
( ) ( )[ ] ( ) ( )[ ]
( ) ( )[ ] ( ) ( )[ ]
( ) ( )
−+=
−ω+−ω−=
−ω+−ω=
00
0000
0000
)(
cossin)(
sincos)(
tteEtptp
tttptttptp
tttptttptp
zz
cycxy
cycxx
(4a)
and crossed configurations
( )( ) ( )[ ]
( ) ( )[ ]
( )
( )( ) ( )[ ]
( ) ( )[ ]
−ω+
+−ωω+=
=
ω−−ω−
−−ωω+=
00
00
0
00
00
cos
sin)(
)(
sin
cos)(
tttp
tteEtptp
tptp
eEtttp
tteEtptp
cz
ccxz
yy
ccz
ccxx
(4b)
Here, t0 and t are, respectively, initial and final moments
of a free flight.
The calculations of the electron coordinates are
easily incorporated into the Monte Carlo scheme. The
following equations are used in our Monte Carlo
procedure for simulation of the electron trajectory in the
coordinate space:
( )
( ) ( ) ( )[ ]
( )
( )[ ]
( ) ( ) ( ) ( )[ ]
( )
( )[ ]
( ) ( )( )
−+
+−+=
−ω
ω
+
+−ω
ω
+
ω
−=
−ω
ω
−
−−ω
ω
+
ω
+=
*2
0
*
000
0*
0
0*
0
*
0
0
0*
0
0*
0
*
0
0
2)(
)(
sin
cos)(
cos
sin)(
mtteE
mtttptztz
tt
m
tp
tt
m
tp
m
tp
tyty
tt
m
tp
tt
m
tp
m
tp
txtx
z
c
c
y
c
c
x
c
x
c
c
y
c
c
x
c
y
(5a)
and
SPQEO, 2018. V. 21, N 4. P. 325-335.
Syngayivska G.I., Korotyeyev V.V., Kochelap V.A. Diffusion properties of electrons in GaN crystals subjected to …
328
( ) ( ) ( )[ ] ( )
( ) ( )[ ] ( )
( ) ( )( )
( ) ( ) ( )[ ] ( )
( ) ( )[ ]
ω
+−ω
ω
+
+
ω
+−ω
ω
ω+
−=
−+=
−
ω
−−ω
ω
+
+
ω
−−ω
ω
ω+
+=
2*0*
0
*
0
0*
0
0
*
000
0*0*
0
*
0
0*
0
0
sin
cos)(
)(
cos
sin)(
c
c
c
z
c
x
c
c
cx
y
c
c
c
z
c
z
c
c
cx
m
eE
tt
m
tp
m
tp
tt
m
eEtp
tztz
mtttptyty
tt
m
eE
tt
m
tp
m
tp
tt
m
eEtp
txtx
(5b)
The equations (5a) and (5b) were used for
determination of electron coordinates in the case of the
parallel and crossed configurations, respectively.
To calculate the components diffusion tensor, Dij,
we used the following equation [27, 39, 40]:
( )( ))()()()(
2
1
trtrtrtr
dt
d
D jjiiij −−= , (6)
where },,{3,2,1, zyxr ji == were calculated according to
equations (5a) or (5b). Angle brackets denote the time
average.
In the zeroth magnetic field and for the parallel
configuration of the fields, E || H, the tensor of the
diffusion coefficients has three non-zero diagonal
components: Dxx, Dyy, Dzz and Dxx = Dyy. (see E-H
orientation in Fig. 1). At HE ⊥ , the tensor Dij have five
non-zero components Dxx, Dyy, Dzz, Dxz and Dzx = Dxz.
Below, we will analyze the components of the tensor Dij
corresponding to the directions transversal to the electric
field.
3. Electrical characteristics at H = 0
In this section we discuss the electron transport
characteristics including the drift velocity, Vd, the
average energy, 〈ε〉, as well as the diagonal components,
Dxx = Dyy, of the diffusion tensor, all as functions of the
electric field at the absence of the magnetic field. The
parameters Dxx and Dyy describe electron diffusion
transversal to the applied field. We present the electric
characteristics for three particular examples, which differ
by the lattice temperature, T0, the concentration of
ionized impurities, Ni, and the electron concentration, Ne.
For the case I, we assume the following parameters:
Ni = 1016 cm–3, Ne = 1015 cm–3 and T0 = 30 K, for the
case II – Ni = 1017 cm–3, Ne = 1016 cm–3 and T0 = 30 K,
and for the case III – Ni = 1016 cm–3 Ne = 1015 cm–3 and
T0 = 300 K. The features of the streaming transport
regime are expected to be well-pronounced for the case I.
Fig. 2. Scattering probability Wtot vs the electron energy ε.
Calculations are performed for three cases described in the text:
case I - solid curve, II - dashed curve, III - dash-dotted curve.
Fig. 2 demonstrates the total scattering probability,
Wtot (ε), as a function of the electron energy, ε, calculated
for three discussed cases. The curves marked by I, II and
III correspond to the cases I, II and III, respectively. The
larger difference between values of Wtot in the passive
(ε < ћω0) and active (ε > ћω0) energy regions is
preferable to realize the well-developed streaming
transport regime (the case I). The large values of the total
scattering probability in the active region correspond to
the intensive optical phonon emission. These values are
two orders larger than those in the passive region, where
the less intensive acoustic phonon and ionized impurity
scatterings occur at T0 = 30 K. Increasing the impurity
concentration (the case II) at a given temperature leads to
increasing Wtot (ε) in the passive region due to the
increase of the electron-impurity scattering. The increase
of Wtot (ε) in the passive region with increasing the
temperature (the case III) at a given impurity
concentration is associated with activation of absorption
by optical phonons.
The field dependences of the drift velocity, Vd (E),
the average electron energy, 〈ε〉, the average energy
correspondent to the transversal electron motion,
( ) *22 2mpp yx +=ε⊥ , and the transversal component of
the diffusion coefficient, Dxx(E), are shown in Fig. 3. As
expected, the characteristic features of the streaming
regime are observed only for the sample I. In the range of
E = 3…10 kV/cm, the drift velocity and the average
energy saturate (see Figs. 3a and 3b) reach one half of the
characteristic velocity */2 00 mV ω= h , and the average
energy approaches to ћω0 / 3, respectively. As seen, for
the case II with larger impurity concentration, a well-
developed streaming regime is not formed. At room
temperature, in the range of E = 3...10 kV/cm, electron
gas remains almost quasi-equilibrium, so Vd (E) shows
linear behaviour, and 〈ε〉(E) is close to its equilibrium
value of 3kBT0 /2.
SPQEO, 2018. V. 21, N 4. P. 325-335.
Syngayivska G.I., Korotyeyev V.V., Kochelap V.A. Diffusion properties of electrons in GaN crystals subjected to …
329
Fig. 3. The drift velocity (a), the average electron energy (b),
the transverse component of the average energy (c) and the
diffusion coefficient (d) vs E at H = 0 for three cases (curves I,
II, III) discussed in the text. V0 = 4·107 cm/s and ћω0 ≈ 92 meV.
The emergence of streaming regime can be clearly
identified by the strong non-monotonic field dependence
of the transversal component of the average electron
energy, 〈ε⊥〉 (see Fig. 3c). This dependence is observed
for the case I. The increase of 〈ε〉 at E up to ~1 kV/cm is
associated with initial heating the electron gas. With
further increasing the field, 〈ε⊥〉 decreases due to
formation of a streaming-like distribution function
elongated along the field direction [20]. This decrease
tends to saturation in the field range 3…10 kV. For the
cases II and III, the streaming is not formed, and 〈ε〉 has a
slightly non-monotonic character.
The field dependences of the transversal-to-E
component of the diffusion coefficient, Dxx(E), (shown in
Fig. 3d) are qualitatively similar to the dependences of
〈ε〉. The specific strong non-monotonic dependence of
Dxx(E) is inherent to the streaming regime, that is realized
for the case I. At E < 500 V, the magnitude of Dxx rapidly
increases from the equilibrium value of 13 cm2/s to the
maximum of about 250 cm2/s. The isotropic spreading in
the transversal direction of electrons in the momentum
space is the main reason for this growth of the diffusion
coefficient. The maximum of Dxx corresponds to the
electric field, at which the rapid spreading of the
distribution function is terminated. In this field, the
essential part of high energy electrons loses its energy
due to emission of optical phonons. With further
increasing E, the streaming-like distribution function
begins to form. In this case, Dxx rapidly decreases and at
E > 5 kV/cm approaches to values 20…25 cm2/s. For the
cases II and III, Dxx(E) slowly decreasing from 50 down
to 30 cm2/s with increasing E from 1 up to 10 kV/cm.
The effect of the magnetic field on behaviour of the
diffusion coefficient is analyzed in the following
sections.
4. Diffusion coefficient at the E || H configuration
It should be noted that for electrons with parabolic
dispersion law, application of the magnetic field along
the electric one has no effect on transport characteristics
Vd (E) and 〈ε⊥〉(E) [38]. It follows from the equations
(3a), (4a), where it is seen that electron motion in
directions along and transverse to fields is uncoupled.
However, the electron diffusion process in the coordinate
space shows the strong dependence on the magnetic
field, H.
Fig. 4 demonstrates the electric field dependences
of the transversal-to-current component of the diffusion
tensor, Dxx(E), calculated for three values of H. Panels
(a), (b) and (c) correspond to the cases I, II and III,
respectively. For comparison, the component Dxx(E) at
Fig. 4. Dependences of Dxx(E) at E || H for the case I – (a); II –
(b); III – (c) at H = 0 (dashed curves), H = 1.1 T (curves 1),
2.3 T (curves 2), 3.4 T (curves 3).
SPQEO, 2018. V. 21, N 4. P. 325-335.
Syngayivska G.I., Korotyeyev V.V., Kochelap V.A. Diffusion properties of electrons in GaN crystals subjected to …
330
Fig. 5. Dependences Dxx(E) at the given values of H = 1.1 T (a),
2.3 T (b), 3.4 T (c). Curves I, II and III correspond to the case I,
II and III, respectively.
H = 0 is shown by the dashed curve. The curves 1, 2 and
3 correspond to three magnitudes of the magnetic field of
1.1, 2.3 and 3.4 T, respectively. Behaviour of the
dependence Dxx(E) in magnetic field has the following
general peculiarities for all the cases: (i) the dependences
of Dxx(E) are non-monotonic with their maximum shifted
to the higher electric fields with increasing H; (ii) the
magnetic field suppresses diffusion in transversal
directions with respect to E and H; (iii) the effect of
magnetic field decreases at higher electric fields.
The magnetic field effect on the electron diffusion
is more essential for the case I, for which the streaming
regime is realized. Even at weak magnetic fields, it is
observed strong suppression of the maximum of
diffusion coefficient, which decreases from the value
~250 cm2/s at H = 0 down to ~ 60 cm2/s at H = 1.1 T.
With further increasing the magnetic field, the maximum
of Dxx progressively decreases and has the values close to
30 cm2/s at H = 2.3 T and approximately 15 cm2/s at
H = 3.4 T. The position of maximum of the diffusion
coefficient correspond to the electric fields of 0.5, 1.5, 3
and 5.5 kV/cm for H = 0, 1.1, 2.3 and 3.4 T, respectively.
For the cases II and III, for which the streaming regime
does not occurs, the magnetic field more weakly
modifies the diffusion coefficient. For example, the
maximum of Dxx decreases only twice from ~60 cm2/s at
H = 0 down to ~30 cm2/s at H = 3.4 T.
Fig. 5 allows us to compare the diffusion coefficient
for these three cases at given magnetic fields. For the
weak magnetic field (H = 1.1 T), there is the range of the
electric fields E = 0...3 kV/cm, where the diffusion
coefficient for case I is higher than that for the cases II
and III (see the panel (a) in Fig. 5). At larger H, Dxx for
the case I is lower than Dxx for the cases II and III in the
studied electric fields (see the panels (b) and (c) in
Fig. 5).
The results of our modelling the diffusion processes
in compensated GaN show that the largest variations of
Dxx(E) with the magnetic field occur in the samples, for
which the streaming electron transport is formed. Both
the electro-gradient and optical measurements (using
LITG techniques) of the diffusion coefficient can provide
additional experimental tools for identification of the
streaming transport regime and hot electron parameters
for GaN crystals.
5. Diffusion coefficient at the E ⊥⊥⊥⊥ H configuration
In this section, we consider the field dependences of
components of the diffusion tensor corresponding to the
transversal motion with respect to the electric field
directions. In the case HE ⊥ , assuming that E and H are
oriented along z-axis and y-axis, respectively (see
Fig. 2b), two components of the diffusion tensor, Dxx and
Dyy, will be analyzed. The Dxx component describes the
diffusion current perpendicular to E and H. The Dyy
component characterizes the diffusion in the direction
parallel to H and perpendicular to E.
As discussed previously [22, 23, 41, 42], the
magnetic field for this configuration strongly affects on
the transport characteristics. It was established that the
streaming transport regime can be destroyed by the
magnetic field, forming a vortex-like distribution
function in the momentum space. The result of the
measurements of current-voltage characteristics depends
on the form of the external circuits. Our transport model
assumes the short-circuited Hall contacts.
In the cases HE ⊥ as well as E || H, behavior of
Dxx(E) depends on material parameters of the cases and
the relation between magnitudes of E and H. The
behavior of the component Dxx has the same general
features (i)-(iii), which are listed in previous section. As
well seen from Fig. 6a, the impact of the magnetic field
on the diffusion is largest for the case I. The dependences
of Dxx(E) show non-monotonic behavior with a
maximum shifted to the region of higher electric fields
with increasing H. However, at a given non-zero H, there
is a region of electric fields, where diffusion is more
intense than at H = 0. This peculiarity was absent in the
case of E || H. Moreover, the maximum of Dxx(E) is more
essentially shifted for the configuration HE ⊥ . For
example, at H = 1.1 T, the maximum of Dxx is realized at
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Syngayivska G.I., Korotyeyev V.V., Kochelap V.A. Diffusion properties of electrons in GaN crystals subjected to …
331
Fig. 6. Dependences Dxx(E) at E⊥H for the case I – (a), II – (b),
III – (c) at H = 0 (dashed curves), H = 1.1 T (curves 1), 2.3 T
(curves 2), 3.4 T (curves 3).
E = 3 kV/cm (for comparison, in parallel configuration –
at E = 1.5 kV/cm). For the cases II and III (panels (b) and
(c) in Fig. 6, respectively), the effect of the magnetic
field is weak, the dependences of Dxx(E) are weakly non-
monotonous with an extended region of the electric field
E = 1…6 kV/cm (at H = 2.3…3.4 T), where Dxx(E) is the
practically constant.
In Fig. 7, the same dependences of Dxx(E) are
grouped accordingly to the given magnetic fields. As
seen, at H = 1.1 T the Dxx component calculated for the
case I exceeds the values of Dxx obtained for the cases II
and III within the range E = 0.1…5 kV/cm. With
increasing H, this range decreases and shifts to the higher
electric fields. For the cases II and III, the field
dependences of Dxx are very similar in wide ranges of E
and H.
The well-pronounced maximum of the diffusion
tensor Dxx component is realized for the case I. It should
be noted that the maximum arises in such fields E and H
at which formation of the specific form of steady-state
distribution function in the momentum space
Fig. 7. Dependences Dxx(E) at the given values of H =1.1 T (a),
2.3 T (b), 3.4 T (c). Curves I, II and III correspond to the case I,
II and III, respectively.
occurs. This distribution function describes the magneto-
transport regime with co-existence of two separated
electron groups (for details, see [22, 42]).
In contrast to the parallel configuration of E and H,
in the crossed configuration components Dxx and Dyy are
not equal. Dyy component describes the diffusion current
along the magnetic field. The electric field dependences
of Dyy at several values of H are shown in Fig. 8. As seen,
behavior of Dyy(E) cardinally differs from Dxx(E). In
general, the increase of H leads to the growth of electron
diffusion in the y-direction for a wide range of the
electric fields. Particularly, for the case I (see panel (a) in
Fig. 8) the amplitude and position of the maximum are
weakly modified with increasing H within the range
1.1…3.4 T. However, at weak magnetic fields, 0…1.1 T,
the amplitude of Dyy maximum is twice increased. With
increasing the electric field, diffusion along the direction
of magnetic field progressively decreases, and at
E > 6 kV/cm it tends to the case corresponding to H = 0.
For the case II, the impact of the magnetic field on the
dependence Dyy(E) is not so essential as for the case I.
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Syngayivska G.I., Korotyeyev V.V., Kochelap V.A. Diffusion properties of electrons in GaN crystals subjected to …
332
Fig. 8. Dependences Dyy(E) at E⊥H for the case I – (a); II – (b);
III – (c) at H = 0 (dashed curves), H = 1.1 T (curves 1), 2.3 T
(curves 2), 3.4 T (curves 3).
The maximum of Dyy is increased by ~30% with
increasing the magnetic field from 0 up to 3.4 T. For the
case III, the magnetic field does not influence on the
dependence Dyy(E).
We found that general behaviour of galvano-
magnetic properties of Dyy component in the diffusion
tensor correlates with behavior of the average energy of
the electron motion in y-direction, 〈εy〉. The results of our
calculations of the electric field dependences, 〈εy〉, for the
three samples are shown in Fig. 9 for three values of H.
6. Conclusions
We have studied the diffusion properties of hot electrons
in bulk-like GaN samples for parallel and crossed
configurations of the applied electric and magnetic fields.
It has been analyzed the field dependences of the
transversal components in the diffusion tensor, which
correspond to electron motion in the directions
perpendicular to the electric field. These results have
been obtained by the Monte Carlo method of
calculations. It was analyzed three types of GaN samples
with the degree of compensation close to 90%. It was
assumed T0 = 30 K, Ni = 1016 cm-3 for the sample I,
T0 = 30 K, Ni = 1017 cm-3 for the sample II, and
T0 = 300 K, Ni = 1016 cm-3 for the sample III). Parameters
Fig. 9. Dependences 〈εy〉(E) for the case I – (a), II – (b), III – (c)
at H = 0 (dashed curves), H = 1.1 T (curves 1), 2.3 T (curves 2),
3.4 T (curves 3).
of the sample I satisfy the requirements for realizing the
streaming transport regime.
We found that a strong impact of magnetic fields on
diffusion properties of the electron gas takes place for the
sample I. In the parallel configuration of E and H, the
electric field dependences of the transversal-to-current
components of the diffusion tensor have non-monotonous
behavior with a maximum. The amplitude and position of
the maxima depend on the magnitudes of magnetic field
and are changed with the increase of magnetic field: the
maximum is decreased, and its position is shifted to the
higher electric fields.
In the crossed configuration of E and H, the
transversal-to-fields component of the diffusion tensor
has similar behaviour. However, the positions of the
maximum are more sensitive to H variation.
The suppression of electron diffusion in transversal
to H direction with the increase of magnetic field is a
general phenomenon that is observed in both
configurations. However, in the crossed configuration,
the magnetic field enhances electron diffusion along H
direction. The electric field dependences of the
longitudinal-to-H component of the diffusion tensor also
have the maximum. Both position and amplitude of the
maximum weakly depend on the magnitudes of H.
SPQEO, 2018. V. 21, N 4. P. 325-335.
Syngayivska G.I., Korotyeyev V.V., Kochelap V.A. Diffusion properties of electrons in GaN crystals subjected to …
333
The parameters of the samples II and III prevent
formation of the streaming transport regime. This is the
main reason of the weak influence of electric and
magnetic fields on diffusion properties of electrons in
these samples. However, the main peculiarities observed
in the field dependences of diffusion processes for the
sample I take place for the samples II and III as well.
We suggest that the streaming transport regime and
related magneto-transport effects can be investigated by
measurements of the diffusion effects of hot electrons in
the electric and magnetic fields. Both the electro-gradient
and optical measurements (using LITG techniques) of the
diffusion coefficient can be used for this purpose. Also,
the knowledge of high-field dependences of the diffusion
coefficient is important for modeling the micron-scale
electronic devices operating in strong electric and
magnetic fields.
Acknowledgments
This work is partially supported by the Ministry of
Education and Science of Ukraine (Project M/24-2018)
and German Federal Ministry of Education and Research
(BMBF Project 01DK17028).
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Authors and CV
Dr. Galina Syngayivska born 1969
in Drohobych (Lviv region, Ukraine),
graduated in physics in 1993, the
Ph.D. degree in solid state physics in
2015 from the V. Lashkaryov Institute
of Semiconductor Physics NAS of
Ukraine. Since 2016, she is the
researcher at the Department of
Theoretical Physics in the V. Lashkaryov Institute of
Semiconductor Physics, NASU. She is the author of more
than 20 publications. Her main research activity is in the
field of Monte Carlo simulation of electron transport in
nanoscale structures and devices.
E-mail: singg@ukr.net
SPQEO, 2018. V. 21, N 4. P. 325-335.
Syngayivska G.I., Korotyeyev V.V., Kochelap V.A. Diffusion properties of electrons in GaN crystals subjected to …
335
Dr. Vadym Korotyeyev born 1977 in
Lutsk (Volyn region, Ukraine),
graduated in physics in 1999, the
Ph.D. degree in solid state physics in
2006 from the V. Lashkaryov Institute
of Semiconductor Physics NAS of
Ukraine. Since 2010, he is the senior
researcher at the Department of
Theoretical Physics in V. Lashkaryov
Institute of Semiconductor Physics, NASU. He is the
author of more than 80 publications. His main research
activity is in the field of electronic transport in nanoscaled
structures and THz optoelectronics.
E-mail: koroteev@ukr.net
Prof., Dr. Viacheslav A. Kochelap
born 1944 in Kyiv (Ukraine),
graduated in theoretical physics in
1966 (Kiev State University), the
Ph.D. degree in solid state physics in
1970 from the Institute of Semi-
conductor Physics NAS of Ukraine.
Since 1987, he is full Professor at the
Department of Theoretical Physics
in the V. Lashkaryov Institute of Semiconductor Physics,
NASU. He is the author of more than 250 publications.
His main research activity is in the field of electronic
transport, fluctuation phenomena and THz-physics of
semiconductors and semiconductor nanoscale devices.
|
| id | nasplib_isofts_kiev_ua-123456789-215329 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1560-8034 |
| language | English |
| last_indexed | 2026-03-23T18:47:48Z |
| publishDate | 2018 |
| publisher | Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України |
| record_format | dspace |
| spelling | Syngaivska, G.I. Koroteev, V.V. Kochelap, V.A. 2026-03-12T08:56:07Z 2018 Diffusion properties of electrons in GaN crystals subjected to electric and magnetic fields / G.I. Syngaivska, V.V. Koroteev, V.A. Kochelap // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2018. — Т. 21, № 4. — С. 325-335. — Бібліогр.: 42 назв. — англ. 1560-8034 PACS: 72.20.Ht, 72.20.Dp, 73.23.-b, 85.35.-p https://nasplib.isofts.kiev.ua/handle/123456789/215329 https://doi.org/10.15407/spqeo21.04.325 We have studied the diffusion coefficient of hot electrons in GaN crystals under moderate electric (1...10 kV/cm) and magnetic (1…4 T) fields. Two configurations, parallel and crossed fields, have been analyzed. The study was carried out for compensated bulk-like GaN samples for various lattice temperatures (30…300 K) and impurity concentrations (10¹⁶…10¹⁷ cm⁻³). We found that at low lattice temperatures and low impurity concentrations, electric-field dependences of the transversal-to-current components of the diffusion tensor are non-monotonic for both configurations, while diffusion processes are mainly controlled by the magnetic field. With increasing the lattice temperature or impurity concentration, the behaviour of the diffusion tensor becomes more monotonous and less affected by the magnetic field. We showed that this behaviour of the diffusion processes is caused by the distinct kinetics of hot electrons in polar semiconductors with strong electron – optical phonon coupling. We have suggested that measurements of the diffusion coefficient of electrons subjected to electric and magnetic fields facilitate the identification of features of different electron transport regimes and the development of more efficient devices and practical applications. This work is partially supported by the Ministry of Education and Science of Ukraine (Project M/24-2018) and German Federal Ministry of Education and Research (BMBF Project 01DK17028). en Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України Semiconductor Physics Quantum Electronics & Optoelectronics Semiconductor physics Diffusion properties of electrons in GaN crystals subjected to electric and magnetic fields Article published earlier |
| spellingShingle | Diffusion properties of electrons in GaN crystals subjected to electric and magnetic fields Syngaivska, G.I. Koroteev, V.V. Kochelap, V.A. Semiconductor physics |
| title | Diffusion properties of electrons in GaN crystals subjected to electric and magnetic fields |
| title_full | Diffusion properties of electrons in GaN crystals subjected to electric and magnetic fields |
| title_fullStr | Diffusion properties of electrons in GaN crystals subjected to electric and magnetic fields |
| title_full_unstemmed | Diffusion properties of electrons in GaN crystals subjected to electric and magnetic fields |
| title_short | Diffusion properties of electrons in GaN crystals subjected to electric and magnetic fields |
| title_sort | diffusion properties of electrons in gan crystals subjected to electric and magnetic fields |
| topic | Semiconductor physics |
| topic_facet | Semiconductor physics |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/215329 |
| work_keys_str_mv | AT syngaivskagi diffusionpropertiesofelectronsingancrystalssubjectedtoelectricandmagneticfields AT koroteevvv diffusionpropertiesofelectronsingancrystalssubjectedtoelectricandmagneticfields AT kochelapva diffusionpropertiesofelectronsingancrystalssubjectedtoelectricandmagneticfields |