Formation of step density shock waves on vicinal NaCl(100) growth surfaces
The morphology of the growth surface near NaCl(100), formed during the pore motion in a crystal due to the temperature gradient, has been studied by the electron microscopic method of vacuum decoration. It is shown that at T = 950 K and Δμ/kΤ = 4·10⁻³, the profile of the vicinal surface in the <1...
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| Date: | 2022 |
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2022
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| Cite this: | Formation of step density shock waves on vicinal NaCl(100) growth surfaces / O.P. Kulyk, O.V. Podshyvalova, O.L. Andrieieva, V.I. Tkachenko, V.A. Gnatyuk, T. Aoki // Problems of Atomic Science and Technology. — 2022. — № 1. — С. 154-160. — Бібліогр.: 36 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1859742583398334464 |
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| author | Kulyk, O.P. Podshyvalova, O.V. Andrieieva, O.L. Tkachenko, V.I. Gnatyuk, V.A. Aoki, T. |
| author_facet | Kulyk, O.P. Podshyvalova, O.V. Andrieieva, O.L. Tkachenko, V.I. Gnatyuk, V.A. Aoki, T. |
| citation_txt | Formation of step density shock waves on vicinal NaCl(100) growth surfaces / O.P. Kulyk, O.V. Podshyvalova, O.L. Andrieieva, V.I. Tkachenko, V.A. Gnatyuk, T. Aoki // Problems of Atomic Science and Technology. — 2022. — № 1. — С. 154-160. — Бібліогр.: 36 назв. — англ. |
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| container_title | Вопросы атомной науки и техники |
| description | The morphology of the growth surface near NaCl(100), formed during the pore motion in a crystal due to the temperature gradient, has been studied by the electron microscopic method of vacuum decoration. It is shown that at T = 950 K and Δμ/kΤ = 4·10⁻³, the profile of the vicinal surface in the <11> direction is represented by monoatomic steps, while in the <10> direction, as the surface curvature increases, there is a grouping of steps with the formation of macrosteps – bunches of elementary steps separated by areas of atomically smooth terraces. The sawtooth dependence of the step density on the longitudinal coordinate is described by a particular solution of the Burgers equation for a shock wave. Data on the parameters of three shock waves and the time of their formation are obtained.
Електронно-мікроскопічним методом вакуумного декорування досліджена морфологія поверхні росту поблизу NaCl(100), що сформувалася під час руху пори в кристалі під дією градієнта температури. Показано, що при T = 950 K and Δμ/kΤ = 4·10⁻³ профіль віцинальної поверхні в напрямку <11> представлений моноатомними сходинками, тоді як у напрямку <10>, зі збільшенням кривизни поверхні, спостерігається групування сходинок з утворенням макросходин – згустків елементарних сходинок, розділених ділянками атомно-гладких терас. Пилкоподібна залежність густини сходин від поздовжньої координати описана частинним розв’язком рівняння Бюргерса для ударної хвилі. Отримано дані про параметри трьох ударних хвиль і час їх формування.
Электронно-микроскопическим методом вакуумного декорирования исследована морфология поверхности роста вблизи NaCl(100), сформировавшейся при движении поры в кристалле под действием градиента температуры. Показано, что при T = 950 K and Δμ/kΤ = 4·10⁻³ профиль вицинальной поверхности в направлении <11> представлен моноатомными ступенями, тогда как в направлении <10>, по мере увеличения кривизны поверхности, наблюдается группирование ступеней с образованием макроступеней – сгустков элементарных ступеней, разделенных участками атомно-гладких террас. Пилообразная зависимость плотности ступеней от продольной координаты описана частным решением уравнения Бюргерса для ударной волны. Получены данные о параметрах трех ударных волн и времени их формирования.
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154 ISSN 1562-6016. ВАНТ. 2022. №1(137)
https://doi.org/10.46813/2022-137-154
UDC 548.52
FORMATION OF STEP DENSITY SHOCK WAVES ON VICINAL
NaCl(100) GROWTH SURFACES
O.P. Kulyk
1,
*, O.V. Podshyvalova
2
, O.L. Andrieieva
1,3
, V.I. Tkachenko
1,3
,
V.A. Gnatyuk
4
, T. Aoki
5
1
V.N. Karazin Kharkiv National University, Kharkiv, Ukraine;
2
National Aerospace University “Kharkov Aviation Institute”, Kharkiv, Ukraine;
3
National Science Center “Kharkov Institute of Physics and Technology”, Kharkiv, Ukraine;
4
V.E. Lashkaryov Institute of Semiconductor Physics of the NAS of Ukraine, Kyiv, Ukraine;
5
Research Institute of Electronics, Shizuoka University, Hamamatsu, Japan
*E-mail: kulykop@gmail.com
The morphology of the growth surface near NaCl(100), formed during the pore motion in a crystal due to the
temperature gradient, has been studied by the electron microscopic method of vacuum decoration. It is shown that at
T = 950 K and = 4·10-3
, the profile of the vicinal surface in the <11> direction is represented by monoatomic
steps, while in the <10> direction, as the surface curvature increases, there is a grouping of steps with the formation
of macrosteps – bunches of elementary steps separated by areas of atomically smooth terraces. The sawtooth
dependence of the step density on the longitudinal coordinate is described by a particular solution of the Burgers
equation for a shock wave. Data on the parameters of three shock waves and the time of their formation are
obtained.
INTRODUCTION
It is known that vicinal surfaces during the growth of
crystals from the vapor phase or solutions are subject to
a certain type of morphological instability – bunching of
steps [14]. The formation of step bunching is a very
serious problem when growing perfect crystals and
obtaining surfaces that are atomically smooth on a
macroscale [57]. On the other hand, such instabilities
lead to the formation of large-scale nanostructured
surfaces, which can be used to obtain low-dimensional
structures actual for various technological applications
[814]. A theoretical description of the nonlinear
processes that result in the development of such kind of
instabilities is very complicated due to a variety of
causes leading to the step bunching in real experimental
conditions (presence of impurities, surface
electromigration effect, Ehrlich-Schwöbel effect, elastic
stress fields, variable macroscopic fields, non-quasi-
static effects, etc.) [1520]. The current state of research
of step bunching, in particular, induced by electric
currents, is presented in the references given in [21],
where it is shown how the general picture of the process
of bunching depends on the short-range repulsive force
between the steps. It is customary to distinguish
between the step bunching as a result of morphological
instability and as a shock in a kinematic wave, when the
flux of steps is determined only by their local density
[3].
The reference equation describing nonlinear waves
in a dissipative nondispersive medium is the Burgers
equation (BE) [22]. Since dissipation is high-frequency,
its effect is strongest where the wave profile changes
most rapidly, i.e. near its front. For this reason, in wave
theory, a steep and thin wave front at the moment when
the steepening process stops is usually called a shock
wave [3]. It is generally accepted that because the shock
corresponds to the location of an abrupt change in
density, let us say a transition from high to low density
at a very short length, shock waves are not actual
bunches, but rather their edges. However, if we talk
about stationary shock waves of the BE, it is worth to
introduce the concept of shock wave amplitude as the
difference between the maximum and minimum values
of the wave profile, as well as the characteristic width of
the front – the longitudinal distance at which the
difference in these values occurs. The coefficient μ at
the second derivative in the BE is an analogue of
viscosity.
The study of kinematic (“shock”) waves of steps on
crystal surfaces was first carried out by Frank [23], and
by Cabrera and Vermilyea [24], who used the results of
the general analysis of kinematic waves done by
Lighthill and Whitham [25]. Later, it was shown in [26]
that during crystal growth from the vapor phase, the
shock wave is the main result of diffusion interaction of
moving steps and exhibits itself as an edge at which the
slope of the vicinal surface changes sharply. A
characteristic feature of shock waves is the presence of
discontinuities in the step density. The density
discontinuity can arise under a wide variety of
perturbations, if only these perturbations result in areas
on the surface where the density of steps has increased
in absolute value.
It is known that not only dissipation, but also
dispersion are among the factors that can stop the
steepening of a wave and prevent it from overturning. In
the general case, the dynamics of the profile of a
macroscopic curved vicinal surface of a crystal growing
from the vapor phase was studied in [27]. In this work,
expressions for the average values of adatom
concentration and the velocity of elementary steps were
obtained by averaging over large spatial intervals. The
nonlinear Korteweg de Vries-Burgers (KVB) equation
https://doi.org/10.46813/2022-137-154
mailto:march@kipt.kharkov.ua
ISSN 1562-6016. ВАНТ. 2022. №1(137) 155
was obtained from the continuity equation for average
values of the adatom concentration and velocity of the
elementary steps, on the condition that the surface
curvature is phenomenologically considered [3, 28].
This equation describes the nonlinear dynamics of
motion of a train of parallel elementary steps on a
macroscopically curved vicinal crystal surface. In a
particular case, the KVB equation transforms into the
BE, which describes the formation and dynamics of
shock waves.
In [29], particular solutions of the BE with zero
boundary conditions were obtained in an analytical
form. It is shown, in particular, that for a shape
parameter of the initial perturbation greater than one, its
amplitude nonmonotonically depends on the spatial
coordinate. Over time, the shock wave does not form,
and the perturbation amplitude decreases exponentially
and tends to zero. A particular solution for the first
mode was used to describe the configuration of
elementary steps with an orientation near <100>,
formed at the base of the cleavage macrostep during the
growth of a NaCl crystal from the vapor phase. It was
shown that the one-dimensional distribution of the step
concentration adequately reflects the shock wave profile
at the decay stage. At small values of the shape
parameter, particular solutions describe the shock wave
formation from initial periodic perturbations.
In this work, the obtained solutions are used to
describe some experimental results associated with the
formation of shock waves of the elementary step density
during the growth of NaCl single crystals from the
vapor phase.
EXPERIMENTAL PROCEDURE
Reasons for the choice of alkali halide crystals, in
particular NaCl, as objects for the study of kinematic
density waves of elementary steps are described in
detail in [27, 29]. The main argument is the possibility
to exclude practically the causes listed in the
introduction that lead to the bunching of steps as a result
of the “morphological instability” of the vicinal surface.
By studying in these crystals the processes of motion
and transformation of the shape of pores and inclusions
of saturated solution, one can obtain important features
of layer-by-layer mass transfer [3033]. Varying as the
only external parameter, the temperature between
opposite crystal faces, one can create unique conditions
for crystallization (decrystallization) at low
supersaturation, which are very difficult to realize in
traditional growth experiments.
The morphology of the growth (evaporation)
surfaces of NaCl single crystals is well studied. Data on
kinetic and thermodynamic characteristics of layer-by-
layer mass transfer on the vicinal surfaces of these
crystals were obtained in a wide temperature range (see
references in [27]). It was shown that the doubled value
of the linear tension of monatomic height steps is less
than the linear tension of double-height steps, which
indicates the thermodynamic stability of the studied
NaCl(100) vicinal surfaces.
Samples with pores were prepared using the known
technique – healing unfinished crack, which consists of
the following. Rectangular 10 10 3 mm plates were
cut from the single crystals, and then they were partially
split parallel to the large faces. After that, the crystal
with an unfinished crack was compressed and annealed
under isothermal conditions at the pre-melting
temperature. While annealing the crack was healing and
closed pores of 1…100 µm in size, faceted by planes
{100} and with rounded vertices, were forming in its
mouth. All pores were in the same plane, and this
allowed one to follow not only individual pores, but also
entire ensembles using transmission optical microscopy.
The pore motion was in the direction of the
temperature gradient T coinciding with the crystal
axis <001> [27]. During the motion, the shape of most
pores became more faceted and took the form of
parallelepipeds elongated in the direction perpendicular
to the temperature gradient. This indicated that
evaporation processes, in addition to the front surface,
occurred on the four lateral pore surfaces, and growth
processes occurred on one, i.e. just the back surface.
The samples with pores were split in a vacuum setup
and the opened pore surfaces were examined using the
electron microscopic technique of vacuum decoration.
The pore motion under the temperature gradient
T in the crystal is caused by the difference in the
chemical potentials of molecules on the front and
back surfaces, which is expressed by the ratio
, where H is the evaporation heat; Z is the
pore linear size in the direction of the gradient. Varying
the value of the temperature gradient, it was possible to
study the processes of evaporation-growth in a wide
range of under- and supersaturation ( ).
Since the growth-evaporation (dissolution) processes
occur in layers in alkali-halide crystals, it is obvious that
layer sources are necessary for the motion of pores
(inclusions of a saturated solution). Such sources for
growth are edges of pores (inclusions). As for
evaporation (dissolution), at low undersaturation, the
only possible layer sources may be the steps formed in
the places where screw dislocations come to the surface
[3032].
Faceted shapes of pores (liquid inclusions) and an
increase in the degree of shape nonisometry in the
perpendicular direction mean that the limiting process
during the pore (inclusion) motion is the evaporation
(dissolution) of the matrix substance, rather than its
transfer through the pore (inclusion) volume. In the case
of saturated solution inclusions, their unchanged shape
during the motion induced by driving forces of different
nature (inhomogeneous density distribution of
dislocations or radiation defects, see [32, 33] and
references therein) indicates the limiting role of the
diffusion process of the matrix substance through the
inclusion volume. In this case, the difference in the
molecules' chemical potential between the front and
back surfaces will be mostly not near the front surface,
but along the entire inclusion length in the direction of
the driving force.
It is clear that the energy, required to embed
molecules into edges on growing surfaces, is much less
than the energy required to form a critical nucleus on a
screw dislocation. This explains the significant
difference in the morphology of the evaporation and
156 ISSN 1562-6016. ВАНТ. 2022. №1(137)
growth surfaces within the pores [27, 31]. While the
evaporation surfaces are practically smooth on an
atomic scale, the growth ones can be represented by
trains of steps whose length is comparable to the pore
size. That makes them a unique object for studying the
kinematic waves of elementary step density on vicinal
surfaces of the investigated crystals. Since the
recrystallization of the matrix material occurs inside the
pore, the effect of impurities on the kinetics of
elementary step motion, which takes place in traditional
growth experiments at low supersaturations, is
practically excluded.
EXPERIMENTAL RESULTS
Fig. 1 shows an area of the pore growth surface
formed at low supersaturation. Taking Z = 13 μm,
∆H = 3.9·10
-19
J [34] at = 1·10
-4
K/m and
T = 950 K, we have = 4·10-3
. The motion of the
train of steps with an orientation near <10> selected for
analysis occurs from the pore edge (outside Fig. 1,
above the area in it) in the 01 direction. The
tangential motion of steps results in the overgrowth of
concentric layers and provides the normal displacement
of the surface.
Fig. 1. TEM image of the growth surface near
NaCl(100), formed during the pore motion under the
temperature gradient (linear steps appeared at the
splitting of the crystal in vacuum)
An important feature of the morphology of the
investigated surface is the difference in its profile for
the <10> and <11> directions. In the <11> direction, the
vicinal surface profile is represented by monoatomic
steps, at least up to those surface curvature values at
which the decoration technique still allows to resolve
individual steps. The fact that the steps are monoatomic
can be concluded from the intersections of the step
trains by slip bands [35], which appear when the crystal
is split in vacuum and the pore surfaces are opened.
Whereas in the <10> direction, as the curvature of the
surface increases, the formation of macrosteps is
observed. In this case, by “macrosteps” we mean
bunches of elementary steps, rather than “true steps”
[26], whose thermodynamic stability is determined by
the anisotropy of the surface energy. It was shown
earlier that vicinal surfaces near NaCl(100) are
thermodynamically stable, i.e. the integration of
monoatomic steps into higher ones is
thermodynamically unfavorable. There are no effects
that can lead to the formation of thermodynamically
stable macrosteps on these surfaces (see [2] and
references therein). Therefore, we use the term “macro-
step” to name a bunch of elementary steps, which is a
part of a kinematic wave or a shock wave of step
density [3, 26].
Analysis of the surface morphology in Fig. 1 shows
that such macrosteps are difficult to distinguish from
elementary steps in the decoration patterns. They are
represented by similar chains of decorating gold
particles, the sizes of which vary from 40 to 100 Å. The
height a of a monoatomic step is 2.81 Å and the number
of elementary steps included in the observed macrosteps
does not exceed one or two dozen. The number of
elementary steps forming a particular macrostep with
the <10> direction can be estimated by the number of
elementary steps branching off from it. To reduce the
error in determining the macrostep heights, the selected
area of the pore surface was digitized in the <01> and
<11> directions. Comparison of the obtained data
allowed a sufficiently reliable reconstruction of the
surface profile in the <01> direction (within 75
elementary steps from the central concentric layer).
The fact that the monoatomic steps do not integrate
to form a “true” macrostep but form a bunch of steps is
also confirmed by direct observations of the
disintegration of steps formed on screw dislocations
with a Burgers vector 2a into two monoatomic ones at
small super-/undersaturation (≤ 10
-2
) [36], i.e. at
conditions when kinetic factors are not so significant.
Therefore, when digitizing the trains of steps in Fig. 2,
the bunch width was determined with an error of the
order of the size of decorating particles. And when
determining the step density in the bunch, we took into
account the width of the adjacent terrace on the side of
the nearest distinguishable step.
A characteristic feature of shock waves described by
the BE particular solutions [29] is a sawtooth profile of
the wave surface. As applied to kinematic density waves
of elementary steps on thermodynamically stable vicinal
surfaces, this means the presence of discontinuities in
step density, accompanied by the formation of relatively
wide atomic-smooth terraces, as well as the
redistribution of elementary steps with the formation of
bunches – macrosteps. Three bunches of steps in Fig. 1,
formed in the immediate vicinity of the pore edge,
satisfy this description.
The dependence of the step concentration
(dimensionless density) averaged over several adjacent
terraces on the longitudinal coordinate in units s (see
Fig. 2) was plotted using the step distribution in
bunches, taking into account the width of adjacent
terraces, obtained due to digitization.
DESCRIPTION OF THE EXPERIMENTAL
RESULTS OF NaCl CRYSTAL GROWTH
FROM THE VAPOR PHASE BY THE BE
PARTICULAR SOLUTION
In [29], a particular BE solution with zero boundary
conditions was used for the first time to carry out a
quantitative analysis of the decay of shock waves. In
this work, a particular BE solution with zero boundary
conditions was used to describe the decay of the profile
of a one-dimensional echelon of elementary steps with
an orientation near <100>, which was formed during the
growth of a NaCl single crystal from the vapor phase at
ISSN 1562-6016. ВАНТ. 2022. №1(137) 157
the base of a macroscopic cleavage step. It is shown that
the distribution of the step concentration with distance
from the initial position of the macrostep adequately
reflects the shock wave profile at the decay stage. The
dimensionless parameters of the shock wave are
determined, and on their basis the estimates of the
characteristic time of its decay are made.
Fig. 2. The structure of the shock waves shown in
Fig. 1: symbols are the experimental values of the
average concentration of steps in the wave (the wave
propagation direction is inverted relative to the step
motion direction); solid lines are the calculation results
based on the particular solution (7) at n=1
To describe the inverse process of shock wave
formation, it is necessary to change the formulation of
the problem in terms of reformatting the boundary
conditions. Since crystal growth occurs from the vapor
phase, a source of adatoms is required. Based on this,
we formulate the problem of crystal growth, using BE to
describe this process.
Let us find the bounded BE solutions |u(x,t)| < ∞ on
the interval 0 ≤ x ≤ L in the time interval 0 ≤ t < t0 <∞:
2
2
u u u
u
t x x
(1)
with boundary conditions:
1
00, 0, ( ) ,u t u L,t A t t f t
(2)
where ,t x are the dimensionless time and coordinate,
respectively; 0 is the dimensionless coefficient of
kinematic viscosity of the medium; 00,A t are
constants, min max( )B f t B is a positive definite
function bounded on both sides. The values of constants
min max,B B will be defined below.
The boundary condition at x = L modelically
describes the source of adatoms that come to the
growing crystal surface from the vapor phase. The
boundedness of the solution |u(x,t)| < ∞ is provided by
the condition 0 maxA t B t D that is feasible for
t0>Bmaxt at a finite value of D.
The Cole-Hopf transformation (CH)
v ,1
, 2
v ,
x t
u x t
x t x
converts the nonlinear
equation (1) into a linear heat equation for the function
v ,x t :
2
2
v , v ,
.
x t x t
t x
(3)
The CH transformation imposes a condition on the
function v ,x t : v , 0x t .
As a result of the CH transformation, the boundary
conditions (2) for equation (3) take the form:
0
v 0, v ,
0, exp ,
2 ( )
t L t A L
C
x x t t f t
(4)
where C is the constant of integration.
Equation (3) has an infinite number of particular
solutions that satisfy the boundary condition (4) for
0x [29]:
2v , cos exp ,n n nx t x t (5)
where
n n L , 1,2,3, ...n .
Particular solutions (5) are determined up to a
positive constant an and a function
2, 2n nx t b t x , where bn is a positive
constant.
Therefore, new functions of the form:
2
2
w , cos exp
2
n n n
n n
x t x t
a b t x
(6)
are also particular solutions of equation (3). It is easy to
verify that solutions (6) satisfy boundary conditions (4).
As a result of the CH transformation we obtain the
particular solutions of the BE:
2
2
2
2 sin exp 2
, ,
cos exp
2
n n n n
n
n n n n
x t b x
u x t
x
x t a b t
(7)
where
2
n nb ,
2 2 1n na b L .
Expression (7) describes an infinite number of BE
particular solutions for different values of the constants.
Let us consider solutions (7) with the lower sign of
the last term of the numerator and denominator. In this
case, there are solutions describing the steepening of the
shock wave profile. Consider such solutions for the
mode n = 1 and values of constants:
2
1 0 1 12 , 2 1 0A b L t a b L . The function ( )f t
in (2) must be specified in the form of
2
1
1
1 exp t
f t b
t
, when the condition
b1 > max (μ
, 1) is met. The inequality b1 > 1 follows
from the condition that the denominator u(L,t) in (2)
tends to zero as t→t0. For the function f(t) we have the
limits of its change: Bmin= b1 - μ
and
2
1 0
max 1 1
0
1 exp t
B b b
t
.
The condition b1 > max (μ
, 1) is fulfilled if we
put, for example,
2
12
1 11 1b e
, as b1 > 1 at
μ
<< 1, and b1 > μ
at μ
> 1. Based on this
assumption, let us determine the characteristic time of
shock wave formation.
158 ISSN 1562-6016. ВАНТ. 2022. №1(137)
The characteristic time of shock wave formation is
estimated from the value of the time when the
denominator (7) tends to zero, but the amplitude of the
disturbance is limited by the condition of the BE
applicability. The denominator feature arises for x = L
and characteristic “explosion” time may be determined
from the equation:
1ln 1 1 exp , (8)
where
2 2
1 1 0, t .
It is easy to see that equation (8) has a simple
solution: μ
= 1, t0 = 1. At critical points from (8) we
have: at 0 the value tends to infinity ; at
the value tends to zero 0 .
From (8) we can determine the characteristic
“explosion” time via the parameter :
1
1
0 ln 1 1 exp .t
(9)
Using (9), it is possible to determine graphically the
dependence of the “explosion” time on the parameter
μ
(Fig. 3).
Fig. 3. The dependence of the “explosion” time
on the parameter 2
1
From Fig. 3 it follows that with an increase of wave
period, the “explosion” time increases. The “explosion”
time also increases with decrease in the coefficient of
kinematic viscosity of the medium at a constant wave
period.
The characteristic “explosion” time of shock wave
will be determined on the basis of the experimental data
presented below.
The obtained experimental data were processed
according to the method previously used to describe the
shock wave decay during the growth of a NaCl single
crystal from the vapor phase [29]. Estimates of the
coefficient μ [27] taking into account the equilibrium
concentration of admolecules on atomic-smooth areas of
the surface ξa0 ~ 10
-6
(according to the data in [34]) with
a known value of λs = 3.37·10
-8
m [27] and the average
value 0 0.8s , obtained by digitization, showed that
μ ~ 1 at q ~ 0.1 (ρ0λs). Exactly of such order is the value
of the parameter v ~ q / (ρ0λs) [29], if we take
q ~ 1/2 L , where L is the average value of the half-
width of the wave front in Fig. 2.
Using the software products Wolfram Mathematica
and Mathcad, the experimental dependences of the steps
concentration of shock waves on their coordinates were
approximated by theoretical dependences at μ = 1.19.
Parameters of three shock waves (Table) were obtained
as a result of processing the experimental data of Fig. 2
on the basis of expression (7) at 1n .
The following features of the shock waves under
study are noteworthy. First, their formation occurs for
the <10> directions and is not observed in the <11>
directions, which allows us to determine the number of
elementary steps included in each wave: 18, 14, and 10,
respectively. This distinction of the surface profiles can
be due to both the difference in the nature of the
thermodynamic interaction between the steps in the
indicated directions [36], and the difference in the
diffusion-kinetic interaction. Indeed, as the step
orientation deviates from the direction of close packing,
the concentration of kinks on the step increases
significantly, reaching a maximum for the orientations
<11>. Consequently, the kinetic coefficients,
characterizing the rate of admolecule incorporation into
the <11> steps, are much greater than the kinetic
coefficients of the <10> steps.
Parameters of the shock waves in Fig. 2
No. 0 s L 1a 1b t
1 0.669 9.90 63 1.12 4.0
2 0.750 6.58 80 1.27 2.0
3 1.111 2.78 130 2.53 0.3
Second, macrosteps, i.e. bunches of elementary
steps, are formed under the condition that the average
(initial) width of the terraces does not exceeds 2λs, i.e.
diffusion fields on the terraces of adjacent steps overlap.
As the degree of overlap increases, the width of the
wave front and the time of its formation decrease.
Meanwhile, since the number of steps involved in the
wave formation decreases, the wave amplitudes differ
slightly.
The faster formation of waves as they approach the
pore edge and the steepening of their profile are caused
by an increase in the average local curvature of the
macroscopic surface as it approaches the edge of the
pore. At that, both the intensity of diffusion interaction
of the steps and the intensity of their thermodynamic
interaction increase [36].
The obtained value of the coefficient μ, as well as
the parameters a1,b1
and the front half-width L, allow us
to estimate the characteristic “explosion” time t0 on the
basis of relation (8), (9). The experimental values of
“explosion” time for shock waves 1, 2, 3 texp are shown
in Table and less than t0.
CONCLUSIONS
The paper presents the results made by electron
microscopy observations of the kinematic waves of
elementary step density on the vicinal growth surface
near NaCl(100), formed during the motion of the pores
in the crystal under the temperature gradient. It is shown
that for the studied growth conditions (T = 950 K and
= 4·10
-3
) the surface topography differs
significantly for the <10> and <11> directions. In the
ISSN 1562-6016. ВАНТ. 2022. №1(137) 159
<11> direction, the vicinal surface profile is represented
by monoatomic steps. Whereas in the <10> direction, as
the surface curvature increases, there is a bunching of
steps with the formation of macrosteps, i.e. bunches of
elementary steps separated by broad areas of atomic-
smooth terraces. Such a sawtooth form of the step
density dependence on the longitudinal coordinate is
characteristic of the shock waves described by the
previously obtained particular solutions of the BE with
zero boundary conditions [29]. Using the BE particular
solution for the first mode and the corresponding values
of the problem parameters to interpret the
experimentally obtained concentration distribution
(dimensionless density) of the steps, we obtained the
data on the parameters of three shock waves. The values
of the formation time of the observed shock waves were
obtained. It was shown that with an increase in the local
surface curvature and, consequently, the intensity of the
diffusion-kinetic (and thermodynamic) interaction of
elementary steps, the time of wave formation decreased.
ACKNOWLEDGEMENTS
This research was partly supported by the
2020 (grants 2066 and 2067) and 2021 (grants 2073 and
2074) Cooperative Research Projects at the Research
Center of Biomedical Engineering (RCBE) adopted as
the 2020 and 2021 Cooperative Research at Research
Institute of Electronics, Shizuoka University, Japan.
REFERENCES
1. T.L. Einstein. Equilibrium shape of crystals:
Handbook of Crystal Growth / T. Nishinaga (Edit).
Amsterdam: “Elsevier”, 2015, v. 1, p. 215-264.
2. N. Akutsu, T. Yamamoto. Rough-smooth
transition of step and surface: Handbook of Crystal
Growth / T. Nishinaga (Edit.) Amsterdam: “Elsevier”,
2015, v. I, p. 265-313.
3. C. Misbah, O. Pierre-Louis, Y. Saito. Crystal
surfaces in and out of equilibrium: A modern view //
Reviews of Modern Physics. 2010, v. 82, N 1, p. 981-
1040.
4. A.A. Chernov. Step bunching and solution flow
// Journal of Optoelectronics and Advanced Materials.
2003, v. 5, N 3, p. 575-587.
5. T. Yamaguchi, K. Ohtomo, S. Sato, N. Ohtani,
M. Katsuno, T. Fujimoto, S. Sato, H. Tsuge, T. Yano.
Surface morphology and step instability on the (0001 )
facet of physical vapor transport-grown 4H-SiC single
crystal boules // Journal of Crystal Growth. 2015,
v. 431, p. 24-31.
6. T. Mitani, N. Komatsu, T. Takahashi, et al.
Effect of aluminum addition on the surface step
morphology of 4H–SiC grown from Si–Cr–C solution //
Journal of Crystal Growth. 2015, v. 423, p. 45-49.
7. A. Gura, G. Bertino, B. Bein, M. Dawber.
Transition regime from step-flow to step-bunching in
the growth of epitaxial SrRuO3 on (001) SrTiO3 //
Applied Physics Letters. 2018, v. 112, N 18, p. 182902-
1-4.
8. H. Morkoc. Handbook of Nitride Semiconductors
and Devices. First ed. New-York: “Wiley-VCH”, 2008,
p. 1257.
9. I. Berbezier, A. Ronda. SiGe nanostructures //
Surface Science Reports. 2009, v. 64, N 2, p. 47-98.
10. I. Goldfarb. Step-mediated size selection and
ordering of heteroepitaxial nanocrystal //
Nanotechnology. 2007, v. 18, N 33, p. 335304-1-7.
11. J. Bao, O. Yasui, W. Norimatsu, K. Matsuda,
M. Kusunoki. Sequential control of step-bunching
during graphene growth on SiC (0001) // Applied
Physics Letters. 2016, v. 109, N 8, p. 081602-1-5.
12. M. Hou, Z. Qin, L. Zhang, T. Han, M. Wang,
F. Xu, X. Wang, T. Yu, Z. Fang, B. Shen. Excitonic
localization at macrostep edges in AlGaN/AlGaN
multiple quantum wells // Superlattices and
Microstructures. 2017, v. 104, p. 397-401.
13. K. Matsuoka, S. Yagi, H. Yaguchi. Growth of
InN/GaN dots on 4H-SiC(0001) 4° off vicinal substrates
by molecular beam epitaxy // Journal of Crystal
Growth. 2017, v. 477, p. 201-206.
14. V.I. Kibets, A.P. Kulik. High-temperature
deformation of copper during oxidation // Powder
Metallurgy and Metal Ceramics. 1995, v. 33, N 5-6,
p. 236-239.
15. J.P. v.d. Eerden, H. Müller-Krumbhaar.
Dynamic coarsening of crystal surfaces by formation of
macrosteps // Physical Review Letters. 1986, v. 57,
N 19, p. 2431-2433.
16. S. Stoyanov. Electromigration induced step
bunching on Si surfaces – how does it depend on the
temperature and heating current direction? // Japanese
Journal of Applied Physics. 1991, v. 30, N 1, p. 1-6.
17. M. Vladimirova, A. De Vita, A. Pimpinelli.
Dimer diffusion as a driving mechanism of the step
bunching instability during homoepitaxial growth //
Physical Review B. 2001, v. 64, N 24, p. 24520-1-6.
18. C. Duport, P. Nozières, J. Villain. New
Instability in Molecular Beam Epitaxy // Physical
Review Letters. 1995, v. 74, N 1, p. 134-137.
19. I. Derényi, C. Lee, A.-L. Barabási. Ratchet
Effect in Surface Electromigration: Smoothing Surfaces
by an ac Field // Physical Review Letters. 1998, v. 80,
N 7, p. 1473-1476.
20. J.B. Keller, H.G. Cohen, G.J. Merchant. The
stability of rapidly growing or evaporating crystals //
Journal of Applied Physics. 1993, v. 73, N 8, p. 3694-
3697.
21. H. Popova, F. Krzyzewski, M.A. Załuska-
Kotur, V. Tonchev. Quantifying the effect of step-step
exclusion on dynamically unstable vicinal surfaces: Step
bunching without macrostep formation // Crystal
Growth and Design. 2020, v. 20, N 11, p. 7246-7259.
22. J.M. Burgers. A mathematical model
illustrating the theory of turbulence // Advances Applied
Mechanics. 1948, v. 1, p. 171-199.
23. F.C. Frank, R.H. Doremus, B.W. Roberts, and
D. Turnbulleds. Growth and Perfection of Crystals.
New York: “John Wiley& Sons”, 1958, p. 411.
24. N. Cabrera, D.A. Vermilyea, B.W. Roberts,
and D. Turnbulleds. Growth and Perfection of Crystals.
New York: “John Wiley & Sons”, 1958, p. 393.
25. M.J. Lighthill, G.B. Whitham. On kinematic
waves. I. Flood movement in long rivers // Proceedings
of the Royal Society of London, Ser. A. 1955,
v. 229(1178), p. 281-316.
https://ui.adsabs.harvard.edu/#search/q=author:%22Sato%2C+Shinya%22&sort=date%20desc,%20bibcode%20desc
https://ui.adsabs.harvard.edu/#search/q=author:%22Tsuge%2C+Hiroshi%22&sort=date%20desc,%20bibcode%20desc
http://refhub.elsevier.com/S0022-0248(17)30714-5/h0005
http://refhub.elsevier.com/S0022-0248(17)30714-5/h0005
http://refhub.elsevier.com/S0022-0248(17)30714-5/h0005
http://refhub.elsevier.com/S0022-0248(17)30714-5/h0005
160 ISSN 1562-6016. ВАНТ. 2022. №1(137)
26. A.A. Chernov. The spiral growth of crystals //
Sov. Phys. Usp. I961, v. 4, N 1, p. 116-148.
27. O.P. Kulyk, V.I. Tkachenko, O.V. Podshy-
valova, V.A. Gnatyuk, T. Aoki. Nonlinear interaction of
macrosteps on vicinal surfaces at crystal growth from
vapor // J. Cryst. Growth. 2020, v. 530, p. 125296-1-7.
28. V.G. Bar'yakhtar, A.E. Borovik, Yu.S. Kaga-
novskii. Formation of macroscopic steps at vicinal
surfaces during crystal growth // JETP Lett. 1988, v. 47,
N 8, p. 474-477.
29. O.L. Andrieieva, V.I. Tkachenko, O.P. Ku-lyk,
O.V. Podshyvalova, V.A. Gnatyuk, T. Aoki.
Application of particular solutions of the Burgers
equation to describe the evolution of shock waves of
density of elementary steps // East European Journal of
Physics. 2021, N 4, p. 59-67.
30. O.P. Kulyk, Y.S. Kaganovskii, V.S. Kruzha-
nov. Motion and shape transformation of pores in NaCl
controlled by step dynamics on their surfaces // 22
nd
European Conference on Surface Science (Praha, Czech
Republic, 7–12 September 2003): Program and CD
Book of Extended Abstracts. 2003, Abstract No 17325.
31. Yu.S. Kaganovskii, V.S. Kruzhanov,
A.P. Kulick. Relaxation of form of nonisometric pores
in single crystals of NaCl // Crystallography. 1989,
v. 34, N 6, p. 921-925.
32. A.P. Kulik, O.V. Podshyvalova, I.G. Mar-
chenko. Radiation-induced motion of liquid inclusions
in alkali halide crystals // Problems of Atomic Science
and Technology. 2019, N 2(120), p. 13-19.
33. O.P. Kulyk, L.A. Bulavin, S.F. Skoromnaya,
V.I. Tkachenko. A model of induced motion of
inclusions in inhomogeneously stressed crystals /
A.R. Varkonyi-Koczy (Eds.) Engineering for Susta-
inable Future. Inter-Academia 2019: Lecture Notes in
Networks and Systems, 2020, v. 101, p. 326-339.
34. E.H. Zimm, J.E. Mayer. Vapor pressures, heats
of vaporization and entropies of some alkali halides // J.
Chem. Phys. 1944, v. 12, N 9, p. 362-369.
35. H. Bethge, K.W. Keller. Zur vollständigen
Beschreibung von Oberflächenstrukturen mit
Stufenhoben atomaren Größenordnung // Optic. 1965,
v. 23, p. 462-471.
36. Yu.S. Kaganovskii, O.P. Kulyk. Linear tension
of mono- and double atomic height steps on (100) NaCl
surface // VII-th European Conference on Surface
Crystallography and Dynamics (ECSCD-7), (Leiden,
The Netherland, 26–29 August 2001): Book of
Abstracts. 2001, p. 51-52.
Article received 27.10.2021
ФОРМИРОВАНИЕ УДАРНЫХ ВОЛН ПЛОТНОСТИ СТУПЕНЕЙ
НА ВИЦИНАЛЬНЫХ ПОВЕРХНОСТЯХ РОСТА NaCl(100)
А.П. Кулик, О.В. Подшивалова, О.Л. Андреева, В.И. Ткаченко, В.А. Гнатюк, Т. Аоки
Электронно-микроскопическим методом вакуумного декорирования исследована морфология
поверхности роста вблизи Na l(100), сформировавшейся при движении поры в кристалле под действием
градиента температуры. Показано, что при T = 950 К и = 4·10
-3
профиль вицинальной поверхности в
направлении <11> представлен моноатомными ступенями, тогда как в направлении <10>, по мере
увеличения кривизны поверхности, наблюдается группирование ступеней с образованием макроступеней –
сгустков элементарных ступеней, разделенных участками атомно-гладких террас. Пилообразная
зависимость плотности ступеней от продольной координаты описана частным решением уравнения
Бюргерса для ударной волны. Получены данные о параметрах трех ударных волн и времени их
формирования.
ФОРМУВАННЯ УДАРНИХ ХВИЛЬ ГУСТИНИ СХОДИН
НА ВІЦИНАЛЬНИХ ПОВЕРХНЯХ РОСТУ NaCl(100)
О.П. Кулик, О.В. Подшивалова, О.Л. Андрєєва, В.І. Ткаченко, В.А. Гнатюк, Т. Аокі
Електронно-мікроскопічним методом вакуумного декорування досліджена морфологія поверхні росту
поблизу Na l(100), що сформувалася під час руху пори в кристалі під дією градієнта температури.
Показано, що при T = 950 К і = 4·10
-3
профіль віцинальної поверхні в напрямку <11> представлений
моноатомними сходинками, тоді як у напрямку <10>, зі збільшенням кривизни поверхні, спостерігається
групування сходинок з утворенням макросходин – згустків елементарних сходинок, розділених ділянками
атомно-гладких терас. Пилкоподібна залежність густини сходин від поздовжньої координати описана
частинним розв’язком рівняння Бюргерса для ударної хвилі. Отримано дані про параметри трьох ударних
хвиль і час їх формування.
http://www.jetpletters.ac.ru/ps/1095/article_16544.shtml
http://www.jetpletters.ac.ru/ps/1095/article_16544.shtml
https://www.scopus.com/sourceid/19700182270?origin=resultslist
https://www.scopus.com/sourceid/19700182270?origin=resultslist
|
| id | nasplib_isofts_kiev_ua-123456789-195921 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1562-6016 |
| language | English |
| last_indexed | 2025-12-01T18:43:04Z |
| publishDate | 2022 |
| publisher | Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| record_format | dspace |
| spelling | Kulyk, O.P. Podshyvalova, O.V. Andrieieva, O.L. Tkachenko, V.I. Gnatyuk, V.A. Aoki, T. 2023-12-08T11:14:56Z 2023-12-08T11:14:56Z 2022 Formation of step density shock waves on vicinal NaCl(100) growth surfaces / O.P. Kulyk, O.V. Podshyvalova, O.L. Andrieieva, V.I. Tkachenko, V.A. Gnatyuk, T. Aoki // Problems of Atomic Science and Technology. — 2022. — № 1. — С. 154-160. — Бібліогр.: 36 назв. — англ. 1562-6016 DOI: https://doi.org/10.46813/2022-137-154 https://nasplib.isofts.kiev.ua/handle/123456789/195921 548.52 The morphology of the growth surface near NaCl(100), formed during the pore motion in a crystal due to the temperature gradient, has been studied by the electron microscopic method of vacuum decoration. It is shown that at T = 950 K and Δμ/kΤ = 4·10⁻³, the profile of the vicinal surface in the <11> direction is represented by monoatomic steps, while in the <10> direction, as the surface curvature increases, there is a grouping of steps with the formation of macrosteps – bunches of elementary steps separated by areas of atomically smooth terraces. The sawtooth dependence of the step density on the longitudinal coordinate is described by a particular solution of the Burgers equation for a shock wave. Data on the parameters of three shock waves and the time of their formation are obtained. Електронно-мікроскопічним методом вакуумного декорування досліджена морфологія поверхні росту поблизу NaCl(100), що сформувалася під час руху пори в кристалі під дією градієнта температури. Показано, що при T = 950 K and Δμ/kΤ = 4·10⁻³ профіль віцинальної поверхні в напрямку <11> представлений моноатомними сходинками, тоді як у напрямку <10>, зі збільшенням кривизни поверхні, спостерігається групування сходинок з утворенням макросходин – згустків елементарних сходинок, розділених ділянками атомно-гладких терас. Пилкоподібна залежність густини сходин від поздовжньої координати описана частинним розв’язком рівняння Бюргерса для ударної хвилі. Отримано дані про параметри трьох ударних хвиль і час їх формування. Электронно-микроскопическим методом вакуумного декорирования исследована морфология поверхности роста вблизи NaCl(100), сформировавшейся при движении поры в кристалле под действием градиента температуры. Показано, что при T = 950 K and Δμ/kΤ = 4·10⁻³ профиль вицинальной поверхности в направлении <11> представлен моноатомными ступенями, тогда как в направлении <10>, по мере увеличения кривизны поверхности, наблюдается группирование ступеней с образованием макроступеней – сгустков элементарных ступеней, разделенных участками атомно-гладких террас. Пилообразная зависимость плотности ступеней от продольной координаты описана частным решением уравнения Бюргерса для ударной волны. Получены данные о параметрах трех ударных волн и времени их формирования. This research was partly supported by the 2020 (grants 2066 and 2067) and 2021 (grants 2073 and 2074) Cooperative Research Projects at the Research Center of Biomedical Engineering (RCBE) adopted as the 2020 and 2021 Cooperative Research at Research Institute of Electronics, Shizuoka University, Japan. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Physics and technology of structural materials Formation of step density shock waves on vicinal NaCl(100) growth surfaces Формування ударних хвиль густини сходин на віцинальних поверхнях росту NaCl(100) Формирование ударных волн плотности ступеней на вицинальных поверхностях роста NaCl(100) Article published earlier |
| spellingShingle | Formation of step density shock waves on vicinal NaCl(100) growth surfaces Kulyk, O.P. Podshyvalova, O.V. Andrieieva, O.L. Tkachenko, V.I. Gnatyuk, V.A. Aoki, T. Physics and technology of structural materials |
| title | Formation of step density shock waves on vicinal NaCl(100) growth surfaces |
| title_alt | Формування ударних хвиль густини сходин на віцинальних поверхнях росту NaCl(100) Формирование ударных волн плотности ступеней на вицинальных поверхностях роста NaCl(100) |
| title_full | Formation of step density shock waves on vicinal NaCl(100) growth surfaces |
| title_fullStr | Formation of step density shock waves on vicinal NaCl(100) growth surfaces |
| title_full_unstemmed | Formation of step density shock waves on vicinal NaCl(100) growth surfaces |
| title_short | Formation of step density shock waves on vicinal NaCl(100) growth surfaces |
| title_sort | formation of step density shock waves on vicinal nacl(100) growth surfaces |
| topic | Physics and technology of structural materials |
| topic_facet | Physics and technology of structural materials |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/195921 |
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