「様式３」Project No.：2077

Formation of a quasiperiodic relief on vicinal surfaces［１］Organization
Project Leader：
Oleksandr Kulyk, Ph.D., Assoc. Prof. (V. N. Karazin
Kharkiv National University, Ukraine)
Representative at RIE：
Toru Aoki, Ph.D., Prof. (Research Institute ofElectronics, Shizuoka University, Japan)
Participants：
Victor Tkachenko, D.Sc. (Nat. Sci. Center"Kharkov Institute for Physics & Technology" ofthe National Academy of Sciences of Ukraine,Ukraine)
Oksana Podshyvalova, Ph.D., Assoc. Prof.
(National Aerospace University “KharkivAviation Institute”, Ukraine)
Volodymyr Gnatyuk, Ph.D., Senior Scientist(V. E. Lashkaryov Institute of Semiconductor
Physics of the National Academy of Sciences ofUkraine, Ukraine)
Olga Lisina, Ph.D. (V. N. Karazin Kharkiv
National University, Ukraine)［２］Research Progress
The aim of the project was to study theprocesses leading to the formation of aquasiperiodic relief on the vicinal surface of thecrystal during layered growth from the vapor. Thestructure and behavior of nonlinear density
waves of elementary steps on thermodynamicallystable vicinal surfaces have been described, in thecase when dissipative effects prevail overdispersion ones. The periodicity is a key
requirement for creation of low-dimensional
structures (quantum wells, quantum wires,quantum dots, etc.) used in optoelectronic andmicroelectronic devices.
The project participants actively collaborated
with each other by providing the information
through electronic media and face-to-face when
the Ukrainian partners O. Kulyk andO. Podshyvalova visited RIE, Hamamatsu inJanuary 2020 (research visits in the frame of the2019 Cooperative Research at Research Center ofBiomedical Engineering). They reported theobtained results, exchanged the data and ideas on
join research in frames of the collaborative project,and discussed the issues of further cooperation
between the partner institutions.
Apart from achieving the research results,
described in the next section, the project
participants were involved in the holding of theinternational scientific forums in Ukraine todiscuss the research work concerning the project
topics and general cooperation issues.
Dr. O. Kulyk presented the project-related
results at The Int. Scientific and Technical Conf.
“Physical and Technical Problems of Energy andTheir Solutions 2019” (Kharkiv, Ukraine, 19 Jun.
2019) [1,2], The 18th Int. Conf. on Global Researchand Education in Engineering for SustainableFuture, Inter-Academia 2019, (Budapest,Hungary 4-7 Sep. 2019) [3] and submitted theabstract with the key research findings to theBook of Abstracts of The Annual Meeting of 2019
Cooperative Research on Biomedical Engineering(Yokohama, Japan, 13 Mar. 2020) [6].
Totally, 6 publications (2 articles and 4abstracts) have been published during the project
implementation [1-6].
［３］Results(３－１) Research results
The averaging method on large intervals of
spatial coordinate change, which neverthelessremained infinitely small, was used to derive theKorteweg-de Vries-Burgers (KdV-B) equation for
an one-dimensional (1D) step model on a vicinal
surface of a crystal [4]. The microscopic approachis based on solving the Burton, Cabrera andFrank equations for the one-dimensional motionof a diffusion-coupled system of elementary steps.
The qualitative picture of the phenomena
depends on the competition of nonlinear processes
with dissipative and dispersion effects
determined by higher derivatives in the KdV-B
equation. In the case when dissipative effects

2u  t , x   2n cos   x   C e   sin   x 
n2tn2

R2, (1)nwhere n  2 n, n  1,2,3... are integers.
Solution (1) for n  1 is visualized as athree-dimensional surface plot (Fig. 2). The
feature of the obtained solution is the possibility
to predict the discontinuity of the step
concentration, which corresponds to the process ofshock kinematic wave formation on an initiallyatomically smooth vicinal surface. In the casen  2 , at some stage of the process, one canobserve a set of alternating shock waves similar
to n  1 , which determines a quasiperiodic
macrostep surface topography.

0.505x0 -0.501020t30Fig. 2. An analytical solution of the Burgers equation
with periodic initial conditions, which demonstrates
the tendency of the wave profile to a sawtooth shape.

3. The described surface topography was
found on the vicinal NaCl surface (near (100)) for
step bunches with orientations [10] and [01]during growth from the vapor phase (Fig. 3).

32u(x, t)sin  n x  CR e n tu(x, t)

prevail over dispersion effects, the KdV-B
equation transforms into the well-known Burgers
equation. Such transformation is possible due tothe smallness of the characteristic parameter,
which is proportional to the ratio of theperturbation wavelength to the dimensionlessdensity (i.e. the concentration of steps in theinitial state of the vicinal surface).
1. Using computer modeling, the numericalsolution of the one-dimensional non-stationary
Burgers equation with periodic initial conditions
was obtained [6]. The equation solving wasperformed with a meshless scheme using themethod of partial solutions and radial basis
functions. The time discretization of theone-dimensional Burgers differential equation
was carried out by the generalized trapezoidal
method. The inverse multiquadric function wasused as radial functions. The solution of theone-dimensional non-stationary Burgers equation
is visualized as a three-dimensional surface plot
(Fig. 1). Qualitative analysis of the solution in theform of three-dimensional plots for individual
time intervals indicates that over time there is anincrease in the slope of the wave profile, which
describes the dimensionless density of steps. In
relation to the normalized profile of the vicinalsurface for the 1D-step model, this means aredistribution of the concentration of steps.

10-1-2-3
241.5t3

120.5

10x0Fig. 1. A numerical solution of the Burgers equation
with periodic initial conditions: u(x, t) is the velocityof the wave profile, x and t are the coordinate andtime, respectively.

2. For quantitative analysis and interpretationof the experimental data, the analytical solutionof the one-dimensional non-stationary Burgers
equation with periodic initial conditions wasobtained for the first time:

Fig. 3. TEM image of the vicinal growth surface near(100) NaCl formed during the pore motion in thetemperature gradient.

Fig. 4. Dependence of the step concentration on thelongitudinal coordinate in the direction [10] (theconcentration of steps for waves 2-5 was determinedby averaging over several adjacent terraces; forconvenient approximation of the results, the same
direction of the waves was chosen).

Concentration

The area shown in Fig. 3 is a part of thegrowth surface of a pore that was moving in thecrystal due to the temperature gradient [4]. Thesupersaturation that occurs inside the pore near
the growth surface as well as the characteristic
parameters of the surface morphology indicatethat the observed kinematic waves should be
described exactly by the Burgers equation (not by
KdV-B). Unfortunately, the used technique ofelectron microscopic vacuum decoration does notallow us to specify the distribution of elementary
steps within the observed macrosteps. However, it
makes possible to determine the number of
elementary steps involved in the formation of
step-bunch structures, which we call macrosteps.
Such determination of the number of elementary
steps is possible because elementary steps with adirection of the <11> type practically do not form
step-bunch
structures.
Apparently,
thismorphological difference is due to the fact that the
kinetic coefficients of the elementary steps of thedirection <11> are much more than ones of the<10> steps.
4. To describe the experimental results with
our analytical solution of the one-dimensional
non-stationary Burgers equation with periodic
initial conditions, we carried out the following: digitized the selected surface area for 75
elementary steps in the [10] and [11] directions; reproduced the topography of the investigatedsurface area;
 basing on the obtained data and known valuesof the length of adatoms diffusion path [4], thedependences of the step concentration on thelongitudinal coordinate in the [10] direction
were constructed (markers in Figs. 4 and 5).
It was shown that waves with small
amplitudes (1 and 2 on the left side in Fig. 4) andwaves with large amplitudes (step-bunch
structures of different heights 3-5 on the rightside in Fig. 4) move towards each other.
5. The experimental dependences of theconcentration of steps on their coordinate wereapproximated by theoretical curves based onsolution (1) (solid lines in Fig. 5) using thesoftware Origin Pro and Matlab. Adequate
agreement between the theoretical calculations
and experiment allowed us to determine thevalues of the characteristic parameters of theobserved shock waves.

Coordinate

Fig. 5. Structure of the shock waves (1 and 2) shownin Fig. 3: marks indicate the values of monatomic
steps concentration; solid curves demonstrate theresult of the computations based on equation (1).

(３－２) Ripple effects and further developments
The obtained research results and calculated
data on the characteristic parameters of theconsidered shock waves open up the possibility ofan analytical solution of the problem on theformation of a quasi-periodic relief onthermodynamically stable vicinal surfaces under
nonequilibrium conditions. In particular, it
becomes possible to describe the kinetic
regularities of the formation of the quasiperiodic
surface profile shown in Fig. 6. As known,
periodicity is the main requirement for obtaining
low-dimensional structures.

Solutions 2019, Proceedings of the International
Scientific and Technical Conference, P. 48, 2019
(Kharkiv, Ukraine, 19 June 2019). (Plenaryreport)(3) O.P. Kulyk, L.А. Bulavin, S.F. Skoromnaya,
V.I. Tkachenko, Model of induced motion ofinclusions in the field of forces of an
inhomogeneously stressed crystal, The 18th

International Conference on Global Research andEducation in Engineering for Sustainable Future,
Inter-Academia 2019, Program and Book ofAbstracts, P. 9-10, 2019. (4-7 September 2019,
Fig. 6. TEM image of a quasiperiodic relief formedon the pore growth surface near (100) NaCl(macrosteps are represented by elementary stepbunches; broken and rectilinear steps intersecting
the step bunches are slip bands that appeared
when the crystal was split and the pores wereopened in vacuum before the procedure ofdecorating with gold particles).
［４］Achievements (List of Publications)(1) O.P. Kulyk, V.I. Tkachenko, O.Yu. Lisina,
V.O. Mikhnych, V.A. Gnatyuk, T. Aoki, Nonlinear
effects of diffusion interaction of steps onthermodynamically stable vicinal surfaces,

Physical and Technical Problems of Energy andTheir Solutions 2019, Proceedings of theInternational Scientific and Technical Conference,
P. 13, 2019 (Kharkiv, Ukraine, 19 June 2019).
(Plenary report)(2) V.M. Kolodyazhny, O.P. Kulyk, O.Yu. Lisina,
Approximate solution of the non-stationary
Burgers-Korteweg de Vries’ problem based on theuse of a generalized Taylor series, Physical andTechnical Problems of Energy and Their

Budapest and Balatonfüred, Hungary).
(4) О.P. Kulyk, V.I. Tkachenko, O.V. Podshyvalova,V.A. Gnatyuk, T. Aoki, Nonlinear interaction ofmacrosteps on vicinal surfaces at crystal growthfrom vapour, Journal of Crystal Growth, Vol. 530,P.
125296-1-7,
Jan.
2020.
DOI:

10.1016/j.jcrysgro.2019.125296(5) O.P. Kulyk, L.А. Bulavin, S.F. Skoromnaya,
V.I. Tkachenko, Model of induced motion ofinclusions in the field of forces of an
inhomogeneously stressed crystal, in: A.R.

Varkonyi-Koczy (ed.) Engineering for Sustainable
Future. Inter-Academia 2019. Lecture Notes in
Networks and Systems (LNNS), Vol. 101, P.
326-3397, 2020, Cham: Springer. DOI:
10.1007/978-3-030-36841-8_32(6) O. Kulyk,
I. Hariachevska,O. Lisina,V. Tkachenko, O. Andrieieva, O. Podshyvalova,
V. Gnatyuk, T. Aoki, Formation of a quasiperiodic
relief on vicinal surfaces, The Annual Meeting of2019 Cooperative Research on Biomedical
Engineering, Book of Abstracts, Abstract No 1-11,
2019, (13 March 2020, Tokyo, Japan), in press.

Page 5：１ column
Travelling Report（Mention each travel by CRP budget.）Name：Oleksandr Kulyk
Affiliation：V. N. Karazin Kharkiv National University, Svobody Sq. 4, Kharkiv 61022, Ukraine
Period of time：04 January – 20 January 2020
Destination：Research Institute of Electronics, Shizuoka University
Purpose： (1) Study of the processes leading to the formation of quasiperiodic relief on crystal vicinal
surfaces under the layer-by-layer growth from the vapour. (2) Participation in the scientific meetings at ResearchInstitute of Electronics, Shizuoka University during the period of stay. (3) Discussion of the research results, preparation ofthe publications and development of the plans of the future research.
Name of receiver：Prof. Toru Aoki

Oleksandr Kulyk
Dr., Assoc. Prof.
V. N. Karazin Kharkiv National University
Kharkiv, Ukraine