「様式３」Project No.：2021

Nonlinear waves of elementary step density on the vicinal
surfaces of growth［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：
Oksana Podshyvalova, Ph.D., Assoc. Prof.
(National Aerospace University “KharkivAviation Institute”, Ukraine)
Volodymyr Gnatyuk, Ph.D., Assoc. Prof. (V. E.
Lashkaryov Institute of Semiconductor Physics ofthe National Academy of Sciences of Ukraine,Ukraine)
Yuri Kaganovskii, Dr., Professor (Bar-IlanUniversity, Israel)［２］Research Progress
The project objective was to study the vicinal
surface evolution of a crystal under thelayer-by-layer growth from the vapour, based onelementary microscopic processes. The structure
and behavior of nonlinear waves of elementary
step density on thermodynamically stable vicinal
surfaces have been described when elementary
step motion is limited by surface mobility ofadsorbed atom but not by the rate of their
attachment (detachment) to the steps. Vicinal
surfaces are important systems which are widelyused as substrates for periodical micro/nano-structured materials (quantum wells, wires,dots, etc.) in opto- and microelectronic devices.
The participants of the project actively
collaborated through electronic media andface-to-face when the Ukrainian partners
O. Kulyk and O. Podshyvalova visited RIE,
Hamamatsu in August 2018 (research visits inthe frame of the 2018 Cooperative Research at
Research Center of Biomedical Engineering).

They reported the achieved results, exchanged
ideas on join work in frames of the cooperative
project, and discussed 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 6th International Scientific andTechnicalConference“ProblemsofInformatization” in Kharkiv, Ukraine (14-16
November 2018) and will present some researchresultsatTheXIX
International
Symposium ”Discrete Singularities Methods inMathematical Physics” (DSMMPh-2019) inOdessa, Ukraine (24-27 June 2019).
［３］Results(３－１) Research results
The profile dynamics of the macroscopic
curved vicinal surface of a crystal, growing fromthe vapor phase, was investigated. The
microscopic approach is based on solving BCF(Burton, Cabrera and Frank) equations for
one-dimensional motion of a diffusion-coupledsystem of elementary steps.
1. The expressions for average values ofadatom concentration and elementary step
velocity were obtained by the method of
averaging over the large variation intervals,
which nevertheless remain infinitely small.
2. The nonlinear Korteweg de Vries – Burgers
equation was derived from the continuity
equation for average values of adatom
concentration and elementary step velocity with aphenomenological account for the boundary
curvature:

This equation describes the nonlinear dynamics
of adatom concentration on a macroscopicallycurved vicinal surface of a crystal (τ is thedimensionless time; Y is the dimensionlesscoordinate,    Y ,  ).
3. The appearance of a monotone shock wave
without oscillations was considered (Fig. 1).

)

3Shock wave amplitude (0.1 Å    3  2  u0   2  0.
Y 3

21

Longitudinalcoordinate (nm)Fig. 3. The structure of the shock waves shown inFig. 2: ∘∘∘ - positions of monoatomic steps; lines calculation results.

The values of the dimensionless amplitude, wave
number and shock wave velocity for three shockwaves from Fig. 3 are presented in Table 1.
Fig. 1. Shock wave structure versus spatial variable
x (in relative units).

4. The parameters of this shock wave weredetermined: the wave numberq0  0.505407239  A0 kT   Va a 2s16and shock wave velocity u0  4 A0 q0 0 s ,where A0 is the dimensionless amplitude of theshock wave, k is the Boltzmann constant, λs is themean diffusion length of an adatom during its
lifetime, σ is the crystal surface tension, Va is theatomic volume, a is the height of elementary steps,
ρ0λs is the average dimensionless density of steps.
5. The theoretical results were confirmed by
the experimental data for alkali halide crystals(Figs. 2 and 3).

Fig. 2. TEM image of the vicinal growth surface near(100) NaCl formed under relaxation of the shape of a
non-isometric pore at isothermal conditions [2]: T =1·103 K; Δμ/kT ≈ 6·10-5.

Wave

0s

A0 108

q0u0 106

1

0.150

6.70880.1225

0.219

23

0.404
0.567

131.080 0.2010
362.352 0.2381

2.609
6.090

Table 1. The values of dimensionless characteristics
of shock waves.

Decreasing in the amplitude, wave number
and velocity of shock waves occurs, first, because
as the average density of elementary steps
becomes lower, the intensity of their diffusioninteraction decreases and, second, the degree ofheterogeneity associated with an average local
macroscopic surface curvature decreases as well.
(３－２) Ripple effects and further developments
Obtained results open an opportunity toanalyze in details the formation of aquasi-periodic relief on thermodynamically stablevicinal surfaces under non-equilibrium conditions.
The periodicity is the main requirement for
low-dimensional structure obtaining.
［４］Achievements (List of Publications)(1) O.P. Kulyk, O.Yu. Lisina, V.I. Tkachenko,
Algorithmization of the search for solutions ofsome nonlinear evolution equations, The 6thInternational Scientific and Technical Conference“Problems of Informatization”, Abstract Book,

Abstract No 4.2.13, P. 137, 2018 (Kharkiv,Ukraine, 14-16 November 2018).
(2) O.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, 18 pages(2019), to be published.
(3) O.P. Kulyk, O.Yu. Lisina, Simulation ofnonlinear
macrostepinteractions,XIX

International Symposium “Discrete Singularities
Methods
inMathematicalPhysics”(DSMMPh-2019), (Odessa, Ukraine, 24-27, June,2019), submitted.

Page 3：１ 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：08 August – 20 August 2018
Destination：Research Institute of Electronics, Shizuoka University
Purpose： (1) Study the processes leading to the formation of macrostep topography on crystal vicinal
surfaces under nonequilibrium conditions. (2) Participation in scientific meetings at Research Institute of Electronics,
Shizuoka University during the period of stay. (3) Discussion of the research results, preparation of the publications anddevelopment 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