Computer Modeling of Depth Distribution of Vacancy Nanoclusters in Ion-Irradiated Materials
Natalia A. Voronova
1,a, Anatoly I. Kupchishin
1,2,b*, Alexander A. Kupchishin
1,c, Akmaral A. Kuatbayeva
2,dand Tat`yana A. Shmygaleva
2,e1Kazakh National Pedagogical University named after Abai, Almaty, Kazakhstan
2Kazakh National University named after al-Farabi, Almaty, Kazakhstan
anatvoronova@mail.ru, bankupchishin@mail.ru, caak1974sasha@mail.ru,
dahamala2017@gmail.com, eshmyg1953@mail.ru
Keywords. Modeling, nanoclusters, ion, vacancy clusters, cascade and probability function, concentration, algorithm, calculation.
Abstract. In this work, the calculation algorithms of cascade and probability functions and vacancy nanoclusters concentration were developed and their calculations for various incident particles in silicon and iron were made.
Introduction
Creation of new materials with unique properties is one of priority research areas, both in physics and engineering. The study of small metal particles properties based on hundreds and thousands atoms is of great interest because of their potential usage as materials or surface nanostructures [1 – 3]. The fact that nanomaterial properties considerably depend on the integral properties of particles is undoubted. The free clusters internal structure study can play a key role in the explanation of their physical or chemical features. Metals and alloys still remain the constructional, tooling and other materials basis [4 – 6]. Important factors in their properties formation are the crystalline grid nanodefects arising, in particular, in case of irradiation. Production of high-quality, dense metal films and substrate coverings can be implemented by ion beam irradiation [7, 8]. Such materials search and development are based on clear understanding of radiation nanodefects creation mechanisms [9]. Earlier we developed the cascade and probability method (CPM) based on simulation processes of particles passing through substance and radiation defects [10, 11] in terms of which this work was carried out.
Calculation Procedure.
For calculation of depth distributions of nanoclusters in the ion-irradiated materials, it was primarily necessary to calculate the cascade and probability functions (CPF). When considering the multichannel interaction processes of charged particles with a solid body it was necessary to take into account, in particular, full energy losses on ionization and atomic excitation in the primary knocked-on atoms generation process.
For the ions forming primary knocked-on atoms, the dependence of approximating function on the energy which in turn depends on penetration depth is presented as follows [1]:
( ) ( )
−
= 1− 1
0 a E0 kh
h
σ
σ
, (1) where σ0,a,k are the approximation ratios connected with interaction range and rate of energy loss, E0 – primary ion energy.The following convenient formula is used for cascade and probability function calculation, [1]:
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, ) ( ln
ln
1 ln ) ln(
)
! ln(
exp ) , , (
0 0
0 0
0 0 0
0
− ′
−
−
− ′ +
+ − ′+
−
− ′
−
×
−
−
′ =
h ak h
kh E kh E n
h h kh E
h k E n ak
n E
h
n h l l l
ψ
(2)
Observations depths were found according to spatial distribution tables of ion-implanted impurities [7]. Further interaction sections calculation was made by Rutherford's equation [1]. The calculated sections values were approximated by formula (1).
CPF calculations were made with a double accuracy throughout all observation depths range.
Concentration of vacancy nanoclusters in silicon and iron during ion beam irradiation [10, 11]
was calculated by formula:
( )
∑ ∫′
′
− − ′
− ′
= −
= −
1
0 2 0 2 1 2
max 2
max
0 2 ( , )exp ( )
) (
)
, ( n
n n
h k
h n
d c c
k d h
h d h E h
E h E
E E
E h E E
c l
ψ l l l
, (3)where Ed is threshold displacement energy, Ec is energy of the primary knocked-on atom (PKA) at which the amount of the displaced atoms equals to Nd atoms number in a zone of spontaneous recombination, E2max is the greatest possible energy acquired by atom, ψn(h′,E0) is modified cascade and probability function, l1( )h′ and l2 are ion- and atom-atomic displacement range, n0, n1 are initial and final values of the result area, k is integer > 1.
Findings and their Discussion
Approximating curves of σ - h dependences for carbon and nitrogen in iron are given in Fig. 1 and 2.
E0 = 1000(1), 800(2), 500(3), 200(4), 100(5) keV;
points are for estimated data, continuous lines are approximations
Fig. 1. Approximations of the modified section of cascade and probability function for carbon in iron
0 2 4 6 8 0.0
0.5 1.0 1.5 2.0
5 4 3 2 1
sigma(
10 E 1)*10,b
h*10-4, cm
E0 = 1000(1), 800(2), 500(3), 200(4), 100(5) keV;
points are for estimated data, continuous lines are approximations
Fig. 2. Approximations of the modified section of cascade and probability function for nitrogen in iron
The calculations analysis shows that approximating curves of the modified interaction sections are well described by formula (1) that allows counting CP function for silicon and iron with high precision. At the same time the theoretical correlation relation fluctuates in the range from 0.99 to 0.9999.
In Figs. 3 and 4, the CPF dependences for carbon and nitrogen in iron on the interaction number are presented. Calculations results show that CPF depending on n and h increase, reaching a maximum, then decrease.
10 20 30 40 50 60 70 80 90
-12 -10 -8 -6 -4 -2 0
lgψ
1 2 3 4 n*10
E0 = 900 keV (1 – 4); h = 1.0×10–4; 2.0×10–4; 3.0×10–4; 4.0×10–4 (cm.) Fig. 3. CPF dependence on interaction number for carbon in iron
2 4 6 8 10 12 14
-12 -10 -8 -6 -4 -2
0 n*102
lgψ
1 2 3 4
E0 = 1000 keV (1 – 4); h = 1.0×10–4; 2.0×10–4; 3.0×10–4; 4.0×10–4 (cm) Fig. 4. CPF dependence on interaction number for nitrogen in iron
Calculations results for profiles by depth distribution of vacancy nanoclusters (1–10 nanometers) are presented in Figs. 5–7.
Ec = 100 keV; E0 = 1000 (1), 800 (2), 500 (3), 200 (4) keV
Fig. 5. Depth distribution profiles of vacancy nanoclusters in the course of ion beam irradiation for nitrogen in silicon
Ec = 100 keV; E0 = 1000 (1), 800 (2), 500 (3), 200(4) keV
Fig. 6. Depth distribution profiles of vacancy nanoclusters in the course of ion beam irradiation for carbon in iron
Ec = 100 keV; E0 = 1000 (1), 800 (2), 500 (3), 200(4) keV
Fig. 7. Depth distribution profiles of vacancy nanoclusters in the course of ion beam irradiation for nitrogen in iron
The factual findings show that with primary particle initial energy reduction the range of the result area is displaced to the right, radiation defects concentration values increase. Depending on penetration depth, the initial and final values of interaction number increase, the range of the result area (n0 n1) also increases and shifts to the right. With increase in threshold energy at the same penetration depth the radiation defects concentration values considerably decrease, the borders of the result area do not change.
Conclusions
1. Selection of approximating values for the modified interaction sections was made, their features were noted. At the same time, the theoretical correlation relation was equal to at least 0.99.
Formulas for the cascade and probability function (CPF) calculation were offered. CPF calculations for the incident carbon and nitrogen ions depending on interaction number in iron were made. With the interaction number increase CPF sharply increased, reached its maximum and further quickly decreased. With increase in ions energy the CPF curves were displaced to the right.
2. The algorithm was developed for the concentration calculation of nanoclusters in the materials irradiated by ions. Concentration behavior features of vacancy nanoclusters depending on primary particle initial energy, penetration depth, threshold energy were established. At the same time, concentration of defects at first slowly increased, then reached its maximum and sharply dropped in the end of an ion path.
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