ISSN 1027-4510, Journal of Surface Investigation. X-ray, Synchrotron and Neutron Techniques, 2008, Vol. 2, No. 2, pp. 301–304. © Pleiades Publishing, Ltd., 2008.
Original Russian Text © T.V. Vakhnii, G.A. Vershinin, I.A. Bozhko, I.A. Kurzina, Yu.P. Sharkeev, T.S. Grekova, 2008, published in Poverkhnost’. Rentgenovskie, Sinkhrotronnye i
Neitronnye Issledovaniya, No. 4, pp. 51–54.
Formation of Concentration Profiles of Implanted Ions
in Metallic Materials under Polyenergetic Implantation
T. V. Vakhniia, G. A. Vershinina, I. A. Bozhkob, I. A. Kurzinab,
Yu. P. Sharkeevb, and T. S. Grekovaa
a
b
Omsk State University, pr. Mira 55-a, Omsk, 644077 Russia
Institute of Strength Physics and Materials Science, Siberian Branch, Russian Academy of Sciences, Tomsk, Russia
Received August 16, 2007
Abstract—A physical and mathematical model of mass transfer in polycrystalline metallic materials under
exposure to ion beams is proposed. Alongside bulk indiffusion from the irradiated surface, diffusion along the
migrating extensive defects interacting with an impurity is considered. For a polyenergetic ion beam, the contribution of bulk indiffusion is presented as an integral over energy of the product of two functions; one of them
describes the energy distribution of ions in a beam and the second represents the implantation profile of
monoenergetic ion beam.
DOI: 10.1134/S1027451008020262
INTRODUCTION
Atom transport is one of the reasons for change in
surface properties of metallic materials under exposure
to the accelerated ion beams. However, the discrepancy
between the projective range of implanted particles and
the diffusant penetration depth into an irradiated sample can reach an order of magnitude or more. The concentration profiles of aluminum ions in nickel and titanium after implantation with a high-intensity ion beam
from a Raduga-5 source may serve as an example (Figs. 1
and 2) [1–4]. The elemental analysis of irradiated samples demonstrates that the dopant penetration depth
exceeds many times the maximum projective range of
ions with an energy of 120 keV in the investigated
materials for high-intensity ion implantation (with
doses up to 2.9 × 1018 ion/cm2). The unambiguous
n, arb. units
1.0
interpretation of this anomalous mass transfer to great
depths is absent in the literature. Therefore, identification of the basic mechanisms and interpretation of the
observed regularities of mass transfer to anomalously
great depths under irradiation of metallic materials with
high-intensity ion beams is an important problem of
contemporary materials science. An understanding of
the nature of the enhanced mass transfer would quicken
significantly the development of novel radiation technologies of materials modification and also allow the
prediction of their properties before and after bombardment.
THEORETICAL MODEL
Concentration profiles of implanted atoms were
described taking into account the following assumpn, arb. units
1.0
(a)
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
100
200
300
400
x, nm
0
100
(b)
200
300
400
x, nm
Fig. 1. Concentration profiles of aluminum ions in nickel: (a) regime 1; (b) regime 2. Dots represent experimental data [2].
301
302
VAKHNII et al.
n, arb. units
1.0
n, arb. units
1.0
(a)
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
500
1000 1500
2000
x, nm
0
(b)
1
2
500
1000
1500
2000
x, nm
Fig. 2. Concentration profiles of aluminum ions in titanium: (a) regime 3; (b) regime 4 (curves 1 and 2 represent experimental results
with and without a film, respectively). Dots represent experimental data [3].
tions: (1) since the beam diameter of charged particles
far exceeds their path length in condensed matters, the
one-dimensional approximation is used; (2) the heattransfer properties of an exposed material are assumed
to be constant for the total time of bombardment; (3) an
ion beam falls perpendicularly to the surface of an
exposed sample which is assumed to be infinitely thick.
Taking into account the sputtering of a target, the
depth distribution of an impurity under monoenergetic
implantation due to the bulk indiffusion into a target
can be described by the equation
∂n ( x, t )
∂ n ( x, t )
∂n ( x, t )
- + V ------------------- + f ( x, t )
------------------- = D -------------------2
∂x
∂t
∂x
The Gaussian function is most commonly used as
f (x, t):
N0
( x – Rp)
- exp – --------------------f ( x, t ) = -------------------- .
2
2π∆R p
2∆R p
2
(5)
The Lomax distribution may also be used [5]:
x 2
f ( x, t ) = K 1 + ⎛ ---⎞
⎝ a⎠
–p
x
exp – v arctan ⎛ ---⎞ ,
⎝ a⎠
(6)
2
(1)
with the following initial and boundary conditions:
n ( x, 0 ) = 0,
∂n ( 0, t )
------------------- = 0,
∂x
lim n ( x, t ) = 0, (2)
x→∞
where D is the diffusion coefficient; f (x, t) is the source
function; V is the sputtering rate related to the sputtering ratio S by the relation V = Sj /(Nq); N is the atomic
concentration of a target; j is the ion current density;
N0 = jt/q is the dose of implanted ions; and q is the elementary charge.
The solution of the boundary-value problem (1)–(2)
is represented in the form
where Rp is the projective range of ions and ∆Rp is the
standard deviation of the range along the x-axis. The
normalization factor K and parameters a, p, and v of the
Lomax distribution are determined from Rp, ∆Rp, and Sk
(where Sk is the asymmetry parameter) by the algorithm
described in the work [5].
The concentration profile of implanted atoms under
polyenergetic implantation can be calculated using the
following integral relation:
E max
n ( x, t ) =
∫ g ( E )n ( x, E ) dE,
0
(7)
E min
t∞
n ( x, t ) =
∫ ∫ f ( ξ, τ )G ( x, ξ, t – τ ) dξ dτ,
(3)
00
where the function G(x, ξ, t) takes the form
1
V (ξ – x) V t
G ( x, ξ, t ) = ----------------- exp -------------------- – -------2D
4D
2 πDt
2
2
2
⎧
( x – ξ)
( x + ξ) ⎫
× ⎨ exp – ------------------ + exp – ------------------- ⎬.
4Dt
4Dt ⎭
⎩
(4)
where g(E) is the energy distribution of incident ions in
a beam and n0(x, E) is the profile of implanted ions corresponding to the monoenergetic beam with the energy E.
The Gaussian function, Lomax distribution, or stationary solution of the (1)–(2) problem may be selected as
n0(x, E). The parameters Rp, ∆Rp, and Sk for different
values of energy and ion–target combinations are presented as tables in the work [5]. The energy distribution
function of particles in a beam g(E) is formed of two
half-Gaussians with dispersions σ1 and σ2 joined at the
point Eav [5]
JOURNAL OF SURFACE INVESTIGATION. X-RAY, SYNCHROTRON AND NEUTRON TECHNIQUES Vol. 2 No. 2 2008
FORMATION OF CONCENTRATION PROFILES OF IMPLANTED IONS
303
Bombardment regimes and parameters used in simulation of concentration profiles
Regimes
System
Average energy,
keV
Dose,
ion/cm2
Diffusion coefficient, cm2/s
Sputtering ratio,
atom/ion
1
2
3
4
Ni–Al
Ni–Al
Ti–Al
Ti–Al
68
68
40
40
2.0 × 1018
1.0 × 1019
6.2 × 1017
1.1 × 1018
7 × 10–13
3 × 10–13
0.5 × 10–13
1.5 × 10–13
20
18
10
15
2
g ( E ) = --------------------------------( σ 1 + σ 2 ) 2π
⎧
2
( E – E av ) ⎞
⎪ exp ⎛⎜ – ----------------------⎟ at E ≤ E av ,
2
⎪
2σ 1 ⎠
⎝
⎪
×⎨
⎪
⎛ ( E – E av ) 2⎞
⎪ exp ⎜ – -----------------------⎟ at E ≥ E av .
2
⎪
2σ 2 ⎠
⎝
⎩
(8)
In the case of treatment of polycrystalline metallic
materials by continuous ion beams (ion implantation
regime), when migrating grain boundaries may be
taken as independent (type B kinetics [6]), i.e., when
the condition Dt + vt < d is valid (where D is the bulk
diffusion coefficient, v is the average migration velocity of grain boundaries, d is the average size of grains,
and t is the exposure time), the concentration profile
along the x-axis directed perpendicular to the bombarded surface can be presented in the form [7]
N ( x, t ) = s p n ( x, t )
+ s 0 n d ( x, t )
v =0
+ s d n d ( x, t )
v ≠ 0.
(9)
Here n(x, t) describes the bulk indiffusion from the
bombarded surface, and nd(x, t)|v = 0 and nd(x, t)|v ≠ 0
describe the diffusion along the immobile and migrating grain boundaries, respectively. The quantities sp, s0,
and sd determine the fractions of contributions to the
concentration profile. The computational algorithm of
the contribution of the migrating grain boundary diffusion interacting with an impurity is given in work [7].
ANALYSIS OF RESULTS
Figure 1 represents the concentration profiles of aluminum in nickel for regimes 1 and 2 of ion implantation
presented in the table. The experimental curves have a
form typical for a monoenergetic beam. To obtain the
best possible fit of the theory and the experiment, the
diffusion coefficient and sputtering ratio were varied
when computing. Concentration values are expressed
in relative units. The best agreement between the theory
(ignoring the diffusion along migrating grain boundaries) and the experiment is obtained when the Lomax
distribution is selected as the source function f(x, t).
Concentration profiles of aluminum in titanium
(Fig. 2) for regimes 3 and 4 of implantation are indicative of the polyenergetic nature of an ion beam. Equations (7)–(8) are used to describe the observed curves.
A relatively better agreement between the computation
and the experiment is obtained if the steady-state solution of Eq. (1) with the source function in the form of
Gaussian (5) is used as n0(x, E).
A coating up to 170 nm thick, containing along with
aluminum significant concentrations of oxygen and
carbon, is formed on the surface of an implanted sample under high-intensity ion implantation with high
doses (regime 4) [3]. Therefore, two experimental profiles of aluminum in titanium, with and without a film,
are presented in Fig. 2b (regime 4). A qualitative agreement between the theory and the experiment is
obtained.
Besides sputtering, the processes such as radiationinduced and thermal diffusion, ion mixing, grain
boundary diffusion, et al. apparently have an impact on
the formation of concentration profile under high-dose
ion implantation. The observed tails of concentration
profiles are explained by the diffusion along the migrating grain boundaries and dislocations (Fig. 3) [7]. Figure 3 shows the calculated concentration profiles of aluminum in titanium after implantation with different
radiation doses ((a) 6.2 × 1017 ion/cm2 and (b) 1.1 ×
1018 ion/cm2) and those measured after bombardment
with 40 keV ions and ion current density of 0.63 mA/cm2
[3]. Taking into account the bulk diffusion describes the
experiment only in the near-surface layer (Fig. 2). The
concentration profile in Fig. 3a is determined by the
contributions of bulk indiffusion (89%), diffusion along
the immobile extensive defects (1%), and diffusion
along the defects migrating with the average velocity
v = 1.83 × 10–7 cm/s (10%). The concentration profile
in Fig. 3b is determined by the contributions of bulk
indiffusion (75%), diffusion along the immobile extensive defects (1%), and diffusion along the defects migrating with the average velocity v = 5.54 × 10–7 cm/s (24%).
The parameters describing the medium and determining the interaction potential between dislocation and
impurity have been taken from reference books [8, 9].
Thus, it follows from the carried out calculations
that taking into account the polyenergetic nature of ion
implantation stipulates an appearance of the maximum
on the concentration profile at large distances from the
bombarded surface (~0.5 µm). Moreover, the inclusion
JOURNAL OF SURFACE INVESTIGATION. X-RAY, SYNCHROTRON AND NEUTRON TECHNIQUES Vol. 2 No. 2 2008
304
VAKHNII et al.
n, arb. units
1.0
n, arb. units
1.0
(a)
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0.5
1.0
1.5
2.0
x, µm
0
0.5
(b)
1.0
1.5 2.0 2.5
x, µm
Fig. 3. Concentration profiles of aluminum ions in titanium after implantation with different radiation dose: (a) 6.2 × 1017 ion/cm2;
(b) 1.1 × 1018 ion/cm2. Dashed line represents the model ignoring diffusion along migrating grain boundaries. Dots represent experimental data [3].
of diffusion along the migrating grain boundaries
allows us to explain the penetration of implanted atoms
into bombarded material towards a greater depth, significantly greater than the projective range of implanted
atoms.
CONCLUSIONS
The study conducted demonstrates that the enhanced
mass transfer in metals and alloys under high-dose ion
implantation can be caused by an increase in the mobility
of diffusing atoms along the migrating structural
extended defects.
Taking into account the polyenergetic nature of ion
implantation stipulates an appearance of the maximum
on the concentration profile at large depths from the
bombarded surface (~0.5 µm).
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JOURNAL OF SURFACE INVESTIGATION. X-RAY, SYNCHROTRON AND NEUTRON TECHNIQUES Vol. 2 No. 2 2008