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Nuclear Instruments and Methods in Physics Research B 95 (1995) 481-484

MiMl B Beam lnteactions

with Materials & Atoms

Relaxation-induced segregation in the theory of collisional mixing

M.G. Stepanova

Keldysh Institute of Applied Mathematics, Russian Acadetq of Sciences. Miusskaya ~1.4, Moscow 125047. Russia

Received 11 August 1994; revised form received 21 November 1994

Abstract A new form of stress relaxation in solid targets under ion bombardment is proposed. The relaxation is accounted for by

the directed part of diffusion fluxes enhanced by the stresses, and may be of nonstoichiometric character. Spatial-temporal composition changes in an ion-bombarded binary target are considered in the framework of the diffusion approximation for collisional mixing with account for the new mechanism of the relaxation. Relaxation-induced segregation in the target is

discussed.

1. Introduction

There is a traditional interest in the literature for simple

phenomenological approaches to describe preferential sput- tering and collisional mixing in two-component targets

under ion bombardment. From the linear transport theory

[I], assuming small relocations of recoils, the density of flux associated with ion-induced collision mixing, JFix( x, t), may be obtained in the diffusion approximation

J,Jyx, t) = - ;(Dni”(x)c,,(x, t))

with

+ Vg”‘“(x)C,(x, t), (1)

yrm”(x) =&$c,(x, x’)(x’-x) dx’, (2)

D”‘i”(x)=0.5J,~~G&, x’)(x’-x)‘dx’, (3)

where LY = 1, 2 denotes different components in the target, C,(x, t) is the concentration of component CK at depth x at time t, C,(x, t)G,(x, x’) dx dx’ is the mean number of atoms (Y relocated from a layer (x, dx) to a layer (x’, dx’) per incident ion at time t, and J, is the density of ion current.

Thus, the concentration C,(x, t) is defined by the equation

;C.( x, t) = - ;(qi”(x, t) +J,‘e’(x, t))

+ v(t);c,(x, t>, (4)

where V(x) is the rate of surface recession due to sputter- ing, and J$(x, t) is the relaxation flux.

The term J,‘“’ (x, t) plays a very important role in the theory of ion mixing. Transport of material by collision cascades under ion bombardment generates density gradi-

ents in the target, so that a relaxation mechanism should be

included in the model to avoid unphysical density changes. It the linear theory of mixing, the homogeneous relaxation

is used, when [l-3]

J:‘(x, t) = -Jan-‘C,(x, t)/=h(x) dx, x

(5)

where R is the mean atomic volume at depth x at time t, and the function h(x) is defined by the condition of

balance

C(JJ’x(x, t)+Jz’(x, t>)=o. (6) a

While the homogeneous relaxation (5) is a commonly

used and straightforward approach, it nevertheless disre- gards the physical mechanism of the relaxation. In Ref. [4],

the radiation-enhanced diffusion was proposed as a possi- ble relaxation mechanism. In this paper, we introduce the

diffusion-like relaxation in Eq. (4) and demonstrate how this relaxation affects spatial-temporal composition changes near a solid surface under ion bombardment.

2. Stress relaxation

Collisional relocation of target atoms generates density gradients in the target [I]. This affects chemical potentials y, and gives rise to driving forces F, = - +,Jax, as well as to directed fluxes, which can be written in the form

[51

Jcl”(x, t) = - $C,(x, t)F,(x, t), (7)

0168-583X/95/$09.50 0 1995 Elsevier Science B.V. All rights reserved SSDI 0168-583X(94)00613-X

482 M.G. Stepanova /Nucl. lnstr. and Meth. in Phys. Res. B 95 (1995) 481-484

where D,* ((Y = 1, 2) are diffusion coefficients of target components. Note that these are the genuine diffusion coefficients, not related with collisional mixing. The driv- ing forces F, and F2 can be written in the following form

[6,71

F,(x, r) = 5,@(x, t), (8)

where 5, are material parameters (5, is proportional to

the relaxation volume related with the component (Y in the target) and @(x, t) is a component-independent function

which represents stress distribution at time moment t. Thus, with the notation D, = 5, D,*, we obtain

.I,“‘(& t) = - ;c,(x, t)@(x, t). (9)

From the condition of balance (6) where Jzix is de-

fined by Eq. (l), one can express the unknown function @(x, t) in terms of Dp, Vd”‘“, and C, , and substitute it

back in Eq. (9). The final result is

Ju”‘“(x, t) +.I,“‘( x, t)

= -&(.x, q&(x, t) + C(x, t)C,(x, t),

(10)

where

ti&, t)=Cp(x, l)oV~mix(~)-dD~i”/dx

-R,, ( Vpmix( x) - d DF’,‘dx)]

&(x, t>

(11)

Cp(x, t)DfiX(x) +C,(x, t)RmpDprniX(x) =

ax, t)R,p + Cfi(X’ t) 7 (12)

with Rmp =D,JDB,andCp=l-C,.

The stress relaxation proposed here allows us to study the effect of Rap = D,/Dp on compositional changes near a target surface under ion bombardment.

Note that, strictly speaking, the term J,“(x, t) = -D,*aC,(x, t)/dx should be added to the expression

Jz”(x, t) +Jz’(x, t) in Eqs. (4) and (6) in so far as we account for the genuine diffusion in the target [5]. How- ever, we neglect it, assuming that D,* < Dfix.

3. Details of the model

Although the most essential features of the approach proposed in this paper were considered above, there still are some details to be discussed.

3.1. Basic equation and boundary conditions

First, let US summarize the model we are going to use. The concentrations C,(x, t) of components ((Y = 1, 2) are defined by the equation

a + V(t)~C&? t),

where ca(x, t) and Da,<x, t) are defined by Eqs. (11) and

(12). The boundary condition at the surface (x = 0) provides

a balance between sputtering of components and the rate

of sputter recession V(t):

V(t)C,(O, t)(c,(o, t>q + C,(O, t>fiJ’

=J,,y,=C,(O, t),

with

(14)

V( t> = J”( C,(O, t)Y$, + C,(O, t)Y;&) 1 (15)

where Y,C and R, are sputtering yield and atomic volume of component (Y.

The boundary condition at depth x = xmax, where the target remains unaffected, is

C&( xm=x, t) = C,(A(t) +xmax, 0) (16)

where C&(x, 0) is the initial condition, A(t) = /;V(T) dr is the eroded depth at time t after the beginning of the ion bombardment, and xmax itself is defined as the depth

where Dp and Vamix vanish, which in our case is about

20 nm, depending on ion energy and angle of incidence.

3.2. Mixing factors

To calculate composition profiles, we need to define the parameters V,“‘“(x) and D:‘“(x), that appear in Eqs. (11) and (12). For this purpose, we obtain the relocation functions G,(x, x’) (see Eqs. (2) and (3)) by numerical solution of master equations for linear collision cascades discussed in Refs. [4,8,9]. For a target with a given compo- sition C,(x), the approach put forward in Ref. [4] enables us to compute mean numbers of recoils originating from a marked layer (x, dx) and deposited per unit time in a unit volume at depth x’. Let us denote these numbers by dl,(x, x’). By definition, there is the following relation between d&(x, x’) and G,(x, x’) dx:

dl,( x, x’)/Ja = C,( x)G,( n, x’) dx. (17)

With the relocation function G,(x, x’) obtained from Eq. (17), we calculate the parameters Vd”‘“(x) and D,“‘“(x)

M.C. Stepanocta / Nucl. Instr. and Meth. in Phys. Res. B 95 (1995) 481-484 483

using Eqs. (2) and (3). The function G,(x, x’) takes no account of sputtered atoms, G,(x, 0) = 0.

4. Relaxation-induced segregation in a binary target

In this section we demonstrate the effect of stress relaxation on compositional changes in a two-component

target under ion bombardment. MoSi, was chosen as

model target, but results discussed below are of a general character and can be applied for a wide number of two-

component compounds.

4.1. Input parameters

We consider sputtering of MoSi, by 4 keV Ar+ ions, at 48” angle of incidence with respect to the surface normal, with the ion current density J, = 5 X 1Or4 cm-?- SC'.

Fig. 1 shows mixing parameters DmiX/JO and VmiX/J,,, for homogeneous, unaffected MoSi,, calculated as dis- cussed in Section 3.2. Note that a very close agreement between VcE and VsTix is a special result rather than a

common rule; for example, our calculations for TaSi, in the framework of a similar approach give V$” about 1.4 times greater than Vg [lo]. The effect of component

masses on values Vam’“, Drix and composition profiles is

Depth x (nm)

al

bl

Depth x [nm)

Fig. 1. Mixing parameters V$A”, V$“’ (a) and D$i, DFI” (b),

normalized to ion current density J,, for homogeneous MoSi,

bombarded by 4 keV Ar+ ions at 48” angle of incidence.

6oJ I 1 : : , : , I , ’ _ ‘-.

55- ;/ ‘\\\ x=0

56 / ‘,

‘. ._---____

____L__I__ . . .

__,. _:-

: -R=l.OO ---R=3.00 ._

‘..__I. -.-- R=0.33 -

3 I 20 io 8 do I do 1 lb0 1 150 Sputter time t [sl

Fig. 2. Dependence of MO surface concentration in MoSi, on

sputter time for different values of the relaxation factor R =

DMO /Ds,.

an important and interesting problem, but its discussion is

beyond the scope of our present research. Together with the mixing parameters, sputtering yields

of components, Y& = 3.03 and Y$ = 5.40, were computed for the unaffected target in the low-fluence limit (details of

such calculations are discussed in Ref. [9]). Sublimation

energres of pure elements, H,, = 6.75, HSi = 4.64, were taken for surface-binding energies.

Composition changes in MoSi, were obtained by nu- merical solution of Eq. (13) with account of Eqs. (11) and

(12). To obtain the values of LIP and Vami” at a given depth x, the spline-interpolation of data presented in Fig. 1 was used, as is shown by the dashed lines.

The relaxation factor R = D,,/Dsi appearing in Eqs. (11) and (12) was treated as a phenomenological parame-

ter. We use three model values, R = 1, R = 0.5 and R = 2.

4.2. Results and discussion

Fig. 2 presents time dependences of MO surface con-

centration, C&O, t), expressed in percent for given values of R. As is evident from the figure, the relaxation factor R = D,,/DSi heavily affects the behavior of surface con-

centrations at the beginning of an ion bombardment. Fig. 2 shows that the ratio R is of relevance for both the shape of

the dependence C,,(O, t), and the time when a stationary surface concentration is reached.

Fig. 3 shows quasi-steady-state MO concentration pro- files (concentration in percent). The profile with R < 1 has

a distinct maximum, and that with R > 1 a minimum. This effect results, first, from the depth dependence of Vmmix, and, second, from the nonstoichiometric character of the relaxation. Collision mixing leads to preferential transfer of recoils from the near-surface region to depths beyond the maximum of Vami”, where contractive stresses are formed. The stresses relax through the mechanism (9), which produces the depletion of the stressed region in the component with greater parameter 0,. More detailed dis- cussions of the relaxation-induced segregation can be found in Ref. [4].

483 M.G. Stepanma / Nucl. Ins&. and Meth. in Phys. Res. B 95 (1995) Gl -&W

1 ,,,,~,,,,;,,l,;l,,, 0

13epth1x0 [nml 15 20

Fig. 3. Quasi-steady state composition profiles in MoSi, for

different values of the relaxation factor R.

The composition profile at R < 1 with a maximum of

MO concentration offers an explanation to the fact that Auger electron spectroscopy of surface composition in

ion-bombarded MoSiz yields more pronounced enrichment in MO when the measurements are performed using high- energy Auger lines (HAES, MO - 2044 eV, Si - 1621 eV), than when low-energy Auger lines are used (LAES, MO - 186 eV, Si - 92 eV). This was found at 3-5 keV

Ar+ bombardment in Ref. [ll], and at 3 and 4 keV in Ref. [12] (the experimental technique was discussed in Ref.

[131). HAES measurements exhibit stronger enrichment in

metal than LAES also in other Mo-Si compounds [14,15],

in CoSi, and CoSi [13], in CrSi and CrSi, [14,15], in TaSi, [16]. In all these compounds the relaxation-induced

segregation can be expected. Another important point is the correlation between the

temporal behavior of surface concentrations and the shape of steady-state profiles, as it can be seen from comparison of Figs. 2 and 3. Thus, following the time dependence of surface concentrations in a compound during ion bombard-

ment, one can draw conclusions about the shape of the steady-state composition profile in the compound. Here,

however, much remains to be learned. Although such a correlation was observed experimentally in Ref. [17] for SIC, there is no evidence that allows general conclusions. Features of the correlation between temporal and in-depth

dependences of composition changes seems to be an inter- esting problem for theoretical and experimental research.

Acknowledgments

This work has been partially supported by the Russian Foundation for Basic Research, grant no. 93-01-01687.

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