Compositional changes in alloys during ion bombardment at elevated temperatures

Nuclear Instruments and Methods in Physics Research B 140 (1998) 75±90

Compositional changes in alloys during ion bombardment at elevated temperatures M.W. Sckerl a

a,1

, N.Q. Lam b, P. Sigmund

a

Physics Department, Odense University, Campusvej 55, DK-5230 Odense M, Denmark Materials Science Division, Argonne National Laboratory, Argonne, IL 60439, USA

b

Received 26 September 1997; received in revised form 21 November 1997

Abstract Near-surface compositional changes in alloys during ion bombardment have been studied theoretically. The employed scheme operates with a stable target, and the e€ects of preferential sputtering, collisional mixing, radiation-enhanced di€usion, and Gibbsian and radiation-induced segregation are allowed for. High-¯uence composition pro®les were determined directly from a nonlinear integro-di€erential equation, after insertion of feasible input, by means of an ecient iteration procedure developed recently. The dependence of the composition pro®le on input parameters such as target temperature and defect mobility has been examined for Ni±Cu, Ni±Ge and Ni±Pd alloys and compared to experimental results. Ó 1998 Elsevier Science B.V.

1. Introduction Ion beam techniques play an exceedingly important role in the analysis, synthesis, and modi®cation of materials and material surfaces. Numerous techniques involving heavy-ion beams in the keV and lower MeV regime imply transport of atoms both within the bombarded material and across the solid±vacuum interface. Such transport may be the central physical process ± e.g., in ion beam mixing or in secondary ion mass spectroscopy ± but usually also has undesired consequences

1 Present address: Mikroelektronik Centret, Danmarks Tekniske Universitet, DK-2800 Lyngby, Denmark.

such as limited depth resolution in analysis, surface roughening and the like. Atomic transport induced by ion bombardment can be classi®ed into a number of processes characterized by widely di€erent scales in length and time. On a time axis one may identify three major categories, a collisional stage, a relaxational stage, and a migrational stage. Prominent collisional processes are mixing, sputtering, and defect formation, the rates of which are governed by the ion current. The leading process in the relaxation stage is considered to be pressure relaxation. Important processes in the migrational stage are radiation-enhanced di€usion, Gibbsian segregation, and radiation-induced segregation, the rates of which are sensitive to target temperature.

0168-583X/98/$19.00 Ó 1998 Elsevier Science B.V. All rights reserved. PII S 0 1 6 8 - 5 8 3 X ( 9 7 ) 0 0 9 2 7 - 0

76

M.W. Sckerl et al. / Nucl. Instr. and Meth. in Phys. Res. B 140 (1998) 75±90

Most pertinent processes are preferential: In a multicomponent medium, di€erent components are transported at di€erent rates, and the di€erence depends on the individual process. This, together with the competition between a considerable number of processes, complicates the analysis. In the past, experimental and theoretical studies tended to focus on a small number of processes at a time. Consequently, numerous experimental ®ndings have remained unexplained [1,2] and the complexity of the problem has become clear only gradually. In a recent review emphasizing sputter processes in multicomponent materials, two of us made an attempt to formulate a theoretical scheme to accomodate all pertinent physical processes [3]. Care was taken to demonstrate the equivalence of that scheme with earlier, less comprehensive theoretical treatments. When only collisional processes are allowed for, the scheme reduces to a formalism developed long ago to describe the sputtering of multicomponent materials in the absence of migration [4]. The scheme also incorporates the essential physics that is known to govern compositional changes at elevated temperatures [2,5,6]. The present work represents a ®rst attempt to apply this scheme to binary alloys over a range of temperatures where the entire variety of processes attains importance and to study their interplay as a function of temperature. For irradiation at low temperatures, i.e. near or below room temperature, collisional processes tend to dominate, and the thickness of the bombardment-induced altered layer is comparable with the damage depth which, in turn, is comparable with the penetration depth of the beam. At elevated temperatures, migrational processes give rise to deeper altered layers. Moreover, Gibbsian segregation tends to drive the alloy surface toward thermodynamic equilibrium with enrichment in a particular component. This can compete with preferential sputtering in depleting the segregating element(s). Conversely, radiation-induced segregation may drive the system away from thermodynamic equilibrium. The centerpiece of the theoretical scheme is a nonlinear integro-di€erential equation [4]. By al-

lowing for pressure relaxation this equation ensures mechanical stability of the target at all ¯uences. The scheme has been applied previously to the treatment of preferential sputtering, collisional mixing, and Gibbsian segregation [4,7]. The present work adds radiation-produced defects and defect-assisted processes. The latter group of processes has previously been studied within a different scheme [5,6]. The present treatment avoids the assumption of a layered medium and di€ers by taking into account pressure relaxation, longrange collisional mixing beyond the di€usion approximation, and a nonvanishing depth of origin of sputtered atoms. A suitable and frequently employed model system for fundamental studies is a semi-in®nite target with a plane surface and an initially homogeneous composition pro®le exposed to a uniform and monochromatic beam. Unless deposition of beam atoms dominates over sputtering, the target will erode. If all action of the beam (direct and indirect) is con®ned to a layer of ®nite thickness, the composition pro®le seen from the instantaneous target surface must reach a stationary state at some ¯uence. This stationary pro®le is a suitable goal for fundamental studies, illustrating the e€ect of various ongoing processes. It is directly calculable since the time variable drops out from the fundamental balance equation. An ecient iteration procedure for solving the resulting system of nonlinear integro-di€erential equations in the depth variable, developed recently [7], has been applied here with proper modi®cation. Model calculations are reported for Ni±Cu, Ni± Ge, and Ni±Pd and compared with experimental ®ndings. The e€ect of implanted beam atoms has been neglected. A part of our results was reported at a recent conference [8]. 2. Basic equations Consider a multicomponent material with a depth-dependent composition pro®le. Let Ni …x; t† be the mean density (number per volume) of species i at depth x and time t. In order to ensure stability of the material we impose the packing condition [4,9]

M.W. Sckerl et al. / Nucl. Instr. and Meth. in Phys. Res. B 140 (1998) 75±90

X

Xi Ni …x; t† ˆ 1;

…1†

i

where Xi is the atomic volume. Beam-induced compositional changes like mixing, sputtering, or defect production initially result in a composition pro®le violating Eq. (1) and must, therefore, be followed by rapid relaxation. It is convenient to distinguish between compositional changes before and after relaxation. Denote the composition change prior to relaxation by   @Ni ˆ Qi : …2† @t unrelaxed Homogeneous ± i.e., nonpreferential ± relaxation is allowed for by a reverse current compensating for density changes [4,9,10]. This yields the composition change after relaxation [4], @Ni …x; t† ˆ Qi …x; t† @t 0 1 Zx X @ ÿ @Ni …x; t† Xj dx0 Qj …x0 ; t†A @x j

…3†

0

independent of the nature of the process governing Qi . It is easily veri®ed that Eq. (3) satis®es Eq. (1). This form of the balance equation includes a translation of the depth coordinate along with sputter erosion to ensure the origin to always coincide with the target surface. The number of components entering the description must, as a minimum, include all chemical species present, i.e., two or three in binary materials, dependent on whether or not incident beam atoms are accounted for [4,11]. Incorporation of defects requires additional species, one for each Table 1 Speci®cation of densities and species needed for description of compositional changes in a binary material. Incident beam atoms not included Density

Species

Na ; Nb NA ; NB Na ; Nb NV Nx Ns

a; b atoms on lattice sites a; b atoms in interstitial positions a; b atoms at sinks Vacancies Vacancies at sinks Intrinsic and radiation-induced sink sites

77

signi®cant type of defect, each of them being assigned an atomic volume [3]. 3. Input 3.1. Diatomic medium The present description of a binary a±b alloy will be based on the nine species listed in Table 1, six of which account for atoms in lattice or interstitial positions or at sinks, two for vacancies in lattice positions or at sinks, and one for sink sites describing loss of defects. Interstitials at sinks are accounted for in Na;b . Implanted beam atoms are disregarded. In ordered alloys and covalent materials it would also be necessary to distinguish between several types of vacancies. Vacancy and interstitial clusters of various sizes can be formed during irradiation. Except for their role as sinks these features have been neglected. 3.2. Sputtering and collisional mixing The description of sputtering and collisional mixing is well established within the present formalism [4,7,9] and may be expressed by …Qi †sput ˆ ÿJ0 ri …x†Ni …x; t† …4† and

Z1

…Qi †mix ˆ J0

dx0 ‰Ni …x0 ; t†Gi …x0 ; x†

0

ÿNi …x; t†Gi …x; x0 †Š;

…5† 0

where J0 is the ion current density and Gi …x; x † a relocation function: d/Gi …x; x0 †dx0 is the probability for an i atom at depth x to be relocated into a layer …x0 ; dx0 † after bombardment by a ¯uence increment d/. The quantity ri …x† is the sputter cross section which is related to the relocation function by Z0 ri …x† ˆ

dx0 Gi …x; x0 †:

…6†

ÿ1

We have employed a model introduced previously [4,7] with Gi …x; x0 † ˆ g…x†Bi eÿbi jxÿx0j

…7†

78

M.W. Sckerl et al. / Nucl. Instr. and Meth. in Phys. Res. B 140 (1998) 75±90

to model both cascade mixing and sputtering, where g…x† scales with the deposited-energy distribution and constants Bi ; bi are to be speci®ed. 3.3. Defect production Production rates of defects are denoted by …Qi †prod ˆ J0 Fi …x†;

…8†

where Fi …x† is the production depth pro®le (species i per ion per unit depth). The pro®les Fi …x† are taken independent of ¯uence and detailed composition and can be computed from cascade theory or from Monte Carlo or molecular dynamics simulation [12]. In the description as it stands, only densities enter and no correlations. Vacancies and interstitials are, therefore, produced independently at their respective rates. Their integrals would be identical in an in®nite medium, Z1 Z1 …9† fFA …x† ‡ FB …x†g dx ˆ FV …x† dx; ÿ1

ÿ1

but di€erences could be expected due to the presence of a surface. There will be deleted a substitutional atom each time an interstitial is created, but the spatial distribution of this negative production (Fa , Fb < 0) is that of the vacancies, Fa …x† ‡ Fb …x† ‡ FV …x†  0:

…10†

Production rates of atoms at sinks are not proportional to J0 and therefore do not come in here. The pro®le Ns of sink sites is uniform initially, but the concentration increases in the damage region during irradiation. Quantitative information on production rates is not available, but experimental evidence on defect clusters [13,14] suggests that the e€ect may be due to defects agglomerating within the cascade region to form immobile clusters which trap point defects. In recent model calculations [13] the size of cascade-produced clusters was found to be in the order of 10 interstitials and 30 vacancies. Assumptions underlying the present treatment will be speci®ed in Section 4.1.

3.4. Di€usion Atoms migrate via exchange with defects. Since particles at sinks are trapped, there are only ®ve di€using species, each being assigned a di€usion current Ji so that @Ji …11† @x according to the equation of continuity. These currents may depend on the concentration gradients of all species, and these dependences are governed by coupling coecients through which di€usion coecients can be expressed [15,16]. Speci®c models of di€usion in dilute alloys have been studied [17,18]. Amongst a variety of migration mechanisms, vacancy migration as well as direct and indirect interstitial (interstitialcy) migration [19] have generally the lowest activation energies. Generally, atom currents are directed along with the interstital current and opposite to the vacancy current. The concept of bound vacancy-solute complexes migrating as distinct entities, which has been used successfully in the modelling of radiation-induced segregation in dilute alloys [17], is not considered in the present scheme for concentrated alloys. There are large uncertainties in the values of di€usion coecients [17]. We therefore pay attention mainly to the qualitative e€ects of defect migration.

…Qi †migr ˆ ÿ

3.4.1. Direct interstitial migration In the direct mechanism an interstitial moves through a crystal by jumping from one interstitial site to another. This mechanism is likely when the interstitial atom is small and ®ts easily into interstitial positions [19]. The migration of a interstitials is here described by a di€usion coecient DA ˆ CA dA

…12†

with 1 …13† dA ˆ k2A ZA mA : 6 Here kA is the jump distance, ZA the coordination number, and mA the e€ective jump frequency. The factor CA allows for surface segregation and will be discussed in Section 3.6 In the bulk, CA ˆ 1.

M.W. Sckerl et al. / Nucl. Instr. and Meth. in Phys. Res. B 140 (1998) 75±90

In principle, dA di€ers from the corresponding expression dB through m, Z and k, but emphasis is laid here on di€erent jump frequencies. mA depends on temperature and is governed by the migration enthalpy HAm ,   HAm : …14† mA ˆ m0 exp ÿ kT The current of a interstitials is given by @…NA DA † @NA @DA ÿ NA …15† ˆ ÿDA @x @x @x and similarly for JB . Here DA is depth dependent only via Ca according to Eq. (12). The equations used previously [20] did not include direct interstitial migration. JA ˆ ÿ

3.4.2. Indirect interstitial migration In the interstitialcy mechanism, the interstitial atoms are in the stable dumbbell con®guration and jump via a combination of translation of the dumbbell center and rotation of the dumbbell axis [21]. In diatomic media, mixed a±b dumbbells form, and the long-range transport of a atoms results from combined caging reorientation and looping jumps [17] or caging and in-place rotation [21]. The details of atomic jumps via an interstitialcy mechanism in concentrated alloys become more complex because several atoms may be involved. However, as a good simpli®cation, one can adopt the concept that bound interstitial-solute complexes migrate as solute interstitials, resulting from the combined in-place rotation of the mixed dumbbell and caging motion of the undersized solute. The quantity of interest is the total number of interstitials NI ˆ NA ‡ NB :

…16†

An equation for NI can be obtained from Eq. (3) by summing over the interstitial species. Here, di€usion of interstitials receives contributions from jumps of both a and b atoms. Therefore the di€usion coecient is given by the sum DI ˆ D0A ‡ D0B ;

…17†

where D0A ˆ CA dA0 Xa Na :

…18†

79

The coecient dA0 is proportional to the jump frequency m0A and given by an expression equivalent to Eq. (13). The current JA of a atoms is written as
@ ÿ JA ˆ ÿ NI D0A ; …19† @x and the current of interstitials is @ …NI DI †: …20† @x Except for the factor CA , D0A and JA are equivalent with expressions used previously [20]. JI ˆ J A ‡ J B ˆ ÿ

3.4.3. Vacancy migration Vacancies migrate via exchange with a and b atoms. The di€usion coecient for vacancies is written as the sum DV ˆ Da ‡ Db

…21†

with Da ˆ Ca da Xa Na ;

…22†

where 1 da ˆ k2a Za ma : …23† 6 Again the factor Ca makes these expressions di€er from the ones previously used [20]. The current Ja of a atoms due to migration of vacancies reads @NV @Da ÿ NV @x @x and the current of vacancies is given by Ja ˆ D a

…24†

@NV @DV ‡ NV : …25† @x @x Note the opposite signs in Eqs. (24) and (25), resulting from Ja ‡ Jb ‡ JV ˆ 0.

JV ˆ ÿDV

3.5. Recombination and annihilation Point defects may annihilate at sinks or may be lost via mutual recombination. A variety of di€erent types of intrinsic or radiation-induced sinks exist for point defects in metals such as line dislocations, grain boundaries, void surfaces, free surfaces, and defect clusters. We need to consider the following processes. 1. Trapping of an a interstitial at a sink results in a production term SA in Qa and a corresponding

80

M.W. Sckerl et al. / Nucl. Instr. and Meth. in Phys. Res. B 140 (1998) 75±90

loss term ÿSA in QA . For interstitialcy migration, primes may have to be added. 2. Trapping of vacancies at sinks results in a production term SV in Qx and a loss term ÿSV in QV . Moreover there are production terms Sa and Sb in Qa and Qb which must satisfy S a ‡ Sb ˆ SV

…26†

and corresponding loss terms ÿSa and ÿSb in Qa and Qb . 3. Recombination between an a interstitial and a vacancy results in a production term Pa in Qa and a corresponding loss term ÿPa in both QA and QV . Sink annihilation is assumed to be di€usion limited and can be described by di€usion-controlled-reaction theory [22]. Annihilation is assumed to occur instantaneously whenever the defect is within a certain radius q of the sink. We have used a chemical reaction rate approximation [23] where the material is approximated by a homogeneous medium; thermal production and loss of point defects occur continuously throughout the medium. All sinks are assumed to be of the same size and strength. Expressions for sink annihilation rates [24] and rate constants are given in Table 2. Here qA and qa are sink capture radii for A atom and vacancy annihilation, respectively. The sink pro®le Ns …x† will increase in the damage region during irradiation. qa is of the order of a lattice constant. Note in particular that kA0 is proportional to Na through Eq. (18). Defect recombination has been treated similarly; instantaneous recombination is assumed whenever an A interstitial is within a certain recombination radius qAV of a vacancy. The proTable 2 Sink annihilation rates and rate constants for three migration mechanisms Migration mechanism Direct interstitial Indirect interstitial Vacancy

Annihilation rate   SA ˆ kA Ns NA ÿ NAequil   SA0 ˆ kA0 Ns NI ÿ NIequil   Sa ˆ ka Ns NV ÿ NVequil

cess is thus governed by the sum of di€usion coef®cients of the two recombining defects. Recombination rates and rate constants [24] are listed in Table 3. 3.6. Gibbsian segregation Gibbsian segregation is incorporated in the diffusion Eqs. (15), (19) and (24) by factors Ca;b and CA;B ; the transport of one component to the surface is preferential if these factors di€er. We have set CA ˆ Ca and similarly for b, and Ca is given by   Va …x† ; …27† Ca ˆ exp kT where V …x† is a segregation potential which vanishes in the bulk and is negative at the surface for the segregating species. This description of surface segregation [7], which di€ers from more common oneor two-layer models [25,26], is well suited for incorporation into the present scheme since it does not require to de®ne discrete atom layers. To derive Vi from existing segregation data we have related the segregation potential to the measured Arrhenius behavior for binary alloys. The surface-layer composition as a function of temperature is known from measurements of ion-surface scattering for various alloys [27±29]. From the resulting Arrhenius plots the enthalpy DH and entropy DS of segregation were estimated. We assume an exponential form of the segregation potential [7] Vi …x† ˆ Ci eÿci x

…28†

with constants Ci and ci adjusted to ensure that the enrichment Ei of species i integrated over all depths,

Rate constant

Table 3 Recombination rates and rate constants for vacancy-interstitial annihilation

kA ˆ 4pqA DA

Migration mechanism

kA0 ˆ 4pqA D0A ka ˆ 4pqa Da

Recombination Rate constant rate

Direct interstitial Pa ˆ ra NA NV Indirect interstitial Pa0 ˆ ra0 NI NV

ra ˆ 4pqAV …Da ‡ DA † ra0 ˆ 4pqAV …Da ‡ D0A †

M.W. Sckerl et al. / Nucl. Instr. and Meth. in Phys. Res. B 140 (1998) 75±90

81

Z1 Ei ˆ

dxfNi …x† ÿ Ni …1†g;

…29†

0

is consistent with what is measured by ion scattering, where Na …x; t† ˆ Na …x; t† ‡ NA …x; t† ‡ Na …x; t†

…30†

is the total concentration of a atoms. Moreover, a close match to the results from the layer model was aimed at. A typical temperature variation of Ci is illustrated in Fig. 1, and Fig. 2 shows the comparison with results from a one-layer model. 4. Solving the balance equation In the following we shall assume the initial pro®les Ni …x; 0† ˆ N i to be depth-independent. Conditions are ful®lled for development of a stationary state at high ¯uence. The resulting composition pro®le has been determined directly from Eq. (3) without solving for the transient behavior. In measurements of chemical composition the total concentration of a atoms, Na …x; t† de®ned by Eq. (30), is of interest. An equation for this quantity is found by summing Eq. (3) over the proper set of species. We have applied an iteration method proposed previously [7] and determined directly the total concentrations Na;b .

Fig. 2. Comparison of the Arrhenius behavior of surface enrichment, Eq. (29), resulting from the potential model, with that of the one-layer model for a Ni±40 at.% Cu alloy. One-layer model [27]: Enrichment calculated with DH ˆ ÿ0:42 eV and [27] DS ˆ ÿ2:60 k. Potential model: Enrichment calculated after ®tting the parameter ci of the Gibbsian segregation potential, Eq. (28).

4.1. Speci®c input A number of additional assumptions were made in the numerical calculations. (1) The parameter bi determining the relocation depth in Eq. (7) was adjusted to the results of molecular-dynamics simulations of sputtering [30], while Bi has been left open as a free parameter to generate a preferential sputter ratio. (2) The indirect interstitial mechanism has been adopted everywhere. This is motivated by the choice of alloys used in the calculations. Alloy systems were chosen where experimental information about steady-state composition and Gibbsian segregation parameters were available. (3) Atomic volumes were chosen according to Xa ˆ XA ˆ Xa ˆ Xb ˆ XB ˆ Xb ˆ XV ˆ X; Xx ˆ 0:

Fig. 1. Temperature variation of the parameter Ci , Eq. (28). The Gibbsian segregation potential was ®tted to measurements for Ni±40 at.% Cu alloys [27].

…31†

The di€erence in atomic volume for the atomic species analyzed is small [31], 10% for Ni±Cu. Local relaxation around lattice defects is ignored. (4) Production rates are assumed to obey the condition Fa ‡ FA ˆ 0, i.e., the di€erence between primary interstitial and vacancy pro®les has been

82

M.W. Sckerl et al. / Nucl. Instr. and Meth. in Phys. Res. B 140 (1998) 75±90

ignored. Moreover, defect production has been assumed to be stoichiometric. For these two reasons, one single depth pro®le characterizes all production of defects. A pro®le obtained from a Monte Carlo calculation [32] for Ni under 3 keV Ne‡ bombardment was adopted and taken equal for both alloy components. Since only freely migrating defects play a role in di€usional processes, the calculated defect production rates are multiplied by a defect production eciency of 0.4, estimated from Figs. 1 and 2 of Ref. [33]. The magnitudes of the cross sections are variable parameters, determined from optimizing partial sputter yields to obtain the best agreement with measured composition pro®les. (5) In the numerical calculation of the relaxation term in Eq. (3), QV is neglected to simplify the numerical procedure. This amounts to ignoring volume expansion by vacancy formation. This appears consistent with the fact that the e€ect of implanted beam atoms ± which is of comparable magnitude at elevated ¯uences ± has not been included presently. (6) Since little information about the production of sinks during irradiation is available we assumed them unbiased and inexhaustible. Their density is taken identical with the damage distribution and assumed to be in steady state. The peak sink concentration, Nsmax , is a variable parameter which is determined by ®tting results of calculations to measured pro®les. Many types of sinks may be produced, but they are all assumed to have the same e€ective defect capture radius. Standard  and q ˆ q ˆ q ˆ q values, qAV ˆ qBV ˆ 7 A A B a b  for the recombination and defect capture raˆ3A dius, respectively, were adopted [34]. The recombination volume is of the order of 100 X, and the sink annihilation volume 20 X. (7) We have used Za ˆ Zb and ka ˆ kb , the jump distance in the a±b solid solution. (8) Di€usion currents must vanish at the surface. This boundary condition ensures that the only nonvanishing ¯ux through the surface is that of sputtered atoms. Di€usion currents were forced to vanish by multiplying the calculated ¯ux with an exponential factor f …x† ˆ 1 ÿ eÿx=L :

…32†

L has been chosen much smaller than the interatomic distance. It was checked that the precise choice did not a€ect the results. 4.2. Numerical procedure In the stationary limit, Eq. (3) is equivalent with [3] Ni ˆ N i  R 1 0 P 0 0 j Xj x dx Ni …x†Qj …x † ÿ Nj …x†Qi …x † Rx P : ‡ 0 0 j Xj 0 dx Qj …x †

…33†

This may be used for determining Ni …x† iteratively since Qi in turn depends on Ni through the respective de®nitions. An alternative version of Eq. (33) is available [7]. Iteration is started from Ni ˆ N i , and the main criterion for a precise solution is stoichiometric sputtering, R1 dx Na …x†ra …x† dx N a R01 ˆ …34† dxNb …x†rb …x† dx N b 0 within an adopted accuracy of 6 10ÿ4 . In practice, iteration produced stable solutions for production terms ± which are composed of integral operators and scalar functions ± while instabilities were hard to avoid when migration terms were included. Therefore the scheme was modi®ed such that each iteration step was split into two, one using Eq. (33) as it stands with primary processes only, followed by a second step where Eq. (33) was solved numerically involving all other processes. Generally, 10±50 iterations were needed to reach the desired precision. Because of the di€erent depth scales of primary and secondary processes, care had to be taken in the choice of the grid. We have chosen a step size Dxk increasing with depth as Dxk ˆ k c Dx0

for k ˆ 1; 2; 3; . . .

…35†

with c ˆ 4. The equations describing the defect concentrations NI and NV , Eq. (33), are nonlinear di€erential equations. We applied the method of Gauss±Seidel iteration [35]. The computation time for one pro®le was typically 1 min on a work station.

M.W. Sckerl et al. / Nucl. Instr. and Meth. in Phys. Res. B 140 (1998) 75±90

4.3. Test of numerical code As a test of the present numerical code we compared solutions with analytic expressions obtained previously [36]. For this comparison we adopted parameters from Fig. 2 in Ref. [36] as input for our code. Only primary processes were included via a relocation cross section G…x; x0 † equivalent to Eq. (7). A comparison between the two results (Fig. 3) shows complete agreement. 5. Results and discussion 5.1. General behavior Calculations have been performed for a model binary alloy a±b in which a atoms segregate to the surface and are sputtered preferentially. The physical parameters used to describe preferential sputtering and Gibbsian segregation are similar to those derived for a Ni±40 at.% Cu solid solution (Table 4). The e€ect of temperature on alloy composition is shown in Fig. 4. A combined linear and logarithmic scale is used for the depth coordinate in Fig. 4 to e€ectively show the compositional variations within the entire altered layer. Below  100 C, collisional processes govern the development of the alloy composition in the altered layer

Fig. 3. Comparison between numerical and analytic solution of Eq. (3). The analytic solution is taken from [36]. Input parameters and units adopted from Fig. 2 there.

83

which extends to a depth approximately equal to the damage range. The higher the temperature the thicker is the altered layer. At 600 C the thickness is about three orders of magnitude larger than the damage depth. It is the extent of compositional gradients to such depths that requires long buildup times to steady state at high temperatures. The gentle humps at large depths result from preferential transport of b atoms into the peak damage region which causes minor enrichment of a atoms at the end of this region. Vacancies are assumed to migrate faster by exchange with b atoms than with a atoms (see also Fig. 8 ). The in¯uence of various processes can be seen in Fig. 5. When only preferential sputtering and mixing are included, a pro®le (PS+MIX) is obtained with a severe depletion of a atoms at the very surface but lessening with increasing depth [7]. When Gibbsian segregation is activated (+GS) the surface layer is enriched with a atoms and preferential sputtering of a atoms is enhanced in the transient state. The combination of continuous feeding and preferential sputtering leads to a concentration spike of surface a atoms followed by a strong subsurface depletion which extends deeply into the interior. The interplay of these pro-

Fig. 4. Steady-state concentration pro®les of a atoms in the altered layer of a binary alloy a±b bombarded with 3 keV Ne‡ at various temperatures. Plotted is the relative abundance Xa Na . Radiation-enhanced di€usion is operational. Migration enthalm m ˆ 0:96 eV and Hb;V ˆ 0:95 eV were adopted to generpies Ha;V ate radiation-induced segregation. All other parameters are considered to apply to a Ni±40 at.% Cu alloy (Table 4). Note the change from linear to logarithmic scale on the x-axis.

84

M.W. Sckerl et al. / Nucl. Instr. and Meth. in Phys. Res. B 140 (1998) 75±90

Table 4 Input parameters for Ni±40% Cu Fixed parameters

From [27]

Ion current density Vacancy migration frequency Interstitial migration frequency Vacancy formation entropy Interstitial formation entropy Interstitial formation enthalpy Interstitial migration enthalpy Cu segregation enthalpy Cu segregation entropy

3.75 ´ 1013 cmÿ2 sÿ1 5 ´ 1013 sÿ1 5 ´ 1012 sÿ1 3k 0 4.0 eV 0.15 eV )0.42 eV )2.6 k

Fitted parameters

Present work

Ref. [27]

Vacancy formation enthalpy HVf m Vacancy migration enthalpy HCu;V m Vacancy migration enthalpy HNi;V Pure-element sputter yield YCu Pure-element sputter yield YNi Peak sink concentration Xs Nsmax

1.55 eV 0.95 eV 0.95 eV 2.5 atoms/ion 2.0 atoms/ion 5.0 ´ 10ÿ2 (100°C) 5.0 ´ 10ÿ3 (300°C) 1.0 ´ 10ÿ3 (400°C) 1.0 ´ 10ÿ4 (500°C) 1.0 ´ 10ÿ5 (700°C)

1.60 eV 0.94 eV 0.95 eV 2.0 atoms/ion 2.0 atoms/ion ) ) ) ) )

cesses has been discussed in detail (for a summary cf. a review by two of us [3]). A similar pro®le (+RIS) is obtained when radiation-induced segregation occurs in the absence of Gibbsian segrega-

Fig. 5. Same as Fig. 4 at 300 C but with various combinations of processes included: Preferential sputtering only (PS), collisional mixing included (MIX), Gibbsian (GS) or radiation-induced segregation (RIS) included. This implies that also radiation-enhanced di€usion is operational. All other parameters as in Fig. 4.

tion. Here b atoms migrate away from the surface by preferential exchange with vacancies m m < Ha;V ). The resulting surface enrichment of (Hb;V a atoms also gives rise to transient nonstoichiometric sputtering, leading to subsurface depletion of this component at high ¯uence. Because of the preferential transport of b atoms into the peak damage region which causes enrichment of a atoms at the end, the extent of the subsurface depletion is smaller than that caused by Gibbsian segregation. The magnitude of subsurface depletion is significantly larger when all processes are in operation (ALL). Note that, although the resulting pro®les can show either surface depletion or enrichment, all pro®les re¯ect stoichiometric sputtering in steady state. 5.2. In¯uence of speci®c processes 5.2.1. Radiation-enhanced di€usion Fig. 6 shows the in¯uence of radiation-induced defects on the composition pro®le. The spatial distribution of the damage pro®le was kept constant

M.W. Sckerl et al. / Nucl. Instr. and Meth. in Phys. Res. B 140 (1998) 75±90

Fig. 6. Importance of bombardment-induced defects on composition pro®les. Labels indicate percentage of calculated rate of defect production. Gibbsian or radiation-induced segregation not included. All other parameters adopted from Table 4.

while the fraction of migrating defects was varied. It is seen that the in¯uence of radiation-enhanced di€usion is substantial even at the 10% level, while the thermal defect concentration at T ˆ 300°C is not sucient to alter the target composition, as it can be seen by comparing curve `0%' with PS+MIX of Fig. 5. It is seen that radiation-enhanced di€usion, jointly with mixing, also contributes to smearing the steep gradients caused by preferential sputtering. With increasing temperature, thermal di€usion gets more important. At some point the thermal concentration of defects dominates the irradiation-produced defect concentration. This transition depends primarily on the vacancy formation enthalpy. Composition pro®les were found insensitive to this parameter at low and very sensitive at high temperatures. 5.2.2. Gibbsian segregation The e€ect of Gibbsian segregation is illustrated in Fig. 7. Here, the depth of the segregation potential is varied from 0% to 100% of the depth derived from data ®tting (Table 4). Gibbsian segregation produces large concentration gradients in the surface region. Even though sputtering was assumed nonpreferential, subsurface depletion is observed. This depletion is caused by the combination of Gibbsian segregation feeding a-atoms to the surface layer and radiation-enhanced di€usion providing a-atoms from deeper layers.

85

Fig. 7. E€ect of Gibbsian segregation on composition pro®les. Only a certain percentage of the ®tted segregation potential Va …0† taken into account (Table 4). Stoichiometric sputtering assumed, Ya …t ˆ 0†=Yb …t ˆ 0† ˆ 3=2. All other parameters as in Fig. 4.

5.2.3. Preferential defect-solute exchange It is known that radiation-induced segregation occurs because of preferential exchange of a particular alloy component with migrating defects [5,6]. In binary alloys the component that di€uses faster by exchange with vacancies or via interstitials will be depleted or enriched, respectively, in regions where there exists an in¯ux of point defects. In our model alloy a±b, radiation-induced segregation causes depletion of b atoms at the bombarded surface because these atoms migrate faster by exchange with vacancies than a atoms. The concentration pro®le is sensitive to the degree of this preferential exchange, as is seen in Fig. 8. Here calculations were carried out for various valm m , keeping Hb;V ˆ 0:95 eV. As the di€erues of Ha;V m m m ence DH ˆ Hb;V ÿ Ha;V increases, the depletion of a atoms in the peak damage region becomes less and less severe, and when DH m P 0:02 eV the curvature of the concentration pro®le in this region m becomes signi®cantly changes sign. When Ha;V m smaller than Hb;V , a atoms migrate faster by preferential exchange with vacancies than b atoms. Hence there will be a net accumulation of a atoms in the peak-damage region where point defects diffuse down their own concentration gradients.

86

M.W. Sckerl et al. / Nucl. Instr. and Meth. in Phys. Res. B 140 (1998) 75±90

Fig. 8. E€ect of the strength of radiation-induced segregation on the steady-state distribution of a atoms in a binary alloy a±b bombarded with 3 keV Ne‡ at 300 C. With all processes inm ®xed at 0.95 eV, concentration pro®les were cluded and Hb;V m between 0.93 and 0.97 calculated for di€erent values of Ha;V eV. All other parameters as in Fig. 4.

5.2.4. Radiation-induced sink density Radiation-induced segregation is driven by persistent ¯uxes of freely migrating defects. Its kinetics depend on the sink density. Concentration pro®les calculated for various values of Nsmax at 300°C are shown in Fig. 9. Subsurface depletion is severe for low sink densities, Nsmax 6 10ÿ2 atom fraction. As the sink density increases, the number of free defects that participate in di€usional processes decreases, and depletion becomes less and less pronounced. In addition, the depression on the concentration pro®le occurs at a shallower depth with higher sink density because the contributions of radiation-enhanced di€usion and radiation-induced segregation are less signi®cant. Note that the density of internal sinks is high at low temperatures (below 400°C) where defect clusters are stable, and decreases rapidly with increasing temperature because of lower clustering probability and more ecient thermal decomposition of cascade remnants. This temperature dependence must be taken into account in comparisons with experimental results. 5.3. Comparison with experiment Several measurements of steady-state compositional pro®les under ion bombardment at elevated

Fig. 9. Same as Fig. 4 at 300 C but with di€erent densities of radiation-induced sinks. The peak density Nsmax was varied between 10ÿ1 and 10ÿ5 atom fraction. All other parameters as in Fig. 4.

temperatures have been reported. The time evolution of surface composition during bombardment at elevated temperature and subsurface concentration pro®les after rapid specimen cooling to room temperature were measured by ion scattering [27± 29] or Auger spectroscopy [37,38]. Because of greater surface sensitivity only ion scattering data are considered here. Information on Gibbsian segregation obtained from the same technique [27±29] is an important input. Results for Ni±40 at.% Cu, Ni±25 at.% Pd and Ni±10 at.% Ge alloys are discussed. Input parameters called `®tted' in Tables 4±6 were found by adjustment to experimental data. Except for the peak concentration of radiation-induced sinks, the same set of parameters has been used to compute pro®les at all temperatures where data are available. Estimates of defect parameters which are dicult to measure directly are derived in this manner. Results for Ni±40 at.% Cu are shown in Fig. 10. In this solid solution, Gibbsian segregation gives rise to Cu enrichment at the surface in the absence of irradiation. The optimal set of parameters deduced from the best ®t is presented in Table 4. These parameters are similar to those reported in previous work [27]. The agreement between calculations and measurements is reasonably good for low to intermediate temperatures but becomes unsatisfactory at 700°C. A possible reason for this discrepancy is that steady state may not have been

M.W. Sckerl et al. / Nucl. Instr. and Meth. in Phys. Res. B 140 (1998) 75±90

Fig. 10. Steady-state concentration pro®les of Cu atoms in a Ni±40 at.% Cu after bombardment with 3 keV Ne‡ at various temperatures. Calculated results (dashed and solid lines) ®tted to experimental data [27].

reached in the experiment, as indicated by the time dependence of the Cu surface concentration during bombardment at [27] 700°C. In addition, since the time required to reach steady state can be very long at high temperatures (e.g.,  106 s at 700°C), the problem of stability of the experimental setup over such time periods may be of concern. A depth pro®le in a Ni±50 at.% Cu alloy bombarded with 2 keV Ne‡ ions at 200°C was calculat-

Fig. 11. Steady-state concentration pro®le of Cu atoms in a Ni± 50 at.% Cu after bombardment with 2 keV Ne‡ at 200 C. The calculated result (solid line) is compared with experimental data [39]. Monte Carlo simulation results (solid triangles) [40] obtained for the same alloy under 3 keV Ar‡ at the same temperature are also included.

87

ed with the same set of physical parameters except for the variables pertinent to ion energy, and compared with the measurement by Swartzfager et al. [39] (Fig. 11). Good agreement is achieved. Monte Carlo simulations of surface composition changes in the same alloy undergoing 3 keV Ar‡ bombardment were reported by Kurokawa et al. [40] who treated the e€ects of Gibbsian segregation and radiation-enhanced di€usion di€erently. Considering a layered medium, they simulated segregation from the second atomic layer into the ®rst by resorting to an adjustable radiation-induced Gibbsian segregation parameter [41]. In addition, the development of the compositional pro®le (i.e., the thickness of the altered layer) was assumed to depend on the ratio of the radiation-enhanced di€usion coecient to the surface recession rate [42], both being empirical quantities. Their simulated pro®le is included in Fig. 11. The di€erence in bombardment energy is considered immaterial at this high temperature. Fig. 12 shows the results of calculations and measurements for Ni±25 at.% Pd exposed to 3 keV Ne‡ bombardment. The optimal set of parameters is summarized and compared with previously obtained values [28] in Table 5. Except for some di€erences in the surface compositions, agreement between the calculated and measured pro®les is remarkable. Comparison was not made for T ˆ 700 C because of diculty in reaching

Fig. 12. Steady-state concentration pro®les of Pd atoms in a Ni±25 at.% Pd after bombardment with 3 keV Ne‡ at various temperatures. Calculated results (dashed and solid curves) ®tted to experimental data [28].

88

M.W. Sckerl et al. / Nucl. Instr. and Meth. in Phys. Res. B 140 (1998) 75±90

Table 5 Input parameters for Ni±25% Pd Fixed parameters

From [28]

Ion current density Vacancy migration frequency Interstitial migration frequency Vacancy formation entropy Interstitial formation entropy Interstitial formation enthalpy Interstitial migration enthalpies Pd segregation enthalpy Pd segregation entropy

3.75 ´ 1013 cmÿ2 sÿ1 2 ´ 1013 sÿ1 5 ´ 1012 sÿ1 3k 0 4.0 eV 0.15 eV )0.245 eV )1.64 k

Fitted parameters

Present work

Ref. [28]

Vacancy formation enthalpy HVf m Vacancy migration enthalpy HPd;V m Vacancy migration enthalpy HNi;V Pure-element sputter yield YPd Pure-element sputter yield YNi Peak sink concentration Xs Nsmax

1.22 eV 1.02 eV 1.01 eV 3.0 atoms/ion 2.0 atoms/ion 3.0 ´ 10ÿ1 (200°C) 5.0 ´ 10ÿ2 (300°C) 5.0 ´ 10ÿ3 (400°C) 1.0 ´ 10ÿ4 (600°C)

1.30 eV 0.9 eV 1.1 eV 2.0 atoms/ion 2.0 atoms/ion 3.5 ´ 10ÿ1 2.0 ´ 10ÿ2 2.0 ´ 10ÿ3 1.0 ´ 10ÿ5

steady state at very high temperatures in the experiment. Fitting was also carried out for a Ni±10 at.% Ge alloy. Even though Ge is oversized it is found to be enriched at the surface due to radiation-induced segregation, and Gibbsian segregation could lead to the formation of an ordered phase on the

surface [29] below 550 C. The measured and best-®tted pro®les are illustrated in Fig. 13. Altered layers are thinner than in the alloys discussed above. The optimal set of parameters obtained is shown in Table 6. Most of these parameters are consistent with those derived in previous work [29]. For the best ®t, however, it was necessary to assume Ge to sputter preferentially in accordance with pure-element sputter yields YGe ˆ 3.0 and YNi ˆ 2.0 atoms/ion. Although the ®t is good for the 300 C pro®le, discrepancies are noticeable at other temperatures. The measured pro®les show signi®cantly stronger subsurface depletion at high temperatures. 6. Summary and outlook

Fig. 13. Steady-state concentration pro®les of Ge atoms in a Ni±10 at.% Ge after bombardment with 3 keV Ne‡ at various temperatures. Calculated results (dashed and solid curves) compared with experimental data [29].

The theoretical scheme applied in this work, based on Eq. (3), has proven to be an ecient tool in the calculation of composition pro®les under ion bombardment. It combines strengths and eliminates weaknesses of two previous schemes which were utilized for ± by and large ± complementary scenarios. One scheme [4,7] emphasized collisional

M.W. Sckerl et al. / Nucl. Instr. and Meth. in Phys. Res. B 140 (1998) 75±90

89

Table 6 Input parameters for Ni±10% Ge Fixed parameters

From [29]

Ion current density Vacancy migration frequency Interstitial migration frequency Vacancy formation entropy Interstitial formation entropy Interstitial formation enthalpy Interstitial migration enthalpies Ge segregation enthalpy Ge segregation entropy

3.75 ´ 1013 cmÿ2 sÿ1 8 ´ 1012 sÿ1 5 ´ 1012 sÿ1 3k 0 4.0 eV 0.15 eV )0.022 eV 1.13 k

Fitted parameters

Present work

Ref. [29]

Vacancy formation enthalpy HVf m Vacancy migration enthalpy HGe;V m Vacancy migration enthalpy HNi;V Pure-element sputter yield YGe Pure-element sputter yield YNi Peak sink concentration Xs Nsmax

1.60 eV 0.96 eV 0.95 eV 3.0 atoms/ion 2.0 atoms/ion 3.0 ´ 10ÿ2 (300°C) 5.0 ´ 10ÿ3 (500°C) 1.0 ´ 10ÿ3 (600°C)

1.50 eV 1.10 eV 1.10 eV 2.0 atoms/ion 2.0 atoms/ion 1.0 ´ 10ÿ1 1.0 ´ 10ÿ3 5.0 ´ 10ÿ4

processes, but it was not obvious until the joint work of two of us [3] how defect-mediated processes could be incorporated feasibly. The other scheme [5,6] was up-to-date with regard to thermal processes, but the description of collisional aspects was rudimentary, forced in part by the lack of a relaxation model. We have concentrated on stationary pro®les here since the elimination of one of the two pertinent variables implies a major reduction of the computational e€ort and, therefore, a more direct test of quantities that are measurable and of practical interest. However, now that the scheme has been tested, ¯uence dependencies are promising targets for further studies. While Figs. 10±13 show promising agreement with experimental data we reiterate that every single curve contains at least one adjustable parameter and that there is considerable uncertainty about the accurate values of almost all others that enter. Some con®dence emerges, though, from the fact that it is the same parameter, the absolute level of the sink concentration, that was used to ®t the curves in all pertinent ®gures. On the other hand it is disturbing that such a signi®cant parameter is so little known.

Acknowledgements This work has been supported by the Faculty of Science of Odense University, the Danish Research Academy, the Danish Natural Science Research Council, and the US Department of Energy (Basic Energy Sciences ± Materials Sciences under contract W-31-109-Eng-38). One of us (MWS) expresses his gratitude to the management and sta€ of the Materials Science Division at Argonne for their hospitality during a one-year's stay. Discussions with H.H. Andersen, W. M oller, and V. Pontikis, as well as valuable advice from E. Christiansen are gratefully acknowledged. References [1] G. Betz, G.K. Wehner, in: R. Behrisch (Ed.), Sputtering by Particle Bombardment, vol. 2, Springer, Berlin, 1983, p. 11. [2] H.H. Andersen, in: J.S. Williams, J.M. Poate (Eds.), Ion Implantation and Beam Processing, Academic Press, New York, 1984, p. 127. [3] P. Sigmund, N.Q. Lam, Mat. Fys. Medd. Dan. Vid. Selsk. 43 (1993) 255. [4] P. Sigmund, A. Oliva, G. Falcone, Nucl. Instr. and Meth. B 194 (1982) 541.

90

M.W. Sckerl et al. / Nucl. Instr. and Meth. in Phys. Res. B 140 (1998) 75±90

[5] N.Q. Lam, H. Wiedersich, J. Nucl. Mater. 103/104 (1981) 433. [6] N.Q. Lam, H. Wiedersich, Nucl. Instr. and Meth. B 18 (1987) 471. [7] P. Sigmund, A. Oliva, Nucl. Instr. and Meth. B 82 (1993) 269. [8] M.W. Sckerl, P. Sigmund, N.Q. Lam, Nucl. Instr. and Meth. B 120 (1996) 221. [9] W.O. Hofer, U. Littmark, Phys. Lett. A 71 (1979) 457. [10] R. Collins, P. Sigmund, J. Appl. Phys. 72 (1992) 1993. [11] U. Littmark, Nucl. Instr. and Meth. B 90 (1990) 202. [12] U. Conrad, H.M. Urbassek, Nucl. Instr. and Meth. B 83 (1993) 125. [13] H. Wiedersich, J. Nucl. Mater. 205 (1993) 40. [14] M.L. Jenkins, M.A. Kirk, W.J. Phythian, J. Nucl. Mater. 205 (1993) 16. [15] A.R. Allnatt, A.B. Lidiard, Atomic Transport in Solids, Cambridge University Press, Cambridge, 1993. [16] R.E. Howard, A.B. Lidiard, J. Phys. Soc. Japan 18 (1963) 405. [17] R.A. Johnson, N.Q. Lam, Phys. Rev. B 13 (1976) 4364. [18] S.M. Murphy, J. Nucl. Mater. 168 (1989) 31. [19] J.R. Manning, Di€usion Kinetics for Atoms in Crystals, Van Nostrand, Princeton, NJ, 1968. [20] H. Wiedersich, N.Q. Lam, in: F.V. Nol®, Jr. (Ed.), Phase Transformations During Irradiation, Applied Science Publishers, Barking, 1983, pp. 1±46. [21] P.H. Dederichs et al., J. Nucl. Mater. 69/70 (1978) 176. [22] M. Smoluchowski, Z. Phys. Chem. 92 (1917) 129. [23] R.W. Ballu, A.H. King, in: F.V. Nol®, Jr. (Ed.), Phase Transformations During Irradiation, Applied Science Publishers, Barking, 1983, pp. 147±187.

[24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42]

T.R. Waite, Phys. Rev. 107 (1957) 436. J. Kirschner, Nucl. Instr. and Meth. B 7/8 (1985) 742. J. du Plessis, E. Taglauer, Surf. Sci. 260 (1992) 355. N.Q. Lam, H.A. Ho€, H. Wiedersich, L.E. Rehn, Surf. Sci. 149 (1985) 517. S. Tang, N.Q. Lam, Surf. Sci. 223 (1989) 179. H.A. Ho€, N.Q. Lam, Surf. Sci. 204 (1988) 233. N.Q. Lam, K. Johannessen, Nucl. Instr. and Meth. B 71 (1992) 371. H.W. King, J. Mater. Sci. 1 (1966) 79. J.P. Biersack, L.G. Haggmark, Nucl. Instr. and Meth. 174 (1980) 257. L.E. Rehn, H. Wiedersich, Mater. Sci. Forum 97 (1992) 43. N.Q. Lam, S.J. Rothman, R. Sizmann, Radiat. E€. 23 (1974) 53. E. Christiansen, Numerisk Analyse, Odense University Press, Odense, 1991. P. Sigmund, Appl. Phys. Lett. 66 (1995) 433. L.E. Rehn, S. Danyluk, H. Wiedersich, Phys. Rev. Lett. 43 (1979) 1764. L.E. Rehn, V.T. Boccio, H. Wiedersich, Surf. Sci. 128 (1985) 37. D.G. Swartzfager, S.B. Ziemecki, M.J. Kelley, J. Vac. Sci. Technol. 19 (1981) 185. A. Kurokawa, H.J. Kang, R. Shimizu, Surf. Sci. 255 (1991) 120. R. Kelly, Nucl. Instr. and Meth. B 39 (1989) 43. P.S. Ho, Surf. Sci. 72 (1978) 253.