Nonlinear effects of diffusion in displacement cascades

Nuclear Instruments North-Holland

and Methods

in Physics Research

B61(1991)

27-37

27

Nonlinear effects of diffusion in displacement cascades Fraqois Rossi a and N.V. Doan b a Centre d’Etudes et Recherches sur les Mat&iaux, 38041 Grenoble Cedex, France ’ Centre d’Etudes et Recherches sur les Materiaux, 91121 Gif sur Yvette, France Received

26 September

1990 and in revised

D$artement

d’Etude des MatMaux,

CENG BP 85X,

Section de Recherches de M&allurgie Physique, Centre d’Etude Nuclkaires de Saclay,

form 4 January

1991

Analytical study of heat diffusion in a solid with n thermal spikes (with n =1 to 4) is performed. It is shown that the spike lifetime depends on the number of spikes happening at the same time and their relative distance. These phenomena are further studied by molecular dynamics simulations where diffusion and displacements of atoms in Cu are studied for 1 spike of energy E and 2E and compared to the results obtained with 2 spikes of energy E. It is shown that the mean square displacement is increased by a factor of approximately 4 when 2 spikes of energy E are prbduced compared to 1 spike of energy E Similarly, the diffusion coefficient and the number of atomic disulacements are increased in the same way. The consequences of this enhancement on diffusion under irradiation are discussed. _

2. Diffusion and thermal spikes

1. Introduction

Recent studies on displacement cascades have led to further development in the basic concepts used in description of diffusion phenomenon under irradiation. First, some discrepancies between the classical theory of radiation enhanced diffusion (RED) [l] and ion beam mixing results have been evidenced [2]. Molecular dynamics (MD) studies performed by Diaz de la Rubia et al. [3,4] and Averback et al. [5] have shown that diffusion was occurring in the volume of the thermal spikes. This volume was similar to a liquid during a few picoseconds. The fact that diffusion in the thermal spike regime occurs through collective atomic motion in “liquid droplets” and not through classical point defects diffusion phenomena, led Rossi et al. [6,7] to propose a new model of ion beam mixing. This model, based on a Monte Carlo technique, described the effect of spike interactions and cascade structure and could explain qualitatively some of the features not described by the classical theory of RED. In this paper, we present in more detail the phenomena involved in thermal spikes and subcascade interactions based on analytical and molecular dynamics (MD) calculations. The idea is to describe what happens when several thermal spikes happen in the same volume and at the same time, and evaluate the change in the temperature field and in diffusion due to this interaction. 0168-583X/91/$03.50

0 1991 - Elsevier Science Publishers

We first describe the general features of cascade and subcascade formation and transition to thermal spikes. The effect of substrate temperature will be mentioned. We will also show the importance of the thermal spikes spatial distribution on diffusion and defects formation. 2.1. Cascades,

subcascades

and thermal

spikes

As an energetic ion penetrates a solid, it produces a series of collisions with target atoms, displacing many of them from their lattice sites. A large fraction of these primary knock-on atoms (PKA) receive sufficient energy to displace other target atoms through secondary recoil events: this is referred to as the displacement cascade. After each collision, the primary ion energy decreases because of nuclear collisions and electronic interactions. The distance between collisions thus decreases until a point where the distance between collision events is of the same order as some interatomic distances. At that point, all the collisions interact and the atoms in this region of the cascade are animated by a collective motion: this region of the cascade is referred to as “thermal spike”. Because of the stochastic nature of the ion trajectories, the thermal spikes are placed at random in the cascade volume and the cascade displacement density, i.e. the number of displacements

B.V. (North-Holland)

F. Rossi, N. V. Doan / Nonlinear effects in displacement cascades

28

per unit volume, is not homogeneous in space: the density is locally very high in the thermal spike and can be very low in other parts of the cascade. However, depending on the stopping power of the target and primary ion energy, the distance between primary collisions can be larger than the size of the cascades initiated by the secondary ions. The cascades thus present a substructure where “subcascades” can be well identified: these subcascades contain the thermal spikes described above. It is expected that the fluctuations in collisions density in the cascade is an important factor on diffusion and defect production. For instance, starting from fractal concepts, Rossi et al. [8] have shown that the onset energy for damage efficiency in a cascade was related to the energy for thermal spike formation. It was shown that the defect production efficiency was decreasing when thermal spikes were formed apart from each other. We will thus evaluate the importance of the spike interaction phenomenon. 2.2. ~~ff~i5n

in the spikes

The role of thermal spikes in diffusion under irradiation has been evidenced recently by molecular dynamics simulations by Diaz de la Rubia et al. [3,4] and Averback et al. [5]. These simulations were performed for Cu and Ni. An important result of these calculations was the indication of transient local melting in the spike core, this melting effect being more important in the case of Cu than Ni. This was first attributed to the lower melting point of Cu, but a second explanation could be the difference in the electron-phonon interaction in the two materials 191.The evidence for melting in these studies was the similarity of the local pair distribution function in the central region of the cascade and the distribution function in the case of the equilibrium liquid. Thermal spikes can thus be assimilated at least qualitatively to “liquid droplets” in a surrounding solid crystal, in which most of the diffusion occurs when delayed effects are not considered. It must be pointed out that these phenomena are different in essence from the classical view of diffusion through point defects. In the case of the thermal spike, all the atoms are animated by a collective motion instead of a direct jump of an interstitial atom or a site exchange between an atom and a vacancy. After resolidification, most of the vacancies are in the core of the spike and the outer shell contains most of the interstitials. These defects will contribute in the delayed regime to the observed macroscopic diffusion.

2.3. Effect of substrate temperature Another MD simulation performed in the case of Cu by Hsieh et al. [lo] shows that the diffusion coefficient in the volume of the thermal spike is comparable to the

diffusion coefficient calculated in liquid Cu. Moreover, diffusion drops dramatically when resolidification of the spike volume is complete. An increase of substrate temperature (up to 700 K, i.e. 0.7T,) does not change the coefficient of diffusion in the spike volume but increases the spike lifetime and its maximum size. The total amount of mixing observed in the crystal was 4 times greater at 700 K than at 0 K. 2.4. Influence of the cascade structure on diffusion The MD simulations thus lead to the conclusion that diffusion in a cascade can be assimilated to diffusion in liquid droplets placed in the cascade volume. It is of interest to know what happens when the distribution density function of thermal spikes in the solid changes, i.e. when overlap or interaction between several spikes occur. This overlap may happen in several cases. First, when the primary ion changes, the PKA energy distribution changes, and thus the spatial distribution of thermal spikes. In dilute cascades (i.e. cascades in a light element initiated by an energetic ion), the thermal spikes are expected to be isolated. When the mass of the target element increases, the average distance between thermal spikes decreases, and intra-cascade overlap can occur. Second, at very high dose rate, two cascades can be initiated at the same place at a time interval smaller than the quench time of the spikes. Here again, an interaction between the spikes can be expected. Third, for a given space distribution of thermal spikes, when the substrate temperature is increased, the size and lifetime of the spikes increase, the average distance between spikes decreases and the overlap probability thus increases. It is expected that the temperature distribution being changed by the simultaneous presence of several spikes, the size and lifetime of each spike will be increased and the diffusion will be enhanced. This could explain some experimental results which cannot be interpreted in the frame of classical theory of diffusion under irradiation or radiation enhanced diffusion 1561. For instance, Thevenard et al. Ill] reported a much more efficient dissolution of precipitates when irradiation is performed with molecular ions than with single ions. Pramanik and Seidman [12] found experimentally a factor of 1.55 between the vacancy production yield when irradiation of W was performed with dimers instead of monomers with the same energy per ion. Andersen and Bay [13] found a dimer to monomer yield ratio of 2.2 in sputtering experiments. Rehn and Okamoto [2], and Rossi et al. [6] reported a change in activation energies in the medium temperature range for ion beam mixing when irradiation was carried out with different ions, the other parameters being constant. In these experiments, the cascade structure was changed and spike interactions could explain the observed results.

F. Rossi, N. V. Doan / Nonlinear effects in displacement cascades 3. Spike interactions Two different approaches can be used to evaluate the changes in diffusion due to thermal spike interactions. The first one is to consider that the crystal is a continuous medium and solve the heat diffusion equation. The second approach is to perform molecular dynamics simulations. Since this method is very computer time consuming we have only performed typical MD calculations on Cu to confirm the qualitative results obtained with the analytical study.

The total amount of mixing (R’) in the spike can be calculated by using the method proposed by Johnson et al. [15], by considering that the total number of jumps per ion in the material is proportional to q/p, and the diffusion distance is proportional to qr:/p, where r, is the characteristic jumping distance and p the density. By assuming that r, is proportional to p-‘/3, that Q is proportional to the absolute value of the cohesive energy of the solid AH,,, [15], one finds for a spherical spike: (R2>s =

3. I. Analytical study

29

K, r

3/5 An8’3r(5n/3) lh?KOCO

2/3

p

S/3

(

,WfoJ)5’3

(3c) The first approach for interactions is to calculate heat conduction equations the calculations performed

studying the thermal spike the spike lifetime from the with an approach similar to by Vineyard [14].

3.1.1. Case of a single spike The model starts from the equation of heat conduction in an isotropic and uniform medium with a conductivity K and heat capacity C [14]: aT

VKVT=C~,

where K and C have the following K = q,Tn-’

form:

C = COT”-I.

The energy E of the spike is initially introduced in the medium and the initial distribution of added internal energy is E s3(r), where s3(r) is the 3-dimensional delta function. For an initial temperature equal to 0, the solution of eq. (2) can be written [14]:

t) =

T(r,

((~T~~~,2)“n exP( si_

(2)

The model supposes that the jumping of the atoms in the spike is an activated process and that the rate of jumping per unit volume can be written A exp( - Q/T), where Q is a normalized activation energy. The total number of jumps 9 can be calculated and one finds, in the case of a spherical spike [14]: ns=

/0

=

004Tr2dr

/0

-Q

mAe

T(r,

3,‘5 Ar~*‘~r(5n/3)

t)dt E

t-1Qn

IO.~~IC,C,~‘~

5’3

’

(3a)

where r(n) is the gamma function of argument n. In the case of a cylindrical spike, the integration gives [14]: An3r(2n)

c

17c=

4TK&,

where spike.

e is the energy

i-1Q”

2

(3b)

’

deposited

per unit length

of the

In the case of a cylindrical

spike:

(34 where K, and K, are constants. These two formulas do not take into account the chemical forces that can act at an interface between two different solids. In this case, a Darken term must be added to the two formulas above [7,15]. This model was compared to experimental values obtained in the literature. A good agreement was found for more than 25 systems tested by Johnson et al. [15] and Rossi et al. [6] when the value of n was taken equal to 1, i.e. when the heat capacities and the thermal conductivities of the solids were taken as constant. Taking another value for n (greater or smaller than 1) leads to a decrease in the correlation coefficient of the regression between the experiments and the model. This point can be explained on the basis of the MD results of Diaz de la Rubia et al. [3,4] and Averback et al. [5]. It is found that diffusion drops dramatically when the temperature in the spike decreases below the melting point of the solid. Thus, the values of heat capacity and conductivity that must be taken into account for calculating diffusion in the spikes are the values corresponding to the liquid phase. These values probably do not vary very much in the temperature range of the liquid spike. 3.1.2. Case of several spikes In the case where several sources of energy are present, for n = 1, eq. (2) is linear and the solution is the sum of the temperature fields created by each source. For instance, when several spikes of energy E, are placed in the space with coordinates r& the temperature field can be represented by: “‘* T(r’ t, = (~~~~)3/2

CE, i

exp(-Clr-rd124KT).

(4)

The integration of eq. (3a) cannot be performed in this case, but a qualitative idea of the characteristics of diffusion can be deduced from the value of the melted

F. Rossi, N. K Doan / Nonlinear

30

effects in displacement cascades

t= 2.5E-14 s

t= 7.5E-14 s

t= 1 .OE-13 s

t= 2.OE-13 s

t= 4.OE-13 s

t= 6.OE-13 s

t= 8.OE-13 s

t= 1.2E-12 s

(_-;a(_>

(I_:i

t= 1.4E-12 s

1=1.6E-12s

t= 1.6E-12s

t= 1 .QE-12 s

I Fig. 1. Evolution

of the liquid volume

in 2 spikes 40 ii apart

as a function

of time. It can be seen that the 2 volumes

coalesce

in a

single “droplet”, allowing slower cooling rate and greater diffusion.

volume of the solid as a function of time. The temperature profile given by eq. (4) can be used to determine numerically the size of the molten volume with time, as well as the thermal spike lifetime, by comparison with the melting temperature of the substrate. However, due to the number of parameters used in the simulation, we can only give qualitative results and we have restricted the study to the cases where the spikes have the same energy E. The general case of a distribution of energy will be discussed in section 4. 3.1.3. Results Calculations have been carried out with eq. (4) with high temperature values of C and K of 1.13 X 10m5 eV AK-’ and 3.74 x lo9 eVAK-‘sl for Cu respectively [16]. The value of the conductivity corresponds to the extrapolated value of the lattice conductivity at the melting point. The value of T, taken was 1100 K in order to allow comparison with the MD simulation, and the spike energy was 3 keV. We have considered that 2 spikes were occurring at the same time. Fig. 7 represents the evolution of the liquid volume for 2 spikes 40 A apart as a function of time. The general features of the results can be deduced from the figure: it can be seen that if the distance x0 between the 2 spikes is small enough, they coalesce in a single molten volume. Since the surface of this volume is less than the surface of the 2 spikes at equal volume, the cooling rate decreases and the spikes lifetime increases, allowing further diffusion

in the spikes. Fig. 2 shows the volume of liquid as a function of time for different distances x0. It can be seen that for the chosen energy, the maximum size of the thermal spike does not depend on the distance x0, and that the spike lifetime is maximum at x0 = 0 and minimum for a distance equal to approximately 50 A

1.2 IO6 ,X,=O,lOA

h

f 7

6.0 IO’

_. w 2.0 105 0.0 10= 0.0

1.0 lcr”

2.0 Id2

3.0 10

Time (s) Fig. 2. Volume of the liquid droplet as function of time for different distances xr, for an initial energy of 1 keV. It can be seen that the maximum size of the thermal spike does not depend on the distance x0 but the spike lifetime is maximum at xa = 0 and ~nimum for a distance equal to appro~mately 30 to 50 A depending on the number of interacting spikes. The curve corresponding to 50 A and single spike can be superimposed after normalization.

F. Rossi, N. V. Doan / nonlinear

effecis in disFlace~ent

31

cascades

100

4.0 lo-l2

90 80 3.0

70

iii

“x L

1

60 50

2.0

40

i.o10-

30

‘2’ 0

20

40

0

80

60

2

4

8

10

12

E &“)

x(&A) Fig. 3. Reduced values of spike lifetime fmax/tO vs spike distance for 1 to 4 spikes interacting. to is the value corresponding to a single 1 keV spike. It can be seen that rmax increases with the number of spikes for a given value x0. Beyond a value x,, close to 50 A in the calculation, the t,, curves join the value of an isolated spike.

and beyond. For that distance, each spike Iifetime is equal to the lifetime of an isolated spike. Of course, these values depend on the spike energy and the thermal properties of the solid. could also be evaluated vs x0 Spike lifetime t,,, when 1 to 4 spikes are interacting. In these cases, the spikes were placed with the same energy at equal distance x0 from each other: for 3 spikes, the spikes were placed at the vertices of an equilateral triangle, for 4 spikes, a square configuration was chosen. Fig. 3 shows the values of f,,, vs x0 for the different cases treated. increases with the number of It can be seen that t,,

Fig. 4. Distance xi (in A) for which no interaction occurs between two spikes vs the spike’s energy. If two spikes are placed at a distance smaller than x,, a coalescence of the molten volume will occur, increasing the diffusion.

spikes for a given value x0. Beyond a value x0 between 30 and 70 A depending on the number of spikes, the t maxcurves join the value of an isolated spike. Since it is expected that diffusion in the molten sphere will depend greatly on its size and lifetime, it is expected that spike interactions and cascade structure can play a role on the observed diffusion. The distance xi for which no interaction occurs between two spikes can be calculated with eq. (4). This distance is such that the maximum temperature reached during the spike lifetime is equal to T,/2. A smaller separation distance of the two spikes would lead to a coalescence of the liquid droplets, and thus to an increase of the spike lifetime and diffusion. The values obtained in the case of Cu are presented on fig. 4.

7--

T-

IV

III

II

V

lOOOeV

-0

6144

2048

8192

i

2048

6144

L._-.-_.’

Fig. 5. Molecular Dynamics simulation cell. The simulation cell is chosen so that no boundary effects perturb the results. The PICA are placed in the box III at a distance no. The cell is divided into 5 boxes containing 6144, 2048, 8192, 2048 and 6144 atoms respectively. The diffusion effects are analyzed in each box independently to check that the cascade extent is well limited inside the box Ill and to get a better evaluation of ( R2).

F. Rossi, N. V. Doan / Nonlinear effects in displacement cascades

32 3.2. Molecular

dynamics

sttiy

In order to verify the above conclusions, MD simulations were carried out on Cu, using the Gibson II Born Mayer potential [17]. This potential was chosen for the purpose of comparison with other MD results performed with the same potential f3,4]. The melting temperature for this potential is between 1000 and 1100 IL Cascades of different energies were simulated in different configurations. In all cases presented, the calculation was carried out with 24576 atoms. The atoms are placed on a fee lattice bordered with periodic boundary conditions, following the method proposed by Tennenbaum and Doan 1181.The simulation eel1 was chosen so that no boundary effects perturb the results. To look more closely at what happens in the cascade core, the cell was divided in 5 boxes containing 6144,2048,8192, 2048 and 6144 atoms respectively, as shown in fig. 5. When a single spike was simulated, the PKA was placed close to the boundary between boxes II and III. When two spikes were simulated they where placed at a distance x0 apart as shown in fig. 5. The direction of the velocity was changed in each of the two cases treated to check its influence. The calculation was carried out in three steps: in the first step, the cascade is introduced and the time step used is 5 x 10-l’ s. The thermalization step uses a At of 5 x lo-l6 s up to 1.025 ps. Then the diffusion step starts with a At of lo-’ s. Fig. 6 shows the total amount of mixing as a function of time ( R2> for a single spike defined by:

where NboX is the number of atoms in each simulation box. This quantity is calculated independently in each

0.0

2.0

Id’

4.0 IOf2

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

No, of spikes 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2

Spike energy

Distance

WI 500 750 1000 1oOo 1000 1500 2000 3000 500 750 750 750 1000 1000 1000

x0 @I

-

65.5 65.5 43.7 22 22 15 11

’ 12

8.0 IO

time (s) Fig. 6. Total amount of mixing as a function of time ( R2) as defined by eq. (6) in the case of a single spike. The values are plotted for each box I to V as indicated in fig. 4. The lines are presented for visual aid only. Two simulations with 2 different initial PKA velocity directions are plotted. It can be seen that in one of the cases, the spikes have propagated into the box number IV. However, the values obtained in box III in the two cases do not seem to be sensitive to the initial PKA velocity direction.

box so as to check that the cascade extent is well defined inside the box III, and to get a better value for the diffusion effects. Two cases were treated with the same energy and 2 different initial velocity directions for the PKA. The values of ( R2) are expressed in 10-l’ cm’/atom. It can be seen in fig. 6 that most of the mixing happens in the central box III. In one of the two cases however, the cascade has expanded beyond the III-IV boxes boundary. This has not changed signifi-

Different cases treated in the MD simulation. The values of (R’) and the number of ~splacem~nts Run

6.0 10’”

are given for a time of 6.025 ps

CR21 IO-“cm2/at

N displa~ments

3.92 6.68 11.28 10.38 10.01 27.85 43.71 113.51 5.82 11.44 20.69 26.84 49.23 45.76 46.26

226 391 638 568 613 1251 1891 4047 355 733 1098 1174 1872 1741 1945

F. Rossi, N. V. Doan / Nonlinear effects in displacement cascades

33

50

45

30

20

10

0 0

1 10-12

2 IO.12

3 10-Q

4 lo-12

5 10-12

6 II+"

7 10-Q

Time (s) Fig. 7. Comparison between the mean square displacement per atom ( R2> induced by a single spike and 2 spikes overlapping, for three values of x0 (II,15 and 22 A). In all the cases, only diffusion in box III is presented. For comparison, the results corresponding to one 2 keV spike are also displayed. It can be seen that the total amount of mixing when 2 spikes overiap is 4 times larger than in the case of a single spike, and the diffusion induced by two 1 keV spikes is comparable to the diffusion induced by one 2 keV spike. These runs were stopped in the two 1 keV spikes case and one 2 keV spike case before complete equilibration for reducing computer time.

cant+ III.

the value of the total

It was

atnotmr

of mixing

in box

found cm the other hand that all the vacanties and interstitials created are located in the. central part of box IJJ.

We have then performed

&e Eaculation

with 2 spikes

of 1 keV placed in the center of the calculation cell with 3 separation distances of x,, equal ta 11, 15 and 22 A respectively. Another wnfiguration with 2 cascades of

Time (s) Fig. 8. Diffusioncoefficient deduced from the avenge displacement distance as displayed in fig. 6. The values are obtained through the formula: D =i ( R2)/6t. We have only shown the results corresponding to 1 value of x0 since no significant difference could be found from the results of fig. 6.

F. Rossi. N. K Doan / Nonlinear effects in displacement cascades

34 2000

+ keV

I---

0

1 to-12

2

3 $0.‘*

10-Q

4 10.12

5 IO-‘*

6 10*12

7

fO-‘*

Time (s) Fig. 9. Number of displacements in a cascade in the case of one 1 keV spike, one 2 keV spike and two 1 keV spikes placed at 11,15 and 22 A from each other. It can be seen that two 1 keV cascades placed at 11 and 22 A produce the same effect as one 2 keV cascade. The result corresponding to a distance xo equal to 15 A gives a smallernumber of displacementswhichcannot be explained by diffusionin other boxes.

750

eV energy was tested with values of xe equal to 22, 44 and 65 A. In the latter case, diffusion happened in

to 1 spike only (box III). For comparison, the results of

the complete simulation box. Table 1 shows the calculation conditions for each case tested. Fig. 7 shows the results of 3 simulations as well as those corresponding

the simulation with 1 cascade of 2 keV energy are also reported in fig. 7. It can be seen that the total amount of mixing is about 4 times larger in the case of 2 spikes than for 1 spike only for a given energy. Besides, we

30

25 ‘;;; nE r b vA

20

“K V

15

10 0

20

40

60 x0

80

100

(A)

Fig. 10. Number of displacements and mean square displacement per atom (R’) induced by 2 spikes of 750 eV versus the spike separation x0. The value of x, (as given by fig. 4) corresponding to this case is indicated on x0 the axis.

E Rossi, A? V. Doan / ~o~~ine~r effects in dispiaceme~r cascades

35

times larger than the effect of 1 cascade of 1 keV alone. However, in one case, the number of displacements is smaller when the distance between the two spikes x0 is increased: when x0 equals 11 or 22 A, the effect of the 2 spikes of 1 keV is equivalent to the effect of a single 0 spike of 2 keV. For a distance x0 equal to 15 A, the number of displacements decreases by 17%. It was checked that the value obtained in this case is not compensated by diffusion in the other boxes. Fig. 10 shows the values of (R’} and the number of displacements obtained in the case of 2 cascades of 750 eV, as a function of x0. These values are compared to twice the values obtained a single 750 eV cascade (represented in fig. 10 by the horizontal dotted line). It can be seen that both values decrease with xc, and the decrease is very steep between 44 and 65 A. The interaction distance obtained is thus larger than the value given by fig.4 (36 A), but has the same order of magnitude. These different values can be compared to the 750 eV cascade size which is equal to 33 Ai: the interaction distance is a little greater than the cascade size. It is worth noting that the values of ( R2) depend on the choice in size and shape of the box because of the non-u~form character of the atomic displacement inside the cascade. Averback et al. [5] and Diaz de la Rubia et al. [3,4] used a sphere centered on the PICA site instead. However, for our purpose of comparing the mixing induced by a single spite and two spikes, our choice of the box III seems very appropriate because it is corroborated by a very close correlation in the varia-

observe that the cascade is not completely thermal&d for the two 1 keV spikes and one 2 keV spike, and this mixing enhancement is thus probably underestimated. The simulation shows that in the case of 1 keV cascades, the mean diffusion obtained for the 3 values of x0 tested is the same, probably because the differences between the different values of xc tested are not large enough, in agreement with theoanalytical result: fig. 4 gives a value of xi equal to 40 A. Fig. 8 shows the corresponding diffusion coefficient defined by D = (R2>/6t in the case of 1 single spike of 1 keV, 2 spikes of 1 keV and 1 single spike of 2 keV. The values for the diffusion coefficient obtained have the same order of magnitude as the values reported by Diaz de la Rubia et al. [3,4] and vary in the same way. It is observed that when the energy increases, the spike lifetime increases considerably. We could not perform the calculation for the two 1 keV spikes and one 2 keV spike until complete equilibration, but the spike lifetime in these cases is expected to be much larger than in the case of a single 1 keV spike. The replacement sequences in the spikes have been studied in the following way. We have evaluated the number of interstiti~s and the number of substitutions created after the end of the thermal spikes. We have considered that the total number of displacements was equal to the sum of these two values. Fig. 9 shows the number of atomic displacements time for different cascade energies. Here again, it can be seen that the effect of 2 cascades of 1 keV energy is about 3 to 4

120

80

60

40

20

0 0

500

1000

1500 E

2000

2500

3000

WI

Fig. 11. Mean square displacement per atom (R’) and number of displacements N per atom induced by a cascade as a function of cascade energy. The regression curves give a E’,9 dependence for ( R2) and a E’.6 dependence for N, in good agreement with the analytical eq. (3~).

36

F. Rossi, N. K Doan / Nonlinear effects in displacement cascades

tion vs time of the calculated values (R’) and the number of atomic displacements observed in the cascade (see fig. 9). Fig. 11 represents the value of (R*) and the number of displacements as a function of energy. A regression made on these values shows that (R*) is proportional to E’,9 with a correlation coefficient equal to 0.996. The same regression made on the number of replacements gives a value of 1.6 for the exponent. These values are in good agreement with the analytical result given by eq. (3~) which gives a value of 1.67 for the exponent of energy. It is however not possible to compare directly the results with the expressions of eq. (3c), because of the several proportionality factors used in the Johnson model.

4. Discussion We can now have a better idea about the consequences of these calculations on ion beam mixing. In a cascade produced by a primary ion of energy E, subcascades will be formed with different energies and in different places in the cascade volume. Depending on the irradiation conditions, several situations can occur. In the case of irradiation of a target with light energetic ions, the cascade will be dilute, i.e. the distance between primary collisions will be much larger than the thermal spike itself. No overlap or spike interaction is expected and when another spike, from another cascade is produced at the same place, one can expect a simple addition of the effects: the total amount of mixing will be the sum of the individual contributions of each spike. In different irradiation conditions, the cascade can be dense: this can happen during irradiation with heavy ions. In that case, the distance between primary collisions can be smaller than a characteristic spike interaction distance, as the one defined in fig. 4. In that case, the diffusion will be greater than the sum of the diffusion produced by the 2 individual spikes, and the diffusion produced by a single spike of energy E, + Ej will be a maximum. This point shows the importance of primary recoil energy spectrum and subcascades spatial distribution. Both depend on the irradiation conditions. Several attempts have been made to characterize the influence of the primary recoil spectrum on the cascade structure [20,21], in the case of a self-ion cascade. The results show that the critical energy for subscascade formation increases with the target mass and density. For instance, critical energy values of 0 to 2.5, 20 keV and 200 keV were found for Al, Cu and Au respectively using fractal geometry considerations [20] and atomic collisions calculations [21]. However, these calculations only take into account the ballistic aspects of the cascade and further work has to be done in order to compare

these results with the thermal diffusion calculations done in the present study. Another case where spike interactions happen is when the irradiation is carried out for instance with molecular ions instead of single ions (e.g. NC instead of N+): the crystal will be submitted to 2 synchronized cascades, and one can expect intercascade spikes overlap. In both cases, one can expect a 2 to 4 fold increase in the total amount of mixing, instead of a simple additivity. This point is in good agreement with the experimental results of Thevenard et al. [ll], Pramanik and Seidman [12], Andersen and Bay [13] and with the results of the analytical study, but smaller than other results reported in the literature [19]. This discrepancy could be explained by the fact that spike interaction probability is very sensitive to the conditions of irradiation and the extent of the interaction effect depends on the properties of the studied system: as mentioned above, the density of a cascade and its structure depend on the primary collisions energy spectrum. For a high energy PKA, the primary collisions will be far apart from each other at the beginning of the cascade. When energy decreases, the average distance between collisions decreases, and spike interaction probability increases. Depending on the depth in the solid, the observed effects of nonlinearity could be different. A transition between the single spike diffusion regime and spike interaction diffusion regime is also expected when, for instance, the temperature increases. MD simulations [lo] have shown that when the substrate temperature is raised, the spike size increases. An increase by a factor of 2 between the radius of a 3 keV spike at 0 and 700 K was found by the MD simulation [lo]. Since the spatial distribution of spikes does not depend much on temperature, the probability of interaction and overlap increases with T. This can explain the differences observed experimentally between the energy activations for ion beam mixing in the thermally assisted regime when mixing is performed with different ions [2,6]. Thus many different possibilities can be observed experimentally, and it is difficult to find general rules to describe the phenomenon quantitatively. The next step of this study is to confirm the general rules to describe the average spike separation and size, to be able to predict the conditions for spike overlap.

5. Conclusion The effect of thermal spike interaction has been studied by two different approaches. First, an analytical study was performed to show the possible effects of thermal spike interactions. It was shown that, for typical thermal properties, each spike lifetime was increased when overlap was taken into account. Moreover, a coalescence of the spike volumes was found. It was

F. Rossi, N. !? Doan / Nonlinear effects in displacement cascades

expected that the average diffusion coefficient would be increased by this coalescence. This point was qualitatively confirmed by MD studies and a quantitative agreement was found between the dependence on energy given by the analytical model and the MD simulation results. Moreover, a non-linear effect was found in the diffusion coefficient. A factor of approximately four was found between the mixing observed for a single spike and two simult~~us spikes of the same energy. This value is in good agreement with some of the values reported in the literature, and the difference with other results can be explained by the fact that the interaction effect is very sensitive to the properties of the system studied. Further studies need to be performed to characterize the structure of the cascade and the conditions of overlap in a real cascade.

Acknowledgements

We would like to thank M. Nastasi and Carl Maggiore from Los Alamos National Laboratory, as well as Gilles Canova from University of Metz and Georges Martin from Centre d’Etudes Nucleaires de Saclay for helpful discussions.

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