Atomic displacement processes in irradiated metals

ELSEVIER

Journal of Nuclear Materials 216 (1994) 49-62

Atomic displacement

processes in irradiated metals R.S. Averback

Department of Materials Science and Engineering, University of Illinois at Urbana-Champaign,

Urbana, IL 61801, USA

Abstract

Irradiation of solids with energetic particles such as electrons, heavy ions and neutrons, results in the displacement of atoms from their lattice sites. In some cases the displaced atoms find metastable interstitial sites in the lattice, leaving vacant sites behind. In other cases, the atoms undergo switching processes with neighbors. No Frenkel defects are produced but atomic disorder is created. In this paper, tlie fundamental theory underlying these processes is presented. Topics include primary recoil spectra, spatial distributions of damage energy, defect production, and ion beam mixing. In addition to the traditional analytical theories of atomic collisions in solids, selected results of computer simulation are presented to elucidate the discussion and to introduce the latest developments in this field.

1. Introduction

In 1958 Seeger sketched his concept of the many processes that take place in an energetic displacement cascade [l], as reproduced in Fig. 1. A primary feature of the cascade, first proposed by Brinkman [2], is the formation of a depleted zone, which is caused by the ejection of atoms from the cascade core. The figure also shows replacement collision sequences (called dynamic crowdions [3]), focusons [4], ion beam mixing (called exchange collisions), and it even indicates channeling-like events. If we add to this picture the ideas concerning thermal spikes outlined by Seitz and Koehler in their classic paper of 1956 [5], and the calculation of Kinchin and Pease for the number of Frenkel pairs created in a cascade [6], a rather complete picture of displacement processes in irradiated solids would be at hand, with it all being formulated before 1960! Indeed, much of the research on atomic displacement processes these past thirty years has been devoted to “filling in the details.” Nevertheless, this has not been an easy task. These details, such as precisely how many defects are created, what are their spatial configuration and properties, how much mixing takes place, do thermal spikes even exist, what are the lengths of replacement collision sequences, and do

phase transformations take place within cascades, have been debated for nearly forty years and still without complete resolution. The difficulties in answering these fundamental questions from the experimental side has been the atomic resolution that is needed to study the structure of defects and the time resolution that is needed to capture the dynamics of events which evolve on time scales of some p&seconds. On the theoretical side has been the difficulty of solving a highly inhomogeneous many body problem on space and time scales where the macroscopic properties of the solid are seemingly not relevant. In this paper, the current level of our understanding of atomic displacement processes is addressed. In some ways, it is rather untimely to be writing this tutorial review, for as it will be seen in this school, major breakthroughs are just now taking place, owing largely to recent advances in computer simulations. Nevertheless, much is already known about displacement processes, and this subject provides the underpinnings for the subsequent topics in this school. We begin this chapter by a description of the energy dissipation of an energetic particle as it comes to rest in a solid. The discussion then turns to how this energy leads to atomic displacements, the detailed structure of the primary state of damage, and finally to atomic mixing.

0022-3115/94/$07.00 0 1994 Elsevier Science B.V. All rights reserved 0022-3115(94)00310-K

SSDI

R.S. Auerback /Journal of Nuclear Materials216 (1994) 49-62

50

2. Deposition of energy 2.1. Primary recoil spectra

The primary recoil spectrum of an energetic particle in a solid refers to the relative number of collisions in which an energy between T and T + dT is transferred from the primary recoil atom to other target atoms. A convenient way to illustrate primary through the normalized function, P(E,T)=;/rTTdT

du( E, T) d

dT

recoil

spectra

>

is

(1)

number of recoils between the minimum displacement energy, Td, and energy T. N is the total number of primary recoils, and da(E, T) is the differential cross section for a particle of energy E to create a recoil of energy, T. Figs. 2a and 2b illustrate recoil spectra for different 1 MeV particles in Cu in both qualitative (Fig. 2a) and quantitative terms (Fig. 2b). Although it is seen that the heavier ions produce more recoils at higher energies than the lighter ones, the differences in Fig. 2b do not at first glance appear large. Half the recoils are produced with energies less than - 60 eV for protons while the same fraction is produced with recoils below 150 eV for Kr ions. Why

which is the fractional

Fig. 1. A schematic two-dimensional diagram of a cascade in Cu created by a fast neutron, after Seeger [I].

the recoil energies are weighted toward low energies in both cases is a consequence of the screened Coulomb potential which governs atomic collisions of charged particles. For an unscreened Coulomb interaction, for example, the probability for a recoil of energy T varies

PRIMARY RECOIL SPECTRA FOR I-MeV PARTICLES

Tl/2

(eV)

I.(

25

3-

1.000 3-

Ne

9,000 FRACTION OF RECOILS WITH ABOVE E,, AND BELOW T

I-t&V ions

Xe

45,000

I02

(a)

I03

ENER(

-Cu

I04

T Ml

Fig. 2. (a) Qualitative representation of the primary recoil spectra for 1 MeV particles in Cu. Circles indicate where energetic recoils occur and their size indicates the amount of energy transferred to the recoil atom. TV,* refers to a “typical” recoil energy which is defined below. (b) Integral primary recoil spectra for 1 MeV particles in Cu. It is the integral fraction of primary recoils between the threshold energy and energy, T.

51

R.S. Averback /Journal of Nuclear Materials 216 (1994) 49-62 as l/T’. For neutrons, the situation is far different. Their primary recoil spectra are approximately described by hard sphere collisions for which the probability of a recoil of energy T is independent of recoil energy, from the threshold displacement energy to the maximum recoil energy.

repulsive force, while at close separations, the positive nuclei penetrate the electron shells causing the potential energy to rapidly increase. If we ignore electron excitations for the moment, E,(T) = T, and Eq. (2) can be integrated in closed form for the differential scattering sections of Eqs. (5), with the results,

2,2. Weighted average recoil spectra

In T - In Tmin WC(T) = In T max- In Tmin

For defect production, it is not the number of recoils of a particular energy that are of direct import, but rather the fractions of defects and damage energy that are produced in recoils of a particular energy. Thus, more insight is gained by considering the “weighted average” recoil spectra, W(E, T) for particles of initial energy, E, as W(E, T) weights the primary recoil spectra by the number of defects, or the damage energy, produced in each recoil, i.e. 1 W( E, T) = ~ E,(E)

TdT / Td

da(E,

T) E,(T),

dT

(2)

where E,(T) is the damage energy created by a recoil of energy T. (Damage energy was defined in the paper by Robinson, and as will be seen, it is proportional to the number of defects produced.) E,(E)

= jTy

dTdg(;;

T, E,(T),

Td is the threshold displacement the maximum recoil energy. Tmax=

4(mP4

E

energy, and T,,

is

(4)

(ml + m2Y

where mi is the atomic mass of atom “i” and the subscripts 1 and 2 refer to projectile and target atoms, respectively. It is instructive to examine W(E, T) for the extremes of Coulomb and hard-sphere interactions. The differential cross sections for these two cases are,

da,,,

?dT=

pm,( Z,Z,e*)* E

dT T*

(64

and T* - T,,$, WC(T) =

(6b)

T,2,

These are illustrated in Fig. 3 for 1 MeV particle irradiations of Cu. The Coulomb potential is a good approximation for proton irradiations while the hard sphere interaction well represents fast neutrons. Notice that protons do not simulate the recoil spectrum of fast neutrons, despite their equal masses. Nearly all of the defects produced by neutron irradiation are in cascades initiated by high energy recoils, whereas for protons most are produced in low-energy recoil events. Other 1 MeV particle irradiations are shown in the figure for comparison. The figure shows that heavy ions, like Kr, provide a far better approximation to neutron irradiation than do light ions, like protons or alpha particles. Consideration of the various primary recoil spectra shown in Fig. 2 would have given little indication of this fact. It has become common practice to characterize the weighted average recoil spectra through the parameter Tl,*. It is the value of T for which W(T) = 0.5, i.e., half of the defects are produced in cascades with energies greater than Tl,* and half in cascades with energies less than T,,,. Unfortunately, no single parameter provides a complete description of the recoil spectrum. For example, TI,*, provides no

1 .oo

0.80

(3

and

G g

0.60 0.40

(5b) They represent extremes because the Coulomb interaction, which extends to infinity, slowly increases as the collision partners approach each other. For the hard sphere interaction, the particles do not interact until they reach the hard sphere radius, at which point the repulsive force goes to infinity. Scattering cross sections for more realistic screened Coulomb potentials lie between the two; at far distances the electrons partially screen the nuclear charge and reduce the

IO’

102

103

lo4

1o6

1o6

T(eV) Fig. 3. Weighted recoil spectra for 1 MeV particles in Cu. Curves representing protons and neutrons (hard sphere), are calculated from Eq. (6a) and Eq. (6b), respectively. W(T) for other particles, Ne and Kr, were calculated using Lindhard cross sections and include electronic excitation.

52

R.S. Averback /Journal

of Nuclear Materials 216 (1994) 49-62

information about the width of the distributions (see Fig. 3). Implicit in this description of the primary recoil spectrum is that the “subcascade” produced by one recoil event does not interact with that of another. Although the primary recoil events are themselves localized at atomic sites, the secondary and higher order recoils emanating from the primary recoil sites create spatially extended subcascades. Whether the subcascades interact, therefore, depends on their spatial extent and their initial separation. This is indicated in the schematic drawing, Fig. 2a. Since the creation of primary recoils is a stochastic process, the distances between recoils of a certain energy is not fixed and only average distances between recoils, or mean free paths, A, can be calculated. The mean free path between recoils of energy greater than T is given by the expression, A(T)

=

1 &,a(E,

T)

dT

Fig. 4 shows the mean free atoms for self-ion irradiations and 1.0 MeV. For 0.3 MeV are very close to each other spectra has little significance.

IO5 t

(X) (AX*) = (X’)-(X)’ (AX’)/(X)* Y*/(AX*)

Skewness

s = (AX~)/(AX~)~/~

Kurtosis

K =

(AX4)/(AX2>’

think of the Au ion with 0.3 MeV as the primary recoil atom, itself, making a single, spatially connected, cascade. Conversely, for primary recoils greater than a certain energy, it would be more natural to include their “secondary” recoil spectra within their primary recoil spectrum. It should be clear from the above discussion that primary recoil spectra lend themselves to some ambiguity with respect to cascades, but generally no serious difficulties arise as long as the possibility for the formation of subcascades is kept in mind. 2.3. Damage

du(E, T)

.

paths of primary recoil of Si, Cu, and Au at 0.3 Au, the energetic recoils so that its primary recoil It is more reasonable to

I

I

Mean depth Variance Relative straggling Transverse relative straggling

(7a)

’

where N,, is the atomic density and the cross section for recoils greater than energy T is, o(E,T)=/TT”“dT

Table 1 Definition of moments

1

MEAN FREE PATH BETWEEN RECOILS OF ENERGY L To

I

/7

distributions

Since ion irradiations have been the principal means of studying radiation damage in recent years, it should be remembered that ions produce defects only near surfaces and very inhomogeneously. In this section a few useful expressions will be provided to estimate the depth distributions of the implanted ions and the damage they create. Unfortunately, these depth distributions cannot be expressed by a simple universal formula. In fact, computer simulations, like TRIM [7], are commonly employed to generate these distributions, although numerical solution of the Boltzmann transport equation provides an alternative means. For either case, it is convenient to find the moments of the distribution; this has been done by Winterbon using Boltzmann transport theory, and the results are tabulated for all possible ion-solid combinations [S] using the definitions in Table 1. Depth distributions can be subsequently generated from an appropriate expansion of the moments. For ion implantation and damage profiles, the Edgeworth expansion is generally employed, i.e.,

F(z) =d%(z) +

$43(Z)

+$f#J4(*)+g&(*)+..., (8) where IO

0.1

I

I 1.0

1

1

I

IO

1 4oq--q

To (keV)

z2 2

i

100

Fig. 4. Mean free paths of primary recoil atoms of 300 keV and 1 MeV, self-ion irradiations of Si, Cu and Au.

--

exp

’ 1

x - (x>

‘=-Jr

Yl=

P3/JPz,

Y2 =P4/(cL2)2

-

3,

R.S. Averback /Journal of Nuclear Materials 216 (1994) 49-62

Table 2 Moments for 10 keV Ge recoils in Ge [8]

c$~= jth derivative of the normalized Gaussian, pi = ith moment of the distribution. For many applications, &s(z) provides a reasonable approximation of the distribution. The volume of a cascade, V,, can be estimated using the second moments of the distribution, v, = ;((6AX)2+

2(sY)2)3’2,

where the “contraction” factors, 6, are included to relate single ion distributions to assembly averages [9]. These contraction factors are necessary because the moments generated by transport theory, or even computer simulation, refer to assembly averages and not the damage distribution of a single ion. The energy density within a cascade is given by, 0,

= ED/K.

53

(10)

Recoil energy 10

keV

En

(X>

/ W2

Y2/ (AX’>

7.5 keV

5.4 nm

0.387

0.509

As an example, consider lo-keV cascades in Ge. Relevant data are compiled in Table 2. The cascade volume and energy density are respectively, V, = 440 nm3 and 0, = 17 eV/nm3 (or 0.33 eV/atom). 2.4. l7zermal spikes An effective temperature estimated by the expression 3k,T,,

in the cascade

(11)

= @o,

Fig. 5. A series of snapshots of atoms in a 10 keV cascade in Au within a cross sectional slab of thickness a,/2,

direction. Each snapshot shows a different instant of time: (a) 0.62 ps, (b) 3.3 ps, (c) 5.0 ps, (d) 11.5 Ref. [ll]).

can be

ps,

(e)

17.7

viewed in the (100) ps,

(f)

23

ps

(from

54

R.S. Averback /Journal

of Nuclear Materials 216 (1994) 49-62

where the Dulong-Petit rule has been used for the lattice heat capacity, i.e., the harmonic approximation. For our example of 10 keV Ge, T,,,, = 1280 K, which is close to the melting temperature of Ge. Consideration of temperature implies that the local heating persists for at least a few lattice vibrations, otherwise defining a lattice temperature has little physical significance. The lifetime of the so-called thermal spike can be estimated by solving the heat equation for energy spreading sperically outward from a initially hot point source. The variance in the temperature profile, R*,is given by

R2=4Dt,

(12)

where t is time and D is the thermal diffusivity, i.e.,

D = K/C, where K is the thermal conductivity and C is the specific heat. The energy per atom in the cascade is

$R~~N~=E~,

(13)

where E is the energy per atom and N, is again the atomic density. Thus, the lifetime, T, is given by

(14) If we use the melting temperature as the “quench” temperature of the cascade, and estimate c = 0.3 eV, and D = lOi* nm*/s (this corresponds to the lattice conductivity, as we assume that electrons do not couple

b

Fig. 6. A series of snapshots of atoms in a 10 keV cascade in NiAl within a cross sectional slab of thickness, ao, viewed in the (100) direction. Each snapshot shows a different instant of time: (a) 0.1 ps, (b) 0.3 ps, (c) 0.5 ps, (d) 1.0 ps, (e) 2.0 ps, (f) 6.0 ps (from Ref.

[121).

55

R.S. Averback /Journal of Nuclear Materials216 (1994) 49-62

the hot phonon system, but see Ref. [lo]) then a cascade of 1 keV has a lifetime of about 1.5 X lo-‘* s, or a few lattice vibration periods. Although the model is crude, it illustrates that thermal spikes can be significant even in rather low-energy cascades. Two points are noteworthy, here. The energy density, as defined in Eq. (lo), is a monotonic decreasing function of energy. This occurs because the volume of the cascade expands faster than the damage energy increases. Notice, however, that even though 0, is a monotonic decreasing function of energy, the nuclear stopping power, S,, is not. S, increases with energy until the reduced energy reaches about 0.3 (see the paper by Robinson, Section 2.3, for the definition of reduced energy), and then it slowly decreases. The maximum in S, for Ge recoils in Ge occurs at about 85 keV, while for Au recoils in Au, the maximum occurs at about 700 keV. The second point is that the lifetime of a cascade is a monotonic increasing function of energy. Thus, if the effects of thermal spikes are to be important in a cascade, the lifetime must be long and the energy density must be high. Although the above discussion provides some physical insight into thermal spikes, the qualitative development employed here should be kept in mind. Various parameters, like the thermal diffusivity, are not well known under spike conditions, and the use of macroscopic concepts when the time and space scales are picoseconds, and nanometers, respectively, is certainly not justified. Other considerations such as supercooling of the liquid during solidification, the high pressure in the cascade, the atomic disorder introduced by the collisional processes, and the role of conduction electrons [lo] all have been neglected in this description. Currently, only molecular dynamics computer simulations can treat these highly inhomogeneous, many-body problems. The application of molecular dynamics for defect production studies is discussed in the paper by Diaz de la Rubia and Bacon; however, we cite a few examples here to clarify our remarks on cascade dynamics and thermal spikes. A clear picture of cascade dynamics is provided by Figs. 5 and 6. They show snapshots of the atomic positions at various instants of time for cascades in NiAl and Au [11,12]. Local melting in the center of the cascades of both metals is clearly observed, but it clearly far more pronounced in Au. The temperature and pressure profiles for Au are illustrated in Fig. 7. In the central core of the cascade, the temperatures far exceed the melting temperature, while the pressures are a few GPa. (For comparison the critical temperature of Au is about 9500 K and its shear modulus is 28 GPa.) In the Au cascade, the local density is greatly reduced; even cavitation is observed. For the NiAl cascade, the thermal spike is not nearly as dramatic. In fact, the complex shape suggests that the melt is confined to the vicinity of the most ener-

, .

1 10'

+-

. I-'-""" \

22

j

6000

$ C

4000

1.0 ps 4.9 ps -7 ll.Sps

2000 0 0

2

4

Distance ($

8

10

8 3 g r

6

f

4 2 0 0

2

4 Dis!ance(ae~

8

10

Fig. 7. (a) Temperature profile for the Au cascade shown in Fig. 5 (from Ref. [ll]). (b) Pressure profile for the Au cascade shown in Fig. 5 (from Ref. [ll]). a, is the lattice parameter.

getic recoils and does not form a homogeneous spherical melt as it did in Au. The difference in these metals, of course, is the higher atomic number in Au and higher melting temperature in NiAl. This cascade in NiAl is also interesting as it shows the efficient disordering in the cascade core where the melt occurred. The effects of thermal spikes on atomic displacement processes and atomic mixing will be discussed below. The main point here has been to provide a visualization of the thermal spike and local melting, and to demonstrate that the dynamics taking place in cascades are indeed complex and not easily modeled by analytical means.

3. Defect production 3.1. Defect production calculations - NRT

The most common means of calculating defect production in metals is by the method of Norgett, Robinson and Torrens (NRT) (13), which is essentially the

R.S. Averback /Journal

56

of Nuclear Materials 216 (1994) 49-62

modified Kinchin-Pease expression, integrated over the primary recoil spectrum of the irradiation particle, i.e., v”(E)

= N,,Ar/rm=

dT

WE,

dT

T ill,”

T) v(T),

(15) 44x)

where VP(E) is the number of Frenkel pairs produced in a material of thickness Ax by a primary particle, p, with initial energy E; [da(E, T)/dT]dT was defined above, and v(T) is the damage function. Usually the modified Kinchin-Pease expression for v(T) is employed in defect calculations,

VKP(T) =

0

for

T< Td,

1

for

Td< T<2ST,,

for

T < 2ST,,

(16)

0.8E,(T) 2E*

(

where E,(T) is the damage energy associated with a recoil of energy, T, Ed is the average displacement energy and Td is the threshold displacement energy. Division of Eq. (15) by NaAx, the number of atoms per unit area within a slab of thickness, Ax, yields the number of times each atom within the target is displaced per unit fluence and is called the “Frenkel-pair cross section,”

dTdu(E, T,

T,,,, (+FP =

/ rmill

dT

(17a)

v(T).

The product of the Frenkel-pair cross section and ion fluence, therefore, yields the number of times each atom is displaced during the irradiation, i.e., the number of displacements per atom (dpa). dpa = @uFp = @jTrn= dTdocd; r0lin

T, v(T).

(I7b)

The use of dpa thus provides a convenient means to normalize the fluence when comparing effects produced by different types of irradiation. Since ions slow down quickly in solids, their energy is not constant, and it becomes necessary to rewrite Eq. (15) using the relation dE dx =N,[S,(E)

+ S,(E)]

so that on substituting

up(E)= /,” is d

(18)

for Ax,

xv(T)

e

(19)

is obtained for the total number of defects produced when an ion is stopped within a material. Here S,(E) and S,(E) are the nuclear and electronic stopping powers, respectively. Calculations of dpa’s in this case

=

OW,(x) ~ 2E

N

,

d 0

(20)

where F,(x) is the amount of damage energy deposited in recoils above the threshold energy by the incident ion per unit distance normal to the surface. Notice that F,(x) is not equivalent to N,&(x) since the former is the energy deposition normal to surface while N,S,(x) is the energy deposition along the trajectory of the ion, and this trajectory may twist and turn. Eq. (8) or computer simulation must be employed for F,(x). It is convenient to think of Eq. (19) in three parts. The outer integral over E describes how the particle slows down in the material, depending only on the stopping powers. These functions are known reasonably well [14]. The differential cross section, which is essentially the primary recoil spectrum, describes the atomic collisions between the particle and the atoms in the target, and it too is accurately known. The last part of Eq. (19) is the damage function; it depends only on the material and is independent of the type of irradiation. Once the damage function has been specified, defect production can be calculated for any type of irradiation for which the primary recoil spectrum is available. Notice that the forms of Eqs. (17) and (19) are convenient for calculating other quantities in addition to defect production. For example, an ion beam mixing function Q(T) can be constructed, as will be shown below, and then substituted for v(T) in Eqs. (17) and (19) to calculate ion beam mixing for any type of irradiation. Thus, Eqs. (17) and (19) provide the quantity of interest averaged over the primary recoil spectrum. 3.2. The damage function The modified Kinchin-Pease expression for the damage function, which is represented by Eq. (161, was discussed in the paper by Robinson, section 2.2. The step-like behavior in the damage function at low energies is certainly unrealistic as the actual v(T) must increase far more gradually. Except for electron or gamma irradiations, where all recoils are close to threshold, the details of the threshold function are unimportant, as only the average displacement energy affects the integrals in Eqs. (17) and (19). The significant feature of the Kinchin-Pease expression is that the number of defects increases linearly with damage energy. That this should be so is not obvious from our description of cascade dynamics, and so we must ques-

,:--dTducdE; T, (Ep+Es (E)1

n

of ions slowing down or stopping within the target are more difficult, since defect production becomes a function of depth. The depth dependent equation for dpa is

RS. Averback /Joumal of Nuclear Materials 216 (1994) 49-62

57

tion whether it is true, what approximations in the theory lead to this result, and finally what is the correct picture of defect production in displacement cascades. 3.3. Experimental determinations

of the damage function

Surprisingly, the damage function of any material was not measured until 1978 [15], and since then it has been carefully measured for only a few additional metals. It has yet to be determined for a non-metal with any reliability. The difficulty in experimentally determining damage functions is two-fold. First, few methods can reveal the number of Frenkel pair defects in a material when their concentrations are low, C,, I lo-’ atom fraction. Second, no radiation sources can produce mono-energetic cascades in the interior of a material which range from some eV to several keV. The principal means of determining the number of defects in metals is by measurements of the electrical resistivity. It yields the defect concentration, C,, through the relation,

AP= PF&FP,

I I I1111

keV cascade on Cu, simulated by molecular dynamics, (after Ref. [16]).

ions. Thus, a damage function measuring the defect production

(21)

where pFP is the resistivity per unit concentration of Frenkel pairs. Unfortunately pFp is not accurately known for many metals; moreover, when defects are produced in clusters, as they are in cascades, pFp becomes dependent on the size and configuration of the cluster. Absolute errors in defect production measurements are therefore probably greater than 30%. The second problem of systematically varying the recoil energy is approached by irradiating with a series of high energy ions with different masses, or by using neutron sources with very different energy spectra. Notice in Figs. 2a and 2b that the recoil energies are lower for irradiations with light ions than with heavy

I

Fig. 9. Final locations of vacancies and interstitial atoms in a 5

I

I Illl1l~

l

I Il11ll~

I-

1.2 7

% d@ =PFP

-

VP(E) Not

can be obtained by rate for various ions,

(22)

and deconvoluting the integral in Eq. (19) to deduce the damage function. Here, t is the thickness of the specimen. This procedure is now standard for elucidating effects of recoil spectrum. Experimentally derived damage function are often expressed in the form v(T) = l(Q+r(T),

(23)

where t(T) is the so called efficiency function, giving the “efficiency” of defect production relative to the Kinchin-Pease model. It is shown in Fig. 8 for Cu as a function of TI,2. The main features are that the efficiency decreases from unity at low energies to about 0.3 for energies above about 5 keV. The constant efficiency at high energies is a consequence of subcascade formation. Increasing the recoil energy above 5-10 keV only increases the number of subcascades, with each subcascade having approximately the same efficiency. The decrease in the damage efficiency at low energies is somewhat more complicated, and a full understanding is only now emerging through use of molecular dynamics computer simulations, as is now discussed. 3.4. Results of molecular dynamics

Fig. 8. Efficiency function for defect production in Cu (Ref. WI).

In Fig. 9 the final locations of the interstitial atoms produced in a 5 keV cascade in Cu are shown [16]. Many of the interstitials were produced by well-defined replacement collision sequences and that all of the

R.S. Auerback /Journal

58

of Nuclear Materials 216 (1994) 49-62

interstitials lie on a shell surrounding the core of the cascade. The number of Frenkel pairs produced in this event was 24, which corresponds to N 0.25~~ and is in good agreement with the experimental value. The reason why the defect production is reduced appears to be a consequence of two effects. The first concerns the initiation of replacement collision sequences. Note that the threshold energy surface is determined from electron-irradiated samples where a small amount of energy is transferred to one lattice atom. The electron does not interact strongly with the other atoms so they remain close to their perfect lattice sites at the time of the recoil event, but allowing for lattice vibrations. Thus, the crystal is perfect during the initiation of the replacement sequence. It is known, however, that replacement sequences are reduced in length by lattice imperfections, including lattice vibrations at elevated temperatures. In the core of the cascade, where the replacement sequences are initiated, the lattice is disturbed by the faster moving recoils. Thus, only replace-

0.3

ps

(a)

ment sequences initiated at the periphery will propagate sufficiently far to create a stable defect [17]. The other reason that defect production is reduced in cascades involves the volume of instability for interstitial-vacancy pairs [15,16,18]. The criterion for the stability of Frenkel pairs in isolation is that the interstitial atom is deposited outside the spontaneous recombination volume of a vacancy. In a cascade, the criterion must be modified to one where the interstitial is deposited outside the melt zone (or agitated region). This latter mechanism was verified by the following experiment. First a sample was irradiated below stage I with protons, creating a high concentration of isolated Frenkel pairs. Subsequently, the sample was irradiated with self ions, so that cascades were superposed on the uniform background of Frenkel pairs. Initially, the concentration of defects decreased with fluence as the thermal spike of each cascade induced recombination of close Frenkel pairs [18]. A final observation in Fig. 9 is that many vacancies and some interstitials form

1

1.01 ps

9.6

ps

Cd)

Fig. 10. Sequence of snapshots of the atomic positions within a cross sectional slab of thickness a,,/2 bombardment of Au (after Ref. [19]).

during

10 keV Au

59

R.S. Auerback /Journal of Nuclear Materials 216 (1994) 49-62

Fig. 11. Complex dislocation structure due to 10 keV Au bombardment of Au. Atoms with potential energies greater than 0.45 eV relative to the perfect crystal potential are shown as they outline the cores of the dislocation structure. Surface atoms are indicated as they also satify this condition (after Ref. [19]).

clusters. Vacancy clustering is believed to occur as part of the solidification process. As the melt resolidifies, the vacancies are dragged along with the liquid-solid interface, similar to zone refining, and precipitate out in the center of the cascade where they can form clusters and even dislocation loops 1161.The formation of the interstitial clusters is less well understood, but it does not appear to be a consequence of loop punching.

center of the cascade, dragging with it free volume, or vacancies. When the concentration of free volume becomes sufficiently high, it condenses into a complex dislocation structure, as shown in Fig. 11. These results find good agreement with the TEM observations reported by JIger and Merkle on 10 keV Bi irradiation of Au in regards to the number of defects in the dislocation structure, the depth of the damage, and the probability for a dislocation loop to form [20]. The precise defect structures in the simulations and experiments are different, most likely because the simulations are performed with the lattice at 0 K while the experiments are performed at 300 K. Relaxation of the defect structure, therefore was possible for the TEM experiments but not the simulations. It is noteworthy that the damage process just described resembles void or crack formation during the casting of metals far more than it does the traditional concepts of atomic collisions in solids as discussed in the introduction.

4. Ion beam mixing The large number of ions set into motion in a displacement cascade gives rise to a “mixing” of atoms in the solid. Three distinct mechanism of mixing can be distinguished, recoil implantation, cascade mixing, and thermal spike mixing; these various contributions are schematically illustrated in Fig. 12.

3.5. Effects of surfaces on defect production 4.1. Recoil implantation

Much of the information about defect production in cascades and defect clustering derive from radiation damage studies using ion irradiations, particularly in connection with transmission electron microscopy. An important question is whether surfaces play a role in the defect production process, both for interpreting ion irradiation studies and for the importance of defect production near surfaces in its own right. Recall that all applications of ion beam modification of materials begins with an energetic particle passing through a surface. Recent computer simulations of 10 keV cascades in Au, suggest that surfaces can play a significant, if not primary, role in damage production in ion-irradiated metals [19]. The results are seen in Fig. 10 where a series of snapshots of atom positions are shown at various instants of time during the cascade evolution. What is observed is that local melting takes place in the near surface region, similar to that observed in the crystal interior. But, the high temperatures and pressures near the surface force hot liquid onto the surface by viscous flow. In the simulation, about 550 atoms flowed to the surface, so that when the local region resolidified, too few atoms were available to fill all lattice sites. Thus, as the melt quenches, the solid-liquid interface moves inward toward the

The first target atom impacted by an energetic particle recoils in the forward direction. I-his process of recoil implantation can be important for precipitate dissolution, for example the re-injection of helium into I

t

N

lo-l4’ +d

I recoil

implantation

coscode

mixing

thermol spike (melting)

Fig. 12. Schematic depiction of the three mechanisms of ion beam mixing: (a) recoil implantation; (b) cascade mixing; and (c) thermal spike mixing.

60

R.S. Averback /Journal of Nuclear Materials 216 (1994) 49-62

the lattice from gas bubbles, or preferential forward recoil of one component of a stoichiometric alloy. It is particularly important for ion implantation. For example, oxygen in thin oxide layers can be driven deep into a target and degrade electrical properties of shallow junction devices. For such a thin surface layer, the distribution of recoil implanted atoms can be approximated by the expression [21], S(x)

= N2’3 jdT

WE, 7’) dT

F(x -x’,

e),

x-(xi)cose =51i erfc

2((Axy> cos2e + 2(yf> sin28)1’2

* (25)

4.2. Cascade miring After several collisions, the momenta of the implanted ion and recoil atoms become randomized so that the relocation of most displaced atoms is a stochastic process. Calculations of the mixing due to the random relocation of atoms are difficult, requiring solution to the Boltzmann transport equation [22]. Accurate results, however, are more easily obtained from computer simulations based on TRIM [23]. The magnitude of cascade mixing can be estimated from the Kinchin-Pease expression as follows. We start with the expression for the mean square displacement of all atoms in the cascade, R2, R2 = nh2,

(26)

where n is the number of atomic jumps within the cascade and A is the jump distance. The number of jumps can be determined using the Kinchin-Pease expression, v=-

or

(24)

T)

0.8Eo (27)

2~5,

0.8/?E, p/P 2E,

(29a)

’

The number of replacements per displacement is not well defined here, but it may represent the number of replacements in a replacement sequence. By dividing both sides of Eq. (29a) by the number of atoms within the cascade volume, N,, the mean square displacement per atom, @‘I2 = R2/N, is obtained as a function of the damage energy per atom, EL = E&N,. For ion irradiations, Eb is readily obtained from Eq. (81, above. Note, F, is the damage energy deposition per unit length normal to the surface and @ is the ion fluence. Thus, the product @F, is the damage energy per unit volume and @F&N,, is the damage energy per atom. Thus (R’)‘=

0.4P@F, NE 0

h2.

Pb)

d

By dividing by @F,,, mixing in different materials can be readily compared on the basis of equal deposition of damage energy. We thus define the mixing parameter, Q = ( R’j2/@FD, where (30) If we employ typical values, No = 0.1 k, p = 4, A2= 8 k, Ed = 10 eV, we obtain Q = 10 k eV. This compares to values of 6 k/eV or 30 d 5/eV obtained using transport theory or TRIM, respectively. Experimental values for several metals are shown in Table 3. For metals that have average to low values of atomic number and relatively high melting temperatures, the cascade mixing theory yields reasonable agreement with experiment, but for those metals which have high atomic numbers and low melting temperatures, the calculated mixing values are far too small. Clearly, the

Table 3 Mixing parameter in several metals [24] Al

Ti

Fe

Z T, (K)

13 933

22 1933

26 1807

Q

120

24

36

&/eV)

Ni

CU

Ag

Pd

AU

Pt

1726

29 1357

47 1235

46 1825

79 1337

78 2045

42

135

450

72

720

114

28

per

(28)

R2=

where F(x -x’, e), the range distribution of a recoil atom, can be obtained from Eq. (8) with a correction for scattering angle, 8, relative to the surface normal, i.e., F(x,

by assuming that there are p “replacements” atomic displacement, i.e.,

R.S. Averback /Journal of Nuclear Materials216 (1994) 49-62

binary collision approximation of cascade mixing is deficient in those materials where thermal spikes are important. 4.3. l%ermal spike mixing Thermal spikes provide another contribution to atomic mixing. Calculations of thermal spike mixing have been performed using the expression, (Ax*)

=

2

/ ““‘d”;T,dT/NV(T,

t)D(T,

t) dt,

(31) where (Ax’) is the variance of a marker plane of atoms located at some depth, x, in the sample; IQ, t) and D(T, t) are the instantaneous volume and diffusion coefficient of the hot cascade at time t, initiated with recoil energy, T. The outer integral sums over the primary recoil spectrum. Since the diffusion coefficient of liquids is much larger than that of solids, most of the thermal-spike mixing takes place in the melt. Uncertainties in the cooling rates and appropriate diffusion coefficients make Eq. (31) difficult to evaluate, but it nevertheless shows that mixing in the thermal spike is large in metals which have low melting temperatures and high energy densities (high atomic number). Eq. (31) can be solved exactly in some limiting cases, as illustrated by Vineyard [25]. He solved the heat equation KV~T

=

CaT/ax,

(32)

assuming as a boundary condition that the energy was initially deposited along a line, i.e. F&z, p) = ES(P) and that the heat capacity and thermal conductivity of the solid could be expressed as K = K,,T~-’ and C = C,T”-‘. For the case II = 1, constant C and K, the temperature is given by,

W,f)=(&)exp($).

(33)

By further assuming that atomic motion is a thermally activated process in the cascade, Vineyard found for the total number of jumps per unit length of spike 5 = im27r, drj “A exp( -AH/kT), 0

where AH is the activation Integration of Eq. (34) yields

enthalpy

(34) for the jump.

61

the damage energy per atom. In addition, the mixing is inversely proportional to the square of the activation enthalpy, AH. For an atomic jumping process, AH is proportional to the melting point of the material. Although Eq. (35) provides the underlying physics of thermal spike mixing and indicates how mixing will vary from one material to another, it suffers from the same shortcomings of the other thermal spike models noted above, namely, it cannot be used for quantitative calculations. Again, molecular dynamics is required. Such calculations have attained good agreement with experiment (see the paper by Diaz de la Rubia and Bacon).

5. Future directions Atomic displacement processes in metals are now understood in rather good detail. A large part of the progress in this field has been made possible by the recent use of computer simulations. Although simulations were first employed for this purpose around 1960 [3], the development of reliable potentials and the enormous improvements in computer capabilities, both in hardware and software, make them both reliable and practical. The simulations make two important contributions: first they provide the atomistic detail needed to understand atomic displacement processes; second they place in the hands of the experimentalist, powerful means to analyze their data, e.g. TRIM. Nevertheless, many important questions remain unanswered: the role of conduction electrons on thermal spikes, the behavior of cascades in intermetallic compounds, concentrated alloys, and amorphous alloys, the effects of interfaces, such as surfaces and second phase particles. Moreover, extremely little is known about defect production processes in non-metals. In short, the understanding of atomic displacement processes is on a very sound basis and understood in some detail in the most simple systems. What is required now is to add the complications of technologically important materials, including both metals and non-metals.

Acknowledgements The author is grateful to the DOE, Basic Energy Sciences under grant DEFG02-91ER45439 for support of this work.

References

Note that in Eq. (35) the number of jumps per atom is not proportional to the damage energy per atom, as found for collisional mixing, but rather to the square of

[l] A. Seeger, Proc. 2nd UN ht. Conf. on Peaceful Uses of Atomic Energy, Geneva, 1958, vol. 6 (United Nations, New York, 1958) p. 250.

62

R.S. Averback /Journal

of Nuclear Materials 216 (1994) 49-62

121 J.A. Brinkman, J. Appl. Phys. 25 (1954) 951; Am. J. Phys.

24 (19.56) 246. [3] J.B. Gibson, A.N. Goland, M. Milgram and G.H. Vineyard, Phys. Rev. 120 (1960) 1229. [4] R.H. Silsbee, J. Appl. Phys. 28 (1957) 1246. [5] F. Seitz and J.S. Koehler, Solid State Phys. 2 (1956) 307. [6] G.H. Kinchin and R.S. Pease, Prog. Phys. 18 (1955) 1. 171J.P. Biersack and L.G. Haggmark, Nucl. Instr. and Meth. 174 (1980) 257. [8] See e.g., K. Winterbon, in Ion Implantation Range and Energy Deposition Distributions, vol. 2 (IFI/Plenum, New York, 1975). [9] P. Sigmund, G.P. Scheidler and G. Roth, Proc. on Solid State Research with Accelerators, ed. A.N. Goland, Brookhaven National Laboratory report BNL 50083 (1968). [lo] C.P. Flynn and R.S. Averback, Phys. Rev. B38 (1988) 7118. [Ill Mai Ghaly and R.S. Averback, unpublished. [12] H. Zhu, R.S. Averback and M. Nastasi, unpublished. [13] M.J. Norgett, M.T. Robinson, and I.M. Torrens, Nucl. Eng. Des. 33 (1974) 50. [14] See e.g., J.F. Ziegler, J.P. Biersack and U. Littmark,

Stopping and Range of Ions in Solids vol. 1 (Pergamon, New York, 1985). 1151 R.S. Averback, R. Benedek and K.L. Merkle, Phys. Rev. B18 (1978) 4156. [16] T. Diaz de la Rubia, R.S. Averback, R. Benedek and W.E. King, Phys. Rev. Lett. 59 (1987) 1930. [17] A.J.E. Foreman, CA. English and W.J. Phythian, Philos. Mag. 66 (1992) 655. [18] R.S. Averback and K.L. Merkle, Phys. Rev. B16 (1977) 3860. [19] Mai Ghaly and R.S. Averback, Phys. Rev. Lett. 72 364 (1994). [20] W. Jiiger and K.L. Merkle, Philos Mag. A44 (1981) 741. [21] R. Kelly, in Ion Beam Modifications of Materials, ed. 0. Aurillio and R. Kelly (Elsevier, Amsterdam, 1984) p. 38. [22] P. Sigmund and A. Gras Marti, Nucl. Instr. and Meth. 182/183 (1982) 25. [23] W. Moller and W. E&stein, Nucl. Instr. and Meth. B7/81 (1985) 645. [24] S.-J. Kim, M.-A. Nicolet, R.S. Averback and D. Peak, Phys. Rev. B37 (1988) 38. [25] G.H. Vineyard, Radiat. Eff. 29 (1976) 245.