New analytical calculation of displacement damage

Nuclear Instruments and Methods 170 (1980) 389-392 © North-Holland Publishing Company

NEW ANALYTICAL CALCULATION OF DISPLACEMENT DAMAGE G.P. MUELLER Radiation-Matter Interactions Branch, Radiation Technology Division, U.S. Naval Research Laboratory, Washington, D.C. 20375, U.S.A.

A new approach to the calculation of displacement damage using analytical transport theory is presented. This approach is a modification of the standard Lindhard, Nielsen, Scharff and Thompsen (LNST) method; the principal difference is that we do n o t neglect the displacement energy and its effect on energy partitioning. In addition, we use a slightly different method of calculating the number o f displacements than is usually used and we adopt a useful approximation for treating the transfer of energies less than the displacement energy. The damage energy and number of displacements produced by an ion of given energy are separately evaluated. After outlining the derivation of the new approach, comparisons are made with the standard (LNST) results. Various interatomic potentials are then used to explore the sensitivity of the radiation damage estimates to the form of the interaction.

ally called the damage energy u(E) = E - r~(E). An important assumption of our model is that bound or captured ions with energies less than Ed suffer no inelastic losses; all of their energy is shared thermally. Fig. 1 summarizes the possible collisions in our model. We see the thermal loss, vacancy production

1. Introduction

One of the few measures of radiation damage in crystals that relates damage at the microscopic level to the gross behavior of the material is the number of Frenkel pairs produced by a given irradiation [1,2]. Even this measure is somewhat inadequate, in that many of the pairs quickly recombine, but the expectation is that the number of vacancies produced by an irradiation is a rough indication of the extent of damage to the material. Consider an amorphous, monatomic, substance of atomic charge Z and mass A. Suppose a displaced lattice atom of energy E collides with another lattice atom, transferring an energy T to it. We assume there is a displacement energy Ed, such that if T < E d , the struck atom stays in place and shares its newly gained energy with the lattice. If T > E d , then the struck atom is displaced and procedes to move through the crystal. Further, if the incoming ion transfers an energy T > E d , but retains an energy E - T < E d , then it replaces the struck atom at the lattice site. We assume an isotropic value for the displacement energy E a and also assume that there is no binding energy associated with the removal of an atom from its lattice site. Given an ion of initial energy E, a portion of that energy labeled ~7(E), will go to exciting the electrons in the solid. The rest of the energy is tradition-

6

--

_~N ~'~. 'N \

4

~Y

\

N NN." ~,x,<,ca ~

-,~

I

N \ N

N

\ N NN

REPLACEMENTS

o

1

1

I

2

4

\

Ix.

I--

6

TIE d

Fig. I. The types of elastic collisions that occur as a function of energy retained and energy transferred. 389

IX. RADIATION DAMAGE

G.P. Mueller / Calculation of displacement damage

390

and replacement regions of the (T, E - T) plane. For an ion of energy E, all of its possible (elastic) collisions lie on a 45 ° line, such as the dashed line shown for E = 5.5Ea. After outlining the formalism that leads to the integral equations for N(E), v(E) and r/(E), we will compare our results with the standard analytical theory of Lindhard, Nielsen, Scharff and Thomsen (LNST) [3]. We will sometimes use reduced units (e =E/EL) for the energy; EL is just EL=2Z2e2/a. The screening length a may be supplied by a theoretical expression [3,4] or it may take some phenomenological value. Details of the derivations as well as more detailed comparisons with other calculations will be made available elsewhere [5].

2. The integral

equations

Early calculations of displacement damage (see ref. 1 for a brief review), calculated the number of displacements N(E) produced by a primary knock-on atom (PKA) of energy E. More recently, estimates have commonly been made by using the expression

N(E) = Kv (E)/(2Ed) ,

(1)

where K ~ 0.8 [6], and the damage energy v(E) is usually taken from the solution of the LNST integral equation [3,7,8]. The advantage of the LNST approach is the inclusion of inelastic (electronic) losses, but they take Ea = O, so that they cannot calculate N(E) directly. We start with the LNST form of integral equation and make several modifications. We retain a finite displacement energy throughout the calculations, and we decompose the elastic cross section into the three parts whose domains are shown in fig. 1. We also modify the integral equation so as to allow separate calculations of the number of vacancies and the number of replacements. Finally, we treat the thermal losses in the same manner that LNST treat inelastic losses, which simplifies solving the integral equation. This approximation is worst for E -~ Ea; in copper, for example, it produces an error in r/(E) of 5% at E = Ea. There are separate Integral equations for rl(E), v(E), N(E), the number of replacements, and for

binding energy losses, if a non-zero binding energy is used. For the remainder of this report, we will concentrate on the first three of these [5].

3. Comparison to LNST

The first point to note in comparing our formalism with that of LNST is that they coincide in the limit that Ea vanishes; our integral equation for ~(E) is identical to the LNST equation under this condition. In fig. 2 we compare our results for the fraction of energy lost to inelastic processes with those of LNST. (In these comparisons, we use the Winterbon, Sigmund and Sanders (WSS) [8] version of the Lindhard, Nielsen and Scharff cross section [9], which is based on the Thomas-Fermi potential.) The solid curve is the LNST equal mass result of Winterbon [I0], plotted as a function of the reduced energy E/EL. The present theory is represented by three curves, obtained by solving our new integral equation using a cubic spline-Gauss quadrature method [11]. In the case of copper the three values of Ea/EL correspond to Ed =0.14 eV, 14 eV and 1400 eV. Even when E a =0.14 eV, there is a marked difference between our results and the Ed = 0 results of the LNST method. The explanation of this effect is that in our formalism, for energies below Ed, certain modes of energy loss are prohibited. Most ions with energies below Ed are tied to lattice sites, either by way of having been a lattice atom that received a subthreshold (thermal) energy transfer or because they arrived at a lattice site as a replacement. In both cases, in our model, they are incapable of losing energy to electronic processes. Changing Ea makes little difference: halving the displacement energy doubles the number of displaced atoms; the total damage energy changes little. In the LNST model ions continue to lose energy electronically until they have zero energy. Our model of low energy losses may or may not be more realistic that that of LNST, but we find it disturbing that a small change in the model conditions has such a large effect on the estimated damage energy. The number of vacancies produced is not affected, of course, by the form of these low energy losses.

G.P. Mueller / Calculation of displacement damage

391

0.1

_

o,o

10-4

I

I

10-3

I

I

I

10-2

10-1

E = E/E L

Fig. 2. Comparison of inelastic energy loss. The solid curve represents the LNST results of Winterbon; the dashed curves represent the present approach for three different values of the displacement energy.

4. Effect o f potential As we have described, the conventional LNST energy partitioning theory is based on the T h o m a s Fermi interaction. One may ask whether the partitioning and displacement damage estimates depend critically on the potential used. While analytical transport theories have used the T h o m a s - F e r m i cross section (usually the WSS form), simulation calculations [12] often use the Moli~re potential [13], which is considered to be more realistic [14]. Upon using the cross section [15] corresponding to the Moli~re potential, we see in fig. 3 the effect o f changing the interaction on energy partitioning. (All o f the curves are based on the formalism o f this paper.) The solid curve represents the results o f using the WSS cross section, with the LNST screening length a = 10.78 pm. The two dashed curves represent the results o f using the cross section based on the Moli~re potential, b o t h for the LNST screening length and the Torrens and Robinson value (7.38 pm) [16]. We see that there are large shifts in the damage energy, first from switching from the T h o m a s - F e r m i to the Moli~re potential, and then another caused b y changing to the smaller screening length.

We can make another comparison o f the effects o f changing the potential be examining the number o f vacancies produced in each case. In fig. 4 we plot the number o f vacancies produced per unit energy [N(E)/E] as a function o f energy. The two potentials that use the 10.78 pm screening length are in rough agreement at all energies, presumably because the two

0.9 --

a

lO.78 Drn

i 0.8 07 i

0.6

a ~

0.5

~

o.4

THOMAS FERMI . . . . a"~7~3~P~n)-- _

0.3

0.1

1.0

10

t

100

ENERGY[E(keVH

Fig. 3. Fraction of energy going into damage energy for three cross sections: the Moli~re cross section with two different screening lenths and the Thomas-Fermi cross section. IX. RADIATION DAMAGE

G.P. Mueller / Calculation of displacement damage

392 7----

[ 15

/

f

\x xx

References

[

-

THOMAS-FERMI

.......

--

MOLIERE

z ~

c,_

i

I 0.1

0.3

I

I

1.0 ENERGY

3.0

10

[E(keV)]

Fig. 4. Number of vacancies produced per unit energy for each of three cross sections; the Molidre cross section with two different screening lengths and the Thomas-Fermi cross section.

potentials coincide at higher energies. Upon comparing figs. 3 and 4, we fail to see any clear relation between damage energy and number of vacancies, relative to changes in the potential. To a fair degree, energy partitioning depends on the low energy-transfer portion of the cross section and displacement damage on a higher energy portion. We thus have a second fault with the use of damage energy to estimate displacement damage: the two quantities have different dependencies on the form of the interatomic potential.

[1] M.T. Robinson, in Radiation induced voids in metals, Atomic Energy Comm. Syrup. Series No. 26, ed. J.W. Corbett and L.C. IannieUo (U.S. Atomic Energy Commision, Oak Ridge, 1972) p. 397. [2] Chr. Lehmann, Interaction of radiation with solids and elementary defect production (North-Holland, New York, 1977). [3] J. Lindhard, V. Nielsen, M. Scharff and P.V. Thomsen, Mat. Fys. Medd. Dan. Vid. Selsk. 33, No. 10 (1963). [410.B. Firsov, Zh. Eksp. Teor. Fiz. 33 (1957) 696 (Soy. Phys. JETP 6 (1958) 534). [5 ] G.P. Mueller, to be published. [6] M.J. Norgett, M.T. Robinson and I.M. Torrens, Nucl. Eng. and Design. 33 (1975) 50. [7] P. Sigmund, M.T. Mathies and D.L. Phillips, Rad. Effects 11 (1970) 39. [8] K.B. Winterbon, P. Sigmund and J.B. Sanders. Mat. Fys. Medd. Dan. Vid. Selsk. 37, No. 14 (1970). [9] J. Lindhard, V. Nielsen and M. Scharff, Mat. Fys. Medd. Dan. Vid. Selsk. 36, No. 10 (1968). [10] K.B. Winterbon, Ion implantation range and energy deposition distributions Vol. 2 (IFI/Plenum, New York, 1975) p. 199. [11] G.P. MueUer and M. Rosen, Naval Res. Lab. Memorandum Report 3942, 5 March 1979. [12] M.T. Robinson and I.M. Torrens, Phys. Rev. B 8 (1974) 5008. [13] G. Moli&e, Z. Naturforsch. 2a (1947) 133. [14] M.T. Robinson, Phys. Rev. 179 (1969) 327. [15] G.P. Mueller, to be published; G.P Mueller, Naval Res. Lab. Rep. 8207, May 1978. [16] I.M. Torrens and M.T. Robinson, ref. 1 p. 739.