Sputtering from elastic-collision spikes - spherical geometry

Nuclear Instruments and Methods 194 (1982) 567-571 North-Holland Publishing Company

SPUTTERING

FROM

ELASTIC-COLLISION

567

SPIKES

- SPHERICAL

GEOMETRY

*

C. C L A U S S E N Fysisk [nstitut, Odense Universitet, DK 5230 Odense hi, Denmark

Elastic-collision spikes in heavy-ion-bombarded metals account for considerably enhanced sputtering yields as compared with predictions of linear collision cascade theory. Using a model proposed by Sigmund, we have previously considered an idealized spike of cylindrical symmetry. This contribution deals with the other extreme, that of an initially spherical spike. Explicit expressions are given for the sputtering yield as well as the spectrum of sputtered particles. It may be concluded that the cylindrical spike model gives the more satisfying description in most cases.

1. Introduction

It is well documented that bombardment of metals and semiconductors by keV heavy ions may create conditions where linear cascade theory is not sufficient to describe the sputtering process. Furthermore, it has been shown [I ], that enhanced yields may be ascribed to elastic-collision spikes, i,e., volumes of a high density of deposited energy with the majority of atoms temporarily in motion. Provided the lifetime of this transiently heated region is large there may be considerable thermal evaporation from the surface. This spike model has previously been used to evaluate sputtering yields and energy spectra for a system with the simplified geometry of a cylindrical spike, surrounding a straight ion track [2]. Good qualitative agreement was obtained between calculated and measured [3] sputtering yields for Sb + , Sb~-, and Sb~ ions bombarding silver. The cylindrical spike model refers to an idealized system where energy is deposited at a constant rate along the ion track. Generally, however, the depth distribution of deposited energy is spatially varying, both in depth and in width (fig. l). It is therefore of interest to consider the other extreme, that of a point-like initial distribution of deposited energy. The evolution of the spike in space and time is then described by a spherically symmetric profile. Sputtering from spherical spikes has previously been considered by Johnson and Evatt [4]. In the same way as Kelly [5], they use the equilibrium vapour pressure to evaluate the number of evaporated atoms during the decay of the spike (i.e. the yield). Following a method proposed by Vineyard [6] they determine a general

* Work supported by the Danish Natural Science Research Council (Statens naturvidenskabelige Forskningsrhd). 0029-554X/82/0000-0000/$02.75 © 1982 North-Holland

expression for the sputtering yield, containing a double integral that was not evaluated. The present model differs from the models mentioned above by the use of an ideal-gas description of the evaporation process [2,7], using the flux at constant volume rather than that at constant (vapour) pressure. By means of Kapitza's procedure [8], the integrations are reduced to a single, numerical integration. Explicit results are given for the present model as applied to the antimony-silver system. Although this model is not optimal in describing sputtering yields under spike conditions it should be interesting to look for effects of the predicted, very large surface temperatures. These may be relevant for, e.g. the study of nonvolatile organic molecules by secondary ion mass spectrometry [9].

deposited energy

sphericalspike

~F(x

cylindricalspike )

X

-

deplh x

Fig. 1. Depth distributions of deposited energy (schematical). Two idealized distributions are readily available for comparison with the "true" distribution, FD(X ). The first is the depth independent distribution, which was used in a previous calculation for the cylindrical spike [2]. The second is the point source distribution, cf. eq. (2), which will be used here for the spherical spike. X. CASCADES AND SPUTTERING

C. Claussen / Sputtering from elastic-collision spikes

568

2. The surface temperature profile

a

Ts(t.f-0)

T°

According to the present model [7] the spike volume contains a dense gas of target atoms moving with average energies up to some eV. It is assumed that the velocity distribution of these atoms is isotropic and Maxwellian [1]. The local temperature T is determined from the density of deposited energy, 8, by the ideal-gas relation # = azNkT, where N is the number density of target atoms and k Boltzmann's constant. Energy transport within the spike is determined by a temperature-dependent thermal diffusivity [7] •(T)-

25 48

1

(kT)

,/

~xt-6 7 05T"

0 l0

t"

(1)

b

T( t, r) --- t-6/7[A - r2/ (14 Xt4/7) ] 2,

E,

(3)

and E is the total deposited energy. It is assumed that the initial temperature profile does not reach the target surface and is located at a depth X below the surface. The determination of a pertinent value for X will be described below. In order to calculate the sputtering yield of a spherical spike we must determine the surface temperature profile corresponding to the bulk profile T(t, r). As the thermal energy spreading may be considered a zeroth order process and sputtering a first order process, we may as a lowest order approximation neglect the energy loss via sputtered particles. Within the thermal description this means that the surface is totally reflecting. This is in ordinary linear heat conduction theory equivalent to using an image source [!1]. By use of the same procedure here, the resulting surface temperature profile is

- ~(t/t*)

4/7(I -{- io2/X2)] 2, (4)

where p is the radial coordinate and

kT*=O.128E/NX 3 , t*=O.135X2/~(T*).

5t"

(2)

where r is the distance from the centre of the spike. The temperature profile thus cuts off to zero at that distance where the term in brackets vanishes. H e r e , X = ~ T - I / 2 , while ,4 : X 3/7(5E/ 321~Nk) 2/7 is a normalization constant determined by

: T*(t/t*)-6/7[~

t

T" ~ ('1'9)

where M i s the atomic mass and A ~- 24, and a "~ 0.219,A are Born-Mayer constants. Solving the equation for heat conduction for a spherical spike in an infinite medium and using the thermal conductivity above, we obtain the temperature profile [10,4]

:

3t"

'/2

Nha 2 ' ~

~Nl~f d3rT( t, r )

21"

i

x

0

2x

This surface temperature profile is shown in fig. 2. Fig.

L

9

Fig. 2. The surface temperature profile. Part a shows the time dependence of the temperature at p=0. Part b gives the radial temperature profile at various times t/t* =0.5, 1, 5. and 10. For Sb + ions bombarding Ag at 5 keV the characteristic values are X~23 ,~, kT* ~'-0.89 eV, and t * - 2 × 10-12 s.

2a shows how the temperature of p -- 0 rises very quickly and then decays very slowly. This behaviour may be compared with that calculated by Kelly [5], where the peak temperature is generally a decaying function of time. Fig. 2b shows correspondingly how the profile expands in space.

3. The sputtering yield The thermal sputtering yield is

Ym= f°°dtlC*~rdp2dP[~(t,P)], to

(6)

0

where t o = (3/7)7/4t * is the time when the temperature enhancement first reaches the surface. The exact value of the lower limit of the integration over time is immaterial as long as it is less than t*, because the decaying part of the distribution is dominant. As in previous work [I,2] the evaporation rate

q~(T) = N(kT/2~rM),/2 e x p ( - U/kT) (5)

3x

(7)

is that of an ideal gas confined by a planar surface potential U.

C. Claussen / Sputtering from elastic-collision spikes After introduction of the temperature as an auxiliary variable [8], the integration over p may be replaced by an integration over temperature:

1

.

.

.

.

569

.

.

.

f~(~)

Yth : ~-2~rS2N( kT" 2)/l~f'tdoftm 'fd('i1"exp(-~)2~rM, o 10-1

(8) where t-~tz/t* and /r.~ T / T * are dimensionless variables, and t o = to~t* = (3/7) 7/4, ]Fm(t") = T~(t, O ) / T * =/'-6/7(42 -- ¼ [ - 4 / 7 ) 2

(9) 10-2

and

(10)

=- U/kT* = 7.838NX3U/E.

After carrying out the integration over 7~ one is left with a single integral over/'[eq. (12) below], which has been evaluated numerically. The thermal sputtering yield is then found to be

Yth = 0.7479N2/aAa

2(E/U)4/3gs(~)

'

10 -3

i

0

1

,

i

2

,

i

A t

3

,

4

I

,

I

5

L I

6

,

I~

7

(11) Fig. 4. The function fs(~'), eq. (14).

where gs(~) ~--0.1644~ 7/3

× f~dt't'( exp[-~/7~m(t')] ,0

-

E,[¢/7"m(;)]) (12)

and

exp(-t)/t.

E,(x) = f ~ d t

The universal function g~(~), which is chosen to have gs(0) = 1, is shown in fig. 3 as computed by GaussChebychev quadrature, [12]. In the same manner one may calculate the differential sputtering yield Yth(E',0') d E ' d2£ ' for atoms emerging with energy ( E ' , d E ' ) and angle (0', d2fl'). Inserting the differential evaporation rate per energy and solid angle equivalent to eq. (7), [2] into eq. (6) we find Yth(E',

1

•

I

'

[

'

t

'

i

'

0')

d E ' d2£ '

-_ 0.2620N2/3~ot2E4/3

E' d E' ( e ' + u ) '°/3

xf~( E' + U 1 cos 8' d2£ ', k

(13)

where the universal function -1

3

lO

OO

_~

f~(~') = 0"0529~'7/ f;o d t t exp[-~'/7"m(/')]

(14)

is shown in fig. 4 as computed by Gauss-Chebychev quadrature. The functions gs and f~ are very similar to the corresponding functions for the cylindrical spike [2]. In both cases the two functions are close to decaying exponentials over a wide range of arguments, the f-functions decaying more slowly than the g-functions.

10 -2

10 -3

,

o

I

1

~

I

2

r

I

3

~

I

4

Fig. 3. The function gs(~), eq. (12).

4. Examples of calculated yields In order to calculate the thermal sputtering yield of a spherical spike we must fix the value of the depth X. X. CASCADES AND SPUTTERING

C Claussen / Sputteringfrom elastic-collisionspikes

570

5. Discussion Y

103

Qr i= 3 In -2..-"

102

F

I

lo

loo

ENERGY PER ATOM (keY)

Fig. 5. Calculated thermal sputtering yields for Sb+ ions bombarding Ag. The dashed curves are the yields calculated according to the cylindrical spike model [2]. The experimental points are from measurements by Thompson and co-workers [31.

This is of course a fluctuating quantity, being determined by the statistics of the energy deposition process, and it is important to take these fluctuations into account [13]. The depth x enters the e~pression for the yield only via the gs-function, and only in the combination ~ -- cX 3 (where c is a constant at fixed energy). The average of the yield over the energy deposition profile is then found by averaging the g~-function, which is approximately *

(gs(4))-~g,((4)) = gs(c
First of all the spherical spike model predicts giant sputtering yields ( ~ 103), especially for the Sb3~ ions. This of course means that the sputtering process is not a very weak perturbation of the system; therefore, some concern about the cooling by sputtering is indicated. This is especially true at the smallest energies (below the yield maxima). Because of this problem the discussion below focuses on selected features for qualitative comparison. There is obviously a strong molecular effect, which for Sb,+ ions gives yield ratios ranging from Yt~'): Yt~2): Yt~3) = 1 : 15 : 50 at 5 keV per atom to 1 : 30 : 300 at 10 keV per atom. This is due to the g~-function which is responsible for the increase in sputtering yield with increasing surface temperature. The yield is a strongly varying function of the angle of incidence 0. This is illustrated in fig. 6, which shows the relative variation of the thermal sputtering yield Yth(O)/Yth(O = 0) as a function of the angle 0. This ratio of yields is the same as the ratio of g~-functions gs[~(O)]/g~(4o), where 4o = 4(0 = 0) is the value of 4 at perpendicular incidence. The function 4(0) may be calculated from the Legendre polynomial expansion of the spatial moments [14]:

4 ( 0 ) = 4o A(')~el(c°s Ol + A(''~P3(c°s O)

(17)

A(i)~ + All)~ where Ao)~, A,i)~ may be determined from the tables in ref. 14 and P~(cos 0) is the lth Legendre polynomial. The strong dependence of the thermal sputtering yield on the angle of incidence is probably the most characteristic feature of the spherical spike model.

(15) Yth(O)

The average ( X 3) is

~o =4

..4-Ax)3> =(x) 3

1+3

(ax:)o (ax3)D)' (x)~ + (x) 3

(16)

which may differ considerably from ( x ) 3 (by a factor 2.5 for intermediate mass ratios, [14]). Examples of thermal sputtering yields calculated according to the expressions above are shown in fig. 5 for the antimony-silver system. At a fixed energy E per atom in a Sb,+ ion, n = 1, 2, 3, the same depth X applies for all n, whereas the total deposited energy is E (') = nE and correspondingly for the 4-parameter 4 (") = 4In.

* It has been pointed out by Winterbon that this need not be the most appropriate average. This point is presently under investigation and will be dealt with in a future publication.

~ --. L 00='

1j(cosO)-% I

I0 °

~

I

200

~

I

500

~

I

40 °

angle ~ of incidence,8

Fig. 6. The relative variation of the thermal sputtering yield with the angle of incidence O. calculated for two values of the ,~-parameter for perpendicular incidence. For comparison the dashed curve is included showing the (cos 0) 5/3 characteristic of sputtering from linear collision cascades [I 5].

C. Claussen / Sputtering from elastic-collisionspikes The maxima of the yields are obviously at very low energies and this seems to be a general feature of the model. Comparing the spherical model and the cylindrical model [2] you may therefore conclude that the latter gives the more adequate description of the meausred yields. I should like to thank Peter Sigmund for his continuing interest and thorough criticism, which has made this work possible. I also acknowledge discussions with my colleagues from the Danish Sputtering Club, especially Hans Henrik Andersen. I thank Bruce Winterbon for his comments on a previous version of this paper. I gratefully appreciate fruitful conversations on numerical computations with E d m u n d Christiansen and Jan Emil Larsen. This work has been made possible through a research grant from the Danish Natural Science Research Council (Statens naturvidenskabelige Forskningsrhd).

References

571

[2] P. Sigmund and C. Claussen, J. Appl. Phys. 52 (1981) 990. [3] S.S. Johar and D.A. Thompson, Surf. Sci. 90 (1979) 319; D.A. Thompson, J. Appl. Phys. 52 (1981) 982. [4] R.E. Johnson and R. Evatt, Rad. Effects 52 (1980) 187. [5] R. Kelly, Rad. Effects 32 (1977) 91. [6] G. Vineyard, Rad. Effects 29 (1976) 245. [7] P. Sigmund, Appl. Phys. Lett. 25 (1974) 169; 27 (1975) 52. [8] P.L. Kapitza, Phil. Mag. 45 (1923) 989. [9] A. Benninghoven and W.K. Sichtermann, Anal. Chem. 50 (1978) 1180. [10] C. Claussen, P. Hansen, S. Hoe, E. Olsen and P. Sigmund, unpublished report, Physics Lab. II, H.C. Orsted Institute (1977); Y.A. Zel'dovich and Yu. P. Raizer, Physics of shock waves and high-temperature hydrodynamic phenomena, Vol. II (Academic, New York, 1967) p. 670. [1 I] J. Crank, The mathematics of diffusion, 2nd ed. (Clarendon, Oxford, 1975) p. 13. [12] M. Abramowitz and I.A. Stegun, eds., Handbook of mathematical functions (Dover Publications, New York, 1965) sect. 25.4. [13] A.R. Oliva-Florio, E.V. Alonso, R.A. Baragiola, J. Ferron and M.M. Jakas, Rad. Effects Lett. 50 (1979) 3. [14] K.B. Winterbon, P. Sigmund and J.B. Sanders, Mat. Fys. Medd. Dan. Vid. Selsk. 37 (1970) no. 14. [15] P. Sigmund, Phys. Rev. 184 (1969) 383; 187 (1969) 768.

[1] P. Sigmund, in: Inelastic ion-surface collisions, ed., N.H. Tolk et al. (Academic Press, New York, 1977) p. 121.

X. CASCADES AND SPUTTERING