IPPJIAM･ 26

ANGULAR DEPENDENCE OF SPUTTERING YIELDS

OF MONATOMIC SOLIDS

yasumichi Yamamural), yukikazu ltikawa2) and Noriaki ltムh3)

Institute of Plasma Physics, Nagoya University

Chikusa-ku, Nagoya 464, Japan

June 1983

Permanent address:

1)Department of Applied Physics, Okayama University of Science

2) Institute of Space andAstronautical Science

3) Departmen'. of Crystalline Materials Science, Faculty of Engineering, Nagoya University

This document is prepared as a preprint of compnation of atomic data for fusion

research sponsored fuuy or partly bythe WP/Nagoya University. This is intended for

future Pubhcation in a JOumalor will be included in a data book after some evaluations

or rearrangements of its contents. This document should not be referred without the

agreement of the au也ors. Enquiries about copyright and reproduction血ould be ad･

dressed to Research lnformation Center, IPPrNagoya Umiversity , Nagoya, Japan.

Abs七rac七

The angular dependence of sputtering yields of light-ion sputtering

and heavy-ion sputtering has been investigated in detail , and the

●

following empirical formula is proposed･.

Y(0)

tf exp卜=(モー1)】 .Y(0)

where 8 is the angle of incidence measured from七be surねce normal.

and t=

i/cos 8. The parameters f and ∑ are adjustable parameters

which are determined by the least-square method so as to fit the present

empirical formula to available experimental data.●

The best-fit parameters and their average values are listed in

七ables′ wllere the value of Oopt is listed土n place of ∑･ Here′ Oopt is

the angle of incidence at the maximum yield and the relation between

8opt and ∑ is simpler i･e-

∑ ≡ fcos Oopt･

The present empirical formula has been compared with available●

experimental data of various ion-target combinations, and it is found

that the agreement is satisfactory in a wide range of the angle of

incidence.

i. =ntroduction

Current needs for sputtering yield at normal incidence and at

oblique incidence have accelerated the experimental measurements of

sputtering yields, particularly by light ions. The extensive uses of

sputtering data for the design Of the fusion reactors required the■

comparison of experimental data with empirical formula and●

the reliable empirical formulae at normal and d･blique incidences.●

■

The empirical formulae for the sputtering yields at normal

incidence have been proposed by Bohdansky et al･l r ･Matsunami et al●2,

and Yamamura et al･3 The compilations of the experimental data for

available combinations of ions and target atoms have been published4,5.

The angular dependence of sputtering yields has been investi9･ated

by many authors6-25 and it was found that its dependence on the angle

of incidence was roughly given by cos-1e for not-too-oblique incidencer8

where a is measured from the surface normal. On the other hand, the

theoretical investigation on the angular dependence of sputtering

yields was done by Sigmuna26r who showed that the normalized yield

y(o)/y(o) had the cos-fo-dependence at not-too-oblique incidence,

where l

the energy and increasing the s･urface binding energy･ For the grazing

aLlgle of incidence, the surface channeling plays an important role and

makes the drop-off of the sputtering yield at the glancing angles.

=t is desired to have some analytical formula describing the angularr

dependence of the sputtering yields in a wide range of the angle of

incidence･ =n this papert a simple formula for the angular dependence

of the norm畠Iized sputtering yield Y(0)/Y(0) is proposedl taking into

account the effect of the surface channeling at large angles, and the

cわmparison of the present empirical formula with the existing

●

experimental data will be made in detail.

■

2. An empirical formula for angular dependence of sputtering yields

Light-ion sputtering is mainly due to collision cascades a.reated by

ions backscattered from the interior of the solidT While the heavy-ion

sputtering is due to collision cascades generated by incoming ions

directly27･ This difference is very important for low-energy sputteringt

especially for the angular dependence of low-energy sputtering.

=n the case of light-ion sputteringp the angular dependence of the

threshold energy does not have a clear minimum. Or! the contrary, in

the case of heavy-ion sputtering, the angular dependence of the threshold

energy has a minimum near 600. The threshold energy of heavy-ion sput-

tering is a decreasing function of the angle of incidence because of

the anisotropIC Velocity distribution of recoil atoms near the surface,●

and for gra2;ing尋ngles of incidence it increases rapidly with increasing

angle of incidence because of surface channeling. (see APPENDIX A).

=n other words, even if the ion energy is less than the threshold

energy at normal incidencel the finite number of target atoms willbe

sputtered in oblique incidence in the case of heavy-ion sputtering.

=t must be noted that the nomali2:ed sputtering yield Y(8)/1'(a) canl not

=

be well defined in such a low energy region.

-2-

･

2.i Angular dependence of light-ion sputtering

Light-ion sputtering is directly connected with the particle reflec-

tion coefficient because iもis due to co.uision cascades generatedby

backscattered ions. F空Cently. Yamamura et a1･ have derived the following

form山a28

y(E)=...｡2且空包竺｣

【 i_ (ヱ虹)舌】2･8 (1)u

E

S

for light-ion sputtering at normalincidence, where FD(E*) is the

deposited energy near the solid surfaceby a backscattered ion, E* is

the average energy of the reflected ion, and RN(F･) is the particle reflection

coefficient of the ion with the incident energy E｡ The- surface binding energy Us is

taken to be equal to the sublimation energy｡ EthlS the threshold energy of the

sputtering at normal incidence.

●

For not-too-oblique incidence, the normalized yield is given by

☆

Y(0) RN(E,8)(2)

Y(0) FD(E☆) RN(E)

where the threshold effect of the light ion sputtering lS neglected because of its

weak angular dependence and Y(0) is the sputtering yield at normal incidence which

is described by Y(E) in eq｡(1)｡ Similarly RN(E) means RN(E,0=0)｡

The angular dependence of the reflection coefficientis roughly

estimated in terns of the rangedistribution4 For very small angles of

incidence･ the ratio RN(E,e)/RN(E) is given by

RN('l7･ 0)

RN(E)

where

O

LcodxF(E.0,Ⅹ)

I:dxF(E･Opx)

･y2㌔

fR=l'苛`

(cos8)-fR ∫ (3)

(4)

and F(E,0,Ⅹ) is the range distribution and the subsrl･rLptR of the

moments means those of the range distribution･The Fi!XPOnent

fR is

nearly equalto 2 for large mass ratio29･

since E*(0) does not depend so largely on the a,Algle of incidence･

-3-

the angula.r dependence of Y(0)/Y(0) is mainly determined by those Of the

particle reflection coefficient, and we can write

Y(8)=

(coso)-fR (5)

Y(0)F

As the angle of incidence increasesl the effect of surface channeling

becomes important. =n order to sputter target atoms at the outermost

layer, the proコeCtile mus七penetrate the first layer of the solid●

surface, and this penetration probability is roughly estimated by

exp(-

NoRo/cosO),

where cT is the hard-sphere collision cross section between a proコeCtile～ 4

I

and a target atom, Ro =N一言is the average lattice conLtlLtant Of the randomI

target, and N is the number density of the target atom･

From above discussions we know that the angular dependence of the

normalized yield will have the form

Y(8)

=七fexp卜∑(モー1)】 ∫ (6)

Y(0)

where t= i/cos a and the parameters f and E are adjustable parameters.

＼

The angle of incidence at the maximum yi'eld is simply given by●

∑

oop七=COS-1(丁) ･ (7)

Using the least-square method, the best-fit parameters to the

present empirical formula are obtained from experimental data19,20,21,24■

and computer results (ACAT)30･ The best-fit values of f and 8opt

are listed in Table 2. =t is very interesting that the best fit f's

are nearly equal to 2 which is a theoretical value of Eq. (4) ･

The best-fit values of f depend slightly on the sublimation energy

and so the ratio of the best-fit f to巧is plottedas a function of

mass ratio M2/Ml in Fig･ 1･ where瀬I and H2 are the atomic masses of

a projectile and a target atomt respectively･ The solid linein Fig･ i

●

corresponds to the average value of f which satisfies thefoil(wing

も

rela七土on:

-4-

√町

≡ o･94 - l･33 × 10-3(M2/Ml) ･

=n Fig･ 2 the best-fit values of Oopt are plotted

function of

al

主

)2 .rl ≡

(8)

aS a

(9)Ro 2Eq

主

where q= (Us/yE )2 corresponds to the cosine of the scattering angle

of a recoil atom which gains the energy Us in a single co11ision･

The LSS reduced energy ｡ is defined by E≡

E/ELF Where

EL=

M1+M2 ZIZ2e2

M2 a

a= 0.4685 (

1 1

)2

zi/3･z13

'･

▼◆-†･I

(10)

(ll)

zl and Z2 are the atomic numbers of a projectile and a target atom,■

respectively･and y≡ 4MIM2/(Ml-2)2† Ml and･M2 being

The solid line in Fig. 2 is a theoretical curve of

eopt=900 - 57･3n

their masses.

oopt31

(12)

which is derived fromthe direct knock-out model. The且greementbetween

the besモーfit values and七he七beoretical results is very good. and so

we can calculate Oopt from the七beore七ical f-mula･

2.2 Angular dependence of heavy-ion sputtering

=n the case of heavy-ion sputtering, one cannot neglect the threshold

effect which is a decreasing function of the angle of incidence for the

not-too-oblique incidence. This means that the normali2:ed yield has

the following angular dependence :

Y(8)

Y(0)

- (cose)-fs [,

去

i - (Eth/E)2cose

1-

(E七h/≡)2

(13)

where the exponent fs is given in the form by Sigmund26 asfollows･･

■

--5-

fs=1+

くY2,｡

.くⅩ,㌔DL

Dー1 】 (14)

for not-too-large angles of incidence, where the subscript D of the

moment means those of the damage distribution.

r

The neccessary condition for the genuine collision cascade near the

solid surface 土s 七ha七tbe proコeCtile must penetrate a few layer of the●

surface. This means that an empirical formula for heavy-ion sputtering

can be expressed by the product of the similar expression to that for

light-ion sputtering and the threshold term of Eq. (13). At present,

the threshold energy is not well established. For simplicity, let us

●

employ an empirical formula with the same functional form as that of

light-ion sputtering, eq.(6), i.e.,

Y(8)

=tf exp卜∑(セー1)】 , (15)

Y(0)

whe工･e 七he exponen七 f will include the threshold effect o王Eq. (13) , and

so it will depend on the ion energy in the low-energy region. This

tendency is much different from light-ion sputtering.

℡be besモーfit parameters to the present formula′ Eq. (15). are obtained

for about 25 ion-target combinations (Table i) , using the least-square

metho(ヨ and are listed 土n ℡able 3. エn the energy

region Where the threshold effect can be neglected, the best-fit values●

of f are consistent with Sigmund fs, a8 Shown in Fig･ 3･

=n Figq 4 the ratios of the best-fit fls to sigmund fs are plotted

主

as a function of ち≡ i - (Eth/E)2･ where Eth is calculated from the

following relation:3

Eth-

l･5斗【l･ l･38 (Ml,M2,h･2 (16,

with h- 0･834 for出2>Ml andh

= 0･18 for M2

1-

;

1 + 2.5 (17)

Equation (17) shows that f is infinite at E = Eth, Which corresponds

to the fact that the normalized sputtering yield Y(e)/Y(0) is not well

defined in the energy region Of E < Eth･●

ェn叫･5 the besモーfit values of 900-

Oopt are plotted a9ains七

中- (エ)3/2 1Ro

ZIZ2

2/3 . ,, 2/3(zlりJ + z2りJ)2 ÷-喜

The parameter ゆis directly connected with the critical angle

よ

中c= (uR(0)/E )2

(18)

(19)

of surface channeling, where uR(0) is the average potential at the

amorphous solid surface which is defined in ÅppF.ND=Ⅹ A. The solid line

in Fig･ 5 means the average value of the best-fit Ooptls which has the

form

oopt=900-286･0中0･45 ～ (20)

3. Compilation of Experimental Data on Angular Dependence of

Sputtering Yields

The angular dependence of sputtering yields of various combinations

has been compiled and stored 土n a computer. ℡be compiled data

are summarized in Table i. When determining the best-fit values of the

adjustable parameters f and ∑. we excluded the experimental data which

do no七have explicit maximum values or which do not bave 七he data at

normal土ncidence.

Table 4 shows the physical constants necessary for calculating the

angular dependece of･ the sputtering yield by means of the present

empirical formula. For extensive use of the present empirical formula,● ●

the parameters f and Tl are Calculated from Eq. (8) and Eq. (9)宕or

● ●

typical proコeCtiles H, D, T and He, which are listed in Tables 5. 6

-7-

7 and 8, where the calculated parameters TI COrreSPOnd to i Rev ion.

For calculations of the angular dependence of heavy-ion sputtering1

26

Sigmund f of m= i/3 and * for i eV ion are also calculated and

listed in Tables 9 through 18, where as the

typical p;ojectiles C4;1･Art Fe･ Nit Cur Kr･ Ⅹe and Fg ions are

selected.

The plots of the angular dependence of sputtering yields for various

combinations of the incident ions and the target atoms are Shown in

Figs. 6 through 58, where Figs. 6 through 24 corres妄)ond to light-ion

sputtering and Figs. 25 through 58 to heavy-ion sputtering. The solid

lines in these figures are best fit curves to the present empirical■

formula, and the solid lines with cross marks show the results calcu-

1ated by putting the average values,of

the best-fit parameters into

each empirical formula.●

工n order to know the absolute yield Y(0). one must calcula七e 七he

sputtering yield at normal incidence. The authors recommend to use

the third Matsunami formula for the sputtering yield at normal incidence,

wb土ch has七he form32

Y(I) = P

where

sn(∈)

i + 0･135Usse(e)

【 - (Eth/E)喜】2･8 ′

EL N

p = 0･042 --α(M2/Ml)Q(Z2)･

RL Us

lRL=

甘a2Ⅳy

′

エ

se(E) =ke2 .

(21)

(22)

(22)

(23)

and sn(c) is the LSS elastic stopping cross section in the reduced unit

and七he following analytical express土on 土s useful for 七be present

purpose=2

3.441yqlog(∈ + 2.718)

sn(E)i + 6.355JE+ a(-1.708 + 6.882ノ官)

-8-

. (24)

The inelastic coefficient k of se(E) is given as33

k = 0･0793 Zll/6

主

(･zIZ.2)2 (Ml + M2)3/2

(z12/3 + z22/3)

. (25)

and the threshold energy Eth for the third Matsunamiformula has the

following empirical relation I

Eth - Us 【1･9 + 3･8(Ml/M2) + 0･314(H2/Ml)l･24】･ (26)

The parameter α(M2/Hl)is represented as

α(H2/Ml)= 0･08 + 0･164 (M2/Ml)0･45+ o･o145 (H2/Ml)l･29

(27)

and Q(Z2) is listed in Table 4 for each element･ For extensive uses

of the third Matsunami formula, the parameters EL, P, Eth, α and k

are listed in Tables 19 through 32 for various ion-target combinations,

where as the projectile H, D, T, ne, C, Ne, A1, Ar', Fe, Ni,Cu, Kr,

■

'Xe and Hg are selected.

ACKNOWLEDGEMENT

The authors would like to､thankPro fs. H.Oechsner and H･L･Bay for

their kind supplies of their experimental data･ The authors are

indebted to Prof. M.Mannami for his kind interest in this work･

The author (Y.Y) alsow弧tS tO thank MTS･ K･Yamamoto, H･Arinobu'

s.Kobayashir Y･Mizuno and Hiss K･Yasuda for their continuous assistance

on this work.

-9-

References

i) J･ Bohdansky, J. Roth, and H.L. Bay, J. App1. Phys. 51, 2861 (1980)

2) N. Hatsunami, y. yamamura, y. ztikawa, N. =toh, Y. Kazumata,

S. Hiyagawa, R. Morita, and R. Shj.mizu,

Radiat. Eff. I'ett. 50, 39 (1980).

3) Y. Yamamura, N. Hatsunami, and N. =toh,

Radiat. Eff. Lett･堕, 83 (1982)･●

4) N. Matsunaml, Y. Yamamura, Y. =tikawa, N. =toh, Y. Kazumata.

S. Miyagawa. R. norita, and R. Shimizu,

Report =PPJ-AM-14,Institute of Plasma Physics,

Nagaya University (1980).

5) J. Both, J. Bohdansky, and W. Ottenberger,

=PP 9/26, max-Plank-=nstitut fur Plasmaphysik (1979).

L

6) G.K. Wehner, J. App1. Phys. 30, 1762 (1959).

7) P.K. Ro1, J.M. Fluit, and J. Kistemaker, Physica 26, 1000 (1960).

8) V.A. Malchanov and V.G. Te1'kovskii,

′

Sovie七Phys･-

Doklady 旦′137 (1961)I

9) D. Almen and G. Bruce, Nuc1. =nstr. Methods ll, 257 (1961).

10) I.I. Dushkov, Ⅴ.A. Malchanov′ Ⅴ.G. Tellchanov, Ⅴ.M. Chicherov,

Sovie七Phys.-

℡echn土cal Pbys. 6. 735 (1962).

ll) C.E. 又amer, A.A. Narasimham, H.R. Reynolds, and J.C. Allred,

J. Appl. Pbys. 35. 1673 (1964).

12) E.S. Mashkova and V.A. Malchanov,

Sov土e七phys.-

℡echn土cal Phys. 9. 1601 (1965).

13) K.B. Cheney and E.T. Pitkin, J. App1. Phys. 36, 3542 (1965).

14) G. Dupp and A. Scharmann, Zeit. Phys. 194. 448 (1966).

15) L.N. Evdokimov and V.A. Malchanov, Can. J. Phys. 46, 779 (1968).

16) H.工smail and A. Sep七土er′

proc. 3rd =nt. Symp･ on Discharges and Electrical Insulation

土n Vacuum. (1968) p 95.

-10-

17) G. Holmen and Q. Almen, Ark. Fys. 40, 429 (1970).

18) A.J. Su皿erS, N.J. Freeman, and N.R. Daly,●

J. Appl. Phys. 42. 4774 (1971).

19) H. Oechsner, Z. Phys. 261, 37 (1973).

20) H.L. Bay and J. Bohdan?ky, App1. Phys. 19, 421 (1979).

21) H.L. Bay, J. Bohdansky, W.0. Hofer, and J. Roth,

Appl. pbys. 21. 327 (1980)

22) W. Kruger, A. Scharmann,打. Afrid, and G. Brauer,

Proc. of Symposium on Sputtering, perchtoldsdorf/Wien,

Austria (1980), p. 28.

23) D.A. Thompson and S.S. Johar, Radiat. Eff. 55, 91 (1981).

24) J. Bohdansky, G.L. Chen, W. Eckstein, J. Both, and B.M.U. Scherzer,

J. Nuc1. Mater. 111&112, 717 (1982).

25) J. Bohdansky, Nuc1. Fusion (to be published).

26) P. Sig7nund, Phys. Rev. 184, 383 (1969).

27) A.W. Weissmann and P. Sigmund, Radiat. Eff. 19, 7 (1973).

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Radia七. Eff. 71, 65 (1983).

29) K.B. Winterbon, P. Sigmund, and J.B. Sanders,

Mat.-Fys. Mead. Dan. Via. Selsk. 37, 14 (1970)

30) W. Takeuchi and Y. Yamamura, Radiat. Eff. 71, 53 (1983).

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S. Miyagawa, K. Morita, and R. Shimizu,

ATOM=C DATA AND NUCLEAR DATA TABLES (to be published).

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Mat.-Fys. Mead. Dan. Vid. Selsk. 33, No.14 (1963).

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35) Y. Yamamura, Radiat. Eff. 55. 49 (1980).

-Il-

APPENDIX A: ANGULAR DEPENDENCE OF THRESHOLD ENERGY OF LIGHT-ION

SPUTTER=NG AND 王王EAVY-ION SPUTTERING

The sputtering mechanism for oblique incidence is separated into three

parts; 1) ℡わe mechanism due to 七be d土rec七knock-ou七 coll土sions w土七h

incident ions at the outermost layer of the solid surface (mechanism

=a dffFig. A1). 2) The mechanism due to collision cascades created by

incoming ions near the solid surface (mechansim =b of Fig. A1)

3) The.mechanism due to collision cascades generated by ions backscatted

from the interior of the solids (mechanism I of Fig. A1).

.UECHA.1【S.iI PIECHA.1(SJl Ⅱ

E)

E

E)

?

∈

/○ i%+kj/

､J.st､､､､､､/ゝ■

Fig. AI Schematic representation of sputtering

mechanism a七obl土que 土nc土dぐnCe

For light-ion sputtering the mechanisms =a and I play an important

role.

The threshold energy of mechanism I is given as34

Us 1

Y ( 1- y)

(Al)

for normal incidence･ For not-too-large angles of incidence, the

factor (i - y) of Eq･ (A1) must be modified. =f only a single colli-

sion process is taken into considerationt the threshold energy will be

a slightly decreasing function ot- the angle of incidence･ For oblique

incidence( howeverr the multiple scattering becomes important and this

effect weaken the angular dependence of the factor (1 - y). Roughly

speaking, the threshold energy of mechanism I is

-12-

Eth

-+ 1-

ycos2(o/2)

(A2)

for not-too-oblique incidencet where the difference of the scattering

angles between in the L system and in the CH system is neglected because

of large mass ratio.

For oblique incidence, an incoming ion must penetrate the first

layer for mechanism I. The average potential at the surface of the

amorphous solids is roughly estimated by

uR(Y,

-苛!芸;;孟zI 2TrdrV(

1.+Rn′∠

.00

0

y- z)2+r2 ). (且3)

where tis the distance from the surface and V(r) is the interatomic

potential. =f one employ the !4oliere potential as the interatomic

poten七土al we have

A

uR(y) =4甘A + 1

EL(‥-)3 z3a α.

1_____二~~~.

~~LJ

Ro i-1

βfw九e re

aS

A = M2/Hl･

α.≡ (0.35. 0.55, 0.10)

1

sinh(%0,ex,(上y,,

a

(A4)

βi= (0･3.1･2･6･0)

･

■

The critical angle for penetration through the first layer is given

中c=

cos-1[ロR(0) エ

】2 , (A5)

whereゆcis measured from the surface norma1･ The critical angle中c

for the incident energy of the order of the threshold energy is about

50o for=-Ni･

For e 700) , the direct knock-out

process become important and the criterion of the sptittering due to

this process is given by31■

-13-

cos8+ 2q < 1

p2 >pl > Rocos8

エ

where q= (Us/岬)2 and the impact parameters pl and p2 are the lower

and the upper lim土七s of 七he impact parameters available for 七he

f

sputtering due to the direct knock-out process.

From the inequality of玉q. (A7) we obtain

Eth･(0)

ロs1

= 二

~Y

siふ4(a/2)

I (A9)

The criterion of Eq. (A8) is not important as compared with surface

channeling for oblique incidence. Namely. in order to knock off an

atom at the outermost layer, an ion must arrive at the surface. This

leads to the following expression for the threshold energy:

Eth(0) ≡UR(Ro/2)

cos2 o

(良lo)

where uR(Ro/2) is the average potentialat thedistance Ro/2･ at which

there exists an atom with a f土ni七e probability ill the case of the

random target. From Eq. (A4),

巨=

tlJ

>一

t3

(亡

tA)Z

t▲J

亡1

■■

l⊃`⊂

OT

tJ}

CE:

I

[=

0 3O 60 90

月NGし∈ OF 【NC柑ENCE (DEGREE)

uR(Ro/2) = UR(0)/2･

Fig. A2

The angular dependence of the●

threshold energies for H + Ni.

mechan土sm Ⅱ

direc七 knock-ou七 process

surface channeling

-14-

Y/

Zn Fig. A2 the threshold energies COrreSPOnding to each mechanism●

are plotted against the angle of incidence. Figure A2 te11s us that

●

the threshold energies COrreSPOnding to the direct knock-out process

can be neglected. Fina11y we get

i - ycos2(a/2)

i + A26.85

Eきh(8)-EL'%'2~~~~~■

■二:∴~:~~~

｣._

cos2o

for not-too-oblique incidence

(Alュ)

for gra2:ing angles of incidence

When the incident energy is very low, sputtered atoms will be elected･●

before 七hey lose the memory of 七he d土rect土on of 七be inciden七ion beam.

This means tbat七he velocity dis七ribu七土on of recoil atoms generated by

incoming heavy-ions is anisotropic. Taking into account the anisotropic

effect of the recoil flux′ Yamamura 5'howed that the angular dependence

of the heavy-ion sput･cering was proportional to cos2 o

35.

For gra2:ing angles of incidence, the surface channeling will play

an important role. =n the case of heavy-ion sputtering, the threshold

●

energy due to surface channeling is given by

E七h(8)

u氏(ym土n)

cos28

(A12)

where ymin is the minimum distance above which the projectile will

be reflected without producing any recoil atom available for sputtering.

For the amorphous solid surface the minimum distance ymin may be of

the order of

ymin+

pm土n ∫ (A13 )

where pmin is the collision diameter for the Moliere potential and

depends on the incident energy. Since the incident energy of present

interest is very low, pmin is roughly given by

pm土n≡

Sa

-15-

where s is the soluti･on of the following transcendental equation!

∈ =exp(-

0.3s)/s .

Let us consider a typical example, i･e., Ar+-Cu. =n this

■

case, s is nearly equal to 15･ =n Fig･ A3 the angular dependence of

threshold energ･y for Ar+ -u is plotted as a function of the angleI

o王土ncidence･ Finally′ we have

H

Eth(0)

Eth(0) cos2o far not-too-oblique incidence

o･3品EL';,3'A14L,

for grazing angles of ir).cidencecos2o

where Eth(0) is the thresholdenergy at normal incidence･ and ymin

is se七to equal to Ro/2 + 15a･

Fig. A3

望he angular dependence

of the threshold energies●

for Ar + Cu.

mechan土sm エ

surface channeling

-16-

I

｢■T

>-

1コ

【亡

▼■

巨2j

uJ

⊂】

t33Ill

LU

【亡

:亡

二=コ

10)

102

0 3D 60 90

RNGLE OF TNCIDENCE tDEGR∈E)

Table 】 工on-target combinations in the present data compilation

References

Target

ュon C AITiFe NiCu Zr馳Mo Pd Ag Ta W Pt Au

wehner (1959)

Roュ et al. (1960)

hrchanov et al. (1961)

AITnen et al. (1961)

Dushikov et al. (1962)

Raner et al. (1964)

Marchanov et al. (1965)

Cheney et al. (1965)

Dupp et al. (1966)

Evdokimov et al. (1968)

工smail et al. (1968)

Hoitnen et al. (1970)

Su1皿rerS et al. (1971)

Oecbsner (1973)

Bay et al. (1979)

Bay e亡al. (1980)

Kruger et al. (1980)

Thonpson et al. (1980)

Bobdansky et al. (1982)

Bohdansky (1983)

嘩g

Ar

Ar

Kr

tie

N

Ne

Ar

Ar

Ar

Xe

Ar

Ne

Ar

Kr

tis

tlg

D

tie

Nb

tie

Ne

Ar

Kr

Xe

ヱ

D

TIC

Ⅱ

Ar

P

Sb

Bi

帽

D

tie

H

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0 0

0

0

0

0

0

0 0 0

0

0

0

0

0 0

0 0 0 0 0

0

0

0

0

0

0

0

0

0

0

0 0 0 0

0

0

0

0

0

0

0

0

0

-17-

Table乏 Best-fit parameters of the present empirical

formula for light-ion sputtering

Best-fit values

Energy =on Target

450 eV

l keV

4 keV

450 eV

l keV

4 keV

I JくeV

IOO eV

500 eV

l keV

4 keV

4 keV

50 keV

1.05.keV

2 keV

8 keV

2 keV

4 keV

l key

4 keV

He

Ni

Ni

Ni

Ni

Ni

Ni

Ni

Ni

Ni

Ni

Ni

H Mo

H Mo

D Mo

He Mo

H Åu

刀 Au

1 Rev D Au

1.62

2.34

2.27

2.19

2.32

2.62

1.88

3.20

3.30

2.50

2.09

1.52

1.88

1.55

2.40

2.80

1.98

2.23

1.14

1.53

1.22

8op七Ref.

74.40

78.30

82.30

78.70

82.9o

84.20

80.40

56.30

66.10

72.10

79.00

80.5o

82.10

25

20

20

20

30

30

30

30

30

30

20

66.50 19

81.80 20

82.00 20

82.00 20

77.30 20

78.00 20

79.50 20

79.20 20

-18-

Table 3 Best-fit parameters of the present empirical●

formula for heavy-ion sputtering

Energy =on Target

Best-fit values

f Oopt

Ref.

30 eV Ar

50 eV Ar

100 eV Ar

500 eV Ar

1 keV Ar

5 keV

20 keV

50 eV

100 eV

500 eV

1 keV

5 keV

t>

50 eV

100 eV

500 eV

1 keV

5 keV

10 keV

200 eV

800 eV

400 eV

800 eV

200 ev

800 eV

400 eV

200 eV

400 eV

800 eV

Ar

Ar

Hg

Hg

Hg

Hg

Hg

Ni

Ni

Ni

Ni

Ni

Ni

Hg

Hg

Hg

Hg

H9

Hg

Hg

Hg

Hg

Hg

Cu

Cu

Cu

Cu

Cu

Cu

Cu

Ni

Ni

Ni

Ni

Ni

Ni

Ni

Ni

Ni

Ni

Ni

Ni

Ni

Fe

Fe

Mo

Mo

℡a

W

W

W

35.4 44.20

9.33 46.10

5.25 43.6o

3.35 49.6o

3.07 55.70

2.66 65.70

2.19 75.00

37.1 47.5¢

15.0 52.5o

5.01 61.20

4.02 63.30

3.56 64.70

22.8 44.2

ll.8 45.5o

3.75 50.50

3.05 56.2o

3.13 64.20

3.37 66.4o

12.8 48.2

8.57 51.00

15.5 57.60

9.53 60.70

26.0 53.00

26.9 50.90

15.0 62.00

5.87 58.30

10.6 52.8o

7.64 56.1o

30

30

30

30

30

30

30

30

30

30

30

30

30

30

30

30

30

30

6

6

6

6

6

6

6

6

6

6

-19-

でable 3 (continuedl

Energy Ion Target

Besモーf主七 values

f Oop七

1.05 keV

1.05 keV

i.05 J{eV

1.05 keV

1.05 keV

1.05 keV

550 eV

1.05 keV

1.55 keV

2.05 keV

1.05 keV

1.05 keV

1.05 keV

1.05 keV

1.05 keV

37 keV

30 keV

9.5 keV

30 keV

30 keV

9.5 keV

Ar

Ar

Ar

Ne

Ar

Kr

Xe

Xe

AI

Ti

Ni

Cu

Cu

Cu

Cu

Ⅹe Cu

Xe Cu

Ar Zr

Ar Pd

Ar Ag

Ar Ta

Ar Au

Ar Cu

Xe Cu

Xe Cu

Xe Mo

Xe W

Xe W

Ref.

2.10 70.20

2.04 68.80

1.72 69.5(I

1.03 70.6o

1.29 69.80

2.19 66.9o

9.40 53.80

4.56 61.50

3.89 64.Oo

2.44 69.00

1.73 67.70

1.85 63.10

1.50 63.70

1.75 67.50

1.50 59.80

1.16 71.9o

1.94 75.00

2.53 69.50

2.58 76.7o

1.33 78.50

1.07 76.Oo

19

19

19

19

19

19

19

19

19

19

19

19

19

19

19

13

13

13

13

13

13

､■~-

-20-

I:]j

蛋

Table 4 Some physical constants for calculations of sputtering yields.

■

Table 5 through 8

The parameters for the angular dependence of light-ion

sputtering.

℡he correspondence between でables and proコeCtj･･1es aret

as follows:

H : Table 5

D : ℡able 6

T : Table 7

He : Table 8

Let us show briefly how to use these tables and suppose the

angular dependence of 4 keV H+ - Ni is desired･

The appropria.te table is Table 5. From Table 5 we have

よ

f = (4.44)2×(o.94 - 0.00133×58.24) = i.82 ,

where figures in italics corresponds to values in Table 5.

The quantity n is inversely proportional to El/4 and so we

get

n-

(÷)i/4×o･1535-0･1085 ･

From Eq･ (12) Oopt can be calculated as follows:

oop七 = 900 -57.3×0.1085= 83.8o.

since ∑ - fcosOop七′ We have

∑ = 1.82 cos83.80= 0.197.

Finally, we have the following angular dependence for 4 keV

H+)Nil

淋-Ⅹl･82exp卜0･197(Ⅹ-i)]･

where x≡ 1/cos 0

-21-

Tables 9 through 18

The parameters for the angulardependence of heavy-ion

sputtering.

℡he correspondence between でables and proコeC七土1esare as

t

follows王

C : ℡able 9

Al ; でable ll

F占: Table 13

c】 : でable 15

Xe : Table 17

Ne : Table 10

Ar

Ni

Xr

Hg

I

●

■

●

●

●

でable 12

でable 14

でable 16

: でable 18

L.e{tuh;;hJ;:i::i;…三≡nhd::ce7o.;s;eta≡:ei｡三a…;:t…:rri:ag:Cu三:…三?ns確suppose the angular

dependence of i keV Ar+- Cu is desired･

The appropriate tableis Table 15･ Fist of all, we must

calculate 七he quanti七y ∈

1 よ

こ= 1-

(Eth/E)富= i- (20･72/1000)2

≡ o･8561･

since fs = =･7=, the parameter fis calculated from Eq･ (17),

f-1.71 X(i+ 2.5xO.1439/0.8561)- 2･43

The quantity 1+is inversely proportional to y@･ which

is

known from Eq. (18). Then, we have

中- 0.1305/価甘- 0･004127 ･

From Eq. (20) we get

oopt = 900 -286xO･0041270･45= 65･80

地e relation ∑≡ fcos Oop七yields

∑ = 2.43cos65.80= 0.9961.

2n the above calculations( figuresin italics are the values

in Tab1(,. 15. Finallyr we get the following angulardependence

for i key Ar+ → Cu(･

Y(8)

印丁≡

Ⅹ2･43exp卜0･9961(Ⅹ

- 1)】･

where x≡ 1/cos8.

-22-

Table 19 through 32-J

The parameters for the third Hatsunami formula,

where the correspondences between Tables and projectiles■

. are as follows:

H

He

Al

Ⅳ土

Ⅹe

でable 19

℡able 22

℡able 25

℡able 28

: Table 31

D ; ℡able 20

C ; ℡able 23

Ar : Table 26

Cu : ℡able 29

Hg : Table 32.

T : Table 21

Ne.･ Table 24

Fe : Table 27

Kr : Table 30

Let us show how to use these tables for calculations of

sputtering yields at normal incidence by the third

Matsunamj. formula. Suppose the sputtering yield of i key

I+ - Ni is desired･ The appropriate table is Table 19･

The reduced energy E for i keV Ar+ → Ni is calculated

like tb土s

∈ - 1000/(β.7ββ × 1000)- 0.3572.

Direct insertion of七bis value土nto Eqs. (23) and (24)

yields

se(e)= 4･35 X何=汚詔~= 2･60

sn(e) =0･408 ･

SinceP= 0･4814 and Eth= 100･6, we have

y= 0.4814 × 0.408/(i + 0.35×4.44×2.60 )

x [ i - (100.6)i ]2･8 = i.33x10-2(atoms/ion),

where U= 4.44 eV is in Table 4, and in the above calculations

S

figures in italics corresponds to those in the present tables･

-23-

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Captions of Figures

Fig. 1 Best-fit values of f for light-ion sputtering, where the ratio

of f to伺is plotted as a function of the mass ratio H2/Hl･

the solid line is calculatedfrom eq.(8).

Fig. 2 Best-fit values of Oopt for light-ion sputtering, where the

best-fit OoptYs are plotted against n and the solid line is

七be 七heoretical curve calculated from Eq. (12).

Fig. 3 The best-fit values of f for relatively high-energy heavy

ion sputtering, where the solid line corresponds to

Sigmund fs of m= i/3･

Fig･ 4 The ratios of the best-fit ffs to sigmund fs as a function

主

of n- i - (Eth//E)2, where the solid line corresponds to

Eq. (17).

Fig. 5 The best-fit values of 900- Oopt as a function of中′ where

the solid line corresponds to

oop七= 900 - 286中0･45･

Figs. 6 through 58

The normalized sputtering yield Y(0)/Y(0) as a function of

the angle of incidence, where the so.lid lines in these

figures are best-fit curves to the presenJ-._ empirical fclrmula,■

and the solid lines with x marks show the results calculated

by putting the average values of f and ∑ into the present

empirical formula. The average parameters of f and ∑ are

easily calculated using the parameters listed in Table 5

through 18.

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