Transport theory of sputtering I: Depth of origin of sputtered atoms

Nuclear Instruments and Methods in Physics Research B 149 (1999) 272±284

Transport theory of sputtering I: Depth of origin of sputtered atoms Zhu Lin Zhang

1

Huainan Institute of Technology, Huainan, Anhui 232001, People's Republic of China Received 10 December 1997; received in revised form 15 September 1998

Abstract Sputter theory employing a sum of two power cross sections has been implemented. Compared with the well known Lindhard power cross section (Vµrÿ1=m ), a sum of two such cross sections can give a much better approximation to the Born±Mayer scattering in the low energy region (m  0.1). By using both one and two power cross sections, we have solved the linear transport equations describing the sputtering problem asymptotically. As usual, electronic stopping is ignored in the analysis. It has further been proved that FalconeÕs theory of the atom ejection process contradicts transport theory. The Andersen±Sigmund relation for partial sputtering yield ratios between two elements in an arbitrary multicomponent target has been derived by both methods. The energy deposited in the target surface layers has been computed for a few typical ion±target combinations. The numerical curves show that both theories generate almost the same results (error <10%) for m ˆ 0.2. It is also shown that, if the sputtering yield equals the corresponding one in SigmundÕs theory, the depth of origin of sputtered atoms must be shorter than in SigmundÕs theory for 0.25 > m P 0. The former even may be only about one half of the latter as long as m ˆ 0. Ó 1999 Elsevier Science B.V. All rights reserved.

1. Introduction Sputtering, the erosion of a solid target during ion bombardment, was discovered experimentally more than a century ago. Both experimental and theoretical interest in this phenomenon has increased constantly for many years, due to remarkable applications in di€erent ®elds. In 1969, by the aid of the Lindhard power cross section [1], Sigmund developed the well1

Corresponding author. Tel.: 554 6640550.

known modern sputtering theory. This theory not only established the theoretical framework to explain the basic aspects of the sputtering process and gave analytical results, but also provided information on the characteristic depth of origin of sputtered atoms [1,2]. In 1981, based on the intrinsic relation between recoil density and sputtering calculations, Falcone and Sigmund proposed an atom ejection theory (or approximation): each recoil atom slows down continuously along a straight line. The reason for this result is the fact that for a homogeneous and isotropic source, any loss of particles due to

0168-583X/99/$ ± see front matter Ó 1999 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 8 - 5 8 3 X ( 9 8 ) 0 0 6 3 4 - X

Z.L. Zhang / Nucl. Instr. and Meth. in Phys. Res. B 149 (1999) 272±284

scattering from a given direction of motion is compensated by an equivalent gain [3]. In this approximation, the authors derived an explicit expression for the distribution of the depth of origin of sputtered atoms. Thus, the mean sputter depth can be estimated. Furthermore, this estimate was generalized to a multicomponent system by Sigmund, Oliva and Falcone without explicit proof in 1982 [4]. In 1986, Kelly and Oliva found that the above conventional  is much larger than characteristic depth (5 A) some simulation results [5]. In order to solve this problem, Falcone proposed a new atom ejection theory (or approximation): each sputtered atom is a recoiling atom set in motion in the direction of the surface without undergoing collisions on its way out [6]. Based on this theory, the author concluded that the characteristic depth of origin of sputtered atoms is equivalent to the low energy collision mean free path [6]. Unfortunately, as is well known, the mean free path vanishes for the Lindhard power cross section. In 1989, Vicanek et al. raised this question (called the Ôdepth of originÕ puzzle in this work) for discussion: ``Why then is the depth of origin of sputtered atoms estimated in Refs. [2,3] about a factor of two larger than that found in Monte Carlo simulations based on a very similar interatomic potential?'' [7]. In Ref. [7], the authors further con®rmed the continuous slowing down approximation for monatomic media. In addition, they proposed that an improved set of constants matching a standard power cross section to the Born±Mayer scattering brings the depth of origin of sputtered atoms into close numerical agreement with results from more realistic simulation models. On the other hand, they made a remark that any adjustment of the constant C0 (or, more precisely, k0 ) in¯uences not only the depth of sputtered atoms but also the total sputtering yield [7]. Therefore, to my knowledge, the Ôdepth of originÕ puzzle as yet remains unsolved. The aim of this paper is to present a sputter theory employing both exact scattering cross section and Lindhard power cross section. At least one intrinsic di€erence may be found, i.e. the depth origin of sputtered atoms in the present theory is

273

about one half the one in SigmundÕs theory, with the total sputtering yield unchanged.

2. Classical scattering theory Let us consider a particle, with mass Mi and energy E, colliding with another particle at rest, with mass Mj . It is well known that, for elastic collision in the classical picture, the di€erential cross section drij …E; T † can be derived, as long as the interaction potential between the two particles is given [7±10]. T is here the energy transfer in the laboratory system. 2.1. Power potential interaction [7±13] For a power potential interaction between the two particles (i and j) Vij …r† ˆ Kij rÿs ;

Kij > 0

…1†

the exact di€erential cross section can be derived, (

drij …E; T † ˆ

Cij Eÿm Zm …X †T ÿ1ÿm dT

for 1 P m > 0;

rij Tmÿ1 dT

for m ˆ 0;

…2†

where 1 P m ˆ 1=s P 0, and the subscript i is dropped. ÿ2

Tm ˆ cij E; cij ˆ 4Mi Mj …Mi ‡ Mj † ;  2m  m cs Kij Mi ; Cij ˆ pm m Mj

…3†

  1 1 s‡1 ; …Table 1†; cs ˆ B 2 2 2  2m 1 2m dg
X 1‡m
Zm …X † ˆ ÿ m cs dX

…4†

and X ˆ T/Tm which is a function of only one variable g, determined by 8   > dq < X ˆ cos2 2g0:5 R 1 p ; 0 g…2ÿq2 †‡d…q2 †y 2ÿs …5† > : s s sÿ2 1 y ˆ gy ‡ 1; d…x† ˆ x ‰1 ÿ …1 ÿ x† Š: Unfortunately, only a few closed forms of Zm (X) can be found analytically: One is the Rutherford scattering (s ˆ m ˆ 1), Z1 (X) ˆ 1, and another is

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Z.L. Zhang / Nucl. Instr. and Meth. in Phys. Res. B 149 (1999) 272±284

the Bohr formula (s ˆ 2, m ˆ 0.5). Besides, two boundary values exist analytically, Zm …0† ˆ 1 and Zm …1†  2m  2 1 2m C…0:5 ‡ m†
ˆ pm cs C…1 ‡ m†

…Table 1†:

In fact, these boundary values are entirely consistent with the errors in the power approximation which were given by Lindhard et al. in Ref. [9]. In general, Eq. (5) can be integrated numerically by using the Gauss±Legendre scheme. The calculated results for Zm (X) form a Data Table for further use, and some numerical results are plotted in Fig. 1. Even if Zm (X) may be calculated to any accuracy, it is still inconvenient to use for some applications. However, we found that Zm (X) can be approximated by Zm …X †  X 1‡m Zm …1†‰…1 ÿ p† ‡ pX ÿq Š for 0:2 > m P 0;

…6†

Zm …X †  1 ‡ pX q

…7†

for 1 P m > 0:1;

where p and q are such parameters that make Eqs. (6) and (7) best ®t the numerical curves of Zm (X). Some calculated data for the p, q parameters in Eqs. (6) and (7) are collected in Table 1 and Table 2, respectively. The corresponding analytical results Eqs. (6) and (7) are plotted in Fig. 1 for comparison. Eqs. (6) and (7) ®t the numerical curves well, if not exactly. In contrast with Eq. (2), the well know Lindhard power cross section was given a long time ago [1,2,8,9], drij …E; T † ˆ Cij Eÿm T ÿ1ÿm dT ;

1PmP0

…8†

for the same power potential interaction Eq. (1). A large discrepancy can be seen between Zm (X) and unity for 0:5 > m P 0 in Fig. 1 [9]. In other words, Eq. (8) cannot properly describe the scattering of a power potential interaction Eq. (1) for m  0 [9]. The present transport theory is based on Eq. (2) or Eqs. (6) and (7), whereas, the one based on Eq. (8) is SigmundÕs theory. [H]Z and [H]P stand for a quantity H in the present and in SigmundÕs theory, respectively. 2.2. Real potential interaction

Fig. 1. The universal function Zm (x) in the di€erential cross section Eq. (2). Thick line: Numerical results calculated by Eqs. (4) and (5); thin line: Approximate analytical results calculated by Eq. (6) in (a) and Eq. (7) in (b) respectively.

For a real potential interaction collision, the approximate classical scattering cross section is derived in two steps: The ®rst step is to approximate the real interaction potential by a power form Eq. (1). This is feasible over limited ranges of r [8]. The second step is to derive the classical

Z.L. Zhang / Nucl. Instr. and Meth. in Phys. Res. B 149 (1999) 272±284

275

Table 1 Calculated results of cs , p, q, Cm and Zm (1) in Eq. (6) S

cs

m

p

q

[Cm ]a

Cm

[Cm ]p

Zm (1)

1 100 50 20 10 5 2

0 0.122 0.173 0.277 0.387 0.533 0.785

0 0.01 0.02 0.05 0.1 0.2 0.5

0 0.03978 0.08447 0.2116 0.3667 0.6059 0.9768

0.8525 0.8388 0.8649 0.9522 1.1093 1.4984

1 0.969 0.939 0.862 0.764 0.617 0.363

1 0.968 0.939 0.862 0.763 0.620 0.363

0.6079 0.604 0.599 0.586 0.563 0.516 0.361

1 93.84 44.67 15.85 6.837 2.835 1.032

Table 2 Calculated results of cs , p, q, Cm and Zm (1) in Eq. (7) S

cs

m

p

q

[Cm ]b

Cm

[Cm ]p

Zm (1)

20 10 7 5 4 3 2

0.277 0.387 0.457 0.533 0.589 0.667 0.785

0.05 0.1 1/7 0.2 0.25 1/3 0.5

13.65 5.327 3.184 1.796 1.125 0.5226 0.0314

0.7503 0.6214 0.6067 0.5820 0.5483 0.5145 0.7764

0.840 0.741 0.687 0.620 0.565 0.488 0.363

0.862 0.763 0.695 0.621 0.565 0.489 0.363

0.586 0.563 0.543 0.516 0.492 0.450 0.361

15.85 6.837 4.353 2.835 2.144 1.523 1.032

scattering cross section from the power potential Eq. (1). It is clear that the error in the presently used cross sections only originates from the ®rst step, while, the error in the simple power cross section comes from both the ®rst and second step. Even so, the simple power cross section could still provide the better approximation due to some possible error cancellations. In 1969, regarding Eq. (8) as exact, Robinson derived the corresponding interatomic potential function by using an inversion procedure [14]. Robinson found that the exact potential approaches the power potential limit Eq. (1) at high energy and large interatomic separation. However, the author also realized that the exact potentials are softer than the asymptotic forms and signi®cantly energy dependent. Both features become more pronounced, as s increases. RobinsonÕs ®ndings are intrinsically consistent with those of the present work. Therefore, for a low energy incidence (m  0), Eq. (8) cannot describe an energy independent collision. As an example, let us consider the Born±Mayer interaction [7]. In this case, the interatomic potential is given by

V …r† ˆ A exp …ÿr=a†  …Aks as †rÿs ; 0:25 > m > 0: Then, Eq. (3) turns out to be  m p Mi 2m Cij ˆ km a2
…2A† ; Mj 2

…9†

2m

where km ˆ 2m…cs ks =2m† . Using Eqs. (2), (8) and (9), it is easy to derive the reduced nuclear-stopping 8 [2] as a function of reduced energy e R1 > < ‰Sn …e†ŠZ ˆ 12 ‰km ŠZ
e1ÿ2m
0 Zm …X †X ÿm dX ; …10† 1 1ÿ2m 1 >
…1ÿm† ; : ‰Sn …e†ŠP ˆ 2 ‰km ŠP
e and the Lindhard scaling function in scattering cross section [2] (  ÿ
1ÿ2m f …t1=2 † Z ˆ ‰km ŠZ
Zm …teÿ2 † t1=2 ; t  X e2 ; …11†  1=2  ÿ
1ÿ2m : f …t † P ˆ ‰km ŠP
t1=2 Matching both [Sn (e)]Z and [Sn (e)]P with the numerical curve of Sn (e) in part (a) of Fig. 2, we obtain, ‰Sn …e†ŠZ ˆ ‰Sn …e†ŠP ˆ 6:59
e0:8 for 10ÿ2 > e > 10ÿ5 :

…12†

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Z.L. Zhang / Nucl. Instr. and Meth. in Phys. Res. B 149 (1999) 272±284

Fig. 2. The Born±Mayer potential scattering cross section approximated by the present theory and by a simple power law theories. (a) Matching ‰Sn …e†ŠZ and ‰Sn …e†ŠP with the numerical curve of Sn …e† in Ref. [7]. (b) Scattering cross section in modi®ed LindhardÕs variables. The present theory with m ˆ 0.1 gives a better approximation than the simple power law theory.

Thus, Eqs. (9), (10) and (12) and Table 1 give all the necessary parameters m ˆ 0:1;

p ˆ 0:3667;

‰km ŠZ ˆ 3:05

and

q ˆ 0:9522;

…13†

‰km ŠP ˆ 12:5:

Inserting these parameters back into Eq. (11) yields ‰f …t1=2 †Šz ˆ 13:2
e0:8 …t1=2
eÿ1 †

3 ÿ1:904


‰1 ‡ 0:579
…t1=2
eÿ1 †

Š;

0:8

‰f …t1=2 †Šp ˆ 12:5
…t1=2 † ; which is plotted in Fig. 2(b) for comparing with the numerical curves of f(t1=2 ). It is clear that very well, for 10ÿ2 P e P 10ÿ5 , [f(t1=2 )]Z ®ts ÿ 1=2them
 does not. In Fig. 2,the numerical while, f t P ÿ
curves Sn …e† and f t1=2 are copied from a and b of Fig. 6 in Ref. [7].

atomic collision cascade. Several distribution functions may be used to describe the collision cascade [15,16]. The slowing-down density Gij (E,E0 ) (Recoiling density Fij (E,E0 ), collision density FijC …E; E0 †), is de®ned as the average number of jatoms moving (recoiling, colliding) per energy interval (E0 ,dE0 ) in the collision cascade. Following a well-known procedure [15±17], the following equations may be derived for Gij ; Fij and FijC X Z 1 ak drik ‰Gij ÿ G0ij ÿ G00kj Š ˆ
dij d…E ÿ E0 †; Nv0 k X

Consider a random, in®nite medium of atomic density N, which consists ofPseveral species of atomic fraction aj …0 6 aj 6 1; aj ˆ 1†, mass Mj and charge Zj: Let an atom of type i with initial energy E slow down in the medium generating an

ak

k

X k

3. Slowing-down, recoil and collision densities

Z

Z ak

…14† drik ‰Fij ÿ Fij0 ÿ Fkj00 ÿ dkj
d…T ÿ E0 †Š ˆ 0; …15† 0

00

drik ‰FijC ÿ FijC ÿ FkjC ÿ dij d…T ÿ E ‡ E0 †

ÿ dkj d…T ÿ E0 †Š ˆ 0;

…16†

where E0 ˆ 12 Mj v20 ; Gij ˆ Gij …E; E0 †; G0ij ˆ Gij …E0 ; E0 † and G00ij ˆ Gij …E00 ; E0 † etc. Obviously, Eqs. (14)±(16) will have to be solved subject to the boundary conditions Gij …E; E0 † ˆ 0

for E < E0 ; and analogous:

Z.L. Zhang / Nucl. Instr. and Meth. in Phys. Res. B 149 (1999) 272±284

3.1. Basic solutions

drij …E; T † ˆ

Qi …E†Aij …X † dX ;

1 P X > h;

0;

otherwise:

…17†

Eq. (14) can be solved by following a standard procedure [15]. Introducing the new variables E ˆ E0 eu ; T ˆ E0 ev in Eq. (14) and taking the Laplace Transform on both sides with respect to u, we obtain a set of algebraic equations. Solving them yields ‰D…s†Šji 1 ~ij …s† ˆ
; G Nv0 E0 Qj …E0 † D…s† where S

bij …s† ˆ aj c


Z

1 0

…18†

S

1 0

c stands for cij ;

dX Aij …X †‰1 ÿ …1 ÿ cX †s Š;

Dij (s) is the element …i; j† of the determinant D(s) ( for i 6ˆ j; Dij …s† ˆ ÿbij …s† P biK …s†‰eiK …s† ÿ diK Š; Dii …s† ˆ K

[D(s)]ij is the algebraic cofactor of the element ~ij …s† is the Laplace transform of (i,j) of D(s), G R ~ij …s† ˆ 1 du eÿSu
Gij …E0 eu ; E0 †. A Gij …E; E0 † : G 0 simple calculation shows that eij …1† ˆ 1:

…20†

By using Eq. (20), following a similar procedure as in Ref. [15], it is easy to show that the highest zero of D(s) is s ˆ 1. Therefore, the asymptotic solutions to Eq. (14) are given by Gj …E0 † ˆ

…23†

where Gj (E0 ) ˆ Gij (E,E0 ), etc. X X Dˆ Dk bkj ekj ; j

k

Dj ˆ ‰D…1†Šjj ; bij ˆ bij …1†; eij ˆ

aj bÿ1 ij

bj ˆ

Z

1 0

X k

bjk ;

dX Aij …X †‰ÿ…cX † ln …cX †

Ci ˆ eÿ1 ii ; …19†

Z

…22†

ÿ …1 ÿ cX † ln …1 ÿ cX †Š;

dX Aij …X †
X ;

eij …s† ˆ aj ‰bij …s†Šÿ1


E Dj
b for E  E0 ; E02 D j E Dj X b …0† for E  E0 ; FjC …E0 † ˆ 2
E0 D k jk

Fj …E0 † ˆ

For a scattering cross section. 

277

Cj Gj E

v0 j…dE0 =dx†j j …G†j E0

for E  E0 : …21†

The same procedure generates the asymptotic solutions to Eqs. (15) and (16)

…24†

bjj Dj Gj the nonstoichiometry factor ‰1Š ˆ
…G†j aj Cj D j…dE=dx†i j ˆ N ‰Sn …E†Šii ; ‰Sn …E†Šij ˆ EQi …E†bij =aj the stopping cross section for an i-atom hitting a j-atom: Comparing Eqs. (22) and (23) with Eq. (21), we obtain the following identities, Wj …E0 † ˆ Lj …E0 †Fj …E0 † ˆ Lcj …E0 †Fjc …E0 †;

…25†

where Wj …E0 † is the atom ¯ux, which is de®ned by Wj …E0 † ˆ v0 Gj …E0 †; Lcj …E0 † is the mean free path of j-atom with energy E0 Lcj …E0 † ˆ

NQj …E0 †

1 P k

bjk …0†

for 1 > h > 0

and Lj (E0 ) is the mean slowing down distance of jatom with energy E0 Lj …E0 † ˆ

1 NQj …E0 †bj

for 1 > h P 0:

3.2. Atom ejection process To explain Eq. (25) in relation to the average depth of origin of sputtered atom, let us derive the

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Z.L. Zhang / Nucl. Instr. and Meth. in Phys. Res. B 149 (1999) 272±284

atom ¯ux produced by a homogeneous and isotropic atom source f(E0 ) dE0 in an in®nite medium. First of all, we note that the atom ¯ux must be homogeneous and isotropic, due to a simple symmetry consideration. Thus, it can be treated as a one dimensional problem (x-axis). To determine the energy spectrum W(E0 ) dE0 of the atom ¯ux, Falcone and Sigmund have proposed two di€erent theories. (i) Falcone±Sigmund theory [3,4,7]. The basic assumption of Falcone±Sigmund theory is that each ¯ux atom must come from some atom sources, and the source atom slows down continuously along a straight line to contribute the atom ¯ux. Thus, all particles in the atom ¯ux W(E0 ) dE0 at the origin O of the x-axis must slow down from an uniformly distributed atom source f(E) dE on the positive (or negative) side of the x-axis, i.e. x2(0,1). The energy loss of each atom is Z E dE dE ; …26† ˆ ÿNSn …E† or x ˆ dx NS n …E† E0 which de®nes a function E ˆ E(E0 ,x). A direct di€erentiation of Eq. (26) yields dx ˆ

…dE†E0 NSn …E†

and

…dE†x dE0 ˆ : Sn …E† Sn …E0 †

density Fj (E0 ) as an atom source, Falcone±Sigmund theory can give a correct atom ¯ux Eq. (25). In other words, any loss of particles due to scattering from a given direction of motion is compensated by an equivalent gain [3]. (ii) FalconeÕs theory [6]. The basic assumption of FalconeÕs theory is that each ¯ux atom must come from some atom sources without undergoing collision on its way. Thus, all particles in the atom ¯ux W(E0 ) dE0 at the origin O of the x-axis must come from a uniformly distributed atom source F(E0 ) dE0 on the positive (or negative) side of the x-axis, i.e. x2(0,1). The energy spectrum of the atom ¯ux must be given by W…E0 † ˆ Lc …E0 †F c …E0 †:

…29†

Therefore, considering the collision density Fjc …E0 † as an atom source , FalconeÕs theory can give a correct atom ¯ux Eq. (25). However, in order to estimate the average depth of origin of the sputtered atoms, Falcone chose the recoil density Fj (E0 ) as an atom source in his analysis, i.e. Wj …E0 † ˆ Lcj …E0 †Fj …E0 †; which contradicts our basic transport theory Eq. (25), because Fjc …E0 † 6ˆ Fj …E0 †:

Then we have: …dE†x
dx ˆ L…E0 †

dE0 …dE†E0 ; E0

…27†

where (dH)w represents the di€erential of H, with W remaining constant. The energy spectrum of the atom ¯ux at the origin O is given by Z 1 f …E†…dE†x dx: …28† W…E0 † ˆ 0

Inserting Eq. (27) into Eq. (28) yields W…E0 † ˆ L…E0 †F …E0 †; where F …E0 † ˆ

1 E0

Z

1 E0

f …E† dE:

If F(E0 ) ˆ f(E0 ), we have f(E0 ) / Eÿ2 0 , which is satis®ed by Eq. (22). Therefore, considering the recoil

3.3. Asymptotic solutions for power potential interaction For power potential interaction Eq. (1), cross section (2) and Eqs. (6)±(8) ®t Eq. (17), thus, all kinds of functions, such as bij …s†; eij …s† etc. can be computed. C de®ned by Eq. (24), is a function of only one variable m, denoted by Cm . Speci®cally we have: (i) Present theory for 1 P m > 0 R1 dg
X 0 Cm ˆ R 1 dg
‰ÿX ln X ÿ …1 ÿ X † ln …1 ÿ X †Š 0 …Tables 1 and 2†; which can be calculated in conjunction with Eq. (5). (ii) Present theory for m ˆ 0

Z.L. Zhang / Nucl. Instr. and Meth. in Phys. Res. B 149 (1999) 272±284

bij …s† ˆ

aj rij cs ; …s ‡ 1†

Inserting Eq. (34) into Eqs. (32) and (33), we obtain the general Andersen±Sigmund relation  2m Wi …E0 † ai Mj ˆ : …35† Wj …E0 † aj Mi

s‡1

eij …s† ˆ cÿsÿ1 ‰…s ‡ 1†c ÿ 1 ‡ …1 ÿ c† Š;  2 1 1 ÿ 1
ln …1 ÿ c† ÿ ln c; eij ˆ ‡ c c C0 ˆ 1

Obviously, this relation is tenable for an arbitrary multicomponent target both in the present theories and in theories utilizing a simple power cross section.

…Table 1†:

(ii) SigmundÕs theory for 1 P m P 0 eij …s† ˆ

Cm ˆ

…s ÿ m† ÿs c m s
‰…1 ÿ c† ÿ 1 ‡ scm
Bc …1 ÿ m; s†Š;

m w…1† ÿ w…1 ÿ m†

279

3.5. Intrinsic di€erences between theories

…Tables 1 and 2† ‰15Š:

3.4. The Andersen±Sigmund relation [15,18] If a power potential Eq. (1) satis®es the following conditions, …30† m1 ˆ m2 ˆ mn  m; kij ˆ kji ; rij ˆ rji ; by using Eqs. (3) and (19), we obtain   bji ai Mj 2m ˆ 1 P m P 0: …31† bij aj Mi

Cm , de®ned in Section 3.3, is calculated and plotted in Fig. 3 for both theories as a function of m for 1 P m P 0: Some of numerical results of Cm are collected in Tables 1 and 2, where ‰Cm Ša and ‰Cm Šb are calculated by Eqs. (6) and (7) respectively. It is clear that [Cm ]z  [Cm ]P for 1 P m > 0.5. However, a great discrepancy can be seen for 0:25 > m P 0. Certainly, this discrepancy will cause some intrinsic di€erences. In order to understand this, it is necessary to restudy the number of recoils [19] and the mean slowing-down distance. For convenience, let us introduce Km ˆ ‰Cm ŠP =‰Cm ŠZ

…36†

If we take the ratio between the ¯uxes of moving atoms of the two species, D and E in Eq. (21) drop out, hence Wi …E0 † Di ˆ : Wj …E0 † Dj

…32†

On the other hand, a direct expansion gives !   X bjr1
br2 r2


brk i Di ˆ A…r1 ; r2 ;


; rk † : birk
brk rkÿ1


br1 j Dj …33† By using Eq. (31), we can derive bjr1 br1 r2


brk i birk brk rkÿ1


br1 j  2m  2m  2m ar Mj a r M r1 a i M rk ˆ 1
2


a j M r1 a r1 M r2 ar k M i  2m ai M j ˆ : …34† aj M i

Fig. 3. Cm , as a function of m. Thick line: Present theory. Thin line: Sigmund theory.

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Z.L. Zhang / Nucl. Instr. and Meth. in Phys. Res. B 149 (1999) 272±284

as a function of m. For 0:5 > m P 0; Km is smaller than unity and decreases with m until m ˆ 0 is reached, where, K0 ˆ 6=p2  0:6079. For simplicity only a monatomic medium will be discussed. C in Eq. (3) and L(E0 ) in Eq. (25), are denoted by Cm and Lm (E0 ) respectively. (i) Number of recoils The number of recoils N  …E0 † was introduced by Urbassek et al. in Ref. [19] as the average number of particles slowing down via collision from an energy E0 > E0 to an energy below E0 : Z E Z E0 N  …E0 † ˆ N dE0 W…E0 † dr…E0 ; T †: E0 ÿE0

E0

The asymptotic formula for N  …E0 † can be derived in both theories, E N …E0 † ˆ Cm E0 

for E  E0 ;

…37†

which makes an opportunity of measuring Cm : In Ref. [19], Urbassek et al. assigned a local power exponent m to the Kr -C stopping cross section: m ˆ 0.12 for Hf±Hf collisions and m ˆ 0.23 for C± C collisions at 10 eV interaction energy. Thus, the number of recoils N generated above 10 eV in a collision cascade initiated by 100 keV Hf ion is predicted by Eq. (37), ‰N  Šz ˆ 7290

‰N  Šp ˆ 5500 for Hf ! Hf;

‰N x Šz ˆ 5890;

‰N  Šp ˆ 5000 for Hf ! C;

where N ˆ N (10 eV). Seeing Fig. 2(c) on page 623 in Ref. [19], we realize that [N ]z reproduce the corresponding data precisely. However, [N ]p is too low. Because N data were generated by using the Kr -C potential interaction in Ref. [19], they naturally provide evidence supporting the point that replacing the actual interatomic potential by a power potential Eq. (1) is a less severe simpli®cation than replacing the actual cross section by a simple power cross section Eq. (8). In general, we have the ratio, ‰N  …E0 †ŠP =‰N  …E0 †ŠZ ˆ Km :

…38†

(ii) Mean slowing- down distance In this case, the atom ¯ux can be derived based on Eq. (21),

W…E0 † ˆ ‰Lm …E0 †Cm Š


E : E02

…39†

Adjusting [Cm ]z makes ‰W…E0 †Šz ˆ ‰W…E0 †Šp , and we get the ratio ‰Lm …E0 †ŠZ =‰Lm …E0 †ŠP ˆ Km :

…40†

As a result, we can see ‰Lm …E0 †ŠZ  ‰Lm …E0 †ŠP

for 1 P m P 0:5;

‰Lm …E0 †ŠZ < ‰Lm …E0 †ŠP

for 0:5 > m P 0:

Thus, at low energies (m  0), the collisions become less penetrating than predicted by SigmundÕs theory [9]. Particularly for m ˆ 0, the cross sections Eqs. (2) and (8) reduce to ‰drŠZ ˆ rEÿ1 dT

and ‰drŠP ˆ C0 T ÿ1 dT ;

…41†

respectively. Substituting Eq. (41) and r ˆ p2 C0 =3

…42†

into Eqs. (39) and (40), the theories give identical atom ¯uxes, but two di€erent mean slowing down distances ‰L0 ŠP ˆ

1 ‰7Š NC0

and

‰L0 Šz  0:6079‰L0 Šp :

…43†

This is exactly the point which results in the Ôdepth of originÕ puzzle.

4. Distribution of deposited energy A set of linear transport equations have been derived for the depth distribution of deposited in Ref. [20]. For energy F…i† (E,h,x) with m1 ˆ m2 ˆ drij …E; T † ˆ Eÿ2m
Aij …X † dX


m; following the standard procedure [20], we obtain the recurrent relations for the moments 8 0 a…i†0 ˆ 1; > > > > P nL h nÿ1 < n n dki
La…K†Lÿ1 a…i†L ˆ …2L‡1† …44† k > > i > > : ‡…L ‡ 1†anÿ1 …K†L‡1 ;  n 2m n where F…i†L …E† ˆ an…i†L
E EN bi , nL Dij is the element (i, j) of the determinant DnL ;

Z.L. Zhang / Nucl. Instr. and Meth. in Phys. Res. B 149 (1999) 272±284

281

  8 nÿ1 ÿn s > DnL bj > ij ˆ ÿaj cij bi > > > R1 > 00 > s > >
0 dX Aij …X †X PL … cos /ij † for i 6ˆ j; > < h P R1 s ˆ ak 0 dX Aik …X † 1 ÿ …1 ÿ cik X † DnL ii > > k >  i > ÿ
> s > > PL cos /0ik ÿ dik … cik X † PL cos /00ik ; > > > : s ˆ 2nm ‡ 1; cos /0ij and cos /00ij were given on p. 22 in Ref. [1]. ‰DnL Šij is the algebraic cofactor of the element …i; j† of DnL , and dijnL ˆ ‰DnL Šij =DnL : By using the approximation in Ref. [20], Zi =Zj ˆ Mi =Mj  qij , we obtain !mÿ1 2=3 1 ‡ q Cij ij …5mÿ2†=3 ˆ q3m
il
qjm 2=3 Clm 1 ‡ qlm for the Thomas±Fermi potential; Cij 5m=2 m=2 ˆ qil
qjm Clm

for the Born±Mayer potential: …45†

According to Eqs. (44) and (45), a small program has been written in BASIC for calculation of the moments anL in both theories . Thus, we have: X an …h† ˆ …2L ‡ 1†PL … cos h†; …46† L

where the subscript (i) has been dropped. Based on the Pade method [21], another small program has been written in BASIC for the reconstruction of the depth distribution a…h; X † from the moments an …h† in Eq. (46). Finally, we have the depth distribution of deposited energy FD …E; h; x† ˆ NSn …E†a…h; X †;

…47†

P where X ˆ x/L(E) and Sn …E† ˆ k ak ‰Sn …E†Šik ; ‰Sn …E†Šik is given by ZBL in Ref. [22]. The energy deposited in the target surface is given by FD …E; h; 0† ˆ N a…h†Sn …E†;

a…h†  a…h; 0†:

…48†

The calculation has been done on the SHARP PC-1500 pocket calculator for a few typical ion± target systems at a perpendicular incidence. The numerical depth distribution functions a…X † ˆ a…0; X † are plotted in Fig. 4. A remarkable result is

Fig. 4. Depth distribution of deposited energy for a target (mass M2 ) bombarded by a projectile (mass M1 ) at perpendicular incidence. The moments are evaluated by using Eqs. (7) and (8) respectively with m ˆ 0.2. The reconstruction of the depth distribution functions is done by Pade method (l/m)f on the basis of (l + m + 1) moments. At least 12 moments are used for the calculation, the computational accuracy being about 10ÿ8 . (a) M2 /M1 ˆ 1.5: l ˆ 5, m ˆ 8. (b) M1 ˆ M2 : l ˆ 4, m ˆ 8 in the simple power law; l ˆ 5, m ˆ 8 in the present theory. (c) Hg‡ ! M0 , M2 /M1  0.5: l ˆ 4, m ˆ 8.

282

Z.L. Zhang / Nucl. Instr. and Meth. in Phys. Res. B 149 (1999) 272±284

that, even though Fig. 1 shows a great discrepancy of Zm (x) between the two theories for m ˆ 0.2, they generate almost the same dimensionless function a ˆ a…0†, the relative error being less than 10%.

which is the number of j-atom recoiling from a 0 0 depth ÿ 0 (x,dx)
at energy …E ; dE † into a solid an2 0 gle X ; d X induced by one incident i-atom, where

5. E€ect of the surface binding energy

e0 ˆ E0 =U0j ; x ˆ XLj ;

For the power potential interaction Eq. (1), within the linear cascade regime [15], Eq. (21) gives the outward particle current Jj dE0 d2 X0 of target atoms of type j [1], Jj …E0 ; X0 † ˆ

Gj Cm cos h0
: FD …E; h; 0†
…G†j 4p E0 …dE0 =dx†j

Taking account of the refraction by the planar surface binding energy U0j [1] gives the sputtered atom ¯ux. Integrating over energy and solid angle, we obtain the partial sputtering yield of component j, Yj ˆ BKj ; where Bÿ1 ˆ …1 ÿ m†…1 ÿ 2m†; Kj …x† ˆ

…49† Kj ˆ Kj …0† and

Gj Cm aj FD … E; h; x†

: …G†j 8 N bjj U0j1ÿ2m

5.1. Sputtering yield ratio If the power potential Eq. (1) satis®es Eq. (30), using Eqs. (32) and (35), it is convenient to derive the partial sputtering yield ratio between two components i and j,  2m  1ÿ2m Y i ai M j U0j ˆ
; …50† Yj aj Mi U0i which is tenable for n P 2 in both theories. 5.2. Depth distribution of origin of sputtered atoms [3,4] The recoil density of the source of sputtered atoms is given by Eq. (22), ÿ
2 ÿ2 …51† d4 Fj …E0 ; X0 ; x† ˆ …e0 † de0 d2 X0 dX Kj XLj ; p

e0 ˆ E0 =U0j ; Lj ˆ U0j2m =N bj :

According to Falcone±Sigmund theory, each recoil atom of type j slows down continuously along a straight line, thus cos h0


dE ˆ ÿN ‰Sn …E†Šj ˆ ÿN bj E1ÿ2m : dx

The energy E0 of a recoil atom with initial energy E0 after having traveled from x to the surface, is given by  1=2m 2mX : …52† e0 ˆ e0 1 ‡ 2m e0 cos h0 Substituting Eq. (52) into Eq. (51) yields the atom ¯ux on the target surface. Taking account of the refraction by the planar surface binding energy U0j gives the sputtered atom ¯ux. Integrating over energy and solid angle as well as depth , we ®nally obtain the partial sputtering yields, Z 1 dX Kj …XLj †H …X †; where Yj ˆ B 0

4 H …X † ˆ
B

Z

1 0

Z dy

y2 0



2mX 2m dt 1 ‡
t y

ÿ1ÿ1=2m …53†

Regardless of the depth dependence in Kj …x†, Eq. (53) is reduced to Eq. (49) by setting x ˆ 0 [4]. This result further con®rms the continuous slowing R1 down approximation [4]. Since 0 dX H …X † ˆ 1, the escape depth of sputtered atoms xj must be given xj ˆ

4…1 ÿ m†Lj …5 ÿ 8m†…1 ÿ 4m†

for 0:25 > m P 0:

…54†

For a monatomic medium, Y in Eq. (49), x and L in Eq. (54), are denoted by Ym , xm and Lm (U0 ) respectively. Eq. (49) is thus reduced to Ym ˆ B‰Lm …U0 †Cm Š


N aSn …E† : 8U0

…55†

Z.L. Zhang / Nucl. Instr. and Meth. in Phys. Res. B 149 (1999) 272±284

Adjusting ‰Cm Šz makes ‰Ym ŠZ ˆ ‰Ym ŠP , and we get the ratio ‰xm ŠZ =‰xm ŠP ˆ Km :

…56†

It is clear that ‰xm ŠZ < ‰xm ŠP for 0:25 > m P 0. Particularly for m ˆ 0, substituting Eqs. (41) and (42) into Eqs. (55) and (56), the theories give the same formula for the total sputtering yield, Y0 ˆ

3 aSn …E†
2 4p C0 U0

‰1Š

…57†

but two di€erent depths of origin of sputtered atoms h i h i h i 0:8 x0 ˆ 0:6079
x0 x0 ˆ and …58† P Z P NC0 which gives us a credible answer to the Ôdepth of originÕ puzzle of Ref. [7], and means that hard sphere collisions are dominant in the low energy (U0 ) cascade. 6. Summary (i) A universal function Zm (x) has been introduced for the ®rst time. With the use of Zm (x), an exact scattering cross section (2) corresponding to a power potential (V µ rÿ1=m ) can be written. In addition, the exact cross section may be approximated by a sum of two power cross sections (6) and (7). Eq. (6) gives a much better approximation to the Born±Mayer interaction scattering in low energy region (m  0.1), than the well known Lindhard power cross section (8). (ii) Without recourse to the exact value of Zm (x), the transport equations (14), (15) and (16) and (44) can be solved analytically. The wellknown Andersen±Sigmund relations (35) and (50) have been derived in both theories for an arbitrary multicomponent target. (iii) With the use of Eqs. (7) and (8), p and q in Table 2, the depth distribution of deposited energy has been computed in both theories. The calculated results show that, even though for m ˆ 0.2, the discrepancy between the cross sections is notable, both theories give almost the same amount of energy deposited in the target surface layers Eq. (48).

283

(iv) Cm , has been calculated in both theories, as a function of m; 1 P m P 0. The results show a big gap between ‰Cm ŠZ and ‰Cm ŠP for 0.25 > m P 0. Particularly for m ˆ 0, we have ‰C0 ŠZ ˆ 1 > ‰C0 ŠP ˆ 6=p2  0:6079. (v) For the sputtered atom ejection process, it has been proved that the Falcone ``mean free path'' theory directly contradicts the transport theory, and been con®rmed that the Falcone±Sigmund ``slow down straight'' theory is tenable for an arbitrary polyatomic medium. (vi) Let the sputtering yield in the present theory equal the corresponding one in SigmundÕs. Then the depth of origin of sputtered atom will be reduced for 0.25 > m P 0, compared with the one in the Sigmund theory. The depth will be reduced by approximate one factor of 1/2, as long as m ˆ 0, or a hard sphere scattering cross section is used, which solves the Ôdepth of originÕ puzzle [7]. (vii) The modern sputtering theory splits the sputtering event into two steps [1]: In the ®rst step, the primary recoil atoms are created by an incident ion. The depth distribution of deposited energy is determined by the high energy recoil atoms, and a simple power scattering cross section (8) with m P 0.2 is suitable for computing the dimensionless function a…h† in Eq. (48). In the second step, a cascade of low energy recoil atoms is generated by the primary recoils, and some of the higher generation recoil atoms are ejected through the target surface. Here, usually m  0. Therefore, ‰drŠP ˆ C0 T ÿ1 dT must be replaced by ‰drŠZ ˆ rTmÿ1 dT in the analysis, because the hard sphere collision is dominant in this low energy (U0 ) cascade. Acknowledgements I would like to thank Dr. K.R. Padmanabhan, Associate Professor in Department of Physics and Astronomy of Wayne State University in USA, without whose initial supervision (1983±1986) this work could not have been done. Special thanks are due to Professor Zhen Hai Feng, ex-president of Huainan Institute of Technology in P.R.C, for providing the necessary conditions to complete this work. I am very grateful to the referee for

284

Z.L. Zhang / Nucl. Instr. and Meth. in Phys. Res. B 149 (1999) 272±284

many critical comments to the manuscript. Also, Professor H.H. Andersen, editor of NIMB, deserves special thanks for his constant encouragement during the revision of the paper and for the attention and criticism paid in reading the manuscript. References [1] P. Sigmund, in: R. Behrisch (Ed.), Sputtering by Particle Bombardment I: Top. Appl. Phys., vol. 47, Springer, Berlin, 1981, p. 9. [2] P. Sigmund, Phys. Rev 184 (1969) 383. [3] G. Falcone, P. Sigmund, App1. Phys. 25 (1981) 307. [4] P. Sigmund, A. Oliva, G. Falcone, Nucl. Instr. and Meth. B 194 (1982) 541. [5] R. Kelly, A. Oliva, Nucl. Instr. and Meth. B 13 (1986) 283. [6] G. Falcone, Phys. Rev. B 33 (1986) 5054; 38 (1988) 6398. [7] M. Vicanek, J.J. Jimenez Rodriguez, P. Sigmund, Nucl. Instr. and Meth. B 36 (1989) 124. [8] P. Sigmund, Rev. Roum. Phys. 17 (1972) 823. [9] J. Lindhard, V. Nielsen, M. Schar€, K. Dan. Vidensk. Selsk. Mat. Fys. Medd. 36 (10) (1968). [10] H. Goldstein, Classical Mechanics, 2nd Edition, AddisonWesley, Reading, MA, 1980.

[11] L.D. Landau, E.M. Lifshitz, Mechanics, 3rd Edition, 1976. [12] H.M. Urbassek, M. Vicanek, Phys. Rev. B 37 (13) (1988) 7256. [13] Z.X. Wang, Introduction to Statistical Mechanics (in Chinese), 2nd Edition, 1978, p. 136. [14] M.T. Robinson, in: D.W. Palmer, M.W. Thompson, P.D. Townsend (Eds.), Proc. Int. Conf. Atomic Collision Phenomena in solids, North Holland, Amsterdam, 1970, p. 66. [15] N. Andersen, P. Sigmund, K. Dan. Vidensk. Selsk. Mat. Fys. Medd. 39 (3) (1974). [16] W. Huang, H.M. Urbassek, P. Sigmund, Phil. Mag. A 52(6) (1985) 753. [17] P. Sigmund, Nucl. Instr. and Meth. B 18 (1987) 375. [18] H.M. Urbassek, U. Conrad, Nucl. Instr. and Meth. B 72 (1993) 151. [19] M. Vicanek, U. Conrad, H.M. Urbassek, Phys. Rev. B 47 (2) (1993) 617. [20] K.B. Winterbon, P. Sigmund, J.B. Sanders, K. Dan. Vidensk. Selsk. Mat. Fys. Medd. 37(14) (1970). [21] U. Littmark, P. Sigmund, J. Phys. D 8 (1975) 241. [22] J.F. Ziegler, J.P. Biersack, U. Littmark, The Stopping and Range of Ions in Solids, Pergamon Press, New York, 1985.