On leave from Faculty of Physical Science, Department of. Electricity and Electronics .... well as crystalline targets. In the absence of atomic mixing, F = 0, eq. (5).
NUCLEAR
INSTRUMENTS
AND M E T H O D S
168 (1980) 3 8 9 - 3 9 4 ;
(~) N O R T H - H O L L A N D
PUBLISHING
CO.
DISTORTION OF DEPTH PROFILES DURING SPUTTERING I. General description of collisional mixing P. SIGMUND and A. GRAS-MARTI*
Physics Institute, Odense University, DK-5230 Odense, Denmark
The present work concerns collisional mixing which is one of the factors limiting the depth resolution of sputter profiling. A general treatment comprising the effects of recoil implantation and cascade mixing on the distortion of an impurity profile in a bulk matrix is presented. The statistics of the distorting events is shown to be a close analogue to that of the stopping of charged particles in random matter, and describable in terms of the Bothe-Landau theory of energy loss; a general expression is given for the Green's function transforming a given impurity profile into an apparent (or distorted) profile. In the diffusion approximation, the present theory yields a well defined shift and smearing of an impurity profile without introduction of a (more or less arbitrary) effective minimum energy for displacement.
1. Introduction Sputtering by ion bombardment is extensively used as a tool for depth profiling in microanalysis of solids 1-4). An ion beam of known fluence is capable of removing material from a solid surface in well defined amounts corresponding to less than a monolayer if desired. Continuous recording of the composition of the surface and/or the flux of sputtered atoms allows in principle to trace a depth profile of the composition of an alloy, a doped sample, etc. In practice, the technique suffers from severe limitations4), part of which consist in lack of knowledge of important parameters, others being of a technical nature that might eventually be overcome. There are also limitations on the depth resolution that are inherent in the sputtering process; they depend on sputtering parameters like the type and energy of the bombarding ion beam, and may thus be optimized by selection of suitable bombardment conditions. It is that latter problem that is addressed in the present paper. Andersen 5) has recently discussed important physical aspects of the depth resolution in sputter profiling and, on the basis of simple, direct estimates, has given guidelines for the choice of sputtering parameters. In Andersen's work, the factor limiting the depth resolution of the sputter technique is the disordering of the sample by the ion beam prior to sputtering. Indeed, for a keV ion beam, the depth of disordering - which is roughly * On leave from Faculty of Physical Science, Department of Electricity and Electronics, University of Madrid, Madrid-3, Spain.
identical with the penetration depth 6) - may be as large as several hundred Angstr6ms, and this substantially exceeds the depth from which atoms are sputtered [typically less than 10 A 7)]. The primary disordering mechanism is collisional mixing 8) which, qualitatively, can be classified into recoil implantation and cascade mixing. The terminology for the latter two effects has been emerging graduallyS). In the following, and in accordance with refs. 5 and 9, the direct displacement of a target atom by a bombarding ion will be called recoil implantation while indirect processes involving other target atoms will be called cascade mixingS). Within this scheme, recoil implantation produces a shift and a broadening of a given initial profile while cascade mixing produces primarily a broadening. In the older literature, e.g. ref. 10, this distinction does not become apparent. It is, however, inconvenient to rigorously distinguish between recoil implantation and cascade mixing as will be seen subsequently. In addition to collisional mixing, one may have to deal with diffusional mixing. Such an effect is relatively easy to incorporate in principle but difficult to quantify because of a rather complete lack of knowledge of the pertinent parameters. A factor somewhat related to collisional and/or diffusional mixing is preferential sputtering, i.e., the fact that the composition of the sputtered flux may differ from that of the pertinent surface layer. It appears that preferential sputtering may cause systematic errors in sputter profiling - if not recognized as being an effect to be corrected for - but need not be a factor limiting the depth resolution V. S P U T T E R P R O F I L I N G AND SIMS
390
P. S I G M U N D
AND
beyond that caused by the factors previously discussed. The bulk of this paper deals with collisional mixing. From the point of view of physical mechanisms, our work offers little beyond what has been discussed previously in more or less detail 8,5,9-~1). Unlike these treatments, however, our description offers a simple and rather general scheme to treat the combined effect of recoil implantation and cascade mixing. In the present paper we show that the description reduces to previous ones s'9) after appropriate specification of the input. Results going beyond those discussed in the literature will be presented in a forthcoming paper II. The present work was motivated by a need for simple analytical predictions of the effect of atomic mixing on sputter profiling, a concern about superficial assumptions introduced in earlier treatments, and a desire to discuss different mixing processes within one unified scheme. The statistics of the distorting events is shown to be a close analogue to that of the stopping of charged particles in random matter, and describable in terms of the BotheLandau theory of energy loss, with recoil implantation producing a shift corresponding to stopping power, and cascade mixing as well as recoil implantation producing a broadening corresponding to straggling. The general conclusions of the theory can be expressed in a rather compact way, by means of the concept of a propagator expressing shift and broadening of profiles as a consequence of ion bombardment in general terms, independent of the particular model adopted for individual mixing processes. The assumption of random medium underlying previous approaches to the problem is avoided initially.
A. G R A S - M A R T I
where Y denotes the sputtering yield in [atoms/ion] and N the number density of target atoms. Since the sputtering yield may vary with fluence (see, for instance, ref. 12) v may depend on ¢. The disordering within the target is described in terms of some relocation function F(x,z), introduced previously 13) such that Nc (x, oh) F (x, z) dx dz (3) is the mean number of impurity atoms per incoming ion relocated from a layer (x, dx) to a layer (x+z, dz). The combined effect of sputter erosion and relocation of impurities due to a fluence increment de is contained in the following balance equation, c(x, qS+d~) = c(x + vddp, q~) +
+ d¢ ~ d z [ c ( x - z , ¢ ) F ( x - z , z ) - c(x, c~) r ( x ,
z)],
(4)
provided that the surface is always located at x = 0. The first term in the integrand on the right-hand side gives the contribution of impurities knocked from x - z to x, while the second term gives the loss in x. For d 0 ~ 0 , eq. (4) reduces to 13)
dz [ c ( x - z, qb) F ( x - z, z) - c(x, c~) F(x, z)].
(5)
In principle, both the relocation function F(x, z) and the sputter rate v depend on the instantaneous profile c(x,O) so that one deals with coupled systems13). This complication can be circumvented
either 1) if the
impurity
concentration
is
small,
c(x, O)~l, or
2. Basic equation Let us consider a semi-infinite target with a plane surface at x = 0, and an impurity profile Co(X) [impurities/matrix atom]. After bombardment with an ion fluence 0 [ions/area], the impurity profile has changed to a profile c(x, ~) because of the combined effect of disordering and sputtering, so that c(x,O) = Co(X). (1) The surface of the target moves because of sputter erosion at a rate v = dx/d¢ = Y/N, (2)
2) if the relative change in composition due to disordering is small. In the former case, target properties are determined essentially by the matrix. In the latter, any dependence of F on v or c may be incorporated approximately as an explicit dependence on 0 instead. As it stands, eq. (5) is rather general; it applies to random as well as crystalline targets. In the absence of atomic mixing, F = 0, eq. (5) yields
f]
(6)
DISTORTION
OF D E P T H
391
PROFILES
that is, the profile at any depth and fluence is merely the result of a translation of the origin of coordinates following the sputtered layer. If the surface concentration is recorded as a function of fluence, one finds
Finally, there is some depth range X that is affected by the ion beam in a single bombardment; this is of the order of the mean penetration depth (or projected range) of the ion. In cases of practical interest, the depth profile Co(X) to be measured extends at least over several c(0,~b) = co d~'v(qY . (7) hundreds of &ngstr6ms. The limitations due to the \Jo nonzero sputter depth AXo are, therefore, quite This suggests the introduction of the quantity acceptable. Because of the necessity of high-fluence bombardment, collisional mixing processes may y = d4,'v(4,') = y(40 (8) however considerably distort the profile. 0 We shall see in the following section that the as the depth variable for the apparent profile. accuracy of possible solutions to eq. (5) depends crucially on whether or not F(x,z) varies rapidly 3. Characteristic depths It is useful to clarify the role of various character- with x. Since the characteristic depths are X for the x-dependence and Axl for the z-dependence it is istic depths involved in the problem (cf. fig. 1). First of all, there is a characteristic depth Axo feasible to make the approximation (9) from which atoms are most likely to emerge in an F(x--z,z) ~- F(x,z) individual sputtering event, dx0 has been esti- in eq. (5). mated 7) to be of the order of 5 A. This is, quite roughly, the range of a matrix atom with an energy 4. General solution representative for sputtered atoms, i.e., a few eV. It is sufficient to solve eq. (5) for a narrow initial Clearly, Ax0 is a lower bound on the depth resolu- distribution corresponding to a &function profile, tion of sputter profiling. i.e. to determine the Green's function of the probNext, there is a depth Ax~ over which collisional lem, where mixing typically occurs in an individual ion bombardment event. The magnitude of this quantity depends on whether we consider recoil implantation or cascade mixing, and on the way how a statistical average is taken. Regardless of the details, Axj is = f dz[a(x-z,x',cb) f(x-z,z) presumed to be greater than Ax0 but smaller than the penetration depth of the ion beam. - G(x, x', c/)) F(x, z)], (10) and /~x t > Ax,
¢//
G(x,x',O) = 6 ( x - x ' ) .
(11)
An arbitrary profile varies, then, according to ION
"~
BEAM
I
c(x, ¢) = j dx' G(x, x', ¢) Co(X') .
It is readily seen that eqs. (1 I) and (12) reduce to eq. (1), and eqs. (10) and (12) to eq. (5). After introduction of a new variable ~ according to
5 S /
~
(12)
~
ean (projected) range
Fig. l. Sketch of important depths in sputter profiling. The x-axis is perpendicular to the sofid surface. FD(X) is a deposited energy distribution. For further explanations see text.
= x+
dqb'v(qb')~ x + y ( 4 ) ,
(13)
eq. (10) reads
L 34, H(~,(a)
= f dz[H(~-z,O) F(~-z-y,z) - H(~, (])) r ( ~ - - y , z)], V. S P U T T E R
PROFILING
(14)
A N D SIMS
392 where
P. S I G M U N D
(~
AND
)
H(~,49) = G\, -y(49),x',49 ,
(15)
and
H(~,O) = f(~-x').
(16)
The dependence on x' is implicit in H but has been dropped in the notation. Solution of eq. (14) with the initial condition (16) is particularly easy in case all dependence of F(x, z) on x can be ignored. Then, eq (14) reduces to
H(~,49) = j" dzF(z)[H(~-z, 49)- H(~,49)], (17) which is well known from the theory of stopping and multiple scattering of charged particles in thin layersW-17). Going through Fourier space, one finds the solution1% U(~, 49) --
A. GRAS-MARTI
Actually, only the surface concentration c(0,0), and hence the value of G(0,x', 0) is of interest. Also, the apparent depth variable is y =y(O) according to eq. (8). Therefore, the apparent profile c(y) -~ c(O,49) (22) is given by t~
c(y) = j dx' G(y,x') Co(X'), with
6(y,x') -~ G(0,x',49) = ua
=--1 f dk exp[ik(y-x')] × 2zr 3 xexp
1 _f dk exp [ik(~-x')]• exp[-49
x
f dzF(z)(1-e-ik~)],
(18)
which is now often called Bothe's formula. Eq. (18) is not of much use except in cases where the profile to be analyzed extends over a depth range ~X. In all other cases, the dependence of F(x, z) on x need to be accounted for. Provided that eq. (9) is valid, the equation to be solved reads
-~ H(~,49) = f dzF(~-y,z)[H(~-z, 49)- H(~,49)], (19) Comparison with eq. (17) suggests the
ansatz
H(~,49) =~--~ dkexp[ik(~-x')] x x exp{-
d49'
dzf(~-y(49'),z) x
x (1-e-i~z)},
(20)
instead of eq. (1 8). Differentiation with respect to shows that within the range of validity of eq. (9), eq. (20) is a solution of eq. (19). Also, eq. (20) satisfies eq. (16). Going back to the Green's function by means of eq. (15), we obtain
G(x,x',49) = ~
dkexp{ik[x-x'+y(49)]} x x exp{-- f~ d49" f dzF(x+y(49)-
- y(49'), z) (1 -e-lkz)}.
(21)
-
o ~-~)
(1- e-lk~)],
dzF(y-y ,z) x (24)
where
v(y') = o [49(y')].
(25)
Eq. (14) becomes much more difficult to solve, once eq. (9) is no longer valid. It is worth-while, therefore, to discuss the limitations of eq. (9) in some more detail. It was mentioned previously that eq. (9) implies that Ax~X. More specifically, it is implied that processes where an atom is dislodged in a single bombardment event over a depth of the order of X are inappreciable. Since the fluence interval for which relocation may occur is of the order of NX/ Y, a given atom has a probability n
0
(23)
NX
n
Y
Y
NX
to be dislodged over a distance - X during the sputter process, n being the mean number of relocat i o n s - X per bombarding ion. Thus, the condition for eq. (9) to be valid is n ,~ Y,
(26)
i.e., the mean number of atoms per bombarding ion that is transported over an appreciable fraction of the penetration depth should be small in comparison with the sputtering yield. Alternatively, a solution to eq. (14) not based on the validity of eq. (9) can be found by means of a perturbation expansion with F being the perturbationS3). One readily finds the solution
393
D I S T O R T I O N OF D E P T H P R O F I L E S
H(~,(b) = 6 ( ~ - x ' ) +
de'
dz[f(~-z-x')
x
Expanding 1 - e -ikz ~ ikzw½k2z 2 ...
(29)
x F(~-z-y',z)-
in eq. (24) we find a propagator
- ~(¢-x') F ( ~ - y ' , z ) ] + ...,
1 [ (y-x'-a)21 G(y,x') - x/(2~ 2) exp 2u 2 _],
(30)
or, going through the steps leading to eq. (24),
dy' G(y,x') = 6 ( y - x ' ) + f Yo V ~ x
f dz[6(y-x'-z)
with ×
a = .Jo v(y')
F(y-y'-z,z)-
- 6(y--x') F ( y - y ' , z ) ] + ....
,z),
(31a)
and (27)
Eqs. (27) and (23) yield
c(y) = co(y) + f 'o ~dy' x x f dz[co(y-z) F(y-y'-z,z)-- co(y) F(y--y',z)].
dz.zF(y-y
(28)
~2 = Jo f ' v(y') dy' f dz. z 2 t ( y - y ' , z ) .
(31b)
Thus, the apparent profile c(y) is shifted relative to Co(X) by a depth-dependent distance a, and smeared by a depth-dependent width ~z (fig. 2). These quantities are analogous to the mean energy loss and straggling, respectively, of particles penetrating a layer of random matterlg). Most important is the case of impurities buried at large depths (>X). We can then write
5. Diffusion approximation It is instructive to evaluate eq. (24) in the diffusion approximation. This implies that atoms are dislodged repeatedly during the sputtering bombardment, typically over a considerably larger distance than the maximum relocation in a single bombardment.
a ~--
1 ~- v(y)
1
dz. z F(y', z)
f fdz"oz oF(y', z) dy',
1 fdz.z2fo
F(y', z) dy'.
~2 ____v(y) _
x
¢(y)
Fig. 2. Evolution of narrow profiles during bombardment (schematic). Upper part: four profiles. Middle: deposited energy distribution defining the range over which damage occurs in a single bombardment. Lower part : distorted profiles as a function of the sputtered layer thickness. Note that the shift and smearing of the profiles increases up to stationary values which have been reached by the two deepest profiles.
(32b)
In case of negligible dependence of the sputtering yield on ion fluence, eq. (30) represents a Gaussian resolution function with parameters a and ~z being determined by the individual disordering event. In the opposite case of an impurity near the surface (dxo~y~X) one obtains
a = ~
y=
(32a)
and correspondingly
c(x,o) = c , ( x )
Jl
dy' o v(y-y')
dz'z
~ v(0)
,f
fo
dy' F(y',z)
dz "z F(0, z),
(33a)
and
~2 =,.~ v(0) Y f dz- Z 2 F(0, z).
(33b)
However, the validity of the diffusion approximation may be more questionable in this casel4-17). V. S P U T T E R P R O F I L I N G AND SIMS
394
P. S I G M U N D
A N D A. G R A S - M A R T I
6. Connection with previous treatments As an application of the present formulation, we shall briefly derive Andersen's 5) expression for the depth resolution, when the corresponding assumptions are incorporated. The dependence of the sputtering yield on ion fluence is ignored; from the definition (3) of the relocation function F, we obtain in case of an isotropic recoil distribution ~8'9)
NF(x, z) =
~ fde'f
dE' F m FD(X)
,
FR(E', e, z), (34)
where m r~ = ~(1)- ~(1-m)'
(34a)
and m is the power in the cross section for recoil generation; q/(x) is the digamma function; FD(X) is the deposited energy distribution function, E' and e' the energy and direction of recoils generated in a cascade, respectively, and FR the projected range distribution of these recoils. Andersen's 5) model is equivalent to a range profile FR(E' , e', z) = 6 ( z - R cos0'), (35) where cos0' is the projection o f e' on the x-axis, and R is a fixed recoil range. Neglecting surface effects, the stationary broadening we find according to eqs. (34), (35) and (32b), c~2 = - - - t 3 Y L (E)min_]
(36)
where v (E) is that part of the incident ion energy E spent in elastic collisions, and (E')m~n a lower limit for E'. The quantity in brackets is the number of recoils generated per ion. Andersen replaced it by the conventional Kinchin-Pease function including replacements ~8) and obtained, therefore, °~z = 0.42R2 v(E)
3
(37)
YEa,err"
This is eq. (7) of ref. 5 except for a spurious factor 1/,/2 in that formula. The assumption of a constant recoil range can easily be dropped in our scheme; introducing the moments over the range distribution as in ref. 6),
F nR ( Et, e )! =- ~ dz.
Zn
"FR(E',e',z)
= ~ (21 + 1) FR. t (E')" P, (cos 0'),
(38)
l
where P; are Legendre polynomials, we get correspondingly
-
r v(e) f Y
2 ,). ( - ~ FR,o(E
(39)
If the coefficients F~,0(E') are proportional 6) to E '4"' with 0 < m ~< 1, the integral only converges for m>¼ when the lower limit is set equal to zero. A similar situation was encountered in ref. 11. The expressions above are in fact rather similar to the ones used by Hofer and Littmark 9) except for the fact that a diffusion limit is not considered in their work. The assumption of an isotropic cascade implies that the shifts predicted by the theory are zero, according to eq. (32a). Non-isotropic effects like recoil implantation and the shifts and broadenings of profiles that it causes will be discussed in part II of this series. Clarifying discussions with U. Littmark are acknowledged. A fellowship from the Danish Ministry of Education, and a kindly arranged leave of absence, made possible the work of one of us, A. GrasMarti, to whom the members of the Physics Institute provided a most pleasant environment.
References 1) H. Lutz and R. Sizmann, Phys. Lett. 5 (1963) 113. 2) R. Kelly, Can. J. Phys. 46 (1968) 473. 3) j.W. Coburn, J. Vac. Sci. Technol. 13 (1976) 1037. 4) j.L. Whitton, in Physics of ionized gases 1978, ed. R.K. Janev, (Institute of Physics, Belgrade) p. 335. 5) H. H. Andersen, Appl. Phys. 18 (1979) 131. 6) K.B. Winterbon, P. Sigmund and J.B. Sanders, Mat. Fys. Medd. Dan. Vid. Selsk. 37, no. 14 (1970). 7) p. Sigmund, Phys. Rev. 184 (1969) 383; ibid. 187 (1969) 768. 8) T. lshitani and R. Shimizu, Appl. Phys. 6 (1975) 241. 9) W.O. Hofer and U. Littmark, Phys. Lett. 71A (1979) 457. 10) R.S. Nelson, Rad. Effects 2 (1969) 47. 11) K.B. Winterbon, in Physics of ionized gases 1978 (Contributed papers) Dubrovnik, Yugoslavia, p, 151. 12) H.H. Andersen, in Physics of ionized gases 1974, ed. V. Vujnovic (Inst. Phys., Zagreb) p. 361. 13) p. Sigmund, J. Appl. Phys., Nov. 1979. 14) W. Bothe, Z. Physik 5 (1921) 63. 15) L.D. Landau, Phys. Z., Soviet-Union 8 (1944) 201. 16) p. Sigmund and K. B. Winterbon, Nucl. Instr. and Meth. 119 (1974) 541, ibid. 125 (1975) 491. 17) j. Lindhard and V. Nielsen, Mat. Fys. Medd. Dan. Vid. Selsk. 38 no. 9 (1971). 18) p. Sigmund, Rev. Roum. Phys. 17 (1972) 823, 969, 1079. 19) N. Bohr, Mat, Fys. Medd. Dan. Vid. Selsk. 18, no. 8 (1948).
