- Email: [email protected]
The diffusion approximation in modelling impurity redistribution during sputtering
cm
Nuclear Instruments and Methods in Physics Research B 115 (1996) 473-477
NIUMI B
Beam Interactions with Materials 8 Atoms ELSEVIER
The diffusion approximation in modelling impurity redistribution during sputtering G. Carter * Deparrment
of Eiecironic and Electrical Engineering, University of Saljord. Suljord MS 4WT, UK
Abstract A method to calculate the moments of the surface concentration-eroded depth profile during sputtering of a solid containing an initially depth distributed concentration profile of equivalent atom markers, within the framework of a diffusion approximation which includes directed atomic drift, incident ion trapping and target atom spatial relaxation processes, is described. Illustrative examples reveal the perturbations introduced by ion trapping and how, in principle, measured moments can be used to reconstruct initial depth profiles. Corrected forms for large eroded depth exponential decay lengths of surface concentration profiles are discussed.
1. Introduction Depth profiling, using surface or near surface analytic probe techniques together with sputter-sectioning, is an important technique for composition assessment of materials. Although they can possess high sensitivity such methods usually suffer from depth resolution limitations because the aggressive process of ion bombardment sputtering can lead to atomic relocation in the bulk of the material and to changes in surface topography. These can lead to real or apparent changes in the depth profiles of the atomic component concentrations it is intended to evaluate and deduction of initial profiles from measured surface concentrations with increasing sputtered depth can be uncertain. Several different theoretical or computational approaches have been developed to aid interpretation of experimental data including solution of the atomic transport equations [ 1,2], evolutionary binary collision approximation computer codes [3] and diffusion approximations to the transport equations [4-61. In this latter area numerical methods have been used to allow solution of the coupled diffusion equations [7] when the solid is polyatomic. These approaches usually attempt to describe the evolution, with increasing sputtered material thickness, of the surface atomic concentration of the species of interest for a given initial depth profile of that species. A delta function at a fixed depth initial profile is frequently assumed. An alternative approach is to predict the moments of the surface concentration-sputter eroded thickness profile and this has been undertaken for an
Tel. +4l
161 745 58%; [email protected]. l
fax
+44
161 745 5119; e-mail
effective constant diffusivity of infinite [4] or finite [5] extent into the solid and for more generally spatially variable effective diffusivity [8]. In none of these treatments were the effects of implantation of the incident ion species or the relaxation of the solid to counteract local densification or rarefaction resulting from atomic relocation included. The present study shows how these processes may be included in a method of calculation of moments developed by Naundorf and Abromeit [8].
2. Moments of the surface eroded depth distribution
concentration-sputter-
For analytical specification a self-similar system is considered in which ions of a specific atomic species irradiate, implant into and sputter a target of the same atomic species containing, initially, a concentration profile C( z, O), of a marker species, again of the same atoms as the target. If irradiation is considered only to generate recoils isotropically within the bulk and sputter the surface then an effective depth dependent diffusivity D(z) and a sputtering velocity ZJ can be defined. In this order of approximation the diffusion equation for C( z, x) where z is depth below the instantaneous surface when a thickness x = /Cl dr has been eroded, can be written [4,5,8]
(1) If however, the irradiation is considered to give rise to a non isotropic, depth dependent drift velocity, v(z), to result in ion stopping and trapping with a probability
0168-583X/96/%15.00 Copyright 0 1996 Elsevier Science B.V. All rights reserved VI. PROJECTILE INDUCED/ASSISTED PROCESSES SSDI 0168-583X(95)01558-2
474
G. Carter/Nucl.
Instr. and Meth. in Phys. Res. B 115 (1996) 473-477
distribution p(z) and the effects of target relaxation to accommodate atomic relocation processes are considered, then Eq. (1) must be replaced by [9,10]:
--
-=
2s
dzp(z)
1 (2)
CT is the total atomic concentration, assumed constant, Ji is the incident ion flux and V is the net erosion velocity of the surface and equal to the difference between the sputtering velocity U - J,Y/C,, and the growth velocity due to ion accumulation v(Ji/CT)jt d z p (z) = Jiq/CT. It is notable that Eq. (2) is independent of the drift velocity, u(z) and can be written as:
(3)
For infinite range diffusivity the boundary conditions for Eqs. (1) and (3) are [4,5,8] DaC/az = 0; z = 0 and C = 0; Z’W. Following Naundorf and Abromeit [8], the moments G,,,(x, z) = 10”d x xmC(x, z) can be defined and successive differentiation of Eqs. (1) or (3) leads to defining ordinary differential equations for the moments defined at z - 0, i.e. G,,jx, 0). From Eq. (1):
= { -C(O,
+ uG,(z)
z)
-{ -mGmm-,(z)
for m-0, (4)
for rn> 1,
and from Eq. (3)
d’Gm( z>
w-&r
+R(z)G,(z)
3. Illustrative examples 3.1. No ion trapping, p(z) = 0; Delta function marker at z = 1 initially
v;-fyz)$+Q(z)E +R(z) c.
D(z)%
ments made to determine optimum values for these parameters. Of course the solutions of Eqs. (4) and (5) are, generally, not simple since for spatially variable D(z) and p(z) they are nonlinear and numerical methods are required. They are, however, more straightforward to solve numerically than the original partial differential equations (l), (2) and (3). In order to illustrate the approach three examples, employing simplifying assumptions to allow tractable analysis are chosen. In each of these D(z) is assumed to be depth independent in the range 0 < z < m.
This is the case examined by Naundorf and Abromeit [8] and Eq. (4) can be integrated easily and the recursion relations used to evaluate the moments. The first two moments, G,(O) and G,(O) can be used to determine the mean, (x) and the variance, u2, of the distribution and are given by:
G,(O)=(x)=/+;, (7)
+ 2$.
These results are different from those quoted by Naundorf and Abromeit [8] but agree identically with those derived [4,5] by Laplace transformation methods of solution of Eq. (1). 3.2. Ion trapping as a delta function at z - r, probability
+QW,z
I,I; Delta function maker at z = 1 initially
WA z)
=(-VC(0, z) for m-0, _ {_ mVG,,,_l(z) for mr
1.
(5)
For both Eqs. (4) and (5) it is readily shown that the following recursion relation holds G,(z = 0) = m/r dzG,,_,(z) with G,(z-0)=/i dz C(x, z). The moments G,,,(z = 0) are the mth order moments of the surface concentration-eroded thickness profile and can be extracted from experimental data. If suitable assumptions, or experimental measurements, are made on the values of D(z), p(z), U and V then, in principle Eqs. (4) or (5) can be solved for different assumptions of the initial marker profile C(0, z) and the predicted moments compared with experimental values and a best fit will yield C(0, z). Alternatively, if C(0, z) is initially specified, for example a delta function, the parameters D(z), p(z), U and V can be varied and comparison of predicted and measured mo-
In this case it is necessary to consider two situations, the first where r < 1 and the second for r > 1. It is now necessary to solve the somewhat more complex hierarchy of Eq. (5) and the results are also more complex. It can be shown that for r < 1; %r (x)_l+;+
OLzea”--le
-I=I+D
V
02-oi
+a-
1, (8)
and D2
a2=3
(1 7
+2;1
02
+2-
WV2
X21(a
V2e-;-U2e-;
(
- 1) + (o - Q2,
>
(9)
G. Carter/Nucl.
Instr. and Meth. in Phys. Res. B 115 (1996) 473-477
where
(x)-l+;+ 1-i
( 1
forr>l;
x
(D
e-u/w-‘)+r-
1
u
1 (10) +a-
1
and u2 ,=c~~[i+~~-l)-~]-r+2(~) Dl +Y+
]+kh
(11)
where $I- r) p=$[SD+(r-l)V]eD 5-r) -;[3D+(r-
l)V]eD
.
It is readily demonstrated that if ion trapping is absent (q=O, u=O and V-U) then cu=l and Eqs. (8) and (10) relax to Eq. (6) while Eqs. (9) and (11) relax to Eq. (7). Although this example is highly idealized it reflects the two important points that the moments (including higher order than presented here) are influenced by the ion collection process and the depth at which this occurs. Consequently, for more generally spatially distributed ion collection the surface concentration-eroded profile will be considerably modified as compared to the case of ion collection absence and this illustrates the necessity of using the more complex description of atomic redistribution given by E$. (2) rather than Eq. (1). However if it is arranged experimentally that r B 1, i.e. by using very high energy ions and/or shallow markers then Eqs. (10) and (11) relax to Eqs. (6) and (7) apart from V replacing U and the effect on the surface concentration profile is only to modify the apparent time (or depth) scale but not to perturb the profile shape. 3.3. No ion trapping, p(z) - 0: Truncated power series initial marker distribution
p(z)
function C(0, z) is defined, e.g. as a delta function, the moments G,,,(x, 0) can be calculated. Comparison of these with experimental values then allows iteration of the values of D(z) and p(z) to obtain optimum match and then to attempt to physically understand the atomic redistribution processes. It is probably more interesting technologically to be able to employ the measured surface concentration-eroded depth profile and its moments to deconvolute an initially unknown concentration distribution C(0, z). In this example the process is demonstrated generally, including the ion trapping process, and examples given for the simpler no trapping situation where it is, reasonably, assumed that the initial concentration distribution may be represented by a power series or depth truncated power series expansion with depth z, i.e. C(0, z) = C~=sunz” or C(0, z) = Lizoa,( z - Z)“. For the infinite extent distribution the general solution of Eq. (S), for m = 0. is of the form G,(z)
D2 $-#e-;-Y?e-;
The preceding examples illustrate how, if D(z) and are specified and if the initial marker distribution
475
n+ I = c a,A,(P, n=O
Q, R)z”,
(12)
where A,(P, Q, R) are determinable functions of P, Q, R evaluated at z - 0 and z = m. Use of the recursion relation then allows G,(O) to be determined as G,(O) = CzXOa,B,(P, Q, R) where B,(P, Q, R) are determinable coefficients. Higher order moments G,(z) and G,,,+ ,(O) can be deduced by successive application of Eq. (5) to lower moments and use of the recursion relation with the general result G,,,(O) = i
anY,,n(P,
n=O
Q> 9
where the coefficients M,,,,(P, Q, R) are determinable, usually by numerical methods once P, Q and R have been specified or assumed. Eq. (13) represents a set of m linear equations in the unknown coefficients a, of the initial distribution function and the M,,,, coefficients are determinable. If the moments, G,(O) are compared to those extracted from experimental data then the coefficients a, can be recovered and the initial profile reconstructed. If P, Q and R are not well known these can be adjusted iteratively until the a, values settle to well behaved values. As an example of the process it is useful to consider the situation in the absence of ion trapping (p(z) - 0) and infinite range uniform diffusivity (D( z) = D) and the depth truncated power series expansion for the initial concentration profile. In this case Eq. (4). in more simplified terms, becomes applicable. For specificity two initial profiles are considered; (i) C(0, z) = aOe-(i-Z)/h with C(0, z) = 0, z > Z and (ii) C(0, z) = a0 + u2( z - Z)’ with C(0, I) =o, z>Z+(-aa,/a2)“2. It is readily shown that the results depend upon the ratio of the “characteristic length”, D/U, of the diffusion process to the truncation depth Z and/or the decay depth X and VI. PROJECTILE INDUCED/ASSISTED
PROCESSES
G. Carter/N&.
476
Instr. and Merh. in Phys. Res. B 115 (1996) 473-477
are somewhat unwieldy to reproduce here in full although readily calculated. An insight into the behaviour is obtained by considering the case where these ratios are small (i.e. relatively slow effective diffusivity). In this situation it is readily deduced, for the first two moments that:
” dzU/D( I 21
Case (i)
1)-Z},
G,(O)_u.h(A(e’-
Cl&)
Z G2(0) = 2a,h
A2e’ - f(Z’ + 2Z) + 2x2)
(14b)
Case (ii) 22
G,(O) = X(6ao
- Z2a,),
( 15a)
23 G2(0)=z(10a,-Z2a2). The independence of these results on the characteristic length D/U is notable. Both Eqs. (14) and (15) depend only upon the unknown coefficients a, (and a,) and upon the decay and/or truncation lengths. If these latter are assumed to be known comparison of only the first two moments with experimental values allows deduction of the coefficients. If, on the other hand, A and Z may not be assumed to be known higher order moments can be deduced to allow independent evaluation of these parameters. If it is required to determine n coefficients (a,) then n + 2 moments should be evaluated and compared with corresponding experimental values. When the characteristic length cannot be assumed small the extended versions of Eqs. (14) and (15) must be used.
4. Exponential tails Experimental studies [ 1I] of surface concentration profiles resulting from an initial planar delta function concentration distribution reveal, quite generally, exponentially decaying tails as a function of eroded depth for large erosion depths. This behaviour has been predicted theoretically [4-6,8] using the effective diffusion approximation for the no ion trapping or restoration atomic flux case, from which it can be shown that the decay length, A, can be represented by: A-’ = R-’
If the diffusivity is depth dependent than the inequality can be genemlised so that the largest decay length 2 D( AZ)/U, results from that part of the depth dependent diffusivity for which
+ (2d/U)-‘,
(16)
where R is the range over which a constant effective diffusivity, D, is active. Thus there are seen to be two characteristic lengths for the redistribution process, R and D/U and the dominant (larger) decay length depends upon the inequality UR/2D 3 1. For small effective diffusivity A -+ 2 D/U which is generally the largest value.
z) z=-2
where
AZ=Z2-Z,.
If U and D(z) are known or can be approximated theoretically A z can be calculated and A evaluated. The earlier analysis of the moments of distributions also revealed the importance of the characteristic length 2D/U and suggests, from the existence of high moments, the operation of an exponential tail. In the case of ion trapping the results of Eqs. (8) to (11) indicate that the characteristic length becomes 2D/V- 2D/(U - v). This form of result may also be deduced from the steady state (large eroded depth) solution of Eq. (2) or, for the no ion trapping case from Eq. (1). The result indicates that any comparison between experimentally measured exponential tail decay lengths and theoretical prediction should be with 2D/V or R which has often been employed as the comparator [ 121or even (C/D/R)‘/’ which has been advocated [ 131. 5. Discussion and conclusions The main purpose of the present work has been to demonstrate how, within the framework of a diffusion-like approximation to ballistically driven atomic redistribution but which also includes directed atomic drift, incident ion implantation and atomic relaxation processes, it is possible to calculate, in principle, the moments of the surface concentration-eroded depth profile resulting from an initial depth distributed concentration profile. It has been shown that this approach reduces to solution of the defining partial differential equations to that of a hierarchial set of ordinary equations for the equivalent atom conditions assumed. Solution of this set will generally require numericai methods because of the depth dependence of diffusivity and ion trapping. However simplifying assumptions about the diffusivity (e.g. depth independence) and the trapping process (e.g. delta function trapping) allow theoretical analysis and have shown the effects, on the moments, of ion trapping. It has also been shown how the moments of the initial profile itself can be derived by comparing experimentally determined momenta of the surface concentration-eroded depth profile with predicted values. At present we refrain from making detailed comparisons with published experimental data since, with one exception, moments have not been calculated nor have detailed estimates of the D(z) or p(z) behaviour been made for systems so far studied. The exception is the work of Macht et al. [14] who measured low order moments for 0: bombardment of delta function markers in Ni and compared them with the predictions of Eqs. (1) and (4), the no ion trapping case and found rather good agreement. This
G. Carrer/Nucl.
Instr. and Meth. in Phys. Res. B I I5 (19961473-477
does not vitiate the preceding analysis of the ion trapping case but indicates that when the sputtering yield (Y) is subs~tially larger than the depth integrated trapping probability c-q s 11, V -+ U and the no trapping analysis is an acceptable approximation. It is, however, advocated that, in future, both more attention should be paid to deduction of moments of measured surface concentration profiles since they allow deduction of initial depth dependent concentration profiles and that the more complete analysis, including ion trapping, should be used. This is p~~cularly true for comparisons of measured and predicted exponential tail decay lengths. Finally it may be noted that while the present analysis has concentrated upon the behaviour resulting from initially established marker profiles, exactly the same methods can be used to describe the evolution of the implanted ion surface concentration by a small adaptation 19,101 of Eqs. (2) and (5). References [I] LT.Littmark and W.O. Hofer, Nucl. Instr. and Meth. 168 ( 1980) 329.
477
[2] P. Sigmund and A. Gras-Marti, Nucl. Instr. and Meth. 168 ( 1980) 389. 131 LA. Peinador, I. Abril, J.J. Jimenez-Rodriquez and A. GrasMarti, Surf. Interf. Anal. 15 (1990) 463. 141 R. Collins and G. Carter, Radiat. Eff. 54 (1981) 325. [51 G. Carter, R. Collins and D.A. Thompson, Radiat. Eff. 55 (1981) 99. [61 B.V. King and I.S.I. Tsong, Ultramicroscopy 14 (1984) 75. 171 R. Badheka, M. Wadsworth, D.G. Armour, J.A. Van den Berg and J.B. Clegg, Surf. Interf. Anal. 15 (1990) 5.50. [81 V. Naundorf and C. Abromeit, Nucl. Instr. and Meth. B 43 (1989) 513. 191G. Carter, M.J. Nobes and I.V. Katardjiev, Nucl. Instr. and Meth B 36 (1989) 404. [lOI G. Carter, IV. Katardjiev and M.J. Nobes, Nucl. Instr. and Metb. B 43 (1989) 149. 1111 K. Wittmaack, in: Practical Surface Analysis, Vol. 2 - Ion and Neutrat Spectroscopy, 2nd Ed., Eds. D. Briggs and M.P. Seah (Wiley, Chichester, 1992) Ch. 3 p. 105. [I21 W. Vanderworst and T. Clarysse, J. Vat. Sci. Technol. B 10 ( 1992) 302. I131 PC. Zalm and C.J. Vriezema, Nucl. Instr. and Meth. B 67 (1992) 49.5. 1141 M.P. Macht, R. Willecke and V. Naundorf, Nucl. Instr. and Meth. B 43 (1989) 507.
VI. PRO~~LE
INDU~ED/ASSIS~D
PROCESSES
Nuclear Instruments and Methods in Physics Research B 115 (1996) 473-477
NIUMI B
Beam Interactions with Materials 8 Atoms ELSEVIER
The diffusion approximation in modelling impurity redistribution during sputtering G. Carter * Deparrment
of Eiecironic and Electrical Engineering, University of Saljord. Suljord MS 4WT, UK
Abstract A method to calculate the moments of the surface concentration-eroded depth profile during sputtering of a solid containing an initially depth distributed concentration profile of equivalent atom markers, within the framework of a diffusion approximation which includes directed atomic drift, incident ion trapping and target atom spatial relaxation processes, is described. Illustrative examples reveal the perturbations introduced by ion trapping and how, in principle, measured moments can be used to reconstruct initial depth profiles. Corrected forms for large eroded depth exponential decay lengths of surface concentration profiles are discussed.
1. Introduction Depth profiling, using surface or near surface analytic probe techniques together with sputter-sectioning, is an important technique for composition assessment of materials. Although they can possess high sensitivity such methods usually suffer from depth resolution limitations because the aggressive process of ion bombardment sputtering can lead to atomic relocation in the bulk of the material and to changes in surface topography. These can lead to real or apparent changes in the depth profiles of the atomic component concentrations it is intended to evaluate and deduction of initial profiles from measured surface concentrations with increasing sputtered depth can be uncertain. Several different theoretical or computational approaches have been developed to aid interpretation of experimental data including solution of the atomic transport equations [ 1,2], evolutionary binary collision approximation computer codes [3] and diffusion approximations to the transport equations [4-61. In this latter area numerical methods have been used to allow solution of the coupled diffusion equations [7] when the solid is polyatomic. These approaches usually attempt to describe the evolution, with increasing sputtered material thickness, of the surface atomic concentration of the species of interest for a given initial depth profile of that species. A delta function at a fixed depth initial profile is frequently assumed. An alternative approach is to predict the moments of the surface concentration-sputter eroded thickness profile and this has been undertaken for an
Tel. +4l
161 745 58%; [email protected]. l
fax
+44
161 745 5119; e-mail
effective constant diffusivity of infinite [4] or finite [5] extent into the solid and for more generally spatially variable effective diffusivity [8]. In none of these treatments were the effects of implantation of the incident ion species or the relaxation of the solid to counteract local densification or rarefaction resulting from atomic relocation included. The present study shows how these processes may be included in a method of calculation of moments developed by Naundorf and Abromeit [8].
2. Moments of the surface eroded depth distribution
concentration-sputter-
For analytical specification a self-similar system is considered in which ions of a specific atomic species irradiate, implant into and sputter a target of the same atomic species containing, initially, a concentration profile C( z, O), of a marker species, again of the same atoms as the target. If irradiation is considered only to generate recoils isotropically within the bulk and sputter the surface then an effective depth dependent diffusivity D(z) and a sputtering velocity ZJ can be defined. In this order of approximation the diffusion equation for C( z, x) where z is depth below the instantaneous surface when a thickness x = /Cl dr has been eroded, can be written [4,5,8]
(1) If however, the irradiation is considered to give rise to a non isotropic, depth dependent drift velocity, v(z), to result in ion stopping and trapping with a probability
0168-583X/96/%15.00 Copyright 0 1996 Elsevier Science B.V. All rights reserved VI. PROJECTILE INDUCED/ASSISTED PROCESSES SSDI 0168-583X(95)01558-2
474
G. Carter/Nucl.
Instr. and Meth. in Phys. Res. B 115 (1996) 473-477
distribution p(z) and the effects of target relaxation to accommodate atomic relocation processes are considered, then Eq. (1) must be replaced by [9,10]:
--
-=
2s
dzp(z)
1 (2)
CT is the total atomic concentration, assumed constant, Ji is the incident ion flux and V is the net erosion velocity of the surface and equal to the difference between the sputtering velocity U - J,Y/C,, and the growth velocity due to ion accumulation v(Ji/CT)jt d z p (z) = Jiq/CT. It is notable that Eq. (2) is independent of the drift velocity, u(z) and can be written as:
(3)
For infinite range diffusivity the boundary conditions for Eqs. (1) and (3) are [4,5,8] DaC/az = 0; z = 0 and C = 0; Z’W. Following Naundorf and Abromeit [8], the moments G,,,(x, z) = 10”d x xmC(x, z) can be defined and successive differentiation of Eqs. (1) or (3) leads to defining ordinary differential equations for the moments defined at z - 0, i.e. G,,jx, 0). From Eq. (1):
= { -C(O,
+ uG,(z)
z)
-{ -mGmm-,(z)
for m-0, (4)
for rn> 1,
and from Eq. (3)
d’Gm( z>
w-&r
+R(z)G,(z)
3. Illustrative examples 3.1. No ion trapping, p(z) = 0; Delta function marker at z = 1 initially
v;-fyz)$+Q(z)E +R(z) c.
D(z)%
ments made to determine optimum values for these parameters. Of course the solutions of Eqs. (4) and (5) are, generally, not simple since for spatially variable D(z) and p(z) they are nonlinear and numerical methods are required. They are, however, more straightforward to solve numerically than the original partial differential equations (l), (2) and (3). In order to illustrate the approach three examples, employing simplifying assumptions to allow tractable analysis are chosen. In each of these D(z) is assumed to be depth independent in the range 0 < z < m.
This is the case examined by Naundorf and Abromeit [8] and Eq. (4) can be integrated easily and the recursion relations used to evaluate the moments. The first two moments, G,(O) and G,(O) can be used to determine the mean, (x) and the variance, u2, of the distribution and are given by:
G,(O)=(x)=/+;, (7)
+ 2$.
These results are different from those quoted by Naundorf and Abromeit [8] but agree identically with those derived [4,5] by Laplace transformation methods of solution of Eq. (1). 3.2. Ion trapping as a delta function at z - r, probability
+QW,z
I,I; Delta function maker at z = 1 initially
WA z)
=(-VC(0, z) for m-0, _ {_ mVG,,,_l(z) for mr
1.
(5)
For both Eqs. (4) and (5) it is readily shown that the following recursion relation holds G,(z = 0) = m/r dzG,,_,(z) with G,(z-0)=/i dz C(x, z). The moments G,,,(z = 0) are the mth order moments of the surface concentration-eroded thickness profile and can be extracted from experimental data. If suitable assumptions, or experimental measurements, are made on the values of D(z), p(z), U and V then, in principle Eqs. (4) or (5) can be solved for different assumptions of the initial marker profile C(0, z) and the predicted moments compared with experimental values and a best fit will yield C(0, z). Alternatively, if C(0, z) is initially specified, for example a delta function, the parameters D(z), p(z), U and V can be varied and comparison of predicted and measured mo-
In this case it is necessary to consider two situations, the first where r < 1 and the second for r > 1. It is now necessary to solve the somewhat more complex hierarchy of Eq. (5) and the results are also more complex. It can be shown that for r < 1; %r (x)_l+;+
OLzea”--le
-I=I+D
V
02-oi
+a-
1, (8)
and D2
a2=3
(1 7
+2;1
02
+2-
WV2
X21(a
V2e-;-U2e-;
(
- 1) + (o - Q2,
>
(9)
G. Carter/Nucl.
Instr. and Meth. in Phys. Res. B 115 (1996) 473-477
where
(x)-l+;+ 1-i
( 1
forr>l;
x
(D
e-u/w-‘)+r-
1
u
1 (10) +a-
1
and u2 ,=c~~[i+~~-l)-~]-r+2(~) Dl +Y+
]+kh
(11)
where $I- r) p=$[SD+(r-l)V]eD 5-r) -;[3D+(r-
l)V]eD
.
It is readily demonstrated that if ion trapping is absent (q=O, u=O and V-U) then cu=l and Eqs. (8) and (10) relax to Eq. (6) while Eqs. (9) and (11) relax to Eq. (7). Although this example is highly idealized it reflects the two important points that the moments (including higher order than presented here) are influenced by the ion collection process and the depth at which this occurs. Consequently, for more generally spatially distributed ion collection the surface concentration-eroded profile will be considerably modified as compared to the case of ion collection absence and this illustrates the necessity of using the more complex description of atomic redistribution given by E$. (2) rather than Eq. (1). However if it is arranged experimentally that r B 1, i.e. by using very high energy ions and/or shallow markers then Eqs. (10) and (11) relax to Eqs. (6) and (7) apart from V replacing U and the effect on the surface concentration profile is only to modify the apparent time (or depth) scale but not to perturb the profile shape. 3.3. No ion trapping, p(z) - 0: Truncated power series initial marker distribution
p(z)
function C(0, z) is defined, e.g. as a delta function, the moments G,,,(x, 0) can be calculated. Comparison of these with experimental values then allows iteration of the values of D(z) and p(z) to obtain optimum match and then to attempt to physically understand the atomic redistribution processes. It is probably more interesting technologically to be able to employ the measured surface concentration-eroded depth profile and its moments to deconvolute an initially unknown concentration distribution C(0, z). In this example the process is demonstrated generally, including the ion trapping process, and examples given for the simpler no trapping situation where it is, reasonably, assumed that the initial concentration distribution may be represented by a power series or depth truncated power series expansion with depth z, i.e. C(0, z) = C~=sunz” or C(0, z) = Lizoa,( z - Z)“. For the infinite extent distribution the general solution of Eq. (S), for m = 0. is of the form G,(z)
D2 $-#e-;-Y?e-;
The preceding examples illustrate how, if D(z) and are specified and if the initial marker distribution
475
n+ I = c a,A,(P, n=O
Q, R)z”,
(12)
where A,(P, Q, R) are determinable functions of P, Q, R evaluated at z - 0 and z = m. Use of the recursion relation then allows G,(O) to be determined as G,(O) = CzXOa,B,(P, Q, R) where B,(P, Q, R) are determinable coefficients. Higher order moments G,(z) and G,,,+ ,(O) can be deduced by successive application of Eq. (5) to lower moments and use of the recursion relation with the general result G,,,(O) = i
anY,,n(P,
n=O
Q> 9
where the coefficients M,,,,(P, Q, R) are determinable, usually by numerical methods once P, Q and R have been specified or assumed. Eq. (13) represents a set of m linear equations in the unknown coefficients a, of the initial distribution function and the M,,,, coefficients are determinable. If the moments, G,(O) are compared to those extracted from experimental data then the coefficients a, can be recovered and the initial profile reconstructed. If P, Q and R are not well known these can be adjusted iteratively until the a, values settle to well behaved values. As an example of the process it is useful to consider the situation in the absence of ion trapping (p(z) - 0) and infinite range uniform diffusivity (D( z) = D) and the depth truncated power series expansion for the initial concentration profile. In this case Eq. (4). in more simplified terms, becomes applicable. For specificity two initial profiles are considered; (i) C(0, z) = aOe-(i-Z)/h with C(0, z) = 0, z > Z and (ii) C(0, z) = a0 + u2( z - Z)’ with C(0, I) =o, z>Z+(-aa,/a2)“2. It is readily shown that the results depend upon the ratio of the “characteristic length”, D/U, of the diffusion process to the truncation depth Z and/or the decay depth X and VI. PROJECTILE INDUCED/ASSISTED
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G. Carter/N&.
476
Instr. and Merh. in Phys. Res. B 115 (1996) 473-477
are somewhat unwieldy to reproduce here in full although readily calculated. An insight into the behaviour is obtained by considering the case where these ratios are small (i.e. relatively slow effective diffusivity). In this situation it is readily deduced, for the first two moments that:
” dzU/D( I 21
Case (i)
1)-Z},
G,(O)_u.h(A(e’-
Cl&)
Z G2(0) = 2a,h
A2e’ - f(Z’ + 2Z) + 2x2)
(14b)
Case (ii) 22
G,(O) = X(6ao
- Z2a,),
( 15a)
23 G2(0)=z(10a,-Z2a2). The independence of these results on the characteristic length D/U is notable. Both Eqs. (14) and (15) depend only upon the unknown coefficients a, (and a,) and upon the decay and/or truncation lengths. If these latter are assumed to be known comparison of only the first two moments with experimental values allows deduction of the coefficients. If, on the other hand, A and Z may not be assumed to be known higher order moments can be deduced to allow independent evaluation of these parameters. If it is required to determine n coefficients (a,) then n + 2 moments should be evaluated and compared with corresponding experimental values. When the characteristic length cannot be assumed small the extended versions of Eqs. (14) and (15) must be used.
4. Exponential tails Experimental studies [ 1I] of surface concentration profiles resulting from an initial planar delta function concentration distribution reveal, quite generally, exponentially decaying tails as a function of eroded depth for large erosion depths. This behaviour has been predicted theoretically [4-6,8] using the effective diffusion approximation for the no ion trapping or restoration atomic flux case, from which it can be shown that the decay length, A, can be represented by: A-’ = R-’
If the diffusivity is depth dependent than the inequality can be genemlised so that the largest decay length 2 D( AZ)/U, results from that part of the depth dependent diffusivity for which
+ (2d/U)-‘,
(16)
where R is the range over which a constant effective diffusivity, D, is active. Thus there are seen to be two characteristic lengths for the redistribution process, R and D/U and the dominant (larger) decay length depends upon the inequality UR/2D 3 1. For small effective diffusivity A -+ 2 D/U which is generally the largest value.
z) z=-2
where
AZ=Z2-Z,.
If U and D(z) are known or can be approximated theoretically A z can be calculated and A evaluated. The earlier analysis of the moments of distributions also revealed the importance of the characteristic length 2D/U and suggests, from the existence of high moments, the operation of an exponential tail. In the case of ion trapping the results of Eqs. (8) to (11) indicate that the characteristic length becomes 2D/V- 2D/(U - v). This form of result may also be deduced from the steady state (large eroded depth) solution of Eq. (2) or, for the no ion trapping case from Eq. (1). The result indicates that any comparison between experimentally measured exponential tail decay lengths and theoretical prediction should be with 2D/V or R which has often been employed as the comparator [ 121or even (C/D/R)‘/’ which has been advocated [ 131. 5. Discussion and conclusions The main purpose of the present work has been to demonstrate how, within the framework of a diffusion-like approximation to ballistically driven atomic redistribution but which also includes directed atomic drift, incident ion implantation and atomic relaxation processes, it is possible to calculate, in principle, the moments of the surface concentration-eroded depth profile resulting from an initial depth distributed concentration profile. It has been shown that this approach reduces to solution of the defining partial differential equations to that of a hierarchial set of ordinary equations for the equivalent atom conditions assumed. Solution of this set will generally require numericai methods because of the depth dependence of diffusivity and ion trapping. However simplifying assumptions about the diffusivity (e.g. depth independence) and the trapping process (e.g. delta function trapping) allow theoretical analysis and have shown the effects, on the moments, of ion trapping. It has also been shown how the moments of the initial profile itself can be derived by comparing experimentally determined momenta of the surface concentration-eroded depth profile with predicted values. At present we refrain from making detailed comparisons with published experimental data since, with one exception, moments have not been calculated nor have detailed estimates of the D(z) or p(z) behaviour been made for systems so far studied. The exception is the work of Macht et al. [14] who measured low order moments for 0: bombardment of delta function markers in Ni and compared them with the predictions of Eqs. (1) and (4), the no ion trapping case and found rather good agreement. This
G. Carrer/Nucl.
Instr. and Meth. in Phys. Res. B I I5 (19961473-477
does not vitiate the preceding analysis of the ion trapping case but indicates that when the sputtering yield (Y) is subs~tially larger than the depth integrated trapping probability c-q s 11, V -+ U and the no trapping analysis is an acceptable approximation. It is, however, advocated that, in future, both more attention should be paid to deduction of moments of measured surface concentration profiles since they allow deduction of initial depth dependent concentration profiles and that the more complete analysis, including ion trapping, should be used. This is p~~cularly true for comparisons of measured and predicted exponential tail decay lengths. Finally it may be noted that while the present analysis has concentrated upon the behaviour resulting from initially established marker profiles, exactly the same methods can be used to describe the evolution of the implanted ion surface concentration by a small adaptation 19,101 of Eqs. (2) and (5). References [I] LT.Littmark and W.O. Hofer, Nucl. Instr. and Meth. 168 ( 1980) 329.
477
[2] P. Sigmund and A. Gras-Marti, Nucl. Instr. and Meth. 168 ( 1980) 389. 131 LA. Peinador, I. Abril, J.J. Jimenez-Rodriquez and A. GrasMarti, Surf. Interf. Anal. 15 (1990) 463. 141 R. Collins and G. Carter, Radiat. Eff. 54 (1981) 325. [51 G. Carter, R. Collins and D.A. Thompson, Radiat. Eff. 55 (1981) 99. [61 B.V. King and I.S.I. Tsong, Ultramicroscopy 14 (1984) 75. 171 R. Badheka, M. Wadsworth, D.G. Armour, J.A. Van den Berg and J.B. Clegg, Surf. Interf. Anal. 15 (1990) 5.50. [81 V. Naundorf and C. Abromeit, Nucl. Instr. and Meth. B 43 (1989) 513. 191G. Carter, M.J. Nobes and I.V. Katardjiev, Nucl. Instr. and Meth B 36 (1989) 404. [lOI G. Carter, IV. Katardjiev and M.J. Nobes, Nucl. Instr. and Metb. B 43 (1989) 149. 1111 K. Wittmaack, in: Practical Surface Analysis, Vol. 2 - Ion and Neutrat Spectroscopy, 2nd Ed., Eds. D. Briggs and M.P. Seah (Wiley, Chichester, 1992) Ch. 3 p. 105. [I21 W. Vanderworst and T. Clarysse, J. Vat. Sci. Technol. B 10 ( 1992) 302. I131 PC. Zalm and C.J. Vriezema, Nucl. Instr. and Meth. B 67 (1992) 49.5. 1141 M.P. Macht, R. Willecke and V. Naundorf, Nucl. Instr. and Meth. B 43 (1989) 507.
VI. PRO~~LE
INDU~ED/ASSIS~D
PROCESSES