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Preferential recoil implantation in polyatomic targets
Nuclear Instruments and Methods in Physics Research B34 (1988) 15-21 North-Holland, Amsterdam
PREFERENTIAL RECOIL IMPLANTATION IN POLYATOMIC TARGETS Peter SIGMUND Fysisk Inst~tut, Odeme U~i~ers~tet,DK-5230 Odense M, Denmark Received 29 February 1988 and in revised form 6 April 1988
Five distinct effects have been identified which contribute
to preference
recoil impl~tation
in a polyatomic material. Their
directions and relative magnitudes each depend on some combination of the following parameters: Masses and atomic numbers of the target atoms; mass and atomic number of the bombarding ion; ion-target and target-target interatomic potential; ion energy (indirectly). High ion energy favors a negative mass effect, i.e., light species are implanted preferenti~ly. A positive mass effect is found at low ion energies, caused by differences in the angular scattering of light and heavy recoil atoms, respectively. The magnitude of this effect as well as the point of transition between positive and negative mass dependence are found to be governed primarily by the shape of the target-target interaction potential. The origin of conflicting conclusions in the theoretical literature on the sign of preferential recoil implantation is located. Estimates are given in the limit of small differences in atomic masses and charge numbers of the constituents of the target.
1. Introduction Conflicting results have been reported in the litesature on the theory of preference recoil ~plantation by ion bombardment of binary targets [l-5]. In brief, refs. [l], [2], and [4] reported preferential implantation of the lighter target species while the opposite conclusion was reached in refs. [3] and [S]. Not even tentative explanations have been offered to resolve this apparent discrepancy. Analytical estimates [1,2] have been based on various simplifications. Simple power cross sections were employed and electronic stopping was neglected. More important, [l] treated a monolayer source in a monoatomic stopping medium. In [2], energy loss was properly treated for a binary medium, but angular deflection of recoil atoms during their slo~ng-doe process was ignored. The latter feature was pointed out to be a possibly significant oversimplification when simulation results became available [3]. However, subsequent work with a different code [4] reproduced the conclusion reached in the analytical estimates. Angular scattering was supposed to be adequately included in either code. A direct comparison between the two codes [5] showed recoil implant profiles of similar magnitudes but opposite sign for supposedly identical systems with regard to composition, ion type, and bombarding energy. Several parameters were varied [5] in order to trace the cause of this discrepancy, but although it has been postulated [4] that “computer simulations of this kind are more powerful than analytical calculations due to their variability in defining parameters, introducing spe0168-583X/88/$03.50 Q Elsevier Science Publishers B.V. (North-robed Physics Pub~s~ng Division)
cial mechanisms, and treating a wide range of problems”, no way out of this frustrating situation has been shown so far.
2.
Essentials
It is evident from ref. 151 that the discrepancy shows up already in the low-fluence profiles. There is, therefore, no reason at this stage to look into the complications caused by high-fluence effects. It is ~plausible that the discrepancy should hinge on electronic stopping. Hence, the effect will be ignored. It is well documented that nonstoichiometric behavior depends on the interaction potential or cross section 121. Here it appears more important to allow for freedom in the choice of parameters determining the cross section than to use an accurate analytical or numerical form. To locate the source of discrepancy, conventional power cross sections seem perfectly adequate [6,7], i.e., da&
E,T)=
CiiE-MT-l--m dT; 0
OsTr
yijE,
otherwise,
(1) where yij=4MiMj(Mi+Mj)-‘,
(la>.
E is the energy of an i-particle hitting a j-particle rest, 2’ is the energy of the recoiling j-atom,
at
16
P. Sigmund / Preferential recoil implantation in polyatomic targets
m an exponent, 0 5 m < 1, depending weakly on the reduced energy [6], and Mj and Mj are atomic masses. In (21, h, is a dimensionless parameter depending on the screening function of the potential, and ai, and A, depend on the interaction potential as well. For m > 4, one may, but need not, use Thomas-Fermi quantities
161 ajj = 0&353(z~‘3
myi: i
+ zjz/“) -1’2,
(3a) (3b)
Aij = ZiZje2,‘aij,
a0 being the Bohr radius, while for m < :, Born-Mayer parameters [7-91 may be more adequate, aij = a z 0.219 A rnr$: i
Ajj = ( ZiZj)‘/”
(4a)
52 eV,
(4b)
where Zi and Zj are nuclear charge numbers. Presumably not all features of the above cross sections need to be important for the issue under consideration. Angular de~e&tion and energy loss of recoil atoms need to be included. Angular deflection and energy loss of the primary ion, although readily incorporated [lo], tend to complicate the issue. If one restricts attention to the relocation of near-surface atoms, this effect may safely be ignored. In order to elucidate the significance of every individual effect, it is convenient to linearize the system by assuming that the constituents of the target have opt& slightly different masses and atomic numbers [ll].
where zi(T, cos 0) is the mean penetration depth of an i-recoil and + the (small) ion fluence. Moreover, Zi=R,(T)
cos 8,
where R, is the projected range following from [12,13], CN,cJadou(E, i = 1, cos (pii= (1-
T)[ R!(E)
+ijRi(E-
-COS
(71 T/E)“z+at,i(T,‘E)(l
- T,‘E)-1’2,
(74 and ~ii=‘i(l-k$%$).
(7b)
If recoil scattering were neglected [Z], one would set cos +ij= 1.
4. Linearization Although eq. (7) is easily solved for power scattering [12,13], the resulting expressions are messy in general due to the occurrence of incomplete beta functions. A very useful simplification was pointed out recently [ll]. If ah target masses and atomic numbers are similar, i.e., if ~i=~+A~j;
AMi
i=l,2...
Z,=Z+AZ,;
AZ, -=cZ,
i=l,
The cross section for relocation into depth (z, dz) of an i-atom located near the target surface and hit by a projectile ion 0 with energy E at normal incidence is doi
= dzpEdeei(E,
T)E;(T,
cos ff, z),
(5)
where E(T, cos 8, z) is the penetration profile of an i-atom recoiling with an energy T at an angle 6 against the inward surface normal in a random homogeneous medium composed of Nj j-atoms per volume, j = 1, 2,. . . . The angle 6 is related to the recoil energy by conservation laws, cos B = ( T/Y,~E)?
Pi = +/z
(9)
since deviations are of second order in A&i/M according to eq. (la). Note, however, that yoi in eq. (6) may differ from 1 since the ion mass MO may differ from the Mi. In view of eq. (9), the solutions of eq. (7) contain only compZete beta functions.
5. The mean relocation depth With the ansatz [32] R, = BiE2”,
(10)
the solution of eq. (7) reads = xA$Cjjildt
t+'[l
- (1-
t)2m CQS+ij],
.i where t = T/E, tion yields
after insertion of (1) and (9). Integra-
(Bi)-1=C~~i~[-~-B(-m,2m+i_) i
doi
=+&@de,,i(E,
(8b)
it is reasonable to restrict attention to effects of first order in AM, and AZ,. This implies that
(&)-r
The profile 4 obeys the usual range equations [12,13]. It is most convenient just to consider the mean relocation depth [21
(8a) 2...
yij z 1 in eq. (7), 3. Basic equations
T)]
T)z,(T,
cos @),
(6)
--ijB(l-m,
;+2m)
I
.
(11)
17
P. Sigmund / Preferentialrecoil implantationin polyatomictargets If angular deflection had been ignored, the contents of the bracket would read [ - l/m - B( - m, 2m + I)]. Insertion of (1) and (10) into (6) yields pi = +C,,iE-“OBi~‘EdT
T-1-m~+2m(T/y,,iE)1’2.
Fromeq. one finds AC,, _
0 w
Here, allowance has been made for two different exponents m, where m, applies to the ion-target interaction and m to the target-target interaction. Eq. (12) illustrates the following sources of nonstoichiometry in the mean implant depth: I) The factor C,,i represents the magnitude of the primary recoil cross section. II) The factor B, represents the magnitude of the recoil range at a given recoil energy and angle. III) The limit of integration yoiE represents the maximum energy transfer in primary collisions, and IV) The factor (T/y,,iE)1/2 represents the recoil angle. Moreover, from eq. (11) it follows that II) splits up into two distinct effects: IIA) The term NjCjj reflects the stopping behavior of the recoil atom, and IIB) The term aij reflects the angular scattering of the recoil atom. The presence of five distinct effects going in different directions would make it quite difficult to ask the right questions in a purely simulational analysis without the guidance of an analytical estimate. Integration of (12) yields [2,10] Pi = (2m - m, + 1/2)-1Coiy~im-moBiE2m-ma,
6.1. Influence of the primary recoil cross section (effect I)
(12’)
where the effects III) and IV) have been collected into one factor yzimPrnogoverned by the kinematics of the primary recoil event.
Co
(2), applied to i+O, AM-
Au,,, 2 p-m0
L M
+
2m,-
AAoi Ao
a0
and m-+mo,
j-i,
(14)
.
At this point, a choice of the potential parameters Aoj and aoi is called for. With the choices (3) or (4), one finds
W 3(1-t
AZ,
-m0>
(zo/z)“‘)
1
z
AM, moM
(1% for m. 2 a, and 3m, AZ,
AGi
AM,
P-J)
M
Co =2~-mo~
for m, < f. Eqs. (15a) and (15b) illustrate the fact that the mass dependence (e.g., the isotope effect) is determined by the exponent m, while the dependence on the charge number is also determined by other potential parameters [ll]. The Z-dependent contribution to eq. (15a) or (15b) can be determined from fig. la, which shows the coefficient of AZ/Z vs m,. Similarly, fig. 2a shows the coefficient of AMJM. It is seen that the atomic-number effect is largely positive, i.e., the heavier species tends to be implanted preferentially due to this particular effect, while the mass effect is negative. Both features are evident from the dependence of the cross section, eqs. (1) and (2) on Zj and Mj. Note that the transition from the low-m0 to the high-m0 region is influenced by other parameters, here Z,/Z, and that it can be either below or above a. 6.2. Influence of the recoil range (effect II)
6. Evaluation
Eq. (11) yields, after linearization,
Eq. (12’) is still exact for a polyatomic target within the adopted model. For small differences in mass and charge between target constituents, linearization yields A Pi/P = AC&C,
+ A B,/B + (2m - m,,) Ayoi/yO,
(13) where APJP is the relative deviation of the mean relocation depth Pi of an i-recoil from the average P, and the three terms on the right-hand side reflect the contributions due to effect I), effects HA) and IIB), and effects III) and IV), respectively. In (13), C, is the average of Cei over all i, properly weighted over target composition, and similarly for ye and B. The three terms in eq. (13) will be evaluated separately in order to elucidate their relative significance.
ABi B
- &
C%ACij J
+ &
.&qaij, J
where the first term represents the effect of stopping (IIA) and the second one the effect of angular deflection (IIB). Evidently, the j-sums yield average values, and hence,
ABi B
AC,
- - 7
V + uai.
(16’)
In (16) and (16’), the abbreviations U= --&B(-m,3/2+2m)
(17a)
and V=B(l-m,
1/2+2m)
have been introduced.
(17b)
18
P. Sigmund / Preferential recoil implantation in polyatomic targets
1.0 Recoil Primary Coefficient
Cross of
Range
Coefficient
Section
01 AZ,/Z
A 2, IZ
-10
Fig. 1. (a) Coefficient of A&/Z in eqs. (15a) and (15b), determining the influence of atomic number differences on preferential recoil implantation through the magnitude of the ion-target interaction cross section. The parameter mO characterizes the power cross section for the ion-target interaction. For small mO, the atomic number of the ion does not enter. Solid lines follow the transition from Born-Mayer-lie interaction at small m,, to screened Coulomb interaction at larger mO. Dashed lines are extrapolations of the small-m, lines to larger mO and vice versa. (b) Coefficient of AZ,/2 in eqs. (19a) and (19b), determining the influence of atomic number differences on preferential recoil implantation through the recoil range. The parameter m characterizes the power cross section for the target-target interaction. Solid and dashed lines as in (a).
Eqs. (2), (7b), and (16’) yield
where again, a choice of the potential parameters a, = CjaijNj/N and A, = CjAijNj/N is called for. With the choices (3) or (4), one finds (19a) for pn2 $, and
atomic-number effect is negative, i.e., the recoil atom with the higher Z is impl~t~ less than average because of its higher stopping power. The atomic-mass effect on the stoppingpower is in the same direction but weaker, while angular defection favors the heavier recoil atom, as was to be expected. Unlike other effects, the magnitude of the angular deflection effect increases witk decreasing m . 6.3. Injluence of primary collision kinematics (effects III and IV)
(19b) for m<$. Augular detection enters through the term containing V/U. According to eqs. (17), this reads m V -= (20) u T(3/2+m) . ++2mr(l - m)r(+ -t 2m) This term would drop out if angular scattering were neglected. Fig. lb shows the dependence of the coefficient of AZ,/2 in eq. (19a) and (19b) on m, and fig. 2b shows the coefficient of AM/M. One may note that the
From (la) one finds
The two factors determining this mass effect are shown separately in figs. 3a and 3b. Evidently, the combined effect may be positive or negative, dependent on the ion mass. 7. Discussion Eq. (13), in conjunction with eqs. (Isa) and (15b), (19a) and (19b), and (21), supplemented by (20), con-
19
P. Sigmund / Preferential recoil implantation in polyatomic targets
1.0 Pnmary
Cross
Coefficient
of
Recod
Section A M;/M
s-m
Range
Coefficient
of AMilM
a 0.0 0.5
m,
1.0
\
-10
Fig. 2. (a) Coefficient of AMi/M in eqs. (15a) and (15b), determining the iufluence of atomic mass differences on preferential recoil impl~tat~on through the ion-target interaction cross section. (b) Coefficient of A~~/~ in eqs. (19a) and (19b), deter~ng the influence of atomic mass differences on preferential recoil implantation through the recoil range (solid line). The dashed line represents the same quantity under the assumption of negligible angular scattering of recoil atoms.
tams all pertinent information. The obvious advantage of the present description is the reduction to relative changes where the crucial parameters that enter are the exponents m O and m for the ion-target and the target-target interaction, respectively.
Primary Factor
Collision
Consider the simple case of mO = m, i.e., a situation where the mass of the bombarding ion does not differ drastically from those of the target atoms. Then, eqs. (15) and (19) lead to the result that the atomic number effects drop out (strictly for 2, = 2, and approximately
Primary
Kinematics
Factor
( 2m -m,)
Cotlision
Kinematics
lM)
(M,-MI/CM
m 1.
1
15
b
0.5 01
10
MJM
0
- 0.5
a 1
Fig. 3. Coefficient of AMi/M through the ion-target
ma
in eq. (21), determiuing the influence of atomic mass differences on preferential recoil implantation collision kinematics. (a) Contour plot of the factor (fm - mo). (b) The factor (Ma - M)/(MO + M). MO is the ion mass.
P. Sigmund / Preferential recoil implantation in polyatomic targets
Total
Effect
Coefficient
of
AMilM
Fig. 4. Coefficient of AM,/M in eq. (22), determining the relative difference in mean recoil implant depth, AP,/P, for the limiting case of MO= M, i.e., self-bombardment. In that situation, mg = m is the proper choice of exponents. The solid curve includes all nonvanishing effects, i.e., the mass dependence of the primary recoil cross section and the stopping and angular scattering of recoil atoms. The dashed line ignores the angular scattering of recoil atoms.
for 2, different from Z). For Ma = M, the primary collision kinematics does not either co~~~bute to a nonsto~c~omet~ in recoil impl~~tion. The combined effect of the primary cross section and the recoil range is then given by AP,/P=
(V/2U-
2m)AMi/M,
(22)
and this result is shown in fig. 4. The most striking feature is the extremely rapid variation with m, including a change in sign at m = 4. Thus, the sign of preferential recoil implantation at low energy must be very sensitive to the interatomic potential, in particular the target-target potential. Inspection of fig. 4 shows that in the absence of angular scattering of recoil atoms, A Pj/P is negative, i.e., lower-mass atoms are r~~-~pl~ted preferentially. This explains the finding in ref. [2], where angular scattering was ignored.
It also follows that for large m, i.e., high recoil energy, APJP is negative even when angular scattering is included. This explains the finding in ref. [l]. Note that both ref. [l] and ref. [2] addressed high recoil energies. Fig. 4 demonstrates that for m = ma and Ma = M, a positive effect must be expected for m -C $, i.e., for steep interaction potentials. The scattering law employed in refs. [3,5] does not seem to have been documented in detail. From the fact that not only Moli&e interaction, but also an inverse square interaction potential was utilized in the low keV range [5], the present author expects that a cutoff procedure may have been incorporated which makes the effective potential that enters the scattering integral very steep for low-energy collisions. Such a procedure was not utilized in [4]. Quite apart from this, the potential employed in ref. [4] is much softer than the Moliere potential. As far as reality is concerned, the present author has little doubt that the interatomic potential governing low-energy motion is sufficiently hard to cause a turnover from negative to positive mass dependence at some low energy. For not too light ions such as argon or xenon 13-51, the turnover may well occur at a bombarding ion energy of several keV. Note in particular that if both the ion-target and the target-target interaction are Born-Mayer-like, where m and m, are of the order of 0.055 [8], there must be expected a preferential recoil implantation effect of the heavier species of the order of AM/M, according to fig. 4. This is a very pronounced mass effect. The present discussion is limited to small differences in Mi and Z, and is, therefore, not immediately applicable to a LuFe alloy, the system discussed extensively in refs. [3-51. Disregarding quadratic terms in A MJM, consider briefly the influence of ion mass and charge. For argon ions on LuFe, MO is significantly smaller than the Mi of the target, hence mO may be somewhat larger than m. Then, the factor 2m - mO occurring in eq. (21) becomes smaller in magnitude although it will not turn negative. At the same time, the second factor, (Ma - M)/( MO + M), is = - 0.49. Hence, primary co& l&ion kinematics causes a shift in the negative direction for argon bombardment. That shift is very small, of the order of -0.1 AM/M or less. For krypton, that effect has opposite sign but its magnitude is even smaller than for argon in the range of ion energies under consideration. For m. # m, the effects determined by charge numbers in eqs. (15b) and (19b) do not cancel out. For argon, where mO > m, the term due to the primary interaction cross section, eq. (15b), dominates and, hence, a positive implantation effect is expected which would be less prono~c~ for xenon bomb~~ent. However, even for argon, the effect is small since it is proportional to the difference mg - m.
P. Sigmund / Preferential recoil implantation in polyatomic targets
8. Conclusions According to fig. 4, AP,/P is very sensitive to m. This makes it somewhat questionable to draw quantitative conclusions on the basis of the power approximation eq. (1) to the atomic scattering cross section, even in case of small mass differences. However, the following conclusions seem justified. (1) Angular scattering of recoil atoms during slowing down is a cause of positive mass dependence of the mean recoil implant depth in a homogeneous alloy, i.e., it would cause the heavier species to be implanted preferentially in the absence of competing processes. (2) This effect increases in importance with decreasing ion energy. (3) The magnitude of this effect depends sensitively on the shape of the potential, in particular the target-target potential. (4) All other effects tend to increase in importance with increasing ion energy. (5) The ion/target mass ratio may affect the details in either direction, but that influence is small except for mass ratios very different from unity. (6) At high energies, the relative importance of angular scattering decreases, and a negative mass effect is predicted, i.e., light atoms are implanted preferentially. (7) Atomic number effects are weaker than mass effects. Discussions with R. Collins, K. Johannessen, and M. Vicanek are gratefully acknowledged.
21
References R. Kelly and J.B. Sanders, Nucl. Instr. and Meth. 132 (1976) 335. PI P. Sigmund, J. Appl. Phys. 50 (1979) 7261. 131 M.L. Roush, O.F. Goktepe, T.D. Andreadis, and F. Davarya, Nucl. Instr. and Meth. 194 (1982) 611; M.L. Roush, T.D. Andreadis, F. Davarya, and O.F. Goktepe, Appl. Surf. Sci. 11/12 (1982) 235; M.L. Roush, F. Davarya, O.F. Goktepe, and T.D. Andreadis, Nucl. Instr. and Meth. 209/210 (1983) 67. t41 W. Miiller and W. Eckstein, Nucl. Instr. and Meth. B2 (1984) 814. PI O.F. Goktepe, T.D. Andreadis, M. Rosen, G.P. Mueller, and M.L. Roush, Nucl. Instr. and Meth. B13 (1986) 434; M. Rosen, G.P. Mueller, M.L. Roush, T.D. Andreadis, and O.F. Goktepe, Nucl. Instr. and Meth. B13 (1986) 439. WI J. Lindhard, V. Nielsen, and M. Scharff, K. Dan. Vidensk. Selsk. Mat. Fys. Medd. 36 (1968) no. 10. [71 N. Andersen and P. Sigmund, K. Dan. Vidensk. Selsk. Mat. Fys. Medd. 39 (1974) no. 3. iSI P. Sigmund, Phys. Rev. 184 (1969) 383. PI H.H. Andersen and P. Sigmund, Nucl. Instr. and Meth. 38 (1965) 238. WI P. Sigmund and A. Gras-Mar& Nucl. Instr. and Meth. 182/183 (1981) 25. VI P. Sigmund, Nucl. Instr. and Meth. B18 (1987) 375. [I21 J. Lindhard, M. Scharff, and H.E. S&i&t, K. Dan. Vidensk. Selsk. Mat. Fys. Medd. 33 (1963) no. 14. 1131 K.B. Winterbon, P. Sigmund, and J.B. Sanders, K. Dan. Vidensk. Selsk. Mat. Fys. Medd. 37 (1970) no. 14.
PI
PREFERENTIAL RECOIL IMPLANTATION IN POLYATOMIC TARGETS Peter SIGMUND Fysisk Inst~tut, Odeme U~i~ers~tet,DK-5230 Odense M, Denmark Received 29 February 1988 and in revised form 6 April 1988
Five distinct effects have been identified which contribute
to preference
recoil impl~tation
in a polyatomic material. Their
directions and relative magnitudes each depend on some combination of the following parameters: Masses and atomic numbers of the target atoms; mass and atomic number of the bombarding ion; ion-target and target-target interatomic potential; ion energy (indirectly). High ion energy favors a negative mass effect, i.e., light species are implanted preferenti~ly. A positive mass effect is found at low ion energies, caused by differences in the angular scattering of light and heavy recoil atoms, respectively. The magnitude of this effect as well as the point of transition between positive and negative mass dependence are found to be governed primarily by the shape of the target-target interaction potential. The origin of conflicting conclusions in the theoretical literature on the sign of preferential recoil implantation is located. Estimates are given in the limit of small differences in atomic masses and charge numbers of the constituents of the target.
1. Introduction Conflicting results have been reported in the litesature on the theory of preference recoil ~plantation by ion bombardment of binary targets [l-5]. In brief, refs. [l], [2], and [4] reported preferential implantation of the lighter target species while the opposite conclusion was reached in refs. [3] and [S]. Not even tentative explanations have been offered to resolve this apparent discrepancy. Analytical estimates [1,2] have been based on various simplifications. Simple power cross sections were employed and electronic stopping was neglected. More important, [l] treated a monolayer source in a monoatomic stopping medium. In [2], energy loss was properly treated for a binary medium, but angular deflection of recoil atoms during their slo~ng-doe process was ignored. The latter feature was pointed out to be a possibly significant oversimplification when simulation results became available [3]. However, subsequent work with a different code [4] reproduced the conclusion reached in the analytical estimates. Angular scattering was supposed to be adequately included in either code. A direct comparison between the two codes [5] showed recoil implant profiles of similar magnitudes but opposite sign for supposedly identical systems with regard to composition, ion type, and bombarding energy. Several parameters were varied [5] in order to trace the cause of this discrepancy, but although it has been postulated [4] that “computer simulations of this kind are more powerful than analytical calculations due to their variability in defining parameters, introducing spe0168-583X/88/$03.50 Q Elsevier Science Publishers B.V. (North-robed Physics Pub~s~ng Division)
cial mechanisms, and treating a wide range of problems”, no way out of this frustrating situation has been shown so far.
2.
Essentials
It is evident from ref. 151 that the discrepancy shows up already in the low-fluence profiles. There is, therefore, no reason at this stage to look into the complications caused by high-fluence effects. It is ~plausible that the discrepancy should hinge on electronic stopping. Hence, the effect will be ignored. It is well documented that nonstoichiometric behavior depends on the interaction potential or cross section 121. Here it appears more important to allow for freedom in the choice of parameters determining the cross section than to use an accurate analytical or numerical form. To locate the source of discrepancy, conventional power cross sections seem perfectly adequate [6,7], i.e., da&
E,T)=
CiiE-MT-l--m dT; 0
OsTr
yijE,
otherwise,
(1) where yij=4MiMj(Mi+Mj)-‘,
(la>.
E is the energy of an i-particle hitting a j-particle rest, 2’ is the energy of the recoiling j-atom,
at
16
P. Sigmund / Preferential recoil implantation in polyatomic targets
m an exponent, 0 5 m < 1, depending weakly on the reduced energy [6], and Mj and Mj are atomic masses. In (21, h, is a dimensionless parameter depending on the screening function of the potential, and ai, and A, depend on the interaction potential as well. For m > 4, one may, but need not, use Thomas-Fermi quantities
161 ajj = 0&353(z~‘3
myi: i
+ zjz/“) -1’2,
(3a) (3b)
Aij = ZiZje2,‘aij,
a0 being the Bohr radius, while for m < :, Born-Mayer parameters [7-91 may be more adequate, aij = a z 0.219 A rnr$: i
Ajj = ( ZiZj)‘/”
(4a)
52 eV,
(4b)
where Zi and Zj are nuclear charge numbers. Presumably not all features of the above cross sections need to be important for the issue under consideration. Angular de~e&tion and energy loss of recoil atoms need to be included. Angular deflection and energy loss of the primary ion, although readily incorporated [lo], tend to complicate the issue. If one restricts attention to the relocation of near-surface atoms, this effect may safely be ignored. In order to elucidate the significance of every individual effect, it is convenient to linearize the system by assuming that the constituents of the target have opt& slightly different masses and atomic numbers [ll].
where zi(T, cos 0) is the mean penetration depth of an i-recoil and + the (small) ion fluence. Moreover, Zi=R,(T)
cos 8,
where R, is the projected range following from [12,13], CN,cJadou(E, i = 1, cos (pii= (1-
T)[ R!(E)
+ijRi(E-
-COS
(71 T/E)“z+at,i(T,‘E)(l
- T,‘E)-1’2,
(74 and ~ii=‘i(l-k$%$).
(7b)
If recoil scattering were neglected [Z], one would set cos +ij= 1.
4. Linearization Although eq. (7) is easily solved for power scattering [12,13], the resulting expressions are messy in general due to the occurrence of incomplete beta functions. A very useful simplification was pointed out recently [ll]. If ah target masses and atomic numbers are similar, i.e., if ~i=~+A~j;
AMi
i=l,2...
Z,=Z+AZ,;
AZ, -=cZ,
i=l,
The cross section for relocation into depth (z, dz) of an i-atom located near the target surface and hit by a projectile ion 0 with energy E at normal incidence is doi
= dzpEdeei(E,
T)E;(T,
cos ff, z),
(5)
where E(T, cos 8, z) is the penetration profile of an i-atom recoiling with an energy T at an angle 6 against the inward surface normal in a random homogeneous medium composed of Nj j-atoms per volume, j = 1, 2,. . . . The angle 6 is related to the recoil energy by conservation laws, cos B = ( T/Y,~E)?
Pi = +/z
(9)
since deviations are of second order in A&i/M according to eq. (la). Note, however, that yoi in eq. (6) may differ from 1 since the ion mass MO may differ from the Mi. In view of eq. (9), the solutions of eq. (7) contain only compZete beta functions.
5. The mean relocation depth With the ansatz [32] R, = BiE2”,
(10)
the solution of eq. (7) reads = xA$Cjjildt
t+'[l
- (1-
t)2m CQS+ij],
.i where t = T/E, tion yields
after insertion of (1) and (9). Integra-
(Bi)-1=C~~i~[-~-B(-m,2m+i_) i
doi
=+&@de,,i(E,
(8b)
it is reasonable to restrict attention to effects of first order in AM, and AZ,. This implies that
(&)-r
The profile 4 obeys the usual range equations [12,13]. It is most convenient just to consider the mean relocation depth [21
(8a) 2...
yij z 1 in eq. (7), 3. Basic equations
T)]
T)z,(T,
cos @),
(6)
--ijB(l-m,
;+2m)
I
.
(11)
17
P. Sigmund / Preferentialrecoil implantationin polyatomictargets If angular deflection had been ignored, the contents of the bracket would read [ - l/m - B( - m, 2m + I)]. Insertion of (1) and (10) into (6) yields pi = +C,,iE-“OBi~‘EdT
T-1-m~+2m(T/y,,iE)1’2.
Fromeq. one finds AC,, _
0 w
Here, allowance has been made for two different exponents m, where m, applies to the ion-target interaction and m to the target-target interaction. Eq. (12) illustrates the following sources of nonstoichiometry in the mean implant depth: I) The factor C,,i represents the magnitude of the primary recoil cross section. II) The factor B, represents the magnitude of the recoil range at a given recoil energy and angle. III) The limit of integration yoiE represents the maximum energy transfer in primary collisions, and IV) The factor (T/y,,iE)1/2 represents the recoil angle. Moreover, from eq. (11) it follows that II) splits up into two distinct effects: IIA) The term NjCjj reflects the stopping behavior of the recoil atom, and IIB) The term aij reflects the angular scattering of the recoil atom. The presence of five distinct effects going in different directions would make it quite difficult to ask the right questions in a purely simulational analysis without the guidance of an analytical estimate. Integration of (12) yields [2,10] Pi = (2m - m, + 1/2)-1Coiy~im-moBiE2m-ma,
6.1. Influence of the primary recoil cross section (effect I)
(12’)
where the effects III) and IV) have been collected into one factor yzimPrnogoverned by the kinematics of the primary recoil event.
Co
(2), applied to i+O, AM-
Au,,, 2 p-m0
L M
+
2m,-
AAoi Ao
a0
and m-+mo,
j-i,
(14)
.
At this point, a choice of the potential parameters Aoj and aoi is called for. With the choices (3) or (4), one finds
W 3(1-t
AZ,
-m0>
(zo/z)“‘)
1
z
AM, moM
(1% for m. 2 a, and 3m, AZ,
AGi
AM,
P-J)
M
Co =2~-mo~
for m, < f. Eqs. (15a) and (15b) illustrate the fact that the mass dependence (e.g., the isotope effect) is determined by the exponent m, while the dependence on the charge number is also determined by other potential parameters [ll]. The Z-dependent contribution to eq. (15a) or (15b) can be determined from fig. la, which shows the coefficient of AZ/Z vs m,. Similarly, fig. 2a shows the coefficient of AMJM. It is seen that the atomic-number effect is largely positive, i.e., the heavier species tends to be implanted preferentially due to this particular effect, while the mass effect is negative. Both features are evident from the dependence of the cross section, eqs. (1) and (2) on Zj and Mj. Note that the transition from the low-m0 to the high-m0 region is influenced by other parameters, here Z,/Z, and that it can be either below or above a. 6.2. Influence of the recoil range (effect II)
6. Evaluation
Eq. (11) yields, after linearization,
Eq. (12’) is still exact for a polyatomic target within the adopted model. For small differences in mass and charge between target constituents, linearization yields A Pi/P = AC&C,
+ A B,/B + (2m - m,,) Ayoi/yO,
(13) where APJP is the relative deviation of the mean relocation depth Pi of an i-recoil from the average P, and the three terms on the right-hand side reflect the contributions due to effect I), effects HA) and IIB), and effects III) and IV), respectively. In (13), C, is the average of Cei over all i, properly weighted over target composition, and similarly for ye and B. The three terms in eq. (13) will be evaluated separately in order to elucidate their relative significance.
ABi B
- &
C%ACij J
+ &
.&qaij, J
where the first term represents the effect of stopping (IIA) and the second one the effect of angular deflection (IIB). Evidently, the j-sums yield average values, and hence,
ABi B
AC,
- - 7
V + uai.
(16’)
In (16) and (16’), the abbreviations U= --&B(-m,3/2+2m)
(17a)
and V=B(l-m,
1/2+2m)
have been introduced.
(17b)
18
P. Sigmund / Preferential recoil implantation in polyatomic targets
1.0 Recoil Primary Coefficient
Cross of
Range
Coefficient
Section
01 AZ,/Z
A 2, IZ
-10
Fig. 1. (a) Coefficient of A&/Z in eqs. (15a) and (15b), determining the influence of atomic number differences on preferential recoil implantation through the magnitude of the ion-target interaction cross section. The parameter mO characterizes the power cross section for the ion-target interaction. For small mO, the atomic number of the ion does not enter. Solid lines follow the transition from Born-Mayer-lie interaction at small m,, to screened Coulomb interaction at larger mO. Dashed lines are extrapolations of the small-m, lines to larger mO and vice versa. (b) Coefficient of AZ,/2 in eqs. (19a) and (19b), determining the influence of atomic number differences on preferential recoil implantation through the recoil range. The parameter m characterizes the power cross section for the target-target interaction. Solid and dashed lines as in (a).
Eqs. (2), (7b), and (16’) yield
where again, a choice of the potential parameters a, = CjaijNj/N and A, = CjAijNj/N is called for. With the choices (3) or (4), one finds (19a) for pn2 $, and
atomic-number effect is negative, i.e., the recoil atom with the higher Z is impl~t~ less than average because of its higher stopping power. The atomic-mass effect on the stoppingpower is in the same direction but weaker, while angular defection favors the heavier recoil atom, as was to be expected. Unlike other effects, the magnitude of the angular deflection effect increases witk decreasing m . 6.3. Injluence of primary collision kinematics (effects III and IV)
(19b) for m<$. Augular detection enters through the term containing V/U. According to eqs. (17), this reads m V -= (20) u T(3/2+m) . ++2mr(l - m)r(+ -t 2m) This term would drop out if angular scattering were neglected. Fig. lb shows the dependence of the coefficient of AZ,/2 in eq. (19a) and (19b) on m, and fig. 2b shows the coefficient of AM/M. One may note that the
From (la) one finds
The two factors determining this mass effect are shown separately in figs. 3a and 3b. Evidently, the combined effect may be positive or negative, dependent on the ion mass. 7. Discussion Eq. (13), in conjunction with eqs. (Isa) and (15b), (19a) and (19b), and (21), supplemented by (20), con-
19
P. Sigmund / Preferential recoil implantation in polyatomic targets
1.0 Pnmary
Cross
Coefficient
of
Recod
Section A M;/M
s-m
Range
Coefficient
of AMilM
a 0.0 0.5
m,
1.0
\
-10
Fig. 2. (a) Coefficient of AMi/M in eqs. (15a) and (15b), determining the iufluence of atomic mass differences on preferential recoil impl~tat~on through the ion-target interaction cross section. (b) Coefficient of A~~/~ in eqs. (19a) and (19b), deter~ng the influence of atomic mass differences on preferential recoil implantation through the recoil range (solid line). The dashed line represents the same quantity under the assumption of negligible angular scattering of recoil atoms.
tams all pertinent information. The obvious advantage of the present description is the reduction to relative changes where the crucial parameters that enter are the exponents m O and m for the ion-target and the target-target interaction, respectively.
Primary Factor
Collision
Consider the simple case of mO = m, i.e., a situation where the mass of the bombarding ion does not differ drastically from those of the target atoms. Then, eqs. (15) and (19) lead to the result that the atomic number effects drop out (strictly for 2, = 2, and approximately
Primary
Kinematics
Factor
( 2m -m,)
Cotlision
Kinematics
lM)
(M,-MI/CM
m 1.
1
15
b
0.5 01
10
MJM
0
- 0.5
a 1
Fig. 3. Coefficient of AMi/M through the ion-target
ma
in eq. (21), determiuing the influence of atomic mass differences on preferential recoil implantation collision kinematics. (a) Contour plot of the factor (fm - mo). (b) The factor (Ma - M)/(MO + M). MO is the ion mass.
P. Sigmund / Preferential recoil implantation in polyatomic targets
Total
Effect
Coefficient
of
AMilM
Fig. 4. Coefficient of AM,/M in eq. (22), determining the relative difference in mean recoil implant depth, AP,/P, for the limiting case of MO= M, i.e., self-bombardment. In that situation, mg = m is the proper choice of exponents. The solid curve includes all nonvanishing effects, i.e., the mass dependence of the primary recoil cross section and the stopping and angular scattering of recoil atoms. The dashed line ignores the angular scattering of recoil atoms.
for 2, different from Z). For Ma = M, the primary collision kinematics does not either co~~~bute to a nonsto~c~omet~ in recoil impl~~tion. The combined effect of the primary cross section and the recoil range is then given by AP,/P=
(V/2U-
2m)AMi/M,
(22)
and this result is shown in fig. 4. The most striking feature is the extremely rapid variation with m, including a change in sign at m = 4. Thus, the sign of preferential recoil implantation at low energy must be very sensitive to the interatomic potential, in particular the target-target potential. Inspection of fig. 4 shows that in the absence of angular scattering of recoil atoms, A Pj/P is negative, i.e., lower-mass atoms are r~~-~pl~ted preferentially. This explains the finding in ref. [2], where angular scattering was ignored.
It also follows that for large m, i.e., high recoil energy, APJP is negative even when angular scattering is included. This explains the finding in ref. [l]. Note that both ref. [l] and ref. [2] addressed high recoil energies. Fig. 4 demonstrates that for m = ma and Ma = M, a positive effect must be expected for m -C $, i.e., for steep interaction potentials. The scattering law employed in refs. [3,5] does not seem to have been documented in detail. From the fact that not only Moli&e interaction, but also an inverse square interaction potential was utilized in the low keV range [5], the present author expects that a cutoff procedure may have been incorporated which makes the effective potential that enters the scattering integral very steep for low-energy collisions. Such a procedure was not utilized in [4]. Quite apart from this, the potential employed in ref. [4] is much softer than the Moliere potential. As far as reality is concerned, the present author has little doubt that the interatomic potential governing low-energy motion is sufficiently hard to cause a turnover from negative to positive mass dependence at some low energy. For not too light ions such as argon or xenon 13-51, the turnover may well occur at a bombarding ion energy of several keV. Note in particular that if both the ion-target and the target-target interaction are Born-Mayer-like, where m and m, are of the order of 0.055 [8], there must be expected a preferential recoil implantation effect of the heavier species of the order of AM/M, according to fig. 4. This is a very pronounced mass effect. The present discussion is limited to small differences in Mi and Z, and is, therefore, not immediately applicable to a LuFe alloy, the system discussed extensively in refs. [3-51. Disregarding quadratic terms in A MJM, consider briefly the influence of ion mass and charge. For argon ions on LuFe, MO is significantly smaller than the Mi of the target, hence mO may be somewhat larger than m. Then, the factor 2m - mO occurring in eq. (21) becomes smaller in magnitude although it will not turn negative. At the same time, the second factor, (Ma - M)/( MO + M), is = - 0.49. Hence, primary co& l&ion kinematics causes a shift in the negative direction for argon bombardment. That shift is very small, of the order of -0.1 AM/M or less. For krypton, that effect has opposite sign but its magnitude is even smaller than for argon in the range of ion energies under consideration. For m. # m, the effects determined by charge numbers in eqs. (15b) and (19b) do not cancel out. For argon, where mO > m, the term due to the primary interaction cross section, eq. (15b), dominates and, hence, a positive implantation effect is expected which would be less prono~c~ for xenon bomb~~ent. However, even for argon, the effect is small since it is proportional to the difference mg - m.
P. Sigmund / Preferential recoil implantation in polyatomic targets
8. Conclusions According to fig. 4, AP,/P is very sensitive to m. This makes it somewhat questionable to draw quantitative conclusions on the basis of the power approximation eq. (1) to the atomic scattering cross section, even in case of small mass differences. However, the following conclusions seem justified. (1) Angular scattering of recoil atoms during slowing down is a cause of positive mass dependence of the mean recoil implant depth in a homogeneous alloy, i.e., it would cause the heavier species to be implanted preferentially in the absence of competing processes. (2) This effect increases in importance with decreasing ion energy. (3) The magnitude of this effect depends sensitively on the shape of the potential, in particular the target-target potential. (4) All other effects tend to increase in importance with increasing ion energy. (5) The ion/target mass ratio may affect the details in either direction, but that influence is small except for mass ratios very different from unity. (6) At high energies, the relative importance of angular scattering decreases, and a negative mass effect is predicted, i.e., light atoms are implanted preferentially. (7) Atomic number effects are weaker than mass effects. Discussions with R. Collins, K. Johannessen, and M. Vicanek are gratefully acknowledged.
21
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PI