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Z1 oscillations in low energy heavy ion ranges
Nuclear I n s t r u m e n t s and Methods 178 (1980) 5 1 7 - 5 2 2 © North-Holland Publishing C o m p a n y
Z 1 OSCILLATIONS IN LOW ENERGY HEAVY ION RANGES
D.J. O'CONNOR, B.W. FARMERY, D. CHIVERS * and M.W. THOMPSON Physics Building, University of Sussex, Falrner, Brighton, BN1 9QH, Sussex, England Received 30 June 1980
Following the observation in an earlier paper o f systematic deviations from a universal r a n g e - e n e r g y relationship, the ranges of a series o f elements from s3I to 92U, implanted at a reduced energy of e = 0.09 in amorphised Si, have been measured and the dimensionless range, p, has been found to exhibit a distinct oscillatory behaviour as a function of Z 1 rather than the constant dependence predicted by any theory based on a T h o m a s - F e r m i model of the atom. These oscillations are believed to result from the modification to the scattering cross section caused by electron shell effects and some evidence for this conclusion arises from a marked difference in trends between the range and the straggling m e a s u r e m e n t s which should be sensitive to the form of the interatomic potential.
1. Introduction
These mechanisms include losses through collisions with electrons, by excitation of electrons in both incident and target atoms and through collisions with the atoms of the target. The first two processes are usually considered together and referred to as the electronic loss, while that due to collisions with the target atoms is termed the nuclear loss. From these processes a stopping power can be defined which is described as an energy loss per unit distance (S) and it is assumed that the two loss processes are independent so that the stopping powers are additive, e.g.
The study of ranges of energetic particles in solids has attracted wide ranging interest both for its technological importance [1,2] and as a tool to improve the understanding of atomic collision processes in solids. Many experimental [3] and theoretical [4,5] studies have added to our knowledge of the general trends in range-energy relationshipg but from recent experimental work, evidence has emerged for a variation of range as a function of atomic number [3] which is not adequately covered by theoretical treatments. This variation is not believed to beassociated with the welI documented Z1 dependence on electronic stopping power [6,7] but appears instead to result from the breakdown of the .Thomas-Fermi model for the description of the a t o m - a t o m interaction [8] at low energies or small scattering angles. The measurements described below are of the range of heavy ions in silicon under conditions which are expected to illustrate the importance of the interatomic potential at relatively low energies.
S = Se + Sn, (1) where Se is the electronic and Sn is the nuclear stopping power. These stopping powers behave differently as a function of energy with the nuclear stopping being dominant at low energies (i.e. energies less than M1 keV, where M 1 is the projectile mass in amu corresponding to e ~ 1 ; see later for definition of e) and the electronic stopping becoming dominant at energies above this level. The inelastic stopping power at low energies is given by: Se = kE 1/2 ,
2. Background
(2)
where k is a constant which depends on the target and projectile species and E is the energy of the incident particle. The nuclear stopping power is given by;
The range of an energetic atom in a solid is governed by the mechanisms by which it loses energy.
Tm
Sn = f
* Chemistry Division, AERE, Harwell, OX11 OR.A, England.
o 517
To(T) d T ,
(3)
518
D.J. O'Connor et al. / Z
where T is the energy transferred to the target atom, o(T) is the energy transfer cross section and Tm is related to the projectile energy by; 4M1M2E
T m =(M 1 + M2)2.
(4)
M1 and M2 are the masses of the projectile and target atoms respectively. The cross section is dependent on the form of the interatomic potential used, and although there are a large number of suggested forms for this potential, few allow an analytical treatment. Lindhard [4,9] has shown that when a power approximation to the T h o m a s - F e r m i potential was assumed [eq. (5)], and the energy and range were reduced to dimensionless quantities [eqs. (6) and (7)], a universal range-energy relationship can be derived.
V(r) - Z1Z-2e2 ks[a--] s , a _ e p -
(5)
s LrJ
Ms a M1 + M 2 Z I Z 2 e 4M1M2
(M1 +M2):
E,
(6)
2 7raZNR ,
(7)
where _
ao
(g)
a - x/(z~, 3 + z~,~,
N is the atomic density of the target, and ks is a constant. The exact form o f the universal function depends on the power of the potential as does, to a lesser extent, the ratio of straggling to the range;
(7;
72 _ _
_
s-
1
s(2s - 1)
(9) '
where Ap is the straggling and 7 2 - (M1 +M2)2
(10)
4M1M2
The agreement between experimental results and a universal range-energy graph was verified in a previous paper [3] in which the results of range measurements in Si and M targets from a number of groups were shown to fall on or near to a universal curve. Deviations from this curve could be attributed to inconsistency between laboratories and techniques, or to a physical effect resulting from a breakdown in the assumptions on which the theory was based. Many of
1
oscillations
these deviations are most pronounced for energies at which the nuclear stopping power is dominant indicating that the assumption of a universal interatomic potential may not be valid. This belief is reinforced by recent measurements o f the differential scattering cross section [10,11] which revealed the importance of electron shell effects on the scattering cross section. It was found that the scattering cross section has a component which has an oscillatory dependence on e sin ½~b rather than being a smooth monotonic function as predicted by the Thomas Fermi model. The peaks in the oscillatory cross section occurred at different values o f e sin ½¢ for different combinations of target atom and incident ion and may be expected to have an effect on the range of slow heavy ions. Some measure of the expected magnitude of this effect can be obtained from the experimental range comparisons mentioned previously [3] in which it was noted that the ranges for Au, Pt, Pb and Bi in Si were in excess of the general trend which defined the universal curve by some tens of percent. In order more thoroughly to investigate the dependence of range on atomic number, a series of ranges have been measured for elements from iodine to uranium at a reduced energy of e = 0.09. This energy was chosen as it was sufficiently low to ensure that nuclear stopping dominated (i.e. e ~ 1), while at the same time ensuring that the range was not so small as to make the relative error in the range measurement large enough to mask the effect we sought to measure.
3. Experimental The Si targets were well polished single crystal slices which were amorphised to a depth greater than the range under investigation by bombardment with 1016 ions/cm 2 of 100 keV Si+ at 7 ° to the normal of the surface [3]. In order to minimise experimental and systematic errors, a portion of each slice was implanted with 120 keV Bi (e = 0.09) to act as a standard against which measured ranges were compared. The existence of the Bi standard ensured that the reproducibility o f measured ranges for an implant between different samples and on different days was to within 2% for the lighter ions and down to 1% for the heavy ions when normalised to the Bi range. The absolute range however could not be determined to better than 7%. All implant doses were 5 × 1014 ions/ Cm
2 .
D.J. O'Connor et al. / Z 1 oscillations
The implant profile, from which the range and straggling were determined, was measured using standard Rutherford backscattering techniques but to improve the depth resolution the target was tilted so that the beam was incident at 30 ° to the surface and the detector was at 20.7 ° to the surface. The implanted targets were also investigated by SIMS profiling [12] and the ranges thus determined exhibited the same features as the RBS results though they were on the average 20% greater.
jjgv
519
0.4
i i I
I -
03
I
J
I
i
i
i
Ill
i I
~1111 I
50
60
70
80
90
Z
4. Results The results presented in table 1 reveal a variation of mean projected range with atomic number which is considerably in excess of the experimental error. The range exhibits a m i n i m u m at Xe and a relatively small value for U, while there is a broad maximum for atomic numbers around Au (fig. 1). The experimental results are compared to the theoretical value of 0.309 calculated by Schi~btt [19] (broken line in fig. 1) which differs little from the range of p = 0.307 predicted by WSS [5]. The greater than average range for elements around Au is in agreement with observations made
Fig. 1. The measured mean projected range for low energy (e = 0.09) heavy ion implants in amorphous silicon. The dashed line is the estimate of Schiq~tt [19].
previously [3] and is also in qualitative agreement with results of a similar set of measurements by Besenbacher [13] at a reduced energy of E = 0.015. Their results exhibited marked oscillations which in general had maxima at the noble metals and minima at the inert gases. One explanation for the observed variation of range is that the effective T h o m a s - F e r m i screening function is different for the various i o n - a t o m combi-
Table 1 Range measurements for implants in Si at E = 0.09 Element
Isotope (ainu)
Energy (keV)
Range (A)
p~-
p-
7Ao/p
f = a'/a [see eq. (11)]
531 54Xe S5Cs 56Ba 57La 58Ce 59pr 62Sm
127 132 133 138 139 140 141 152 153 159 165 169 175 181 187 193 197 205 208 209 238
45.0 47.5 49.0 51.0 53.0 54.0 56.0 63.0 65.0 70.0 75.0 79.0 84.0 90.0 96.0 102.0 107.0 114.0 117.0 120.0 153.0
251 244 267 266 276 281 286 319 335 350 370 384 416 430 452 469 488 493 495 501 553
0.326 0.307 0.331 0.322 0.327 0.331 0.330 0.343 0.355 0.355 0.360 0.363 0.379 0.376 0.380 0.380 0.385 0.373 0.369 0.368 0.349
0.365 0.343 0.369 0.358 0.363 0.368 0.366 0.378 0.391 0.389 0.394 0.396 0.412 0.408 0.411 0.411 0.415 0.402 0.397 0.396 0.373
0.38 0.55 0.40 0.37 0.39 0.40 0.38 0.37 0.36 0.36 0.38 0.39 0.40 0.39 0.37 0.37 0.40 0.38 0.38 0.39 0.36
0.961 1.005 0.950 0.970 0.958 0.950 0.952 0.925 0.901 0.901 0.892 0.886 0.858 0.863 0.856 0.856 0.848 0.868 0.875 0.877 0.913
63Eu 65Tb
67Ho 69Tm 71Lu 73Ta 7SRe
771r 79Au 81T1 82pb 83Bi 92U
520
D.J. O'Connor et al. / Z 1 oscillations
[Xe
~Af i f
S~2
I
! i
'
I
'1
![
, I I , ' ~ ; I
~=31
element. The former has already been mentioned [10] and can only be further studied by computer simulation [8,14] if detailed information on the interatomic potential becomes available. The adjustment of the scaling parameters however does lend itself to simple analysis, and the most common method is to alter the screening length by including a "correction factor" [ 14,15 ], thus t
5O
(11)
a =fa,
I 6'0
70
80
90
Fig. 2. The measured reduced relative straggling compared to values predicted by power potentials.
nations. If this were the case and we sought to verify that this difference in interatomic potential manifested itself as a change in the power of the power approximation then we would expect from eq. (9) to see a similar trend in the relative straggling to that exhibited by the .reduced range except that 7 A p / p should be less sensitive to s. The relatice straggling results are presented in fig. 2 and all, except Xe, are in" agreement with that predicted by power potentials. The anomalously high value for Xe is attributed to diffusion mechanisms as the first measurement (within 24 h of implantation) yield a value of 7 A # / o = 0.44 but later measurements (days or weeks later) yield values in the region of 0.55. Although the relative straggling is only weakly dependent on the potential (see table 2) and the relative experimental error is larger, there does not seem to be the same general trend that is evident in the reduced range. This could indicate that either the power approximation is not an adequate description of the potential at these energies or that the scaling parameters [eqs. (6)-(8)] may differ from element to Table 2 The variation of ~'Ao/o at E = 0.09 for different interatomic potentials [4,8]. Potential
3.Ap/p
Thomas-Fermi Moli~re Bohr Free electron (Kr-C) Free electron (average) r-2 r-3
0.365 0.326 0.284 0.326 0.331 0.408 0.365
where a is the screening length given by eq. (8), a' is the effective screening length a n d f i s the "correction factor". The screening length correction factor in table 1 is calculated assuming that the range-energy relationship obeys a power law [ 1 6 - 1 8 ] of the form; (12)
p acE 2Is ,
thus
S
f= [po/0] 2(s-1),
(13)
where O is the reduced range determined using the screening length given by eq. (8) and Po is taken from the tables of SchiCtt [19] for e = 0.09 and k = 0.115. The power of the potential, s, is assumed to have the value of 3 as this is in agreement with empirical determinations of the range-energy relationship leading to O c~ e 2/3. The correction factors presented in table 1 should only be regarded as a guide to the magnitude of "correction" necessary to obtain agreement between theory and experiment, as the true range depends on the form of the interatomic potential about which little is certain. It should also be noted that the variation of correction factor with atomic number is not in agreement with previously reported work [14,15]. The range presented in fig. 1, the mean projected range, is less than the total path length of the particle which is the parameter most easily determined analytically. Although the difference between the two is normally regarded as negligible when M1 >>M2, it is not quite so small an effect when comparing small differences in the range for different values of M1. If we use the relationship [4], s2/l plop = 1 + 4(2s - 1)'
(14)
where p is the total path length, pp is the measured mean projected range and/a =M2/M1, then a comparison of the variation of total path length with Z can be made.
D.J. O'Connor et al. / Z 1 oscillations
I
04
I
b
I
I
I
I
~
I III
i I
t q l I I I
f 0.3-
s0
6'0
7~
80
9~
Z
Fig. 3. The estimated total path length from fig. 1 and eq. (14) using s = 3.
The deduced values of total path length presented in table 1 and fig. 3 are for s = 3, and it can be seen that the oscillation as a function o f Z1 is still evident though slightly reduced in magnitude. The important observation though is that once the correction is made, all values o f the reduced range are in excess of the theoretical estimates by at least 10%.
521
The use of a Bi standard has improved the relative accuracy ( 1 - 2 % ) of the range determinations but the absolute accuracy is limited by the Bi range accuracy which is estimated to be 7%. These range measurements can be quoted to greater absolute accuracy if they can be normalised to any one in the set which is measured to greater absolute precision. Attention should be drawn to the anomalously wide Xe distribution which has been attributed to diffusion processes. This is important as it brings into question the validity of the range measurements for that element and underlines the importance of correlating straggling measurements to range measurements whenever possible to ensure that o t h e r physical effects are not creating anomalous ranges. We are grateful to Metallurgy Division, AERE Harwell for sponsoring this work and for a fellowship for one o f us (J.O.C.). We also wish to thank S. Pugh, S. Nelson and A. Marwick for valuable discussions and advice. We appreciate the assistance o f Mr. C. Jackson (AERE, Harwell) with the SIMS examination o f these samples.
References 5. Conclusions The determination o f the Z dependence of reduced range at constant reduced energy has revealed an oscillatory behaviour which cannot be predicted from current theoretical models. This oscillatory behaviour, when viewed in conjunction with recent multiple scattering and differential scattering cross section [10] measurements, suggests that the current statistical models for the atom, used for determining the interatomic potential, are not adequate to describe atomic collision processes at energies below e = 0.1. These models, which exhibit a smooth monotonic variation of potential parameters with increasing atomic ~number, do not properly account for the electronic shell effects which may become significant at low energies. Although the adjustment of the potential can be achieved by adjusting the screening length this is not really satisfactory as it may have to be an energy dependent adjustment unless the oscillations in the electronic stopping coincide with those in the nuclear stopping. As there is yet no reliable way of predicting from other parameters what the correction should be, it must remain an empirical fitting parameter.
[1 ] J.W. Mayer, L. Eriksson and J.A. Davies, Ion implantation (Academic Press, New York, 1970). [2] G. Dearnaley, J.H. Freeman, R.S. Nelson and J. Stephen, Ion implantation (North-Holland Publ. Co., Amsterdam, 1973). [3] J.L. Combasson, B.W. Faxmery, D. McCulloch, G.W. Neilson and M.W. Thompson, Rad. Eff. 36 (1978) 149 and references mentioned therein. [4] J. Lindhard, M. Scharff and H.E. Schi4tt, Kgl. Dansk. Vid. Selsk. Matt. Fys. Medd. 33 (1963) no. 14. [5] K.B. Winterbon, P. Sigmund and J.B. Sanders, Kgl. Dansk. Vid. Selsk. Matt. Fys. Medd. 37 (1970) no. 14. [6] F.H. Eisen, Can. J. Phys. 46 (i968) 561. [7] L. Eriksson, J.A. Davies and P. Jespersgard, Phys. Rev. 161 (1967) 219. [8] W.D. Wilson, L.G. Haggemark and J.P. Biersack, Phys. Rev. B15 (1977) 2458. [9] J. Lindhaxd, V. Nielsen and M. Scharff, Kgl. Dansk. Vid. Selsk. Matt. Fys. Medd. 36 (1968) no. 10. [10] P. Loftager, F. Besenbacher, O.S. Jensen and V.S. Sorensen, Phys. Rev. A20 (1979) 1143. [11] H. Knudsen, F. Besenbacher, J. Heinemeier and P. Hvelplund, Phys. Rev. A13 (1976) 2095. [12] A.D. Marwick and R.C. Piller, Rad. Eft. 33 (1977) 245. [13] F. Besenbacher, J. B~ttiger, T. Laursen, P. Loftager and W. M611er, 8th Int. Conf. on Atomic collisions in solids, Hamilton, Canada; Nucl. Instr. and Meth. 170 (1980) 183. [14] M. Bister, M. Hautala and M. J~ntti, Rad. Eft. 42 (1979) 201.
522
D.J. O'Connor et al. / Z 1 oscillations
[15] D.J. O'Connor and R.J. MacDonald, Rad. Eft. 34 (1977) 247. [16] H. Oetzmann, A. Feuerstein, H. Grahmam and S. Kalbitzer, Phys. Lett. 55A (1975) 170. [17] A. Feuerstein, S. Kabitzer and H. Oetzmann, Phys. Lett. 51A (1975) 165.
[18] W.A. Grant, J.S. Williams and D. Dodds, Proc. Karlsruhe Conf. on Surface analysis (Plenum Press, New York, 1976). [19] H.E. SchiCtt, Kgl. Dansk. Vid. Selsk. Matt. Fys. Medd. 35 (1966) no. 9.
Z 1 OSCILLATIONS IN LOW ENERGY HEAVY ION RANGES
D.J. O'CONNOR, B.W. FARMERY, D. CHIVERS * and M.W. THOMPSON Physics Building, University of Sussex, Falrner, Brighton, BN1 9QH, Sussex, England Received 30 June 1980
Following the observation in an earlier paper o f systematic deviations from a universal r a n g e - e n e r g y relationship, the ranges of a series o f elements from s3I to 92U, implanted at a reduced energy of e = 0.09 in amorphised Si, have been measured and the dimensionless range, p, has been found to exhibit a distinct oscillatory behaviour as a function of Z 1 rather than the constant dependence predicted by any theory based on a T h o m a s - F e r m i model of the atom. These oscillations are believed to result from the modification to the scattering cross section caused by electron shell effects and some evidence for this conclusion arises from a marked difference in trends between the range and the straggling m e a s u r e m e n t s which should be sensitive to the form of the interatomic potential.
1. Introduction
These mechanisms include losses through collisions with electrons, by excitation of electrons in both incident and target atoms and through collisions with the atoms of the target. The first two processes are usually considered together and referred to as the electronic loss, while that due to collisions with the target atoms is termed the nuclear loss. From these processes a stopping power can be defined which is described as an energy loss per unit distance (S) and it is assumed that the two loss processes are independent so that the stopping powers are additive, e.g.
The study of ranges of energetic particles in solids has attracted wide ranging interest both for its technological importance [1,2] and as a tool to improve the understanding of atomic collision processes in solids. Many experimental [3] and theoretical [4,5] studies have added to our knowledge of the general trends in range-energy relationshipg but from recent experimental work, evidence has emerged for a variation of range as a function of atomic number [3] which is not adequately covered by theoretical treatments. This variation is not believed to beassociated with the welI documented Z1 dependence on electronic stopping power [6,7] but appears instead to result from the breakdown of the .Thomas-Fermi model for the description of the a t o m - a t o m interaction [8] at low energies or small scattering angles. The measurements described below are of the range of heavy ions in silicon under conditions which are expected to illustrate the importance of the interatomic potential at relatively low energies.
S = Se + Sn, (1) where Se is the electronic and Sn is the nuclear stopping power. These stopping powers behave differently as a function of energy with the nuclear stopping being dominant at low energies (i.e. energies less than M1 keV, where M 1 is the projectile mass in amu corresponding to e ~ 1 ; see later for definition of e) and the electronic stopping becoming dominant at energies above this level. The inelastic stopping power at low energies is given by: Se = kE 1/2 ,
2. Background
(2)
where k is a constant which depends on the target and projectile species and E is the energy of the incident particle. The nuclear stopping power is given by;
The range of an energetic atom in a solid is governed by the mechanisms by which it loses energy.
Tm
Sn = f
* Chemistry Division, AERE, Harwell, OX11 OR.A, England.
o 517
To(T) d T ,
(3)
518
D.J. O'Connor et al. / Z
where T is the energy transferred to the target atom, o(T) is the energy transfer cross section and Tm is related to the projectile energy by; 4M1M2E
T m =(M 1 + M2)2.
(4)
M1 and M2 are the masses of the projectile and target atoms respectively. The cross section is dependent on the form of the interatomic potential used, and although there are a large number of suggested forms for this potential, few allow an analytical treatment. Lindhard [4,9] has shown that when a power approximation to the T h o m a s - F e r m i potential was assumed [eq. (5)], and the energy and range were reduced to dimensionless quantities [eqs. (6) and (7)], a universal range-energy relationship can be derived.
V(r) - Z1Z-2e2 ks[a--] s , a _ e p -
(5)
s LrJ
Ms a M1 + M 2 Z I Z 2 e 4M1M2
(M1 +M2):
E,
(6)
2 7raZNR ,
(7)
where _
ao
(g)
a - x/(z~, 3 + z~,~,
N is the atomic density of the target, and ks is a constant. The exact form o f the universal function depends on the power of the potential as does, to a lesser extent, the ratio of straggling to the range;
(7;
72 _ _
_
s-
1
s(2s - 1)
(9) '
where Ap is the straggling and 7 2 - (M1 +M2)2
(10)
4M1M2
The agreement between experimental results and a universal range-energy graph was verified in a previous paper [3] in which the results of range measurements in Si and M targets from a number of groups were shown to fall on or near to a universal curve. Deviations from this curve could be attributed to inconsistency between laboratories and techniques, or to a physical effect resulting from a breakdown in the assumptions on which the theory was based. Many of
1
oscillations
these deviations are most pronounced for energies at which the nuclear stopping power is dominant indicating that the assumption of a universal interatomic potential may not be valid. This belief is reinforced by recent measurements o f the differential scattering cross section [10,11] which revealed the importance of electron shell effects on the scattering cross section. It was found that the scattering cross section has a component which has an oscillatory dependence on e sin ½~b rather than being a smooth monotonic function as predicted by the Thomas Fermi model. The peaks in the oscillatory cross section occurred at different values o f e sin ½¢ for different combinations of target atom and incident ion and may be expected to have an effect on the range of slow heavy ions. Some measure of the expected magnitude of this effect can be obtained from the experimental range comparisons mentioned previously [3] in which it was noted that the ranges for Au, Pt, Pb and Bi in Si were in excess of the general trend which defined the universal curve by some tens of percent. In order more thoroughly to investigate the dependence of range on atomic number, a series of ranges have been measured for elements from iodine to uranium at a reduced energy of e = 0.09. This energy was chosen as it was sufficiently low to ensure that nuclear stopping dominated (i.e. e ~ 1), while at the same time ensuring that the range was not so small as to make the relative error in the range measurement large enough to mask the effect we sought to measure.
3. Experimental The Si targets were well polished single crystal slices which were amorphised to a depth greater than the range under investigation by bombardment with 1016 ions/cm 2 of 100 keV Si+ at 7 ° to the normal of the surface [3]. In order to minimise experimental and systematic errors, a portion of each slice was implanted with 120 keV Bi (e = 0.09) to act as a standard against which measured ranges were compared. The existence of the Bi standard ensured that the reproducibility o f measured ranges for an implant between different samples and on different days was to within 2% for the lighter ions and down to 1% for the heavy ions when normalised to the Bi range. The absolute range however could not be determined to better than 7%. All implant doses were 5 × 1014 ions/ Cm
2 .
D.J. O'Connor et al. / Z 1 oscillations
The implant profile, from which the range and straggling were determined, was measured using standard Rutherford backscattering techniques but to improve the depth resolution the target was tilted so that the beam was incident at 30 ° to the surface and the detector was at 20.7 ° to the surface. The implanted targets were also investigated by SIMS profiling [12] and the ranges thus determined exhibited the same features as the RBS results though they were on the average 20% greater.
jjgv
519
0.4
i i I
I -
03
I
J
I
i
i
i
Ill
i I
~1111 I
50
60
70
80
90
Z
4. Results The results presented in table 1 reveal a variation of mean projected range with atomic number which is considerably in excess of the experimental error. The range exhibits a m i n i m u m at Xe and a relatively small value for U, while there is a broad maximum for atomic numbers around Au (fig. 1). The experimental results are compared to the theoretical value of 0.309 calculated by Schi~btt [19] (broken line in fig. 1) which differs little from the range of p = 0.307 predicted by WSS [5]. The greater than average range for elements around Au is in agreement with observations made
Fig. 1. The measured mean projected range for low energy (e = 0.09) heavy ion implants in amorphous silicon. The dashed line is the estimate of Schiq~tt [19].
previously [3] and is also in qualitative agreement with results of a similar set of measurements by Besenbacher [13] at a reduced energy of E = 0.015. Their results exhibited marked oscillations which in general had maxima at the noble metals and minima at the inert gases. One explanation for the observed variation of range is that the effective T h o m a s - F e r m i screening function is different for the various i o n - a t o m combi-
Table 1 Range measurements for implants in Si at E = 0.09 Element
Isotope (ainu)
Energy (keV)
Range (A)
p~-
p-
7Ao/p
f = a'/a [see eq. (11)]
531 54Xe S5Cs 56Ba 57La 58Ce 59pr 62Sm
127 132 133 138 139 140 141 152 153 159 165 169 175 181 187 193 197 205 208 209 238
45.0 47.5 49.0 51.0 53.0 54.0 56.0 63.0 65.0 70.0 75.0 79.0 84.0 90.0 96.0 102.0 107.0 114.0 117.0 120.0 153.0
251 244 267 266 276 281 286 319 335 350 370 384 416 430 452 469 488 493 495 501 553
0.326 0.307 0.331 0.322 0.327 0.331 0.330 0.343 0.355 0.355 0.360 0.363 0.379 0.376 0.380 0.380 0.385 0.373 0.369 0.368 0.349
0.365 0.343 0.369 0.358 0.363 0.368 0.366 0.378 0.391 0.389 0.394 0.396 0.412 0.408 0.411 0.411 0.415 0.402 0.397 0.396 0.373
0.38 0.55 0.40 0.37 0.39 0.40 0.38 0.37 0.36 0.36 0.38 0.39 0.40 0.39 0.37 0.37 0.40 0.38 0.38 0.39 0.36
0.961 1.005 0.950 0.970 0.958 0.950 0.952 0.925 0.901 0.901 0.892 0.886 0.858 0.863 0.856 0.856 0.848 0.868 0.875 0.877 0.913
63Eu 65Tb
67Ho 69Tm 71Lu 73Ta 7SRe
771r 79Au 81T1 82pb 83Bi 92U
520
D.J. O'Connor et al. / Z 1 oscillations
[Xe
~Af i f
S~2
I
! i
'
I
'1
![
, I I , ' ~ ; I
~=31
element. The former has already been mentioned [10] and can only be further studied by computer simulation [8,14] if detailed information on the interatomic potential becomes available. The adjustment of the scaling parameters however does lend itself to simple analysis, and the most common method is to alter the screening length by including a "correction factor" [ 14,15 ], thus t
5O
(11)
a =fa,
I 6'0
70
80
90
Fig. 2. The measured reduced relative straggling compared to values predicted by power potentials.
nations. If this were the case and we sought to verify that this difference in interatomic potential manifested itself as a change in the power of the power approximation then we would expect from eq. (9) to see a similar trend in the relative straggling to that exhibited by the .reduced range except that 7 A p / p should be less sensitive to s. The relatice straggling results are presented in fig. 2 and all, except Xe, are in" agreement with that predicted by power potentials. The anomalously high value for Xe is attributed to diffusion mechanisms as the first measurement (within 24 h of implantation) yield a value of 7 A # / o = 0.44 but later measurements (days or weeks later) yield values in the region of 0.55. Although the relative straggling is only weakly dependent on the potential (see table 2) and the relative experimental error is larger, there does not seem to be the same general trend that is evident in the reduced range. This could indicate that either the power approximation is not an adequate description of the potential at these energies or that the scaling parameters [eqs. (6)-(8)] may differ from element to Table 2 The variation of ~'Ao/o at E = 0.09 for different interatomic potentials [4,8]. Potential
3.Ap/p
Thomas-Fermi Moli~re Bohr Free electron (Kr-C) Free electron (average) r-2 r-3
0.365 0.326 0.284 0.326 0.331 0.408 0.365
where a is the screening length given by eq. (8), a' is the effective screening length a n d f i s the "correction factor". The screening length correction factor in table 1 is calculated assuming that the range-energy relationship obeys a power law [ 1 6 - 1 8 ] of the form; (12)
p acE 2Is ,
thus
S
f= [po/0] 2(s-1),
(13)
where O is the reduced range determined using the screening length given by eq. (8) and Po is taken from the tables of SchiCtt [19] for e = 0.09 and k = 0.115. The power of the potential, s, is assumed to have the value of 3 as this is in agreement with empirical determinations of the range-energy relationship leading to O c~ e 2/3. The correction factors presented in table 1 should only be regarded as a guide to the magnitude of "correction" necessary to obtain agreement between theory and experiment, as the true range depends on the form of the interatomic potential about which little is certain. It should also be noted that the variation of correction factor with atomic number is not in agreement with previously reported work [14,15]. The range presented in fig. 1, the mean projected range, is less than the total path length of the particle which is the parameter most easily determined analytically. Although the difference between the two is normally regarded as negligible when M1 >>M2, it is not quite so small an effect when comparing small differences in the range for different values of M1. If we use the relationship [4], s2/l plop = 1 + 4(2s - 1)'
(14)
where p is the total path length, pp is the measured mean projected range and/a =M2/M1, then a comparison of the variation of total path length with Z can be made.
D.J. O'Connor et al. / Z 1 oscillations
I
04
I
b
I
I
I
I
~
I III
i I
t q l I I I
f 0.3-
s0
6'0
7~
80
9~
Z
Fig. 3. The estimated total path length from fig. 1 and eq. (14) using s = 3.
The deduced values of total path length presented in table 1 and fig. 3 are for s = 3, and it can be seen that the oscillation as a function o f Z1 is still evident though slightly reduced in magnitude. The important observation though is that once the correction is made, all values o f the reduced range are in excess of the theoretical estimates by at least 10%.
521
The use of a Bi standard has improved the relative accuracy ( 1 - 2 % ) of the range determinations but the absolute accuracy is limited by the Bi range accuracy which is estimated to be 7%. These range measurements can be quoted to greater absolute accuracy if they can be normalised to any one in the set which is measured to greater absolute precision. Attention should be drawn to the anomalously wide Xe distribution which has been attributed to diffusion processes. This is important as it brings into question the validity of the range measurements for that element and underlines the importance of correlating straggling measurements to range measurements whenever possible to ensure that o t h e r physical effects are not creating anomalous ranges. We are grateful to Metallurgy Division, AERE Harwell for sponsoring this work and for a fellowship for one o f us (J.O.C.). We also wish to thank S. Pugh, S. Nelson and A. Marwick for valuable discussions and advice. We appreciate the assistance o f Mr. C. Jackson (AERE, Harwell) with the SIMS examination o f these samples.
References 5. Conclusions The determination o f the Z dependence of reduced range at constant reduced energy has revealed an oscillatory behaviour which cannot be predicted from current theoretical models. This oscillatory behaviour, when viewed in conjunction with recent multiple scattering and differential scattering cross section [10] measurements, suggests that the current statistical models for the atom, used for determining the interatomic potential, are not adequate to describe atomic collision processes at energies below e = 0.1. These models, which exhibit a smooth monotonic variation of potential parameters with increasing atomic ~number, do not properly account for the electronic shell effects which may become significant at low energies. Although the adjustment of the potential can be achieved by adjusting the screening length this is not really satisfactory as it may have to be an energy dependent adjustment unless the oscillations in the electronic stopping coincide with those in the nuclear stopping. As there is yet no reliable way of predicting from other parameters what the correction should be, it must remain an empirical fitting parameter.
[1 ] J.W. Mayer, L. Eriksson and J.A. Davies, Ion implantation (Academic Press, New York, 1970). [2] G. Dearnaley, J.H. Freeman, R.S. Nelson and J. Stephen, Ion implantation (North-Holland Publ. Co., Amsterdam, 1973). [3] J.L. Combasson, B.W. Faxmery, D. McCulloch, G.W. Neilson and M.W. Thompson, Rad. Eff. 36 (1978) 149 and references mentioned therein. [4] J. Lindhard, M. Scharff and H.E. Schi4tt, Kgl. Dansk. Vid. Selsk. Matt. Fys. Medd. 33 (1963) no. 14. [5] K.B. Winterbon, P. Sigmund and J.B. Sanders, Kgl. Dansk. Vid. Selsk. Matt. Fys. Medd. 37 (1970) no. 14. [6] F.H. Eisen, Can. J. Phys. 46 (i968) 561. [7] L. Eriksson, J.A. Davies and P. Jespersgard, Phys. Rev. 161 (1967) 219. [8] W.D. Wilson, L.G. Haggemark and J.P. Biersack, Phys. Rev. B15 (1977) 2458. [9] J. Lindhaxd, V. Nielsen and M. Scharff, Kgl. Dansk. Vid. Selsk. Matt. Fys. Medd. 36 (1968) no. 10. [10] P. Loftager, F. Besenbacher, O.S. Jensen and V.S. Sorensen, Phys. Rev. A20 (1979) 1143. [11] H. Knudsen, F. Besenbacher, J. Heinemeier and P. Hvelplund, Phys. Rev. A13 (1976) 2095. [12] A.D. Marwick and R.C. Piller, Rad. Eft. 33 (1977) 245. [13] F. Besenbacher, J. B~ttiger, T. Laursen, P. Loftager and W. M611er, 8th Int. Conf. on Atomic collisions in solids, Hamilton, Canada; Nucl. Instr. and Meth. 170 (1980) 183. [14] M. Bister, M. Hautala and M. J~ntti, Rad. Eft. 42 (1979) 201.
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D.J. O'Connor et al. / Z 1 oscillations
[15] D.J. O'Connor and R.J. MacDonald, Rad. Eft. 34 (1977) 247. [16] H. Oetzmann, A. Feuerstein, H. Grahmam and S. Kalbitzer, Phys. Lett. 55A (1975) 170. [17] A. Feuerstein, S. Kabitzer and H. Oetzmann, Phys. Lett. 51A (1975) 165.
[18] W.A. Grant, J.S. Williams and D. Dodds, Proc. Karlsruhe Conf. on Surface analysis (Plenum Press, New York, 1976). [19] H.E. SchiCtt, Kgl. Dansk. Vid. Selsk. Matt. Fys. Medd. 35 (1966) no. 9.