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Cascade simulation of the crystal orientation dependence of sputtering and lattice damage of single crystal copper by irradiation wity 100 keV copper ions
360
Nuclear Instruments and Methods in Physics Research B18 (1987) 360-364 North-Holland, Amsterdam
CASCADE SIMULATION OF THE CRYSTAL ORIENTATION DEPENDENCE OF SPU'ITERING AND LATrICE DAMAGE OF SINGLE CRYSTAL COPPER BY I R R A D I A T I O N W I T H 100 keV C O P P E R I O N S G.P. M U E L L E R 17, Mervine R O S E N 1), W.A. F R A S E R x,2), J.A. S P R A G U E 1), P.R. M A L M B E R G 1~, J.M. L A M [ I E R T 1,3), P.A. T R E A D O 1,3) a n d G.W. R E Y N O L D S 1) t) Naoal Research Laborato~. , Washington, DC 20375-5000, USA 2) K.M. Sciences, 6200 Meadow Wood, Reno, N V 89502. USA 3j Georgetown University, Washington, DC 20057, USA
The sputtering yield, joint polar and azimuthal angular distributions and the energy distribution of atoms sputtered from copper single crystals by 100 keV copper ions has been calculated as a function of crystal orientation using the cascade simulation code MARLOWE. The calculated yields and angular distributions are compared with experimental results for ions incident parallel to a (110) close-packed axial channelling direction, parallel to a (100) plane, and along a pseudo-random direction. Significant channeling effects are exhibited in both calculations and experiment. Strong peaking in the backscattered direction is calculated and observed for the axial irradiations alone. Primary range distributigns and damage profiles are calculated for the three crystal orientations and compared with lattice damage measurements. The distributions of sputtered ions in both energy and angle are presented. 1. Introduction and calculational model
Our purpose is to examine the relative effects of crystal orientation on lattice damage and sputtering yields when single crystals of copper are irradiated with 100 keV copper ions. We will compare experimental measurements of these effects with computer simulations. A companion paper [1] to this one describes the experiments in detail. We will only quote the results of the experiments here, while concentrating on the computer simulations. The calculations were performed using the cascade simulation code MARLOWE (Version 12) of Robinson and Torrens [2]. Additional analysis of the sputtering results was performed by an auxiliary code written by one of us (W.A.F.). To represent the interaction between the copper ions we used the (universal) Molirre potential with the screening parameter set [3] to 0.738 nm. The inelastic, electronic losses are represented by an equal mixture of the local and nonlocal loss mechanisms in MARLOWE. We have labeled the three crystal orientations, described below, as the axial, planar, and off-axis orientations. For the range calculations we used 10000 incident ions for each orientations; for the damage results we used 1000, 300, and 300, respectively, for the axial, planar, and off-axis orientation. In both those sets of calculations the cutoff energy, below which ions were not followed, was set to 15 eV, which is of the order of the displacement energy in copper. For the sputtering calculations we simulated the
entire crystal by a finite slab with a thickness of twenty lattice spacings (a=0.3615 nm). Because sputtering calculations require a much lower cutoff energy, it is prohibitively expensive to follow the incident ions down to rest. We reasoned that no significant number of chains of collisions lead back to the surface from deeper than twenty lattice spacings into the crystal [4]. We tested this assumption by making a computer run with a slab of thickness thirty lattice spacings; there was no difference in the sputtering results. We should emphasize that MARLOWE treats each cascade as a new process; the damage done to the crystal by one incident ion is not retained when the next cascade is started. All of the MARLOWE simulations should be compared to the low fluence limit of the experimental results. Robinson [5] discusses the differences that may arise due to higher fluences.
2. Crystal orientations To describe the orientations used in these studies, we define a z-axis pointing into the crystal along a (110) direction, an x-axis along a (100) direction, and the y-axis along a (110) direction. Three crystal/beam orientations were considered. The axial crystal was cut so that its front surface was a (1"10) surface; the x- and y-axes of our coordinate system lie in the surface and the z-axis is normal (into) the surface. In this orientation, one of the large (110) axial channels, one of the (100) planar channels, and other higher order channels
361
G.P. Mueller et a L / Cascade simulation
run perpendicular to the surface. If one takes the plane of the surface just described and rotates it by 15 ~ about the x-axis and then recuts the crystal along the rotated plane, one obtains our "planar" orientation. In this form, the (110) axial channel is no longer collinear with the incoming (perpendicular) beam, but the (100) planar channel is still open to the beam. In order to obtain the "off-axis" orientation, for which no major channel is open, we tilted the planar crystal by 7.5 ~ about the (0, cos 15 ~ sin 15 ~) axis that lies in the surface of the planar crystal. The off-axis crystal was not recut, but was physically rotated relative to the incoming beam; for this case the beam is not normal to the surface. For the sputtering measurements, a catcher foil was used at one azimuthal angle; it covered a range in azimuth of about 15 ~ . In the coordinate system just described, it was placed at an azimuthal angle of 90 ~. This had the effect that the (110) channel that was perpendicular to the surface in the axial case was pointed away from the catcher foil in the planar and off-axis cases. Ideally the ion beam would be perfectly collimated and normally incident. We determined that the actual beam was tilted 1.5 ~ away from the normal about the y-axis of our coordinate system. For the off-axis orientation, therefore, the beam was 6.0 ~ off the (001) planar channel. In addition, the beam had a half-angle divergence of between 0.26 ~ and 0.40 ~. For the computer simulations we used a value of 0.33 ~ In order to avoid excessive repetition of the same phrasing, whenever we compare results for the three orientations, we will present them in the order axial, planar, and off-axis.
;i I
AXIAL -I
--
-- PLANAR
......
OFF A X I S
\\ 100
200
I 300
OEPTH(nm)
Fig. 2. Damage distributions form MARLOWE for the axial, planar, and off-axis orientations; the damage is represented by the density of vacancies produced in the targets.
211, 52 and 36 nm, with statitistical error of - 1 % . This comparison clearly demonstrates the effects of axial channeling. As a measure of damage to the crystal, we plot in fig. 2 the distribution of vacancies produced by the three simulations. If one allows recombinations a recombination radius can be specified in MARLOWE - the absolute number of vacancies is less for each case but the relative numbers do not shift. We also determined the damage energy, which is that portion of the energy of the incoming ion that is not transferred to electrons or to lattice atoms below the displacement threshhold. These results are similar to those shown in the vacancy plot. More detailed comparisons with the experimental range and damage results appear in a companion paper [1].
3. Range and damage results
Fig. 1 shows the calculated range distributions for the three crystal orientations; the average ranges are
SYMBOL I
I
.....
ORIENTATION
AVERAGE RANGE
AXIAL
211 nrn
PLANAR
52 n m
OFF A X I S
36 n m
,,\ ',\ t
100
200 RANGEinml
300
400
Fig. 1. Range distributions from MARLOWE for the axiS, planar, and off-axis orientations.
4. Sputtering results
The experimentally determined yields for the three orientations were 1.76, 5.19, and 8.27, with ratios of yields of 1 : 3.0 : 4.7. The large increase in yield as one goes to the less channeled irradiations is consistent with the increased amount of energy deposited near the surface in those irradiations. In fact, the ratios of the damage energy produced in the first 25 atomic layers in our three simulations are 1:4.4:9.0, which bears a rough relationship to the observed yields. While the interatomic potentials used in MARLOWE are certainly adequate for simulating the higher energy particle and energy deposition processes, they lack the necessary subtlety to represent the low energy (few eV) processes that are involved in sputtering. (The ejected particle energy specturm is sharply peaked at very low energies.) To compensate, in part, for the relatively crude potentials used in the calculations, we use the I. THEORY AND COMPUTER SIMULATION
362
G.P. Mueller et al. /Cascadesimulation 1.0
12
A
AXIAL CASE
10
g o
O
EXPERIMENTAL DATA
z
9
I(,.) <~
0.1
MARLOWE
/
t/
9 >.-
i.-,-
a. N 10 -2
_u k--
Or) I---
O O O O
,,7 10-3
0
100
200 ENERGY (eV)
300
o oo o
Fig. 3. Sputtered energy spectra for the axial, planar, and off-axis orientations. The width of the band is measure of the statistical uncertainty; all three spectra lie within the band. surface binding energy as a parameter [5], whose purpose is to improve the simulation of the escape of low energy ions from the surface. We obtain the sputtering yield in these calculations as a function of this surface binding energy. When we simultaneously fit the results for all three orientations with the experimental data, we obtained a value of B = 2.9 eV, for which the three yields are 1.66, 5.61, and 8.09, with statistical errors of - 10%. We should mention that we noticed some sensitivity of the sputtering yield on the cutoff and displacement energies (both 3 eV). We think that is also due to the relative inappropriateness of the potential used at low energies. Fig. 3 shows the ejected particle energy spectrum using the above value of the surface binding energy. The band shown in the figure reflects the statistical uncertainty at the higher energies; at low energies the statistics are quite good. It is interesting to observe that all orientations lead to essentiallly identical spectra. Despite large differences in the nature of the cascades, the ejected energy spectrum seems to be independent of these differences.
0Go
30 ~ 60- o ~ POLAR ANGLE 0
~
S0o
Fig. 4. Angular distribution of sputtered particles for the axial orientation.
0
PLANAR CASE o
~
9 is
EXPERIMENTAL DATA
MARLOWE
~//
o
/
>_z
o
~
5
o~
5. A n g u l a r d i s t r i b u t i o n s
0
I 00
Figs. 4-6 show the angular distributions of the sputtering yields for the three crystal orientations. The widths of the bands representing the simulation results
I
I
30*
I 60 ~
I 90 ~
POLAR ANGLE 0
Fig. 5. Angular distribution of sputtered particles for the planar orientation.
G.P. Mueller et al. / Cascade simulation
40
i
0
i
0 00 3O
~
o
J
i
i
OFFAXISCASE O EXPERIMENTALDATA
o
MARLOWE
q
L0///
z20
lO
0o
30 ~
60 ~
90 ~
POLARANGLE0 Fig. 6. Angular distribution of sputtered particles for the off-axis orientation.
reflect the statistical uncertainties. We do not indicate the error bars on the experimental points; they were not large enough to alter any of our conclusions. Recall, however, that these yields were measured over a narrow range of azimuthal angles and that the total yield as a function of polar angle was determined by assuming azimuthal symmetry. As we will see, this assumption is somewhat questionable in the planar and off-axis cases. Because the total yield estimates of the simulations are dependent on the surface binding energy, whereas the shapes of the yield curves are not, we chose to present the three cases using the values of the surface binding energy that reproduce the total yield of the experiments for that case. (These values are 2.65, 3.20 and 2.80 eV.) U p o n examining these figures we can immediately draw two conclusions: The gross behavior of the experimental results are reproduced by the simulations; and, despite the fact that the three sputtered energy spectra are virtually identical, the axial angular distribution differs from the other two, which have similar shapes. Most noticeably, both experiment and simulation indicate a leveling off the yield curve between 20 ~ and 40 ~ for the axial case. We also note that in all three cases the simulations have a somewhat lower yield at smaller angles and a somewhat higher yield at larger angles. This may be due to the fact that the simulated yield curves are sums over all azimuthal angles, whereas the experimental yields were taken at a particular azimuth. A possible explanation for the different shapes of the three yield curves is that as the orientation of the crystal
363
changes, the channels that are open to the surface shift. If one looks at the large (110) axial channel that is perpendicular to the surface in the axial case, and would be expected to enhance the angular yield in the (0 ~ undefined) (polar angle, azimuthal angle) direction, one finds in the planar and off-axis cases that this channel exits in the (15 ~ , 270 ~ ) direction. Similarly, the (111) planar channel that exits in the ( 0 0 - 2 ~ , 0 0 - 3 6 0 ~ ) range in the axial" case, appears in the (12~ ~, 2000-320 ~ ) range for the planar and off-axis orientations. Thus, based on just these two channels, we expect a significant shift of yield from 0 ~ to 15 ~ in comparing the axial case to the other two. We see just such a shift upon examining figs. 4 - 6 ; There is a strong peak at 0 ~ in the axial case and in the other two cases there is a peak at 15 ~ and a lessened yield at 0 ~ In p o l a r / a z i m u t h plots obtained from the simulations we find a strong peak centered at (15 ~ , 270 ~ ) in both the planar and off-axis cases. There are similar shifts of open channels at other polar angles in the 2 5 o - 4 0 ~ range, but our simulation statistics are not good enough, as yet, to draw meaningful distinctions. We think that a channeling peak in the sputtering yield at about 400 may be responsible for the plateau in the axial yield curve. The question, then, is why the experimental results do not show the same behavior in the '0~-15 ~ region. This is due to the fact that all of the experimental measurements were made at only one azimuth, where the simulations indicated no 15 ~ peaking.
6. Conclusions We noted earlier that the sputtering energy spectra were identical for the three orientations, independent of the direction of the beam. We expect that a more detailed examination of the question of sputtering angular distribution would indicate that they, too, were essentially identical - that most of the apparent differences are due to a shifting of the crystal relative to the coordinate system in which the measurements and simulations are made. When examined in the natural coordinate system of the crystal, we expect the angular distributions to be essentially the same. To this end we plan more simulations wherein we specifically keep track of which channels are contributing to the sputtering, and also plan several more experiments wherein we measure the yield over a range of azimuthal angles. Another minor effect that needs examining is the effect of the surface binding energy assessment on channeled sputtering. The assessment is really a m o m e n t u m assessment, with only the momentum in the direction perpendicular to the surface being affected. In the case of open channels, this may be an unphysical way to use the surface binding energy parameter. I. THEORY AND COMPUTER SIMULATION
364
G.P. Mueller et al. / C a s c a d e simulation
References [1] J.A. Sprague et al., Proc. of IBMM'86, Catania, Italy, Nucl. Instr. and Meth. B19/20 (1987) in press. [2] M.T. Robinson and I.M. Torrens, Phys. Rev. B9 (1974) 5008. [3] I.M. Torrens and M.T. Robinson, in: Interatomic Poten-
tials and Simulation of Lattice Defects, eds., P.C. Gehlen, J.R. Beeler Jr and R.I. Jaffee (Plenum, New York, 1972) p. 423. [4] M. Rosen, G.P. Mueller and W.A. Fraser, Nucl. Instr. and Meth. 209/210 (1983) 63. [5] M.T. Robinson, in: Topics in Applied physics, Vol. 47, ed., R. Behrisch (Springer, New York, 1981) p. 77.
Nuclear Instruments and Methods in Physics Research B18 (1987) 360-364 North-Holland, Amsterdam
CASCADE SIMULATION OF THE CRYSTAL ORIENTATION DEPENDENCE OF SPU'ITERING AND LATrICE DAMAGE OF SINGLE CRYSTAL COPPER BY I R R A D I A T I O N W I T H 100 keV C O P P E R I O N S G.P. M U E L L E R 17, Mervine R O S E N 1), W.A. F R A S E R x,2), J.A. S P R A G U E 1), P.R. M A L M B E R G 1~, J.M. L A M [ I E R T 1,3), P.A. T R E A D O 1,3) a n d G.W. R E Y N O L D S 1) t) Naoal Research Laborato~. , Washington, DC 20375-5000, USA 2) K.M. Sciences, 6200 Meadow Wood, Reno, N V 89502. USA 3j Georgetown University, Washington, DC 20057, USA
The sputtering yield, joint polar and azimuthal angular distributions and the energy distribution of atoms sputtered from copper single crystals by 100 keV copper ions has been calculated as a function of crystal orientation using the cascade simulation code MARLOWE. The calculated yields and angular distributions are compared with experimental results for ions incident parallel to a (110) close-packed axial channelling direction, parallel to a (100) plane, and along a pseudo-random direction. Significant channeling effects are exhibited in both calculations and experiment. Strong peaking in the backscattered direction is calculated and observed for the axial irradiations alone. Primary range distributigns and damage profiles are calculated for the three crystal orientations and compared with lattice damage measurements. The distributions of sputtered ions in both energy and angle are presented. 1. Introduction and calculational model
Our purpose is to examine the relative effects of crystal orientation on lattice damage and sputtering yields when single crystals of copper are irradiated with 100 keV copper ions. We will compare experimental measurements of these effects with computer simulations. A companion paper [1] to this one describes the experiments in detail. We will only quote the results of the experiments here, while concentrating on the computer simulations. The calculations were performed using the cascade simulation code MARLOWE (Version 12) of Robinson and Torrens [2]. Additional analysis of the sputtering results was performed by an auxiliary code written by one of us (W.A.F.). To represent the interaction between the copper ions we used the (universal) Molirre potential with the screening parameter set [3] to 0.738 nm. The inelastic, electronic losses are represented by an equal mixture of the local and nonlocal loss mechanisms in MARLOWE. We have labeled the three crystal orientations, described below, as the axial, planar, and off-axis orientations. For the range calculations we used 10000 incident ions for each orientations; for the damage results we used 1000, 300, and 300, respectively, for the axial, planar, and off-axis orientation. In both those sets of calculations the cutoff energy, below which ions were not followed, was set to 15 eV, which is of the order of the displacement energy in copper. For the sputtering calculations we simulated the
entire crystal by a finite slab with a thickness of twenty lattice spacings (a=0.3615 nm). Because sputtering calculations require a much lower cutoff energy, it is prohibitively expensive to follow the incident ions down to rest. We reasoned that no significant number of chains of collisions lead back to the surface from deeper than twenty lattice spacings into the crystal [4]. We tested this assumption by making a computer run with a slab of thickness thirty lattice spacings; there was no difference in the sputtering results. We should emphasize that MARLOWE treats each cascade as a new process; the damage done to the crystal by one incident ion is not retained when the next cascade is started. All of the MARLOWE simulations should be compared to the low fluence limit of the experimental results. Robinson [5] discusses the differences that may arise due to higher fluences.
2. Crystal orientations To describe the orientations used in these studies, we define a z-axis pointing into the crystal along a (110) direction, an x-axis along a (100) direction, and the y-axis along a (110) direction. Three crystal/beam orientations were considered. The axial crystal was cut so that its front surface was a (1"10) surface; the x- and y-axes of our coordinate system lie in the surface and the z-axis is normal (into) the surface. In this orientation, one of the large (110) axial channels, one of the (100) planar channels, and other higher order channels
361
G.P. Mueller et a L / Cascade simulation
run perpendicular to the surface. If one takes the plane of the surface just described and rotates it by 15 ~ about the x-axis and then recuts the crystal along the rotated plane, one obtains our "planar" orientation. In this form, the (110) axial channel is no longer collinear with the incoming (perpendicular) beam, but the (100) planar channel is still open to the beam. In order to obtain the "off-axis" orientation, for which no major channel is open, we tilted the planar crystal by 7.5 ~ about the (0, cos 15 ~ sin 15 ~) axis that lies in the surface of the planar crystal. The off-axis crystal was not recut, but was physically rotated relative to the incoming beam; for this case the beam is not normal to the surface. For the sputtering measurements, a catcher foil was used at one azimuthal angle; it covered a range in azimuth of about 15 ~ . In the coordinate system just described, it was placed at an azimuthal angle of 90 ~. This had the effect that the (110) channel that was perpendicular to the surface in the axial case was pointed away from the catcher foil in the planar and off-axis cases. Ideally the ion beam would be perfectly collimated and normally incident. We determined that the actual beam was tilted 1.5 ~ away from the normal about the y-axis of our coordinate system. For the off-axis orientation, therefore, the beam was 6.0 ~ off the (001) planar channel. In addition, the beam had a half-angle divergence of between 0.26 ~ and 0.40 ~. For the computer simulations we used a value of 0.33 ~ In order to avoid excessive repetition of the same phrasing, whenever we compare results for the three orientations, we will present them in the order axial, planar, and off-axis.
;i I
AXIAL -I
--
-- PLANAR
......
OFF A X I S
\\ 100
200
I 300
OEPTH(nm)
Fig. 2. Damage distributions form MARLOWE for the axial, planar, and off-axis orientations; the damage is represented by the density of vacancies produced in the targets.
211, 52 and 36 nm, with statitistical error of - 1 % . This comparison clearly demonstrates the effects of axial channeling. As a measure of damage to the crystal, we plot in fig. 2 the distribution of vacancies produced by the three simulations. If one allows recombinations a recombination radius can be specified in MARLOWE - the absolute number of vacancies is less for each case but the relative numbers do not shift. We also determined the damage energy, which is that portion of the energy of the incoming ion that is not transferred to electrons or to lattice atoms below the displacement threshhold. These results are similar to those shown in the vacancy plot. More detailed comparisons with the experimental range and damage results appear in a companion paper [1].
3. Range and damage results
Fig. 1 shows the calculated range distributions for the three crystal orientations; the average ranges are
SYMBOL I
I
.....
ORIENTATION
AVERAGE RANGE
AXIAL
211 nrn
PLANAR
52 n m
OFF A X I S
36 n m
,,\ ',\ t
100
200 RANGEinml
300
400
Fig. 1. Range distributions from MARLOWE for the axiS, planar, and off-axis orientations.
4. Sputtering results
The experimentally determined yields for the three orientations were 1.76, 5.19, and 8.27, with ratios of yields of 1 : 3.0 : 4.7. The large increase in yield as one goes to the less channeled irradiations is consistent with the increased amount of energy deposited near the surface in those irradiations. In fact, the ratios of the damage energy produced in the first 25 atomic layers in our three simulations are 1:4.4:9.0, which bears a rough relationship to the observed yields. While the interatomic potentials used in MARLOWE are certainly adequate for simulating the higher energy particle and energy deposition processes, they lack the necessary subtlety to represent the low energy (few eV) processes that are involved in sputtering. (The ejected particle energy specturm is sharply peaked at very low energies.) To compensate, in part, for the relatively crude potentials used in the calculations, we use the I. THEORY AND COMPUTER SIMULATION
362
G.P. Mueller et al. /Cascadesimulation 1.0
12
A
AXIAL CASE
10
g o
O
EXPERIMENTAL DATA
z
9
I(,.) <~
0.1
MARLOWE
/
t/
9 >.-
i.-,-
a. N 10 -2
_u k--
Or) I---
O O O O
,,7 10-3
0
100
200 ENERGY (eV)
300
o oo o
Fig. 3. Sputtered energy spectra for the axial, planar, and off-axis orientations. The width of the band is measure of the statistical uncertainty; all three spectra lie within the band. surface binding energy as a parameter [5], whose purpose is to improve the simulation of the escape of low energy ions from the surface. We obtain the sputtering yield in these calculations as a function of this surface binding energy. When we simultaneously fit the results for all three orientations with the experimental data, we obtained a value of B = 2.9 eV, for which the three yields are 1.66, 5.61, and 8.09, with statistical errors of - 10%. We should mention that we noticed some sensitivity of the sputtering yield on the cutoff and displacement energies (both 3 eV). We think that is also due to the relative inappropriateness of the potential used at low energies. Fig. 3 shows the ejected particle energy spectrum using the above value of the surface binding energy. The band shown in the figure reflects the statistical uncertainty at the higher energies; at low energies the statistics are quite good. It is interesting to observe that all orientations lead to essentiallly identical spectra. Despite large differences in the nature of the cascades, the ejected energy spectrum seems to be independent of these differences.
0Go
30 ~ 60- o ~ POLAR ANGLE 0
~
S0o
Fig. 4. Angular distribution of sputtered particles for the axial orientation.
0
PLANAR CASE o
~
9 is
EXPERIMENTAL DATA
MARLOWE
~//
o
/
>_z
o
~
5
o~
5. A n g u l a r d i s t r i b u t i o n s
0
I 00
Figs. 4-6 show the angular distributions of the sputtering yields for the three crystal orientations. The widths of the bands representing the simulation results
I
I
30*
I 60 ~
I 90 ~
POLAR ANGLE 0
Fig. 5. Angular distribution of sputtered particles for the planar orientation.
G.P. Mueller et al. / Cascade simulation
40
i
0
i
0 00 3O
~
o
J
i
i
OFFAXISCASE O EXPERIMENTALDATA
o
MARLOWE
q
L0///
z20
lO
0o
30 ~
60 ~
90 ~
POLARANGLE0 Fig. 6. Angular distribution of sputtered particles for the off-axis orientation.
reflect the statistical uncertainties. We do not indicate the error bars on the experimental points; they were not large enough to alter any of our conclusions. Recall, however, that these yields were measured over a narrow range of azimuthal angles and that the total yield as a function of polar angle was determined by assuming azimuthal symmetry. As we will see, this assumption is somewhat questionable in the planar and off-axis cases. Because the total yield estimates of the simulations are dependent on the surface binding energy, whereas the shapes of the yield curves are not, we chose to present the three cases using the values of the surface binding energy that reproduce the total yield of the experiments for that case. (These values are 2.65, 3.20 and 2.80 eV.) U p o n examining these figures we can immediately draw two conclusions: The gross behavior of the experimental results are reproduced by the simulations; and, despite the fact that the three sputtered energy spectra are virtually identical, the axial angular distribution differs from the other two, which have similar shapes. Most noticeably, both experiment and simulation indicate a leveling off the yield curve between 20 ~ and 40 ~ for the axial case. We also note that in all three cases the simulations have a somewhat lower yield at smaller angles and a somewhat higher yield at larger angles. This may be due to the fact that the simulated yield curves are sums over all azimuthal angles, whereas the experimental yields were taken at a particular azimuth. A possible explanation for the different shapes of the three yield curves is that as the orientation of the crystal
363
changes, the channels that are open to the surface shift. If one looks at the large (110) axial channel that is perpendicular to the surface in the axial case, and would be expected to enhance the angular yield in the (0 ~ undefined) (polar angle, azimuthal angle) direction, one finds in the planar and off-axis cases that this channel exits in the (15 ~ , 270 ~ ) direction. Similarly, the (111) planar channel that exits in the ( 0 0 - 2 ~ , 0 0 - 3 6 0 ~ ) range in the axial" case, appears in the (12~ ~, 2000-320 ~ ) range for the planar and off-axis orientations. Thus, based on just these two channels, we expect a significant shift of yield from 0 ~ to 15 ~ in comparing the axial case to the other two. We see just such a shift upon examining figs. 4 - 6 ; There is a strong peak at 0 ~ in the axial case and in the other two cases there is a peak at 15 ~ and a lessened yield at 0 ~ In p o l a r / a z i m u t h plots obtained from the simulations we find a strong peak centered at (15 ~ , 270 ~ ) in both the planar and off-axis cases. There are similar shifts of open channels at other polar angles in the 2 5 o - 4 0 ~ range, but our simulation statistics are not good enough, as yet, to draw meaningful distinctions. We think that a channeling peak in the sputtering yield at about 400 may be responsible for the plateau in the axial yield curve. The question, then, is why the experimental results do not show the same behavior in the '0~-15 ~ region. This is due to the fact that all of the experimental measurements were made at only one azimuth, where the simulations indicated no 15 ~ peaking.
6. Conclusions We noted earlier that the sputtering energy spectra were identical for the three orientations, independent of the direction of the beam. We expect that a more detailed examination of the question of sputtering angular distribution would indicate that they, too, were essentially identical - that most of the apparent differences are due to a shifting of the crystal relative to the coordinate system in which the measurements and simulations are made. When examined in the natural coordinate system of the crystal, we expect the angular distributions to be essentially the same. To this end we plan more simulations wherein we specifically keep track of which channels are contributing to the sputtering, and also plan several more experiments wherein we measure the yield over a range of azimuthal angles. Another minor effect that needs examining is the effect of the surface binding energy assessment on channeled sputtering. The assessment is really a m o m e n t u m assessment, with only the momentum in the direction perpendicular to the surface being affected. In the case of open channels, this may be an unphysical way to use the surface binding energy parameter. I. THEORY AND COMPUTER SIMULATION
364
G.P. Mueller et al. / C a s c a d e simulation
References [1] J.A. Sprague et al., Proc. of IBMM'86, Catania, Italy, Nucl. Instr. and Meth. B19/20 (1987) in press. [2] M.T. Robinson and I.M. Torrens, Phys. Rev. B9 (1974) 5008. [3] I.M. Torrens and M.T. Robinson, in: Interatomic Poten-
tials and Simulation of Lattice Defects, eds., P.C. Gehlen, J.R. Beeler Jr and R.I. Jaffee (Plenum, New York, 1972) p. 423. [4] M. Rosen, G.P. Mueller and W.A. Fraser, Nucl. Instr. and Meth. 209/210 (1983) 63. [5] M.T. Robinson, in: Topics in Applied physics, Vol. 47, ed., R. Behrisch (Springer, New York, 1981) p. 77.