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Reference stopping cross sections for 30–600 keV protons in silicon
Nuclear Instruments and Methods in Physics Research B33 (1988) 133-137 North-Holland, Amsterdam
~FE~NCE
STOPPING
CROSS SECTIONS
133
FOR 30-600
keV PROTONS
IN SILICON
P. MERTENS Hahn-Meitner-Institus, Berlin, Germany
P. BAUER Johannes-Kepler-Vniversitiit, Linz, Austria
The stopping cross sections of protons were determined by using transmission and backscattering techniques at equivalent target foils. Over the whole energy range compared (30 to 370 keV) both methods deliveredthe same data within f 1%. The uncertainty in the derivedcross sections amounts to * 2%,
1. Introduction Proton stopping cross sections are widely accepted as a reliable data base for the scaling of stopping cross sections for the heavier ions. Accordingly, they play an important role in determining implantation ranges and radiation damage profiles. Nevertheless, even for a material of high technological priority like silicon, there could only be relatively few and largely scattering experimental data included in the tabulation by Andersen and Ziegler [l]. The semiempirical fit function given there for silicon does not exhibit any close relation to the experiments. On the other hand, it could be demonstrated for the rare earth elements [2] that the interpolated fit functions were often closer to reality in cases, where no measurements for the specific element fixed the function to data points of unclear quality. Obviously the procedures applied for interpolation between different materials in ref. [l] were quite adequate. The Andersen-Ziegler (AZ) tables strikingly revealed the severe deficiencies of some experimental methods. The data obtained by foil transmission (T) were especially subjected to large scatter. Rutherford backscattering measurements (RBS) seemed to be less influenced by these problems, which largely depended on the material investigated. Accordingly, the majority of the data accumulated after the publishing of ref. [l] was measured by RBS-techniques. In fig. 1 recent experimental results are depicted together with the AZ semiempirical function. Sirotinin et al. [3] and Ktihrt et al. [4] applied RBS for solid targets, while Santry and Werner [5] performed transmission experiments for thick silicon films (77-137 pg/cm2). Compared to the AZ compilation the scatter of the data is largely reduced. The restriction of some RBS-applications to high energies becomes obvious. For the transmission measure0168-583X/88/$03.50 Q Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
ments of Santry and Werner no systematic difference from the RBS-values can be detected. The use of thick targets, where fewer problems with surface oxide contamination are to be expected, in connection with the availability of high energy projectiles results in a reduction of systematic experimental errors. Generally, high energy measurements appear to be less troublesome, as in these experiments errors can be detected more easily by reference to the Bethe [6] theory. Measurements at energies around the stopping maximum and below remained relatively rare and exhibited larger deviations from each other. Consequently it is the goal of this work, to cover the low and medium energy range by experiments, which make use of the special virtues of either the backscattering technique (widely insensitive to target hazards) and the transmission method (large useable range in energy and ion species). As a result of the small errors implied (+ 2%) these stopping cross
7’ 25
/ 100 energy
Ike'/1
Fig. 1. Proton stopping cross sections (in lO_” silicon:
data
published after Andersen-Ziegler
the completion tables.
III. ENERGY
eV cm2) in of
LOSS/SCATTERING
the
134
sections theory.
P. Mertens, P. Bauer / Reference stopping cross sections may serve as a reference
for application
and
2. Experimental It has been the first aim of the cooperative program between the Hahn-Meitner-Institut in Berlin and the Johannes-Kepler-Universitlt in Linz to investigate, if the huge data scatter in the existing experimental data was caused by methodical problems of trans~ssion and backscattering experiments. In order to check the methods, identical targets seemed to be desirable. But the first attempt soon made clear that foils once irradiated and exposed to atmosphere thereafter could deliver different energy losses in a consecutive experiment [7]. Thus it was decided to perform all further measurements with equivalent targets, i.e. with targets simultaneously produced at neighbouring positions in the evaporation unit. The obvious special virtues alleged above are matched by the inherent drawbacks of transmission and backscattering: the need for self-supporting foils and sensitivity to surface contaminations (T), poor energy resolution of the surface barrier detector commonly applied and the need for sophisticated and time consu~ng computer programs to evaluate the energy loss from a backscattering spectrum of a foil (RBS). Before adequate solutions can be found for these problems, no general agreement can be expected with different target materials for the results of both the methods. For RBS, detector systems have become available with a particle resolution of < 3 keV (for the one used in Linz see Geretschllger [S]), as well as computer programs which derive the spectra halfwidth to a precision of typically 200 eV (Semrad [9]). For transmission, a foil set method was employed, which gave stopping cross sections independent of surface contaminants or systematic additive errors in the foil thickness calibration by weight. Separate articles have been dedicated to transmission and backscattering as utilized by us at the moment [lO,ll]. Therefore only some catchwords will be given in the following. Backscattering: 40-700 keV protons and deuterons, energy reproducibility 500 eV, detector resolution 3 keV, target chamber cooled by LN,, pressure lo-’ Torr, ion current 1 nA (at 40 kev) to 120 nA (at 700 keV), measuring time 100 to 2000 s, separate production of control targets. Tra~miss~an~ 20-380 keV protons, energy stability better 30 eV, electrostatic double focusing energy analyzer with AE/E < 10F3, target chamber evacuated by turbomolecular pumps to better than 2 x lo-* Torr, ion current less than 1 nA, measuring time = 40 s, foils produced by e-gun without crucible at deposition rates
of 3-10 A/s, microbalance
thickness calibration by means of external at 1 cm’ reference targets.
The foil set method mentioned above uses self-supporting foils of typically 300-2000 A thickness. The stopping cross section is derived from the slope of the straight line A E( d) (see below). In order to reduce the influence of an individual target’s error in thickness calibration, this method is desirable; in the presence of a common oxide layer on the targets, it is necessary. Additive errors in thickness calibration like an oxide layer contribute to the intercept AE(d = 0) and leave the stopping cross section unchanged. Thus, by means of the foil set method transmission can become as insensitive to surface contaminants as is backscattering. The method can only be successfully applied, if all targets of a set exhibit the same additive calibration errors (for example due to changes in atmospheric pressure and humidity) and an oxide layer of equal thickness. An indispensable prerequisite therefore is the evaporation of the foil targets within one run. As the production of self-supporting silicon films is highly troublesome, there is no real chance to reasonably make use of the method in the usual way. The crucial step in the procedure is the floating in distilled water. This was circumvented here. Instead, the silicon layers were deposited on thin carbon foils (200-400 A), which were previously caught on rings (2 mm internal diameter). The energy loss in the carbon carriers was determined in addition to that of the compound foils.
3. Evaluation of stopping cross sections The evaluation of the RBS measurements was performed as usual. The 11-parameter fit of the S&peaks in the spectrum worked very reliably. In addition to the experiments included here, an earlier run had been performed, for which the silicon layers were deposited on very resistant but inhomogeneous carbon foils (deposited on betaine). Additionally, they were heavily oxidized. In spite of these shortcomings, the results obtained did not differ from the ones presented here. They have not been considered here as the foils were too inhomogeneous to be useable in transmission experiments. A typical RBS spectrum is shown in fig. 2. From RBS-spectra like this the amount of oxygen on a target’s surface was derived (typically 2 X 1016 atoms/cm2). It turned out that within experimental errors the oxide layers had the same thickness for all targets used in the foil set. The application of the foil set method was therefore justified. Only minor (= 1 at.% oxygen) bulk contamination was indicated. When the stopping cross sections were to be evaluated by the foil set method, the individual energy losses had to refer to the same average ion energy in the
P. Mertens, P. Bauer / Reference stopping cross sections 0.5 1.
20 -
135
r-
0.4
680 keV protons 7
ANO)
0.2
2,
protons --->
silicon
set of 9 foila(270 - 1635 A)
*
FL 0 !! 8
0.0
.c-o.* z 3 b-O.4 E
-0.7
0
128
256
512
6LO
768
896
100
200
300
400
450
[keV]
channel
energy
Fig. 2. RBS-spectrum of 680 keV protons for a 1540 A silicon layer deposited on a 500 A carbon foil, taken after completion of the RBS-measurements (carbon build-up: = 25 A).
Fig. 4. Energies at intercept d = 0 for the foil set used for the stopping power measurements by the transmission method.
foil. Experimentally, however, these average energies are different. The thicker a foil, the smaller the average energy is for the constant energy of the impinging ions. Therefore some correction has to be provided which equally eliminates the additive energy loss in the carbon carrier. On the basis of the measured energies of incoming and outgoing ions, the average energies were calculated by utilizing the functional dependence given by the AZ semiempirical fit. Yet even for the lowest energies the values resulting from these procedures never differed significantly from the ones obtained by averaging the energies at the foil entrance and exit. Thus the latter might have been used as well. As the deposited silicon layer was in the path of the incoming beam, the energy loss measured at the uncoated carbon carrier had to be transformed to a reduced energy. Again, this was done by means of the AZ function. Alternatively, the experimental cross sections measured for a single
target could have been used. Finally, after the energy losses in the carbon foil had been subtracted from the ones in the compound target, these energy loss differences were AZ-transformed to a common average energy (the one in a foil of 800 A thickness). Having been evaluated in this way, the energy losses of the foil set were used for a linear regression to derive the stopping cross section S. Some examples for the data base of this linear regression are given in fig. 3. The corrected experimental energy losses AE (in keV) are plotted versus the foil thickness d (in angstrom) as resulting from the weight calibration. Deviations of the data points from a straight line are mainly due to statistical errors in the weighing procedure. Thus the figure demonstrates first of all that this calibration did not work too badly (the weight differences involved ranged from 7 to 40 pg!). Influences of additive calibration errors or surface oxide layers can hardly be detected (see fig. 4). The intercept energies at d = 0 are near zero. If they are the result of additive deviations in the weight calibration, the intercept will exhibit the same energy dependence as the stopping cross section [ll]. Fig. 4 confirms this expectation. Naturally, the experimental errors in AE implied for very low ion energies are larger than they are for the higher energies. This is because the foils here cannot be reliably aligned to the beam by means of a phosphorus screen. But even the deviating intercept for the 40 keV ions is not reflected by the data points in fig. 3. Thus it can be stated that the intercepts as resulting from linear regression are a sensitive measure for the precision of an experiment.
25
.”
1..“““”
BE = S * d + AE,
foil
thickness
[A]
Fig. 3. Energy loss measurements for a foil set: thickness from weight calibration, energy losses corrected for carbon carrier and transformed to a common average energy.
4. Results and discussion Stopping cross sections have usually been derived from measurements of single foils. The former discusIII. ENERGY
LOSS/SCATTERING
136
P. Mertens, P. Bauer / Reference slopping cross sections
10
* 0
100
200 energy
300
400
430
[kd]
Fig. 5. Proton stopping cross sections (in lo-” eV cm’) in silicon for single foils as resulting from the transmission method (together with the Andersen-Ziegler fit).
sion on the so-called thickness dependence of the stopping power reflects this troublesome practice. Except for small effects like pre-equilibrium stopping [12] the problems envisaged were related to errors in thickness calibration and to the presence of surface contaminants. As a consequence experimental stopping cross sections derived from thin and thick foils could differ considerably (see for examples ref. [ll]). If, on the other hand, measurements for thin and thick foils appear to agree reasonably well, the absence of these problems may be deduced. It is testified by fig. 5 that silicon is a material of this type. No systematic differences can be observed for the absolute values and energy dependence of the stopping cross section. The results of the backscattering experiment and the transmission experiment for the foil set are accumulated in fig. 6. Perfect agreement within mostly _t 1% is shown, at 600 keV the proton RBS data approach the AZ-function to 1%. Whereas the fit function has its
5 10
100 energy
1000
I keV I
Fig. 6. Proton stopping cross sections (in lo-l5 eV cm2) in silicon: backscattering and foil set transmission data together with the Andersen-Ziegler fit (full line), broken line: graphical interpolation of our data for aluminum.
maximum at about 80 keV, the maximum of the experimental data is reached at about 57 keV. At the maximum and below the semiempirical fit is too low by typically 12%. The relative position of our data and those depicted in fig. 1 can be judged by comparison with the AZ-function, which is included in figs. 1 and 6. Although the errors quoted are higher than ours (Kiihrt et al.: 5%, Santry and Werner: +4%, Sirotinin: 3% statistical) the overall agreement is much better. Over the whole energy range the data of Kihrt et al. [4] correspond to ours by better than 2.5%. This equally holds for the measurements of Santry and Werner [5] for proton energies above 120 keV. Also the stopping cross sections given by Sirotinin [3] are located within this margin between 100 and 300 keV. Only at the limits of his energy range (80 keV, 400 keV) are the cross sections too low by 6%. Facing this large extent of agreement it has to be concluded that the experiments deliver realistic stopping cross sections for protons within *2%. This is actually the experimental error attributed to our data. The promising agreement of results obtained by a variety of experimental methods for silicon does in no way guarantee similar findings for other materials. For materials exhibiting, for example, large crystallites and pronounced surface or bulk contaminants, larger deviations cannot be excluded. At the beginning of our cooperative program for backscattering and transmission, the goal was agreement within + 3%. This margin proved to be a realistic limit of precision. Even working with equivalent targets minor differences in the foils investigated have to be considered. In spite of being stored and transported in vacuum vessels an increase of hydrogen in the bulk can be neither excluded nor detected by RBS methods. The achieved precision, however, defines a margin not yet reached by semiempirical tabulations or theory. Our results (see fig. 6) show a stopping cross section maximum of 25.7 X lo-l5 eV cm2 at a proton energy of 57 keV. With increasing energy the stopping cross section decreases rapidly. In comparison, for Al the stopping cross section maximum is 22.2 X lo-l5 eV cm2 at the same energy E,, [13], while at 500 keV the stopping values of Al and Si are almost identical. To get some insight into the stopping processes involved, we compared our data to BEA calculations [13]. The results of this theory for Si and Al [7,13] exceed the experimental data by just 4-8% in our energy range and are thus a good basis for the interpretation. In this model, at about 300-400 keV the contribution of the core electrons to the stopping process has its maximum, yielding for Si about 40% (for Al about 60%) of the total stopping cross section. Around the maximum, the contribution of the core electrons is small (- 2 X lo-l5 eV cm2 for Si and - 6 x lo-l5 eV cm2 for Al) with a very weak energy dependence. Therefore, the shape of the
P. Mertens, P. Bauer / Reference stopping cross sections
stopping cross section here is almost entirely dominated by the electron gas. The electron gas densities are quite similar in both cases (in Si, the higher number of valence electrons per roughly compensates for the larger atomic volume). Thus, the (almost) equal values of E,,, for Al and Si are quite reasonable. The height of the cross sections around the maximum is a result of combined core and valence electron contributions.
References [l] H.H. Andersen and J.F. Ziegler, The Stopping and Ranges of Ions in Matter, vol. 3 (Pergamon, New York, 1977). [2] Th. Krist and P. Mertens, Nucl. Instr. and Meth. 218 (1983) 790. [3] E.I. Sirotinin, A.F. Tulinov, V.A. Khodyrev and V.N. Mizgulin, Nucl. Instr. and Meth. B4 (1984) 337.
137
[4] E. Kthrt, K. Lenkeit and F. Taubner, Phys. Status Solidi (a)66 (1981) K131. [5] D.C. Santry and R.D. Werner, Nucl. Instr. and Meth. 188 (1981) 211. [6] H.A. Bethe, Ann. Phys. 5 (1930) 325; Z. Phys. 76 (1932) 293. [7] P. Mertens, P. Bauer and D. Semrad, Nucl. Instr. and Meth. B15 (1986) 91. [S] M. Geretschhager, Nucl. Instr. and Meth. 204 (1983) 479. [9] D. Semrad and P. Bauer, Nucl. Instr. and Meth. 149 (1978) 159. [lo] P. Bauer, Nucl. Instr. and Meth. B27 (1987) 301. [ll] P. Mertens, Nucl. Instr. and Meth. B27 (1987) 315. [12] Th. Krist, P.J. Scanlon, P. Mertens, K.M. Barfoot, I. Reid and C.H. Cheng, Nucl. Instr. and Meth. B14 (1986) 179. [13] D. Semrad, P. Bauer and P. Mertens, Nucl. Instr. and Meth. B15 (1986) 86. [14] E. Kiihrt and R. Wedell, Phys. Status Solidi B116 (1983) 585.
III. ENERGY
LOSS/SCATTERING
~FE~NCE
STOPPING
CROSS SECTIONS
133
FOR 30-600
keV PROTONS
IN SILICON
P. MERTENS Hahn-Meitner-Institus, Berlin, Germany
P. BAUER Johannes-Kepler-Vniversitiit, Linz, Austria
The stopping cross sections of protons were determined by using transmission and backscattering techniques at equivalent target foils. Over the whole energy range compared (30 to 370 keV) both methods deliveredthe same data within f 1%. The uncertainty in the derivedcross sections amounts to * 2%,
1. Introduction Proton stopping cross sections are widely accepted as a reliable data base for the scaling of stopping cross sections for the heavier ions. Accordingly, they play an important role in determining implantation ranges and radiation damage profiles. Nevertheless, even for a material of high technological priority like silicon, there could only be relatively few and largely scattering experimental data included in the tabulation by Andersen and Ziegler [l]. The semiempirical fit function given there for silicon does not exhibit any close relation to the experiments. On the other hand, it could be demonstrated for the rare earth elements [2] that the interpolated fit functions were often closer to reality in cases, where no measurements for the specific element fixed the function to data points of unclear quality. Obviously the procedures applied for interpolation between different materials in ref. [l] were quite adequate. The Andersen-Ziegler (AZ) tables strikingly revealed the severe deficiencies of some experimental methods. The data obtained by foil transmission (T) were especially subjected to large scatter. Rutherford backscattering measurements (RBS) seemed to be less influenced by these problems, which largely depended on the material investigated. Accordingly, the majority of the data accumulated after the publishing of ref. [l] was measured by RBS-techniques. In fig. 1 recent experimental results are depicted together with the AZ semiempirical function. Sirotinin et al. [3] and Ktihrt et al. [4] applied RBS for solid targets, while Santry and Werner [5] performed transmission experiments for thick silicon films (77-137 pg/cm2). Compared to the AZ compilation the scatter of the data is largely reduced. The restriction of some RBS-applications to high energies becomes obvious. For the transmission measure0168-583X/88/$03.50 Q Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
ments of Santry and Werner no systematic difference from the RBS-values can be detected. The use of thick targets, where fewer problems with surface oxide contamination are to be expected, in connection with the availability of high energy projectiles results in a reduction of systematic experimental errors. Generally, high energy measurements appear to be less troublesome, as in these experiments errors can be detected more easily by reference to the Bethe [6] theory. Measurements at energies around the stopping maximum and below remained relatively rare and exhibited larger deviations from each other. Consequently it is the goal of this work, to cover the low and medium energy range by experiments, which make use of the special virtues of either the backscattering technique (widely insensitive to target hazards) and the transmission method (large useable range in energy and ion species). As a result of the small errors implied (+ 2%) these stopping cross
7’ 25
/ 100 energy
Ike'/1
Fig. 1. Proton stopping cross sections (in lO_” silicon:
data
published after Andersen-Ziegler
the completion tables.
III. ENERGY
eV cm2) in of
LOSS/SCATTERING
the
134
sections theory.
P. Mertens, P. Bauer / Reference stopping cross sections may serve as a reference
for application
and
2. Experimental It has been the first aim of the cooperative program between the Hahn-Meitner-Institut in Berlin and the Johannes-Kepler-Universitlt in Linz to investigate, if the huge data scatter in the existing experimental data was caused by methodical problems of trans~ssion and backscattering experiments. In order to check the methods, identical targets seemed to be desirable. But the first attempt soon made clear that foils once irradiated and exposed to atmosphere thereafter could deliver different energy losses in a consecutive experiment [7]. Thus it was decided to perform all further measurements with equivalent targets, i.e. with targets simultaneously produced at neighbouring positions in the evaporation unit. The obvious special virtues alleged above are matched by the inherent drawbacks of transmission and backscattering: the need for self-supporting foils and sensitivity to surface contaminations (T), poor energy resolution of the surface barrier detector commonly applied and the need for sophisticated and time consu~ng computer programs to evaluate the energy loss from a backscattering spectrum of a foil (RBS). Before adequate solutions can be found for these problems, no general agreement can be expected with different target materials for the results of both the methods. For RBS, detector systems have become available with a particle resolution of < 3 keV (for the one used in Linz see Geretschllger [S]), as well as computer programs which derive the spectra halfwidth to a precision of typically 200 eV (Semrad [9]). For transmission, a foil set method was employed, which gave stopping cross sections independent of surface contaminants or systematic additive errors in the foil thickness calibration by weight. Separate articles have been dedicated to transmission and backscattering as utilized by us at the moment [lO,ll]. Therefore only some catchwords will be given in the following. Backscattering: 40-700 keV protons and deuterons, energy reproducibility 500 eV, detector resolution 3 keV, target chamber cooled by LN,, pressure lo-’ Torr, ion current 1 nA (at 40 kev) to 120 nA (at 700 keV), measuring time 100 to 2000 s, separate production of control targets. Tra~miss~an~ 20-380 keV protons, energy stability better 30 eV, electrostatic double focusing energy analyzer with AE/E < 10F3, target chamber evacuated by turbomolecular pumps to better than 2 x lo-* Torr, ion current less than 1 nA, measuring time = 40 s, foils produced by e-gun without crucible at deposition rates
of 3-10 A/s, microbalance
thickness calibration by means of external at 1 cm’ reference targets.
The foil set method mentioned above uses self-supporting foils of typically 300-2000 A thickness. The stopping cross section is derived from the slope of the straight line A E( d) (see below). In order to reduce the influence of an individual target’s error in thickness calibration, this method is desirable; in the presence of a common oxide layer on the targets, it is necessary. Additive errors in thickness calibration like an oxide layer contribute to the intercept AE(d = 0) and leave the stopping cross section unchanged. Thus, by means of the foil set method transmission can become as insensitive to surface contaminants as is backscattering. The method can only be successfully applied, if all targets of a set exhibit the same additive calibration errors (for example due to changes in atmospheric pressure and humidity) and an oxide layer of equal thickness. An indispensable prerequisite therefore is the evaporation of the foil targets within one run. As the production of self-supporting silicon films is highly troublesome, there is no real chance to reasonably make use of the method in the usual way. The crucial step in the procedure is the floating in distilled water. This was circumvented here. Instead, the silicon layers were deposited on thin carbon foils (200-400 A), which were previously caught on rings (2 mm internal diameter). The energy loss in the carbon carriers was determined in addition to that of the compound foils.
3. Evaluation of stopping cross sections The evaluation of the RBS measurements was performed as usual. The 11-parameter fit of the S&peaks in the spectrum worked very reliably. In addition to the experiments included here, an earlier run had been performed, for which the silicon layers were deposited on very resistant but inhomogeneous carbon foils (deposited on betaine). Additionally, they were heavily oxidized. In spite of these shortcomings, the results obtained did not differ from the ones presented here. They have not been considered here as the foils were too inhomogeneous to be useable in transmission experiments. A typical RBS spectrum is shown in fig. 2. From RBS-spectra like this the amount of oxygen on a target’s surface was derived (typically 2 X 1016 atoms/cm2). It turned out that within experimental errors the oxide layers had the same thickness for all targets used in the foil set. The application of the foil set method was therefore justified. Only minor (= 1 at.% oxygen) bulk contamination was indicated. When the stopping cross sections were to be evaluated by the foil set method, the individual energy losses had to refer to the same average ion energy in the
P. Mertens, P. Bauer / Reference stopping cross sections 0.5 1.
20 -
135
r-
0.4
680 keV protons 7
ANO)
0.2
2,
protons --->
silicon
set of 9 foila(270 - 1635 A)
*
FL 0 !! 8
0.0
.c-o.* z 3 b-O.4 E
-0.7
0
128
256
512
6LO
768
896
100
200
300
400
450
[keV]
channel
energy
Fig. 2. RBS-spectrum of 680 keV protons for a 1540 A silicon layer deposited on a 500 A carbon foil, taken after completion of the RBS-measurements (carbon build-up: = 25 A).
Fig. 4. Energies at intercept d = 0 for the foil set used for the stopping power measurements by the transmission method.
foil. Experimentally, however, these average energies are different. The thicker a foil, the smaller the average energy is for the constant energy of the impinging ions. Therefore some correction has to be provided which equally eliminates the additive energy loss in the carbon carrier. On the basis of the measured energies of incoming and outgoing ions, the average energies were calculated by utilizing the functional dependence given by the AZ semiempirical fit. Yet even for the lowest energies the values resulting from these procedures never differed significantly from the ones obtained by averaging the energies at the foil entrance and exit. Thus the latter might have been used as well. As the deposited silicon layer was in the path of the incoming beam, the energy loss measured at the uncoated carbon carrier had to be transformed to a reduced energy. Again, this was done by means of the AZ function. Alternatively, the experimental cross sections measured for a single
target could have been used. Finally, after the energy losses in the carbon foil had been subtracted from the ones in the compound target, these energy loss differences were AZ-transformed to a common average energy (the one in a foil of 800 A thickness). Having been evaluated in this way, the energy losses of the foil set were used for a linear regression to derive the stopping cross section S. Some examples for the data base of this linear regression are given in fig. 3. The corrected experimental energy losses AE (in keV) are plotted versus the foil thickness d (in angstrom) as resulting from the weight calibration. Deviations of the data points from a straight line are mainly due to statistical errors in the weighing procedure. Thus the figure demonstrates first of all that this calibration did not work too badly (the weight differences involved ranged from 7 to 40 pg!). Influences of additive calibration errors or surface oxide layers can hardly be detected (see fig. 4). The intercept energies at d = 0 are near zero. If they are the result of additive deviations in the weight calibration, the intercept will exhibit the same energy dependence as the stopping cross section [ll]. Fig. 4 confirms this expectation. Naturally, the experimental errors in AE implied for very low ion energies are larger than they are for the higher energies. This is because the foils here cannot be reliably aligned to the beam by means of a phosphorus screen. But even the deviating intercept for the 40 keV ions is not reflected by the data points in fig. 3. Thus it can be stated that the intercepts as resulting from linear regression are a sensitive measure for the precision of an experiment.
25
.”
1..“““”
BE = S * d + AE,
foil
thickness
[A]
Fig. 3. Energy loss measurements for a foil set: thickness from weight calibration, energy losses corrected for carbon carrier and transformed to a common average energy.
4. Results and discussion Stopping cross sections have usually been derived from measurements of single foils. The former discusIII. ENERGY
LOSS/SCATTERING
136
P. Mertens, P. Bauer / Reference slopping cross sections
10
* 0
100
200 energy
300
400
430
[kd]
Fig. 5. Proton stopping cross sections (in lo-” eV cm’) in silicon for single foils as resulting from the transmission method (together with the Andersen-Ziegler fit).
sion on the so-called thickness dependence of the stopping power reflects this troublesome practice. Except for small effects like pre-equilibrium stopping [12] the problems envisaged were related to errors in thickness calibration and to the presence of surface contaminants. As a consequence experimental stopping cross sections derived from thin and thick foils could differ considerably (see for examples ref. [ll]). If, on the other hand, measurements for thin and thick foils appear to agree reasonably well, the absence of these problems may be deduced. It is testified by fig. 5 that silicon is a material of this type. No systematic differences can be observed for the absolute values and energy dependence of the stopping cross section. The results of the backscattering experiment and the transmission experiment for the foil set are accumulated in fig. 6. Perfect agreement within mostly _t 1% is shown, at 600 keV the proton RBS data approach the AZ-function to 1%. Whereas the fit function has its
5 10
100 energy
1000
I keV I
Fig. 6. Proton stopping cross sections (in lo-l5 eV cm2) in silicon: backscattering and foil set transmission data together with the Andersen-Ziegler fit (full line), broken line: graphical interpolation of our data for aluminum.
maximum at about 80 keV, the maximum of the experimental data is reached at about 57 keV. At the maximum and below the semiempirical fit is too low by typically 12%. The relative position of our data and those depicted in fig. 1 can be judged by comparison with the AZ-function, which is included in figs. 1 and 6. Although the errors quoted are higher than ours (Kiihrt et al.: 5%, Santry and Werner: +4%, Sirotinin: 3% statistical) the overall agreement is much better. Over the whole energy range the data of Kihrt et al. [4] correspond to ours by better than 2.5%. This equally holds for the measurements of Santry and Werner [5] for proton energies above 120 keV. Also the stopping cross sections given by Sirotinin [3] are located within this margin between 100 and 300 keV. Only at the limits of his energy range (80 keV, 400 keV) are the cross sections too low by 6%. Facing this large extent of agreement it has to be concluded that the experiments deliver realistic stopping cross sections for protons within *2%. This is actually the experimental error attributed to our data. The promising agreement of results obtained by a variety of experimental methods for silicon does in no way guarantee similar findings for other materials. For materials exhibiting, for example, large crystallites and pronounced surface or bulk contaminants, larger deviations cannot be excluded. At the beginning of our cooperative program for backscattering and transmission, the goal was agreement within + 3%. This margin proved to be a realistic limit of precision. Even working with equivalent targets minor differences in the foils investigated have to be considered. In spite of being stored and transported in vacuum vessels an increase of hydrogen in the bulk can be neither excluded nor detected by RBS methods. The achieved precision, however, defines a margin not yet reached by semiempirical tabulations or theory. Our results (see fig. 6) show a stopping cross section maximum of 25.7 X lo-l5 eV cm2 at a proton energy of 57 keV. With increasing energy the stopping cross section decreases rapidly. In comparison, for Al the stopping cross section maximum is 22.2 X lo-l5 eV cm2 at the same energy E,, [13], while at 500 keV the stopping values of Al and Si are almost identical. To get some insight into the stopping processes involved, we compared our data to BEA calculations [13]. The results of this theory for Si and Al [7,13] exceed the experimental data by just 4-8% in our energy range and are thus a good basis for the interpretation. In this model, at about 300-400 keV the contribution of the core electrons to the stopping process has its maximum, yielding for Si about 40% (for Al about 60%) of the total stopping cross section. Around the maximum, the contribution of the core electrons is small (- 2 X lo-l5 eV cm2 for Si and - 6 x lo-l5 eV cm2 for Al) with a very weak energy dependence. Therefore, the shape of the
P. Mertens, P. Bauer / Reference stopping cross sections
stopping cross section here is almost entirely dominated by the electron gas. The electron gas densities are quite similar in both cases (in Si, the higher number of valence electrons per roughly compensates for the larger atomic volume). Thus, the (almost) equal values of E,,, for Al and Si are quite reasonable. The height of the cross sections around the maximum is a result of combined core and valence electron contributions.
References [l] H.H. Andersen and J.F. Ziegler, The Stopping and Ranges of Ions in Matter, vol. 3 (Pergamon, New York, 1977). [2] Th. Krist and P. Mertens, Nucl. Instr. and Meth. 218 (1983) 790. [3] E.I. Sirotinin, A.F. Tulinov, V.A. Khodyrev and V.N. Mizgulin, Nucl. Instr. and Meth. B4 (1984) 337.
137
[4] E. Kthrt, K. Lenkeit and F. Taubner, Phys. Status Solidi (a)66 (1981) K131. [5] D.C. Santry and R.D. Werner, Nucl. Instr. and Meth. 188 (1981) 211. [6] H.A. Bethe, Ann. Phys. 5 (1930) 325; Z. Phys. 76 (1932) 293. [7] P. Mertens, P. Bauer and D. Semrad, Nucl. Instr. and Meth. B15 (1986) 91. [S] M. Geretschhager, Nucl. Instr. and Meth. 204 (1983) 479. [9] D. Semrad and P. Bauer, Nucl. Instr. and Meth. 149 (1978) 159. [lo] P. Bauer, Nucl. Instr. and Meth. B27 (1987) 301. [ll] P. Mertens, Nucl. Instr. and Meth. B27 (1987) 315. [12] Th. Krist, P.J. Scanlon, P. Mertens, K.M. Barfoot, I. Reid and C.H. Cheng, Nucl. Instr. and Meth. B14 (1986) 179. [13] D. Semrad, P. Bauer and P. Mertens, Nucl. Instr. and Meth. B15 (1986) 86. [14] E. Kiihrt and R. Wedell, Phys. Status Solidi B116 (1983) 585.
III. ENERGY
LOSS/SCATTERING