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Displacement energy for various ions in particle detector materials
Nuclear Instruments and Methods in Physics Research A 449 (2000) 277}287
Displacement energy for various ions in particle detector materials A. Chilingarov*, D. Lipka, J.S. Meyer, T. Sloan Department of Physics, University of Lancaster, Lancaster LA1 4YB, UK Received 19 September 1999; received in revised form 9 November 1999; accepted 2 December 1999
Abstract The total displacement energy or total non-ionising energy loss has been calculated for a variety of ions during their slowing down to rest in the detector materials carbon, silicon and gallium arsenide. The calculations, based on the theory of Lindhard et al., have been performed using a Monte Carlo method and a simple parameterisation of the results is presented. Such a parameterisation will simplify considerably the future computation of the di!erential non-ionising energy loss by fast particles in particle detectors. 2000 Elsevier Science B.V. All rights reserved. PACS: 07.77!n; 85.30.!z Keywords: Lindhard; Silicon; Diamond; Gallium arsenide; Detectors
1. Introduction As a charged particle moves through a material it loses energy either by electronic or by nuclear collisions. The latter results in the displacement of atoms from the lattice and the energy loss due to this process is called the non-ionising energy loss (NIEL). The deterioration in performance due to radiation damage in charged particle detectors made from silicon (Si) and gallium arsenide (GaAs) has been shown to be proportional to this displacement energy deposited [1,2]. The di!erential NIEL (dE/dx) for fast charged particles is much smaller L than the usual ionisation loss (dE/dx) . It is di$ cult to measure and usually it is calculated.
* Corresponding author. Tel.: #44-1524-594627; fax: #441524-844037. E-mail address: [email protected] (A. Chilingarov).
One of the di$culties in such calculations is that the required full di!erential cross section for the primary particle interaction with the nucleus of the media is often poorly known. Another problem arises from the fact that the recoil nuclei or their fragments produced in the nuclear interaction cascade induce both ionising and non-ionising energy losses. The range of these secondary ions is usually short compared to the thickness of the material for which NIEL calculations are made. Therefore, for any type of secondary ion in a given material one needs to know the amount (or fraction) of the energy that goes to NIEL as a function of the ion's initial energy (the so-called Lindhard partitioning). This calculation is far from trivial because in many cases the recoil fragments from the primary interaction are able to produce a whole cascade of further recoils and for all of these the energy sharing between ionising and non-ionising energy losses must be properly taken into account. In the past such
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calculations have been made analytically using various approximations [3}5]. In this paper we address the problem of the recoil nuclear fragments, using Monte Carlo calculations of the total displacement energy (or the total NIEL) deposited as an ion slows down to the end of its range. The calculations were performed using the theory of Lindhard et al. [6], adopting their approximations A}D. Previous calculations [3}5] needed to adopt even further approximations to solve the integral equations in the theory. The Monte Carlo technique, reported here, avoids the need for such further approximations so that our results are more accurate than the previous calculations. This is the "rst time such calculations have been made in a systematic and identical manner for ions ranging in mass from the proton to the maximum allowed mass for the three widely used semiconductor detector media C (Diamond), Si and GaAs. A parameterisation of our results by an analytic formula is given. Such a parameterisation will considerably simplify future calculations of the di!erential NIEL for fast particles. 2. The slowing down of ions For ions passing through a medium the relative importance of the processes of ionisation and atomic displacement changes rapidly with the kinetic energy of the incident particle. For high particle velocities the energy loss to atomic electrons is dominant, but as the velocity approaches that of the electrons in the atom the energy loss to atomic displacement (or NIEL) becomes more important. Therefore, both processes must be calculated to simulate the slowing down of a particle in the material and to compute the total displacement energy deposited. These calculations are described in this section. In the following Z , m (Z , m ) refer to the atomic number and the mass number of the incident particle (material). 2.1. Electronic stopping For fast particles the energy loss to electrons is given by the Bethe}Bloch formula.
dE dx
4peZ 1 2m cbc n " ln !1 m c b I(1!b)
(1)
where e and m are the charge and mass of the electron, n is the number of electrons per unit volume in the material, b is the relative velocity of the incident particle, c"1/(1!b and I is the ionisation potential of the material. The latter can be parameterised by [7] I(Z )"(12Z #7) for Z 413 or (9.76Z # 58.8Z\ ) eV for Z '13. For low energies when the velocity of the incident particle is comparable or less than the velocity of the electrons in the atom, this formula is no longer valid. The energy loss is then proportional to the particle velocity v [6].
Z Z v "Nm 8pea (2) M Z v M where N is the number of atoms per unit volume, m +Z (see Ref. [6]), a is the Bohr radius M of the hydrogen atom and Z "Z #Z . This formula holds for v(Z ) v where M v "e/2e h (+2.2;10 cm/s) is the electron veM locity in the classical lowest Bohr orbit of the hydrogen atom. At intermediate energies between the validity of Eqs. (1) and (2) we employ a polynomial function as described in the appendix. The accuracy of this procedure can be assessed from Fig. 1 which shows the dependence on b of (dE/dx) for Si ions in Si and Ge ions in GaAs. If the rise at low b and the fall at high b coming from Eqs. (1) and (2) were extrapolated to join discontinuously the results would not be dissimilar to the polynomial interpolation shown in this "gure. dE dx
2.2. Nuclear stopping There are three di!erent processes in which the incident particle can transfer energy to the lattice nuclei. These are Rutherford scattering, elastic and inelastic nuclear interactions. Our purpose is to calculate the displacement energy for slow moving ions with a range less than the thickness of a typical particle detector (a few hundred microns). At such energies Rutherford scattering dominates the inter Here germanium (Ge) is taken to be the average of Ga and As to simulate an ion in its own GaAs medium.
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Fig. 1. Dependence of (dE/dx) for Si ions in Si and Ge ions in GaAs on normalised velocity b. The rise at small b is described by Eq. (2) and the fall at large b by Eq. (1). The curved region between comes from the polynomial approximation described in the appendix.
actions which produce NIEL and the e!ects of other nuclear reactions can be neglected. However, the calculations presented, which include only Rutherford scattering, were extended to higher energies to constrain the parameterisation of the results. In the Lindhard}Schar! theory of Rutherford scattering of an ion of kinetic energy, E, the dimensionless energy variable e"(a/eZ Z ) (m /(m #m ))E is used where a"0.885a Z\ is M the Thomas}Fermi screening distance. The scattering cross section can then be transformed into a function of a single argument t"e¹/¹ "e sin F [6]
dt dp"pa f (t ) (3) 2t where h is the scattering angle and ¹, ¹ are the
energy transferred to the nucleus and its kinematic maximum ("4m m E/(m #m )).
The function f (t ) depends on the potential energy distribution of the atom. With f (t )" , which corresponds to a Coulomb potential, Rthe cross-section is reduced to the standard Rutherford scattering formula. However, to account for the screening of the nucleus by the atomic electrons, a Thomas}Fermi}Firsov potential is used since the standard Rutherford formula is only correct when screening of the nucleus is negligible i.e. in close collisions which correspond to larger scattering angles. For the Thomas}Fermi}Firsov potential f (t ) has been computed numerically [6]. We use here the analytic representation of the data [8] f (t )"jt [1#(2jt ) ]\ with j"1.309. (4)
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The cross-sections described by Eq. (3) were used to calculate the Rutherford scattering probabilities in the Monte Carlo simulation described in Section 3. The energy imparted to a recoiling nucleus was then calculated from the scattering angle of the collision. 3. Calculation of the displacement energies The total NIEL (or displacement energies) for ions were computed using a Monte Carlo program which simulated the losses due to Rutherford scattering (Section 2.2) and simultaneously calculated the ionisation energy loss (Section 2.1). The calculations have been performed for a variety of ions in the particle detector media C, Si and GaAs. In order to calculate the displacement energies, the incident ion was followed in small steps through the material. In each step, the ionisation energy loss, assumed to be continuous, was computed and the discontinuous energy loss due to Rutherford scattering was simulated. In the latter case the energy of each nucleus recoiling from both primary and secondary interactions was stored. The Rutherford scattering process was simulated down to a lower recoil energy which was taken to be the minimum energy, E , to dislodge an atom from the lattice. For Si two di!erent values E (Si)"12.9 eV and E (Si)"21 eV can be found in the literature [9]. The "rst of these values has been used in these calculations and the second value has been used to estimate their accuracy. The value for C is E (C)"50 eV [10] and for GaAs is E (GaAs)"10 eV [11]. The results are in sensitive to the exact values of this threshold energy, e.g., the two di!erent values of the threshold energy for Silicon change the computed displacement energies by about 10%. To achieve accurate simulation the step thickness was chosen to keep the Rutherford scattering probability to below 10% per step. The steps were continued until the kinetic energy of the primary ion decreased below the cut-o! energy which is the energy for which the maximum energy transfer is equal to the threshold energy, E . The process was then repeated for all the stored recoiling nuclei taking the sum of the residual energies of all recoils (excluding the incident ion) below the cut-o! to be the total displace-
ment energy. For almost all recoils the residual energy was close to the threshold energy. From this, the reason for the insensitivity of the displacement energy to the assumed threshold energy can be understood. It can be seen that the total displacement energy is approximately equal to the product of the number of recoils and the cut-o! energy. Hence, a lower (higher) cut-o! energy leads to a greater (lower) number of recoils so that the change in the product is quite small. In general, the number of recoils is very large and the running time of the program is long, especially for high particle energies. For low-energy particles, when the running time of the program is shorter, the Monte Carlo results can be averaged over many incident particles but for higher energies the calculation can only be done for a small number. However, the statistical accuracy of the results is good at higher energies because each particle produces such a large number of recoils that the calculation for a single particle is already an average itself. The computed displacement energies (or total NIEL) for some of the incident ions in C, Si and GaAs are shown in Figs. 2, 3 and 4, respectively. In Figs. 5 and 6 the total NIEL for protons and a particles, respectively, is shown for the three detector materials. The error bars on the points (which are sometimes smaller than the size of the symbols) show the statistical uncertainties on the simulation. The curves in the "gures are the results of the parameterisation described below.
4. Parameterisation of the displacement energy To parameterise the displacement energy variation with ion energy for ions in the detector materials C,Si and GaAs shown in Figs. 2}6 we use a function of the form mE NIEL" (5) 1#mE/E where m is the slope at low energy and E is the saturation level at high energy. The quantities m and E are further parameterised 1 m"M #0.1 (6) (E/E )@
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281
Fig. 2. Displacement energy for several ions in C as a function of the incident ion kinetic energy. The smooth curves show the results of the parameterisation described in the text.
E b"h#i log (E/E )#d log E E Q E "E . (7) E The factor 0.1 in m is to prevent the value of m becoming very small at higher energies. Each parameter M , h, i, d, E and s is further para meterised as a function of the charge and atomic mass numbers of the ions and the materials in the form
x a#ex#b exp ! c
(8)
where x is normally the mass number of the ion, m . However, for the parameters M and E in carbon
such a simple mass dependence was not smooth enough and had to be modi"ed to the form x"m (Z /Z )N. Here the normalisation factor Z was taken to be 4, the approximate average value of Z of the ions for which the simulation in carbon was made (excluding the proton and a particle). The parameters were chosen to minimise the value of s between the "t and the calculated values. The results of the "t are shown as the smooth curves in Figs. 2}6 and the normalised s are given in Table 1. A range of masses of the incident ions starting from the proton was considered as shown in Table 1. The single set of parameters given in Table 2 was su$cient to describe the total NIEL for all ions heavier than a particles. Where e or p is not included in the list of
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Fig. 3. The same as Fig. 2 for several ions in Si.
parameters in Table 2 it is taken to be zero. The optimal values of all the parameters, M , h, i, d, E and s, for protons and a-particles were found to be signi"cantly di!erent from the extrapolation from heavy ions to their values of m and Z by Eq. (8). Therefore the optimal values of these parameters for those particles are presented separately in Table 3. The number of degrees of freedom (NDF) was 6 and 7 for the "ts to the proton and a-particle calculations, respectively. For heavier ions the further parameterisation through Eq. (8) links ions of di!erent masses. Hence, the e!ective number of free parameters is smaller and the number of degrees of freedom is larger and is typically 15 per ion. It can be seen from Table 1 that the values of s per degree of freedom (s/NDF) is usually close
to unity. This implies that the parameterisation works well within the statistical errors of the simulation which are a few per cent. In the worst cases of protons and a particles in C and GaAs, the s/NDF would be reasonable if the statistical errors are roughly doubled (see Figs. 5 and 6). This implies that the parameterisation is accurate to within 5% in these cases. The calculations were made down to energies of 0.1 keV. Below this energy the parameterisation becomes an extrapolation rather than an interpolation and so its validity cannot be guaranteed. At higher energies (above 10 MeV per nucleon) the elastic and inelastic nuclear interactions (not considered here) give additional contributions to the NIEL. Hence the displacement energies computed
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283
Fig. 4. The same as Fig. 2 for several ions in GaAs.
using the parameterisation represent a lower limit at such higher energies.
5. Comparison with other calculations The results for Si and GaAs have been compared with the curves published in Refs. [3,5] and with those in carbon in Ref. [4]. Fig. 7 shows a sample of the published curves as dashed lines compared to the results of the parameterisation of our calculations (the solid lines). The curves for the ions with the highest and lowest atomic mass were selected from each publication for each material. There is agreement everywhere within 35% except for the calculations of He in carbon where our results di!er from those of Ref. [4] by up to a factor of 3 at
the highest energies. The agreement for Si and GaAs ions in their own material is within 10%. As in the work reported here, each of the existing published calculations makes use of the Lindhard theory as a starting point and the approximations therein [6]. However, the existing published works use di!erent approximations to solve the integral equations to obtain numerical results e.g. Burke et al. [3] use Ziegler's approximation [12]. Since ours is a Monte Carlo approach we do not need to make approximations to achieve numerical solutions. It should be noted that the original Lindhard paper treats the case of lighter ions separately from the ions in their own material. Whereas in our Monte Carlo such lighter ions are treated in an identical manner to the ions in their own material. Hence, ours is a more general treatment. Since our
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Fig. 5. Displacement energy for protons in three materials as a function of their kinetic energies. The smooth curves show the results of the parameterisation described in the text.
work agrees within &10% with other computations for Si and GaAs ions in their own media and ours is a general method for all atomic masses, m , we believe that the accuracy of our simulation is probably better than 10% over the whole range of m . There is a rather large discrepancy for low mass ions in C between our calculations and those of Lazanu et al. [4]. To investigate this further we have made a consistency check of our calculations for low m ions by comparing the total NIEL for protons from our simulation with a calculation starting from the di!erential NIEL values computed by Summers et al. [9]. The di!erential NIEL values from the latter were integrated over the
range of the protons taking into account all ionisation losses to obtain values of the total NIEL. These values were then compared to the results from our Monte Carlo program. The comparison was made for protons at several energies below 10 MeV kinetic energy for both Si and GaAs. In this energy range Rutherford scattering dominates the interactions producing NIEL mostly by atoms recoiling in their own material for which there is good agreement between all the calculations of the displacement energy. The crosssection formula in our simulation for this comparison was identical to that used by Summers et al. [9]. The total NIEL from this simulation for protons agreed with that derived from the integral over
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285
Fig. 6. The same as Fig. 5 for a-particles.
Table 1 Table of "tted ions with normalized s between the data and the parameterisation Material
C
Ion
s/NDF
C B B
1.07 0.36 0.91
Be Be Li Li Li He H * * *
1.07 0.90 0.93 1.00 0.77 2.84 2.76 * * *
Material
Si
Ion
s/NDF
Si Mg Na
1.61 0.38 0.95
Ne O N B Be Li He H * *
2.35 0.89 0.92 1.02 1.02 0.62 0.31 1.55 * *
Material
GaAs
Ion
s/NDF
Ge Zn Cu
1.76 0.58 0.45
Ni Fe Mn Ca S Mg O Be He H
0.50 1.06 0.64 0.59 1.11 1.10 1.23 1.37 4.13 3.33
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Table 2 Table of parameters for heavy ions Parameter
Height
Carbon
M
a b c e p
0.5916 !0.5980 2.870 !0.00322 0.0
0.8462 !0.3560 62.370 !0.00295 0.0
h
a b c e
0.4944 !0.3993 4.108 0.0 ! 0.0183 0.406 2.94 0.0
!0.0992 0.0 1.0 0.005026
0.00086 !2.0096 3.18 0.00025
i
a b c
!0.06793 !43.304 0.7
0.0073 !0.241 10.09
!0.01442 !0.849 4.87
d
a b c
!0.031 !1.710 1.59
!0.0023 !0.138 5.63
!0.00311 !0.262 5.14
E
a b c e p
!294.18 keV 305.48 keV 12.98 19.637 keV 0.0
!679.63 keV 689.51 keV 16.92 35.042 keV 0.0
s
a b c e Z
!2.0 keV 0.0 keV 1.0 2.578 keV 0.0050 0.1209 3.80 0.0 4.0
0.0 0.2126 6.67 0.0 *
0.0318 6.225 2.0 !0.000396 *
Table 3 Table of the parameters for protons and a-particles Particle Height Carbon
Silicon
Gallium arsenide
p
M h i d E s
0.003 0.0269 !1.14 !1.14 0.13 0.36 !0.11 0.012 0.0395 keV 0.103 keV 0.2913 0.339
0.0081 !1.69 0.87 !0.099 0.19 keV 0.347
a
M h i d E s
0.396 0.170 0.011 !0.151 3.96 keV 0.075
0.359 !0.028 !0.178 !0.214 5.19 keV 0.189
0.427 !0.047 !0.201 !0.125 6.30 keV 0.1147
Silicon
Gallium arsenide
the di!erential NIEL to better than 15% over most of the energy range with discrepancies up to 30% for energies below 10 keV. This con"rms that our simulation program for light ions is giving reliable values of the displacement energies. It should be noted that in the data presented in Fig. 5 the cross sections calculated as described in Section 2.2 were used which include the e!ects of atomic screening by electrons. These e!ects are rather important but they are not included in the cross section used in Ref. [9]. On the other hand, we do not include the e!ects of spin in our cross sections, which are included in Ref. [9], because these e!ects are quite small. 6. Conclusions Monte Carlo calculations of the displacement energy or total NIEL for a wide variety of ions
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287
a polynomial is used [7] of the form
dE dx
"Ab#Bb#Cb#D (9) where A, B, C, D are derived from the four simultaneous equations obtained by equating (dE/dx) and d(dE/dx) /db at b and b for Eqs. (1), (2) and (9). This gives
A" (W #k! W) @ @ @ B" (3(b #b ) W!(b #2b ) W !*yk) @ @ @ W !@ @ W ) C" (b (b #2b )k#@ @ >@ @ @ @ D" [ (b y (3b !b )#b y (b !3b )) @ @ !b b (b k#b W )] @ where *b"b !b , y , y are the dE/dx values computed from Eqs. (2) and (1) at velocities b and b , respectively,*y"y !y , k" [2pC8 n(c ! @ A K ln(K A @A ))] and c "1/(1!b . '\@ Fig. 7. Comparison of the displacement energies from our Monte Carlo (solid lines) with other calculations (dashed lines) for ions with various values of A and Z in (a) gallium arsenide [3], (b) silicon [5], and (c) carbon [4].
in three semiconductor detector materials have been made using Lindhard theory. A simple parameterisation of the results is given. This parameterisation will simplify considerably future computations of the di!erential NIEL from fast particles such as pions, protons and neutrons. Acknowledgements We wish to thank Dr. T.J. Brodbeck for his assistance with "tting. Appendix. Ionisation loss at intermediate energies The intermediate region, in which neither the Bethe}Bloch formula (Eq. (1)) nor the Lindhard formula (Eq. (2)) is valid, is de"ned by the velocity range b )b)b , where b "Max[aZ / (1#aZ ), (2Z #1)/400], b "aZ /(1#aZ ) and a is the "ne structure constant. In this region
References [1] A. Chilingarov, H. Feick, E. Fretwurst, G. Lindstrom, S. Roe, T. Schulz, Nucl. Instr. and Meth. A 360 (1995) 432. [2] A. Chilingarov, J.S. Meyer, T. Sloan, Nucl. Instr. and Meth. A 395 (1997) 35. [3] E.A. Burke et al., IEEE Trans. Nucl. Sci. NS-34 (1987) 1220. [4] I. Lazanu, S. Lazanu, E. Borchi, M. Bruzzi, Nucl. Instr. and Meth. A 406 (1998) 259. [5] G.W. Simons, J.M. Denney, R.G. Downing, Phys. Rev. 129 (1963) 2454. [6] J. Lindhard, V. Nielsen, M. Schar!, P.V. Thomsen, Mat. Fys. Medd. Dan. Vid. Selsk. 33 (10) (1963) 1. [7] J.D. Lewin, Range-energy relations for hypothetical charged particles, Rutherford Laboratory, RL-77 126/A (1977). [8] M.A. Kumakhov, F.F. Komarov, Energy loss and ion ranges in solids, Gordon and Breach, London, 1981, pp. 21}38. [9] G.P. Summers, E.A. Burke, P. Shapiro, S.R. Messenger, R.J. Walters, IEEE Trans. Nucl. Sci. NS-40 (1993) 1372. [10] J.W. Corbett, N.B. Urli, Radiation E!ects in Semiconductors, 1976, Institute of Physics, Conference Series No. 31, 1977. [11] A.L. Barry, R. Maxseiner, R. Wojcik, M.A. Briere, D. Braeunig, IEEE Trans. Nucl. Sci. NS-37 (1990) 1726. [12] J.F. Ziegler, J.P. Biersack, U. Littmark, The Stopping and Range of Ions in Solids, Vol. 1, Pergamon Press, New York, 1985.
Displacement energy for various ions in particle detector materials A. Chilingarov*, D. Lipka, J.S. Meyer, T. Sloan Department of Physics, University of Lancaster, Lancaster LA1 4YB, UK Received 19 September 1999; received in revised form 9 November 1999; accepted 2 December 1999
Abstract The total displacement energy or total non-ionising energy loss has been calculated for a variety of ions during their slowing down to rest in the detector materials carbon, silicon and gallium arsenide. The calculations, based on the theory of Lindhard et al., have been performed using a Monte Carlo method and a simple parameterisation of the results is presented. Such a parameterisation will simplify considerably the future computation of the di!erential non-ionising energy loss by fast particles in particle detectors. 2000 Elsevier Science B.V. All rights reserved. PACS: 07.77!n; 85.30.!z Keywords: Lindhard; Silicon; Diamond; Gallium arsenide; Detectors
1. Introduction As a charged particle moves through a material it loses energy either by electronic or by nuclear collisions. The latter results in the displacement of atoms from the lattice and the energy loss due to this process is called the non-ionising energy loss (NIEL). The deterioration in performance due to radiation damage in charged particle detectors made from silicon (Si) and gallium arsenide (GaAs) has been shown to be proportional to this displacement energy deposited [1,2]. The di!erential NIEL (dE/dx) for fast charged particles is much smaller L than the usual ionisation loss (dE/dx) . It is di$ cult to measure and usually it is calculated.
* Corresponding author. Tel.: #44-1524-594627; fax: #441524-844037. E-mail address: [email protected] (A. Chilingarov).
One of the di$culties in such calculations is that the required full di!erential cross section for the primary particle interaction with the nucleus of the media is often poorly known. Another problem arises from the fact that the recoil nuclei or their fragments produced in the nuclear interaction cascade induce both ionising and non-ionising energy losses. The range of these secondary ions is usually short compared to the thickness of the material for which NIEL calculations are made. Therefore, for any type of secondary ion in a given material one needs to know the amount (or fraction) of the energy that goes to NIEL as a function of the ion's initial energy (the so-called Lindhard partitioning). This calculation is far from trivial because in many cases the recoil fragments from the primary interaction are able to produce a whole cascade of further recoils and for all of these the energy sharing between ionising and non-ionising energy losses must be properly taken into account. In the past such
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calculations have been made analytically using various approximations [3}5]. In this paper we address the problem of the recoil nuclear fragments, using Monte Carlo calculations of the total displacement energy (or the total NIEL) deposited as an ion slows down to the end of its range. The calculations were performed using the theory of Lindhard et al. [6], adopting their approximations A}D. Previous calculations [3}5] needed to adopt even further approximations to solve the integral equations in the theory. The Monte Carlo technique, reported here, avoids the need for such further approximations so that our results are more accurate than the previous calculations. This is the "rst time such calculations have been made in a systematic and identical manner for ions ranging in mass from the proton to the maximum allowed mass for the three widely used semiconductor detector media C (Diamond), Si and GaAs. A parameterisation of our results by an analytic formula is given. Such a parameterisation will considerably simplify future calculations of the di!erential NIEL for fast particles. 2. The slowing down of ions For ions passing through a medium the relative importance of the processes of ionisation and atomic displacement changes rapidly with the kinetic energy of the incident particle. For high particle velocities the energy loss to atomic electrons is dominant, but as the velocity approaches that of the electrons in the atom the energy loss to atomic displacement (or NIEL) becomes more important. Therefore, both processes must be calculated to simulate the slowing down of a particle in the material and to compute the total displacement energy deposited. These calculations are described in this section. In the following Z , m (Z , m ) refer to the atomic number and the mass number of the incident particle (material). 2.1. Electronic stopping For fast particles the energy loss to electrons is given by the Bethe}Bloch formula.
dE dx
4peZ 1 2m cbc n " ln !1 m c b I(1!b)
(1)
where e and m are the charge and mass of the electron, n is the number of electrons per unit volume in the material, b is the relative velocity of the incident particle, c"1/(1!b and I is the ionisation potential of the material. The latter can be parameterised by [7] I(Z )"(12Z #7) for Z 413 or (9.76Z # 58.8Z\ ) eV for Z '13. For low energies when the velocity of the incident particle is comparable or less than the velocity of the electrons in the atom, this formula is no longer valid. The energy loss is then proportional to the particle velocity v [6].
Z Z v "Nm 8pea (2) M Z v M where N is the number of atoms per unit volume, m +Z (see Ref. [6]), a is the Bohr radius M of the hydrogen atom and Z "Z #Z . This formula holds for v(Z ) v where M v "e/2e h (+2.2;10 cm/s) is the electron veM locity in the classical lowest Bohr orbit of the hydrogen atom. At intermediate energies between the validity of Eqs. (1) and (2) we employ a polynomial function as described in the appendix. The accuracy of this procedure can be assessed from Fig. 1 which shows the dependence on b of (dE/dx) for Si ions in Si and Ge ions in GaAs. If the rise at low b and the fall at high b coming from Eqs. (1) and (2) were extrapolated to join discontinuously the results would not be dissimilar to the polynomial interpolation shown in this "gure. dE dx
2.2. Nuclear stopping There are three di!erent processes in which the incident particle can transfer energy to the lattice nuclei. These are Rutherford scattering, elastic and inelastic nuclear interactions. Our purpose is to calculate the displacement energy for slow moving ions with a range less than the thickness of a typical particle detector (a few hundred microns). At such energies Rutherford scattering dominates the inter Here germanium (Ge) is taken to be the average of Ga and As to simulate an ion in its own GaAs medium.
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Fig. 1. Dependence of (dE/dx) for Si ions in Si and Ge ions in GaAs on normalised velocity b. The rise at small b is described by Eq. (2) and the fall at large b by Eq. (1). The curved region between comes from the polynomial approximation described in the appendix.
actions which produce NIEL and the e!ects of other nuclear reactions can be neglected. However, the calculations presented, which include only Rutherford scattering, were extended to higher energies to constrain the parameterisation of the results. In the Lindhard}Schar! theory of Rutherford scattering of an ion of kinetic energy, E, the dimensionless energy variable e"(a/eZ Z ) (m /(m #m ))E is used where a"0.885a Z\ is M the Thomas}Fermi screening distance. The scattering cross section can then be transformed into a function of a single argument t"e¹/¹ "e sin F [6]
dt dp"pa f (t ) (3) 2t where h is the scattering angle and ¹, ¹ are the
energy transferred to the nucleus and its kinematic maximum ("4m m E/(m #m )).
The function f (t ) depends on the potential energy distribution of the atom. With f (t )" , which corresponds to a Coulomb potential, Rthe cross-section is reduced to the standard Rutherford scattering formula. However, to account for the screening of the nucleus by the atomic electrons, a Thomas}Fermi}Firsov potential is used since the standard Rutherford formula is only correct when screening of the nucleus is negligible i.e. in close collisions which correspond to larger scattering angles. For the Thomas}Fermi}Firsov potential f (t ) has been computed numerically [6]. We use here the analytic representation of the data [8] f (t )"jt [1#(2jt ) ]\ with j"1.309. (4)
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The cross-sections described by Eq. (3) were used to calculate the Rutherford scattering probabilities in the Monte Carlo simulation described in Section 3. The energy imparted to a recoiling nucleus was then calculated from the scattering angle of the collision. 3. Calculation of the displacement energies The total NIEL (or displacement energies) for ions were computed using a Monte Carlo program which simulated the losses due to Rutherford scattering (Section 2.2) and simultaneously calculated the ionisation energy loss (Section 2.1). The calculations have been performed for a variety of ions in the particle detector media C, Si and GaAs. In order to calculate the displacement energies, the incident ion was followed in small steps through the material. In each step, the ionisation energy loss, assumed to be continuous, was computed and the discontinuous energy loss due to Rutherford scattering was simulated. In the latter case the energy of each nucleus recoiling from both primary and secondary interactions was stored. The Rutherford scattering process was simulated down to a lower recoil energy which was taken to be the minimum energy, E , to dislodge an atom from the lattice. For Si two di!erent values E (Si)"12.9 eV and E (Si)"21 eV can be found in the literature [9]. The "rst of these values has been used in these calculations and the second value has been used to estimate their accuracy. The value for C is E (C)"50 eV [10] and for GaAs is E (GaAs)"10 eV [11]. The results are in sensitive to the exact values of this threshold energy, e.g., the two di!erent values of the threshold energy for Silicon change the computed displacement energies by about 10%. To achieve accurate simulation the step thickness was chosen to keep the Rutherford scattering probability to below 10% per step. The steps were continued until the kinetic energy of the primary ion decreased below the cut-o! energy which is the energy for which the maximum energy transfer is equal to the threshold energy, E . The process was then repeated for all the stored recoiling nuclei taking the sum of the residual energies of all recoils (excluding the incident ion) below the cut-o! to be the total displace-
ment energy. For almost all recoils the residual energy was close to the threshold energy. From this, the reason for the insensitivity of the displacement energy to the assumed threshold energy can be understood. It can be seen that the total displacement energy is approximately equal to the product of the number of recoils and the cut-o! energy. Hence, a lower (higher) cut-o! energy leads to a greater (lower) number of recoils so that the change in the product is quite small. In general, the number of recoils is very large and the running time of the program is long, especially for high particle energies. For low-energy particles, when the running time of the program is shorter, the Monte Carlo results can be averaged over many incident particles but for higher energies the calculation can only be done for a small number. However, the statistical accuracy of the results is good at higher energies because each particle produces such a large number of recoils that the calculation for a single particle is already an average itself. The computed displacement energies (or total NIEL) for some of the incident ions in C, Si and GaAs are shown in Figs. 2, 3 and 4, respectively. In Figs. 5 and 6 the total NIEL for protons and a particles, respectively, is shown for the three detector materials. The error bars on the points (which are sometimes smaller than the size of the symbols) show the statistical uncertainties on the simulation. The curves in the "gures are the results of the parameterisation described below.
4. Parameterisation of the displacement energy To parameterise the displacement energy variation with ion energy for ions in the detector materials C,Si and GaAs shown in Figs. 2}6 we use a function of the form mE NIEL" (5) 1#mE/E where m is the slope at low energy and E is the saturation level at high energy. The quantities m and E are further parameterised 1 m"M #0.1 (6) (E/E )@
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Fig. 2. Displacement energy for several ions in C as a function of the incident ion kinetic energy. The smooth curves show the results of the parameterisation described in the text.
E b"h#i log (E/E )#d log E E Q E "E . (7) E The factor 0.1 in m is to prevent the value of m becoming very small at higher energies. Each parameter M , h, i, d, E and s is further para meterised as a function of the charge and atomic mass numbers of the ions and the materials in the form
x a#ex#b exp ! c
(8)
where x is normally the mass number of the ion, m . However, for the parameters M and E in carbon
such a simple mass dependence was not smooth enough and had to be modi"ed to the form x"m (Z /Z )N. Here the normalisation factor Z was taken to be 4, the approximate average value of Z of the ions for which the simulation in carbon was made (excluding the proton and a particle). The parameters were chosen to minimise the value of s between the "t and the calculated values. The results of the "t are shown as the smooth curves in Figs. 2}6 and the normalised s are given in Table 1. A range of masses of the incident ions starting from the proton was considered as shown in Table 1. The single set of parameters given in Table 2 was su$cient to describe the total NIEL for all ions heavier than a particles. Where e or p is not included in the list of
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Fig. 3. The same as Fig. 2 for several ions in Si.
parameters in Table 2 it is taken to be zero. The optimal values of all the parameters, M , h, i, d, E and s, for protons and a-particles were found to be signi"cantly di!erent from the extrapolation from heavy ions to their values of m and Z by Eq. (8). Therefore the optimal values of these parameters for those particles are presented separately in Table 3. The number of degrees of freedom (NDF) was 6 and 7 for the "ts to the proton and a-particle calculations, respectively. For heavier ions the further parameterisation through Eq. (8) links ions of di!erent masses. Hence, the e!ective number of free parameters is smaller and the number of degrees of freedom is larger and is typically 15 per ion. It can be seen from Table 1 that the values of s per degree of freedom (s/NDF) is usually close
to unity. This implies that the parameterisation works well within the statistical errors of the simulation which are a few per cent. In the worst cases of protons and a particles in C and GaAs, the s/NDF would be reasonable if the statistical errors are roughly doubled (see Figs. 5 and 6). This implies that the parameterisation is accurate to within 5% in these cases. The calculations were made down to energies of 0.1 keV. Below this energy the parameterisation becomes an extrapolation rather than an interpolation and so its validity cannot be guaranteed. At higher energies (above 10 MeV per nucleon) the elastic and inelastic nuclear interactions (not considered here) give additional contributions to the NIEL. Hence the displacement energies computed
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Fig. 4. The same as Fig. 2 for several ions in GaAs.
using the parameterisation represent a lower limit at such higher energies.
5. Comparison with other calculations The results for Si and GaAs have been compared with the curves published in Refs. [3,5] and with those in carbon in Ref. [4]. Fig. 7 shows a sample of the published curves as dashed lines compared to the results of the parameterisation of our calculations (the solid lines). The curves for the ions with the highest and lowest atomic mass were selected from each publication for each material. There is agreement everywhere within 35% except for the calculations of He in carbon where our results di!er from those of Ref. [4] by up to a factor of 3 at
the highest energies. The agreement for Si and GaAs ions in their own material is within 10%. As in the work reported here, each of the existing published calculations makes use of the Lindhard theory as a starting point and the approximations therein [6]. However, the existing published works use di!erent approximations to solve the integral equations to obtain numerical results e.g. Burke et al. [3] use Ziegler's approximation [12]. Since ours is a Monte Carlo approach we do not need to make approximations to achieve numerical solutions. It should be noted that the original Lindhard paper treats the case of lighter ions separately from the ions in their own material. Whereas in our Monte Carlo such lighter ions are treated in an identical manner to the ions in their own material. Hence, ours is a more general treatment. Since our
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Fig. 5. Displacement energy for protons in three materials as a function of their kinetic energies. The smooth curves show the results of the parameterisation described in the text.
work agrees within &10% with other computations for Si and GaAs ions in their own media and ours is a general method for all atomic masses, m , we believe that the accuracy of our simulation is probably better than 10% over the whole range of m . There is a rather large discrepancy for low mass ions in C between our calculations and those of Lazanu et al. [4]. To investigate this further we have made a consistency check of our calculations for low m ions by comparing the total NIEL for protons from our simulation with a calculation starting from the di!erential NIEL values computed by Summers et al. [9]. The di!erential NIEL values from the latter were integrated over the
range of the protons taking into account all ionisation losses to obtain values of the total NIEL. These values were then compared to the results from our Monte Carlo program. The comparison was made for protons at several energies below 10 MeV kinetic energy for both Si and GaAs. In this energy range Rutherford scattering dominates the interactions producing NIEL mostly by atoms recoiling in their own material for which there is good agreement between all the calculations of the displacement energy. The crosssection formula in our simulation for this comparison was identical to that used by Summers et al. [9]. The total NIEL from this simulation for protons agreed with that derived from the integral over
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Fig. 6. The same as Fig. 5 for a-particles.
Table 1 Table of "tted ions with normalized s between the data and the parameterisation Material
C
Ion
s/NDF
C B B
1.07 0.36 0.91
Be Be Li Li Li He H * * *
1.07 0.90 0.93 1.00 0.77 2.84 2.76 * * *
Material
Si
Ion
s/NDF
Si Mg Na
1.61 0.38 0.95
Ne O N B Be Li He H * *
2.35 0.89 0.92 1.02 1.02 0.62 0.31 1.55 * *
Material
GaAs
Ion
s/NDF
Ge Zn Cu
1.76 0.58 0.45
Ni Fe Mn Ca S Mg O Be He H
0.50 1.06 0.64 0.59 1.11 1.10 1.23 1.37 4.13 3.33
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Table 2 Table of parameters for heavy ions Parameter
Height
Carbon
M
a b c e p
0.5916 !0.5980 2.870 !0.00322 0.0
0.8462 !0.3560 62.370 !0.00295 0.0
h
a b c e
0.4944 !0.3993 4.108 0.0 ! 0.0183 0.406 2.94 0.0
!0.0992 0.0 1.0 0.005026
0.00086 !2.0096 3.18 0.00025
i
a b c
!0.06793 !43.304 0.7
0.0073 !0.241 10.09
!0.01442 !0.849 4.87
d
a b c
!0.031 !1.710 1.59
!0.0023 !0.138 5.63
!0.00311 !0.262 5.14
E
a b c e p
!294.18 keV 305.48 keV 12.98 19.637 keV 0.0
!679.63 keV 689.51 keV 16.92 35.042 keV 0.0
s
a b c e Z
!2.0 keV 0.0 keV 1.0 2.578 keV 0.0050 0.1209 3.80 0.0 4.0
0.0 0.2126 6.67 0.0 *
0.0318 6.225 2.0 !0.000396 *
Table 3 Table of the parameters for protons and a-particles Particle Height Carbon
Silicon
Gallium arsenide
p
M h i d E s
0.003 0.0269 !1.14 !1.14 0.13 0.36 !0.11 0.012 0.0395 keV 0.103 keV 0.2913 0.339
0.0081 !1.69 0.87 !0.099 0.19 keV 0.347
a
M h i d E s
0.396 0.170 0.011 !0.151 3.96 keV 0.075
0.359 !0.028 !0.178 !0.214 5.19 keV 0.189
0.427 !0.047 !0.201 !0.125 6.30 keV 0.1147
Silicon
Gallium arsenide
the di!erential NIEL to better than 15% over most of the energy range with discrepancies up to 30% for energies below 10 keV. This con"rms that our simulation program for light ions is giving reliable values of the displacement energies. It should be noted that in the data presented in Fig. 5 the cross sections calculated as described in Section 2.2 were used which include the e!ects of atomic screening by electrons. These e!ects are rather important but they are not included in the cross section used in Ref. [9]. On the other hand, we do not include the e!ects of spin in our cross sections, which are included in Ref. [9], because these e!ects are quite small. 6. Conclusions Monte Carlo calculations of the displacement energy or total NIEL for a wide variety of ions
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a polynomial is used [7] of the form
dE dx
"Ab#Bb#Cb#D (9) where A, B, C, D are derived from the four simultaneous equations obtained by equating (dE/dx) and d(dE/dx) /db at b and b for Eqs. (1), (2) and (9). This gives
A" (W #k! W) @ @ @ B" (3(b #b ) W!(b #2b ) W !*yk) @ @ @ W !@ @ W ) C" (b (b #2b )k#@ @ >@ @ @ @ D" [ (b y (3b !b )#b y (b !3b )) @ @ !b b (b k#b W )] @ where *b"b !b , y , y are the dE/dx values computed from Eqs. (2) and (1) at velocities b and b , respectively,*y"y !y , k" [2pC8 n(c ! @ A K ln(K A @A ))] and c "1/(1!b . '\@ Fig. 7. Comparison of the displacement energies from our Monte Carlo (solid lines) with other calculations (dashed lines) for ions with various values of A and Z in (a) gallium arsenide [3], (b) silicon [5], and (c) carbon [4].
in three semiconductor detector materials have been made using Lindhard theory. A simple parameterisation of the results is given. This parameterisation will simplify considerably future computations of the di!erential NIEL from fast particles such as pions, protons and neutrons. Acknowledgements We wish to thank Dr. T.J. Brodbeck for his assistance with "tting. Appendix. Ionisation loss at intermediate energies The intermediate region, in which neither the Bethe}Bloch formula (Eq. (1)) nor the Lindhard formula (Eq. (2)) is valid, is de"ned by the velocity range b )b)b , where b "Max[aZ / (1#aZ ), (2Z #1)/400], b "aZ /(1#aZ ) and a is the "ne structure constant. In this region
References [1] A. Chilingarov, H. Feick, E. Fretwurst, G. Lindstrom, S. Roe, T. Schulz, Nucl. Instr. and Meth. A 360 (1995) 432. [2] A. Chilingarov, J.S. Meyer, T. Sloan, Nucl. Instr. and Meth. A 395 (1997) 35. [3] E.A. Burke et al., IEEE Trans. Nucl. Sci. NS-34 (1987) 1220. [4] I. Lazanu, S. Lazanu, E. Borchi, M. Bruzzi, Nucl. Instr. and Meth. A 406 (1998) 259. [5] G.W. Simons, J.M. Denney, R.G. Downing, Phys. Rev. 129 (1963) 2454. [6] J. Lindhard, V. Nielsen, M. Schar!, P.V. Thomsen, Mat. Fys. Medd. Dan. Vid. Selsk. 33 (10) (1963) 1. [7] J.D. Lewin, Range-energy relations for hypothetical charged particles, Rutherford Laboratory, RL-77 126/A (1977). [8] M.A. Kumakhov, F.F. Komarov, Energy loss and ion ranges in solids, Gordon and Breach, London, 1981, pp. 21}38. [9] G.P. Summers, E.A. Burke, P. Shapiro, S.R. Messenger, R.J. Walters, IEEE Trans. Nucl. Sci. NS-40 (1993) 1372. [10] J.W. Corbett, N.B. Urli, Radiation E!ects in Semiconductors, 1976, Institute of Physics, Conference Series No. 31, 1977. [11] A.L. Barry, R. Maxseiner, R. Wojcik, M.A. Briere, D. Braeunig, IEEE Trans. Nucl. Sci. NS-37 (1990) 1726. [12] J.F. Ziegler, J.P. Biersack, U. Littmark, The Stopping and Range of Ions in Solids, Vol. 1, Pergamon Press, New York, 1985.