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An analytical formula for high energy ion ranges
Materials Science and Engineering, B I 5 (1992) 187-191
187
An analytical formula for high energy ion ranges J. Martan*, N. H e c k i n g t a n d W. R. F a h r n e r Chair of Electronic Devices, Fern Universitiit, PO Box 940, W-5800 Hagen (Germany) (Received June 10, 1992)
Abstract Analytical expressions for ion ranges and range straggles of ions implanted into a solid state target are presented. The calculations are extended to the high energy case (up to about 1 GeV). It is assumed that the ions lose their energy only by electronic collisions as described by the Bethe and Lindhart-Scharff-SchiStt formulae. Analytical results of calculations are compared with experimental data and other numerical computations (TRIM, PRAL and MARLOWE codes). Good agreement is obtained.
1. Introduction
2. Ion-target interaction at high energies
High energy ion implantation is beginning to play an increasing role not only in semiconductor technology but also in other fields of material processing (ion implantation in metals and insulators) [1]. Megaelectronvolt (MeV) ions help enhance and reduce respectively diffusion processes in the production of n-type metal-oxide-semiconductor (n-MOS) and complementary MOS (CMOS) devices, simulate singleevent upsets [2] and improve the radiation hardness of electronic devices [3]. MeV implantation combined with the effect of radiation-enhanced annealing is a means of avoiding post-implantation annealing [4]. There is currently widespread interest in the use of high energy ion beams for simulating material behaviour in fission and fusion reactors (see e.g. ref. 5). Production of hard silicon-carbide-like material during MeV implantation was recently observed [6]. The nature of the interaction between fast ions and solids differs considerably from that of slow ions. In the first case inelastic collisions dominate while the second case is controlled by nuclear interactions. The path of high energy projectiles is nearly straight with small angle deflections. The concentration profiles are asymmetric with a characteristic central peak. Analytical expressions for the mean values of the total paths R and their straggles o are presented in this paper.
The slowing down of ions in solids (target material) is due mainly to elastic and inelastic collisions with target atoms. In this paper we focus only on the high energy regime; hence the inelastic (high energy) collisions combined with their electronic stopping powers are examined. To analyse this problem, it is convenient to divide the energy of ions into three ranges: high energy range (orbital velocity of an ion shell electron is lower than the ion velocity), low energy range (orbital velocity of an ion shell electron is higher than the ion velocity) and intermediate range. Energy losses of high energy ions are described by the Bethe formula [7] s
-f z 2 N l n - meVl
where N is the atomic density of the target material, zl and z2 are the atomic numbers of the ion and target material respectively, v~ is the velocity of the ion, e is the electron charge, rne is the electron mass, I= eRz2 is the mean ionization energy according to the ThomasFermi atom model and eR = 11.5 eV is the Rydberg energy. Expression (1) can be presented in a form more convenient for analytical calculations: NA
ScB= T * On leave from: Politechnika Wroclawska Instytut Technologii Elektronowej, PL-50-372 Wroclaw, Poland. tPresent address: Deutsche Agentur fiir Raumfahrtangelegenheiten G.m.b.H., Koblenzer Str. 112, W-5300 Bonn, Germany. 0921-5107/92/$5.00
(1)
In(bE)
(2)
where
A = 2 . 3 7 x 106 Z2Zl2MD b = 1.922 x 10 -4 (1/MlZ2) , M 1 is the atomic mass of the ion, E (eV) is the energy of the ion and N (at. ]k-a) is the atomic density of the target material. © 1992 - Elsevier Sequoia. All fights reserved
J. Martan et al.
188
/
A nalytical Jormula Jbr iott ranges
For the low energy regime the electronic stopping power is described by the well-known LindhardtScharff-Schiott (LSS) formula (3)
Set.ss= N C K K E 1/2
Y, Yl
38.3ztT/6z2
/ / /
100
Y1--- ~ " / / ~ " ~ " , , ..~" "y.~
1
1
-¢
The mean total range value is approximately the same as the projected range, R ~- Rp: E
R=~
dE' So(K)
(5)
Inserting (2) and (3) into (4), we find
R =~
E
6041320200
400
600
800
1000
X
Fig. 1. Graph of function y =x/lnx and its approximation.
(4)
& &Lss Se.
1 !
z -
80
and c~ is a correction factor. Varelas and Biersack [8] have proposed an expression to describe the electronic stopping power for an Intermediate range of ion energy, namely a sum of the reciprocals of eqns. (2) and (3):
1
C8;'7 +1,84
120
K =(z12/3 +z22/3)3/2MI 1/2103/2
-
Y~= 0,037x
---
160
X--L-[nX
140
where
1
y=
l
dE'
E
1
c~KE, U2
dE'
o A In(bE')
(6)
The feature of high energy ion profiles is a central peak. It is caused by an electronic energy loss straggling, which, especially for energetic light ions, exceeds the nuclear energy loss straggling by far. It is a general observation that for heavy ions with reduced energy e ~, 1 and with S~ > SN the correlation Qe > QN is also true, where Qe and QN are the electronic and nuclear energy loss straggling respectively. The range straggle is calculated (for Se > SN) on the basis of the relation E
2 ( Q~+ QN or
O" =
J
Se 3
dE
(10)
0 2 E 1/2 =
1 +
R Nc~K ~ 0
(
z'
J 7--7 dz'
lnz
(7)
where z = bE. The function y=zllnz can be approximated for z in the range 2.72~
For high energy ions both Q~ and QN are independent of energy and are described by the formulae [9, 10] Qe = 4~z12z2 e4N Z2
QN= Q~ (1 + "' 2 m " ' /1m )
Taking into account expressions (8 ) and ( 10 ),(12), it can be shown that
2 Qe+QN i (_~b(O.337zO.877+1.84)
2.72
0
(12)
(8)
The approximation quality in this range of z is better than 5%. Figure 1 presents the graphs of functions Yl and y. Taking into account that f
(11)
2.72
z dz=3.0535 Inz
b 1/2 13 + cK--K~zla] dz
we find the final analytical equation from expressions (7) and (8) for the mean total path of high energetic ions as 2E t/2
1
R = NcrK +,ZT~,21vao[0"1795(bE)L877 + 1.84(bE)-3.126] (9)
(13)
It is assumed that bEmin = 2.72 (the range 0-2.72 can be ignored for very high energy). On the basis of eqn. (13) we can write a 2-" Q Qe+~ b
(zl +z2 +z3 +z4)
(14)
J. Martan et aL / Analytical formula for ion ranges
189
I
~urnl
I000 I
p31 ------ Si
100 -----
this work (eq.9 .c k =1,35)
[ 11,12,13] \
10
1,0
"[16.17] 1~5] 0,1
IL
10~
105
106
107
1~
E {eV]
Fig. 2. Results of range calculations for P* ions implanted in a silicon target compared with experimental data and numerical computationsfor various energies.
where
3. Results and discussion
1 z] --A---~ (1"0510-2z3'631 + 0"2277z2'754+ l'8237zlS77
In Figs. 2 and 3 the results of range calculations for B + and P+ ions into a silicon target are compared with experimental [11] and numerical (TRIM [12] and M A R L O W E [13]) data for different energies. It is seen that for high energies the agreement with the analytical calculations is very good. Differences appear at lower energies where the nuclear stopping power begins to play an important role. Experimental and calculated ranges coincide at energies above 10 MeV for phosphorus and above 1 MeV for boron. This reflects the fact that the electronic stopping power dominates over the nuclear one at much lower energies for lighter ions. A last remark is required concerning the range straggle. The straggle of the projected range is larger than that of the total path length. The large energy transfers between colliding particles reduce the total
+ 6.23z - 32.847) 1 Z2 = A2b3/2cr K
(0.15z 2'254+ 2.7z 1'377
+ 20.28zt/2 _ 45.587) 1
z3-A(crK)~ (1.15Z °'877+ 5.52 lnz -8.286)
z4=(crK)3
1.21
2
J. Martan et al.
190
/
Analytical formula for ion ranges
[pm]
10000
[18]
811
1000
[11,12] ~/~[13]
[221
100' -----
this work (eq g,ck = 1.35)
x~-.119]
10"
[18]
[16]
[15]
[16,171
164
10s
106
10T
100
E [eV]
=-
Fig. 3. Results of range calculations for B ÷ ions implanted in a silicon target compared with experimental data and numerical
computations for various energies. TABLE 1. Total range-straggling values achieved with the help of PRAL and TRIM codes and eqns. (13) and (14). At low energies no data are given for boron and arsenic since SN > Se System and energy (MeV)
o (#m) from PRAL [14]
0.48 14.6 219 822 4100
o (/~m) from Qe + QN, eqn. (10)[14]
o (#m) experiment
Eqn.(14)
Eqn, (13)
0.35 9.69 160.3 554.9 1935
0.35 10.2 161 549 1903
1 10 50 100 200
B ÷ " Si
1 10 50 100 200
0.1823 0.4637 3.82 11.95 39.47
0,174 0.252 1.26 2.07 12.1
0.08 0.1338 0.6146 1.693 5.1
0.108 0.58 1.65 5.25
0.1069 0.6146 1.74 5.43
1 10 50 100 200
0.1249 0.3538 0.486 0.61 0.9
0.159 0.425
0.1657 0.3716 0.41 0.42 0.45
0.25 0.3 0.35
0.2516 0.303 0.35
0.503 0.529
0.3 9.95 143.5 463.3 1514
This work
H ÷ ~ Si
As ÷ -*Si
0.83 34.23 574.7 1971 6827
o (/~m) from TRIM [12]
0.145 [241
0.4 [25]
J. Martan et al.
/
Analytical formula for ion ranges
ion range and the projection on the original direction of flight by the corresponding direction cosine [14]. Hence larger values of straggling are obtained from e.g. the P R A L and T R I M codes than from Qe and QN alone. Table 1 presents a comparison of the total range-straggling results computed by means of P R A L [14] and T R I M [12] codes with the results achieved on the basis of eqn. (14).
4. Conclusions The main advantage of our approach is seen in its speed of calculation. Neither computers nor programmable calculators are required. A simple calculator with the power ("y") function is sufficient. T h e time required to calculate the boron ranges according to eqn. (13) is about 1 s. This should be compared with the 2 h required for the same problem when the T R I M 2D code is used. Our approach holds for all ion-target systems. However, eqns. (9) and (14) become inaccurate at lower energies. The critical energy is given by the condition S e = S N. Above this energy the agreement with other calculations and with experiments is better than 4%.
Acknowledgments The authors are grateful to Dipl.-Phys. D. Borchert, University of Hagen for discussions and performing T R I M simulations.
References 1 D.C. Ingrain, NucL Instrum. Methods B, 12 (1985) 161. 2 A.B. Campbell, W. J. Stapor, R. Koga and W. A. Kolasinski, IEEE Trans. Nucl. Sci., NS-32 (1958)4150. 3 M. R. Wordeman, R. H. Dermard and G. A. Sai-Malasz, Int. Electron Dev. Meet., Washington, DC, 7-9 December, 1981, IEEE, New York, 1981.
191
4 R. G. Elliman, S. T. Johnson, J. S. Williams and A, P. Pogary, Nucl. Instrum. Methods B, 7-8 ( 1985) 310. 5 Proc. llth. Int. Syrup. on Effects of Radiation on Materials, Scotsdale, AZ, 28-30 June, 1982, ASTM, Philadelphia, PA, 1982, 48 pp. 6 T. Venkalesan, T. Wolf, D. Allora, B. J. Wilkens and G. N. Taylor, Appl. Phys. Lett., 43(1983)934. 7 H.A. Bethe, Ann. Phys. (Lpz.), 5 (1930) 325. 8 C. Varelas and J. Biersack, Nucl. lnstrum. Methods, 79 (1970) 213. 9 N. Bohr, K. Dan. Vidensk. Selsk. Mat. Fys. Meddr., 18 (8) (1948) 1. 10 J. F. Ziegler, J. Biersack and U. Littmark, Stopping and Ranges of Ions in Matter, Pergamon, New York, 1985. 11 W. R. Fahrner, K. G. Oppermann and T. Harms, Phys. Status Solidi A, 123(1991) 109. 12 J. P. Biersack and L. G. Haggmark, Nucl. Instrum. Methods, 174(1980) 257. 13 M. T. Robinson and I. M. Torrens, Phys. Rev. B, 9 (1974) 5008. 14 J.P. Biersack, Nucl. lnstrum. Methods B, 35(1988)205. 15 H. Ryssel and I. Ruge, Ion Implantation, Teubner, Stuttgart, 1978. 16 B. Smith, Ion Implantation Range Data for Silicon and Germanium Device Technologies, Research Studies Press, Forest Grove, OR, 1977. 17 J. F. Gibbons, W. S. Johnson and S. W. Mylroie, Projected Range Statistics, Dowden, Hutchinson and Ross, Stroudsburg, PA, 1975. 18 U. Littmark and J. E Ziegler, in J. E Ziegler (ed.), Range Distribution for Energetic Ions in All Elements, Vol. 6, Pergamon, New York, 1980, p. 153. 19 E W. Martin, Nucl. Instrum. Methods, 72 (1969) 223. 20 J. R. Ferretti, W. R. Fahrner and D. Br~iunig, IEEE Trans. Nucl. Sci., NS-26 (1979) 4828. 21 W. R. Fahrner, K. Heidemann and P. Schfttle, Phys. Status Solidi, 70 (1982) 463. 22 K. G. Oppermaim and W. R. Fahrner, in G. Bentini (ed.), Proc. MRS Conf., Strasbourg, May-June 1988, Elsevier Sequoia, Lausanne, 1989, p. 75. 23 N. La Ferla, A. Di Franeo, E. Rimini, G. Ciavola and G. Ferla, Mater. Sci. Eng. B, 2 (1989) 69. 24 L. Wang, Thesis, Free University of Berlin, 1990, p. 142, 25 H. Wong, E. Deng, N. W. Cheung, P. K. Chu, E. M. Strathman and M. D. Strathman, Nucl. lnstrum. Methods B, 21 (1987) 447.
187
An analytical formula for high energy ion ranges J. Martan*, N. H e c k i n g t a n d W. R. F a h r n e r Chair of Electronic Devices, Fern Universitiit, PO Box 940, W-5800 Hagen (Germany) (Received June 10, 1992)
Abstract Analytical expressions for ion ranges and range straggles of ions implanted into a solid state target are presented. The calculations are extended to the high energy case (up to about 1 GeV). It is assumed that the ions lose their energy only by electronic collisions as described by the Bethe and Lindhart-Scharff-SchiStt formulae. Analytical results of calculations are compared with experimental data and other numerical computations (TRIM, PRAL and MARLOWE codes). Good agreement is obtained.
1. Introduction
2. Ion-target interaction at high energies
High energy ion implantation is beginning to play an increasing role not only in semiconductor technology but also in other fields of material processing (ion implantation in metals and insulators) [1]. Megaelectronvolt (MeV) ions help enhance and reduce respectively diffusion processes in the production of n-type metal-oxide-semiconductor (n-MOS) and complementary MOS (CMOS) devices, simulate singleevent upsets [2] and improve the radiation hardness of electronic devices [3]. MeV implantation combined with the effect of radiation-enhanced annealing is a means of avoiding post-implantation annealing [4]. There is currently widespread interest in the use of high energy ion beams for simulating material behaviour in fission and fusion reactors (see e.g. ref. 5). Production of hard silicon-carbide-like material during MeV implantation was recently observed [6]. The nature of the interaction between fast ions and solids differs considerably from that of slow ions. In the first case inelastic collisions dominate while the second case is controlled by nuclear interactions. The path of high energy projectiles is nearly straight with small angle deflections. The concentration profiles are asymmetric with a characteristic central peak. Analytical expressions for the mean values of the total paths R and their straggles o are presented in this paper.
The slowing down of ions in solids (target material) is due mainly to elastic and inelastic collisions with target atoms. In this paper we focus only on the high energy regime; hence the inelastic (high energy) collisions combined with their electronic stopping powers are examined. To analyse this problem, it is convenient to divide the energy of ions into three ranges: high energy range (orbital velocity of an ion shell electron is lower than the ion velocity), low energy range (orbital velocity of an ion shell electron is higher than the ion velocity) and intermediate range. Energy losses of high energy ions are described by the Bethe formula [7] s
-f z 2 N l n - meVl
where N is the atomic density of the target material, zl and z2 are the atomic numbers of the ion and target material respectively, v~ is the velocity of the ion, e is the electron charge, rne is the electron mass, I= eRz2 is the mean ionization energy according to the ThomasFermi atom model and eR = 11.5 eV is the Rydberg energy. Expression (1) can be presented in a form more convenient for analytical calculations: NA
ScB= T * On leave from: Politechnika Wroclawska Instytut Technologii Elektronowej, PL-50-372 Wroclaw, Poland. tPresent address: Deutsche Agentur fiir Raumfahrtangelegenheiten G.m.b.H., Koblenzer Str. 112, W-5300 Bonn, Germany. 0921-5107/92/$5.00
(1)
In(bE)
(2)
where
A = 2 . 3 7 x 106 Z2Zl2MD b = 1.922 x 10 -4 (1/MlZ2) , M 1 is the atomic mass of the ion, E (eV) is the energy of the ion and N (at. ]k-a) is the atomic density of the target material. © 1992 - Elsevier Sequoia. All fights reserved
J. Martan et al.
188
/
A nalytical Jormula Jbr iott ranges
For the low energy regime the electronic stopping power is described by the well-known LindhardtScharff-Schiott (LSS) formula (3)
Set.ss= N C K K E 1/2
Y, Yl
38.3ztT/6z2
/ / /
100
Y1--- ~ " / / ~ " ~ " , , ..~" "y.~
1
1
-¢
The mean total range value is approximately the same as the projected range, R ~- Rp: E
R=~
dE' So(K)
(5)
Inserting (2) and (3) into (4), we find
R =~
E
6041320200
400
600
800
1000
X
Fig. 1. Graph of function y =x/lnx and its approximation.
(4)
& &Lss Se.
1 !
z -
80
and c~ is a correction factor. Varelas and Biersack [8] have proposed an expression to describe the electronic stopping power for an Intermediate range of ion energy, namely a sum of the reciprocals of eqns. (2) and (3):
1
C8;'7 +1,84
120
K =(z12/3 +z22/3)3/2MI 1/2103/2
-
Y~= 0,037x
---
160
X--L-[nX
140
where
1
y=
l
dE'
E
1
c~KE, U2
dE'
o A In(bE')
(6)
The feature of high energy ion profiles is a central peak. It is caused by an electronic energy loss straggling, which, especially for energetic light ions, exceeds the nuclear energy loss straggling by far. It is a general observation that for heavy ions with reduced energy e ~, 1 and with S~ > SN the correlation Qe > QN is also true, where Qe and QN are the electronic and nuclear energy loss straggling respectively. The range straggle is calculated (for Se > SN) on the basis of the relation E
2 ( Q~+ QN or
O" =
J
Se 3
dE
(10)
0 2 E 1/2 =
1 +
R Nc~K ~ 0
(
z'
J 7--7 dz'
lnz
(7)
where z = bE. The function y=zllnz can be approximated for z in the range 2.72~
For high energy ions both Q~ and QN are independent of energy and are described by the formulae [9, 10] Qe = 4~z12z2 e4N Z2
QN= Q~ (1 + "' 2 m " ' /1m )
Taking into account expressions (8 ) and ( 10 ),(12), it can be shown that
2 Qe+QN i (_~b(O.337zO.877+1.84)
2.72
0
(12)
(8)
The approximation quality in this range of z is better than 5%. Figure 1 presents the graphs of functions Yl and y. Taking into account that f
(11)
2.72
z dz=3.0535 Inz
b 1/2 13 + cK--K~zla] dz
we find the final analytical equation from expressions (7) and (8) for the mean total path of high energetic ions as 2E t/2
1
R = NcrK +,ZT~,21vao[0"1795(bE)L877 + 1.84(bE)-3.126] (9)
(13)
It is assumed that bEmin = 2.72 (the range 0-2.72 can be ignored for very high energy). On the basis of eqn. (13) we can write a 2-" Q Qe+~ b
(zl +z2 +z3 +z4)
(14)
J. Martan et aL / Analytical formula for ion ranges
189
I
~urnl
I000 I
p31 ------ Si
100 -----
this work (eq.9 .c k =1,35)
[ 11,12,13] \
10
1,0
"[16.17] 1~5] 0,1
IL
10~
105
106
107
1~
E {eV]
Fig. 2. Results of range calculations for P* ions implanted in a silicon target compared with experimental data and numerical computationsfor various energies.
where
3. Results and discussion
1 z] --A---~ (1"0510-2z3'631 + 0"2277z2'754+ l'8237zlS77
In Figs. 2 and 3 the results of range calculations for B + and P+ ions into a silicon target are compared with experimental [11] and numerical (TRIM [12] and M A R L O W E [13]) data for different energies. It is seen that for high energies the agreement with the analytical calculations is very good. Differences appear at lower energies where the nuclear stopping power begins to play an important role. Experimental and calculated ranges coincide at energies above 10 MeV for phosphorus and above 1 MeV for boron. This reflects the fact that the electronic stopping power dominates over the nuclear one at much lower energies for lighter ions. A last remark is required concerning the range straggle. The straggle of the projected range is larger than that of the total path length. The large energy transfers between colliding particles reduce the total
+ 6.23z - 32.847) 1 Z2 = A2b3/2cr K
(0.15z 2'254+ 2.7z 1'377
+ 20.28zt/2 _ 45.587) 1
z3-A(crK)~ (1.15Z °'877+ 5.52 lnz -8.286)
z4=(crK)3
1.21
2
J. Martan et al.
190
/
Analytical formula for ion ranges
[pm]
10000
[18]
811
1000
[11,12] ~/~[13]
[221
100' -----
this work (eq g,ck = 1.35)
x~-.119]
10"
[18]
[16]
[15]
[16,171
164
10s
106
10T
100
E [eV]
=-
Fig. 3. Results of range calculations for B ÷ ions implanted in a silicon target compared with experimental data and numerical
computations for various energies. TABLE 1. Total range-straggling values achieved with the help of PRAL and TRIM codes and eqns. (13) and (14). At low energies no data are given for boron and arsenic since SN > Se System and energy (MeV)
o (#m) from PRAL [14]
0.48 14.6 219 822 4100
o (/~m) from Qe + QN, eqn. (10)[14]
o (#m) experiment
Eqn.(14)
Eqn, (13)
0.35 9.69 160.3 554.9 1935
0.35 10.2 161 549 1903
1 10 50 100 200
B ÷ " Si
1 10 50 100 200
0.1823 0.4637 3.82 11.95 39.47
0,174 0.252 1.26 2.07 12.1
0.08 0.1338 0.6146 1.693 5.1
0.108 0.58 1.65 5.25
0.1069 0.6146 1.74 5.43
1 10 50 100 200
0.1249 0.3538 0.486 0.61 0.9
0.159 0.425
0.1657 0.3716 0.41 0.42 0.45
0.25 0.3 0.35
0.2516 0.303 0.35
0.503 0.529
0.3 9.95 143.5 463.3 1514
This work
H ÷ ~ Si
As ÷ -*Si
0.83 34.23 574.7 1971 6827
o (/~m) from TRIM [12]
0.145 [241
0.4 [25]
J. Martan et al.
/
Analytical formula for ion ranges
ion range and the projection on the original direction of flight by the corresponding direction cosine [14]. Hence larger values of straggling are obtained from e.g. the P R A L and T R I M codes than from Qe and QN alone. Table 1 presents a comparison of the total range-straggling results computed by means of P R A L [14] and T R I M [12] codes with the results achieved on the basis of eqn. (14).
4. Conclusions The main advantage of our approach is seen in its speed of calculation. Neither computers nor programmable calculators are required. A simple calculator with the power ("y") function is sufficient. T h e time required to calculate the boron ranges according to eqn. (13) is about 1 s. This should be compared with the 2 h required for the same problem when the T R I M 2D code is used. Our approach holds for all ion-target systems. However, eqns. (9) and (14) become inaccurate at lower energies. The critical energy is given by the condition S e = S N. Above this energy the agreement with other calculations and with experiments is better than 4%.
Acknowledgments The authors are grateful to Dipl.-Phys. D. Borchert, University of Hagen for discussions and performing T R I M simulations.
References 1 D.C. Ingrain, NucL Instrum. Methods B, 12 (1985) 161. 2 A.B. Campbell, W. J. Stapor, R. Koga and W. A. Kolasinski, IEEE Trans. Nucl. Sci., NS-32 (1958)4150. 3 M. R. Wordeman, R. H. Dermard and G. A. Sai-Malasz, Int. Electron Dev. Meet., Washington, DC, 7-9 December, 1981, IEEE, New York, 1981.
191
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