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Determination of the coefficients of the Pearson distribution describing the concentration of implanted dopants in metals
Materials Science and Engineering, 69 (1985) 21-23
21
Determination of the Coefficients of the Pearson Distribution Describing the Concentration of Implanted Dopants in Metals* J. MARTAN and A. MULAK
Technical University of Wroc~aw, ul. Janiszewskiego 11/1 7, 50-372 Wroc~aw (Poland) (Received September 17, 1984)
ABSTRACT
The real profile o f implanted dopants in a target differs from a symmetrical gaussian distribution and shows both skewness and kurtosis. The distribution can be approximated by, for example, a Pearson distribution. Determination o f the coefficients in the Pearson function requires knowledge o f the four m o m e n t s o f the function. These m o m e n t s can be evaluated only by numerical calculations. In this paper a simple method for the calculation o f these m o m e n t s is proposed. It is assumed that the ions dissipate their energy only in elastic collisions. This assumption allows analytical formulae for the m o m e n t s o f the distribution function to be calculated. Good agreement with the results obtained by other researchers for Rp and A Rp was obtained. Only the value o f ~ differs from the value given by other researchers. The simple formula (only two parameters) which was used for the approximation o f Sn'(e) can explain this difference.
1. INTRODUCTION K n o w l e d g e o f the d o p a n t c o n c e n t r a t i o n d i s t r i b u t i o n in m e t a l targets is necessary for analysis o f t h e i n f l u e n c e of ion i m p l a n t a t i o n o n metal properties. A c c o r d i n g to the classical t h e o r y o f i n t e r a c t i o n of ions with solid targets (the Lindhard-Scharff-Schi¢,btt t h e o r y ) the c o n c e n t r a t i o n profile is gaussian. E x p e r i m e n t s a n d M o n t e Carlo n u m e r i c a l s i m u l a t i o n s h o w t h a t t h e gaussian f u n c t i o n does n o t always give a precise description. T h e real d i s t r i b u t i o n s s h o w b o t h skewness
a n d kurtosis. T o take these f a c t o r s into consideration it is necessary t o use m o r e complic a t e d distributions, e.g. the Pearson distribution. T o calculate this d i s t r i b u t i o n the first f o u r m o m e n t s o f the f u n c t i o n d i s t r i b u t i o n s h o u l d be k n o w n . In this p a p e r a simple calculation m e t h o d for these m o m e n t s is proposed. T h e ion energy and the ratio o f the ion mass t o the target mass are c h o s e n such t h a t t h e t o t a l ion energy loss is due t o nuclear stopping. I o n i m p l a n t a t i o n m e t a l l u r g y falls w i t h i n this regime.
2. THEORY T h e Pearson d i s t r i b u t i o n is described by the f u n c t i o n f(x) [1] given b y
T h e c o n s t a n t K is o b t a i n e d f r o m the constraint
f f(x) dx = 1 A k n o w l e d g e o f the central m o m e n t s p , (n = 1, 2, 3, 4) o f t h e f u n c t i o n d i s t r i b u t i o n allows the c o n s t a n t s a, q a n d v t o be calculated.
q=
3(~ -- ~ -- 1)
+1
2fi - - 3"), - - 6
w h e r e ~ = p4/P2 ~ (kurtosis). r( r - - 2)~,I/2
( 1 6 ( r - - I ) - - ~ ( r - - 2)2} '/2 w h e r e ~i/2 = p3/p23/2 (skewness).
*Paper presented at the International C o n f e r e n c e on Surface M o d i f i c a t i o n o f Metals by Ion Beams, Heidelberg, F.R.G., September 17-21, 1984. 0025-5416/85/$3.30
a =
16(r -- 1) -- 3'(r -- 2) 2
(D Elsevier Sequoia/Printed in The N e t h e r l a n d s
22 where r = 2q -- 2. According to Gibbons [2],
Z1Z2e2(MI + M2)
= 2.8 + 2.472 In order to calculate the values of the central moments p , it is necessary to know the noncentral moments m, (n = 1, 2, 3, 4) [1]. On the basis of the theory of random movement [3, 4] and using methods of statistical mathematics the m o m e n t s mn can be obtained in the following way:
Ei exp(--2v)
dE
(1)
E o
Eo exp(--2T2) . ? e x p ( + 2rl) me = 2! ~P-;2J ~ dE1 f Sn(E2) o E2 (2) , Ei exp(--2rs)
.j
m3=3
0
E0 exp(+ 2r2)
(3)
Eo E0 m 4 : 4! ~Iexp(--gT4) dE4 Jv~./e~l_,~:3 ) d E 3 × J ~n (~-J3) 2 Sn(E4) o E4 E o
Ef e x p ( - 2 v 2 ) dE2 ~exp(+2~'l) dE1 (4) ×J ~ J Sn(E1) E3 E2 where
U,n( _l T = -~ \Eo] M2 Ma Sn(E) is the nuclear stopping power, M1 and M2 are the atomic masses of the ion and the target material, Eo is the initial energy of the ion and E is the ion energy. The stopping power can be written as
where
e =PE
D =
a:
47raZ1Z2e2M1N M1 + M2 0.8853ao (Z12/3 -~ Z22/3)1/2
ao is the Bohr radius (5.3 × 10 -9 cm), Z1 and Z2 are the atomic numbers of the ion and the target material and N is the atomic density of the target material. If {S,'(e)} -1 is approximated by the function Ae B and then eqns. (1)-(4) are integrated, the moments rn. can be calculated. For A = 1.66 and B = -- 0.216 the approximation accuracy is about 6% for e in the range 0.01 ~< e ~< 0.2. Hence
/Ttl --
A (u/2 + B + 1)DP
e0B+I
m2 --
mlA B+I e0 DP(B + 1)
m3
3!Am2 2(3B + u/2 + 3)DP e°B+l
E3
~(exp(--2~-l) dE1 × J Sn(E1) E2
Sn(E) = DSn'(e)
aM2
p=
As the values p , are known, it is now possible to describe the following characteristic parameters of the distribution of the implanted ions: the mean projected range of ions is given by
Rp =/21 the range straggling is given by ARp = p21/2 the skewness is given by
/~32 7
p23/2
and the kurtosis is given by
g4 p22
3. DISCUSSION The results calculated with the help of this technique have been compared with the results of other researchers [1]. Values of Rp, ARp and 7 for gold ions implanted into copper in the energy range 1 5 - 3 0 0 keV (corre-
23
ARp
Rp
iJk) 150
300
t / I 200
100
100
50
r"
I
/, I
J
r" i
L
i
l I
i
t I II
0,01
0.1
(a)
O.Ol
I
I
i
i i i i i
I
o.1
(b)
~, 1/z
fl
Q4. 0
o.
o
~.,
o.1
_1 ¸
-2-
(c) Fig. 1. The dependence of (a) the mean projected range Rp, (b) the range straggling ARp and (c) the skewness 7 on the reduced energy e for implantation of gold ions into copper: , from ref. 1 ; - - - , present work.
s p o n d i n g to decreasing e in the range 0.01 e ~ 0.2) were c o m p a r e d . T h e results o f the calculations are p r e s e n t e d in Fig. 1. F o r example, the c o n s t a n t s a, q and v were calculated and the following values were o b t a i n e d : a = 1 . 2 1 X 1 0 -2, q = 2.55 and u = 0.44 f o r E -- 200 keV. It can be seen t h a t o n l y 7 differs f r o m the results o b t a i n e d b y o t h e r researchers. T h e f a c t t h a t the a p p r o x i m a t i n g f u n c t i o n c o n t a i n s o n l y t w o p a r a m e t e r s A and B and this limits the exactness o f our calculations (particularly for t h e higher m o m e n t s ) s h o u l d be t a k e n into a c c o u n t . T h e m e t h o d , h o w e v e r , allows for simple and fast e s t i m a t i o n o f the c h a r a c t e r o f the c o n c e n t r a t i o n distribut i o n o f various d o p a n t s in a target material. T h e calculations m a y be p e r f o r m e d using a
simple p o c k e t calculator and no n u m e r i c a l calculations are necessary.
REFERENCES 1 A. Burenkov, F. Komarov, A. Kumahov and M. Temkin, Tablicy Parametrov Prostranstvennovo
Razpredelenia Ionno-implantirovanyh Primesej, Lenina, Minsk, 1980. 2 J. F. Gibbons, in T. S. Moss (ed.), Handbook on Semiconductors, Vol. 3, North-Holland, Amsterdam, 1980. 3 H. Ryssel and H. Glawischnig, Ion Implantation Techniques, Springer, Berlin, 1982. 4 M. Kac, Some Stochastics Problems in Physics and Mathematics, Magnolia Petroleum Company, Dallas, TX, 1957.
21
Determination of the Coefficients of the Pearson Distribution Describing the Concentration of Implanted Dopants in Metals* J. MARTAN and A. MULAK
Technical University of Wroc~aw, ul. Janiszewskiego 11/1 7, 50-372 Wroc~aw (Poland) (Received September 17, 1984)
ABSTRACT
The real profile o f implanted dopants in a target differs from a symmetrical gaussian distribution and shows both skewness and kurtosis. The distribution can be approximated by, for example, a Pearson distribution. Determination o f the coefficients in the Pearson function requires knowledge o f the four m o m e n t s o f the function. These m o m e n t s can be evaluated only by numerical calculations. In this paper a simple method for the calculation o f these m o m e n t s is proposed. It is assumed that the ions dissipate their energy only in elastic collisions. This assumption allows analytical formulae for the m o m e n t s o f the distribution function to be calculated. Good agreement with the results obtained by other researchers for Rp and A Rp was obtained. Only the value o f ~ differs from the value given by other researchers. The simple formula (only two parameters) which was used for the approximation o f Sn'(e) can explain this difference.
1. INTRODUCTION K n o w l e d g e o f the d o p a n t c o n c e n t r a t i o n d i s t r i b u t i o n in m e t a l targets is necessary for analysis o f t h e i n f l u e n c e of ion i m p l a n t a t i o n o n metal properties. A c c o r d i n g to the classical t h e o r y o f i n t e r a c t i o n of ions with solid targets (the Lindhard-Scharff-Schi¢,btt t h e o r y ) the c o n c e n t r a t i o n profile is gaussian. E x p e r i m e n t s a n d M o n t e Carlo n u m e r i c a l s i m u l a t i o n s h o w t h a t t h e gaussian f u n c t i o n does n o t always give a precise description. T h e real d i s t r i b u t i o n s s h o w b o t h skewness
a n d kurtosis. T o take these f a c t o r s into consideration it is necessary t o use m o r e complic a t e d distributions, e.g. the Pearson distribution. T o calculate this d i s t r i b u t i o n the first f o u r m o m e n t s o f the f u n c t i o n d i s t r i b u t i o n s h o u l d be k n o w n . In this p a p e r a simple calculation m e t h o d for these m o m e n t s is proposed. T h e ion energy and the ratio o f the ion mass t o the target mass are c h o s e n such t h a t t h e t o t a l ion energy loss is due t o nuclear stopping. I o n i m p l a n t a t i o n m e t a l l u r g y falls w i t h i n this regime.
2. THEORY T h e Pearson d i s t r i b u t i o n is described by the f u n c t i o n f(x) [1] given b y
T h e c o n s t a n t K is o b t a i n e d f r o m the constraint
f f(x) dx = 1 A k n o w l e d g e o f the central m o m e n t s p , (n = 1, 2, 3, 4) o f t h e f u n c t i o n d i s t r i b u t i o n allows the c o n s t a n t s a, q a n d v t o be calculated.
q=
3(~ -- ~ -- 1)
+1
2fi - - 3"), - - 6
w h e r e ~ = p4/P2 ~ (kurtosis). r( r - - 2)~,I/2
( 1 6 ( r - - I ) - - ~ ( r - - 2)2} '/2 w h e r e ~i/2 = p3/p23/2 (skewness).
*Paper presented at the International C o n f e r e n c e on Surface M o d i f i c a t i o n o f Metals by Ion Beams, Heidelberg, F.R.G., September 17-21, 1984. 0025-5416/85/$3.30
a =
16(r -- 1) -- 3'(r -- 2) 2
(D Elsevier Sequoia/Printed in The N e t h e r l a n d s
22 where r = 2q -- 2. According to Gibbons [2],
Z1Z2e2(MI + M2)
= 2.8 + 2.472 In order to calculate the values of the central moments p , it is necessary to know the noncentral moments m, (n = 1, 2, 3, 4) [1]. On the basis of the theory of random movement [3, 4] and using methods of statistical mathematics the m o m e n t s mn can be obtained in the following way:
Ei exp(--2v)
dE
(1)
E o
Eo exp(--2T2) . ? e x p ( + 2rl) me = 2! ~P-;2J ~ dE1 f Sn(E2) o E2 (2) , Ei exp(--2rs)
.j
m3=3
0
E0 exp(+ 2r2)
(3)
Eo E0 m 4 : 4! ~Iexp(--gT4) dE4 Jv~./e~l_,~:3 ) d E 3 × J ~n (~-J3) 2 Sn(E4) o E4 E o
Ef e x p ( - 2 v 2 ) dE2 ~exp(+2~'l) dE1 (4) ×J ~ J Sn(E1) E3 E2 where
U,n( _l T = -~ \Eo] M2 Ma Sn(E) is the nuclear stopping power, M1 and M2 are the atomic masses of the ion and the target material, Eo is the initial energy of the ion and E is the ion energy. The stopping power can be written as
where
e =PE
D =
a:
47raZ1Z2e2M1N M1 + M2 0.8853ao (Z12/3 -~ Z22/3)1/2
ao is the Bohr radius (5.3 × 10 -9 cm), Z1 and Z2 are the atomic numbers of the ion and the target material and N is the atomic density of the target material. If {S,'(e)} -1 is approximated by the function Ae B and then eqns. (1)-(4) are integrated, the moments rn. can be calculated. For A = 1.66 and B = -- 0.216 the approximation accuracy is about 6% for e in the range 0.01 ~< e ~< 0.2. Hence
/Ttl --
A (u/2 + B + 1)DP
e0B+I
m2 --
mlA B+I e0 DP(B + 1)
m3
3!Am2 2(3B + u/2 + 3)DP e°B+l
E3
~(exp(--2~-l) dE1 × J Sn(E1) E2
Sn(E) = DSn'(e)
aM2
p=
As the values p , are known, it is now possible to describe the following characteristic parameters of the distribution of the implanted ions: the mean projected range of ions is given by
Rp =/21 the range straggling is given by ARp = p21/2 the skewness is given by
/~32 7
p23/2
and the kurtosis is given by
g4 p22
3. DISCUSSION The results calculated with the help of this technique have been compared with the results of other researchers [1]. Values of Rp, ARp and 7 for gold ions implanted into copper in the energy range 1 5 - 3 0 0 keV (corre-
23
ARp
Rp
iJk) 150
300
t / I 200
100
100
50
r"
I
/, I
J
r" i
L
i
l I
i
t I II
0,01
0.1
(a)
O.Ol
I
I
i
i i i i i
I
o.1
(b)
~, 1/z
fl
Q4. 0
o.
o
~.,
o.1
_1 ¸
-2-
(c) Fig. 1. The dependence of (a) the mean projected range Rp, (b) the range straggling ARp and (c) the skewness 7 on the reduced energy e for implantation of gold ions into copper: , from ref. 1 ; - - - , present work.
s p o n d i n g to decreasing e in the range 0.01 e ~ 0.2) were c o m p a r e d . T h e results o f the calculations are p r e s e n t e d in Fig. 1. F o r example, the c o n s t a n t s a, q and v were calculated and the following values were o b t a i n e d : a = 1 . 2 1 X 1 0 -2, q = 2.55 and u = 0.44 f o r E -- 200 keV. It can be seen t h a t o n l y 7 differs f r o m the results o b t a i n e d b y o t h e r researchers. T h e f a c t t h a t the a p p r o x i m a t i n g f u n c t i o n c o n t a i n s o n l y t w o p a r a m e t e r s A and B and this limits the exactness o f our calculations (particularly for t h e higher m o m e n t s ) s h o u l d be t a k e n into a c c o u n t . T h e m e t h o d , h o w e v e r , allows for simple and fast e s t i m a t i o n o f the c h a r a c t e r o f the c o n c e n t r a t i o n distribut i o n o f various d o p a n t s in a target material. T h e calculations m a y be p e r f o r m e d using a
simple p o c k e t calculator and no n u m e r i c a l calculations are necessary.
REFERENCES 1 A. Burenkov, F. Komarov, A. Kumahov and M. Temkin, Tablicy Parametrov Prostranstvennovo
Razpredelenia Ionno-implantirovanyh Primesej, Lenina, Minsk, 1980. 2 J. F. Gibbons, in T. S. Moss (ed.), Handbook on Semiconductors, Vol. 3, North-Holland, Amsterdam, 1980. 3 H. Ryssel and H. Glawischnig, Ion Implantation Techniques, Springer, Berlin, 1982. 4 M. Kac, Some Stochastics Problems in Physics and Mathematics, Magnolia Petroleum Company, Dallas, TX, 1957.