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The asymmetry of projected range distributions in amorphous solids
Nuclear Instruments and Methods in Physics Research B 82 (1993) 522-527 North-Holland
The asymmetry of projected range distributions
NM B
Beam Interactions with Materials 8 Atoms
in amorphous solids
M.Yu. Barabanenkov Institute of Microelectronics Technology and High Purity Materials, 142432 Chernogolouka, Moscow Region, Russian Federation
Received 6 February 1992 and in revised form 18 August 1992
Physical reasons are considered to explain the asymmetry of range distributions of implanted ions in amorphous solids. It is established that the appearance of negative asymmetry is due to the prevalence of ion electronic energy loss in the slowing down process. Nuclear stopping makes a positive contribution to the asymmetry value. A simple analytical expression for the asymmetry value is obtained. This allows extension of the possibilities of the well known program DIMUS to describe the asymmetrical range distributions of energetic particles.
2. Skewness analysis
1. Introduction
It is well known that, in general, the depth distribution of implanted ions in amorphous solids has an asymmetrical shape (see, e.g. ref. [l]). The asymmetry becomes apparent by the observation that the most probable ion ranges do not coincide with the average ranges. In other words, a Gaussian-like function does not give the best fit to the experimentally measured distributions. Moreover, as up-to-date microelectronic devices become smaller, predictions of the implanted ions depth profiles are required [2-41. Numerous works [5-111 beginning with the classical LSS theory [12] have been devoted to calculate the projected range CR,) of energetic ions in solids, the standard deviation (AR,) of the range as well as the skewness (Sk) and the kurtosis of the implanted ion distribution. But up to this writing no one has investigated the physics of the skewness and its dependence on various parameters. The main purpose of the present paper is to analyze in detail the dependence of the skewness on the projectile energy as well as upon the ratio of the projectile mass to the target atom mass. A simple formula for the skewness was obtained in the small-angle approximation. In section 2 the asymmetry analysis is given. In section 3 the approximate solution of the Boltzmann transport equation is considered. A formula for the calculation of the skewness is derived also. In the appendix the program for skewness computation is presented.
Correspondence to: Dr. M.Yu. Barabanenkov, Institute of Microelectronics Technology and High Purity Materials, 142432 Chernogolovka, Russian Federation.
0168-583X/93/$06.00
Earlier [13] we have shown, based on a numerical scheme to solve the Boltzmann transport equation [9], that the skewness Sk depends on the relationship between the ions’ nuclear and electronic energy losses in the slowing down process. The negative value of the skewness of the implanted ion concentration distribution results from the prevalence of electronic over nuclear stopping. The nuclear losses make a positive contribution to the skewness value. In particular, Sk = 0 holds approximately if the two energy losses are equal [13]. The solid line in fig. 1 passes through the abscissa at the point designated E,. The depth profile of ions implanted with this energy has a Gaussian-like shape. On the other hand, it is at this energy that the nuclear and electronic ion energy losses in the target are roughly equal. This assertion requires more rigorous study, because if it is true, among other things the equation Sk(E, M,, M2) = 0, unattainable for the solution, may be exchanged for the well studied equation CdE/dxJelastic = = (d E/dX)inelastic. Here we use the following designations: E is the ion energy, M, and M2 are the ion and target atom masses, respectively, x is a coordinate along the target depth. The implicit functions method offers the possibility of obtaining the dependence of the S(E) = (S, - &J/S, modulus upon the ion and target atom mass ratios provided the value of the ion range distribution skewness is zero (Sk = 0). Here the stopping cross section per atom in the elastic and inelastic interactions is denoted by S, and S,, respectively. So, the relationship between the skewness and the initial ion energy is obtained from range tables [9]. A number of methods have been proposed to calculate 6(E) as the energy function (see e.g. refs.
0 1993 - Elsevier Science Publishers B.V. All rights reserved
M. Yu. Barabanenkov /Asymmetric range distributions
523
ion scattering angle in laboratory binary collision with a target atom
ENERGY.
coordinates
for a
Le:'
The moments marked with two indexes are linked to the central moments pk of the distribution by the following relations
cL,=p:,
+p,’ + 3P,’ - (Pi)‘,
/.L* =
P:(Po’+
p3=gp;+;p:Fig. 1. Dependence of the skewness of nitrogen ion range distributions on implantation energy [13]. The solid line represents data from range tables [9]. The dashed line depicts the numerical prediction of the skewness with the assumption that nuclear energy losses are negligible; asterisks depict skewness values calculated by neglecting inelastic interactions.
2P3
- 2(Pg3.
Finally, the skewness of the distribution the formula Sk = /+/bz)
30
(4)
is defined by
.
(5)
[14,X]). Then the function Sk = Sk(S) was constructed based on the functions Sk = Sk(E) and 6 = S(E). Fig. 2 shows precisely these points Sk(s) = 0.
3. The skewness calculation With the aim of obtaining an analytical expression for the skewness let us consider the Boltzmann transport equation in the form given in ref. [9] (see also ref. [8]) for the probability P(r, E) for the ion to be slowed down to zero energy in a target volume d3r near the point with radius vector r. A description is given in ref. [16] to solve this equation by the moments method. Here we write the final form of the coupled equations for the moments Pl which already depend only on the ion energy -n)Pi;i
(m + l)(m
-m(m
fn
1-j ,;
,,,,,(,,,,,,
-tN
T,maxdce(P,n(E-
/0
T,) -Pl),
-Pi)
_-__-_ -----~ II c 0.4
m
C/Y - 0.2
(1)
t
~ *+o+PQ* *
= 4r/omPm(r,
E)r”+’
dr.
+
+
*PQ
P o*
b
where, according to the definition, P,“(E)
a,;
4
0.5
20.3
= (2m + l)N~T”maXd~~( PL( E - T,)P,(q)
, ,,,,,, ,, ,,.,, ,,,,,,,,,,,,,, 2 MT /Mp3
\
+ l)P,“I
i
0
(2)
On the right-hand side of eq. (1) N is the atomic target density, T,,e is the energy transfer to the target recoil atoms and target electrons, respectively, dg”,, are the differential scattering cross sections, P,(q) = P&OS $1 is the Legendre polynomial of order m and @ is the
“~~~“~‘~~‘.~~~I~‘.,~““I”“““‘i”““~~” 4 1 3 2
5
Fig. 2. The modulus of the relative difference of inelastic and elastic energy losses (S(E)J = /(Se- S,)/S, as a function of the ratio of the ion to the target atom mass. The ion energy E is chosen so that the skewness of the range distributions is zero (E = E *). The target substances are: (a) (0) boron, (A) neon, ( X) silicon; (b) (+) calcium, (0 ) copper, (* ) iron.
M. Yu. Barabanenkou /Asymmetric range distributions
524
1.o
Let us apply the small-angle approximation in the elastic ion-target atom interaction. The integral describing electronic stopping is given in the usual fashion [12] Jinel = /
‘+dcrJ
P;( E - T,) - Pi) z - g&,
0.5
cg
0
0.0
(6)
where S, = /duJ,. The small-angle approximation elastic collisions has the physical meaning that inequality T, =z E holds [ll]. The validity of this proximation will be discussed below. As long as nuclear energy transfer T,, is small compared to E may expand in terms of T,,/E and obtain &,=
q=cos@=l+Aq,
-;$,
for the apthe we
-0.5
-1.0
Fig. 3. Relative differences between the inelastic and elastic energy losses of a function of ion energy for a few M ratios of target-to-ion mass.
(7)
dP,” P;(E-T,)=P;(E)--&,,
P,(q) =f’,,,(l+ AT) z P,(l)
+ 5 dq
zl+“p”
7-i
AT +
Aq. dq
g=l
Here, M = MJM, and the condition P,,,(l) = 1 is used for any index m. The derivative of the Legendre polynomials with respect to the directional cosine is taken from Legendre’s equation [17] together with the fact that the first polynomial derivatives cannot have singularities at the point q = 1
dP,
m(m + 1)
z.__,=
2
and g(E) denotes the right-hand side of eq. (11). A normal numerical method can be applied to calculate the integral (12). But we use an additional simplification to obtain the analytical expression for the skewness. According to ref. [15] the ion nuclear stopping in the high energy region depends weakly on the ion energy S, = Const. ln(E)/E. Furthermore, 6(E) is also a slowly varying function of the energy, provided the ion mass is greater than the target atom mass (M + 0) (see fig. 3). Therefore, in the first approximation we can pass S, and S through the integral sign in eq. (12). In this case, eq. (11) permits a simple solution. For example, the average projected range (moment Pi> is now directly accessible, as easily as other moments (see fig. 4) 6
(10)
.
(13)
NS, 3M+8+2S’
Substituting eqs. (6)-(10) into eq. (1) yields an ordinary set of differential equations dP; dE
m(2m + 1) + 8+46/(m+l)
Mp” E m
= -&{(m-n)Pl;:-m(l+n /(m P;(E=O)=O,
+ l))P/1:]{2
+s/(m
+ l))-‘,
P,o=l.
The solution of this equation ref. [18])
(11) has the form (see, e.g.,
P,“(E) = e 0.4~ 0.01
where F(E)
0.1
ENERGY,
=m(2m
+ 1)Mk’E,[8
+ 4tTirn
+ l)] ’
1
10
MeV
Fig. 4. Ratios of the average projected ranges predicted by eq. (13) and by the DIMUS program [20] for the following ions implanted in silicon: B (curve 11, As (2), Sb (3) and Au (4).
M. Yu. Barabanenkou /Asymmetric range distriixhons
525
Table 1 Data from ref. [9]. (+) corresponds to the skewness of the ion range distributions passing through zero within the implantation energy region 1 keV to 1 MeV. (-1 indicates negative skewness within the same energy interval Target
Ion Hf
6‘2C
7l4N 160 $Si $Ti $Fe :;Ge 9% %qsAg ‘“Nd M) lWAu 79
-2.0
b
Fig. 5. Comparison of the skewness predicted by eq. (14) with tabulated data [9] (solid lines). (a) Silicon implanted with the following ions: (1, A) B, (2, 0) N, (3, 0) Si, (4, +) V; (b) iron implanted with the following ions: (1, +) N, (2, 0) Ne, (3, A) P, (4, 0) As.
Finally, the skewness
becomes,
through
eqs. (4) and (5)
Sk = $[ 1 - 3l/*((384
+ 504s + 11613~+ 20S3)
x\l(M+2+6)@4+6+S)}{(M+8+S) ~[(M+6+2S)(6+2S)-2S]~‘*)-~]. Fig. 5 shows the skewness value predicted together with tabulated data [9].
-
-
He+
Li+
B+
5
+
+
f + f
f f + .+ f f
* + + + f f + + -
-
-
-
ion masses, but it quickly reaches saturation at the level 1S 1+ 0.33. This constant deviation may be easily taken into consideration. The present research allows us to explain physically the sign of the skewness of the implanted ion depth profile (see table 1). Fig. 6 shows the zero skewness (Sk = 0) curves for a few target materials. The area above the curve corresponds to the negative skewness of range distributions. The area below the curve corresponds to the positive skewness value. Let us consider, for example, silicon irradiated by protons. Proton implantation at 100 eV produces equal elastic and inelastic energy losses in silicon [19]. Typical implantation energies are usually more than 10 keV. Thus, the accelerated protons lose the greater part of their energy in electronic interactions with silicon atoms. Ac-
1000
(14) by eq. (14)
4. Discussion
Sk>0
Fig. 2 shows the correlation between two typical ion implantation energies; the former, EC, corresponds to
the Gaussian-like shape of the implanted ion depth profile, and the latter energy E * is the solution of the equation CdE/dx),, = CdE/dxIine,. The approximate equality E * = EC holds the less the ion mass is relative to the target atom mass. The value S deviates for low
Fig. 6. Zero skewness curves for a number of ions (MI) and for a few target substances CM,): curve 1 = C, 2 = Si, 3 = Ti.
h4. Yu. Barabanenkov /Asymmetric range distributions
526
The skewness value recovers from a fault that occurs in the average projected range. Initially this may seem rather unexpected. The explanation is probably that the skewness depends weakly on the ions average projected range (the moment P:). The relative skewness variation with respect to the moment Pj, through eqs. (4) and (5), may be written as
cording to the results of the present paper it is this which is responsible for the negative asymmetry of proton range distributions in silicon. Let us turn our attention to the second part of this paper and discuss the applications of eqs. (13) and (14). Eq. (13) was derived, firstly, in the small-angle approximation and, secondly, the assumption was made that S, and 6 depend weakly on the ion energy. The smallangle approximation in elastic scattering has the following applicability condition [II] (see also ref. [19])
1
a(ln Sk) 6Pi
Sk=--
a(ln Pi)
P: .
It may now be shown numerically that the logarithmic derivative is significantly less than unity. For example, the relative error of the average projected range of boron in silicon is 50% (see fig. 4) in the cases when the implantation energy is below 200 keV. In this case the logarithmic derivative is 0.2. Hence the skewness error will be 10%.
(15)
E > 2 x 1042M;Z/“,
6Sk
where E is measured in MeV and m, is the electron mass. The numerical estimates for M2 = 28 (silicon) are the following: E(B) > 100 keV, E(As) > 14 keV, E(Sb1 > 9 keV, E(Au) > 6 keV. In fig. 4 the average projected range predictions by eq. (13) are seen to give quite a good representation of the analogous calculations by the DIMUS [20] program in the higher energy region than predicted only by the small-angle approximation (15). It is this energy region where the functions S, and 6 are slowly varying functions. We shall make this dependence explicit in the high, but less than the relativistic ion, energy region by writing S, = Const, ln(E)/E [ll]. Hence, the variable 6 = ) Se/S, - 1 I = I Const ,/Const, - 1 I is approximately constant. Note that the latter assumption holds the heavier the ion mass and the lower the energy. Here we would like to restate that the analytic expression for the skewness is of prime importance in section 3 of this paper. Fig. 5 shows a comparison of the skewness prediction by eq. (14) and the values given in tables [9].
5. Conclusions
In closing we remark that a connection is established between the character of the ion energy losses in the slowing down process and the asymmetry of the range distributions in the case when channeling is negligible. The results obtained allow us to use the experimental depth profiles of the implanted ion concentration as a visual information source for the predominant mechanism of these ion energy losses. A formula is derived to compute the skewness, enabling the possibility of the DIMUS program to be extended to describe the asymmetrical range distributions of accelerated particles.
Appendix program uses
(TURBO
bar-skewness; crt,
PASCAL)
dos; ion
const
= ’
21 =
‘; ;
Ml=
target z2=26;
;
var sk,m,eps,a,energy, i
:
Function
order
begin
Ck begin
CK:=l;
(k,n order:=
Function var
sel,snl,reld:
real;
integer;
se
:
real;
(energy
:real):reaL; exp(n*ln(k));
:real):real;
end;
CE
=keVI
= M2=28.09;
‘SILICON
‘;
n=Se22;
M. Yu. Barabanenkov
/Asymmetric
range distributions
527
se:=CK*38.8*order(zl,7/6)*z2*n/order(order(zl,2/3)+order(z2,2
/3) ,3/2)/order(m1,1/2)*order(energy,l/2)xle-24; Function
sn(energy:real):real;
CeV/A)
end;
CE=keV>
begin a:=0.468~/(order(z1,0.23)+order(z2,0.23)); eps:=
69.4*a*m2/zl/z2/(ml+m2)*energy;
sn:=181*zl*z2*n*a/(l+m2/ml)*0.5*ln(l+eps~ /(eps+0.10718*order(eps,0.37544))*1e-24; BEGIN
end;
part1
clrscr; writeln
m:=m2/ml; ( 'input
se1 :=se(energy);
energy snl:=
CkeVl
'1;
snlenergy);
readln reld:=
(energy); (sel-snl)/
snl; Sk:=3/5-order~3,3/2~/5*~384+504*reld+116*reld*reldt2O* reld*reld*reld)/order(((m+6+2*reld)*(6+2*reld~Z*reld), 3/2~*order~~m+2+reld~*~m+6+reld),1/2)/(m~8~reld~; writein(lSkewness=
',Sk);
Acknowledgements
The author is grateful for helpful discussions with Prof. V.N. Mordkovich and Prof. V.S. Remizovich. References [l] H. Ryssel and I. Ruge, Ion Implantation (B.C. Teubner Stuttgart, 1987). [2] Process and Device Simulation for MOS-VLSI Circuits, eds. P. Antognetti, D.A. Antoniadis, R.W. Dutton and W.G. Oldham (Martinus Nijhoff, Boston, 1983). [3] B.J. Scaly, Mater. Sci. Technol. 4 (1988) 500. [4] E.A. Maydell-Ondrusz and I.H. Wilson, Thin Solid Films 114 (1984) 357. [51 J.F. Gibbons, W.S. Johnson and S.W. Mylroie, Projected Range Statistics (Stroudsburg, Pennsylvania, 1975). [6] D.K. Brice, Ion Implantation Range and Energy Deposition Distribution, vol. 1. High Incident Energies (Plenum, New York, 1975). [7] K.B. Winterbon, Ion Implantation Range and Energy Deposition Distribution, vol. 2. Low Incident Energies (Plenum, New York, 1975). [8] U. Littmark and J.F. Ziegler, Phys. Rev. A23 (1981) 64.
end.
[9] A.F. Burenkov, F.F. Komarov, M.A. Kumakhov and M.M. Temkin, Spatial Distributions of Energy Deposited in the Cascade of Atomic Collisions in Solids (Energoatomizdat, Moscow, 1985). [lo] K.B. Winterbon, Nucl. Instr. and Meth. B17 (1986) 193. [ll] VS. Remizovich, D.B. Rogozkin and M.I. Ryazanov, Fluctuations of Charged Particles Ranges (Energoatomizdat, Moscow, 1988). [12] J. Lindhard, M. Scharff and H.E. Schiott, K. Dan. Vidensk. Selsk. Mat. Fys. Medd. 33 (1963) 4. [13] M. Yu. Barabanenkov, Surface 21 (1992) 21 (in Russian). [14] O.B. Firsov, Sov. Phys. JETP 36 (1959) 1076. [15] W.D. Wilson, L.C. Haggmark and J.P. Biersack, Phys. Rev. B15 (1977) 2458. [16] K.B. Winterbon, P. Sigmund and J.B. Sanders, K. Dan. Vidensk. Selsk. Mat. Fys. Medd. 37 (1970) 1. [17] E. Jahnke and F. Emde, Tables of Functions (Dover Publications, New York, 1945). [18] E. Kamke, Reference Book on Differential Equations (Nauka, Moscow, 1971) p. 294. [19] N.F. Mott and H.S.W. Massey, The Theory of Atomic Collisions (Clarendon, Oxford, 196%. [20] J.P. Biersack and J.F. Ziegler, in: Ion Implantation Techniques, Springer Series in Electrophysics, vol. 10, eds. H. Ryssel and H. Glawischnig (Springer, Berlin, 1982).
The asymmetry of projected range distributions
NM B
Beam Interactions with Materials 8 Atoms
in amorphous solids
M.Yu. Barabanenkov Institute of Microelectronics Technology and High Purity Materials, 142432 Chernogolouka, Moscow Region, Russian Federation
Received 6 February 1992 and in revised form 18 August 1992
Physical reasons are considered to explain the asymmetry of range distributions of implanted ions in amorphous solids. It is established that the appearance of negative asymmetry is due to the prevalence of ion electronic energy loss in the slowing down process. Nuclear stopping makes a positive contribution to the asymmetry value. A simple analytical expression for the asymmetry value is obtained. This allows extension of the possibilities of the well known program DIMUS to describe the asymmetrical range distributions of energetic particles.
2. Skewness analysis
1. Introduction
It is well known that, in general, the depth distribution of implanted ions in amorphous solids has an asymmetrical shape (see, e.g. ref. [l]). The asymmetry becomes apparent by the observation that the most probable ion ranges do not coincide with the average ranges. In other words, a Gaussian-like function does not give the best fit to the experimentally measured distributions. Moreover, as up-to-date microelectronic devices become smaller, predictions of the implanted ions depth profiles are required [2-41. Numerous works [5-111 beginning with the classical LSS theory [12] have been devoted to calculate the projected range CR,) of energetic ions in solids, the standard deviation (AR,) of the range as well as the skewness (Sk) and the kurtosis of the implanted ion distribution. But up to this writing no one has investigated the physics of the skewness and its dependence on various parameters. The main purpose of the present paper is to analyze in detail the dependence of the skewness on the projectile energy as well as upon the ratio of the projectile mass to the target atom mass. A simple formula for the skewness was obtained in the small-angle approximation. In section 2 the asymmetry analysis is given. In section 3 the approximate solution of the Boltzmann transport equation is considered. A formula for the calculation of the skewness is derived also. In the appendix the program for skewness computation is presented.
Correspondence to: Dr. M.Yu. Barabanenkov, Institute of Microelectronics Technology and High Purity Materials, 142432 Chernogolovka, Russian Federation.
0168-583X/93/$06.00
Earlier [13] we have shown, based on a numerical scheme to solve the Boltzmann transport equation [9], that the skewness Sk depends on the relationship between the ions’ nuclear and electronic energy losses in the slowing down process. The negative value of the skewness of the implanted ion concentration distribution results from the prevalence of electronic over nuclear stopping. The nuclear losses make a positive contribution to the skewness value. In particular, Sk = 0 holds approximately if the two energy losses are equal [13]. The solid line in fig. 1 passes through the abscissa at the point designated E,. The depth profile of ions implanted with this energy has a Gaussian-like shape. On the other hand, it is at this energy that the nuclear and electronic ion energy losses in the target are roughly equal. This assertion requires more rigorous study, because if it is true, among other things the equation Sk(E, M,, M2) = 0, unattainable for the solution, may be exchanged for the well studied equation CdE/dxJelastic = = (d E/dX)inelastic. Here we use the following designations: E is the ion energy, M, and M2 are the ion and target atom masses, respectively, x is a coordinate along the target depth. The implicit functions method offers the possibility of obtaining the dependence of the S(E) = (S, - &J/S, modulus upon the ion and target atom mass ratios provided the value of the ion range distribution skewness is zero (Sk = 0). Here the stopping cross section per atom in the elastic and inelastic interactions is denoted by S, and S,, respectively. So, the relationship between the skewness and the initial ion energy is obtained from range tables [9]. A number of methods have been proposed to calculate 6(E) as the energy function (see e.g. refs.
0 1993 - Elsevier Science Publishers B.V. All rights reserved
M. Yu. Barabanenkov /Asymmetric range distributions
523
ion scattering angle in laboratory binary collision with a target atom
ENERGY.
coordinates
for a
Le:'
The moments marked with two indexes are linked to the central moments pk of the distribution by the following relations
cL,=p:,
+p,’ + 3P,’ - (Pi)‘,
/.L* =
P:(Po’+
p3=gp;+;p:Fig. 1. Dependence of the skewness of nitrogen ion range distributions on implantation energy [13]. The solid line represents data from range tables [9]. The dashed line depicts the numerical prediction of the skewness with the assumption that nuclear energy losses are negligible; asterisks depict skewness values calculated by neglecting inelastic interactions.
2P3
- 2(Pg3.
Finally, the skewness of the distribution the formula Sk = /+/bz)
30
(4)
is defined by
.
(5)
[14,X]). Then the function Sk = Sk(S) was constructed based on the functions Sk = Sk(E) and 6 = S(E). Fig. 2 shows precisely these points Sk(s) = 0.
3. The skewness calculation With the aim of obtaining an analytical expression for the skewness let us consider the Boltzmann transport equation in the form given in ref. [9] (see also ref. [8]) for the probability P(r, E) for the ion to be slowed down to zero energy in a target volume d3r near the point with radius vector r. A description is given in ref. [16] to solve this equation by the moments method. Here we write the final form of the coupled equations for the moments Pl which already depend only on the ion energy -n)Pi;i
(m + l)(m
-m(m
fn
1-j ,;
,,,,,(,,,,,,
-tN
T,maxdce(P,n(E-
/0
T,) -Pl),
-Pi)
_-__-_ -----~ II c 0.4
m
C/Y - 0.2
(1)
t
~ *+o+PQ* *
= 4r/omPm(r,
E)r”+’
dr.
+
+
*PQ
P o*
b
where, according to the definition, P,“(E)
a,;
4
0.5
20.3
= (2m + l)N~T”maXd~~( PL( E - T,)P,(q)
, ,,,,,, ,, ,,.,, ,,,,,,,,,,,,,, 2 MT /Mp3
\
+ l)P,“I
i
0
(2)
On the right-hand side of eq. (1) N is the atomic target density, T,,e is the energy transfer to the target recoil atoms and target electrons, respectively, dg”,, are the differential scattering cross sections, P,(q) = P&OS $1 is the Legendre polynomial of order m and @ is the
“~~~“~‘~~‘.~~~I~‘.,~““I”“““‘i”““~~” 4 1 3 2
5
Fig. 2. The modulus of the relative difference of inelastic and elastic energy losses (S(E)J = /(Se- S,)/S, as a function of the ratio of the ion to the target atom mass. The ion energy E is chosen so that the skewness of the range distributions is zero (E = E *). The target substances are: (a) (0) boron, (A) neon, ( X) silicon; (b) (+) calcium, (0 ) copper, (* ) iron.
M. Yu. Barabanenkou /Asymmetric range distributions
524
1.o
Let us apply the small-angle approximation in the elastic ion-target atom interaction. The integral describing electronic stopping is given in the usual fashion [12] Jinel = /
‘+dcrJ
P;( E - T,) - Pi) z - g&,
0.5
cg
0
0.0
(6)
where S, = /duJ,. The small-angle approximation elastic collisions has the physical meaning that inequality T, =z E holds [ll]. The validity of this proximation will be discussed below. As long as nuclear energy transfer T,, is small compared to E may expand in terms of T,,/E and obtain &,=
q=cos@=l+Aq,
-;$,
for the apthe we
-0.5
-1.0
Fig. 3. Relative differences between the inelastic and elastic energy losses of a function of ion energy for a few M ratios of target-to-ion mass.
(7)
dP,” P;(E-T,)=P;(E)--&,,
P,(q) =f’,,,(l+ AT) z P,(l)
+ 5 dq
zl+“p”
7-i
AT +
Aq. dq
g=l
Here, M = MJM, and the condition P,,,(l) = 1 is used for any index m. The derivative of the Legendre polynomials with respect to the directional cosine is taken from Legendre’s equation [17] together with the fact that the first polynomial derivatives cannot have singularities at the point q = 1
dP,
m(m + 1)
z.__,=
2
and g(E) denotes the right-hand side of eq. (11). A normal numerical method can be applied to calculate the integral (12). But we use an additional simplification to obtain the analytical expression for the skewness. According to ref. [15] the ion nuclear stopping in the high energy region depends weakly on the ion energy S, = Const. ln(E)/E. Furthermore, 6(E) is also a slowly varying function of the energy, provided the ion mass is greater than the target atom mass (M + 0) (see fig. 3). Therefore, in the first approximation we can pass S, and S through the integral sign in eq. (12). In this case, eq. (11) permits a simple solution. For example, the average projected range (moment Pi> is now directly accessible, as easily as other moments (see fig. 4) 6
(10)
.
(13)
NS, 3M+8+2S’
Substituting eqs. (6)-(10) into eq. (1) yields an ordinary set of differential equations dP; dE
m(2m + 1) + 8+46/(m+l)
Mp” E m
= -&{(m-n)Pl;:-m(l+n /(m P;(E=O)=O,
+ l))P/1:]{2
+s/(m
+ l))-‘,
P,o=l.
The solution of this equation ref. [18])
(11) has the form (see, e.g.,
P,“(E) = e 0.4~ 0.01
where F(E)
0.1
ENERGY,
=m(2m
+ 1)Mk’E,[8
+ 4tTirn
+ l)] ’
1
10
MeV
Fig. 4. Ratios of the average projected ranges predicted by eq. (13) and by the DIMUS program [20] for the following ions implanted in silicon: B (curve 11, As (2), Sb (3) and Au (4).
M. Yu. Barabanenkou /Asymmetric range distriixhons
525
Table 1 Data from ref. [9]. (+) corresponds to the skewness of the ion range distributions passing through zero within the implantation energy region 1 keV to 1 MeV. (-1 indicates negative skewness within the same energy interval Target
Ion Hf
6‘2C
7l4N 160 $Si $Ti $Fe :;Ge 9% %qsAg ‘“Nd M) lWAu 79
-2.0
b
Fig. 5. Comparison of the skewness predicted by eq. (14) with tabulated data [9] (solid lines). (a) Silicon implanted with the following ions: (1, A) B, (2, 0) N, (3, 0) Si, (4, +) V; (b) iron implanted with the following ions: (1, +) N, (2, 0) Ne, (3, A) P, (4, 0) As.
Finally, the skewness
becomes,
through
eqs. (4) and (5)
Sk = $[ 1 - 3l/*((384
+ 504s + 11613~+ 20S3)
x\l(M+2+6)@4+6+S)}{(M+8+S) ~[(M+6+2S)(6+2S)-2S]~‘*)-~]. Fig. 5 shows the skewness value predicted together with tabulated data [9].
-
-
He+
Li+
B+
5
+
+
f + f
f f + .+ f f
* + + + f f + + -
-
-
-
ion masses, but it quickly reaches saturation at the level 1S 1+ 0.33. This constant deviation may be easily taken into consideration. The present research allows us to explain physically the sign of the skewness of the implanted ion depth profile (see table 1). Fig. 6 shows the zero skewness (Sk = 0) curves for a few target materials. The area above the curve corresponds to the negative skewness of range distributions. The area below the curve corresponds to the positive skewness value. Let us consider, for example, silicon irradiated by protons. Proton implantation at 100 eV produces equal elastic and inelastic energy losses in silicon [19]. Typical implantation energies are usually more than 10 keV. Thus, the accelerated protons lose the greater part of their energy in electronic interactions with silicon atoms. Ac-
1000
(14) by eq. (14)
4. Discussion
Sk>0
Fig. 2 shows the correlation between two typical ion implantation energies; the former, EC, corresponds to
the Gaussian-like shape of the implanted ion depth profile, and the latter energy E * is the solution of the equation CdE/dx),, = CdE/dxIine,. The approximate equality E * = EC holds the less the ion mass is relative to the target atom mass. The value S deviates for low
Fig. 6. Zero skewness curves for a number of ions (MI) and for a few target substances CM,): curve 1 = C, 2 = Si, 3 = Ti.
h4. Yu. Barabanenkov /Asymmetric range distributions
526
The skewness value recovers from a fault that occurs in the average projected range. Initially this may seem rather unexpected. The explanation is probably that the skewness depends weakly on the ions average projected range (the moment P:). The relative skewness variation with respect to the moment Pj, through eqs. (4) and (5), may be written as
cording to the results of the present paper it is this which is responsible for the negative asymmetry of proton range distributions in silicon. Let us turn our attention to the second part of this paper and discuss the applications of eqs. (13) and (14). Eq. (13) was derived, firstly, in the small-angle approximation and, secondly, the assumption was made that S, and 6 depend weakly on the ion energy. The smallangle approximation in elastic scattering has the following applicability condition [II] (see also ref. [19])
1
a(ln Sk) 6Pi
Sk=--
a(ln Pi)
P: .
It may now be shown numerically that the logarithmic derivative is significantly less than unity. For example, the relative error of the average projected range of boron in silicon is 50% (see fig. 4) in the cases when the implantation energy is below 200 keV. In this case the logarithmic derivative is 0.2. Hence the skewness error will be 10%.
(15)
E > 2 x 1042M;Z/“,
6Sk
where E is measured in MeV and m, is the electron mass. The numerical estimates for M2 = 28 (silicon) are the following: E(B) > 100 keV, E(As) > 14 keV, E(Sb1 > 9 keV, E(Au) > 6 keV. In fig. 4 the average projected range predictions by eq. (13) are seen to give quite a good representation of the analogous calculations by the DIMUS [20] program in the higher energy region than predicted only by the small-angle approximation (15). It is this energy region where the functions S, and 6 are slowly varying functions. We shall make this dependence explicit in the high, but less than the relativistic ion, energy region by writing S, = Const, ln(E)/E [ll]. Hence, the variable 6 = ) Se/S, - 1 I = I Const ,/Const, - 1 I is approximately constant. Note that the latter assumption holds the heavier the ion mass and the lower the energy. Here we would like to restate that the analytic expression for the skewness is of prime importance in section 3 of this paper. Fig. 5 shows a comparison of the skewness prediction by eq. (14) and the values given in tables [9].
5. Conclusions
In closing we remark that a connection is established between the character of the ion energy losses in the slowing down process and the asymmetry of the range distributions in the case when channeling is negligible. The results obtained allow us to use the experimental depth profiles of the implanted ion concentration as a visual information source for the predominant mechanism of these ion energy losses. A formula is derived to compute the skewness, enabling the possibility of the DIMUS program to be extended to describe the asymmetrical range distributions of accelerated particles.
Appendix program uses
(TURBO
bar-skewness; crt,
PASCAL)
dos; ion
const
= ’
21 =
‘; ;
Ml=
target z2=26;
;
var sk,m,eps,a,energy, i
:
Function
order
begin
Ck begin
CK:=l;
(k,n order:=
Function var
sel,snl,reld:
real;
integer;
se
:
real;
(energy
:real):reaL; exp(n*ln(k));
:real):real;
end;
CE
=keVI
= M2=28.09;
‘SILICON
‘;
n=Se22;
M. Yu. Barabanenkov
/Asymmetric
range distributions
527
se:=CK*38.8*order(zl,7/6)*z2*n/order(order(zl,2/3)+order(z2,2
/3) ,3/2)/order(m1,1/2)*order(energy,l/2)xle-24; Function
sn(energy:real):real;
CeV/A)
end;
CE=keV>
begin a:=0.468~/(order(z1,0.23)+order(z2,0.23)); eps:=
69.4*a*m2/zl/z2/(ml+m2)*energy;
sn:=181*zl*z2*n*a/(l+m2/ml)*0.5*ln(l+eps~ /(eps+0.10718*order(eps,0.37544))*1e-24; BEGIN
end;
part1
clrscr; writeln
m:=m2/ml; ( 'input
se1 :=se(energy);
energy snl:=
CkeVl
'1;
snlenergy);
readln reld:=
(energy); (sel-snl)/
snl; Sk:=3/5-order~3,3/2~/5*~384+504*reld+116*reld*reldt2O* reld*reld*reld)/order(((m+6+2*reld)*(6+2*reld~Z*reld), 3/2~*order~~m+2+reld~*~m+6+reld),1/2)/(m~8~reld~; writein(lSkewness=
',Sk);
Acknowledgements
The author is grateful for helpful discussions with Prof. V.N. Mordkovich and Prof. V.S. Remizovich. References [l] H. Ryssel and I. Ruge, Ion Implantation (B.C. Teubner Stuttgart, 1987). [2] Process and Device Simulation for MOS-VLSI Circuits, eds. P. Antognetti, D.A. Antoniadis, R.W. Dutton and W.G. Oldham (Martinus Nijhoff, Boston, 1983). [3] B.J. Scaly, Mater. Sci. Technol. 4 (1988) 500. [4] E.A. Maydell-Ondrusz and I.H. Wilson, Thin Solid Films 114 (1984) 357. [51 J.F. Gibbons, W.S. Johnson and S.W. Mylroie, Projected Range Statistics (Stroudsburg, Pennsylvania, 1975). [6] D.K. Brice, Ion Implantation Range and Energy Deposition Distribution, vol. 1. High Incident Energies (Plenum, New York, 1975). [7] K.B. Winterbon, Ion Implantation Range and Energy Deposition Distribution, vol. 2. Low Incident Energies (Plenum, New York, 1975). [8] U. Littmark and J.F. Ziegler, Phys. Rev. A23 (1981) 64.
end.
[9] A.F. Burenkov, F.F. Komarov, M.A. Kumakhov and M.M. Temkin, Spatial Distributions of Energy Deposited in the Cascade of Atomic Collisions in Solids (Energoatomizdat, Moscow, 1985). [lo] K.B. Winterbon, Nucl. Instr. and Meth. B17 (1986) 193. [ll] VS. Remizovich, D.B. Rogozkin and M.I. Ryazanov, Fluctuations of Charged Particles Ranges (Energoatomizdat, Moscow, 1988). [12] J. Lindhard, M. Scharff and H.E. Schiott, K. Dan. Vidensk. Selsk. Mat. Fys. Medd. 33 (1963) 4. [13] M. Yu. Barabanenkov, Surface 21 (1992) 21 (in Russian). [14] O.B. Firsov, Sov. Phys. JETP 36 (1959) 1076. [15] W.D. Wilson, L.C. Haggmark and J.P. Biersack, Phys. Rev. B15 (1977) 2458. [16] K.B. Winterbon, P. Sigmund and J.B. Sanders, K. Dan. Vidensk. Selsk. Mat. Fys. Medd. 37 (1970) 1. [17] E. Jahnke and F. Emde, Tables of Functions (Dover Publications, New York, 1945). [18] E. Kamke, Reference Book on Differential Equations (Nauka, Moscow, 1971) p. 294. [19] N.F. Mott and H.S.W. Massey, The Theory of Atomic Collisions (Clarendon, Oxford, 196%. [20] J.P. Biersack and J.F. Ziegler, in: Ion Implantation Techniques, Springer Series in Electrophysics, vol. 10, eds. H. Ryssel and H. Glawischnig (Springer, Berlin, 1982).