CHINESE JOURNAL OF PHYSICS
VOL. 50, NO. 4
August 2012
Total Electron Scattering Cross Sections of SiH4 and PH3 Molecules in the
Energy Range from 10 eV to 5000 eV
Xiao-Ming Tan1, ∗ and Xue-Mei Liu2
1
2
School of Physics, Ludong University, Yantai 264025, P. R. China
Reproductive Medicine Center, Yantai Yuhuangding Hospital, Yantai 264000, P. R. China
(Received April 20, 2011; Revised December 30, 2011)
The total cross sections for electron scattering from SiH4 and PH3 molecules in the energy
range from 10 eV to 5000 eV are calculated with the additivity rule method. The present
results are compared with the available experimental and theoretical data. It shows that the
additivity rule method is very successful for calculating the total cross sections of electron
scattering from SiH4 and PH3 molecules in the energy range from 200 eV to 5000 eV. The
total electron cross sections for SiH4 from 1000 eV to 5000 eV, and for PH3 from 10 eV to
5000 eV are calculated based on the additivity rule method.
PACS numbers: 34.80.Bm, 34.80.-i
I. INTRODUCTION
The total cross sections for electron scattering from gaseous molecules are necessary in
high resolution plasma processing, simulating of the glow discharge, and the semiconductor
industry [1]. For example, the total electron scattering cross sections for SiH4 molecule are
important for understanding the behavior of mono-silane plasma and for the deposition of
hydrogenated amorphous silicone films [2, 3]. The total cross sections for electron scattering
from the PH3 molecule are indispensable in the semiconductor industry, astrophysics, and
atmospheric physics, because the PH3 molecule exists in some planetary atmospheres and
interstellar atmospheres, and is a suitable doping agent in the processing of materials for
opto-electronics and for the realization of atomic-scale devices for quantum computing [1, 4].
The total cross section measurements for electron scattering from SiH4 have been made by
Ariyasinghe et al. from 90 eV to 3500 eV [4], Zecca et al. from 75 eV to 4000 eV [5],
and Sueoka et al. below 400 eV [6]. Very recently, the total cross sections for electron
scattering from the PH3 molecule have been measured by Ariyasinghe et al. from 90 eV to
3500 eV [4] and by Szmytkowski et al. only below 370 eV [1]. Above 4000 eV, there are no
experimental data for the total cross sections of electron scattering from the two molecules.
Theoretical total cross sections for the SiH4 and PH3 molecules have been obtained by Jain
and Baluja [7] in the energy range from 10 eV to 5000 eV based on the spherical complex
optical potential method.
In theory, electron-molecule scattering is a more complex problem than the corresponding electron-atom scattering due to the multi-center nature, the lack of a center
∗
Electronic address: [email protected]
http://PSROC.phys.ntu.edu.tw/cjp
573
c 2012 THE PHYSICAL SOCIETY
OF THE REPUBLIC OF CHINA
TOTAL ELECTRON SCATTERING CROSS . . .
574
VOL. 50
of symmetry, and nuclear motion. Many approaches have been proposed and developed,
such as the R-matrix method [8–12], the spherical complex optical potential method [13],
the Schwinger variational method [14], the Bethe-Born theory [15], the close-coupling
method [16], and the additivity rule (AR) method [17–19]. Among these approaches, the
additivity rule is a relatively simple but effective one, especially for smaller molecules.
The additivity rule method is based on the assumption that anisotropic electron-molecule
interactions do not play an important role in shaping up the total cross sections of the
intermediate- and high-energy electron-molecule collisions. Thus, the total cross section for
a molecule is the sum of the total cross sections for the constituent atoms. Raj [18] made
the first application of the additivity rule to obtain the elastic cross sections for electron
scattering from simple molecules. Joshipura and Patel [19] and Sun et al. [20] also employed
the additivity rule to obtain the total cross sections (elastic and inelastic) for electron scattering with simple diatomic and triatomic molecules, and proved that the additivity rule
is proper for the calculation of the total cross sections for electron scattering from simple
and smaller molecules. The calculations by use of the additivity rule have been done for
SiH4 only below 1000 eV [21] and have not been done for PH3 at any energy. In this paper,
we shall employ the additivity rule and the complex optical potential to calculate the total
cross sections of electron scattering by SiH4 and PH3 molecules in the energy range from
10 eV to 5000 eV.
II. THEORY
In the additivity rule model [18], molecule orbits can be described by the sum of the
valence orbits of all the atoms present in the molecule. As a result, the total cross section
of electron-molecule scattering is written as the sum of the total cross sections of atoms.
Thus the total cross section QT for a molecule is given by
N
N
j=1
j=1
X j
X
4π
4π
qT (E),
fj (θ = 0) =
ImfM (θ = 0) =
Im
QT =
k
k
(1)
where qTj and fj are the total cross section due to the jth atom of the molecule and the
complex scattering amplitude for constituent atoms of the molecule, respectively. The qTj
of Eq. (1) for the jth atom is obtained by the method of partial waves:
qTj
lmax
2 2 π X
j
= 2
,
(2l + 1) 1 − sl + 1 − sjl k
(2)
l=0
where sjl is the lth complex scattering matrix element of the jth atom, which is related to
the partial wave phase shift as sjl = exp(2iδlj ). The limit lmax is taken, which is enough
to generate the higher partial-wave contributions until a convergence of less than 0.5% is
achieved in the total cross section calculation. To obtain sjl we solve the following radial
VOL. 50
XIAO-MING TAN AND XUE-MEI LIU
equation:
2
d
l(l + 1)
2
+ k − 2Vopt −
ul (r) = 0
dr 2
r2
575
(3)
under the boundary condition
ul (kr) ∼ kr[jl (kr) − inl (kr)] + sl kr[jl (kr) + inl (kr)],
(4)
where jl and nl are spherical Bessel and Neumann functions separately. The atom is
replaced by the complex optical potential
Vopt = Vs (r) + Ve (r) + Vp (r) + iVa (r).
(5)
It incorporates all the important physical effects. Presently the static potential Vs (r) for
the electron-atom system is calculated from the well-known Hartree-Fock atomic wave functions [22]. The exchange potential Ve (r) has a semi-classical energy-dependent form due
to Riley and Truhlar [23]. Zhang et al. [24] gives a smooth form at all r for the polarization potential Vp (r), which has the correct asymptotic form at large r and approaches the
free-electron-gas correlation potential [25] in the near-target region. The imaginary part of
the optical potential Va (r) is the absorption potential, which represents approximately the
combined effect of all the inelastic channels. The absorption potential was derived from a
quasifree-scattering model by Staszawska et al. [26], and then modified by Jiang et al. [27].
Here, the absorption potential of Jiang et al. is adopted.
III. RESULTS AND DISCUSSION
Using the additivity rule, we have calculated the total cross sections for electron
scattering from SiH4 and PH3 molecules in the energy range from 10 eV to 5000 eV. The
present results are listed in Table I and compared with the available experimental and
theoretical data shown in Figures 1 and 2.
TABLE I: The present results of the total cross sections for electron
scattering from SiH4 and PH3 in units of 10−20 m2 .
Electron energy (eV)
SiH4
PH3
10
64.82
53.63
20
42.30
35.55
30
33.85
28.99
40
29.91
24.50
50
27.32
21.59
Continued . . .
576
TOTAL ELECTRON SCATTERING CROSS . . .
Electron energy (eV)
SiH4
PH3
60
25.18
19.76
70
23.33
18.41
80
21.75
17.29
90
20.38
16.31
100
19.18
15.43
200
12.11
10.17
300
8.80
7.59
400
6.86
6.02
500
5.60
4.97
600
4.72
4.22
700
4.07
3.67
800
3.58
3.24
900
3.19
2.90
1000
2.88
2.62
1100
2.63
2.39
1200
2.42
2.20
1300
2.24
2.04
1400
2.09
1.90
1500
1.96
1.78
1600
1.85
1.67
1700
1.75
1.58
1800
1.66
1.50
1900
1.58
1.42
2000
1.51
1.36
2100
1.44
1.30
2200
1.39
1.24
2300
1.33
1.19
2400
1.28
1.15
2500
1.24
1.11
2600
1.19
1.07
Continued . . .
VOL. 50
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XIAO-MING TAN AND XUE-MEI LIU
Electron energy (eV)
SiH4
PH3
2700
1.16
1.03
2800
1.12
1.00
2900
1.08
0.97
3000
1.05
0.94
3100
1.02
0.91
3200
0.99
0.88
3300
0.97
0.86
3400
0.94
0.84
3500
0.92
0.82
3600
0.89
0.79
3700
0.87
0.78
3800
0.85
0.76
3900
0.83
0.74
4000
0.81
0.72
4100
0.79
0.71
4200
0.78
0.69
4300
0.76
0.68
4400
0.74
0.66
4500
0.73
0.65
4600
0.71
0.64
4700
0.70
0.62
4800
0.69
0.61
4900
0.67
0.60
5000
0.66
0.59
577
The present results for the total cross sections for electron scattering from SiH4 are
shown in Figure 1 with the available experimental and theoretical data. From it, we can see
that the present results are in good agreement with the experimental data above 200 eV. For
example, the differences between the present results and the measurements of Ariyasinghe
et al. [4], Zecca et al. [5], and Sueoka et al. [6] are only 5.46% at 500 eV, 1.91% at 800
eV, and 6.02% at 300 eV, respectively. The differences between the present results and
the experimental data are larger at lower energies, especially below 200 eV. Fortunately,
the difference decreases quickly as the incident energy of electron increases. The present
results are in good accord with the theoretical data based on the spherical complex optical
TOTAL ELECTRON SCATTERING CROSS . . .
578
VOL. 50
60
e-SiH
Ariyasinghe et al.
4
Zecca et al.
Total cross sections (10
-20
2
m )
Present results
50
Sueoka et al.
40
Jain and Baluja
Garcia and Manero
30
20
10
0
0
1000
2000
3000
4000
5000
Electron Energy (eV)
FIG. 1: Total cross sections for SiH4 . Solid line: present results. Experimental data: Ariyasinghe
et al. [4], Zecca et al. [5], Sueoka et al. [6]. Theoretical data: Jain and Baluja [7]. Empirical data:
Garcia and Manero [28].
50
Ariyasinghe et al.
Total cross sections (10
-20
2
m )
Present results
40
Szmytkowski et al.
e-PH
Jain and Baluja
3
Garcia and Manero
30
20
10
0
0
1000
2000
3000
4000
5000
Electron Energy (eV)
FIG. 2: Total cross sections for PH3 . Solid line: present results. Experimental data: Ariyasinghe et
al. [4], Szmytkowski et al. [1]. Theoretical data: Jain and Baluja [7]. Empirical data: Garcia and
Manero [28].
VOL. 50
XIAO-MING TAN AND XUE-MEI LIU
579
potential method of Jain and Baluja [7] above 800 eV, and in good agreement with the
results of the empirical formula of Garcia and Manero [28] in the whole overlapping energy
range. Above 4000 eV, there are no experimental data, so the present results are compared
with the theoretical data of Jain and Baluja [7] and the results of the empirical formula of
Garcia and Manero [28]. Good agreements are obtained.
In Figure 2, the present results for PH3 are compared with the measurements of
Ariyasinghe et al. [4] and Szmytkowski et al. [1] in the energy range from 10 eV to 3500
eV. From Figure 2, we can see that the present results are in good accord with the experimental data of Ariyasinghe et al. [4] above 200 eV and Szmytkowski et al. [1] above
20 eV. For example, the differences between the present results and the experimental data
of Ariyasinghe et al. [4] and Szmytkowski et al. [1] are 3.55% at 300 eV and 1.84% at 50
eV, respectively. Above 3500 eV, there are no experimental data for PH3 , so the present
results are compared with the theoretical results calculated by Jain and Baluja [7] using
the spherical complex optical potential method and the data of the empirical formula of
Garcia and Manero [28]. Good agreements between them are also obtained.
IV. CONCLUSIONS
In this paper, using the additivity rule model, we have calculated the total cross
sections for electron scattering from the SiH4 and PH3 molecules in the energy range from
10 eV to 5000 eV. The present calculations are also compared with the experimental and
theoretical data. From our studies, we have the following conclusions:
(1) The additivity rule method is very successful for calculating the total cross sections
of electron scattering from SiH4 and PH3 molecules in the energy range from 200 eV to
5000 eV. Above 4000 eV for SiH4 and above 3500 eV for PH3 , there are no experimental
data, so the present results can give a reference for further experiments.
(2) Jiang et al. have calculated the total cross sections for SiH4 by use of the additivity
rule method only below 1000 eV. To our knowledge, there are no total cross sections for
PH3 from the additivity rule method at any energy up to now. So, the total electron
cross sections for SiH4 from 1000 eV to 5000 eV, and for PH3 from 10 eV to 5000 eV are
calculated based on the additivity rule method for the first time.
(3) Larger differences between the present results and the measurements exist at lower
energies, especially below 200 eV. This shows that the additivity rule method is not very
proper for calculating the total cross sections for SiH4 and PH3 molecules below 200 eV.
This may be because no molecular geometry is involved in the additivity rule method and
electron–molecule scattering problem is reduced to electron-atom scattering problem. Since
the molecular geometry is not considered in the additivity rule method, the differences may
be caused by it at lower energies. Of course, more experimental and theoretical investigations for electron scattering from the SiH4 and PH3 molecules are needed to elucidate the
present calculations.
580
TOTAL ELECTRON SCATTERING CROSS . . .
VOL. 50
Acknowledgements
This work was supported by the National Natural Science Foundation of China
(11204121), the Joint Special Fund of the Natural Science Foundation of Shandong Province
(ZR2011AL021), the National Natural Science Foundation of China (11074104) and the
Discipline Construction Fund of Ludong University.
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