# Electron-Collision-Induced Dissociative Ionization Cross Sections for Silane

Authors: Satyendra Pal; Neeraj Kumar; Anshu
Journal: Advances in Physical Chemistry
 (2009)

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## Abstract

Secondary electron energy and angle dependent differential cross sections for the production of cationsSiHn+ (n=0–3), H2+ and H+ resulting from dissociative ionization of SiH4by electron collision have been evaluated at fixed incident electron energies of 100 and 200 eV. The semiempirical formulation of Jain and Khare which requires the oscillator strength data as a major input has been employed. In the absence of experimental data for differential cross sections, the corresponding derived integral partial and total ionization cross sections in the energy range varying from ionization threshold to 1000 eV revealed a satisfactory agreement with the available experimental and theoretical data. We have also evaluated the ionization rate coefficients on the basis of calculated partial ionization cross sections and Maxwell-Boltzmann energy distributions.


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## Body

## 1. Introduction

This work is a part of our project on electron impact
ionization on technological molecules in low-energy regimes. Our aim is to
determine the differential and integral cross sections corresponding to the
production of molecular and atomic cations in electron impact ionization of theSiH4 molecule, which is widely used in plasma deposition of silicon containing thin films. Electron ionization cross sections of SiH4 are needed for the modeling of charge carrier balance in plasma and gas phase
media [1, 2]. There is dearth to elucidate the atomic and molecular properties
and their interaction with photons and electrons. Photoabsorption and
photoionization studies of silane and its radicals have already been made by
Gallagher et al. [3], Johnson III et al. [4], and
Cooper et al.  [5, 6]. The experimental determination of partial and total electron ionization cross sections includes those of Perrin et al. [7], Chatham et al. [8], Krishnakumar and Srivastava [9], and Basner et al. [10]. Haaland [11] estimated
the partial ionization cross sections for the formation of Si containing radicals by scaling the data of
Chatham et al. [8] and the
differential data of Morrison and Traeger [12] to his absolute values at 50 eV.  From a theoretical standpoint, calculations for SiH4 are particularly challenging. The rigorous quantum mechanical approach for the calculations for molecules is limited to
the application of simple molecules. Contrary to it, there now exist the binary
encounter Bethe formalism by Ali et al. [13], the semiempirical formalismby Khare et al. [14], DM-formalism by  Deutsch et al. [15], and complex potential model calculations by Joshipura et al. [16, 17].This paper reports the spectrum of the single differential
cross sections with the energy of secondary electron produced in the ionization
of silane molecule by electron collision at fixed incident electron energies of
100 and 200 eV, employing a semiempirical  formalism based on Jain and Khare
approach [18–21]. At these fixed incident electron energies, double
differential cross sections as a function of angle and secondary electron
energy have also been calculated. This is the only formulation that enables us
to evaluate the energy and angle dependent cross sections for molecules corresponding to the formation of cations in electron-molecule collisions. To the best of our knowledge, no experimental and/or theoretical data is available for comparison to the present calculations for differential cross sections. Thus the corresponding derived integral cross sections in terms of the partial
ionization cross sections leading to the formation of various cations SiHn+ (n= 0–3), H2+, and H+ through dissociation of
SiH4 by electron collision are compared with the available
experimental and theoretical data. The present results alongwith the total
ionization cross sections show good agreement with the experimental and
theoretical results. In addition, we have calculated the ionization rate
coefficients corresponding to the produced cations using the computed
ionization cross sections and Maxwell-Boltzmann distribution for the electrons
as a function of electron temperature/energy. Instead of cross sections itself,
ionization rate coefficients corresponding to the produced cations are more
important in various plasma applications, gas discharge, and flowing afterglow
studies [22, 23].


## 2. Theoretical

The present calculations are carried out using the modified
semiempirical formalism developed by  Pal et al. (see discussion in
[18–21]). In brief, the single differential cross sections in the
complete solid angle (Ω=4π=∫2πsinθⅆθ) as a function of secondary electron
energy ε corresponding to the production of  ith type of ion in the
ionization of a molecule by incident electron of energy E is given by (1)Qi(E,W)=a02RE[(1−ε(E−Ii))RWdfi(W,0)dWln[1+Ci(E−Ii)]+RESi(E−Ii)(ε03+εi3)(ε−ε2(E−ε)+ε3(E−ε)2)]×∫2πsinθdθ, where W(=ε+Ii) is defined as energy loss suffered by the
incident electron. Ii,a0,ε0,Ci,Si, and R are the ionization threshold for the production of ith type of
ion, the Bohr radius, energy parameter, collision parameter, number of
ionizable electrons, and the Rydberg constant, respectively.In the present
formulation, the dipole oscillator strengthsdfi/dW are the key
parameters. The oscillator strength is directly proportional to the
photoionization cross section [3]. We have used partial photoionization cross section data set in the energy range from 12 to 52 eV provided by Brion et al. [6]
using (e,e) spectroscopy. The accuracy of the determined
oscillator strength scales was estimated to be better than ±5%. In the photon
energy range 52–180 eV, we have used their measured total valence photoabsorption
oscillator strength data [5], and for higher photon energy range W> 180 eV the
same were extrapolated by Thomas-Reiche-Kuhn (TRK) sum rule (within ±10% error
bars) (see, e.g., [5, 6]). The total photoabsorption cross sections have been distributed into ionic fragments considering the constant ionization
efficiency to be 1.0 above the dipole breakdown limit of ~25 eV. However, its
evaluation is possible quantum mechanically using the suitable wave functions
and transition probabilities corresponding to the production of cations. In
case of dissociative ionization of polyatomic molecule SiH4, we have
no reliable probabilities corresponding to different dissociative ionization
processes. The collision parameter Ci (=0.08621/eV) and energy
parameter ε0(=30eV) are evaluated as for other polyatomic
molecules 	[18–21]. The vertical onsets or the ionization potentials
corresponding to the various cations are also given alongwith the
photoionization measurements 	[5, 6]. In the present evaluations of cross sections,
the estimated uncertainty is more or less the same as for the measurement of
photoionization cross sections.The double differential cross sections as a function of energy and angle were evaluated by the differentiation of (1) with respect to the solid angle Ω as follows:(2)Qi(E,W,θ)=dQi(E,W)dΩ.The double differential cross sections are angular dependant in all the scattering geometries, and hence the oscillator strength must be angle dependent. In this context, we have used the angular oscillator strengths that were derived in the optical limit (Bethe regime) where angular-momentum-transferK→0 (e.g., see [19] for a detailed
discussion):(3)dfi(W,0,θ)dWdΩ=14πdfi(W,0)dW[1+β2(3cos2θ−1)], where β is an energy dependent asymmetric parameter.
Its evaluation is difficult due to the lack of wave functions of molecular ions
in ground and excited states. In valance shell ionization of SiH4,
we have computed β as the ratio of the Bethe spectral transitions
Si(W) to the dipole matrix squared Mi2(W) [24, 25]. The oscillator strength appeared in (1)
is simply a derived form of (3)
in the forward scattering corresponding
K→0 and θ→0.The partial ionization cross section
is obtained by the integration of the energy dependent single differential
cross sections (1) over the entire energy loss as follows:(4)Qi(E)=∫IiEQi(E,W)dW, and the counting or total ionization cross section is obtained by (5)QT(E)=∑iQi(E).In plasma processes, the ionization rate coefficients are
important quantities which are determined by using our calculated partial and
total ionization cross sections and Maxwell-Boltzmann distribution of
temperature/energy [26, 26] as follows:(6)Ri=∫−∞+∞4π(12πmkT)3/2me(−E/kT)Qi(E)EdE, where k,T, and m are the Boltzmann constant, absolute temperature, and
mass of the electron, respectively.


## 3. Results and Discussion

The electronic configuration of the ground state of silane isK2L8(3a1)2(2t2)6A11.
Ionization of the outermost 2t2 electrons results in a degenerate T22 ion state, which is subject to the Jahn-Teller effect. Experimentally, 2t2 electron ionization or the parent ion formation, observed by Berkowitz et al. [27] at adiabatic ionization potential, is the best evidence for the photogeneration of
a bound SiH4+ species. The amount of molecular ions
detected experimentally were approximately two orders of magnitude lower than
the intensities of the dissociative ionization products SiHn+ (n = 0–3). On the other hand, due to the instability of SiH4+,
it decomposes into other stable ions. The present calculations for cross
sections are based on the experimental data for the oscillator strengths
(photoionization cross sections) [5, 6]. The calculations for differential and
integral cross sections were carried out for the produced cations SiHn+ (n = 0–3), H2+, and H+ through electron
dissociation of SiH4.The partial single differential cross sections as a function
of secondary electron energy in terms of energy lossW at fixed incident
electron energies of 100 and 200 eV are presented in Figure 1.  More qualitative results are also presented in the Platzman plot of Y(W). The Y(W)  parameter is the ratio of the calculated differential cross section and the Rutherford cross section with energy loss W
in the dipole energy range. Qualitatively, Y(W) corresponds to the effective
number of electrons participating in ionizing collisions. In the present formulation (1), the first Born-Bethe part for slow secondary electron, corresponds to the growing contribution of the dipole-allowed interaction (known as glancing collision) and resembles the photoionization cross section. The second part accounts for the electron exchange effect and corresponds to the nondipole part that defines knock-on collisions. For fast secondary electrons, it is an adaptation of the Rutherford cross sections for the free electrons.(a) Single differential cross sections (in the units of10−20cm2/eV) as a function of energy loss W (=Ii+ε), for the production of cations from electron impact ionization of SiH4 at constant electron impact energy of 100 eV. (b) Same as (a) but at E=200eV. (c) The solid lines show the trends of the calculated ratios of cross sections in Platzman plot Y(W) in half energy range at 100 and 200 eV.
(a)(b)(c)In Figure2, we show the behavior of double differential
cross sections with angle varying from 10° to 180° at constant secondary electron
energies of 10 and 20eV and fixed primary electron energies of 100 and 200eV.
The calculations as a function of energy loss W, at constant angles of 30° and 60° at the same incident electron
energies, are presented in Figure 3. The 3D profiles of the total double cross
sections as a function of secondary electron energy (in the range of 5eV to
W/2) and angle (10° to 180°) at 100 and  200eV are also shown
in Figure 4. To the best of our knowledge, no experimental data is available to
compare the present results for differential cross sections. However, the qualitative
behavior of the cross section is the same as for other molecules investigated
[18–21]. The energy dependent cross sections are symmetric at W/2 where the energies of primary and the secondary electrons are almost equal. The figures clearly show the weight contribution of the molecular and atomic cations. The cross sections for molecular ions are much larger than the atomic ions. In dipole-induced breakdown scheme for the photoionization of silane molecule, the major ions SiH3+ and SiH2+ are produced from 2t2−1 state, and the H+ and H2+ ions are produced
from 3a1−1 state, while the production of the SiH+ and Si+ ions comes from the contribution of the both states. In the
threshold energy range, the atomic photoionization cross sections include the
contribution of the structures and many body states produced near onsets [5, 6] which are reflected in the present calculations for energy dependent differential cross sections.(a) The double differential cross sections (in the units of10−20cm2/eV-sr) as a function of angle θ at (ε=10eV and E=100eV) corresponding to the production of various cations from electron impact ionization of SiH4. (b) Same as (a) but at (ε=20eV and E=100eV). (c) Same as (a) but at (ε = 10 eV and E=200eV). (d) Same as (a) but at (ε=20eV and E=200eV).
(a)(b)(c)(d)(a) The double differential cross sections (in the units of10−20cm2/eV-sr) as a function of energy loss W at (θ=30° and E=100eV) for the production of cations from electron impact ionization of SiH4. (b) Same as (a) but at (θ=60° and E=100eV).  (c) Same as (a) but at (θ=30° and E=200eV). (d) Same as (a) but at (θ=60° and E=200eV).
(a)(b)(c)(d)(a)  3D profiles of the double differential cross sections (in the units of10−20cm2/eV-sr) as a function of secondary electron energy ε and angle θ at E=100eV. (b) Same as (a) but at E=200eV.
(a)(b)Because of the lack of experimental
data for differential cross sections, the corresponding derived partial cross
sections and their sum (the total ionization cross sections), from ionization
threshold to 1000 eV, become important. The numerical values of our calculated
partial and the total ionization cross sections are presented in Table1. In Figure 5, we have presented the comparison of our  partial ionization cross sections
with the established experimental data sets of Chatham et al. [8], Krishnakumar and Srivastava [9], and Basner et al. [10]. It is noted that for the SiHn+ (n = 0–3) ions, our results are in good agreement with the experimental data [8, 10] within their composite error bars (The
measurements of Chatham et al., Krishnakumar and Srivastava, and Basner et al.
account the experimental uncertainties 30%–40%, 15%, and 10%–20%, resp.). On the other hand the experimental data of Haaland [11] which is a scaled value of Chatham et al. [8] and the Morrison and Traeger [12] are lower by a factor of two to three than the present calculations as well as the other experimental data sets [8–10]. Hence for the sake of reasonable shape and size of the figure, the data
of Haaland is not shown in the figure. For the H2+ and H+ ions, a considerable disagreement with the experimental data has been noticed.
However, our calculations for H2+ agree well with recent
data of Basner et al., while the data set of Chatham (not shown) is much higher
than the present calculations and the data of Basner et al. The data of Chatham
et al. is about 3 and 4 times higher than Basner et al. and our calculations
for H+, respectively. No identification of H+ and H2+ ions was made in earlier investigations [7, 11, 12]. In case of all dissociative
fragment ions, the experimental data of Krishnakumar and Srivastava [9] show disagreement with our calculations as well as other experimental data [8, 10] in the energy range from 40 to 500 eV. Nevertheless, considering that this is a comparison with the absolute data where error bars of these sets of data are easily in the 10% to 40% regime (in particular taking into account that the calculations are depending on the accuracy of the experimental input
parameters), the agreement is acceptable.Table 1
Table for the partial ionization cross sections forSiH4 by electron impact.
E (eV)Qi(E)(10−16cm2)SiH3+SiH2+SiH+Si+H2+H+Total200.711.000.140.172.02301.391.810.430.340.010.044.02401.632.100.560.450.020.114.87501.712.180.600.490.020.155.15601.702.160.600.500.030.165.16701.662.110.590.500.030.175.05801.612.030.570.480.030.174.89901.541.950.540.470.030.174.701001.481.870.520.450.030.174.511251.341.690.460.410.020.164.071501.221.530.410.370.020.143.692001.031.290.340.310.020.123.112500.891.120.290.260.020.112.693000.790.990.260.230.010.092.374000.640.810.210.190.010.081.935000.550.680.170.160.010.061.636000.480.600.150.140.010.061.427000.420.530.130.120.010.051.268000.380.480.120.110.010.041.149000.350.440.110.100.010.041.03Partial ionization cross sections (in the units of10−16cm2) for electron impact ionization of SiH4 (designated with solid lines) in comparison with the experimental data designated by ×—Chatham et al. [8], —Krishnakumar and Srivastava [9], and —Basner et al. [10].
(a)(b)(c)(d)(e)(f)Figure6 shows good agreement of the partial cross sections when added up to a total cross section with the experimental data sets [8–10] along with the experimental data of Perrin et al. [7] which is available only at 100 eV and theoretical cross section data sets [13, 16, 17] in the complete energy
range covered in the calculations. Recently, Malcolm and Yeager [28] have reviewed the BEB model [13] employing the more accurate ionization potential evaluation from Hartree-Fock theorem using cc-pVDZ and cc-pVTZ basis sets with
multiconfigurational self-consistent field.  This approach is a viable method for calculating electron impact ionization cross sections for systems where
Koopman’s theorem is known to be unreliable, or no experimental data is
available. However, in case of this open channel molecule SiH4, these
calculations for total ionization cross sections do not have a significant
effect on the data derived from BEB (see [28] for detailed discussion). The data of Krishnakumar and Srivastava [9] for total cross sections show similar trends as for dissociative ionization cross sections. The total ionization cross sections reported by Haaland [11] are much smaller than all the measured data sets and the calculations. Hence for the sake of brevity, we have not shown the same data [11] in the comparison.Figure 6
Presently calculated total ionization cross sections (in the units of10−16cm2) for electron impact ionization of SiH4 (designated by solid line) in comparison with the various experimental data sets designated by —Perrin et al. [7], ×—Chatham et al. [8], —Krishnakumar and Srivastava [9], and —Basner et al. [10] and theoretical data sets designated by  BEB calculations [13, 28] and —Joshipura et al. [16, 17].The present calculations for partial and total ionization
cross sections satisfy the necessary consistency checks to access their
consistency and reliability. The consistency checks are derived from the fact
that the total electron impact ionization cross section (i) is equal to the
charged-weighted sum of the partial ionization cross sections and (ii) may be
obtained by integration of the differential cross sections over secondary
electron energies and angles. The former condition is used in the summation
method for calibration purposes and the later fact in the use of Platzman
plots. Both relationships allow one to check the reliability of the absolute
magnitude and the energy dependence of ionization cross sections under
consideration.  In the low-energy limit
close to the onset of ionization, the shapes of the partial and total electron
impact ionization cross section curves are governed by a threshold law, which is usually expressed in the form ofQ(E)~(E−Ii)Z, where Z is the
charge state of the ion. The precise shape of the cross section in this region
is especially important in determination (by extrapolation) of respective
ionization thresholds, to compare with those derived by other means [29–31].In relation to the applications, in particularly to plasma
processes, ionization rate coefficients are rather more desirable than
ionization cross sections. We have evaluated a set of ionization rate
coefficients as a function of electron temperature in the units of energy for
the individual cations produced in electron collision with theSiH4 molecule. The calculations are made using the calculated ionization cross
sections and Maxwell-Boltzmann energy distribution, and the results are presented
in Figure 7 along with Table 2.Table 2
Table for the ionization rate coefficients (in the
units of10−20cm3/s) corresponding to the formation of
cations in electron dissociative ionization of SiH4.
E (eV)SiH3+SiH2+SiH+Si+H2+H+Total206.228.700.811.1516.873043.4256.6911.369.480.110.50121.5640111.66143.3634.3528.420.774.45323.0150204.56260.6566.5255.542.0012.21601.4755257.82327.7784.9871.462.7917.27762.0860314.43399.05104.5588.533.6722.97933.2170434.79550.47145.91125.095.6835.811297.7580560.64708.68188.81163.447.8849.921679.3890688.18868.92231.94202.3010.1964.662066.18100814.631027.73274.36240.7612.5279.582449.581251112.681401.83373.04330.9218.15115.553352.171501369.771724.34456.55407.8623.10147.204128.822001732.032178.41570.58514.0630.22192.615217.902501899.272387.60619.23560.6133.63214.405714.743001911.952402.83617.83561.0934.09217.375745.164001659.732085.17528.62481.8629.72189.534974.645001270.811596.30400.20365.5822.74145.033800.66600900.731131.32281.06257.0916.08102.552688.84700605.90760.96187.62171.7810.7968.771805.81800392.47492.89120.73110.626.9644.411168.08900247.07310.2875.5769.274.3727.87734.43Figure 7
Ionization rate coefficients as a function of the temperature forSiH4.


## 4. Conclusion

The calculations for
differential cross sections as a function of secondary electron energy and
angle at fixed impinging electron energy, corresponding to the formation of various
cations in electron impact dissociative ionization of theSiH4 molecule, have been carried out by employing a semiempirical formalism based on
the Jain-Khare approach. The calculations were made for the production of various
cations produced via dissociative electron ionization of SiH4.
In absence of any data for differential cross sections, the corresponding
derived partial ionization cross sections revealed a good agreement with
available experimental data sets. However in case of minor H+ and H2+ ions, some discrepancy was noticed. The ionization rate coefficients, a
key parameter in plasma modeling, have been evaluated using a Maxwell energy
distribution. The present evaluations for electron ionization cross sections
and rate coefficients provide a contribution to the knowledge of various plasma
processes.



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*Source: 309292-2009-05-10.xml*