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Photoionization cross-sections and asymmetry parameters for GeH4 in the VUV region
Journal of Electron Spectroscopy and Related Phenomena 135 (2004) 149–153
Photoionization cross-sections and asymmetry parameters for GeH4 in the VUV region A.S. dos Santos a , L.E. Machado a,∗ , L.M. Brescansin b , E.M. Nascimento c , M.-T. Lee d a
d
Departamento de F´ısica, Universidade Federal de São Carlos (UFSCar), Via Washington Luiz km 235, São Carlos 13565-905, SP, Brazil b Instituto de F´ısica “Gleb Wataghin”, UNICAMP, Campinas 13083-970, SP, Brazil c Instituto de F´ısica, UFBa, Salvador 40210-340, BA, Brazil Departamento de Qu´ımica, Universidade Federal de São Carlos (UFSCar), São Carlos 13565-905, SP, Brazil Received 9 January 2004; received in revised form 5 March 2004; accepted 5 March 2004 Available online 30 April 2004
Abstract We report calculated cross-sections and photoelectron angular distributions for photoionization out of the two outermost orbitals of GeH4 for photon energies ranging from near threshold to 50 eV. The iterative Schwinger variational method in the exact static-exchange level is used to obtain the continuum photoelectron orbitals. Our calculated results show a remarkable similarity between the photoionization properties of the outermost valence orbitals of silane and germane. © 2004 Elsevier B.V. All rights reserved. PACS: 33.80.−b; 33.80.Eh Keywords: Photoionization cross-sections; Photoionization asymmetry parameters; GeH4 ; SVIM
1. Introduction XH4 (X = C, Si, Ge) molecules are important in various areas of application such as environmental physics [1] and semiconductor technology [2,3]. In particular, germane is widely used in chemical vapor deposition for the fabrication of microcrystalline and amorphous Six Ge1−x thin films which have been demonstrated to be potentially important for solar cells [4,5] and transistors [6,7]. It has been observed that the introduction of Ge into the silicon-based films lowers the optical band gap. However, some studies have shown that the addition of germanium to the silicon-based films results in a drastic deterioration of their electric properties [8,9]. As a consequence, the improvement of such properties in SiGe alloys has become subjects of many recent researches [10–12]. In addition, germane has been detected ∗ Corresponding author. Tel.: +55-16-2608226/214; fax: +55-16-2614835. E-mail address: [email protected] (L.E. Machado).
0368-2048/$ – see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.elspec.2004.03.001
in the atmospheres of Jupiter and Saturn [13,14] and is expected to be of major importance in the photochemistry of the atmospheres of these planets. Despite that, studies on photoabsorption and photoionization of GeH4 have received little attention, both experimentally and theoretically in the last three decades. Besides the pioneering works of Pullen et al. [15] and Potts and Price [16] on the photoelectron spectra of germane, a recent work [17] was published, reporting interesting data on the vacuum ultraviolet (VUV) photochemistry of methane, silane and germane. To our knowledge, only one article reported measured VUV photoabsorption and photoionization cross-sections for germane [18]. Even so, the photoionization cross-sections reported in that paper are restricted to a very narrow energy range close to the threshold. From the theoretical point of view, studies on photon interaction with this molecule have also been scarce. The electronic structure of germane, as well as the Jahn–Teller distortion of the ground-state GeH4 + , was studied by
150
A.S. dos Santos et al. / Journal of Electron Spectroscopy and Related Phenomena 135 (2004) 149–153
Caballol et al. [19], Kudo and Nagase [20], and Frey and Davidson [21]. Nevertheless, to our knowledge, there is no theoretical article on photoionization of germane available in the literature. In this work, we report a theoretical study on the photoionization of germane in the VUV region. More specifically, photoionization cross-sections (σ) and asymmetry parameters (β) for photoionization from the two outermost valence orbitals are reported, for photon energies ranging from near the ionization thresholds up to 50 eV. Due to the lack of photoionization data for GeH4 , our σ and β are compared with those of XH4 (X = C, Si) molecules, when available. The present study makes use of the exact static-exchange (SE) potential, whereas the Schwinger variational iterative method (SVIM) [22] is used to solve the continuum Lippmann–Schwinger equation. The paper is organized as follows. In Section 2, we briefly review the method used for obtaining the photoelectron orbitals along with some numerical details of the calculation. In Section 3, we present our calculated σ and β, and finally, in Section 4, we summarize our conclusions.
2. Theory and calculation Details of the method have been given elsewhere [22,23] and only its essential aspects will be outlined here. The photoelectron differential cross-sections averaged over molecular orientations are given by: dσ (L,V) σ (L,V) (L,V) [1 + β ˆ = P2 ( cos θ)], k dΩkˆ 4π
(1)
where σ (L,V) is the total photoionization cross-section, obtained with the length (L) or velocity (V ) form of the dipole moment operator. For non-linear molecules, σ (L,V) is given by: σ (L,V) =
4π2 E pµ(L,V) 2 |Ilhv | , 3c
(2)
pµlhv
where E is the photon energy, p one of the irreducible representations (IR) of the symmetry group of the molecule and h distinguishes between different bases for the same IR corresponding to the same value of l. The index µ labels components of vectors belonging to the same IR and v designates components of the dipole moment operator. (L,V) The quantity β ˆ appearing in Eq. (1) can be written as: k
(L,V) k
βˆ
=
3 5
1 pµ(L,V) 2 | pµlhv |Ilhv
p µ
p µ
(1100|20)
pµlhvmmv p µ l h v m mv
pµ(L,V)
are the partial-wave components of the dywhereas Ilhv (L,V) namical coefficients I : k,nˆ 1/2 4π pµ(L,V) pµ ˆ p µ (L,V) I = Ilhv Xlh (k)X1vv v (n), ˆ (4) k,nˆ 3 pµlhv
where (L) k,nˆ
I
(−) k
= (k)1/2 Ψi | r · n|Ψ ˆ ,
and (V) k,nˆ
I
=
(k)1/2 (−) · n|Ψ
Ψi |∇ ˆ . k E
(6)
pµ
In Eq. (4), Xlh are the symmetry-adapted functions [24] that are related to the usual spherical harmonics by: pµ pµ Xlh (ˆr ) = blhm Ylm (ˆr ), (7) m
whereas in Eqs. (5) and (6) Ψi is the target ground state (−) wave function, Ψ the final state (incoming-wave normalk ized) wave function of the system (ion plus photoelectron), nˆ represents the unit vector in the direction of polarization of the radiation and k the photoelectron momentum. In the present study, Ψi is a one-determinant wave function calculated in the Hartree–Fock level. The final molecular state is described by a single electronic configuration in which the ionic orbitals are constrained to be identical to those of the initial ground state. In this approximation, the photoelectron orbital φk ( r ) is a solution of the one-electron Schrödinger equation: + 1 k2 ]φ ( r ) = 0, [− 1 ∇ 2 + VN−1 ( r ; R) (8) 2
2
k
where VN−1 is the SE potential of the molecular ion and φk ( r ) satisfies appropriate boundary conditions. To proceed, Eq. (8) is rewritten in an integral form, the Lippmann–Schwinger equation, and is solved using an iterative procedure based on the Schwinger variational principle. This method provides continuum wave functions that are shown to converge to the exact solutions for a given projectile–target interaction potential [25]. The numerical calculation of the scattering wave functions was performed using our SVIM codes, developed for molecular systems with symmetries reducible to C2v . The potential field of the 2 T2 (3t−1 2 ) ionic state was constructed assuming that the C2v -reduced components (A1 , B1 and B2 ) of the t2 hole orbital were equally depopulated. In this case, the cross-sections were evaluated by adding up the contributions from the C2v -reduced components of the t2 orbital, while the asymmetry parameters were obtained as an average over pµ(L,V) pµ(L,V) Il h v
(−1)m −mv Ilhv
∗
pµ p µ
blhm bl h m
v v × blvm b v v [(2l + 1)(2l + 1)]1/2 (11 − mv mv |2M )(l l00|20)(l l − m m|2 − M ), v lvm v
(5)
(3)
A.S. dos Santos et al. / Journal of Electron Spectroscopy and Related Phenomena 135 (2004) 149–153
those components taking the corresponding cross-sections as weights. All matrix elements arising in the SVIM calculation were evaluated using a single-center expansion truncated at lmax = 18 and all allowed h values, i.e. h ≤ l, were retained for a given l. In order to verify the convergence in the partial-wave expansion, some test calculations were carried out also with lmax = 20. No significant differences were noticed in the calculated cross-sections with these two sets of cutoff parameters, showing that all the calculations had already converged. The convergence in the iterative procedure was also studied. All the final results shown below were converged within six iterations. We have used the values of 11.34, 11.80 and 12.26 eV for the IPs of the three Jahn–Teller components of the 3t2 orbital of GeH4 , and the value of 18.40 eV for the IP of the 4a1 orbital [16].
3. Results and discussion In Fig. 1(a) and (b), we present our calculated results for σ and β, respectively, obtained in both dipole length (DL) and dipole velocity (DV) forms, for the photoionization out
151
of the 3t2 orbital of GeH4 for photon energies ranging from near threshold up to 50 eV. The same quantities, but for the 4a1 orbital are shown in Fig. 2(a) and (b). In Fig. 1(a), our calculated σ show a step-like sharp rise near threshold, arising from successive vibronic contributions associated to the Jahn–Teller distortion of the ground GeH4 + ionic state. The difference between the DL and DV results, seen in Figs. 1 and 2, are known to be generally due to the neglect of both electronic correlation in the target wave function and multichannel coupling effects in the photoelectron wave function. The influence of the former has been studied by Lucchese et al. [22] and that of the interchannel effects was recently studied by Cacelli et al. [26]. These authors have shown that the difference between the DL and DV results is significantly reduced by the inclusion of these effects. In general, the agreement between our calculated βL and βV is better than that between σ L and σ V . Unfortunately, no experimental and/or theoretical results on the photoionization of GeH4 in this range of energy is available in the literature. For this reason, in Fig. 3, we compare our calculated σ L and βL results for the 3t2 orbital of GeH4 with the corresponding calculated data for the photoionization of the outermost t2 orbitals of CH4 [27] and SiH4 [23] as a function of the photoelectron kinetic energy. The same type of comparison is also made in Fig. 4, but for the outermost a1 orbitals. The experimental σ of Brion [28], derived from the
(a) (a)
(b) Fig. 1. Photoionization cross-sections (a) and asymmetry parameters (b) + for the lowest 2 T2 (3t−1 2 ) state of GeH4 . Solid line presents results in DL form and dashed line presents results in DV form.
(b) Fig. 2. Same as Fig. 1, but for the lowest 2 A1 (4a1−1 ) state of GeH4 + .
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A.S. dos Santos et al. / Journal of Electron Spectroscopy and Related Phenomena 135 (2004) 149–153
(a)
(a)
(b)
(b)
Fig. 3. Photoionization cross-sections (a) and asymmetry parameters (b) for the lowest 2 T2 states of CH4 + , SiH4 + and GeH4 + . Solid line presents DL results for the 3t2 orbital of GeH4 , dashed line presents DL results for the 1t2 orbital of CH4 [27], short-dashed line presents DL results for the 2t2 orbital of SiH4 [23], and full circles present experimental results of Brion for silane [28].
Fig. 4. Photoionization cross-sections (a) and asymmetry parameters (b) for the lowest 2 A1 states of CH4 + , SiH4 + and GeH4 + . Solid line presents DL results for the 4a1 orbital of GeH4 , dashed line presents DL results for the 2a1 orbital of CH4 [30], short-dashed line presents DL results for the 3a1 orbital of SiH4 [23], and full circles present experimental results of Brion for silane [28].
total photoabsorption data of Cooper et al. [29] for silane are also included for comparison in both figures. For the t2 orbital a remarkable similarity among the calculated σ is seen in Fig. 3(a), particularly between the results of SiH4 and GeH4 . It is interesting to notice that the experimental σ of Brion [28] for silane agree quite well with our calculated σ L for germane. This good agreement comes in support of the speculation made in our previous work [23] that the photoionization of the valence t2 orbitals of the XH4 (X = C, Si, Ge, Sn, Pb) molecules would be similar. In Fig. 4, a general good agreement is also seen between the calculated σ and β for the photoionization out of the outermost a1 orbitals in silane and germane. In addition, the calculated photoionization cross-sections for silane agree well with the experimental data of Brion [28], except at photon energies near threshold, where the latter show a decreasing behavior towards the threshold, in contrast with the calculation. The physical origin of this discrepancy is still unclear. Also, a qualitative disagreement is observed between σ L and βL for these molecules and those for methane [30]. Specifically, it is seen in Fig. 4(b) that the values of β, calculated by Stener and Decleva [30] for the photoionization out of the 2a1 orbital of CH4 , are nearly 2 in the entire energy
range. This behavior suggests the atomic character of this orbital.
4. Conclusions We have reported cross-sections and asymmetry parameters for the photoionization of the two outermost valence orbitals of GeH4 , in both DL and DV forms. To our knowledge, this is the first theoretical study that reports ab initio results of these physical quantities for photoionization of germane at the exact SE level. Also, an evident similarity was observed between the photoionization properties of the 1t2 orbital of methane, the 2t2 orbital of silane and the 3t2 orbital of germane. Indeed, the cross-sections as well as the asymmetry parameters, are very close both in shape and magnitude, which may indicate that the photoionization of the outermost valence t2 orbitals of XH4 (X = C, Si, Ge, Sn and Pb) molecules would be similar. On the other hand, although a very clear similarity was observed between the photoionization properties of the 3a1 orbital of silane and the 4a1 orbital of germane, this was not the case for the 2a1 orbital of methane. This fact shows that the observed dis-
A.S. dos Santos et al. / Journal of Electron Spectroscopy and Related Phenomena 135 (2004) 149–153
crepancy is possibly due to the atomic character of the 2a1 orbital of methane. Acknowledgements This research was partially supported by Brazilian Agencies CNPq, FAPESP, CAPES/PADCT and FAEP/ UNICAMP. References [1] K. Takahashi, M. Konagai (Eds.), Handbook of Photoexcited Processing, Science Forum, Tokyo, Japan, 1987 (in Japanese). [2] The Committee of Electronic Materials and Technology (Ed.), Report of Investigation and Research on New Electronic Materials XIII, Report of Investigation on Photoexcited Processing Technology 2, Nippon Denshi Kogyo Shinko Kyokai, Tokyo, Japan, 1987 (in Japanese). [3] F.A. Houle, T.F. Deutsch, R.M. Osgood Jr. (Eds.), Extended Abstracts: Laser Chemical Processing of Semiconductor Devices, Material Research Society, Pittsburg, 1984. [4] S. Guha, Y. Yang, P. Nath, M. Hack, Appl. Phys. Lett. 49 (1986) 218. [5] J. Meier, R. Flückinger, H. Keppner, A. Shah, Appl. Phys. Lett. 65 (1994) 860. [6] J.I. Woo, H.J. Lim, J. Jang, Appl. Phys. Lett. 65 (1994) 1664. [7] Y. Chen, K. Pangal, J.C. Sturm, S. Wargner, J. Non-Cryst. Solids 266–269 (2000) 1274. [8] K.D. Mackenzie, J.R. Eggert, D.J. Leopold, Y.M. Li, S. Lin, W. Paul, Phys. Rev. B 31 (1985) 2198. [9] M.E. Gueurnier, J.P. Kleider, R. Brüggemann, S. Lebib, P. Roca i Cabarrocas, R. Meaudre, B. Canut, J. Appl. Phys. 92 (2002) 4959.
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[10] S. Miyazaki, H. Takahashi, H. Yamashita, N. Narasaki, M. Hirose, J. Non-Cryst. Solids 299–302 (2002) 148. [11] G.H. Bauer, F. Voigt, R. Carius, M. Krause, R. Brüggemann, T. Unold, J. Non-Cryst. Solids 299–302 (2002) 153. [12] J. Braun, H. Rauscher, R.J. Behm, Surf. Sci. 531 (2003) 265. [13] B.P. Bezard, P. Drossart, E. Lellouch, J.-P. Maillard, Astrophys. J. 346 (1989) 509. [14] P.G.J. Irwin, F.W. Taylor, R.W. Carlson, K.H. Baines, A. Weir, P. Cameron Smith, S. Calcutt, T. Encrenaz, P. Drossart, M. Ross Serote, E. Lellouch, Adv. Space Res. 23 (1999) 1623. [15] B.P. Pullen, T.A. Carlson, W.E. Moddeman, G.K. Schweitzer, W.E. Bull, F.A. Grimm, J. Chem. Phys. 53 (1970) 768. [16] A.W. Potts, W.C. Price, Proc. R. Soc. Lond. Ser. A 326 (1972) 165. [17] P.A. Cook, M.N.R. Ashfold, Y.-J. Jee, K.-H. Jung, S. Harich, X. Yang, Phys. Chem. Chem. Phys. 3 (2001) 1848. [18] C.Y.R. Wu, F.Z. Chen, D.L. Judge, J. Chem. Phys. 99 (1993) 1531. [19] R. Caballol, J.A. Català, J.M. Problet, Chem. Phys. Lett. 130 (1986) 278. [20] T. Kudo, S. Nagase, Chem. Phys. Lett. 148 (1988) 73. [21] R.F. Frey, E.R. Davidson, J. Chem. Phys. 89 (1988) 4227. [22] R.R. Lucchese, G. Raseev, V. McKoy, Phys. Rev. A 25 (1982) 2572. [23] L.E. Machado, M.-T. Lee, L.M. Brescansin, J. Chem. Phys. 110 (1999) 7228. [24] P.G. Burke, N. Chandra, F.A. Gianturco, J. Phys. B 5 (1972) 2212. [25] R.R. Lucchese, D.K. Watson, V. McKoy, Phys. Rev. A 21 (1980) 421. [26] I. Cacelli, R. Moccia, R. Montuoro, Phys. Rev. A 63 (2000) 012512. [27] M. Braunstein, V. McKoy, L.E. Machado, L.M. Brescansin, M.A.P. Lima, J. Chem. Phys. 89 (1988) 2998. [28] C.E. Brion, 1999, private communication. [29] G. Cooper, G.R. Burton, W.F. Chan, C.E. Brion, Chem. Phys. 196 (1995) 293. [30] M. Stener, P. Decleva, J. Electron Spectrosc. Relat. Phenom. 104 (1999) 135.
Photoionization cross-sections and asymmetry parameters for GeH4 in the VUV region A.S. dos Santos a , L.E. Machado a,∗ , L.M. Brescansin b , E.M. Nascimento c , M.-T. Lee d a
d
Departamento de F´ısica, Universidade Federal de São Carlos (UFSCar), Via Washington Luiz km 235, São Carlos 13565-905, SP, Brazil b Instituto de F´ısica “Gleb Wataghin”, UNICAMP, Campinas 13083-970, SP, Brazil c Instituto de F´ısica, UFBa, Salvador 40210-340, BA, Brazil Departamento de Qu´ımica, Universidade Federal de São Carlos (UFSCar), São Carlos 13565-905, SP, Brazil Received 9 January 2004; received in revised form 5 March 2004; accepted 5 March 2004 Available online 30 April 2004
Abstract We report calculated cross-sections and photoelectron angular distributions for photoionization out of the two outermost orbitals of GeH4 for photon energies ranging from near threshold to 50 eV. The iterative Schwinger variational method in the exact static-exchange level is used to obtain the continuum photoelectron orbitals. Our calculated results show a remarkable similarity between the photoionization properties of the outermost valence orbitals of silane and germane. © 2004 Elsevier B.V. All rights reserved. PACS: 33.80.−b; 33.80.Eh Keywords: Photoionization cross-sections; Photoionization asymmetry parameters; GeH4 ; SVIM
1. Introduction XH4 (X = C, Si, Ge) molecules are important in various areas of application such as environmental physics [1] and semiconductor technology [2,3]. In particular, germane is widely used in chemical vapor deposition for the fabrication of microcrystalline and amorphous Six Ge1−x thin films which have been demonstrated to be potentially important for solar cells [4,5] and transistors [6,7]. It has been observed that the introduction of Ge into the silicon-based films lowers the optical band gap. However, some studies have shown that the addition of germanium to the silicon-based films results in a drastic deterioration of their electric properties [8,9]. As a consequence, the improvement of such properties in SiGe alloys has become subjects of many recent researches [10–12]. In addition, germane has been detected ∗ Corresponding author. Tel.: +55-16-2608226/214; fax: +55-16-2614835. E-mail address: [email protected] (L.E. Machado).
0368-2048/$ – see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.elspec.2004.03.001
in the atmospheres of Jupiter and Saturn [13,14] and is expected to be of major importance in the photochemistry of the atmospheres of these planets. Despite that, studies on photoabsorption and photoionization of GeH4 have received little attention, both experimentally and theoretically in the last three decades. Besides the pioneering works of Pullen et al. [15] and Potts and Price [16] on the photoelectron spectra of germane, a recent work [17] was published, reporting interesting data on the vacuum ultraviolet (VUV) photochemistry of methane, silane and germane. To our knowledge, only one article reported measured VUV photoabsorption and photoionization cross-sections for germane [18]. Even so, the photoionization cross-sections reported in that paper are restricted to a very narrow energy range close to the threshold. From the theoretical point of view, studies on photon interaction with this molecule have also been scarce. The electronic structure of germane, as well as the Jahn–Teller distortion of the ground-state GeH4 + , was studied by
150
A.S. dos Santos et al. / Journal of Electron Spectroscopy and Related Phenomena 135 (2004) 149–153
Caballol et al. [19], Kudo and Nagase [20], and Frey and Davidson [21]. Nevertheless, to our knowledge, there is no theoretical article on photoionization of germane available in the literature. In this work, we report a theoretical study on the photoionization of germane in the VUV region. More specifically, photoionization cross-sections (σ) and asymmetry parameters (β) for photoionization from the two outermost valence orbitals are reported, for photon energies ranging from near the ionization thresholds up to 50 eV. Due to the lack of photoionization data for GeH4 , our σ and β are compared with those of XH4 (X = C, Si) molecules, when available. The present study makes use of the exact static-exchange (SE) potential, whereas the Schwinger variational iterative method (SVIM) [22] is used to solve the continuum Lippmann–Schwinger equation. The paper is organized as follows. In Section 2, we briefly review the method used for obtaining the photoelectron orbitals along with some numerical details of the calculation. In Section 3, we present our calculated σ and β, and finally, in Section 4, we summarize our conclusions.
2. Theory and calculation Details of the method have been given elsewhere [22,23] and only its essential aspects will be outlined here. The photoelectron differential cross-sections averaged over molecular orientations are given by: dσ (L,V) σ (L,V) (L,V) [1 + β ˆ = P2 ( cos θ)], k dΩkˆ 4π
(1)
where σ (L,V) is the total photoionization cross-section, obtained with the length (L) or velocity (V ) form of the dipole moment operator. For non-linear molecules, σ (L,V) is given by: σ (L,V) =
4π2 E pµ(L,V) 2 |Ilhv | , 3c
(2)
pµlhv
where E is the photon energy, p one of the irreducible representations (IR) of the symmetry group of the molecule and h distinguishes between different bases for the same IR corresponding to the same value of l. The index µ labels components of vectors belonging to the same IR and v designates components of the dipole moment operator. (L,V) The quantity β ˆ appearing in Eq. (1) can be written as: k
(L,V) k
βˆ
=
3 5
1 pµ(L,V) 2 | pµlhv |Ilhv
p µ
p µ
(1100|20)
pµlhvmmv p µ l h v m mv
pµ(L,V)
are the partial-wave components of the dywhereas Ilhv (L,V) namical coefficients I : k,nˆ 1/2 4π pµ(L,V) pµ ˆ p µ (L,V) I = Ilhv Xlh (k)X1vv v (n), ˆ (4) k,nˆ 3 pµlhv
where (L) k,nˆ
I
(−) k
= (k)1/2 Ψi | r · n|Ψ ˆ ,
and (V) k,nˆ
I
=
(k)1/2 (−) · n|Ψ
Ψi |∇ ˆ . k E
(6)
pµ
In Eq. (4), Xlh are the symmetry-adapted functions [24] that are related to the usual spherical harmonics by: pµ pµ Xlh (ˆr ) = blhm Ylm (ˆr ), (7) m
whereas in Eqs. (5) and (6) Ψi is the target ground state (−) wave function, Ψ the final state (incoming-wave normalk ized) wave function of the system (ion plus photoelectron), nˆ represents the unit vector in the direction of polarization of the radiation and k the photoelectron momentum. In the present study, Ψi is a one-determinant wave function calculated in the Hartree–Fock level. The final molecular state is described by a single electronic configuration in which the ionic orbitals are constrained to be identical to those of the initial ground state. In this approximation, the photoelectron orbital φk ( r ) is a solution of the one-electron Schrödinger equation: + 1 k2 ]φ ( r ) = 0, [− 1 ∇ 2 + VN−1 ( r ; R) (8) 2
2
k
where VN−1 is the SE potential of the molecular ion and φk ( r ) satisfies appropriate boundary conditions. To proceed, Eq. (8) is rewritten in an integral form, the Lippmann–Schwinger equation, and is solved using an iterative procedure based on the Schwinger variational principle. This method provides continuum wave functions that are shown to converge to the exact solutions for a given projectile–target interaction potential [25]. The numerical calculation of the scattering wave functions was performed using our SVIM codes, developed for molecular systems with symmetries reducible to C2v . The potential field of the 2 T2 (3t−1 2 ) ionic state was constructed assuming that the C2v -reduced components (A1 , B1 and B2 ) of the t2 hole orbital were equally depopulated. In this case, the cross-sections were evaluated by adding up the contributions from the C2v -reduced components of the t2 orbital, while the asymmetry parameters were obtained as an average over pµ(L,V) pµ(L,V) Il h v
(−1)m −mv Ilhv
∗
pµ p µ
blhm bl h m
v v × blvm b v v [(2l + 1)(2l + 1)]1/2 (11 − mv mv |2M )(l l00|20)(l l − m m|2 − M ), v lvm v
(5)
(3)
A.S. dos Santos et al. / Journal of Electron Spectroscopy and Related Phenomena 135 (2004) 149–153
those components taking the corresponding cross-sections as weights. All matrix elements arising in the SVIM calculation were evaluated using a single-center expansion truncated at lmax = 18 and all allowed h values, i.e. h ≤ l, were retained for a given l. In order to verify the convergence in the partial-wave expansion, some test calculations were carried out also with lmax = 20. No significant differences were noticed in the calculated cross-sections with these two sets of cutoff parameters, showing that all the calculations had already converged. The convergence in the iterative procedure was also studied. All the final results shown below were converged within six iterations. We have used the values of 11.34, 11.80 and 12.26 eV for the IPs of the three Jahn–Teller components of the 3t2 orbital of GeH4 , and the value of 18.40 eV for the IP of the 4a1 orbital [16].
3. Results and discussion In Fig. 1(a) and (b), we present our calculated results for σ and β, respectively, obtained in both dipole length (DL) and dipole velocity (DV) forms, for the photoionization out
151
of the 3t2 orbital of GeH4 for photon energies ranging from near threshold up to 50 eV. The same quantities, but for the 4a1 orbital are shown in Fig. 2(a) and (b). In Fig. 1(a), our calculated σ show a step-like sharp rise near threshold, arising from successive vibronic contributions associated to the Jahn–Teller distortion of the ground GeH4 + ionic state. The difference between the DL and DV results, seen in Figs. 1 and 2, are known to be generally due to the neglect of both electronic correlation in the target wave function and multichannel coupling effects in the photoelectron wave function. The influence of the former has been studied by Lucchese et al. [22] and that of the interchannel effects was recently studied by Cacelli et al. [26]. These authors have shown that the difference between the DL and DV results is significantly reduced by the inclusion of these effects. In general, the agreement between our calculated βL and βV is better than that between σ L and σ V . Unfortunately, no experimental and/or theoretical results on the photoionization of GeH4 in this range of energy is available in the literature. For this reason, in Fig. 3, we compare our calculated σ L and βL results for the 3t2 orbital of GeH4 with the corresponding calculated data for the photoionization of the outermost t2 orbitals of CH4 [27] and SiH4 [23] as a function of the photoelectron kinetic energy. The same type of comparison is also made in Fig. 4, but for the outermost a1 orbitals. The experimental σ of Brion [28], derived from the
(a) (a)
(b) Fig. 1. Photoionization cross-sections (a) and asymmetry parameters (b) + for the lowest 2 T2 (3t−1 2 ) state of GeH4 . Solid line presents results in DL form and dashed line presents results in DV form.
(b) Fig. 2. Same as Fig. 1, but for the lowest 2 A1 (4a1−1 ) state of GeH4 + .
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(a)
(a)
(b)
(b)
Fig. 3. Photoionization cross-sections (a) and asymmetry parameters (b) for the lowest 2 T2 states of CH4 + , SiH4 + and GeH4 + . Solid line presents DL results for the 3t2 orbital of GeH4 , dashed line presents DL results for the 1t2 orbital of CH4 [27], short-dashed line presents DL results for the 2t2 orbital of SiH4 [23], and full circles present experimental results of Brion for silane [28].
Fig. 4. Photoionization cross-sections (a) and asymmetry parameters (b) for the lowest 2 A1 states of CH4 + , SiH4 + and GeH4 + . Solid line presents DL results for the 4a1 orbital of GeH4 , dashed line presents DL results for the 2a1 orbital of CH4 [30], short-dashed line presents DL results for the 3a1 orbital of SiH4 [23], and full circles present experimental results of Brion for silane [28].
total photoabsorption data of Cooper et al. [29] for silane are also included for comparison in both figures. For the t2 orbital a remarkable similarity among the calculated σ is seen in Fig. 3(a), particularly between the results of SiH4 and GeH4 . It is interesting to notice that the experimental σ of Brion [28] for silane agree quite well with our calculated σ L for germane. This good agreement comes in support of the speculation made in our previous work [23] that the photoionization of the valence t2 orbitals of the XH4 (X = C, Si, Ge, Sn, Pb) molecules would be similar. In Fig. 4, a general good agreement is also seen between the calculated σ and β for the photoionization out of the outermost a1 orbitals in silane and germane. In addition, the calculated photoionization cross-sections for silane agree well with the experimental data of Brion [28], except at photon energies near threshold, where the latter show a decreasing behavior towards the threshold, in contrast with the calculation. The physical origin of this discrepancy is still unclear. Also, a qualitative disagreement is observed between σ L and βL for these molecules and those for methane [30]. Specifically, it is seen in Fig. 4(b) that the values of β, calculated by Stener and Decleva [30] for the photoionization out of the 2a1 orbital of CH4 , are nearly 2 in the entire energy
range. This behavior suggests the atomic character of this orbital.
4. Conclusions We have reported cross-sections and asymmetry parameters for the photoionization of the two outermost valence orbitals of GeH4 , in both DL and DV forms. To our knowledge, this is the first theoretical study that reports ab initio results of these physical quantities for photoionization of germane at the exact SE level. Also, an evident similarity was observed between the photoionization properties of the 1t2 orbital of methane, the 2t2 orbital of silane and the 3t2 orbital of germane. Indeed, the cross-sections as well as the asymmetry parameters, are very close both in shape and magnitude, which may indicate that the photoionization of the outermost valence t2 orbitals of XH4 (X = C, Si, Ge, Sn and Pb) molecules would be similar. On the other hand, although a very clear similarity was observed between the photoionization properties of the 3a1 orbital of silane and the 4a1 orbital of germane, this was not the case for the 2a1 orbital of methane. This fact shows that the observed dis-
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