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First-principle theoretical study on the electronic properties of SiO2 models with hydrogenated impurities and charges
Applied Surface Science 216 (2003) 463–470
First-principle theoretical study on the electronic properties of SiO2 models with hydrogenated impurities and charges Kentaro Doi, Koichi Nakamura, Akitomo Tachibana* Depertment of Engineering Physics and Mechanics, Kyoto University, Kyoto 606-8501, Japan
Abstract We adopt first-principle approach to calculate potential energy surfaces for a-quartz and -cristobalite silicon dioxide (SiO2) with hydrogenated impurities in the neutral, positive, and negative charged states. It is expected that H atoms in SiO2 have an important role to generate leakage current paths. In this paper, we calculate stable and unstable positions of a migrating H atom and discuss electrons or holes trapping mechanism. These results clarify that the dynamics of the H atom in the neutral, negative, and positive charged states should be different from one another, and therefore, the mechanism of the dielectric break down of SiO2 also depends on the charged environment. Based on the regional density functional theory, we have calculated the effective charge tensor density of the migrating H atom in the neutral state. # 2003 Elsevier Science B.V. All rights reserved. PACS: 71.15.Mb; 72.20.-i; 71.55.Ht Keywords: Breakdown of SiO2; Hydrogenated impurity; Charge trapping; First-principle calculation; Potential energy surfaces; Molecular dynamics
1. Introduction Thin SiO2 films have an important role in a metaloxide–semiconductor field-effect transistor (MOSFET). A MOS capacitor can get high performance because of the thin films. On the other hand, the thinner the films become, the faster those get to dielectric breakdown. It is reported that dielectric breakdown occurred because at first electrons or holes were trapped in SiO2 under high electric field stress [1–3] and next stress-induced leakage current (SILC) increased [4]. DiMaria and Stasiak reported that charge trapping increased in proportion to injected *
Corresponding author. Tel.: þ81-75-753-5184; fax: þ81-75-753-5184. E-mail address: [email protected] (A. Tachibana).
charges and electric field [2], and that H atoms which terminated Si dangling bonds were released by hot electrons and they made Si/SiO2 interface unstable state, as a result SiO2 got to breakdown [3,5]. Other experimental results also concluded that hot electrons or holes in SiO2 caused of dielectric breakdown and the replacement of hydrogen with deuterium was effective to reduce hot electron degradation effect [6,7]. Yokozawa and Miyamoto calculated O vacancies in a SiO2 model and hydrogen dynamics in SiO2 [8,9]. In their reports, they calculated stable configurations of impurities in SiO2 and concluded that hydrogen would be released by charges which came from Si substrate to SiO2, and that Cl atoms, NH and OH molecules were stable at position where they terminated Si dangling bonds. Bunson et al. calculated stability of Hþ in a-quartz SiO2 and they concluded
0169-4332/03/$ – see front matter # 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0169-4332(03)00407-0
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K. Doi et al. / Applied Surface Science 216 (2003) 463–470
Hþ does not simply bonded to the bridging oxygen sites but also it preferred to the point surrounded by several neighboring oxygen atoms [10]. These authors indicated that dynamics of H or Hþ atoms in SiO2 were one of the reasons for dielectric breakdown. The finite-temperature electronic structures of SiO2 thin film with the interstitial H atom under the external electric field were reported in our previous paper [11]. In this paper, we calculate, with first-principle calculation, potential energy surfaces (PESs) of an H atom in a-quartz and -cristobalite SiO2 bulk and make clear the stable and unstable states in charged crystals. In order to display the driving force of the electromigration, we have calculated the effective charge tensor densities for the migrating H atom and investigated the response of the force to the external electric field in both of static and dynamic aspects.
2. Method of calculations First-principle energy calculations of the periodic boundary interface models were carried out by means of supercell approximation techniques with Periodic Regional DFT program package [12]. Fig. 1 shows examples of a-quartz and -cristobalite SiO2 bulk models with an H atom. The a-quartz unit cell consists
of Si3O6H in a hexagonal lattice where lattice con˚ and c ¼ 5:405 A ˚ . The a-crisstants are a ¼ 4:913 A tobalite unit cell consists of Si4O8H in a tetragonal ˚ and lattice where lattice constants are a ¼ 4:978 A ˚ c ¼ 6:948 A. We prepared four types of charged models; neutral, singly positive (þ1), doubly positive (þ2), and negative (1), for a-quartz and -cristobalite, respectively. The positive, þ1 and þ2, models are taken away one and two electrons from the unit cells. On the other hand, the negative models are doped one electron into the unit cells. We assume that external electric field and leakage current respond to the electronic state of the crystal by giving local charge due to electrons or holes. An electronic wave function is expanded by plane waves with 50 Ry energy cutoff and energy eigenvalues are calculated by the density functional theory with Perdew–Wang expression for the exchange-correlation energy [13,14]. The effect of inner core electrons for each atom is replaced by Troullier–Martins-type pseudopotentials [15]. For calculations of the positive and negative charged states, we combined the cutoff of radius (30 bohr) for electrostatic potential with the simple Ewald scheme. Distribution of valence electrons is decided by Fermi–Dirac-distribution function. For the electronic state under the external electric field ~ Eext ð~ rÞ, we solved the Kohn–Sham equation with the effective
˚ and c ¼ 5:405 A ˚ , and (b) a-cristobalite Fig. 1. SiO2 bulk model and the injected H atom: (a) a-quartz SiO2 with lattice constants a ¼ 4:913 A ˚ and c ¼ 6:948 A ˚ . The most stable points of the injected H atom in each charged system are shown by ‘‘*’’ for the neutral with a ¼ 4:978 A state, ‘‘&’’ for the singly positive state, ‘‘^’’ for the doubly positive state, and ‘‘~’’ for the negative state. Relative energies are standardized by the energy minimum on the z ¼ 0 plane for each of charged states.
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Fig. 2. PES curves of the migrating H atom along z-axis in (a) a-quartz and (b) a-cristobalite charged models. The energy minima of each zplane are picked up and connected for the neutral, singly positive, doubly positive, and negative states, respectively.
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potential including the term due to the external electric field n~Eext ð~ rÞ [11,12,16]. 3. Results and discussion 3.1. Energy barrier for migration of H atom We investigated the PESs and potential energy barrier for an H atom which migrates in each of
charged models. Fig. 2 shows PES curves of the migrating H atom in a-quartz and -cristobalite charged models along the z-axis in Fig. 1. We picked up the energy minima of each z-plane and connected them. Relative energies are standardized by the energy minimum on the z ¼ 0 plane for each of charged states. Stable positions of the H atom change according to charge environment. In a-quartz and -cristobalite, the stable points in the neutral and negative states along zaxis are identical or very close to each other as shown
Fig. 3. PESs of z-plane on which the most stable position of the migrating H atom (‘‘*’’) lies in each of a-quartz charged models: (a) ˚ plane in the neutral state, (b) z ¼ 1:802 A ˚ plane in the singly positive state, and (c) z ¼ 2:703 A ˚ plane in the negative state. z ¼ 2:703 A Potential energy (eV) is relatively measured from the most stable point on each plane.
K. Doi et al. / Applied Surface Science 216 (2003) 463–470
in Fig. 1, but these stable points are different from those in the positive states. Figs. 3 and 4 show PESs of z-plane on which the most stable position of the H atom lies in a-quartz and -cristobalite, respectively. It is clarified that the potential energy minimum is surrounded by higher potential energy barriers within the z-plane than barriers along the z-direction in each of a-quartz and -cristobalite bulk systems. Therefore, the mobility for migration of the H atom in a-quartz and -cristobalite is quite higher in the neutral state
467
rather than in the positive and negative charged states, where the H atom is tightly bound at the potential energy minimum as shown in Fig. 2. It is also found that the migration of the H atom in a-cristobalite takes place more easily as compared with the migration in aquartz. In particular, PES of the neutral a-cristobalite bulk system is much flatter than those of other systems as shown in Figs. 2(b) and 4(a), and accordingly, it is expected that the mobility of migration is very high in the neutral a-cristobalite system.
Fig. 4. PESs of z-plane on which the most stable position of the migrating H atom (‘‘*’’) lies in each of a-cristobalite charged models: (a) ˚ plane in the neutral state, (b) z ¼ 3:474 A ˚ plane in the singly positive state, and (c) z ¼ 0:000 A ˚ plane in the negative state. z ¼ 0:000 A Potential energy (eV) is measured from the most stable point on each plane.
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Fig. 5 shows the change of the band structures for a-quartz and -cristobalite bulk systems by the injection of H atom in the neutral state. In the band structures with H atom, new energy band originated by the H atom appears around the Fermi energy level eF. Due to the existence of the interstitial H atom at the most stable position, the energy band gaps, which are 5.951 and 5.736 eV for pure a-quartz and cristobalite, are reduced to 3.073 and 3.863 eV, respectively.
3.2. Effective charge of migrating H atom in non-equilibrium state Recently, new concept for electromigration has been given by Tachibana [17] using the non-relativistic limit of quantum electrodynamics (QED). In this article, the application for the migrating H atom in the a-cristobalite SiO2 bulk system shall be performed. The effective charge tensor density of the migrating H $ atom, Z H ð~ rÞ, is defined as [17] $ Z H ð~ rÞeNH ð~ rÞ
¼
@~ FHS ð~ rÞ @~ Eext ð~ rÞ
(1)
where e is the charge of electron, ~ Eext ð~ rÞ the external electric field, NH ð~ rÞ the position probability density designating the position ~ r at which the impurities are most likely to be found, and ~ FHS ð~ rÞ is the electromigration force density of migrating H atom given by mH
@~ SH ð~ rÞ ¼ ~ FHS ð~ rÞ @t
(2)
with the probability $ flux density ~ SH ð~ rÞ and the mass of H atom mH [17]. Z H ð~ rÞ is further decomposed as follows: $ Z H ð~ rÞ
$
¼ ZH þ Z H wind ð~ rÞ
$ rÞ Z H wind ð~
$
(3) $
¼ Z H dynamic wind ð~ rÞ þ Z H static wind ð~ rÞ
(4)
$
ZH may be called the direct charge, and Z H dynamic wind ð~ rÞ $ and Z H static wind ð~ rÞ are the dynamic and static wind charge tensor density, respectively [17].
Fig. 5. Band structures of a-quartz and -cristobalite models in the neutral state: (a) pure crystal model and (b) H-injected model of aquartz, and (c) pure crystal model and (d) H-injected model of acristobalite. The injected H atom in (b) and (d) is located on the most stable point referred to Fig. 1. eF denotes the Fermi energy.
Fig. 6. Wave packet in the dynamics simulation for a-cristobalite in the neutral state: (a) initial wave packet and (b) propagated one at the t ¼ 39; 400 au (953 fs). Dots represent that positions where density of wave packet is over 0.01.
K. Doi et al. / Applied Surface Science 216 (2003) 463–470
$
469
$
˚ plane of a-cristobalite Fig. 7. Component vectors of Z H static wind ð~ rÞ (upper left) and Z H dynamic wind ð~ rÞ (lower right) on the points at z ¼ 3:474 A in the neutral state.
We have selected an excited-state configuration with a finite flux with carrying out the SCF procedure under the constraint that the Fermi surface is forced to shift [17,18]. The calculus of variation with the constraint in this manner is based on the regional density functional theory [19] encompassing non-equilibrium states [20], chemical potentials [21–23], and QED [24–26]. The shift of the Fermi surface was represented by taking k-points with a shift of 0.15 fractional coordinate for the component of reciprocal lattice vector to the h 1 1 0i direction of uniform external h 1 1 0i electric field E ext . The component vector of $ Z H static wind ð~ rÞ has been calculated by numerical differential of the Hellmann–Feynman force of migrating h1 1 0i H atom ~ FHHF ð~ rÞ with respect to Eext as follows: $ 1 @~ FHHF ð~ rÞ Z H static wind ð~ rÞ ¼ (5) h1 1 0i e @~ Eext by means of the conventional five-point formula with the interval width of 9:7 103 au (5:0 109 V/m). For the migrating H atom, we have examined quantum mechanical wave-packet propagation by using the Bloch functions of electrons fcj ð~ re ;~ rÞg. The procedures of wave-packet propagation are given in [17,18]. Fig. 6 shows the initial Gaussian wave packet, which is expanded by 5 1 2 plane waves, and propagated wave packet after 39,400 au (953 fs) with initial wave-packet core momentum of 1 kcal/mol
along h1 1 0i direction. The displacement of the migrating atom changes the electronic state around the atom. The change of the$electronic state under the rÞ. We have electric field bring about Z H dynamic wind ð~ $ calculated Z H dynamic wind ð~ rÞ by means of the wave packet in Fig. 6(b) according to Eqs. (39) and (41) in [17], where we approximate the transport relaxation time as 100 au and brackets for the electronic wave functions, hcj ð~ re ;~ rÞjð@cj ð~ re ;~ rÞÞ=ð@xk Þi, were estimated by the numerical differentiation with respect to the displacement of the H atom xk with correction of extraordinary components [17,18]. Fig. 7 shows $ $ Z H static wind ð~ rÞ and Z H dynamic wind ð~ rÞ on the ˚ plane of a-cristobalite, the bisecting zz ¼ 3:474 A plane of the unit cell shown$in Fig. 6(b), in the neutral state. We can observe that Z H static wind ð~ rÞ exactly displays the response of the force due to the static electronic properties on the migrating atom to the $ external electric field. In addition, Z H dynamic wind ð~ rÞ vividly reflects the effect of the electric field on the wave packet through the electronic state.
4. Conclusion In this paper, we calculated PESs of a-quartz and cristobalite bulk models which contain an interstitial H atom in the neutral, positive, and negative charged
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states. We have clarified that injected charge changes the electronic structure vividly, and that the H atom is bound more tightly in the negative and positive states rather than in the neutral state. Experimental [2,3,5,7] and theoretical [8–10] results indicate that an H atom, released by injection of hot electron into SiO2, transports to Si/SiO2 interface and damages it. On the contrary, a-quartz and -cristobalite models in our calculations do not contain O vacancies and Si dangling bonds, so neutral environment with the H atom would be unstable excessively. H atom would make these systems more stable if crystals have vacancies or dangling bonds, and the dynamics of the H atom in the crystal with vacancies and dangling bonds rouses our interest in the mechanism of dielectric breakdown. The effective charge tensor densities of the migrating $ H atom, the static wind$charge Z H static wind ð~ rÞ and the dynamic wind charge Z H dynamic wind ð~ rÞ, can be calculated based on the regional density functional theory [17–26]. These charge tensor densities explain the response of the force on the migrating atom to the external electric field in static and dynamic aspects, respectively.
Acknowledgements This work was supported by Grant-in-Aid for Scientific Research from the Ministry of Education, Science and Culture of Japan, for which the authors express their gratitude.
References [1] Y. Nissan-Cohen, J. Shappir, D. Frohman-Bentchkowsky, J. Appl. Phys. 60 (1986) 2024. [2] D.J. DiMaria, J.W. Stasiak, J. Appl. Phys. 65 (1989) 2342. [3] D.J. DiMaria, E. Cartier, D. Arnold, J. Appl. Phys. 73 (1993) 3367. [4] P. Riess, G. Ghibaudo, G. Pananakakis, Appl. Phys. Lett. 75 (1999) 3871.
[5] D.J. DiMaria, E. Cartier, J. Appl. Phys. 78 (1995) 3883. [6] D.J. DiMaria, J.H. Stathis, Appl. Phys. Lett. 71 (1997) 3230. [7] K. Vauheusden, W.L. Warren, R.A.B. Devine, D.M. Fleetwood, J.R. Schwank, M.R. Shaneyfelt, P.S. Winokur, Z.J. Lemnios, Nature 386 (1997) 587. [8] A. Yokozawa, Y. Miyamoto, Appl. Phys. Lett. 73 (1998) 1122. [9] A. Yokozawa, Y. Miyamoto, J. Appl. Phys. 88 (2000) 4542. [10] P.E. Bunson, M. DiVentra, S.T. Pantelides, R.D. Schrimph, K.F. Galloway, IEEE Trans. Nucl. Sci. 46 (1999) 1568. [11] K. Doi, K. Nakamura, A. Tachibana, in: S. Ohmi, et al. (Ed.), Extended Abstracts of International Workshop on Gate Insulator 2001, Business Center for Academic Societies, Tokyo, Japan, 2001, pp. 148–151: therein, the direction of external electric field in Fig. 1 is erroneous, the reverse direction is correct. [12] Periodic Regional DFT Program Package, Version 1, Tachbana Lab., Kyoto University, Kyoto, 2001. [13] J.P. Perdew, Y. Wang, Phys. Rev. B 45 (1992) 13244. [14] J.P. Perdew, J.A. Chevary, S.H. Vosko, K.A. Jackson, M.A. Pederson, D.J. Singh, C. Fiolhais, Phys. Rev. B 46 (1992) 6671. [15] N. Troullier, J.L. Martins, Phys. Rev. B 43 (1991) 1993. [16] S. Egami, K. Nakamura, A. Tachibana, in: S. Ohmi, et al. (Ed.), Extended Abstracts of International Workshop on Gate Insulator 2001, Business Center for Academic Society, Tokyo, Japan, 2001, pp. 234–237: therein, the direction of external electric field in Fig. 1 is erroneous, the reverse direction is correct. [17] A. Tachibana, in: S.P. Baker (Ed.), Stress Induced Phenomena in Metallization, American Institute of Physics, New York, 2002, pp. 105–116. [18] K. Doi, K. Iguchi, K. Nakamura, A. Tachibana, Phys. Rev. B, in press. [19] A. Tachibana, Int. J. Quant. Chem. Symp. 21 (1987) 181. [20] A. Tachibana, Int. J. Quant. Chem. 35 (1989) 361. [21] A. Tachibana, R.G. Parr, Int. J. Quant. Chem. 41 (1995) 527. [22] A. Tachibana, Int. J. Quant. Chem. 57 (1996) 423. [23] A. Tachibana, Theor. Chem. Acc. 102 (1999) 188. [24] A. Tachibana, J. Chem. Phys. 115 (2001) 3497. [25] A. Tachibana, in: K.D. Sen (Ed.), Review in Modern Quantim Chemistry: A Celebration of the Contributions of Robert Parr, World Scientific, Singapore, 2002, pp. 1327–1366 (Chapter 45). [26] A. Tachibana, in: E. Brandas, E. Kryachko (Eds.), Fundamental Perspectives in Quantum Chemistry: A Tribute to the Memory of Per-Olov Lo¨ wdin, Kluwer Academic Publishers, Dordrecht, 2003.
First-principle theoretical study on the electronic properties of SiO2 models with hydrogenated impurities and charges Kentaro Doi, Koichi Nakamura, Akitomo Tachibana* Depertment of Engineering Physics and Mechanics, Kyoto University, Kyoto 606-8501, Japan
Abstract We adopt first-principle approach to calculate potential energy surfaces for a-quartz and -cristobalite silicon dioxide (SiO2) with hydrogenated impurities in the neutral, positive, and negative charged states. It is expected that H atoms in SiO2 have an important role to generate leakage current paths. In this paper, we calculate stable and unstable positions of a migrating H atom and discuss electrons or holes trapping mechanism. These results clarify that the dynamics of the H atom in the neutral, negative, and positive charged states should be different from one another, and therefore, the mechanism of the dielectric break down of SiO2 also depends on the charged environment. Based on the regional density functional theory, we have calculated the effective charge tensor density of the migrating H atom in the neutral state. # 2003 Elsevier Science B.V. All rights reserved. PACS: 71.15.Mb; 72.20.-i; 71.55.Ht Keywords: Breakdown of SiO2; Hydrogenated impurity; Charge trapping; First-principle calculation; Potential energy surfaces; Molecular dynamics
1. Introduction Thin SiO2 films have an important role in a metaloxide–semiconductor field-effect transistor (MOSFET). A MOS capacitor can get high performance because of the thin films. On the other hand, the thinner the films become, the faster those get to dielectric breakdown. It is reported that dielectric breakdown occurred because at first electrons or holes were trapped in SiO2 under high electric field stress [1–3] and next stress-induced leakage current (SILC) increased [4]. DiMaria and Stasiak reported that charge trapping increased in proportion to injected *
Corresponding author. Tel.: þ81-75-753-5184; fax: þ81-75-753-5184. E-mail address: [email protected] (A. Tachibana).
charges and electric field [2], and that H atoms which terminated Si dangling bonds were released by hot electrons and they made Si/SiO2 interface unstable state, as a result SiO2 got to breakdown [3,5]. Other experimental results also concluded that hot electrons or holes in SiO2 caused of dielectric breakdown and the replacement of hydrogen with deuterium was effective to reduce hot electron degradation effect [6,7]. Yokozawa and Miyamoto calculated O vacancies in a SiO2 model and hydrogen dynamics in SiO2 [8,9]. In their reports, they calculated stable configurations of impurities in SiO2 and concluded that hydrogen would be released by charges which came from Si substrate to SiO2, and that Cl atoms, NH and OH molecules were stable at position where they terminated Si dangling bonds. Bunson et al. calculated stability of Hþ in a-quartz SiO2 and they concluded
0169-4332/03/$ – see front matter # 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0169-4332(03)00407-0
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K. Doi et al. / Applied Surface Science 216 (2003) 463–470
Hþ does not simply bonded to the bridging oxygen sites but also it preferred to the point surrounded by several neighboring oxygen atoms [10]. These authors indicated that dynamics of H or Hþ atoms in SiO2 were one of the reasons for dielectric breakdown. The finite-temperature electronic structures of SiO2 thin film with the interstitial H atom under the external electric field were reported in our previous paper [11]. In this paper, we calculate, with first-principle calculation, potential energy surfaces (PESs) of an H atom in a-quartz and -cristobalite SiO2 bulk and make clear the stable and unstable states in charged crystals. In order to display the driving force of the electromigration, we have calculated the effective charge tensor densities for the migrating H atom and investigated the response of the force to the external electric field in both of static and dynamic aspects.
2. Method of calculations First-principle energy calculations of the periodic boundary interface models were carried out by means of supercell approximation techniques with Periodic Regional DFT program package [12]. Fig. 1 shows examples of a-quartz and -cristobalite SiO2 bulk models with an H atom. The a-quartz unit cell consists
of Si3O6H in a hexagonal lattice where lattice con˚ and c ¼ 5:405 A ˚ . The a-crisstants are a ¼ 4:913 A tobalite unit cell consists of Si4O8H in a tetragonal ˚ and lattice where lattice constants are a ¼ 4:978 A ˚ c ¼ 6:948 A. We prepared four types of charged models; neutral, singly positive (þ1), doubly positive (þ2), and negative (1), for a-quartz and -cristobalite, respectively. The positive, þ1 and þ2, models are taken away one and two electrons from the unit cells. On the other hand, the negative models are doped one electron into the unit cells. We assume that external electric field and leakage current respond to the electronic state of the crystal by giving local charge due to electrons or holes. An electronic wave function is expanded by plane waves with 50 Ry energy cutoff and energy eigenvalues are calculated by the density functional theory with Perdew–Wang expression for the exchange-correlation energy [13,14]. The effect of inner core electrons for each atom is replaced by Troullier–Martins-type pseudopotentials [15]. For calculations of the positive and negative charged states, we combined the cutoff of radius (30 bohr) for electrostatic potential with the simple Ewald scheme. Distribution of valence electrons is decided by Fermi–Dirac-distribution function. For the electronic state under the external electric field ~ Eext ð~ rÞ, we solved the Kohn–Sham equation with the effective
˚ and c ¼ 5:405 A ˚ , and (b) a-cristobalite Fig. 1. SiO2 bulk model and the injected H atom: (a) a-quartz SiO2 with lattice constants a ¼ 4:913 A ˚ and c ¼ 6:948 A ˚ . The most stable points of the injected H atom in each charged system are shown by ‘‘*’’ for the neutral with a ¼ 4:978 A state, ‘‘&’’ for the singly positive state, ‘‘^’’ for the doubly positive state, and ‘‘~’’ for the negative state. Relative energies are standardized by the energy minimum on the z ¼ 0 plane for each of charged states.
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Fig. 2. PES curves of the migrating H atom along z-axis in (a) a-quartz and (b) a-cristobalite charged models. The energy minima of each zplane are picked up and connected for the neutral, singly positive, doubly positive, and negative states, respectively.
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potential including the term due to the external electric field n~Eext ð~ rÞ [11,12,16]. 3. Results and discussion 3.1. Energy barrier for migration of H atom We investigated the PESs and potential energy barrier for an H atom which migrates in each of
charged models. Fig. 2 shows PES curves of the migrating H atom in a-quartz and -cristobalite charged models along the z-axis in Fig. 1. We picked up the energy minima of each z-plane and connected them. Relative energies are standardized by the energy minimum on the z ¼ 0 plane for each of charged states. Stable positions of the H atom change according to charge environment. In a-quartz and -cristobalite, the stable points in the neutral and negative states along zaxis are identical or very close to each other as shown
Fig. 3. PESs of z-plane on which the most stable position of the migrating H atom (‘‘*’’) lies in each of a-quartz charged models: (a) ˚ plane in the neutral state, (b) z ¼ 1:802 A ˚ plane in the singly positive state, and (c) z ¼ 2:703 A ˚ plane in the negative state. z ¼ 2:703 A Potential energy (eV) is relatively measured from the most stable point on each plane.
K. Doi et al. / Applied Surface Science 216 (2003) 463–470
in Fig. 1, but these stable points are different from those in the positive states. Figs. 3 and 4 show PESs of z-plane on which the most stable position of the H atom lies in a-quartz and -cristobalite, respectively. It is clarified that the potential energy minimum is surrounded by higher potential energy barriers within the z-plane than barriers along the z-direction in each of a-quartz and -cristobalite bulk systems. Therefore, the mobility for migration of the H atom in a-quartz and -cristobalite is quite higher in the neutral state
467
rather than in the positive and negative charged states, where the H atom is tightly bound at the potential energy minimum as shown in Fig. 2. It is also found that the migration of the H atom in a-cristobalite takes place more easily as compared with the migration in aquartz. In particular, PES of the neutral a-cristobalite bulk system is much flatter than those of other systems as shown in Figs. 2(b) and 4(a), and accordingly, it is expected that the mobility of migration is very high in the neutral a-cristobalite system.
Fig. 4. PESs of z-plane on which the most stable position of the migrating H atom (‘‘*’’) lies in each of a-cristobalite charged models: (a) ˚ plane in the neutral state, (b) z ¼ 3:474 A ˚ plane in the singly positive state, and (c) z ¼ 0:000 A ˚ plane in the negative state. z ¼ 0:000 A Potential energy (eV) is measured from the most stable point on each plane.
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Fig. 5 shows the change of the band structures for a-quartz and -cristobalite bulk systems by the injection of H atom in the neutral state. In the band structures with H atom, new energy band originated by the H atom appears around the Fermi energy level eF. Due to the existence of the interstitial H atom at the most stable position, the energy band gaps, which are 5.951 and 5.736 eV for pure a-quartz and cristobalite, are reduced to 3.073 and 3.863 eV, respectively.
3.2. Effective charge of migrating H atom in non-equilibrium state Recently, new concept for electromigration has been given by Tachibana [17] using the non-relativistic limit of quantum electrodynamics (QED). In this article, the application for the migrating H atom in the a-cristobalite SiO2 bulk system shall be performed. The effective charge tensor density of the migrating H $ atom, Z H ð~ rÞ, is defined as [17] $ Z H ð~ rÞeNH ð~ rÞ
¼
@~ FHS ð~ rÞ @~ Eext ð~ rÞ
(1)
where e is the charge of electron, ~ Eext ð~ rÞ the external electric field, NH ð~ rÞ the position probability density designating the position ~ r at which the impurities are most likely to be found, and ~ FHS ð~ rÞ is the electromigration force density of migrating H atom given by mH
@~ SH ð~ rÞ ¼ ~ FHS ð~ rÞ @t
(2)
with the probability $ flux density ~ SH ð~ rÞ and the mass of H atom mH [17]. Z H ð~ rÞ is further decomposed as follows: $ Z H ð~ rÞ
$
¼ ZH þ Z H wind ð~ rÞ
$ rÞ Z H wind ð~
$
(3) $
¼ Z H dynamic wind ð~ rÞ þ Z H static wind ð~ rÞ
(4)
$
ZH may be called the direct charge, and Z H dynamic wind ð~ rÞ $ and Z H static wind ð~ rÞ are the dynamic and static wind charge tensor density, respectively [17].
Fig. 5. Band structures of a-quartz and -cristobalite models in the neutral state: (a) pure crystal model and (b) H-injected model of aquartz, and (c) pure crystal model and (d) H-injected model of acristobalite. The injected H atom in (b) and (d) is located on the most stable point referred to Fig. 1. eF denotes the Fermi energy.
Fig. 6. Wave packet in the dynamics simulation for a-cristobalite in the neutral state: (a) initial wave packet and (b) propagated one at the t ¼ 39; 400 au (953 fs). Dots represent that positions where density of wave packet is over 0.01.
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$
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$
˚ plane of a-cristobalite Fig. 7. Component vectors of Z H static wind ð~ rÞ (upper left) and Z H dynamic wind ð~ rÞ (lower right) on the points at z ¼ 3:474 A in the neutral state.
We have selected an excited-state configuration with a finite flux with carrying out the SCF procedure under the constraint that the Fermi surface is forced to shift [17,18]. The calculus of variation with the constraint in this manner is based on the regional density functional theory [19] encompassing non-equilibrium states [20], chemical potentials [21–23], and QED [24–26]. The shift of the Fermi surface was represented by taking k-points with a shift of 0.15 fractional coordinate for the component of reciprocal lattice vector to the h 1 1 0i direction of uniform external h 1 1 0i electric field E ext . The component vector of $ Z H static wind ð~ rÞ has been calculated by numerical differential of the Hellmann–Feynman force of migrating h1 1 0i H atom ~ FHHF ð~ rÞ with respect to Eext as follows: $ 1 @~ FHHF ð~ rÞ Z H static wind ð~ rÞ ¼ (5) h1 1 0i e @~ Eext by means of the conventional five-point formula with the interval width of 9:7 103 au (5:0 109 V/m). For the migrating H atom, we have examined quantum mechanical wave-packet propagation by using the Bloch functions of electrons fcj ð~ re ;~ rÞg. The procedures of wave-packet propagation are given in [17,18]. Fig. 6 shows the initial Gaussian wave packet, which is expanded by 5 1 2 plane waves, and propagated wave packet after 39,400 au (953 fs) with initial wave-packet core momentum of 1 kcal/mol
along h1 1 0i direction. The displacement of the migrating atom changes the electronic state around the atom. The change of the$electronic state under the rÞ. We have electric field bring about Z H dynamic wind ð~ $ calculated Z H dynamic wind ð~ rÞ by means of the wave packet in Fig. 6(b) according to Eqs. (39) and (41) in [17], where we approximate the transport relaxation time as 100 au and brackets for the electronic wave functions, hcj ð~ re ;~ rÞjð@cj ð~ re ;~ rÞÞ=ð@xk Þi, were estimated by the numerical differentiation with respect to the displacement of the H atom xk with correction of extraordinary components [17,18]. Fig. 7 shows $ $ Z H static wind ð~ rÞ and Z H dynamic wind ð~ rÞ on the ˚ plane of a-cristobalite, the bisecting zz ¼ 3:474 A plane of the unit cell shown$in Fig. 6(b), in the neutral state. We can observe that Z H static wind ð~ rÞ exactly displays the response of the force due to the static electronic properties on the migrating atom to the $ external electric field. In addition, Z H dynamic wind ð~ rÞ vividly reflects the effect of the electric field on the wave packet through the electronic state.
4. Conclusion In this paper, we calculated PESs of a-quartz and cristobalite bulk models which contain an interstitial H atom in the neutral, positive, and negative charged
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states. We have clarified that injected charge changes the electronic structure vividly, and that the H atom is bound more tightly in the negative and positive states rather than in the neutral state. Experimental [2,3,5,7] and theoretical [8–10] results indicate that an H atom, released by injection of hot electron into SiO2, transports to Si/SiO2 interface and damages it. On the contrary, a-quartz and -cristobalite models in our calculations do not contain O vacancies and Si dangling bonds, so neutral environment with the H atom would be unstable excessively. H atom would make these systems more stable if crystals have vacancies or dangling bonds, and the dynamics of the H atom in the crystal with vacancies and dangling bonds rouses our interest in the mechanism of dielectric breakdown. The effective charge tensor densities of the migrating $ H atom, the static wind$charge Z H static wind ð~ rÞ and the dynamic wind charge Z H dynamic wind ð~ rÞ, can be calculated based on the regional density functional theory [17–26]. These charge tensor densities explain the response of the force on the migrating atom to the external electric field in static and dynamic aspects, respectively.
Acknowledgements This work was supported by Grant-in-Aid for Scientific Research from the Ministry of Education, Science and Culture of Japan, for which the authors express their gratitude.
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