- Email: [email protected]
A quantum approach to photodarkening in chalcogenides
Journal of Non-Crystalline Solids 351 (2005) 1582–1585 www.elsevier.com/locate/jnoncrysol
A quantum approach to photodarkening in chalcogenides
q
Jai Singh *, I.-K. Oh School of Engineering and Logistics, B-41, Faculty of Technology, Charles Darwin University, Darwin, NT 0909, Australia Received 26 February 2005
Abstract We present a microscopic quantum mechanical theory to calculate the reduction in the band gap energy due to illumination in chalcogenides. It is shown that photodarkening occurs due to the strong-carrier lattice interaction leading to pairing of the excited charge carriers. Results demonstrate similar behavior as observed in some experiments. Ó 2005 Elsevier B.V. All rights reserved.
1. Introduction Reduction in the optical band gap due to illumination observed in amorphous chalcogenides (a-Chs) is called photodarkening (PD) [1–3], which has been known to be metastable until recently. This means that PD disappears by annealing but remains even if the illumination is stopped. Much efforts have been devoted to understanding the phenomenon in the last two decades [3], however, no model has been successful in resolving all issues in all materials exhibiting PD. Another photostructural change that occurs in a-Ch is the volume expansion (VE) due to illumination [3]. Although VE and PD are not linearly related, it is demonstrated by Shimakawa et al. [4] that they are related. They have also suggested that both PD and VE occur due to the excessive repulsive Coulomb force among the photoexcited electrons occupying the localized tail states. Accordingly,
q This paper was originally presented at the 20th International Conference on Amorphous and Microcrystalline Semiconductors – Science and Technology (ICAMS 20), Campos do Jorda˜o, Sa˜o Paulo, Brazil, Aug. 25–29, 2003. * Corresponding author. Tel.: +61 (08) 8946 6811; fax: +61 (08) 8946 6366. E-mail address: [email protected] (J. Singh).
0022-3093/$ - see front matter Ó 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.jnoncrysol.2005.03.041
the repulsive force between the layers of a-Chs is considered responsible for VE and the same force is assumed to induce an in-plane slip motion that causes PD. Recently, the same group [5] have also observed the transient PD in a-As2S3, a-As2Se3 and a-Se, which disappears as soon as the illumination is stopped. That means the material reverses back to normal once the illumination is stopped. They have found in a-As2Se3 films that the transient peak of PD is dominant about 60% at 300 K but at a lower temperature of 50 K it reduces to 30% only. A photoassisted site switching (PASS) is suggested for the variation of PD with time, where the fractional growth in the number of PD sites is expressed as a stretched exponential function. It is not clearly understood what causes two types of PD, one that is metastable and gets reversed by annealing and the other that reverses back after stopping the illumination. Moreover, the repulsive Coulomb force model is qualitative and provides no estimates of the magnitude of the repulsive forces required for causing PD and VE. It is not known how much electrostatic repulsive Coulomb force among the excited electrons is required to push two atoms farther apart from their initial positions, whether it relates to in-plane or inter-layer movements. For causing movements in an atomic network, one requires the involvement of lattice vibrations, which surprisingly none of the models on PD and VE has given any account to. However, the involvement of
J. Singh, I.-K. Oh / Journal of Non-Crystalline Solids 351 (2005) 1582–1585
lattice vibrations has recently been considered [6] in inducing photostructural changes in glassy semiconductors. In 1988, Singh [7] discovered a drastic reduction in the band gap due to pairing of charge carriers in excitons and exciton–lattice interaction in non-metallic crystalline solids. The magnitude of reduction in the band gap varies with the magnitude of exciton–phonon interaction which is different in different materials, softer the structure stronger is the carrier–phonon interaction. It has been established that due to the planar structure [3], the carrier–phonon interaction in a-Chs is stronger than in a-Si:H, which satisfies the condition for the occurrence of AndersonÕs negative-U [8] in a-Chs but not in a-Si:H. This is the basis of the hole pairing model for the creation of light-induced-defects in a-Chs [3]. It may be noted that the concept of negative-U has been applied to photodarkening before [10–12], but to our knowledge no quantitative theory has been developed, not for the volume expansion any way. In this paper, based on HolsteinÕs [9] approach, the energy eigenvalues of positively and negatively charged polarons and paired charge carriers, created by excitations due to illumination, are calculated. It is found that the energies of the excited electron (negative charge) and hole (positive charge) polarons gets lowered due to the carrier–phonon interaction. Also the like excited charge carriers can get paired because of the negative-U effect [8] caused by the strong carrier–lattice interaction and the energy of such paired states is also gets lowered. Thus, the hole polaronic state and paired hole states overlap with the lone pair and tail states in a-Chs, which expands the valence band and reduces the band gap energy and hence causes PD. Formation of polarons as well as pairing of holes increases the bond length on which such localizations occur, which causes VE. The energy eigenvalues of such polarons and pairing of charge carriers are calculated. Only the main results are presented and details will be published elsewhere.
2. Theory Based on HolsteinÕs [9] approach, we consider here a model amorphous solid in the form of a linear chain of atoms. The electronic Hamiltonian in such a chain can be written as a sum of charge carrier, phonon, and charge carrier–phonon interaction energy operators as: b ¼H b el þ H b ph þ H b I [3]. H In a-Chs, as the lone pair orbitals overlap with the valence band, the combined band becomes much wider than the conduction band. As the effective mass of charge carriers is inversely proportional to the corresponding bandwidth, the hole effective mass becomes smaller than the electron effective mass in a-Chs. As a result, holes move faster than electrons in a-Chs. How-
1583
ever, in the tail states the charge carriers get localized. Considering the effect of strong carrier–phonon interaction, hole may become a positive charge polaron and an electron may become a negative charged polaron. Further more, two excited electrons and two excited holes can get paired and localized on a bond due to the effect of negative-U, which has already been established [8,3]. Pairing of holes on a bond breaks the bond and creates a pair of dangling bonds, which is used to explain the creation of light-induced defects in a-Ch [3]. Let us first consider the case of exciting a pair of electron and hole, not necessarily an exciton although that possibility exists. The secular equation of such a pair of excited charge carriers is written as: " W 1 C l ¼ EelþL Elh AelþL xlþL Ahl xl 1 X þ Mx2 x2m þ U 12 2 m
!# C l J e ðC lþLþ1 þ C lþL1 Þ
þ J h ðC lþ1 þ C ll Þ;
ð1Þ
where W1 is the energy eigenvalue of such an excited pair, Cl is the probability amplitude coefficient, Ele and Emh represent the energy of an electron at site l and that of a hole at m, respectively, Ael and Ahl are electron-phonon and hole-phonon coupling coefficients, U12 is the Coulomb interaction between the excited pair, Je and Jh are the energy of transfer of electron and hole, respectively, between nearest neighbors. M is the atomic mass and x is the frequency of vibration of nearest neighbor atoms. L is the distance between the excited pair and x is the bond length between nearest neighbors. Minimizing the energy with respect x, we obtain from Eq. (1) as [3] W 1 ¼ E0e E0h þ U 12 2ðJ e J h Þ Eep Ehp ; E0e
ð2Þ
E0h
where and are energies of the excited electron and hole, respectively, without the lattice, and Coulomb interactions between the excited charge carriers, and !2 2 1 Ail Eip ¼ ; i ¼ e; h; ð3Þ 48J i Mx2 which is the polaron binding energy by which the electron and hole energies get lowered due to the carrier– phonon interaction. The lowering of a hole energy means that it moves upward and overlaps more with the lone-pair orbitals, which narrows the band gap. Magnitudes of the polaron energy will be calculated in the next section. Next we consider that two pairs of electrons and holes are excited in the linear chain such that the pair of holes get localized on the same bond and two electrons become two polarons. The energy eigenvalue of such an excited state is obtained as [3] W 2h ¼ 2E0e 2E0h þ U 012 2ðJ e J h Þ 2Eep Ehh ;
ð4Þ
1584
J. Singh, I.-K. Oh / Journal of Non-Crystalline Solids 351 (2005) 1582–1585
where U 012 is the total Coulomb interaction among the excited charge carriers and !2 2 1 Ahl . ð5Þ Ehh ¼ 6J h Mx2 The bond length of a bond on which such a pairing occurs becomes twice as large as when only a polaron is formed and it is given by xhh p ¼
2Aep C p C p : Mx2
ð6Þ
The energy of two excitations without the charge carrier–phonon interaction, can be written as W 20 ¼
2E0e
2E0h
þ
U 012
2ðJ e J h Þ.
ð7Þ
Thus, the energy of a pair of excitations with two holes localized on a bond is lowered by DE given by DE ¼ W 20 W 2h ¼ 2Eep þ Ehh .
ð8Þ
Likewise, the energy eigenvalue for the case where a pair of excited electrons are localized on an anti-bonding orbital of a bond and two holes are like positive charge polarons can be written as W 2e ¼ 2E0e 2E0h þ U 012 2ðJ e J h Þ 2Ehp Eee ;
ð9Þ
where Eee can be obtained from Eq. (9) replacing subscript h by e. It may be noted here that unlike the case of pairing holes, pairing of electrons on a bond does not break the bond.
3. Results We have shown that energetically pairing of like charge carriers is favorable because the energy of such an excited state is lower. Formation of polarons also lowers the energy of the excited states but the effect is less than pairing. It is shown that, when a single hole is localized on a bond, the condition of the minimum energy is given by xð0Þ p in Eq. (6.30) of Ref. [3] which is a half of xhh in Eq. (6). Therefore, the bond on which p the two holes get localized gets enlarged to twice the size as compared to when a single hole localized. This can explain the photoinduced volume expansion in a-Chs. Depending on the material, DE can be in the range of a fraction of an eV to a few eV. We have estimated DE in As2S3 as follows: The energy of lattice vibration of a bond can be written as [3] 2
EðqÞ ¼ E0 12Mx2 ðq q0 Þ ;
For As2S3, the phonon energy of symmetric mode is 344 cm1 [13]. Using this and applying ToyozawaÕs criterion [14,15] of strong carrier–phonon interaction as Eep P Je, we get Je = 5.82 meV from Eq. (3), which gives Eep = 5.82 meV and Ehh = 46.56 meV and DE = 58.32 meV. The band gap of As2S3 is 2.32 eV [16], which gives DE ¼ 1.7%. This agrees quite well with the observed Eg reduction in band gap of about 2% [4].
4. Discussions It is shown that the like excited charge carriers get paired on a bond due to strong carrier–lattice interactions in a-Chs, because energetically such an excited state is more stable. Thus, the energy of paired holes on a bond moves up further in the lone-pair orbitals and tail states, which expands the valence bands. A similar situation occurs by pairing of electrons in the anti-bonding orbitals that lowers the conduction mobility edge down. These two effects together reduce the band gap. The reduction calculated here is about 1.7%, which agrees quite well with about 2% observed experimentally. It has been established [3] that pairing of holes on a bond breaks the bond as soon as two excited holes get localized on it. The bond breaks due to the removal of covalent electrons and a pair of dangling bonds are created. This is the essence of the pairing-hole theory of creating light induced defects in a-Chs. Such light-induced defects are reversible by annealing. However, pairing of excited electrons does not break a bond it only reduces the band gap and such an excited state will be reversed back to the normal after switching off the illumination. This can be applied to explain the metastable and transient PD. The former occurs due to pairing of holes that breaks the bond and that cannot be recovered by stopping the illumination. It remains metastable. The latter occurs due to pairing of electrons and formation of polarons which reverse back to normal after the illumination is stopped. Usually the transient PD is observed more in percentage than metastable PD. This is because there are three processes contributing to the transient PD, pairing of electrons, formation of positive charge polarons and negative charge polarons, in comparison with only one channel of pairing holes contributing to metastable PD.
ð10Þ
where E0 and q0 are the energy and the interaction coordinate at the minimum of the vibrational energy, respectively. The vibrational force along the interaction coordinate can be obtained as A ¼
Using this in Eqs. (6.32) of Ref. [3] and (9), we obtain 2 2 2 2 2 2 Mx q0 Mx q ¼ 481 , and EJhhh ¼ 16 J h 0 . Je
Eep Je
oE oq
q¼0
¼ Mx2 q0 .
5. Conclusion It is shown quantum mechanically that it is the strong carrier–lattice interaction in a-Chs that causes all three
J. Singh, I.-K. Oh / Journal of Non-Crystalline Solids 351 (2005) 1582–1585
phenomena, creation of the light-induced defects, PD and VE. Acknowledgments This work is supported by the Australian Research CouncilÕs Large Grants (2000–2003) and IREX (2001– 2003) schemes. References [1] Ke. Tanaka, Phys. Rev. B 57 (1998) 5163. [2] K. Shimakawa, A.V. Kobolov, S.R. Elliott, Adv. Phys. 44 (1995) 475. [3] J. Singh, K. Shimakawa, Advances in Amorphous Semiconductors, Taylor & Francis, London, 2003.
1585
[4] K. Shimakawa, N. Yoshida, A. Ganjoo, Y. Kuzakawa, J. Singh, Phil. Mag. Lett. 77 (1998) 153. [5] A. Ganjoo, K. Shimakawa, K. Kitano, E.A. Davis, J. Non-Cryst. Solids 299–302 (2002) 917. [6] M.I. Klinger, V. Halpern, F. Bass, Phys. Stat. Sol. (b) 230 (2002) 39. [7] J. Singh, Chem. Phys. Lett. 149 (1988) 447. [8] P.W. Anderson, Phys. Rev. Lett. 34 (1975) 953. [9] T. Holstein, Ann. Phys. 8 (1959) 325. [10] N.F. Mott, E.A. Davis, R.A. Street, Philos. Mag. 32 (1975) 961. [11] R.A. Street, N.F. Mott, Phys. Rev. Lett. 35 (1975) 1293. [12] M. Kastner, D. Adler, H. Fritzsche, Phys. Rev. Lett. 37 (1976) 1504. [13] Ke. Tanaka, S. Gohda, A. Odajima, Solid State Commun. 56 (1985) 899. [14] Y. Toyozawa, J. Phys. Soc. Jpn. 50 (1981) 1861. [15] J. Singh, N. Itoh, Appl. Phys. A 51 (1990) 427. [16] K. Morigaki, Physics of Amorphous Semiconductors, WorldScientific, London, 1999.
A quantum approach to photodarkening in chalcogenides
q
Jai Singh *, I.-K. Oh School of Engineering and Logistics, B-41, Faculty of Technology, Charles Darwin University, Darwin, NT 0909, Australia Received 26 February 2005
Abstract We present a microscopic quantum mechanical theory to calculate the reduction in the band gap energy due to illumination in chalcogenides. It is shown that photodarkening occurs due to the strong-carrier lattice interaction leading to pairing of the excited charge carriers. Results demonstrate similar behavior as observed in some experiments. Ó 2005 Elsevier B.V. All rights reserved.
1. Introduction Reduction in the optical band gap due to illumination observed in amorphous chalcogenides (a-Chs) is called photodarkening (PD) [1–3], which has been known to be metastable until recently. This means that PD disappears by annealing but remains even if the illumination is stopped. Much efforts have been devoted to understanding the phenomenon in the last two decades [3], however, no model has been successful in resolving all issues in all materials exhibiting PD. Another photostructural change that occurs in a-Ch is the volume expansion (VE) due to illumination [3]. Although VE and PD are not linearly related, it is demonstrated by Shimakawa et al. [4] that they are related. They have also suggested that both PD and VE occur due to the excessive repulsive Coulomb force among the photoexcited electrons occupying the localized tail states. Accordingly,
q This paper was originally presented at the 20th International Conference on Amorphous and Microcrystalline Semiconductors – Science and Technology (ICAMS 20), Campos do Jorda˜o, Sa˜o Paulo, Brazil, Aug. 25–29, 2003. * Corresponding author. Tel.: +61 (08) 8946 6811; fax: +61 (08) 8946 6366. E-mail address: [email protected] (J. Singh).
0022-3093/$ - see front matter Ó 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.jnoncrysol.2005.03.041
the repulsive force between the layers of a-Chs is considered responsible for VE and the same force is assumed to induce an in-plane slip motion that causes PD. Recently, the same group [5] have also observed the transient PD in a-As2S3, a-As2Se3 and a-Se, which disappears as soon as the illumination is stopped. That means the material reverses back to normal once the illumination is stopped. They have found in a-As2Se3 films that the transient peak of PD is dominant about 60% at 300 K but at a lower temperature of 50 K it reduces to 30% only. A photoassisted site switching (PASS) is suggested for the variation of PD with time, where the fractional growth in the number of PD sites is expressed as a stretched exponential function. It is not clearly understood what causes two types of PD, one that is metastable and gets reversed by annealing and the other that reverses back after stopping the illumination. Moreover, the repulsive Coulomb force model is qualitative and provides no estimates of the magnitude of the repulsive forces required for causing PD and VE. It is not known how much electrostatic repulsive Coulomb force among the excited electrons is required to push two atoms farther apart from their initial positions, whether it relates to in-plane or inter-layer movements. For causing movements in an atomic network, one requires the involvement of lattice vibrations, which surprisingly none of the models on PD and VE has given any account to. However, the involvement of
J. Singh, I.-K. Oh / Journal of Non-Crystalline Solids 351 (2005) 1582–1585
lattice vibrations has recently been considered [6] in inducing photostructural changes in glassy semiconductors. In 1988, Singh [7] discovered a drastic reduction in the band gap due to pairing of charge carriers in excitons and exciton–lattice interaction in non-metallic crystalline solids. The magnitude of reduction in the band gap varies with the magnitude of exciton–phonon interaction which is different in different materials, softer the structure stronger is the carrier–phonon interaction. It has been established that due to the planar structure [3], the carrier–phonon interaction in a-Chs is stronger than in a-Si:H, which satisfies the condition for the occurrence of AndersonÕs negative-U [8] in a-Chs but not in a-Si:H. This is the basis of the hole pairing model for the creation of light-induced-defects in a-Chs [3]. It may be noted that the concept of negative-U has been applied to photodarkening before [10–12], but to our knowledge no quantitative theory has been developed, not for the volume expansion any way. In this paper, based on HolsteinÕs [9] approach, the energy eigenvalues of positively and negatively charged polarons and paired charge carriers, created by excitations due to illumination, are calculated. It is found that the energies of the excited electron (negative charge) and hole (positive charge) polarons gets lowered due to the carrier–phonon interaction. Also the like excited charge carriers can get paired because of the negative-U effect [8] caused by the strong carrier–lattice interaction and the energy of such paired states is also gets lowered. Thus, the hole polaronic state and paired hole states overlap with the lone pair and tail states in a-Chs, which expands the valence band and reduces the band gap energy and hence causes PD. Formation of polarons as well as pairing of holes increases the bond length on which such localizations occur, which causes VE. The energy eigenvalues of such polarons and pairing of charge carriers are calculated. Only the main results are presented and details will be published elsewhere.
2. Theory Based on HolsteinÕs [9] approach, we consider here a model amorphous solid in the form of a linear chain of atoms. The electronic Hamiltonian in such a chain can be written as a sum of charge carrier, phonon, and charge carrier–phonon interaction energy operators as: b ¼H b el þ H b ph þ H b I [3]. H In a-Chs, as the lone pair orbitals overlap with the valence band, the combined band becomes much wider than the conduction band. As the effective mass of charge carriers is inversely proportional to the corresponding bandwidth, the hole effective mass becomes smaller than the electron effective mass in a-Chs. As a result, holes move faster than electrons in a-Chs. How-
1583
ever, in the tail states the charge carriers get localized. Considering the effect of strong carrier–phonon interaction, hole may become a positive charge polaron and an electron may become a negative charged polaron. Further more, two excited electrons and two excited holes can get paired and localized on a bond due to the effect of negative-U, which has already been established [8,3]. Pairing of holes on a bond breaks the bond and creates a pair of dangling bonds, which is used to explain the creation of light-induced defects in a-Ch [3]. Let us first consider the case of exciting a pair of electron and hole, not necessarily an exciton although that possibility exists. The secular equation of such a pair of excited charge carriers is written as: " W 1 C l ¼ EelþL Elh AelþL xlþL Ahl xl 1 X þ Mx2 x2m þ U 12 2 m
!# C l J e ðC lþLþ1 þ C lþL1 Þ
þ J h ðC lþ1 þ C ll Þ;
ð1Þ
where W1 is the energy eigenvalue of such an excited pair, Cl is the probability amplitude coefficient, Ele and Emh represent the energy of an electron at site l and that of a hole at m, respectively, Ael and Ahl are electron-phonon and hole-phonon coupling coefficients, U12 is the Coulomb interaction between the excited pair, Je and Jh are the energy of transfer of electron and hole, respectively, between nearest neighbors. M is the atomic mass and x is the frequency of vibration of nearest neighbor atoms. L is the distance between the excited pair and x is the bond length between nearest neighbors. Minimizing the energy with respect x, we obtain from Eq. (1) as [3] W 1 ¼ E0e E0h þ U 12 2ðJ e J h Þ Eep Ehp ; E0e
ð2Þ
E0h
where and are energies of the excited electron and hole, respectively, without the lattice, and Coulomb interactions between the excited charge carriers, and !2 2 1 Ail Eip ¼ ; i ¼ e; h; ð3Þ 48J i Mx2 which is the polaron binding energy by which the electron and hole energies get lowered due to the carrier– phonon interaction. The lowering of a hole energy means that it moves upward and overlaps more with the lone-pair orbitals, which narrows the band gap. Magnitudes of the polaron energy will be calculated in the next section. Next we consider that two pairs of electrons and holes are excited in the linear chain such that the pair of holes get localized on the same bond and two electrons become two polarons. The energy eigenvalue of such an excited state is obtained as [3] W 2h ¼ 2E0e 2E0h þ U 012 2ðJ e J h Þ 2Eep Ehh ;
ð4Þ
1584
J. Singh, I.-K. Oh / Journal of Non-Crystalline Solids 351 (2005) 1582–1585
where U 012 is the total Coulomb interaction among the excited charge carriers and !2 2 1 Ahl . ð5Þ Ehh ¼ 6J h Mx2 The bond length of a bond on which such a pairing occurs becomes twice as large as when only a polaron is formed and it is given by xhh p ¼
2Aep C p C p : Mx2
ð6Þ
The energy of two excitations without the charge carrier–phonon interaction, can be written as W 20 ¼
2E0e
2E0h
þ
U 012
2ðJ e J h Þ.
ð7Þ
Thus, the energy of a pair of excitations with two holes localized on a bond is lowered by DE given by DE ¼ W 20 W 2h ¼ 2Eep þ Ehh .
ð8Þ
Likewise, the energy eigenvalue for the case where a pair of excited electrons are localized on an anti-bonding orbital of a bond and two holes are like positive charge polarons can be written as W 2e ¼ 2E0e 2E0h þ U 012 2ðJ e J h Þ 2Ehp Eee ;
ð9Þ
where Eee can be obtained from Eq. (9) replacing subscript h by e. It may be noted here that unlike the case of pairing holes, pairing of electrons on a bond does not break the bond.
3. Results We have shown that energetically pairing of like charge carriers is favorable because the energy of such an excited state is lower. Formation of polarons also lowers the energy of the excited states but the effect is less than pairing. It is shown that, when a single hole is localized on a bond, the condition of the minimum energy is given by xð0Þ p in Eq. (6.30) of Ref. [3] which is a half of xhh in Eq. (6). Therefore, the bond on which p the two holes get localized gets enlarged to twice the size as compared to when a single hole localized. This can explain the photoinduced volume expansion in a-Chs. Depending on the material, DE can be in the range of a fraction of an eV to a few eV. We have estimated DE in As2S3 as follows: The energy of lattice vibration of a bond can be written as [3] 2
EðqÞ ¼ E0 12Mx2 ðq q0 Þ ;
For As2S3, the phonon energy of symmetric mode is 344 cm1 [13]. Using this and applying ToyozawaÕs criterion [14,15] of strong carrier–phonon interaction as Eep P Je, we get Je = 5.82 meV from Eq. (3), which gives Eep = 5.82 meV and Ehh = 46.56 meV and DE = 58.32 meV. The band gap of As2S3 is 2.32 eV [16], which gives DE ¼ 1.7%. This agrees quite well with the observed Eg reduction in band gap of about 2% [4].
4. Discussions It is shown that the like excited charge carriers get paired on a bond due to strong carrier–lattice interactions in a-Chs, because energetically such an excited state is more stable. Thus, the energy of paired holes on a bond moves up further in the lone-pair orbitals and tail states, which expands the valence bands. A similar situation occurs by pairing of electrons in the anti-bonding orbitals that lowers the conduction mobility edge down. These two effects together reduce the band gap. The reduction calculated here is about 1.7%, which agrees quite well with about 2% observed experimentally. It has been established [3] that pairing of holes on a bond breaks the bond as soon as two excited holes get localized on it. The bond breaks due to the removal of covalent electrons and a pair of dangling bonds are created. This is the essence of the pairing-hole theory of creating light induced defects in a-Chs. Such light-induced defects are reversible by annealing. However, pairing of excited electrons does not break a bond it only reduces the band gap and such an excited state will be reversed back to the normal after switching off the illumination. This can be applied to explain the metastable and transient PD. The former occurs due to pairing of holes that breaks the bond and that cannot be recovered by stopping the illumination. It remains metastable. The latter occurs due to pairing of electrons and formation of polarons which reverse back to normal after the illumination is stopped. Usually the transient PD is observed more in percentage than metastable PD. This is because there are three processes contributing to the transient PD, pairing of electrons, formation of positive charge polarons and negative charge polarons, in comparison with only one channel of pairing holes contributing to metastable PD.
ð10Þ
where E0 and q0 are the energy and the interaction coordinate at the minimum of the vibrational energy, respectively. The vibrational force along the interaction coordinate can be obtained as A ¼
Using this in Eqs. (6.32) of Ref. [3] and (9), we obtain 2 2 2 2 2 2 Mx q0 Mx q ¼ 481 , and EJhhh ¼ 16 J h 0 . Je
Eep Je
oE oq
q¼0
¼ Mx2 q0 .
5. Conclusion It is shown quantum mechanically that it is the strong carrier–lattice interaction in a-Chs that causes all three
J. Singh, I.-K. Oh / Journal of Non-Crystalline Solids 351 (2005) 1582–1585
phenomena, creation of the light-induced defects, PD and VE. Acknowledgments This work is supported by the Australian Research CouncilÕs Large Grants (2000–2003) and IREX (2001– 2003) schemes. References [1] Ke. Tanaka, Phys. Rev. B 57 (1998) 5163. [2] K. Shimakawa, A.V. Kobolov, S.R. Elliott, Adv. Phys. 44 (1995) 475. [3] J. Singh, K. Shimakawa, Advances in Amorphous Semiconductors, Taylor & Francis, London, 2003.
1585
[4] K. Shimakawa, N. Yoshida, A. Ganjoo, Y. Kuzakawa, J. Singh, Phil. Mag. Lett. 77 (1998) 153. [5] A. Ganjoo, K. Shimakawa, K. Kitano, E.A. Davis, J. Non-Cryst. Solids 299–302 (2002) 917. [6] M.I. Klinger, V. Halpern, F. Bass, Phys. Stat. Sol. (b) 230 (2002) 39. [7] J. Singh, Chem. Phys. Lett. 149 (1988) 447. [8] P.W. Anderson, Phys. Rev. Lett. 34 (1975) 953. [9] T. Holstein, Ann. Phys. 8 (1959) 325. [10] N.F. Mott, E.A. Davis, R.A. Street, Philos. Mag. 32 (1975) 961. [11] R.A. Street, N.F. Mott, Phys. Rev. Lett. 35 (1975) 1293. [12] M. Kastner, D. Adler, H. Fritzsche, Phys. Rev. Lett. 37 (1976) 1504. [13] Ke. Tanaka, S. Gohda, A. Odajima, Solid State Commun. 56 (1985) 899. [14] Y. Toyozawa, J. Phys. Soc. Jpn. 50 (1981) 1861. [15] J. Singh, N. Itoh, Appl. Phys. A 51 (1990) 427. [16] K. Morigaki, Physics of Amorphous Semiconductors, WorldScientific, London, 1999.