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Photoexcitation-induced processes in amorphous semiconductors
Applied Surface Science 248 (2005) 50–55 www.elsevier.com/locate/apsusc
Photoexcitation-induced processes in amorphous semiconductors Jai Singh * School of Engineering and Logistics, Charles Darwin University, Darwin, NT 0909, Australia Available online 22 March 2005
Abstract Theories for the mechanism of photo-induced processes of photodarkening (PD), volume expansion (VE) in amorphous chalcogenides are presented. Rates of spontaneous emission of photons by radiative recombination of excitons in amorphous semiconductors are also calculated and applied to study the excitonic photoluminescence in a-Si:H. Results are compared with previous theories. # 2005 Elsevier B.V. All rights reserved. PACS: 78.55.Qr; 74.81.Bd; 78.66.Jg Keywords: Photo-induced processes; Photodarkening; Volume expansion; Radiative recombination; Amorphous semiconductors
1. Introduction Amorphous semiconductors are used in fabricating many opto-electronic devices such as solar cells, sensors, large area thin film transistors (TFT), X-ray image detectors, memory storage discs, modulators, etc., and hence, have many industrial applications. Most of these devices operate on the principle of first creating electron–hole pairs by optical excitations or injections and then their separation and collection or their radiative recombination. On one hand, structures of such semiconductors do not have any long-range * Tel.: +61 8 89 466 811; fax: +61 8 89 466 366. E-mail address: [email protected].
order, and hence, tend to hinder the motion of charge carriers. On the other hand, the lack of long-range periodicity gives rise to several new phenomena, which do not occur in crystalline solids. These phenomena are, for example, photo-induced creation of dangling bonds (DB), which leads to the well known Staebler–Wronski effect (SWE) in hydrogenated amorphous silicon (a-Si:H), anomalous Hall effect, photodarkening (PD) and volume expansion (VE) in amorphous chalcogenides (a-Chs), etc. [1]. Some of these new phenomena are used in new frontier technologies, for example, future DVDs are likely to use the phenomena of photodarkening and volume expansion for storing information in their optical memory. Light emitting devices of amorphous
0169-4332/$ – see front matter # 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.apsusc.2005.03.031
J. Singh / Applied Surface Science 248 (2005) 50–55
organic materials use the concept of radiative recombination of excitons in a-semiconductors. Xray imaging is based on the creation of electron–hole pairs in a-Chs by X-ray photons and their collection and detection, etc. In amorphous chalcogenides, there are two types of metastabilities: one is defect-related, e.g., the creation of dangling bonds due to illumination, and the other is structure-related, i.e., photostructural changes, such as volume expansion or contraction and photodarkening or photobleaching. Earlier, the origins of defectrelated and structure-related metastabilities were thought to be different because of their different annealing behaviour [1]. However, recently, photostructural changes in a-Chs have been simulated with first-principles type molecular dynamics (MD) [2] as well as by ab initio molecular orbital calculations [3]. These microscopic calculations suggest that some of coordination defects are involved in the structural transformations under optical excitations. Thus, a careful study of the mechanism of creation of lightinduced metastable defects (LIMD) is still required. The reduction in the optical band gap due to illumination observed in amorphous chalcogenides is called photodarkening [1,4,5], which has been known to be metastable until recently because it disappears by annealing but remains even if the illumination is stopped. Many attempts have been made at understanding the phenomenon in the last two decades but no model has been successful in resolving all issues observed in materials exhibiting PD. Another photostructural change that occurs in a-Chs is the volume expansion due to illumination [1]. Although VE and PD are not linearly related, it was demonstrated by Shimakawa et al. [6] 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, 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 [7] has also observed the transient PD in a-As2S3, a-As2Se3 and aSe, which disappears as soon as the illumination is stopped. It is not clearly understood what causes two types of PD, metastable and transient. Moreover, the repulsive Coulomb force model is qualitative and provides no estimates of the magnitude of the
51
repulsive forces required for causing PD and VE. For causing movements in an atomic network, one requires lattice motion, which surprisingly none of the models on PD and VE have given any account of. However, the involvement of lattice vibrations has recently been considered [8] in inducing photostructural changes in glassy semiconductors. In 1988, Singh discovered a large reduction in the band gap [9] due to pairing of charge carriers in excitons and exciton–lattice interaction in nonmetallic crystalline solids. The magnitude of the reduction in the band gap varies with the magnitude of exciton–phonon interaction which is different for different materials. The softer the structure, the stronger is the carrier–phonon interaction. It has been established that due to the planar structure [1], the carrier–phonon interaction in a-Chs is stronger than in a-Si:H, which satisfies the condition for Anderson’s negative-U [10] in a-Chs but not in a-Si:H. It may be noted that the concept of negative-U has been applied to photodarkening before [11–13], but to the author’s knowledge no quantitative theory has been developed at least not for VE. Some controversy has recently arisen on the magnitude of the radiative lifetime in the photoluminescence of a-Si:H. Using time-resolved spectroscopy (TRS), Wilson et al. [14] have observed PL peaks with radiative lifetimes in the nanosecond (ns), microsecond (ms) and millisecond (ms) time ranges in a-Si:H at a temperature of 15 K. In contrast to this, using the quadrature frequency resolved spectroscopy (QFRS), other groups [15–17] have observed only a double peak structure PL in a-Si:H at liquid helium temperature. One peak appears at a short time in the ms range and the other in the ms range. Using the effective mass approach, a theory for the excitonic states in amorphous semiconductors has been developed by Singh et al. [18] and the occurrence of the double peak structure has successfully been explained for both aSi:H and a-Ge:H. In the present paper, based on Holstein’s [19] approach, the energy eigenvalues of positively and negatively charged polarons and paired charge carriers, created by illumination, are calculated. It is found that the energies of the excited electron (negative charge) and hole (positive charge) polarons decrease due to the carrier–phonon interaction. Also, the energy of the paired charge carriers decreases.
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J. Singh / Applied Surface Science 248 (2005) 50–55
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. For photoluminescence, using the first order perturbation theory, rates of spontaneous emission of photons due to the radiative recombination of excitons are calculated at the thermal equilibrium. Assuming that the maximum of the calculated rates and that of the observed PL intensity occur at the same photon energy, the radiative lifetime of excitons is calculated from the inverse of the maximum rates.
2. Photo-induced changes in a-Ch Following Holstein’s approach for a linear chain [19], it is shown that a bond gets stretched when a hole is localized (hole–polaron) on it due to the strong hole–phonon interaction and the bond becomes weaker. The hole energy is lowered by the polaron binding energy. The expression for the bond length xhp with one localized hole on it is [1]: xhp ¼
Ahp Cp C p Mv2
;
(1)
where Ahp is the force of vibration of a pth bond with a localized hole on it and C p is the probability amplitude coefficient (C p is its complex conjugate) for a hole being localized on the bond. M is the atomic mass and v is the frequency of vibration in the Einstein approximation. A weak bond with a localized hole facilitates capturing another hole again due to the strong hole– phonon interaction and then the bond, denoted by xhh p, h. increases to double its size, i.e., xhh ¼ 2x p p Pairing of holes on a weak bond results into ‘‘bond breaking’’ due to removal of covalent electrons and two dangling bonds are created. The gained energy DEhh due to a pair of holes localized on a bond and two electrons localized elsewhere as polarons in a simplified linear chain is obtained as [1]: DEhh ¼ 2Ee p þ Ehh ; ðMv2 Q20 Þ2
(2)
1 is the lattice relaxation where Ee p ¼ 48 T energy due to localization of an electron (polaron
binding energy); here Q0 is the reaction co-ordinate at the minimum of the vibrational potential and T is the carrier transfer matrix element between the nearest ðMv2 Q20 Þ2 neighbours. Ehh ¼ 16 is the lattice relaxation T energy due to localization of a pair of holes (bipolaron binding energy). DEhh is then obtained as: DEhh ¼
5 ½Mv2 Q20 2 : 24 T
(3)
A lattice with strong carrier–phonon interaction induces pairing of both like charge carriers, and therefore, excited electrons can also become paired. The energy of such a paired electron state with the two holes localized as polarons is also lowered by DEee = 2Eh p + Eee, where Eh p is the lattice relaxation energy due to localization of a hole and Eee that due to localization of a pair of electrons. For simplification, assuming that electron–phonon and hole–phonon interactions are equal, one gets DEh p = DEe p and DEhh = DEee. The effect of DEhh and DEee on the energy band gap is to expand the valence band upward and conduction band downward, and hence, narrowing the band gap and producing photodarkening. It may be noted that the pairing of electrons on a bond does not break the bond and it is not very probable that in a pair of excitons, both holes and both electrons will be paired because the effect of strong carrier–phonon interaction drives unlike charge carriers far apart. Therefore, here we consider only the three most probable possibilities: (1) all four charge carriers in a pair of excitons become four individual polarons, (2) two holes are paired and two electrons remain as two polarons and (3) two electrons are paired and two holes remain as polarons. A bond gets broken only in the possibility (2) and it will remain broken even after switching-off the illumination. However, the effect of the possibilities (1) and (3) will disappear after the illumination is switched-off and all excitons have either recombined radiatively or non-radiatively. Thus, only possibility (2) contributes to the metastable photodarkening and possibilities (1) and (3) to transient photodarkening. The pairing of holes on a bond breaks the bond causing an increase in interatomic separation. The localization of a hole on a weak bond also increases the length of the bond. These two effects contribute to the increase in volume by illumination.
J. Singh / Applied Surface Science 248 (2005) 50–55
3. Photoluminescence Four possibilities for the excitonic radiative recombination are considered: (i) both excited electrons and holes are in their extended states, (ii) electrons are in the extended and holes in tail states, (iii) electrons are in tail and hole in extended states and (iv) both are in their tail states. There are two different forms of electron–photon, and hence, exciton–photon transition matrix elements used for amorphous solids [1]. Using these two different forms and applying Fermi’s golden rule, two different forms of rates of spontaneous emission for the possibilities (i)–(iii) are obtained under thermal equilibrium as: Rs p1 ¼
e2 Lmx ð hv E 0 Þ 2 4e0 h3 n2 ð hvÞ ðhv E0 Þ exp Qð h v E0 Þ kB T
(4)
and Rs p2 ¼
m3x e2 a2ex hvð h v E0 Þ 2 2p2 e0 n2 h7 vrA ð hv E0 Þ exp Qðhv E0 Þ; kB T
(5)
where E0 ¼ Ec Ev is the energy difference between an excited pair of electron and hole prior to their recombination, e the electronic charge, e0 the permittivity of vacuum, L the average bond length in a sample, mx and aex the excitonic reduced mass and Bohr radius, respectively, n the refractive index of the material, v the coordination number of the valence electrons per atom, rA the atomic mass density per unit volume and £v is the energy of the emitted photon. Q(£v E0) is a step function used to indicate that there is no radiative recombination for £v < E0 and kB is the Boltzmann constant. For possibility (iv), where both electrons and holes have relaxed down in the tail states before their radiative recombination, their envelope wave functions become localized [1]. Therefore, the transition matrix element changes and then the corresponding two rates of spontaneous emission are obtained as: Rspti ¼ Rs pi exp ð2te0 aex Þ;
i ¼ 1; 2;
(6)
where the subscript spt stands for the emipspontaneous ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ssion from tail-to-tail states, te0 ¼ 2me ðEc Ee Þ= h with me being the effective mass of electron and
53
aex = (5mea0)/(4mx) is the excitonic Bohr radius in the tail states, where m is the reduced mass of electron in the hydrogen atom, e the static dielectric constant and ˚ is the Bohr radius. a0 = 0.529 A
4. Results To estimate the photodarkening, we use the phonon energy = 344 cm1 for the symmetric stretching mode of AsS3/2 units [1] and applying Toyozawa’s criteria [1] of strong electron–phonon interaction as Ee p T, we get T = 12.3 meV and DEhh = 0.12 eV for a-As2S3, which agrees well with the value of 0.16 eV estimated from experiments [1]. The effective masses of electron and hole, required for calculating the rates in Eqs. (4)–(6), have recently been derived for amorphous solids [1]. Accordingly, one gets different effective masses for a charge carrier in its extended and tail states, and for sp3 hybrid systems the electron effective mass is found to be the same as the hole effective mass. Thus, in a sample of aSi:H with 1 at.% weak bonds contributing to the tail states, we get the effective mass of a charge carrier in the extended states as mex ¼ mhx ¼ 0:34 me and in the tail states as met ¼ mht ¼ 7:1 me . For determining E0, it is assumed that the peak of the observed PL intensity occurs at the same energy as that of the rate of spontaneous emission obtained in Eqs. (4) and (5). The PL intensity as a function of 9v has been measured in a-Si:H [14,17,20]. From these measurements, the photon energy corresponding to the PL peak maximum can be determined. By comparing the experimental energy thus obtained with the energy corresponding to the maximum of the rate of spontaneous emission, one can determine E0. Wilson et al. [14] have measured the PL intensity as a function of the emission energy in a-Si:H at 15 K, Stearns [20] at 20 K and Aoki et al. [17] at 3.7 K. The values of Emx estimated from these three measurements at 3.7, 15 and 20 K are obtained as 1.360, 1.428 and 1.450 eV, respectively. The values of Emx are well below the mobility edge Ec by about 0.44 eV at 3.7 K and 0.4 eV at 15 and 20 K. This means that most excited charge carriers have relaxed down below their mobility edges in these experiments. Using these values of Emx and the corresponding lattice temperature in Eqs. (4) and (5), the values of E01 and E02 are found to be the same,
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J. Singh / Applied Surface Science 248 (2005) 50–55
i.e., E01 = E01 = E0 = 1.359, 1.398 and 1.447 eVat 3.7, 15 and 20 K, respectively. For calculating the maximum rate for the possibility (i), the required excitonic reduced mass is obtained as mx = 0.17me and the excitonic Bohr radius as 4.67 nm. For the possibilities (ii) and (iii), where one of the charge carriers of an exciton is in its extended states and the other in its tail states, we get mex = 0.32me and the corresponding excitonic Bohr radius is 2.5 nm. The rates for all three of these possibilities calculated from Eqs. (4) and (5) are found to be of the same order of magnitude 108 s1 at 15– 20 K and 107 s1 at 3.7 K. For the possibility (iv), ˚ we get mex = 3.55me, aex = 2.23 A and te0 = 1.29 1010 m1. Using these in Eq. (6), we get rates one to two orders of magnitude lower than those of possibilities (i)–(iii). The corresponding radiative lifetime, ti = 1/Rs pi for the possibilities (i)–(iii) is found in the ns time range at 15–20 K and in the ms range at 3.7 K. For the possibility (iv), the radiative lifetime is much slower in the ms range.
5. Discussion It is shown that the excited like charge carriers pair 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 antibonding orbitals that lowers the conduction mobility edge. These two effects together reduce the band gap. The reduction calculated here in a-As2S3 is about 0.12 eV, which agrees quite well with 0.16 eV estimated experimentally. It has been established [1] that pairing of holes on a bond breaks the bond as soon as two excited holes get localized on it. This is the essence of the pairing-hole theory of creating light-induced defects in a-Chs, which 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 the illumination off. 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 at higher temperatures. 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. The breaking of bonds expands the interatomic separation, and hence, can expand the volume of a flexible structure like those of a-Chs. Thus, all three processes, bond breaking, photodarkening and volume expansion, occur in a flexible structure due to very strong carrier–phonon interaction. For the photoluminescence, rates of spontaneous emission due to radiative recombinations of excitons are derived and found to be independent of the excitation density but they increase as the PL energy increases, and hence, the radiative lifetime becomes shorter, which is quite consistent with the observed results [16,17]. We have calculated different values of E0 at different values of Emx. Such a change in E0 can only be possible in amorphous solids, which do not have well defined energy gap, and therefore, the excited charge carriers can relax down to different E0 through the four different possibilities. According to the above results of the present theory and taking into account the Stokes shift of about 0.4 eV, the radiative lifetime measured in the ns range by Wilson et al. [14] at 15 K, by Stearns [20] at 20 K, and that in the ms range by Aoki et al. [17] at 3.7 K may be attributed to the radiative recombination of singlet excitons through the possibility (ii). Wilson et al. [14] and Aoki et al. [17] have also observed slower lifetimes but in the ms range, which may be attributed to the radiative recombination of geminate pairs through the possibility (iv). As an exciton relaxes to the tail states, its Bohr radius is retained but not its excitonic motion due to localization. It becomes a geminate pair. Thus, in a-Si:H two types of geminate pairs are possible: (i) from excitons and (ii) from other excited electrons and holes. Latter are likely to have larger separation, and hence, slower radiative lifetime. Such slower lifetime has recently been observed [17].
J. Singh / Applied Surface Science 248 (2005) 50–55
Acknowledgements This work is supported by the ARC large Grant (2000–2003) and ARC IREX (2001–2003) Grant Schemes. References [1] J. Singh, K. Shimakawa, Advances in Amorphous .Semiconductors, Taylor & Francis, London/New York, 2003 [2] D.A. Jun Li, Drabold, Phys. Rev. Lett. 85 (2000) 2785. [3] T. Uchino, D.C. Clary, S.R. Elliott, Phys. Rev. Lett. 85 (2000) 3305. [4] K. Shimakawa, A. Kolobov, R. Elliott, Adv. Phys. 44 (1995) 475. [5] Ke. Tanaka, Phys. Rev. B 57 (1998) 5163. [6] K. Shimakawa, N. Yoshida, A. Ganjoo, Y. Kuzakawa, J. Singh, Philos. Mag. Lett. 77 (1998) 153. [7] A. Ganjoo, K. Shimakawa, K. Kitano, E.A. Davis, J. NonCryst. Solids 299–302 (2002) 917.
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[8] M.I. Klinger, V. Halpern, F. Bass, Phys. Status Solidi b 230 (2002) 39. [9] J. Singh, Chem. Phys. Lett. 149 (1988) 447. [10] P.W. Anderson, Phys. Rev. Lett. 34 (1975) 953. [11] N.F. Mott, E.A. Davis, R.A. Street, Philos. Mag. 32 (1975) 961. [12] R.A. Street, N.F. Mott, Phys. Rev. Lett. 35 (1975) 1293. [13] M. Kastner, D. Adler, H. Fritzsche, Phys. Rev. Lett. 37 (1976) 1504. [14] B.A. Wilson, P. Hu, J.P. Harbison, T.M. Jedju, Phys. Rev. Lett. 50 (1983) 1490. [15] R. Stachowitz, M. Schubert, W. Fuhs, J. Non-Cryst. Solids 227–230 (1998) 190. [16] S. Ishii, M. Kurihara, T. Aoki, K. Shimakawa, J. Singh, J. NonCryst. Solids 266–269 (1999) 721. [17] T. Aoki, T. Shimizu, D. Saito, K. Ikeda, Presented at the 13th International School of Condensed Matter Physics (13 ISCMP), 30 August–3 September, 2004, Varna, Bulgaria, and references therein, J. Optoelectron. Adv. Mater. 7 (2005) 137. [18] J. Singh, T. Aoki, K. Shimakawa, Philos. Mag. B 82 (2002) 855. [19] T. Holstein, Ann. Phys. 8 (1959) 325. [20] D.G. Stearns, Phys. Rev. B 30 (1984) 6000.
Photoexcitation-induced processes in amorphous semiconductors Jai Singh * School of Engineering and Logistics, Charles Darwin University, Darwin, NT 0909, Australia Available online 22 March 2005
Abstract Theories for the mechanism of photo-induced processes of photodarkening (PD), volume expansion (VE) in amorphous chalcogenides are presented. Rates of spontaneous emission of photons by radiative recombination of excitons in amorphous semiconductors are also calculated and applied to study the excitonic photoluminescence in a-Si:H. Results are compared with previous theories. # 2005 Elsevier B.V. All rights reserved. PACS: 78.55.Qr; 74.81.Bd; 78.66.Jg Keywords: Photo-induced processes; Photodarkening; Volume expansion; Radiative recombination; Amorphous semiconductors
1. Introduction Amorphous semiconductors are used in fabricating many opto-electronic devices such as solar cells, sensors, large area thin film transistors (TFT), X-ray image detectors, memory storage discs, modulators, etc., and hence, have many industrial applications. Most of these devices operate on the principle of first creating electron–hole pairs by optical excitations or injections and then their separation and collection or their radiative recombination. On one hand, structures of such semiconductors do not have any long-range * Tel.: +61 8 89 466 811; fax: +61 8 89 466 366. E-mail address: [email protected].
order, and hence, tend to hinder the motion of charge carriers. On the other hand, the lack of long-range periodicity gives rise to several new phenomena, which do not occur in crystalline solids. These phenomena are, for example, photo-induced creation of dangling bonds (DB), which leads to the well known Staebler–Wronski effect (SWE) in hydrogenated amorphous silicon (a-Si:H), anomalous Hall effect, photodarkening (PD) and volume expansion (VE) in amorphous chalcogenides (a-Chs), etc. [1]. Some of these new phenomena are used in new frontier technologies, for example, future DVDs are likely to use the phenomena of photodarkening and volume expansion for storing information in their optical memory. Light emitting devices of amorphous
0169-4332/$ – see front matter # 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.apsusc.2005.03.031
J. Singh / Applied Surface Science 248 (2005) 50–55
organic materials use the concept of radiative recombination of excitons in a-semiconductors. Xray imaging is based on the creation of electron–hole pairs in a-Chs by X-ray photons and their collection and detection, etc. In amorphous chalcogenides, there are two types of metastabilities: one is defect-related, e.g., the creation of dangling bonds due to illumination, and the other is structure-related, i.e., photostructural changes, such as volume expansion or contraction and photodarkening or photobleaching. Earlier, the origins of defectrelated and structure-related metastabilities were thought to be different because of their different annealing behaviour [1]. However, recently, photostructural changes in a-Chs have been simulated with first-principles type molecular dynamics (MD) [2] as well as by ab initio molecular orbital calculations [3]. These microscopic calculations suggest that some of coordination defects are involved in the structural transformations under optical excitations. Thus, a careful study of the mechanism of creation of lightinduced metastable defects (LIMD) is still required. The reduction in the optical band gap due to illumination observed in amorphous chalcogenides is called photodarkening [1,4,5], which has been known to be metastable until recently because it disappears by annealing but remains even if the illumination is stopped. Many attempts have been made at understanding the phenomenon in the last two decades but no model has been successful in resolving all issues observed in materials exhibiting PD. Another photostructural change that occurs in a-Chs is the volume expansion due to illumination [1]. Although VE and PD are not linearly related, it was demonstrated by Shimakawa et al. [6] 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, 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 [7] has also observed the transient PD in a-As2S3, a-As2Se3 and aSe, which disappears as soon as the illumination is stopped. It is not clearly understood what causes two types of PD, metastable and transient. Moreover, the repulsive Coulomb force model is qualitative and provides no estimates of the magnitude of the
51
repulsive forces required for causing PD and VE. For causing movements in an atomic network, one requires lattice motion, which surprisingly none of the models on PD and VE have given any account of. However, the involvement of lattice vibrations has recently been considered [8] in inducing photostructural changes in glassy semiconductors. In 1988, Singh discovered a large reduction in the band gap [9] due to pairing of charge carriers in excitons and exciton–lattice interaction in nonmetallic crystalline solids. The magnitude of the reduction in the band gap varies with the magnitude of exciton–phonon interaction which is different for different materials. The softer the structure, the stronger is the carrier–phonon interaction. It has been established that due to the planar structure [1], the carrier–phonon interaction in a-Chs is stronger than in a-Si:H, which satisfies the condition for Anderson’s negative-U [10] in a-Chs but not in a-Si:H. It may be noted that the concept of negative-U has been applied to photodarkening before [11–13], but to the author’s knowledge no quantitative theory has been developed at least not for VE. Some controversy has recently arisen on the magnitude of the radiative lifetime in the photoluminescence of a-Si:H. Using time-resolved spectroscopy (TRS), Wilson et al. [14] have observed PL peaks with radiative lifetimes in the nanosecond (ns), microsecond (ms) and millisecond (ms) time ranges in a-Si:H at a temperature of 15 K. In contrast to this, using the quadrature frequency resolved spectroscopy (QFRS), other groups [15–17] have observed only a double peak structure PL in a-Si:H at liquid helium temperature. One peak appears at a short time in the ms range and the other in the ms range. Using the effective mass approach, a theory for the excitonic states in amorphous semiconductors has been developed by Singh et al. [18] and the occurrence of the double peak structure has successfully been explained for both aSi:H and a-Ge:H. In the present paper, based on Holstein’s [19] approach, the energy eigenvalues of positively and negatively charged polarons and paired charge carriers, created by illumination, are calculated. It is found that the energies of the excited electron (negative charge) and hole (positive charge) polarons decrease due to the carrier–phonon interaction. Also, the energy of the paired charge carriers decreases.
52
J. Singh / Applied Surface Science 248 (2005) 50–55
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. For photoluminescence, using the first order perturbation theory, rates of spontaneous emission of photons due to the radiative recombination of excitons are calculated at the thermal equilibrium. Assuming that the maximum of the calculated rates and that of the observed PL intensity occur at the same photon energy, the radiative lifetime of excitons is calculated from the inverse of the maximum rates.
2. Photo-induced changes in a-Ch Following Holstein’s approach for a linear chain [19], it is shown that a bond gets stretched when a hole is localized (hole–polaron) on it due to the strong hole–phonon interaction and the bond becomes weaker. The hole energy is lowered by the polaron binding energy. The expression for the bond length xhp with one localized hole on it is [1]: xhp ¼
Ahp Cp C p Mv2
;
(1)
where Ahp is the force of vibration of a pth bond with a localized hole on it and C p is the probability amplitude coefficient (C p is its complex conjugate) for a hole being localized on the bond. M is the atomic mass and v is the frequency of vibration in the Einstein approximation. A weak bond with a localized hole facilitates capturing another hole again due to the strong hole– phonon interaction and then the bond, denoted by xhh p, h. increases to double its size, i.e., xhh ¼ 2x p p Pairing of holes on a weak bond results into ‘‘bond breaking’’ due to removal of covalent electrons and two dangling bonds are created. The gained energy DEhh due to a pair of holes localized on a bond and two electrons localized elsewhere as polarons in a simplified linear chain is obtained as [1]: DEhh ¼ 2Ee p þ Ehh ; ðMv2 Q20 Þ2
(2)
1 is the lattice relaxation where Ee p ¼ 48 T energy due to localization of an electron (polaron
binding energy); here Q0 is the reaction co-ordinate at the minimum of the vibrational potential and T is the carrier transfer matrix element between the nearest ðMv2 Q20 Þ2 neighbours. Ehh ¼ 16 is the lattice relaxation T energy due to localization of a pair of holes (bipolaron binding energy). DEhh is then obtained as: DEhh ¼
5 ½Mv2 Q20 2 : 24 T
(3)
A lattice with strong carrier–phonon interaction induces pairing of both like charge carriers, and therefore, excited electrons can also become paired. The energy of such a paired electron state with the two holes localized as polarons is also lowered by DEee = 2Eh p + Eee, where Eh p is the lattice relaxation energy due to localization of a hole and Eee that due to localization of a pair of electrons. For simplification, assuming that electron–phonon and hole–phonon interactions are equal, one gets DEh p = DEe p and DEhh = DEee. The effect of DEhh and DEee on the energy band gap is to expand the valence band upward and conduction band downward, and hence, narrowing the band gap and producing photodarkening. It may be noted that the pairing of electrons on a bond does not break the bond and it is not very probable that in a pair of excitons, both holes and both electrons will be paired because the effect of strong carrier–phonon interaction drives unlike charge carriers far apart. Therefore, here we consider only the three most probable possibilities: (1) all four charge carriers in a pair of excitons become four individual polarons, (2) two holes are paired and two electrons remain as two polarons and (3) two electrons are paired and two holes remain as polarons. A bond gets broken only in the possibility (2) and it will remain broken even after switching-off the illumination. However, the effect of the possibilities (1) and (3) will disappear after the illumination is switched-off and all excitons have either recombined radiatively or non-radiatively. Thus, only possibility (2) contributes to the metastable photodarkening and possibilities (1) and (3) to transient photodarkening. The pairing of holes on a bond breaks the bond causing an increase in interatomic separation. The localization of a hole on a weak bond also increases the length of the bond. These two effects contribute to the increase in volume by illumination.
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3. Photoluminescence Four possibilities for the excitonic radiative recombination are considered: (i) both excited electrons and holes are in their extended states, (ii) electrons are in the extended and holes in tail states, (iii) electrons are in tail and hole in extended states and (iv) both are in their tail states. There are two different forms of electron–photon, and hence, exciton–photon transition matrix elements used for amorphous solids [1]. Using these two different forms and applying Fermi’s golden rule, two different forms of rates of spontaneous emission for the possibilities (i)–(iii) are obtained under thermal equilibrium as: Rs p1 ¼
e2 Lmx ð hv E 0 Þ 2 4e0 h3 n2 ð hvÞ ðhv E0 Þ exp Qð h v E0 Þ kB T
(4)
and Rs p2 ¼
m3x e2 a2ex hvð h v E0 Þ 2 2p2 e0 n2 h7 vrA ð hv E0 Þ exp Qðhv E0 Þ; kB T
(5)
where E0 ¼ Ec Ev is the energy difference between an excited pair of electron and hole prior to their recombination, e the electronic charge, e0 the permittivity of vacuum, L the average bond length in a sample, mx and aex the excitonic reduced mass and Bohr radius, respectively, n the refractive index of the material, v the coordination number of the valence electrons per atom, rA the atomic mass density per unit volume and £v is the energy of the emitted photon. Q(£v E0) is a step function used to indicate that there is no radiative recombination for £v < E0 and kB is the Boltzmann constant. For possibility (iv), where both electrons and holes have relaxed down in the tail states before their radiative recombination, their envelope wave functions become localized [1]. Therefore, the transition matrix element changes and then the corresponding two rates of spontaneous emission are obtained as: Rspti ¼ Rs pi exp ð2te0 aex Þ;
i ¼ 1; 2;
(6)
where the subscript spt stands for the emipspontaneous ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ssion from tail-to-tail states, te0 ¼ 2me ðEc Ee Þ= h with me being the effective mass of electron and
53
aex = (5mea0)/(4mx) is the excitonic Bohr radius in the tail states, where m is the reduced mass of electron in the hydrogen atom, e the static dielectric constant and ˚ is the Bohr radius. a0 = 0.529 A
4. Results To estimate the photodarkening, we use the phonon energy = 344 cm1 for the symmetric stretching mode of AsS3/2 units [1] and applying Toyozawa’s criteria [1] of strong electron–phonon interaction as Ee p T, we get T = 12.3 meV and DEhh = 0.12 eV for a-As2S3, which agrees well with the value of 0.16 eV estimated from experiments [1]. The effective masses of electron and hole, required for calculating the rates in Eqs. (4)–(6), have recently been derived for amorphous solids [1]. Accordingly, one gets different effective masses for a charge carrier in its extended and tail states, and for sp3 hybrid systems the electron effective mass is found to be the same as the hole effective mass. Thus, in a sample of aSi:H with 1 at.% weak bonds contributing to the tail states, we get the effective mass of a charge carrier in the extended states as mex ¼ mhx ¼ 0:34 me and in the tail states as met ¼ mht ¼ 7:1 me . For determining E0, it is assumed that the peak of the observed PL intensity occurs at the same energy as that of the rate of spontaneous emission obtained in Eqs. (4) and (5). The PL intensity as a function of 9v has been measured in a-Si:H [14,17,20]. From these measurements, the photon energy corresponding to the PL peak maximum can be determined. By comparing the experimental energy thus obtained with the energy corresponding to the maximum of the rate of spontaneous emission, one can determine E0. Wilson et al. [14] have measured the PL intensity as a function of the emission energy in a-Si:H at 15 K, Stearns [20] at 20 K and Aoki et al. [17] at 3.7 K. The values of Emx estimated from these three measurements at 3.7, 15 and 20 K are obtained as 1.360, 1.428 and 1.450 eV, respectively. The values of Emx are well below the mobility edge Ec by about 0.44 eV at 3.7 K and 0.4 eV at 15 and 20 K. This means that most excited charge carriers have relaxed down below their mobility edges in these experiments. Using these values of Emx and the corresponding lattice temperature in Eqs. (4) and (5), the values of E01 and E02 are found to be the same,
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J. Singh / Applied Surface Science 248 (2005) 50–55
i.e., E01 = E01 = E0 = 1.359, 1.398 and 1.447 eVat 3.7, 15 and 20 K, respectively. For calculating the maximum rate for the possibility (i), the required excitonic reduced mass is obtained as mx = 0.17me and the excitonic Bohr radius as 4.67 nm. For the possibilities (ii) and (iii), where one of the charge carriers of an exciton is in its extended states and the other in its tail states, we get mex = 0.32me and the corresponding excitonic Bohr radius is 2.5 nm. The rates for all three of these possibilities calculated from Eqs. (4) and (5) are found to be of the same order of magnitude 108 s1 at 15– 20 K and 107 s1 at 3.7 K. For the possibility (iv), ˚ we get mex = 3.55me, aex = 2.23 A and te0 = 1.29 1010 m1. Using these in Eq. (6), we get rates one to two orders of magnitude lower than those of possibilities (i)–(iii). The corresponding radiative lifetime, ti = 1/Rs pi for the possibilities (i)–(iii) is found in the ns time range at 15–20 K and in the ms range at 3.7 K. For the possibility (iv), the radiative lifetime is much slower in the ms range.
5. Discussion It is shown that the excited like charge carriers pair 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 antibonding orbitals that lowers the conduction mobility edge. These two effects together reduce the band gap. The reduction calculated here in a-As2S3 is about 0.12 eV, which agrees quite well with 0.16 eV estimated experimentally. It has been established [1] that pairing of holes on a bond breaks the bond as soon as two excited holes get localized on it. This is the essence of the pairing-hole theory of creating light-induced defects in a-Chs, which 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 the illumination off. 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 at higher temperatures. 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. The breaking of bonds expands the interatomic separation, and hence, can expand the volume of a flexible structure like those of a-Chs. Thus, all three processes, bond breaking, photodarkening and volume expansion, occur in a flexible structure due to very strong carrier–phonon interaction. For the photoluminescence, rates of spontaneous emission due to radiative recombinations of excitons are derived and found to be independent of the excitation density but they increase as the PL energy increases, and hence, the radiative lifetime becomes shorter, which is quite consistent with the observed results [16,17]. We have calculated different values of E0 at different values of Emx. Such a change in E0 can only be possible in amorphous solids, which do not have well defined energy gap, and therefore, the excited charge carriers can relax down to different E0 through the four different possibilities. According to the above results of the present theory and taking into account the Stokes shift of about 0.4 eV, the radiative lifetime measured in the ns range by Wilson et al. [14] at 15 K, by Stearns [20] at 20 K, and that in the ms range by Aoki et al. [17] at 3.7 K may be attributed to the radiative recombination of singlet excitons through the possibility (ii). Wilson et al. [14] and Aoki et al. [17] have also observed slower lifetimes but in the ms range, which may be attributed to the radiative recombination of geminate pairs through the possibility (iv). As an exciton relaxes to the tail states, its Bohr radius is retained but not its excitonic motion due to localization. It becomes a geminate pair. Thus, in a-Si:H two types of geminate pairs are possible: (i) from excitons and (ii) from other excited electrons and holes. Latter are likely to have larger separation, and hence, slower radiative lifetime. Such slower lifetime has recently been observed [17].
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Acknowledgements This work is supported by the ARC large Grant (2000–2003) and ARC IREX (2001–2003) Grant Schemes. References [1] J. Singh, K. Shimakawa, Advances in Amorphous .Semiconductors, Taylor & Francis, London/New York, 2003 [2] D.A. Jun Li, Drabold, Phys. Rev. Lett. 85 (2000) 2785. [3] T. Uchino, D.C. Clary, S.R. Elliott, Phys. Rev. Lett. 85 (2000) 3305. [4] K. Shimakawa, A. Kolobov, R. Elliott, Adv. Phys. 44 (1995) 475. [5] Ke. Tanaka, Phys. Rev. B 57 (1998) 5163. [6] K. Shimakawa, N. Yoshida, A. Ganjoo, Y. Kuzakawa, J. Singh, Philos. Mag. Lett. 77 (1998) 153. [7] A. Ganjoo, K. Shimakawa, K. Kitano, E.A. Davis, J. NonCryst. Solids 299–302 (2002) 917.
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