- Email: [email protected]
Electron trapping by excited microvoids does not give rise to a Staebler–Wronski effect
ARTICLE IN PRESS
Physica B 358 (2005) 181–184 www.elsevier.com/locate/physb
Electron trapping by excited microvoids does not give rise to a Staebler–Wronski effect Harald Overhof Theoretical Physics, Faculty of Science, University of Paderborn, D 33098 Paderborn, Germany Received 29 November 2004; received in revised form 6 January 2005; accepted 6 January 2005
Abstract It is shown that the model of electron trapping by excited microvoids recently proposed by Kru¨ger and Sax (Physica B 353 (2004) 263) does not lead to the phototransport features that are characteristic for the Staebler–Wronski effect. According to that model, a photoinduced degradation, if present at all, would show a characteristic time constant of the order of a few milliseconds rather than the experimentally observed 1000 h. r 2005 Elsevier B.V. All rights reserved. PACS: 61.72.Cc; 72.40.+4; 72.80.Ng Keywords: Staebler–Wronski effect; Amorphous silicon
1. Introduction The application of amorphous hydrogenated silicon, a-Si:H for solar cells is strongly impeded by the Staebler–Wronski effect (SWE) [1,2], i.e. the reduction of the photoconductivity upon extended exposure to light. This reduction saturates, but is persistent and cannot be reversed except by thermal annealing. A microscopic understanding of this effect could be a first step towards a strategy to reduce the SWE and, therefore, a large body of theoretical and experimental works on the Tel.: +49 5251 602334; fax: +49 5251 603435.
E-mail address: [email protected] (H. Overhof).
SWE has accumulated in the last 25 years (for a review, see e.g. [5]). A detailed microscopic model for this effect, however, is still missing. The conventional explanation of the SWE [3,4] starts with some structural defects as e.g. weak bonds or Si–H bonds. In the virgin state of the sample and in the annealed state (SW state A), these defects are electrically inactive. By the direct or indirect interaction with light, these defects are somehow activated and form additional recombination centers. These additional recombination centers give rise to the decreased photocarrier lifetime or mobility. Assuming a small cross section for the defect activation process (e.g. the breaking of a weak bond), this model can readily
0921-4526/$ - see front matter r 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2005.01.142
ARTICLE IN PRESS 182
H. Overhof / Physica B 358 (2005) 181–184
account for the long time constant (1000 h) for the photoinduced degradation as opposed to the rather short ( 107 s) photoresponse time. After extended exposure to light, all the structural defects are activated and the degradation of the photocurrent saturates (SW state B). As elucidated in Ref. [6], these models do not quite satisfactorily account for all properties of the additional ‘‘dangling bond’’ paramagnetic centers that are observed after photodegradation. However, the conventional models of the SWE effect can easily be constructed to satisfactorily describe the slow rise and the thermal annealing of the activated structural defects. Recently, Kru¨ger and Sax [6] have proposed an alternative SWE process that starts from a microscopic defect model. In their theory, excited microvoid centers are considered to act as traps for the photoexcited electrons. These authors present a complex set of rate equations for which solutions in the two limiting cases are presented, the case of a sample without photodegradation, and for a sample where the degradation has saturated. The essential point in the Kru¨ger and Sax model is some neutral defect A (the microscopic nature of this defect is of no concern for our present purposes) with an excited state A that can be excited by light of the same photon energy as the light that gives rise to the photoconductivity. When capturing a photoelectron, the excited neutral A defect is transformed into some negatively charged defect state B ; which by internal conversion transforms into the metastable negative state B. This capturing of photoelectrons into metastable B states is considered to be the reason for the SWE, according to Kru¨ger and Sax [6]. In this communication, we show that the model presented by Kru¨ger and Sax [6] cannot explain the observed SWE. First, we show that the rate equations and the numbers for the rate constants given in Table 1 of Kru¨ger and Sax [6] are inconsistent. Next, we show that, even for a consistent set of rate equations and rate constants, the model itself does not even provide the asymptotic solutions that are required to explain the SWE. In order to show this, we do not have to solve the full set of rate equations. Instead, we
restrict our discussion to the implications of Eqs. (KS-2), (KS-3) of Kru¨ger and Sax [6] using the values for the rate constants, etc. as given in Table 1 of their paper. Since the time constant for the Staebler–Wronski effect is of the order of 1000 h, while the photocurrent relaxation time is of the order of 107 s; the system under study can always be considered to be in a quasistationary state. We start from Eq. (KS-3) of Kru¨ger and Sax [6]: d ½A ¼ k3 ½A k2 ½A ½e k4 ½A : dt
(1)
In the SW state A, there are no B states according to Ref. [6]. Before the light exposure is started, there are also no A states. Hence, at the beginning of the illumination, Eq. (KS-3) reduces to d ½A ¼ k3 ½A: dt
(2)
Using ½A ¼ 1016 cm3 and k3 ¼ 1:2 1021 s1 from Table 1 of Ref. [6] (in Table 1 of Ref. [6], the initial concentration of A-states is denoted by w0 ), we find the rather horrible rate of 1:2 1037 cm3 s1 : This shows already that the set of rate equations of Ref. [6] is incomplete: the constant k3 must be replaced by some term that is proportional to intensity of the exciting light. Furthermore, it must be considered in the rate equation that the generation rate G of the photocarriers competes with the A!A process. We disregard this inconsistency assuming that Eq. (2) simply means that the quasiequilibrium value of ½A is reached immediately as the sample is exposed to light. For the SW state A, (before the photodegradation) the conduction band photoelectron density according to Kru¨ger and Sax (Ref. [6] Eq. (KS28)) is 6:25 1013 cm3 : Assuming ½A to be some fraction x of [A], we obtain with k2 ¼ 1:2 108 cm3 s1 k2 ½A ½e ¼ x 7:25 1021 cm3 s1 while with k4 ¼ 1:2 1018 s1 ; k4 ½A ¼ x 1:2 1034 cm3 s1 : Hence, we may safely leave out the second term of the r.h.s. of Eq. (1) and obtain for the
ARTICLE IN PRESS H. Overhof / Physica B 358 (2005) 181–184
stationary state ½A ¼
k4 ½A ¼ 103 ½A : k3
Hence, in the presence of light x ’ 1 and practically all A states are excited. With this reduction of [A], the first term on the r.h.s. of Eq. (1) is still horrible (corresponding to a power of 3:5 1015 W cm3 for a photon energy of 2 eV). If we insert this result for x into Eq. (KS-2) of Kru¨ger and Sax [6] which reads d ½e ¼ G k1 ½hþ ½e k2 ½A ½e ; dt
(3)
we obtain for the stationary case k1 ½e ½hþ ¼ G (this relation was used in Ref. [6] for the determination of the electron and hole densities), but since k2 ½A ½e ¼ 7:25xG; more photoelectrons would be captured than created. This result shows once more that the rate equations as well as the rate constants k1 to k4 are not consistent and must be reconsidered. The model presented by Kru¨ger and Sax differs from the conventional SWE models in that, in Kru¨ger and Sax, the photoelectrons are trapped by the additional photogenerated centers (in the conventional models, these act as additional recombination centers, rather than as traps). We now show that such a model does not give rise to the experimentally observed SWE. To this aim, we assume that all rate constants can be chosen in a consistent way such that for the SW state A, there is a certain number of A-states, with ½A ¼ x½A (we leave the value of x open), with a photoelectron density as given above, and without B states. In the light-soaked SW state B, the parameters are assumed to result in values for the conduction band electron densities, for [A], and also for ½A that are reduced by a factor of 100 with respect to the SW state A. Therefore, in the light-soaked case the density of negatively charged B-states must be large, comparable to the value of [A] in SW state A. For reasons of charge neutrality, the number of
183
holes must be close to [B] and these holes must somehow be trapped, because in the experiment, the photoconductivity is by electrons. From Eq. (3), we see that in the SW state B the second term on the r.h.s is reduced by a factor of 104 with respect to the SW state A, while the third term is reduced by a factor of 100. But since the generation rate for photoelectrons, G, is not altered by the SW effect, there can be no stationary solution of Eq. (3) with these densities. Furthermore, if the third term on the r.h.s. of Eq. (3), i.e. the capture of the photoelectrons by the A excited defects, is considered to reduce the photoelectron density by a factor of 100 in the Staebler–Wronski state B, then virtually all photoexcited electrons must be captured by this process. It follows that for virtually every photoelectron–hole pair also one A ! A excitation must take place. Hence the full A ! A ! B ! B ! A cycle is run at a rate similar to G. With G ¼ 1021 cm3 s1 ; as given by Kru¨ger and Sax [6, Table 1] this process must not take more time than about 105 s in order to avoid a bottleneck which would lead to a pileup of one of the intermediate states. Usually, a final quasiequilibrium is established in a reaction cycle once all the relevant consecutive reaction steps have been completed a few times. Thus, the Staebler–Wronski effect should saturate after 103 s at most. In the model of Kru¨ger and Sax, there is no mechanism that would cause the 1000 h time constant of the Staebler–Wronski effect. We might alternatively consider the second term on the r.h.s. of Eq. (3) to be the dominant term. Within the framework of the Kru¨ger–Sax model, this would follow naturally when assuming that both free and trapped holes could recombine directly. But this assumption would turn the model of Kru¨ger and Sax into a conventional model for the Staebler–Wronski effect: as a byproduct of the photoexcitation with low probability and hence a long time constant, some B defect centers are created, maybe via the A ! A ! B ! B process, which capture the photoexcited holes and thus act as a recombination center for the photoelectrons without being transformed into the A centers.
ARTICLE IN PRESS 184
H. Overhof / Physica B 358 (2005) 181–184
References [1] D.L. Staebler, C.R. Wronski, Appl. Phys. Lett. 31 (1977) 292. [2] D.L. Staebler, C.R. Wronski, J. Appl. Phys. 51 (1980) 3262.
[3] M. Stutzmann, W.B. Jackson, C.C. Tsai, Phys. Rev. B 32 (1985) 23. [4] H.M. Branz, Phys. Rev. B 59 (1999) 5498; H.M. Branz, Phys. Rev. B 60 (1999) 7725. [5] H. Fritzsche, Ann. Rev. Mater. Res. 31 (2001) 47. [6] T. Kru¨ger, A.F. Sax, Physica B 353 (2004) 263.
Physica B 358 (2005) 181–184 www.elsevier.com/locate/physb
Electron trapping by excited microvoids does not give rise to a Staebler–Wronski effect Harald Overhof Theoretical Physics, Faculty of Science, University of Paderborn, D 33098 Paderborn, Germany Received 29 November 2004; received in revised form 6 January 2005; accepted 6 January 2005
Abstract It is shown that the model of electron trapping by excited microvoids recently proposed by Kru¨ger and Sax (Physica B 353 (2004) 263) does not lead to the phototransport features that are characteristic for the Staebler–Wronski effect. According to that model, a photoinduced degradation, if present at all, would show a characteristic time constant of the order of a few milliseconds rather than the experimentally observed 1000 h. r 2005 Elsevier B.V. All rights reserved. PACS: 61.72.Cc; 72.40.+4; 72.80.Ng Keywords: Staebler–Wronski effect; Amorphous silicon
1. Introduction The application of amorphous hydrogenated silicon, a-Si:H for solar cells is strongly impeded by the Staebler–Wronski effect (SWE) [1,2], i.e. the reduction of the photoconductivity upon extended exposure to light. This reduction saturates, but is persistent and cannot be reversed except by thermal annealing. A microscopic understanding of this effect could be a first step towards a strategy to reduce the SWE and, therefore, a large body of theoretical and experimental works on the Tel.: +49 5251 602334; fax: +49 5251 603435.
E-mail address: [email protected] (H. Overhof).
SWE has accumulated in the last 25 years (for a review, see e.g. [5]). A detailed microscopic model for this effect, however, is still missing. The conventional explanation of the SWE [3,4] starts with some structural defects as e.g. weak bonds or Si–H bonds. In the virgin state of the sample and in the annealed state (SW state A), these defects are electrically inactive. By the direct or indirect interaction with light, these defects are somehow activated and form additional recombination centers. These additional recombination centers give rise to the decreased photocarrier lifetime or mobility. Assuming a small cross section for the defect activation process (e.g. the breaking of a weak bond), this model can readily
0921-4526/$ - see front matter r 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2005.01.142
ARTICLE IN PRESS 182
H. Overhof / Physica B 358 (2005) 181–184
account for the long time constant (1000 h) for the photoinduced degradation as opposed to the rather short ( 107 s) photoresponse time. After extended exposure to light, all the structural defects are activated and the degradation of the photocurrent saturates (SW state B). As elucidated in Ref. [6], these models do not quite satisfactorily account for all properties of the additional ‘‘dangling bond’’ paramagnetic centers that are observed after photodegradation. However, the conventional models of the SWE effect can easily be constructed to satisfactorily describe the slow rise and the thermal annealing of the activated structural defects. Recently, Kru¨ger and Sax [6] have proposed an alternative SWE process that starts from a microscopic defect model. In their theory, excited microvoid centers are considered to act as traps for the photoexcited electrons. These authors present a complex set of rate equations for which solutions in the two limiting cases are presented, the case of a sample without photodegradation, and for a sample where the degradation has saturated. The essential point in the Kru¨ger and Sax model is some neutral defect A (the microscopic nature of this defect is of no concern for our present purposes) with an excited state A that can be excited by light of the same photon energy as the light that gives rise to the photoconductivity. When capturing a photoelectron, the excited neutral A defect is transformed into some negatively charged defect state B ; which by internal conversion transforms into the metastable negative state B. This capturing of photoelectrons into metastable B states is considered to be the reason for the SWE, according to Kru¨ger and Sax [6]. In this communication, we show that the model presented by Kru¨ger and Sax [6] cannot explain the observed SWE. First, we show that the rate equations and the numbers for the rate constants given in Table 1 of Kru¨ger and Sax [6] are inconsistent. Next, we show that, even for a consistent set of rate equations and rate constants, the model itself does not even provide the asymptotic solutions that are required to explain the SWE. In order to show this, we do not have to solve the full set of rate equations. Instead, we
restrict our discussion to the implications of Eqs. (KS-2), (KS-3) of Kru¨ger and Sax [6] using the values for the rate constants, etc. as given in Table 1 of their paper. Since the time constant for the Staebler–Wronski effect is of the order of 1000 h, while the photocurrent relaxation time is of the order of 107 s; the system under study can always be considered to be in a quasistationary state. We start from Eq. (KS-3) of Kru¨ger and Sax [6]: d ½A ¼ k3 ½A k2 ½A ½e k4 ½A : dt
(1)
In the SW state A, there are no B states according to Ref. [6]. Before the light exposure is started, there are also no A states. Hence, at the beginning of the illumination, Eq. (KS-3) reduces to d ½A ¼ k3 ½A: dt
(2)
Using ½A ¼ 1016 cm3 and k3 ¼ 1:2 1021 s1 from Table 1 of Ref. [6] (in Table 1 of Ref. [6], the initial concentration of A-states is denoted by w0 ), we find the rather horrible rate of 1:2 1037 cm3 s1 : This shows already that the set of rate equations of Ref. [6] is incomplete: the constant k3 must be replaced by some term that is proportional to intensity of the exciting light. Furthermore, it must be considered in the rate equation that the generation rate G of the photocarriers competes with the A!A process. We disregard this inconsistency assuming that Eq. (2) simply means that the quasiequilibrium value of ½A is reached immediately as the sample is exposed to light. For the SW state A, (before the photodegradation) the conduction band photoelectron density according to Kru¨ger and Sax (Ref. [6] Eq. (KS28)) is 6:25 1013 cm3 : Assuming ½A to be some fraction x of [A], we obtain with k2 ¼ 1:2 108 cm3 s1 k2 ½A ½e ¼ x 7:25 1021 cm3 s1 while with k4 ¼ 1:2 1018 s1 ; k4 ½A ¼ x 1:2 1034 cm3 s1 : Hence, we may safely leave out the second term of the r.h.s. of Eq. (1) and obtain for the
ARTICLE IN PRESS H. Overhof / Physica B 358 (2005) 181–184
stationary state ½A ¼
k4 ½A ¼ 103 ½A : k3
Hence, in the presence of light x ’ 1 and practically all A states are excited. With this reduction of [A], the first term on the r.h.s. of Eq. (1) is still horrible (corresponding to a power of 3:5 1015 W cm3 for a photon energy of 2 eV). If we insert this result for x into Eq. (KS-2) of Kru¨ger and Sax [6] which reads d ½e ¼ G k1 ½hþ ½e k2 ½A ½e ; dt
(3)
we obtain for the stationary case k1 ½e ½hþ ¼ G (this relation was used in Ref. [6] for the determination of the electron and hole densities), but since k2 ½A ½e ¼ 7:25xG; more photoelectrons would be captured than created. This result shows once more that the rate equations as well as the rate constants k1 to k4 are not consistent and must be reconsidered. The model presented by Kru¨ger and Sax differs from the conventional SWE models in that, in Kru¨ger and Sax, the photoelectrons are trapped by the additional photogenerated centers (in the conventional models, these act as additional recombination centers, rather than as traps). We now show that such a model does not give rise to the experimentally observed SWE. To this aim, we assume that all rate constants can be chosen in a consistent way such that for the SW state A, there is a certain number of A-states, with ½A ¼ x½A (we leave the value of x open), with a photoelectron density as given above, and without B states. In the light-soaked SW state B, the parameters are assumed to result in values for the conduction band electron densities, for [A], and also for ½A that are reduced by a factor of 100 with respect to the SW state A. Therefore, in the light-soaked case the density of negatively charged B-states must be large, comparable to the value of [A] in SW state A. For reasons of charge neutrality, the number of
183
holes must be close to [B] and these holes must somehow be trapped, because in the experiment, the photoconductivity is by electrons. From Eq. (3), we see that in the SW state B the second term on the r.h.s is reduced by a factor of 104 with respect to the SW state A, while the third term is reduced by a factor of 100. But since the generation rate for photoelectrons, G, is not altered by the SW effect, there can be no stationary solution of Eq. (3) with these densities. Furthermore, if the third term on the r.h.s. of Eq. (3), i.e. the capture of the photoelectrons by the A excited defects, is considered to reduce the photoelectron density by a factor of 100 in the Staebler–Wronski state B, then virtually all photoexcited electrons must be captured by this process. It follows that for virtually every photoelectron–hole pair also one A ! A excitation must take place. Hence the full A ! A ! B ! B ! A cycle is run at a rate similar to G. With G ¼ 1021 cm3 s1 ; as given by Kru¨ger and Sax [6, Table 1] this process must not take more time than about 105 s in order to avoid a bottleneck which would lead to a pileup of one of the intermediate states. Usually, a final quasiequilibrium is established in a reaction cycle once all the relevant consecutive reaction steps have been completed a few times. Thus, the Staebler–Wronski effect should saturate after 103 s at most. In the model of Kru¨ger and Sax, there is no mechanism that would cause the 1000 h time constant of the Staebler–Wronski effect. We might alternatively consider the second term on the r.h.s. of Eq. (3) to be the dominant term. Within the framework of the Kru¨ger–Sax model, this would follow naturally when assuming that both free and trapped holes could recombine directly. But this assumption would turn the model of Kru¨ger and Sax into a conventional model for the Staebler–Wronski effect: as a byproduct of the photoexcitation with low probability and hence a long time constant, some B defect centers are created, maybe via the A ! A ! B ! B process, which capture the photoexcited holes and thus act as a recombination center for the photoelectrons without being transformed into the A centers.
ARTICLE IN PRESS 184
H. Overhof / Physica B 358 (2005) 181–184
References [1] D.L. Staebler, C.R. Wronski, Appl. Phys. Lett. 31 (1977) 292. [2] D.L. Staebler, C.R. Wronski, J. Appl. Phys. 51 (1980) 3262.
[3] M. Stutzmann, W.B. Jackson, C.C. Tsai, Phys. Rev. B 32 (1985) 23. [4] H.M. Branz, Phys. Rev. B 59 (1999) 5498; H.M. Branz, Phys. Rev. B 60 (1999) 7725. [5] H. Fritzsche, Ann. Rev. Mater. Res. 31 (2001) 47. [6] T. Kru¨ger, A.F. Sax, Physica B 353 (2004) 263.