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The light intensity exponent of the minority carrier lifetime and the mobility gap states in a-Si:H
Journal of Non-Crystalline Solids 227–230 Ž1998. 206–210
The light intensity exponent of the minority carrier lifetime and the mobility gap states in a-Si:H I. Balberg b
a,)
, Y. Lubianiker a , L. Fonseca b, S.Z. Weisz
b
a Racah Institute of Physics, Hebrew UniÕersity, Jerusalem 91904, Israel Department of Physics, UniÕersity of Puerto Rico, San Juan 00931, Puerto Rico
Abstract We recently gave physical arguments that the recombination in a single type of dangling bond centers will yield a zero value for the light intensity exponent of the minority carriers, l. We then concluded that the observed l / 0 value in a-Si:H indicates the presence of other states in its mobility gap. In this paper, we substantiate those arguments by an analytic proof, by computer simulations and by considering our recent experimental findings on the temperature dependence of l. In particular, we show that in an n-type photoconductor the dangling bond acts as a single acceptor-like recombination center. q 1998 Elsevier Science B.V. All rights reserved. Keywords: Light intensity exponent; Minority carriers; Mobility gap
1. Introduction The ability to measure the ambipolar diffusion length in hydrogenated amorphous silicon, a-Si:H, has provided another source of information regarding the state distribution in this material w1–5x. However, one parameter which can be derived from the corresponding measurements has hardly been discussed w1,2,5x. This is the light intensity exponent, l, which is associated with the minority carrier recombination time, t h , by the definition, t h A G l, where G is the carrier generation rate. Recently, we have applied w1x a Rose-like w6x argument w7x to find the above-mentioned information from the dependence of l on various controllable parameters. This application )
Corresponding author. Fax: q972-2 658 4437; e-mail: [email protected].
yielded the conclusion that l is a unique parameter in the sense that it can provide a qualitative test for the presence of recombination centers other than a single type of dangling bonds. In particular, we concluded that a l / 0 is an indication that at least one other type of recombination center must be active. Following this conclusion we, suggested that in a-Si:H, one must assume the presence of other types of states. Such states are bandtails w8–10x andror other types of dangling bonds w2,4x. A priori, the well-known w8–10x existence of bandtails would immediately suggest that we should have l / 0 and thus this test is trivial. However, it is expected w9,10x that above room temperature, the recombination rate in the bandtails is diminished and thus the l / 0 phenomenon is informative, indicating other active recombination processes. This makes the l / 0 test a conclusive tool for state distribution spectroscopy for
0022-3093r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved. PII: S 0 0 2 2 - 3 0 9 3 Ž 9 8 . 0 0 0 4 9 - 0
I. Balberg et al.r Journal of Non-Crystalline Solids 227–230 (1998) 206–210
semiconductors in general and for amorphous semiconductors in particular. Following these considerations and since we gave w1x only qualitative arguments in support of this test, it appeared important to substantiate the above argument by an analytical proof and by model simulations. This need was further amplified when the relevance of the test to a-Si:H was questioned. The doubts came following an analytical argument w5x that there is a possible ‘curious situation’ for which a l / 0 value can be found for a single type of dangling bonds in a-Si:H. In the present paper, we also disprove that claim by analytical, numerical and experimental results. The structure of this paper is as follows. In Section 2, we derive analytically the expression for the lifetime of both charge carriers in the case of a single acceptor-like level showing that it is the same as the one obtained for the single type of dangling bonds. Then we determine the corresponding values of l. In Section 3, we apply the model to a-Si:H and show that it cannot produce the l / 0 result claimed in Ref. w5x. Finally, in Section 4, we discuss experimental results and present our main conclusion.
2. Analytical derivation of the light intensity exponents The four parameters which can be derived from phototransport measurements are the electron and hole mobility-lifetime products, mcte , m ht h , and their corresponding exponents, g and l w1–5x. The light intensity exponents, g and l, are defined by te A G gy1 and t h A G l. We note in passing that in other papers the parameter l was defined as 2S w1,3x or f h w2x. The dependence of these four parameters on Ec y E F has been used w2,3,5x for deriving information on the state distribution in a-Si:H. Here, Ec is the conduction band edge and E F is the dark Žequilibrium. Fermi level. We turn then to prove that in an n-type semiconductor, the single dangling bond acts as a single acceptor-like recombination center. For this, we assume that n frpf ) 1, where n f and pf are the steady state concentrations of free electrons and holes under light excitation. Since the p-type photoconductor
207
yields a symmetric situation, we do not discuss it here. Let us start our analytic derivation with the simplest version of the Shockley–Read analysis for a deeply lying acceptor-like state w6,7x. For the electrons in this system we assume a capture coefficient, sno , and for the holes a capture coefficient, spy. If the concentration of these states is Na , a fraction of them, fyNa , is occupied by electrons and a fraction of them, Ž1 y fy. Na , is occupied by holes. Under steady state conditions the recombination rates for electrons and holes must be equal, so that: n f Ž sno Na .Ž1 y fy. s pf Ž spy Na . fy. Using the definition of the ‘deep trapping times’ w7x tno ' 1rŽ sno Na . Ž y . and ty p ' 1r sp Na we obtain then that the steady state kinetics is governed by the relation: n f Ž 1 y fy . rtno s pf fyrty p .
Ž 1.
Eq. Ž1. is exactly the expression obtained in Ref. w5x for the n-type photoconductor Ž n f ) pf . in which the recombination center is the multilevel, occupationcorrelated, dangling bond. In particular, it is important to note Žsee below. that this result is found in Ref. w5x for both the high and low n frpf cases. This finding and the fact that Eq. Ž1. was derived here using the single level Shockley–Read model justifies, a posteriori, our argument w1x that the dangling bond acts as a single level-like recombination center. In turn it disproves the claim in Ref. w5x that the low n frpf case Žthe ‘curious situation’. ‘does not exist in the classical Shockley–Read recombination scheme’. Hence, to make Eq. Ž1. pertinent to the case of a single type of dangling bonds, all that has to be done is simply replace Na by the concentration of dangling bonds, Ndb . Let us now discuss g and l which are relevant to the recombination via acceptor-like states. For this purpose, we repeat briefly the argument given by Rose w6x Žfor g . and by Balberg w1x and Balberg and Lubianiker w3x Žfor l., but, this time, in relation to Eq. Ž1.. Consider then two distinct situations under the condition Žboth assumed in Refs. w1,5x. that Na 4 n f , pf . The first case is when fyf 1 y fy, 0.5. In this case w1x, the increase of G and thus the increase of n f and pf will hardly effect the electron occupation
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of the recombination centers, and their recombinay tion times, tnorŽ1 y fy. and ty p rf , will be independent of G. We will have then that g s 1 and l s 0. The other extreme case is when fyQ 1 so that Ž1 y fy. Na R n f . In this case, the charge neutrality restriction will not allow n f to increase with G without a corresponding increase of Ž1 y fy. , so that the condition n f s Ž1 y fy. Na must be obeyed. If so, since the steady state recombination rate equals G, i.e., Žsee Eq. Ž1.. G A n f Ž1 y fy. Na , we get that G A n 2f and thus g s 1r2. On the other hand, fyNa is essentially a constant and since G A pf fyNa we get that l s 0. The transition from the first case to the other can be obtained w1x by moving E F towards Ec . Consequently, the value of g shifts from 1 to 1r2 with increase of EF , while l s 0 is maintained throughout the increase of EF . In principle, it is possible Žsee Section 3. that, even if n f ) pf , under an extreme ratio of the recombination capture coefficients, we will still have a situation where fy< 1 so that the charge neutrality condition is pf s fyNa . This ‘curious situation’ is noted in Ref. w5x in which Ž1 y fy. Na is practically a constant, and G A pf fyNa . Hence, under the requirements of charge neutrality we get that G A pf2 and that G A n f , yielding l s y1r2 and g s 1. It is important to note then, that in general we will neÕer have a situation where g / 1 and l / 0 simultaneously, since, mathematically, we cannot have that both fy and Ž1 y fy. much smaller than unity, and, physically, since the charge neutrality condition can affect the concentration of only one of the charge carriers.
3. Application of the analytical derivation to aSi:H The central argument of Ref. w5x was that ‘in reality’ the ‘curious situation’ described above may account for the observation of a l / 0 in undoped a-Si:H. We will show now that this is very unlikely for a-Si:H, by using the very recombination parameters suggested in Ref. w5x. We note that these parameters are most favorable Žwithin the range of known w8–10x a-Si:H parameters. for the ‘curious situation’ in a-Si:H.
In principle, the general ‘curious situation’ may occur under the special combination of the deep trapping times given by w5x:
tnortpo < n frpf < tnorty p ,
Ž 2.
where tpo is the deep trapping time of holes in the neutral dangling bonds. The right-hand inequality is already the case implied by Eq. Ž1. for the ‘curious’ case of fy< 1 y fy. As for the left-hand side inequality, it is commonly agreed that in a-Si:H, tnortpo f 1. Hence, the particular way of writing w5x unity as tnortpo does not reflect the multilevel nature of the dangling bond but rather the a priori assumption that we have an n-type photoconductor Ž n f ) pf .. Hence, the simple single acceptor level-like nature of the problem is maintained, in the ‘curious situation’. Turning to the numerical evaluation of the right hand side inequality we note that tnorty p derived from the experimental data by Ref. w5x Žand others w8–10x. is Ž Ž .. that the ‘curious tnorty p s 100. This means Eq. 2 situation’ should correspond to n frpf f 10. Now, for the common G s 10 19 cmy3 sy1 the authors of Ref. w5x find that the maximum value of pf is smaller than 4 = 10 11 cmy3 . Accordingly, n f must be smaller than 10 13 cmy3. Since Ndb Žas given in Refs. w5,8x. is larger than 10 16 cmy3 , we obtain that Ndb rn f , Ndb rpf ) 10 3. Returning to the above analysis of g and l as given in Section 2, we remember, that in order for the charge neutrality condition to affect the carrier concentration Žyielding g / 1 or l / 0. we must have that either n f f Ž1 y fy. Ndb or pf f fy Ndb Žthe ‘curious situation’.. In view of those requirements, we conclude from the above discussion that for a-Si:H, fyrŽ1 y fy. s 10y3 . Examining Eq. Ž1., we see that even if n f s pf Žfavorable conditions for the ‘curious case’. the tnorty p s 100 will yield that fyrŽ1 y fy. f 10y2 rather than the above observed value of 10y3 . We saw in Section 2 that as long as n f - Ž1 y fy. Ndb or pf - fy Ndb , the carrier concentrations do not affect the occupation of the recombination centers and the lifetime. Hence, eÕen under the above very favorable parameters for the ‘curious case’, we get that g s 1 and l s 0 which in turn, explains why in the numerical simulations of the single type of dangling bonds model w2x one always finds the l s 0 result for all reasonable parameters
I. Balberg et al.r Journal of Non-Crystalline Solids 227–230 (1998) 206–210
209
havior is analogous to that of the ‘n-type’ material. It is clearly seen that for the ‘n-type’ regime the decreasing n frpf ratio does not yield the ‘curious situation’ Ž l / 0. claimed in Ref. w5x and that a rather simple n-type to p-type transition w1x takes place. This l s 0 finding proves then that the ‘curious situation’ is not ‘relevant’ to a-Si:H and that a single type of dangling bonds cannot account for the experimental l / 0 observation in undoped a-Si:H.
4. Discussion and conclusions
Fig. 1. The calculated optically excited free electron and free hole concentrations Ža., and their corresponding light intensity exponents Žb., as a function of the Fermi level position, for the single type of dangling bonds model. Note that g and l are associated here with the electrons and holes, respectively. For the ‘n-type’ material Ž Ec y E F F 0.8 eV. these are majority and minority carriers, respectively.
of a-Si:H. The interpretation w5x of the l variation as a transition between the ‘normal’ case of l s 0 to the ‘curious situation’ of l s 1r2 is therefore unfounded in a-Si:H, and one must resort to a model that has at least one other type of recombination center in addition to the single level dangling bond centers. To further illustrate the above point, we show in Fig. 1 the results of our own numerical computations of the Ec y E F dependence of n f , pf , g and l, when the parameters mentioned above were utilized. The other parameters, less critical for the present work, were taken from Refs. w2,8x and the numerical calculations were done using the dangling bond statistics of Ref. w8x. We show these results in order to affirm the results of Morgado w2x as well as Žunlike Morgado. to present them in terms of n f and pf . The ‘n-type’ cases discussed above are given in Fig. 1 for the interval of Ec y E F F 0.8 eV. For the larger Ec y E F values the material is ‘p-type’ and the be-
Following the results of Section 3, it is obvious that our test w1x of l / 0 is enough to establish the existence of states other than a single type of dangling bonds, but it is not enough for the determination of which of the more complex models is the correct one. On the other hand the dependence of l on ‘external’ parameters, such as Ec y E F or the temperature, can yield some new conclusions regarding the state distribution in a-Si:H. For example, we expect w9,10x that the role of the recombination in the bandtails diminishes with increasing temperature. According to our l / 0 test, we expect then that in this material we will find a l ™ 0 behavior with increasing T. Indeed, we have experimentally found w11x this expected behavior. This finding yields then further support to the applicability of our test for a-Si:H. It also shows that this test can yield information that is otherwise obtained by comparison of the experimental results with detailed numerical computations w9,10x.
Acknowledgements This work was supported in part by the Enrique Berman Solar Energy Research Fund and in part by NSF EPSCoR Grant OSR No. 94-52893 and NASA grant No. NCCW-0088.
References w1x I. Balberg, J. Appl. Phys. 75 Ž1994. 914. w2x E. Morgado, J. Non-Cryst. Solids 164–166 Ž1993. 627. w3x I. Balberg, Y. Lubianiker, Phys. Rev. B 48 Ž1993. 8709.
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w4x G. Schumm, C.-D. Abel, G.H. Bauer, J. Non-Cryst. Solids 137–138 Ž1991. 351. w5x J. Hubin, A.V. Shah, E. Sauvain, P. Pipoz, J. Appl. Phys. 78 Ž1995. 6050. w6x A. Rose, Concepts in Photoconductivity and Allied Problems, Wiley-Interscience, New York, 1963. w7x R.S. Crandall, in: J.I. Pankove ŽEd.., Semiconductors and Semimetals, Academic Press, New York, Vol. 21, Part C, 1986, p. 245.
w8x F. Vaillant, D. Jousse, Phys. Rev. B 34 Ž1986. 4088. w9x M.Q. Tran, Philos. Mag. 72 Ž1995. 35. w10x R. Bruggemann, in: J.M. Marshal, N. Kirov ŽEds.., Photoconductive Properties in Future Directions, in Thin Film Science and Technology, World Scientific, Singapore, 1997. w11x Y. Lubianiker, I. Balberg, L. Fonseca, S.Z. Weisz, Mater. Res. Soc. Symp. Proc., 1998, in press.
The light intensity exponent of the minority carrier lifetime and the mobility gap states in a-Si:H I. Balberg b
a,)
, Y. Lubianiker a , L. Fonseca b, S.Z. Weisz
b
a Racah Institute of Physics, Hebrew UniÕersity, Jerusalem 91904, Israel Department of Physics, UniÕersity of Puerto Rico, San Juan 00931, Puerto Rico
Abstract We recently gave physical arguments that the recombination in a single type of dangling bond centers will yield a zero value for the light intensity exponent of the minority carriers, l. We then concluded that the observed l / 0 value in a-Si:H indicates the presence of other states in its mobility gap. In this paper, we substantiate those arguments by an analytic proof, by computer simulations and by considering our recent experimental findings on the temperature dependence of l. In particular, we show that in an n-type photoconductor the dangling bond acts as a single acceptor-like recombination center. q 1998 Elsevier Science B.V. All rights reserved. Keywords: Light intensity exponent; Minority carriers; Mobility gap
1. Introduction The ability to measure the ambipolar diffusion length in hydrogenated amorphous silicon, a-Si:H, has provided another source of information regarding the state distribution in this material w1–5x. However, one parameter which can be derived from the corresponding measurements has hardly been discussed w1,2,5x. This is the light intensity exponent, l, which is associated with the minority carrier recombination time, t h , by the definition, t h A G l, where G is the carrier generation rate. Recently, we have applied w1x a Rose-like w6x argument w7x to find the above-mentioned information from the dependence of l on various controllable parameters. This application )
Corresponding author. Fax: q972-2 658 4437; e-mail: [email protected].
yielded the conclusion that l is a unique parameter in the sense that it can provide a qualitative test for the presence of recombination centers other than a single type of dangling bonds. In particular, we concluded that a l / 0 is an indication that at least one other type of recombination center must be active. Following this conclusion we, suggested that in a-Si:H, one must assume the presence of other types of states. Such states are bandtails w8–10x andror other types of dangling bonds w2,4x. A priori, the well-known w8–10x existence of bandtails would immediately suggest that we should have l / 0 and thus this test is trivial. However, it is expected w9,10x that above room temperature, the recombination rate in the bandtails is diminished and thus the l / 0 phenomenon is informative, indicating other active recombination processes. This makes the l / 0 test a conclusive tool for state distribution spectroscopy for
0022-3093r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved. PII: S 0 0 2 2 - 3 0 9 3 Ž 9 8 . 0 0 0 4 9 - 0
I. Balberg et al.r Journal of Non-Crystalline Solids 227–230 (1998) 206–210
semiconductors in general and for amorphous semiconductors in particular. Following these considerations and since we gave w1x only qualitative arguments in support of this test, it appeared important to substantiate the above argument by an analytical proof and by model simulations. This need was further amplified when the relevance of the test to a-Si:H was questioned. The doubts came following an analytical argument w5x that there is a possible ‘curious situation’ for which a l / 0 value can be found for a single type of dangling bonds in a-Si:H. In the present paper, we also disprove that claim by analytical, numerical and experimental results. The structure of this paper is as follows. In Section 2, we derive analytically the expression for the lifetime of both charge carriers in the case of a single acceptor-like level showing that it is the same as the one obtained for the single type of dangling bonds. Then we determine the corresponding values of l. In Section 3, we apply the model to a-Si:H and show that it cannot produce the l / 0 result claimed in Ref. w5x. Finally, in Section 4, we discuss experimental results and present our main conclusion.
2. Analytical derivation of the light intensity exponents The four parameters which can be derived from phototransport measurements are the electron and hole mobility-lifetime products, mcte , m ht h , and their corresponding exponents, g and l w1–5x. The light intensity exponents, g and l, are defined by te A G gy1 and t h A G l. We note in passing that in other papers the parameter l was defined as 2S w1,3x or f h w2x. The dependence of these four parameters on Ec y E F has been used w2,3,5x for deriving information on the state distribution in a-Si:H. Here, Ec is the conduction band edge and E F is the dark Žequilibrium. Fermi level. We turn then to prove that in an n-type semiconductor, the single dangling bond acts as a single acceptor-like recombination center. For this, we assume that n frpf ) 1, where n f and pf are the steady state concentrations of free electrons and holes under light excitation. Since the p-type photoconductor
207
yields a symmetric situation, we do not discuss it here. Let us start our analytic derivation with the simplest version of the Shockley–Read analysis for a deeply lying acceptor-like state w6,7x. For the electrons in this system we assume a capture coefficient, sno , and for the holes a capture coefficient, spy. If the concentration of these states is Na , a fraction of them, fyNa , is occupied by electrons and a fraction of them, Ž1 y fy. Na , is occupied by holes. Under steady state conditions the recombination rates for electrons and holes must be equal, so that: n f Ž sno Na .Ž1 y fy. s pf Ž spy Na . fy. Using the definition of the ‘deep trapping times’ w7x tno ' 1rŽ sno Na . Ž y . and ty p ' 1r sp Na we obtain then that the steady state kinetics is governed by the relation: n f Ž 1 y fy . rtno s pf fyrty p .
Ž 1.
Eq. Ž1. is exactly the expression obtained in Ref. w5x for the n-type photoconductor Ž n f ) pf . in which the recombination center is the multilevel, occupationcorrelated, dangling bond. In particular, it is important to note Žsee below. that this result is found in Ref. w5x for both the high and low n frpf cases. This finding and the fact that Eq. Ž1. was derived here using the single level Shockley–Read model justifies, a posteriori, our argument w1x that the dangling bond acts as a single level-like recombination center. In turn it disproves the claim in Ref. w5x that the low n frpf case Žthe ‘curious situation’. ‘does not exist in the classical Shockley–Read recombination scheme’. Hence, to make Eq. Ž1. pertinent to the case of a single type of dangling bonds, all that has to be done is simply replace Na by the concentration of dangling bonds, Ndb . Let us now discuss g and l which are relevant to the recombination via acceptor-like states. For this purpose, we repeat briefly the argument given by Rose w6x Žfor g . and by Balberg w1x and Balberg and Lubianiker w3x Žfor l., but, this time, in relation to Eq. Ž1.. Consider then two distinct situations under the condition Žboth assumed in Refs. w1,5x. that Na 4 n f , pf . The first case is when fyf 1 y fy, 0.5. In this case w1x, the increase of G and thus the increase of n f and pf will hardly effect the electron occupation
208
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of the recombination centers, and their recombinay tion times, tnorŽ1 y fy. and ty p rf , will be independent of G. We will have then that g s 1 and l s 0. The other extreme case is when fyQ 1 so that Ž1 y fy. Na R n f . In this case, the charge neutrality restriction will not allow n f to increase with G without a corresponding increase of Ž1 y fy. , so that the condition n f s Ž1 y fy. Na must be obeyed. If so, since the steady state recombination rate equals G, i.e., Žsee Eq. Ž1.. G A n f Ž1 y fy. Na , we get that G A n 2f and thus g s 1r2. On the other hand, fyNa is essentially a constant and since G A pf fyNa we get that l s 0. The transition from the first case to the other can be obtained w1x by moving E F towards Ec . Consequently, the value of g shifts from 1 to 1r2 with increase of EF , while l s 0 is maintained throughout the increase of EF . In principle, it is possible Žsee Section 3. that, even if n f ) pf , under an extreme ratio of the recombination capture coefficients, we will still have a situation where fy< 1 so that the charge neutrality condition is pf s fyNa . This ‘curious situation’ is noted in Ref. w5x in which Ž1 y fy. Na is practically a constant, and G A pf fyNa . Hence, under the requirements of charge neutrality we get that G A pf2 and that G A n f , yielding l s y1r2 and g s 1. It is important to note then, that in general we will neÕer have a situation where g / 1 and l / 0 simultaneously, since, mathematically, we cannot have that both fy and Ž1 y fy. much smaller than unity, and, physically, since the charge neutrality condition can affect the concentration of only one of the charge carriers.
3. Application of the analytical derivation to aSi:H The central argument of Ref. w5x was that ‘in reality’ the ‘curious situation’ described above may account for the observation of a l / 0 in undoped a-Si:H. We will show now that this is very unlikely for a-Si:H, by using the very recombination parameters suggested in Ref. w5x. We note that these parameters are most favorable Žwithin the range of known w8–10x a-Si:H parameters. for the ‘curious situation’ in a-Si:H.
In principle, the general ‘curious situation’ may occur under the special combination of the deep trapping times given by w5x:
tnortpo < n frpf < tnorty p ,
Ž 2.
where tpo is the deep trapping time of holes in the neutral dangling bonds. The right-hand inequality is already the case implied by Eq. Ž1. for the ‘curious’ case of fy< 1 y fy. As for the left-hand side inequality, it is commonly agreed that in a-Si:H, tnortpo f 1. Hence, the particular way of writing w5x unity as tnortpo does not reflect the multilevel nature of the dangling bond but rather the a priori assumption that we have an n-type photoconductor Ž n f ) pf .. Hence, the simple single acceptor level-like nature of the problem is maintained, in the ‘curious situation’. Turning to the numerical evaluation of the right hand side inequality we note that tnorty p derived from the experimental data by Ref. w5x Žand others w8–10x. is Ž Ž .. that the ‘curious tnorty p s 100. This means Eq. 2 situation’ should correspond to n frpf f 10. Now, for the common G s 10 19 cmy3 sy1 the authors of Ref. w5x find that the maximum value of pf is smaller than 4 = 10 11 cmy3 . Accordingly, n f must be smaller than 10 13 cmy3. Since Ndb Žas given in Refs. w5,8x. is larger than 10 16 cmy3 , we obtain that Ndb rn f , Ndb rpf ) 10 3. Returning to the above analysis of g and l as given in Section 2, we remember, that in order for the charge neutrality condition to affect the carrier concentration Žyielding g / 1 or l / 0. we must have that either n f f Ž1 y fy. Ndb or pf f fy Ndb Žthe ‘curious situation’.. In view of those requirements, we conclude from the above discussion that for a-Si:H, fyrŽ1 y fy. s 10y3 . Examining Eq. Ž1., we see that even if n f s pf Žfavorable conditions for the ‘curious case’. the tnorty p s 100 will yield that fyrŽ1 y fy. f 10y2 rather than the above observed value of 10y3 . We saw in Section 2 that as long as n f - Ž1 y fy. Ndb or pf - fy Ndb , the carrier concentrations do not affect the occupation of the recombination centers and the lifetime. Hence, eÕen under the above very favorable parameters for the ‘curious case’, we get that g s 1 and l s 0 which in turn, explains why in the numerical simulations of the single type of dangling bonds model w2x one always finds the l s 0 result for all reasonable parameters
I. Balberg et al.r Journal of Non-Crystalline Solids 227–230 (1998) 206–210
209
havior is analogous to that of the ‘n-type’ material. It is clearly seen that for the ‘n-type’ regime the decreasing n frpf ratio does not yield the ‘curious situation’ Ž l / 0. claimed in Ref. w5x and that a rather simple n-type to p-type transition w1x takes place. This l s 0 finding proves then that the ‘curious situation’ is not ‘relevant’ to a-Si:H and that a single type of dangling bonds cannot account for the experimental l / 0 observation in undoped a-Si:H.
4. Discussion and conclusions
Fig. 1. The calculated optically excited free electron and free hole concentrations Ža., and their corresponding light intensity exponents Žb., as a function of the Fermi level position, for the single type of dangling bonds model. Note that g and l are associated here with the electrons and holes, respectively. For the ‘n-type’ material Ž Ec y E F F 0.8 eV. these are majority and minority carriers, respectively.
of a-Si:H. The interpretation w5x of the l variation as a transition between the ‘normal’ case of l s 0 to the ‘curious situation’ of l s 1r2 is therefore unfounded in a-Si:H, and one must resort to a model that has at least one other type of recombination center in addition to the single level dangling bond centers. To further illustrate the above point, we show in Fig. 1 the results of our own numerical computations of the Ec y E F dependence of n f , pf , g and l, when the parameters mentioned above were utilized. The other parameters, less critical for the present work, were taken from Refs. w2,8x and the numerical calculations were done using the dangling bond statistics of Ref. w8x. We show these results in order to affirm the results of Morgado w2x as well as Žunlike Morgado. to present them in terms of n f and pf . The ‘n-type’ cases discussed above are given in Fig. 1 for the interval of Ec y E F F 0.8 eV. For the larger Ec y E F values the material is ‘p-type’ and the be-
Following the results of Section 3, it is obvious that our test w1x of l / 0 is enough to establish the existence of states other than a single type of dangling bonds, but it is not enough for the determination of which of the more complex models is the correct one. On the other hand the dependence of l on ‘external’ parameters, such as Ec y E F or the temperature, can yield some new conclusions regarding the state distribution in a-Si:H. For example, we expect w9,10x that the role of the recombination in the bandtails diminishes with increasing temperature. According to our l / 0 test, we expect then that in this material we will find a l ™ 0 behavior with increasing T. Indeed, we have experimentally found w11x this expected behavior. This finding yields then further support to the applicability of our test for a-Si:H. It also shows that this test can yield information that is otherwise obtained by comparison of the experimental results with detailed numerical computations w9,10x.
Acknowledgements This work was supported in part by the Enrique Berman Solar Energy Research Fund and in part by NSF EPSCoR Grant OSR No. 94-52893 and NASA grant No. NCCW-0088.
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