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Non-Langevin recombination in disordered materials with random potential distributions
Solid State Communications.Vol. 103, No. 9. pp. 541-543, 1997 0 1997 Else&r Science Ltd Printed in Great Britain. All tights reserved 0038-1098/97 $17.00+.00
Pergamon
PII: SOO38-1098(W)OO233-0
NON-LANGEVIN RECOMBINATION IN DISORDERED MATERIALS WITH RANDOM POTENTIAL DISTRIBUTIONS G.J. Adriaenssens and V.I. Arkhipov* Laboratorium voor Halfgeleiderfysica, Katholieke Universiteit Leuven, Celestijnenlaan 2OOD, B-3001 Heverlee-Leuven, Belgium (Received and accepted 26 February 1997 by D. Van Dyck)
A concept of random spatial fluctuations in the potential landscape of disordered semiconductors is employed to explain the effect of suppressed charge carrier recombination in these materials. The rate of bimolecular recombination is shown to be anomalously low due to spatial separation of electrons and holes in the fluctuating potential landscape. Photoexcited carriers can avoid geminate recombination due to the effect of a sufficiently strong local electric field coupled to the potential fluctuations. 0 1997 Elsevier Science Ltd Keywords: A. disordered systems, A. semiconductors, D. electronic states (localized), D. recombination and trapping.
1. INTRODUCTION A diffusion controlled mechanism of carrier recombination is one of the basic characteristic features of many amorphous materials with carrier mobilities of -1 cm2 V-’ s-’ or less [l-3]. For such values of the mobility the recombination rate must be completely determined by the time required for electrons and holes to diffuse close enough together for the final annihilation event to take place. This must be the case for initial recombination of thermalized carriers excited by the same photon (geminate recombination), as well as for recombination of carriers which have been generated or injected independently (bimolecular recombination). These two processes are described by the Onsager and Langevin models of recombination, respectively. In a variety of disordered materials, predictions of both models are found to be in agreement with experimental data (see e.g. [ 1, 41). However, it was recently shown by Tyutnev et al. [5] and by Arkhipov et al. [6] that in some polymers the rate of bimolecular recombination is much less than that
* On leave from the Moscow Engineering Physics Institute, Kashirskoye shosse 3 1, Moscow 115409, Russia.
predicted by the Langevin theory. Introduction of another recombination process cannot solve this puzzle, since the Langevin mechanism represents the lowest possible rate of carrier annihilation. Moreover, it should be noted that, for most of these materials, the geminate recombination remains, nevertheless, governed by the Onsager model, even though this model is based on essentially the same assumptions as the Langevin theory. In hydrogenated amorphous silicon (a-Si : H), both geminate and bimolecular recombination rates are found to be much less than what is predicted by the models of diffusion-controlled carrier annihilation [7,8]. In the present work we attempt to explain such effects of suppressed recombination in terms of carrier kinetics within a spatially random potential landscape that is typical for disordered materials. Intrinsic potential fluctuations were already mentioned by Schiff [33 as a possible explanation for suppressed recombination in a-Si : H. 2. THE MODEL The concept of the random potential landscape in disordered semiconductors is illustrated in Fig. 1. Assuming the characteristic length of the spatial fluctuations 1 to be larger than the carrier localization radius a, the random potential landscape can be thought of as
541
542
NON-LANGEVIN
RECOMBINATION
IN DISORDERED
MATERIALS
Vol. 103, No. 9
A > kT, the average density of electrons (n)(f) and that of holes (p)(t) are related to n,,(r) and PO(r) by (3)
Y
I
Fig. 1. Spatially separated distributions of electrons and holes over a random potential landscape in disordered materials. spatially separated systems of variously-shaped “valleys” for electrons and holes (see Fig. 1). These valleys, together with the “passes” which connect them, form two independent percolation networks for transport of electrons and holes. Since the two types of carriers are practically always well-separated, bimolecular recombination can occur only due to the rather weak overlap of the electron and hole densities, thus implying a suppressed recombination rate even within the framework of standard diffusion-controlled models. To estimate the effect of fluctuations on the bimolecular recombination kinetics we consider a simple model of regular parallel potential channels (see Fig. 2) U,(x) = U,, + A cos (27rx/l), U,,(x) = U,,, + A cos (2ux/Z),
(1)
In the following we assume that the initial densities of electrons and holes, no(O) and pa(O), are not very high compared to the inverse characteristic length of the fluctuations: no(O) + Ee3 andpa + lP3. For 15 10 nm the latter inequalities imply no, p. 5 lo’* cme3, each of which are reasonable values for a-Si : H [9]. Under these conditions recombination events will be sufficiently rare to allow the equilibrium spatial distributions of carriers that are described by equation (2) to be restored each time some carriers are eliminated by recombination. The rate of bimolecular recombination, v,(x, r), is governed by the equation
= -Rexp where R is the recombination constant. Since the rate of recombination does not depend upon the coordinate, and spatial distributions of electrons and holes are not changed in time, the recombination equation to be used for comparison with experimental observations is the one for the average density of carriers rather than for their spatial distributions. Averaging carrier densities over the coordinate and using equation (3) yields
where x is the spatial coordinate, U, and U,, are the energies of the mobility edges for electrons and holes, with Uk and Uw being their average values and the mobility gap width AU = U, - U,,, being constant and A is the characteristic amplitude of the potential fluctuations. The time-dependent spatial distributions of the electron and hole densities, n(x, t) and p(x, t), are modulated by the potential oscillations according to
equation (5) for the average carrier densities has the form of a traditional equation describing the rate of bimolecular recombination. However, the effective value of the recombination constant R, given by
n(x, t) = rr&) exp ( - A[1 + cos (27rx/l)]/kT},
&=RFexp
exp ( - A[1 - cos (2?rx/l)]/kT},
p(x, t) =P&)
(2)
where no andpo are the densities of electrons and holes at the respective potential minima, k is the Boltzmann constant and T the temperature. Under the condition UA U&_
n 1
Uoll
n
Fig. 2. The model of regular spatial distributions electron and hole potentials.
of the
d(&) _ _
R
dr
21rA exp kT
27rA
can be much less than its microscopic value R determined by the Langevin theory. To obtain Rd = 10u3R, which is experimentally observed in a-Si : H at room temperature, one must accept A = 0.13 eV. The above value of A should be considered as a lowest estimate since it is calculated for a model system of parallel channels which is more favourable for carrier transport than it is for recombination. Within a realistic random potential landscape, valley regions containing electrons and holes will be separated by passes which must be overcome in transport by both types of carriers. These passes will probably be the regions of most effective recombination since the joint densities of both electrons and holes are relatively high there. Since the
Vol. 103, No. 9
NON-LANGEVIN
RECOMBINATION
occurrence of such regions makes recombination more probable than outlined above, a higher amplitude of fluctuations A will be required to get the same effective value of the recombination constant. 3. DISCUSSION Two important features of equation (6) should be pointed out. Firstly, the effective recombination constant strongly increases with increasing temperature. This effect is observed in both a-Si : H and polymers [5, 8, lo]. Note that, in disordered materials, a distribution of potential barrier heights must exist rather than a single value A. This will lead to a temperature somewhat different from the dependence of R, exponential one predicted by equation (6). The second important fact is that the effective recombination constant does not depend upon the characteristic length of the fluctuations 1. The latter is, nevertheless, very important when the problem of geminate recombination is considered. The occurrence of a random potential landscape for charge carriers implies also the occurrence of a local intrinsic electric field Fi. A characteristic value of the field can be estimated as F; = 4Alel. Assuming 1~ 5 nm practically all yields Fi z lo6 V cm-‘. Consequently, geminate electron-hole pairs are excited in regions of rather strong intrinsic electric field. Estimating the critical field F, at which the quantum yield of carrier photogeneration, 9, reaches 1, as F, = e/4?rsosri, one obtains 3 X lo5 V cm-’ in a-Si : H with the dielectric permittivity F = 12 and for geminate pair with the initial separation r0 = 2 nm. Thus, potential fluctuations yield a fluctuating intrinsic electric field which is strong enough to provide complete separation of practically all geminate pairs created by photoexcitation. This explains the puzzle of the anomalously high, temperatureand field-independent quantum yield, n = 1, of carrier photogeneration in a-Si : H. Note that, within the framework of the present model, a suppressed geminate recombination requires both a sufficiently high amplitude and a relatively short characteristic length of the potential fluctuations. Only the former condition is necessary for a suppressed bimolecular recombination. Therefore, in materials with a suppressed geminate recombination, the rate of bimolecular recombination must also be anomalously low. However, since low geminate recombination is not a necessary consequence of a low bimolecular one, there are polymeric materials in which the Onsager model of geminate recombination
IN DISORDERED
MATERIALS
543
works well but the bimolecular recombination is strongly suppressed. It may be pointed out that the model which we have used for our calculations (see Fig. 2), is reminiscent of the ones that have been used for the electronic bands in earlier studies of doping superlattices [ll, 121. The separation of electrons and holes, which we ascribe to the random potential in a disordered system, is of course there by design in the superlattice structures. In conclusion, a model for suppressed recombination in disordered materials has been formulated. The model takes into consideration random spatial fluctuations of the potential landscape that lead to spatially separated percolation networks for electrons and holes, with a low average carrier recombination rate as a consequence. Rather strong fluctuations of the intrinsic electric field, coupled to the potential fluctuations, can prevent photoexcited carriers from recombining geminately. Numeric estimates yield reasonably good agreement with existing experimental data. Acknowledgements-VIA. acknowledges a fellowship from Katholieke Universiteit Leuven. This work was supported by the Fonds voor Wetenschappelijk Onderzoek-Vlaanderen. REFERENCES 1.
2. 3. 4. 5.
10. 11. 12.
Mott, N.F. and Davis, E.A., Electronic Processes in Non-Crystalline Materials, 2nd edn. Clarendon, Oxford, 1979. Ries, B., Bassler, H., Schonherr, G., Silver, M. and Snow, E., J. Non-Cryst. Solids, 66, 1984, 243. Schiff, E.A., J. Non-Cryst. Solids, 190, 1995, 1. Hughes, R.C., J. Chem. Phys., 58, 1973, 2212. Tyutnev, A.P., Karpechin, A.I., Boev, S.G., Saenko, V.S. and Pozhidaev, E.D., Phys. Status Solidi (a), 132, 1992, 163. Arkhipov, V.I., Perova, I.A. and Rudenko, A.I., Int. J. Electron., 72, 1992, 99. Carasco, F. and Spear, W.E., Phil. Mug., B47, 1983,495. Juska, G., Kocka, J., Viliunas, M. and Arlauskas, K., J. Non-Cryst. Solids, 164-166, 1993, 579. Overhof, H. and Thomas, P., Electronic Transport in Hydrogenated Amorphous Semiconductors. Springer Verlag, Berlin, 1989. Conrad, K.A. and Schiff, E.A., Solid State Commun., 60,1986,291. Hundhausen, M., Ley, L. and Carius, R., Phys. Rev. Lett., 53, 1984, 1598. Dohler, G.H., J. Non-Cryst. Solids, 77&78, 1985, 1041.
Pergamon
PII: SOO38-1098(W)OO233-0
NON-LANGEVIN RECOMBINATION IN DISORDERED MATERIALS WITH RANDOM POTENTIAL DISTRIBUTIONS G.J. Adriaenssens and V.I. Arkhipov* Laboratorium voor Halfgeleiderfysica, Katholieke Universiteit Leuven, Celestijnenlaan 2OOD, B-3001 Heverlee-Leuven, Belgium (Received and accepted 26 February 1997 by D. Van Dyck)
A concept of random spatial fluctuations in the potential landscape of disordered semiconductors is employed to explain the effect of suppressed charge carrier recombination in these materials. The rate of bimolecular recombination is shown to be anomalously low due to spatial separation of electrons and holes in the fluctuating potential landscape. Photoexcited carriers can avoid geminate recombination due to the effect of a sufficiently strong local electric field coupled to the potential fluctuations. 0 1997 Elsevier Science Ltd Keywords: A. disordered systems, A. semiconductors, D. electronic states (localized), D. recombination and trapping.
1. INTRODUCTION A diffusion controlled mechanism of carrier recombination is one of the basic characteristic features of many amorphous materials with carrier mobilities of -1 cm2 V-’ s-’ or less [l-3]. For such values of the mobility the recombination rate must be completely determined by the time required for electrons and holes to diffuse close enough together for the final annihilation event to take place. This must be the case for initial recombination of thermalized carriers excited by the same photon (geminate recombination), as well as for recombination of carriers which have been generated or injected independently (bimolecular recombination). These two processes are described by the Onsager and Langevin models of recombination, respectively. In a variety of disordered materials, predictions of both models are found to be in agreement with experimental data (see e.g. [ 1, 41). However, it was recently shown by Tyutnev et al. [5] and by Arkhipov et al. [6] that in some polymers the rate of bimolecular recombination is much less than that
* On leave from the Moscow Engineering Physics Institute, Kashirskoye shosse 3 1, Moscow 115409, Russia.
predicted by the Langevin theory. Introduction of another recombination process cannot solve this puzzle, since the Langevin mechanism represents the lowest possible rate of carrier annihilation. Moreover, it should be noted that, for most of these materials, the geminate recombination remains, nevertheless, governed by the Onsager model, even though this model is based on essentially the same assumptions as the Langevin theory. In hydrogenated amorphous silicon (a-Si : H), both geminate and bimolecular recombination rates are found to be much less than what is predicted by the models of diffusion-controlled carrier annihilation [7,8]. In the present work we attempt to explain such effects of suppressed recombination in terms of carrier kinetics within a spatially random potential landscape that is typical for disordered materials. Intrinsic potential fluctuations were already mentioned by Schiff [33 as a possible explanation for suppressed recombination in a-Si : H. 2. THE MODEL The concept of the random potential landscape in disordered semiconductors is illustrated in Fig. 1. Assuming the characteristic length of the spatial fluctuations 1 to be larger than the carrier localization radius a, the random potential landscape can be thought of as
541
542
NON-LANGEVIN
RECOMBINATION
IN DISORDERED
MATERIALS
Vol. 103, No. 9
A > kT, the average density of electrons (n)(f) and that of holes (p)(t) are related to n,,(r) and PO(r) by (3)
Y
I
Fig. 1. Spatially separated distributions of electrons and holes over a random potential landscape in disordered materials. spatially separated systems of variously-shaped “valleys” for electrons and holes (see Fig. 1). These valleys, together with the “passes” which connect them, form two independent percolation networks for transport of electrons and holes. Since the two types of carriers are practically always well-separated, bimolecular recombination can occur only due to the rather weak overlap of the electron and hole densities, thus implying a suppressed recombination rate even within the framework of standard diffusion-controlled models. To estimate the effect of fluctuations on the bimolecular recombination kinetics we consider a simple model of regular parallel potential channels (see Fig. 2) U,(x) = U,, + A cos (27rx/l), U,,(x) = U,,, + A cos (2ux/Z),
(1)
In the following we assume that the initial densities of electrons and holes, no(O) and pa(O), are not very high compared to the inverse characteristic length of the fluctuations: no(O) + Ee3 andpa + lP3. For 15 10 nm the latter inequalities imply no, p. 5 lo’* cme3, each of which are reasonable values for a-Si : H [9]. Under these conditions recombination events will be sufficiently rare to allow the equilibrium spatial distributions of carriers that are described by equation (2) to be restored each time some carriers are eliminated by recombination. The rate of bimolecular recombination, v,(x, r), is governed by the equation
= -Rexp where R is the recombination constant. Since the rate of recombination does not depend upon the coordinate, and spatial distributions of electrons and holes are not changed in time, the recombination equation to be used for comparison with experimental observations is the one for the average density of carriers rather than for their spatial distributions. Averaging carrier densities over the coordinate and using equation (3) yields
where x is the spatial coordinate, U, and U,, are the energies of the mobility edges for electrons and holes, with Uk and Uw being their average values and the mobility gap width AU = U, - U,,, being constant and A is the characteristic amplitude of the potential fluctuations. The time-dependent spatial distributions of the electron and hole densities, n(x, t) and p(x, t), are modulated by the potential oscillations according to
equation (5) for the average carrier densities has the form of a traditional equation describing the rate of bimolecular recombination. However, the effective value of the recombination constant R, given by
n(x, t) = rr&) exp ( - A[1 + cos (27rx/l)]/kT},
&=RFexp
exp ( - A[1 - cos (2?rx/l)]/kT},
p(x, t) =P&)
(2)
where no andpo are the densities of electrons and holes at the respective potential minima, k is the Boltzmann constant and T the temperature. Under the condition UA U&_
n 1
Uoll
n
Fig. 2. The model of regular spatial distributions electron and hole potentials.
of the
d(&) _ _
R
dr
21rA exp kT
27rA
can be much less than its microscopic value R determined by the Langevin theory. To obtain Rd = 10u3R, which is experimentally observed in a-Si : H at room temperature, one must accept A = 0.13 eV. The above value of A should be considered as a lowest estimate since it is calculated for a model system of parallel channels which is more favourable for carrier transport than it is for recombination. Within a realistic random potential landscape, valley regions containing electrons and holes will be separated by passes which must be overcome in transport by both types of carriers. These passes will probably be the regions of most effective recombination since the joint densities of both electrons and holes are relatively high there. Since the
Vol. 103, No. 9
NON-LANGEVIN
RECOMBINATION
occurrence of such regions makes recombination more probable than outlined above, a higher amplitude of fluctuations A will be required to get the same effective value of the recombination constant. 3. DISCUSSION Two important features of equation (6) should be pointed out. Firstly, the effective recombination constant strongly increases with increasing temperature. This effect is observed in both a-Si : H and polymers [5, 8, lo]. Note that, in disordered materials, a distribution of potential barrier heights must exist rather than a single value A. This will lead to a temperature somewhat different from the dependence of R, exponential one predicted by equation (6). The second important fact is that the effective recombination constant does not depend upon the characteristic length of the fluctuations 1. The latter is, nevertheless, very important when the problem of geminate recombination is considered. The occurrence of a random potential landscape for charge carriers implies also the occurrence of a local intrinsic electric field Fi. A characteristic value of the field can be estimated as F; = 4Alel. Assuming 1~ 5 nm practically all yields Fi z lo6 V cm-‘. Consequently, geminate electron-hole pairs are excited in regions of rather strong intrinsic electric field. Estimating the critical field F, at which the quantum yield of carrier photogeneration, 9, reaches 1, as F, = e/4?rsosri, one obtains 3 X lo5 V cm-’ in a-Si : H with the dielectric permittivity F = 12 and for geminate pair with the initial separation r0 = 2 nm. Thus, potential fluctuations yield a fluctuating intrinsic electric field which is strong enough to provide complete separation of practically all geminate pairs created by photoexcitation. This explains the puzzle of the anomalously high, temperatureand field-independent quantum yield, n = 1, of carrier photogeneration in a-Si : H. Note that, within the framework of the present model, a suppressed geminate recombination requires both a sufficiently high amplitude and a relatively short characteristic length of the potential fluctuations. Only the former condition is necessary for a suppressed bimolecular recombination. Therefore, in materials with a suppressed geminate recombination, the rate of bimolecular recombination must also be anomalously low. However, since low geminate recombination is not a necessary consequence of a low bimolecular one, there are polymeric materials in which the Onsager model of geminate recombination
IN DISORDERED
MATERIALS
543
works well but the bimolecular recombination is strongly suppressed. It may be pointed out that the model which we have used for our calculations (see Fig. 2), is reminiscent of the ones that have been used for the electronic bands in earlier studies of doping superlattices [ll, 121. The separation of electrons and holes, which we ascribe to the random potential in a disordered system, is of course there by design in the superlattice structures. In conclusion, a model for suppressed recombination in disordered materials has been formulated. The model takes into consideration random spatial fluctuations of the potential landscape that lead to spatially separated percolation networks for electrons and holes, with a low average carrier recombination rate as a consequence. Rather strong fluctuations of the intrinsic electric field, coupled to the potential fluctuations, can prevent photoexcited carriers from recombining geminately. Numeric estimates yield reasonably good agreement with existing experimental data. Acknowledgements-VIA. acknowledges a fellowship from Katholieke Universiteit Leuven. This work was supported by the Fonds voor Wetenschappelijk Onderzoek-Vlaanderen. REFERENCES 1.
2. 3. 4. 5.
10. 11. 12.
Mott, N.F. and Davis, E.A., Electronic Processes in Non-Crystalline Materials, 2nd edn. Clarendon, Oxford, 1979. Ries, B., Bassler, H., Schonherr, G., Silver, M. and Snow, E., J. Non-Cryst. Solids, 66, 1984, 243. Schiff, E.A., J. Non-Cryst. Solids, 190, 1995, 1. Hughes, R.C., J. Chem. Phys., 58, 1973, 2212. Tyutnev, A.P., Karpechin, A.I., Boev, S.G., Saenko, V.S. and Pozhidaev, E.D., Phys. Status Solidi (a), 132, 1992, 163. Arkhipov, V.I., Perova, I.A. and Rudenko, A.I., Int. J. Electron., 72, 1992, 99. Carasco, F. and Spear, W.E., Phil. Mug., B47, 1983,495. Juska, G., Kocka, J., Viliunas, M. and Arlauskas, K., J. Non-Cryst. Solids, 164-166, 1993, 579. Overhof, H. and Thomas, P., Electronic Transport in Hydrogenated Amorphous Semiconductors. Springer Verlag, Berlin, 1989. Conrad, K.A. and Schiff, E.A., Solid State Commun., 60,1986,291. Hundhausen, M., Ley, L. and Carius, R., Phys. Rev. Lett., 53, 1984, 1598. Dohler, G.H., J. Non-Cryst. Solids, 77&78, 1985, 1041.