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Kinetics of distant-pair recombination: Application to amorphous silicon
Physica 117B & 118B (1983) 902-904 North-Holland PublishingCompany
902
KINETICS OF DISTANT-PAIR RECOMBINATION: APPLICATION TO AMORPHOUS SILICON D.J. Dunstan
Angewandte Physik Johannes Kepler Universit~t 4040 Linz, Austria
We present an exact solution for the decay of randomly distributed excited carriers, and find that a monomoleculah distant-pair approximation is valid over a wide range of excitation powers. Under r e p e t i t i v e l y pulsed or continuous excitation, an accumulated pair population is present, the result of which is that the decay curve varies at high excitation power but is independent of excitation power at low power. In consequence, time-resolved experiments cannot distinguish between distant-pair and geminate recombination, and frequency-resolved methods must be used. We find that distant-pair recombination provides a quantitative explanation of the experimental results in a-Si:H
2. Theory
i. Introduction
The radiative recombination of independently trapped electrons and holes is an important mechanism of luminescence in semiconductors. Because, howeveh the l i f e t i m e T depends on the electron-hole separation r as T(r) = ToeXp(ar) the kinetics of such systems are far from simple. Thomas, Hopfield and Augustyniak (1) obtained exaet solutions assuming uncompensated donor-acceptor recombination with the centres at random in the crystal, but for the fully compensated case they obtained only an approximate solution, in which the decay of the luminescence went as t -2 at long times. Because of their Hartree t r e a t m e n 6 this solution is only applicable to high temperatures. In amorphous semiconduetors, most studies of kinetics have been made in intrinsie materla6 so exact solutions have not been available. It has often been assumed that the kinetics are either bimolecular or geminated according to the excitation intensity; the dependence of the decay curve on the excitation power distinguishing the two regimes (2,3). We show here that this is not the case; that an intermediate regime must be distinguished in which the recombination is monomoleeular and the decay curve may or may not vary with the excitation p o w e h according to the conditions. It is this intermediate regime that we refer to as distant-pair recombination. Conventional time-resolved spectroscopy (TRS) cannot distinguish betweeen distant-pair and geminate recombination; we show that it is necessary to use frequency-resolved spectroscopy (FRS) (4). We compare our theoretical results with the behaviour of the 1.4eV photoluminescence (PL) band of a-SitH, and we find that this band, previously attributed to geminate recombination (2), in fact displays the behaviour predicted quantitatively by the distant-pair model. For reasons of space, we can only outline the methods of calculation of the kinetics. A full account is given elsewhere (5).
0 378-4363/83/0000-0000/$03.00 © 1983 North-Holland
2.1
Pulsed Excitation
We assume that a single pulse at t=0 creates a density poof carriers of each sign, which are trapped at random throughout the sample. A t high temperature, diffusion maintains the randomness of the spatial distribution and we need only find O(t) to obtain the decay curves; at low t e m p e r a t u r e with no diffusion the spatial distribution becomes anticlustered as decay proceeds. For the high temperature case, the probability that there is a hole trapped in the shell of radius r to r+dr around an e l e c t r o n at t i m e t is P(r,t)dr = 4rrr2p(t)dr
and the probability of recombination during the time interval t to t+dt is P(t)dt = fr~°oP(r,t)lr;lexp(-izr)drdt = 8 w p (t)Tol~Jdt But i~(t)=-p(t)P(t), so p(t)=
Oo/(8rrpor'ola'zt+l)
and the luminescence intensity I(t)=-i~(t) is, at long times, proportional to t -z, in agreement with the result of Thomas et al (1), For the low temperature cased we set the diffusion rates to zero, and consider the evolution of the initially random distributions, We start by making a monomoleeular approximation-" that an electron (hole) recombines only with the nearest available hole (electron); by 'available' we mean one that itself, by the same criterion, is not destined to recombine elsewhere (over a smaller separation). This may be described as the 'nearest available neighbour' approximation; when it is valid, recombination is monamolecular. Then we define P(r)dr as the probability that the nearest available neighbour of any given carrier is at a distance betweeen r and r+dr; clearly r 1-fu=~°(u)du is the probability that a carrier does not recombine at a distance less than r. SoD considering an electron, the probability that there is a hole at a distance r to r+dr, and that neither it nor the electron recombine elsewhere (i.e. have nearer
D.J. Dunstan / Kinetics of distant-pair recombination
.-.
903
" ' P" ' r" ' I"' ' I"' ' I"' ' I"' ' r" ' P" ' P" ' 1 - G:lO24/:llt !2 18 1!
1028
~uEIO 24
~ 0.5
•-= ~ 10 2 0 n W
0
lns
lps Time
1ms after
pulse
ls t
Modulation
Figure i: Decay curves after a single excitation pulse of intensity are shown, for the exact calculation from P(t), Eqn 3 (full curves), and for the distantpair model using P(r), Eqn 1 (dotted curves). The parameter values are marked. Where the dotted and full curves coincide, the distant-pair monomolecular model is valid; at higher carrier densities recombination is bimolecular.
available neighbours), so that they must recombine with each other, is r
P(r)dr = Zmr 2podr(1- ~oP(U)dU) 2
l
and this equation may be solved by a Neumann series to give
p(r) : a~m2po(l+ ~ r 3 p o ) -2
Z
We may now cheek the range of v a l i d i t y of the nearest available neighbour approximation, and hence the range of monomolecular recombination, by a comparison with an exact solution for the present case, that includes the possibility of bimolecular recombination. Instead of P(r)dr, we consider P(t)dt, the probability that a carrier recombines in the time interval t to t+dt. Following the same method as for Eqn 1, but making no approximation, we obtain
P(t) = (1-fntP(u)du)2f:411r2poT'l(r)e-t/T(r)dr
"
1MHz
3
which we solve numerically. The decay curves I(t) can be c a l c u l a t e d from P(r) and from In(t), end are shown in Figure 1. Comparison of the two sets of curves in Fig.1 shows that the approximation of Eqn 1 breaks down only at very high carrier densities, when recombination is bimoleeular. Over the entire range where the dotted and full curves of Fig.1 coincide, recombination is distant-pair (until at sufficiently low excitation power, the geminate regime is reached). We now consider the effect of r e p e t i t i v e l y pulsed excitation, as in TRS, in the distant-pair regime. An important feature of Eqn 2 is that the density of pairs with separation r,poP(r), is independent ofpo at large r (r>~po3). This means that if we have a random distribution of carriers, and we add some more to it, we do not change the number of pairs of large
•
lkHz
1Hz
frequency
v
Figure 2: Theoretical results of F-RS quadrature measurements on a distant-pair system. G is marked and the other parameters are as in F i g . l . Experimental data from Ref.4 for gd a-Si:H is shown; values of G are 1018(x), 102°(0) and 1021'Scm'3s"1 (+). All spectra are normalised to a peak height of unity.
separation. Also, if we remove the pairs of small separation (e.g. by radiative recombination) and then add carriers again at random, we still do not change the number of pairs of large separation. We define a c r i t i c a l separation re, at which the pair l i f e t i m e equals the pulse repetition period to, and we find that there is a population of pairs of separation r > r c , whose density is v i r t u a l l y independent of the pulse intensity or repetition rate. For a detailed analysis, see Ref.5; here we approximate P1 b~' integrating poP(r) from r c to co, obtaining (with pc>re ~ ) with r c =In(to/To)Or"1 4 P1= ( 34 ~ c3)-1 If an excitation pulse introduces a density of carriers po>>p1, clearly P1 will be of no significance and the decay will vary with pc as shown in F i g . l . But if the pulse introduces a density p o < P l , the new carriers cannot form pairs of separation greater than r c (as they would in the absence of P1); instead the new pairs have separations less than r c , characteristic of a random distribution of density PI. The subsequent decay curve is characteristic of P1 and not of Pc, and so does not vary as the excitation power is changed. This contrast in the behaviour of the decay curve for pulse densities above and below the 'permanent' pair density P1 has been taken to demonstrate geminate recombination at low pulse densities (3); it is in fact a direct consequence of distant-pair monomolecular recombination. 2.Z
Frequency-Resolved Spectroscopy (FRS)
L i f e t i m e analysis may be carried out by making measurements in the time domain (TRS), as considered in Section 2.1, or in the frequency domain (FRS). It is important to note that the information given by TRS and by FRS is not necessarily equivalent. For a full discussion, see Ref.6. Here, we shall illustrate the
904
D.J. Dunstan / Kinetics of distant-pair recombination
difference with the results of the FRS method on the distant-pair system. FRS is carried out under continuous excitation, which carries a small intensity modulation at an angular frequency ~. If lock-in detection set in quadrature (w/2) to the excitation is used to detect the emission, only those components with lifetimes close to 1/~ are observed, the response function being (6) S(~) = (~T+(~T)'I) "I and a frequency sweep will give a signal which represents directly the lifetime distribution present under continuous excitation. The l i f e t i m e distribution of the distant-pair system is calculated (5) as for the pulse-train problem of Section 2.1, letting tG.0, and the convolution with the FRS response function is shown in Fig.2. Note that as the e x c i t a t i o n rate Q is reduced, the l i f e t i m e distribution continues to shift to longer times. If it were observed to cease to shift at some value of Q, that would be clear evidence of geminate recombination; evidence that, as we have seen in Section 2.1, a TRS experiment cannot give. The experimental data in Fig.2, for gd a-Si:H (4), shows some discrepancies with the theoretical curves. The shift with changes in G shows, however, that distant-pair recombination is applicable. The discrepancies show, perhaps, that FRS is a more sensitive technique than TRS for studying such systems. 3.
Application to Amorphous Silicon
We have recently shown, using FRS, that the 1./4eV PL of a-Si:H is due to distant-pair recombination (4), and that the evidence previously thought to indicate a geminate model (]) is in fact q u a n t i t a t i v e l y consistent with the predictions of the distant-pair model (5). We saw, in Section 2.1, that the decay curve in TRS is independent of the e x c i t a t i o n density aPo when Pc
Although generally linear, the PL emission intensity can have a superlinear dependence on the excitation intensity, in some samples (I0). Attempts have been m a d e to interpret this phenomenon in terms of bimolecular recombination, and to deduce structural differences between samples (ii); such models m a y be inappropriate. In optically detected magnetic resonance (ODMR), a resonance of either carrier of a pair will give a quenching signal in a geminate system (123, but an enhancing signal in a distant-pair system (13). Clearly, then, the correct interpretation of the O D M R signals relies critically on a correct identification of the kinetics (14). Finally, we note that a model in which carriers are trapped at random implies greater diffusion lengths than does the geminate model, in which carriers are trapped close to their geminate partner. This may require new models of the electronic structure of amorphous semiconductors, in which longrange fluctuations of the band-edges are of greater significance than localised band-tail states (9,15,16). We wish to acknowledge many valuable discussions with Dr F. Boulitrop and Dr S.P. Depinna. References 1.
D.C. Thomas, J.J. Hopfield and W.M. Augustyniak, Phys. Rev. 14...0_0(1965) A202
2.
R.A. Street, Adv. Phys. 3__00(19813593
3.
C. Tsang and R.A. Street, Phys. Rev. B19 (1979) 3027
4.
D.3. Dunstan, S.P. Depinna J. Phys. CI__55(1982) L425
5.
D.J. Ounstan, Phil. Mag. in press
and
B.C. Cavenett,
6.
S.P. Depinna and D.3. Dunstan, to be published
7.
R.A. Street and D.K. Biegelsen, to be published
8.
R.W. Collins, M.A. Paesler and W. Paul, Sol. St. Comm. 34 (1980) 833
9.
F. Boulitrop, D.3. Dunstan and A. ChenevasPaule, Phys. Rev. B2._55(1982) in press
10. R.A. Street, Phys. Rev. B23 (1981) 861 11. R.W. Collins and W. Paul, Phys. Rev. B25 (1982) 5263 12. D.K. Biegelsen, J.C. Knights, R.A. Street, C. Tsang and R.M. White, Phil. Mag. B37 (1978) 477 13. D.J. Dunstan and J.J. Davies, J. Phys. C12 (1979) 2927 14. S.P. Depinna, B.C. Cavenett, I.(3. Austin, T.M. Searle, M.3. Thompson, 3. Allison and P.G. LeComber, Phil. Mag. to be published 15. 3.M. Dusseau, Thesis (1980) Montpellier 16. D.3. Dunstan, Sol. St. Comm. in press
902
KINETICS OF DISTANT-PAIR RECOMBINATION: APPLICATION TO AMORPHOUS SILICON D.J. Dunstan
Angewandte Physik Johannes Kepler Universit~t 4040 Linz, Austria
We present an exact solution for the decay of randomly distributed excited carriers, and find that a monomoleculah distant-pair approximation is valid over a wide range of excitation powers. Under r e p e t i t i v e l y pulsed or continuous excitation, an accumulated pair population is present, the result of which is that the decay curve varies at high excitation power but is independent of excitation power at low power. In consequence, time-resolved experiments cannot distinguish between distant-pair and geminate recombination, and frequency-resolved methods must be used. We find that distant-pair recombination provides a quantitative explanation of the experimental results in a-Si:H
2. Theory
i. Introduction
The radiative recombination of independently trapped electrons and holes is an important mechanism of luminescence in semiconductors. Because, howeveh the l i f e t i m e T depends on the electron-hole separation r as T(r) = ToeXp(ar) the kinetics of such systems are far from simple. Thomas, Hopfield and Augustyniak (1) obtained exaet solutions assuming uncompensated donor-acceptor recombination with the centres at random in the crystal, but for the fully compensated case they obtained only an approximate solution, in which the decay of the luminescence went as t -2 at long times. Because of their Hartree t r e a t m e n 6 this solution is only applicable to high temperatures. In amorphous semiconduetors, most studies of kinetics have been made in intrinsie materla6 so exact solutions have not been available. It has often been assumed that the kinetics are either bimolecular or geminated according to the excitation intensity; the dependence of the decay curve on the excitation power distinguishing the two regimes (2,3). We show here that this is not the case; that an intermediate regime must be distinguished in which the recombination is monomoleeular and the decay curve may or may not vary with the excitation p o w e h according to the conditions. It is this intermediate regime that we refer to as distant-pair recombination. Conventional time-resolved spectroscopy (TRS) cannot distinguish betweeen distant-pair and geminate recombination; we show that it is necessary to use frequency-resolved spectroscopy (FRS) (4). We compare our theoretical results with the behaviour of the 1.4eV photoluminescence (PL) band of a-SitH, and we find that this band, previously attributed to geminate recombination (2), in fact displays the behaviour predicted quantitatively by the distant-pair model. For reasons of space, we can only outline the methods of calculation of the kinetics. A full account is given elsewhere (5).
0 378-4363/83/0000-0000/$03.00 © 1983 North-Holland
2.1
Pulsed Excitation
We assume that a single pulse at t=0 creates a density poof carriers of each sign, which are trapped at random throughout the sample. A t high temperature, diffusion maintains the randomness of the spatial distribution and we need only find O(t) to obtain the decay curves; at low t e m p e r a t u r e with no diffusion the spatial distribution becomes anticlustered as decay proceeds. For the high temperature case, the probability that there is a hole trapped in the shell of radius r to r+dr around an e l e c t r o n at t i m e t is P(r,t)dr = 4rrr2p(t)dr
and the probability of recombination during the time interval t to t+dt is P(t)dt = fr~°oP(r,t)lr;lexp(-izr)drdt = 8 w p (t)Tol~Jdt But i~(t)=-p(t)P(t), so p(t)=
Oo/(8rrpor'ola'zt+l)
and the luminescence intensity I(t)=-i~(t) is, at long times, proportional to t -z, in agreement with the result of Thomas et al (1), For the low temperature cased we set the diffusion rates to zero, and consider the evolution of the initially random distributions, We start by making a monomoleeular approximation-" that an electron (hole) recombines only with the nearest available hole (electron); by 'available' we mean one that itself, by the same criterion, is not destined to recombine elsewhere (over a smaller separation). This may be described as the 'nearest available neighbour' approximation; when it is valid, recombination is monamolecular. Then we define P(r)dr as the probability that the nearest available neighbour of any given carrier is at a distance betweeen r and r+dr; clearly r 1-fu=~°(u)du is the probability that a carrier does not recombine at a distance less than r. SoD considering an electron, the probability that there is a hole at a distance r to r+dr, and that neither it nor the electron recombine elsewhere (i.e. have nearer
D.J. Dunstan / Kinetics of distant-pair recombination
.-.
903
" ' P" ' r" ' I"' ' I"' ' I"' ' I"' ' r" ' P" ' P" ' 1 - G:lO24/:llt !2 18 1!
1028
~uEIO 24
~ 0.5
•-= ~ 10 2 0 n W
0
lns
lps Time
1ms after
pulse
ls t
Modulation
Figure i: Decay curves after a single excitation pulse of intensity are shown, for the exact calculation from P(t), Eqn 3 (full curves), and for the distantpair model using P(r), Eqn 1 (dotted curves). The parameter values are marked. Where the dotted and full curves coincide, the distant-pair monomolecular model is valid; at higher carrier densities recombination is bimolecular.
available neighbours), so that they must recombine with each other, is r
P(r)dr = Zmr 2podr(1- ~oP(U)dU) 2
l
and this equation may be solved by a Neumann series to give
p(r) : a~m2po(l+ ~ r 3 p o ) -2
Z
We may now cheek the range of v a l i d i t y of the nearest available neighbour approximation, and hence the range of monomolecular recombination, by a comparison with an exact solution for the present case, that includes the possibility of bimolecular recombination. Instead of P(r)dr, we consider P(t)dt, the probability that a carrier recombines in the time interval t to t+dt. Following the same method as for Eqn 1, but making no approximation, we obtain
P(t) = (1-fntP(u)du)2f:411r2poT'l(r)e-t/T(r)dr
"
1MHz
3
which we solve numerically. The decay curves I(t) can be c a l c u l a t e d from P(r) and from In(t), end are shown in Figure 1. Comparison of the two sets of curves in Fig.1 shows that the approximation of Eqn 1 breaks down only at very high carrier densities, when recombination is bimoleeular. Over the entire range where the dotted and full curves of Fig.1 coincide, recombination is distant-pair (until at sufficiently low excitation power, the geminate regime is reached). We now consider the effect of r e p e t i t i v e l y pulsed excitation, as in TRS, in the distant-pair regime. An important feature of Eqn 2 is that the density of pairs with separation r,poP(r), is independent ofpo at large r (r>~po3). This means that if we have a random distribution of carriers, and we add some more to it, we do not change the number of pairs of large
•
lkHz
1Hz
frequency
v
Figure 2: Theoretical results of F-RS quadrature measurements on a distant-pair system. G is marked and the other parameters are as in F i g . l . Experimental data from Ref.4 for gd a-Si:H is shown; values of G are 1018(x), 102°(0) and 1021'Scm'3s"1 (+). All spectra are normalised to a peak height of unity.
separation. Also, if we remove the pairs of small separation (e.g. by radiative recombination) and then add carriers again at random, we still do not change the number of pairs of large separation. We define a c r i t i c a l separation re, at which the pair l i f e t i m e equals the pulse repetition period to, and we find that there is a population of pairs of separation r > r c , whose density is v i r t u a l l y independent of the pulse intensity or repetition rate. For a detailed analysis, see Ref.5; here we approximate P1 b~' integrating poP(r) from r c to co, obtaining (with pc>re ~ ) with r c =In(to/To)Or"1 4 P1= ( 34 ~ c3)-1 If an excitation pulse introduces a density of carriers po>>p1, clearly P1 will be of no significance and the decay will vary with pc as shown in F i g . l . But if the pulse introduces a density p o < P l , the new carriers cannot form pairs of separation greater than r c (as they would in the absence of P1); instead the new pairs have separations less than r c , characteristic of a random distribution of density PI. The subsequent decay curve is characteristic of P1 and not of Pc, and so does not vary as the excitation power is changed. This contrast in the behaviour of the decay curve for pulse densities above and below the 'permanent' pair density P1 has been taken to demonstrate geminate recombination at low pulse densities (3); it is in fact a direct consequence of distant-pair monomolecular recombination. 2.Z
Frequency-Resolved Spectroscopy (FRS)
L i f e t i m e analysis may be carried out by making measurements in the time domain (TRS), as considered in Section 2.1, or in the frequency domain (FRS). It is important to note that the information given by TRS and by FRS is not necessarily equivalent. For a full discussion, see Ref.6. Here, we shall illustrate the
904
D.J. Dunstan / Kinetics of distant-pair recombination
difference with the results of the FRS method on the distant-pair system. FRS is carried out under continuous excitation, which carries a small intensity modulation at an angular frequency ~. If lock-in detection set in quadrature (w/2) to the excitation is used to detect the emission, only those components with lifetimes close to 1/~ are observed, the response function being (6) S(~) = (~T+(~T)'I) "I and a frequency sweep will give a signal which represents directly the lifetime distribution present under continuous excitation. The l i f e t i m e distribution of the distant-pair system is calculated (5) as for the pulse-train problem of Section 2.1, letting tG.0, and the convolution with the FRS response function is shown in Fig.2. Note that as the e x c i t a t i o n rate Q is reduced, the l i f e t i m e distribution continues to shift to longer times. If it were observed to cease to shift at some value of Q, that would be clear evidence of geminate recombination; evidence that, as we have seen in Section 2.1, a TRS experiment cannot give. The experimental data in Fig.2, for gd a-Si:H (4), shows some discrepancies with the theoretical curves. The shift with changes in G shows, however, that distant-pair recombination is applicable. The discrepancies show, perhaps, that FRS is a more sensitive technique than TRS for studying such systems. 3.
Application to Amorphous Silicon
We have recently shown, using FRS, that the 1./4eV PL of a-Si:H is due to distant-pair recombination (4), and that the evidence previously thought to indicate a geminate model (]) is in fact q u a n t i t a t i v e l y consistent with the predictions of the distant-pair model (5). We saw, in Section 2.1, that the decay curve in TRS is independent of the e x c i t a t i o n density aPo when Pc
Although generally linear, the PL emission intensity can have a superlinear dependence on the excitation intensity, in some samples (I0). Attempts have been m a d e to interpret this phenomenon in terms of bimolecular recombination, and to deduce structural differences between samples (ii); such models m a y be inappropriate. In optically detected magnetic resonance (ODMR), a resonance of either carrier of a pair will give a quenching signal in a geminate system (123, but an enhancing signal in a distant-pair system (13). Clearly, then, the correct interpretation of the O D M R signals relies critically on a correct identification of the kinetics (14). Finally, we note that a model in which carriers are trapped at random implies greater diffusion lengths than does the geminate model, in which carriers are trapped close to their geminate partner. This may require new models of the electronic structure of amorphous semiconductors, in which longrange fluctuations of the band-edges are of greater significance than localised band-tail states (9,15,16). We wish to acknowledge many valuable discussions with Dr F. Boulitrop and Dr S.P. Depinna. References 1.
D.C. Thomas, J.J. Hopfield and W.M. Augustyniak, Phys. Rev. 14...0_0(1965) A202
2.
R.A. Street, Adv. Phys. 3__00(19813593
3.
C. Tsang and R.A. Street, Phys. Rev. B19 (1979) 3027
4.
D.3. Dunstan, S.P. Depinna J. Phys. CI__55(1982) L425
5.
D.J. Ounstan, Phil. Mag. in press
and
B.C. Cavenett,
6.
S.P. Depinna and D.3. Dunstan, to be published
7.
R.A. Street and D.K. Biegelsen, to be published
8.
R.W. Collins, M.A. Paesler and W. Paul, Sol. St. Comm. 34 (1980) 833
9.
F. Boulitrop, D.3. Dunstan and A. ChenevasPaule, Phys. Rev. B2._55(1982) in press
10. R.A. Street, Phys. Rev. B23 (1981) 861 11. R.W. Collins and W. Paul, Phys. Rev. B25 (1982) 5263 12. D.K. Biegelsen, J.C. Knights, R.A. Street, C. Tsang and R.M. White, Phil. Mag. B37 (1978) 477 13. D.J. Dunstan and J.J. Davies, J. Phys. C12 (1979) 2927 14. S.P. Depinna, B.C. Cavenett, I.(3. Austin, T.M. Searle, M.3. Thompson, 3. Allison and P.G. LeComber, Phil. Mag. to be published 15. 3.M. Dusseau, Thesis (1980) Montpellier 16. D.3. Dunstan, Sol. St. Comm. in press