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Photoluminescence decay affected by variable range hopping in band tail in amorphous As2S3
Solid State Communications, Vol.53,No.2 pp.]25-128, Printed in Great Britain.
]985.
0038-1098/85 $3.00 + .00 Pergamon Press Ltd.
PHOTOLUMINESCENCE DECAY AFFECTED BY VARIABLE RANGE HOPPING IN BAND TAlL IN AMORPHOUS As2S 3 K. Murayama and T. Ninomiya Department of Physics, The University of Tokyo, Tokyo 113, Japan (Received 8 October 1984 by W. Sasaki)
Photoluminescence decay was observed with various excitation energies and at various temperatures in amorphous As2S 3. The decay at high temperature or with the high excitation energy had the t-~/2 long time behavior. This decay is concluded to arise from the combination of the tunneling recombination and the variable range hopping of carriers in the band tail.
Photoluminescence in amorphous semiconductors is usually interpreted as due to recombination of excitons or tunneling recombination of electron-hole pairs. Slow decay luminescence observed at liquid helium temperature in arsenic chalcogenide glasses [1,2] is a typical example of the radiative recombination of localized spin triplet excitons, which is verified by optically detected ESR [3,4]. It is shown by the authors [2] and Ngai and Murayama [5] that the decay of the slow decay luminescence is empirically described with the Williams-Watts decay function d exp-(t/T) B, where T is an [6,7]. f(t) = - ~-{
hours and air-quenched to room temperature. The sample was held in a liquid helium or helium gas for photoluminescence measurement. It was excited with 10 nsec light pulses from an excimer-pumped dye laser. Scattered laser light was eliminated using a thallium bromoiodide filter. The photoluminescence was monocromatized with an interference filter and was detected with a S-i type photomultiplier. Signals were processed using a transient waveform recorder (Digital memory DM901 lwatsu) whose output was transferred to a signal averager (Spectrum analyzer SM2100A lwatsu). The results were averaged with a constant ratio of the time resolution (At) to time of measurement (t) of 1:10 (4t/t = 0.I). The sample were excited with light pulses at an energy of 2.38eV in the Urbach tail of the absorption edge. The photoluminescence decay curves observed at the peak energy (1.16eV) of the photoluminescence spectrum [10] at 4.2, 49, 79, 109 and 142K are shown in double logarithmic plots in Fig.1. The three shoulders of the photoluminescence decay curve observed at 4.2K show that the decay can be divided into three components [2 I. The component observed from 10 u to 2x10-1sec is called the fast decay luminescence. The component observed from 2×10 -7 to 10-5sec is called the intermediate decay luminescence. The component observed from 10 -5 to 10-2sec is called the slow decay luminescence. The decay of the fast decay luminescence had the t-3/2 behavior as indicated by solid lines a [11]. The decay curves of the intermediate and the slow decay luminescence could be fitted with the W-W decay function as indicated by solid lines b in Fig. 1. The values of T and B of the intermediate and the slow decay luminescence are 2.0x10-6sec (B=0.8) and 1.0x10-4sec (B=0.6), respectively. The decay curve of the slow decay luminescence observed at temperature lower than about 20K had the form of the W-W decay function. This suggests that the slow decay luminescence observed at the lower temperature than 20K is due to the radiative recombination of the localized spin triplet excitons.
effective decay time and B is a constant between 0 and I. The simple exponential decay function is included in the W-W decay function as the special case of B = I. Photoluminescence due to the tunneling recombination is observed in amorphous hydrogenated Si (a-Si:H). The nonexponential photoluminescence decay in a-Si:H is interpreted as due to the distribution of the initial separation of electron-hole pairs [8]. It is, further, pointed out by Hong et al [9] that the photoluminescence arising from the tunneling recombination is affected by the diffusion of carriers (electrons or holes) and then the decay curve has the t-3/2 long time behavior. We measured the photoluminescence decay with various excitation energies and at various temperatures in a-As2S 3 to investigate the dynamic process of the diffusion and the recombination of photo-excited carriers. The photoluminescence decay observed near liquid helium temperature or with the low excitation energy had the form of the W-W decay function. However, the photoluminescence decay with the t-3/2 long time behavior was observed at the temperature higher than 36K or with the high energy excitation. It will be concluded that the decay is due to the combination of the tunneling recombination and the variable range hopping of carriers in the band tail. Samples of bulk a-As2S 3 were prepared from elemental As and S, each of five nines purity. The sample was sealed in evaquated quartz ampoules held in a rocking furnace at 600°C for 24 125
PHOTOLUMINESCENCE
Vol. 53, No. 2
DECAY AFFECTED
127
BY VARIABLE RANGE HOPPING
sary for the electron to diffuse apart by ~-I. When T R > t > tc~, the decay function l(t) is given by
a -A%S 3 107 ~ .
no
.t . 3/2
i-E-)
I(t) ~
,
8/f~ 0
= v0 exp(-BT
3B
r h =~--~T
4 ),
(6)
m+ I 4
(7)
3-m I
w h = ~ kBT
4
,
(8)
where B = 3(~) 3/4 {
= .
eY
,%
This shows that the photoluminescence decay has the t -3/2 behavior for long times, which is observed in the fast decay luminescence and the slow decay luminescence at the temperature higher than 36K. The decay function l(t) obtained by calculating equation (4) with a computer is shown by solid lines c in Fig.1. The photoluminescence decay curves observed at the higher temperature than 36K can be beautifully fitted with l(t). The arrows on the photoluminescence decay curves mark the hopping time t . The hopping times t~ at 49, 79, 109 and I~2K are 3.2xi0 -5, 1.2xi0 -5, 6.4xi0 -6 and 4.3x 10-6sec, respectively. The hopping of electrons in the band tail is considered to be of variable range type. If the form of the density of states in the band tail is of the form N(E) = C(E-EA )m where C is a constant and E A is an energy of the band edge, the hopping probability l)h, the average hopping length r h and energy w h will then vary with the temperature T as [14] m+ I Vh
Ex = 2.38 ¢V
(5)
~E)m~3 ~N(E )mmk m+1 c
-} '
(9)
u0 is a hopping frequency at the high temperature, E c is a mobility edge of the conduction band and bE (= Ec-EA) is a range of localized states in the band tail. Now, we assume that the form of the density of states in the band tail in amorphous semiconductors does not differ much from that in crystal. Namely, we assume that the band tail has the parabolic density of states (m=I/2) as illustrated in Fig.2. Then, the hopping time t~ which is obtained from the photoluminescence decay is given by 1 _ 2 D 2 1 ~-- = = ~ (~ ~h r )
3 v0B2T-3/4 128
-BT -3/8 e
(10)
-1 3/4 The quantity t~ T is plotted as a function of T -3/8 in Fig.3. The data points in Fig.3 are almost on a straight line. The values of B and Vn obtained from the slope and the value of t-IT 374 at T-3/8=0 are 36 K 378 and 8.Sx107sec -I respectively. The average hopping length rh and
u) v I'--
106
10 5
I
,
0.16
I
~18
I
I
~20
!
I
~22
I
I
0.24
I
I
I
~26
"r 3/8 ( F 3/8 ) Fig. 3.
Temperature dependence of the hopping time. The value of t~IT 3/4 is plotted as a function of T -3/~. The solid line shows the fit to the theory.
energy w h calculated with equations (7) and (8) for values of ~-I=12A and T=I00K are 29A and 14 meV, respectively. The small value of v 0 compared with the phonon frequency (1012sec -I) is considered to be responsible for the discrepancy between the effective Bohr radius of the localized state and the wave length of the phonon which assists the variable range hopping [14]. Up to this point, the discussion of the photoluminescence decay was developed assuming that the hole was trapped at the deep state and the electron diffused by the hopping. However, it is the same even if we assume that the electron is trapped at the deep state and the hole diffuses by the hopping. It cannot be identified from this method whether an electron diffuses or a hole does. The diffusion constant and the drift mobility calculated from the obtained B and v0 in zero electric field at room temperature are 7.2 ×10-9cm2/sec and 2.8×10-7cm2/Vsec, respectively. It is observed by Owen and Robertson [15] from the measurements of the photoconductivity that the zero-field hole mobility is 5×10-7cm2/V sec in a-As2Se 3. The drift mobility obtained from the photoluminescence decay shows the value close to that in a-As2Se 3. We estimates the quantity B using some reasonable parameters. For AE = 0.2eV, I/~ = 12A and N(E) = 3×1021cm-3eV-1, the value at 100K is 37K 3/8. The estimated B is close to that obtained from the experiments. The excitation energy dependence of the effective decay time T and the hopping time t~ at 77K are shown in Fig.4. The photoluminescence decay observed with the excitation energy higher than 2.35eV could be fitted with the decay function given by equation (4). The h o p -
PHOTOLUMINESCENCE
]28
DECAY AFFECTED
' 16 4
i
o-As2S 3 77K EL= 1.16eV
(0.s)
I/)
"-.(0.4) "~. --~
. . . . .
4,-
--~-_
_.
16s
Fig. 4.
I
I
2.1
2.2
I
I
I
2.3 2.4 2.5 Exc. E n e r g y ( W )
I
I
2.6
2.7
Excitation energy dependence of the effective decay time and the hopping time. The numbers noted in parentheses show the values of B of the W-W decay function.
ping time t~ obtained is 1.9x10-5sec. For the photoluminescence observed with the excitation energy lower than 2.35eV, the decay curve could be well fitted with the W-W decay function rather than the equation (4). The effective decay time obtained is about 3x10-5sec. These show that the excitation with the energy higher than 2.35eV can create carriers in the shallow localized states in the band tail where the electron or hole diffuses by the hopping until the geminate recombination occurs. The excitation energy lower than 2.35eV is not enough to overwhelm the Coulomb interaction between an electron (hole) is bounded at a localized hole (electron) by the Coulomb interaction and an localized spin triplet exciton is formed. The explanation for the excitation energy dependence of the photoluminescence decay is supported by the optical band gap of 2.45eV at 77K [16] which is close to 2.35eV.
BY VARIABLE RANGE HOPPING
Vol. 53, No. 2
We showed in this paper that the effect of the variable range hopping in the band tail was observed in the photoluminescence decay at the higher temperature than 36K in a-As2S 3. The drift mobility estimated from the photoluminescence decay has a reasonable value in comparison with that obtained from the measurements of the photoconductivity. The photoluminescence due to the recombination of localized spin triplet excitons is observed only at the temperature lower than 2OK. The photoluminescence decay affected by the variable range hopping is observed at 36K. It is considered that the energy level of a triplet exciton is close to that of an electron-hole pair because of the potential fluctuation produced by random structure so that the exciton is thermally ionized even at the low temperature of 36K. -3/2 Finally, the t long time behavior was also observed for the fast decay luminescence. The hopping time of carriers inducing the fast decay luminescence is estimated to be shorter than 10-8sec from the decay curve. This suggests that another path in which carriers diffuse 1000 times faster than those inducing the slow decay luminescence exists in a-As2S 3. The technique of pico-second measurements will be required to understand the fast path. In summary, we obtained the form of the photoluminescence decay by solving the equation of the motion of the photo-excited carriers which diffused by the variable range hopping in the band tail and recombined by the tunneling process. The equation of the motion has essentially the same form as that used for the photoconduction. Development of this method will lead us to the unified understanding of the photoconduction and photoluminescence in amorphous semiconductors. We will discuss elsewhere the photoluminescence decay in a-Si:H and other chalcogenide glasses.
Acknowledgement We thank K. Osawa and A. Yamaguchi their help in the experiments.
for
References [I] [2] [3]
[4]
[5] [6] [7] [8] [9]
G.S. Higashi & M.A. Kastner, J. Phys. C12, L821 (1979). K. Murayama & T. Ninomiya, Jpn. J. Appl. Phys. 21, L512 (1982). S. Depinna & B.C. Cavenett, Phys. Rev. Lett. 48, 556 (1982); Phil. Mag. B46, 71 (1982) T. Tada, H. Suzuki, K. Murayama & T. Ninomiya, to be published in AIP conference proceeding of Optical Effects in Amorphous Semiconductors, Utah, 1984. K.L. Ngai & K. Murayama, Physica B+C 117/ 118, 980 (1983). G. Williams & D.C. Watts, Trans. Faraday Soc. 66, 80 (1970). K.L. Ngai, Comments Solid State Physics 9, 127 (1979). R.A. Street, Adv. Phys. 30, 593 (1981). K.M. Hong, J. Noolandi & R.A. Street, Phys. Rev. B23, 2967 (1981).
[10] [11]
[12] [13] [14]
[15] [16]
R.A. Street, Adv. Phys. 25, 397 (1976). The decay curve of the fast decay luminescence was fitted with the W-W decay function in Ref.[2]. However, the present detailed experiments show that the curve has the form of the t-3/2 behavior rather than that of the W-W decay function. R.A. Street & N.F. Mott, Phys. Rev. Lett. 35, 1293 (1975). L. Onsager, J. Chem. Phys. 2, 599 (1934); Phys. Rev. 54, 554 (1938). N.F. Mott & E.A. Davis, Electronic Processes in Non-crystalline Materials, 2nd Ed., Clarendon Press, Oxford, 1979. A.E. Owen & J.M. Robertson, J. Non-cryst. Solids 2, 40 (1970). F. Kosek & J. Tauc: Czech. J. Phys. B20, 94 (1970).
]985.
0038-1098/85 $3.00 + .00 Pergamon Press Ltd.
PHOTOLUMINESCENCE DECAY AFFECTED BY VARIABLE RANGE HOPPING IN BAND TAlL IN AMORPHOUS As2S 3 K. Murayama and T. Ninomiya Department of Physics, The University of Tokyo, Tokyo 113, Japan (Received 8 October 1984 by W. Sasaki)
Photoluminescence decay was observed with various excitation energies and at various temperatures in amorphous As2S 3. The decay at high temperature or with the high excitation energy had the t-~/2 long time behavior. This decay is concluded to arise from the combination of the tunneling recombination and the variable range hopping of carriers in the band tail.
Photoluminescence in amorphous semiconductors is usually interpreted as due to recombination of excitons or tunneling recombination of electron-hole pairs. Slow decay luminescence observed at liquid helium temperature in arsenic chalcogenide glasses [1,2] is a typical example of the radiative recombination of localized spin triplet excitons, which is verified by optically detected ESR [3,4]. It is shown by the authors [2] and Ngai and Murayama [5] that the decay of the slow decay luminescence is empirically described with the Williams-Watts decay function d exp-(t/T) B, where T is an [6,7]. f(t) = - ~-{
hours and air-quenched to room temperature. The sample was held in a liquid helium or helium gas for photoluminescence measurement. It was excited with 10 nsec light pulses from an excimer-pumped dye laser. Scattered laser light was eliminated using a thallium bromoiodide filter. The photoluminescence was monocromatized with an interference filter and was detected with a S-i type photomultiplier. Signals were processed using a transient waveform recorder (Digital memory DM901 lwatsu) whose output was transferred to a signal averager (Spectrum analyzer SM2100A lwatsu). The results were averaged with a constant ratio of the time resolution (At) to time of measurement (t) of 1:10 (4t/t = 0.I). The sample were excited with light pulses at an energy of 2.38eV in the Urbach tail of the absorption edge. The photoluminescence decay curves observed at the peak energy (1.16eV) of the photoluminescence spectrum [10] at 4.2, 49, 79, 109 and 142K are shown in double logarithmic plots in Fig.1. The three shoulders of the photoluminescence decay curve observed at 4.2K show that the decay can be divided into three components [2 I. The component observed from 10 u to 2x10-1sec is called the fast decay luminescence. The component observed from 2×10 -7 to 10-5sec is called the intermediate decay luminescence. The component observed from 10 -5 to 10-2sec is called the slow decay luminescence. The decay of the fast decay luminescence had the t-3/2 behavior as indicated by solid lines a [11]. The decay curves of the intermediate and the slow decay luminescence could be fitted with the W-W decay function as indicated by solid lines b in Fig. 1. The values of T and B of the intermediate and the slow decay luminescence are 2.0x10-6sec (B=0.8) and 1.0x10-4sec (B=0.6), respectively. The decay curve of the slow decay luminescence observed at temperature lower than about 20K had the form of the W-W decay function. This suggests that the slow decay luminescence observed at the lower temperature than 20K is due to the radiative recombination of the localized spin triplet excitons.
effective decay time and B is a constant between 0 and I. The simple exponential decay function is included in the W-W decay function as the special case of B = I. Photoluminescence due to the tunneling recombination is observed in amorphous hydrogenated Si (a-Si:H). The nonexponential photoluminescence decay in a-Si:H is interpreted as due to the distribution of the initial separation of electron-hole pairs [8]. It is, further, pointed out by Hong et al [9] that the photoluminescence arising from the tunneling recombination is affected by the diffusion of carriers (electrons or holes) and then the decay curve has the t-3/2 long time behavior. We measured the photoluminescence decay with various excitation energies and at various temperatures in a-As2S 3 to investigate the dynamic process of the diffusion and the recombination of photo-excited carriers. The photoluminescence decay observed near liquid helium temperature or with the low excitation energy had the form of the W-W decay function. However, the photoluminescence decay with the t-3/2 long time behavior was observed at the temperature higher than 36K or with the high energy excitation. It will be concluded that the decay is due to the combination of the tunneling recombination and the variable range hopping of carriers in the band tail. Samples of bulk a-As2S 3 were prepared from elemental As and S, each of five nines purity. The sample was sealed in evaquated quartz ampoules held in a rocking furnace at 600°C for 24 125
PHOTOLUMINESCENCE
Vol. 53, No. 2
DECAY AFFECTED
127
BY VARIABLE RANGE HOPPING
sary for the electron to diffuse apart by ~-I. When T R > t > tc~, the decay function l(t) is given by
a -A%S 3 107 ~ .
no
.t . 3/2
i-E-)
I(t) ~
,
8/f~ 0
= v0 exp(-BT
3B
r h =~--~T
4 ),
(6)
m+ I 4
(7)
3-m I
w h = ~ kBT
4
,
(8)
where B = 3(~) 3/4 {
= .
eY
,%
This shows that the photoluminescence decay has the t -3/2 behavior for long times, which is observed in the fast decay luminescence and the slow decay luminescence at the temperature higher than 36K. The decay function l(t) obtained by calculating equation (4) with a computer is shown by solid lines c in Fig.1. The photoluminescence decay curves observed at the higher temperature than 36K can be beautifully fitted with l(t). The arrows on the photoluminescence decay curves mark the hopping time t . The hopping times t~ at 49, 79, 109 and I~2K are 3.2xi0 -5, 1.2xi0 -5, 6.4xi0 -6 and 4.3x 10-6sec, respectively. The hopping of electrons in the band tail is considered to be of variable range type. If the form of the density of states in the band tail is of the form N(E) = C(E-EA )m where C is a constant and E A is an energy of the band edge, the hopping probability l)h, the average hopping length r h and energy w h will then vary with the temperature T as [14] m+ I Vh
Ex = 2.38 ¢V
(5)
~E)m~3 ~N(E )mmk m+1 c
-} '
(9)
u0 is a hopping frequency at the high temperature, E c is a mobility edge of the conduction band and bE (= Ec-EA) is a range of localized states in the band tail. Now, we assume that the form of the density of states in the band tail in amorphous semiconductors does not differ much from that in crystal. Namely, we assume that the band tail has the parabolic density of states (m=I/2) as illustrated in Fig.2. Then, the hopping time t~ which is obtained from the photoluminescence decay is given by 1 _ 2 D 2 1 ~-- = = ~ (~ ~h r )
3 v0B2T-3/4 128
-BT -3/8 e
(10)
-1 3/4 The quantity t~ T is plotted as a function of T -3/8 in Fig.3. The data points in Fig.3 are almost on a straight line. The values of B and Vn obtained from the slope and the value of t-IT 374 at T-3/8=0 are 36 K 378 and 8.Sx107sec -I respectively. The average hopping length rh and
u) v I'--
106
10 5
I
,
0.16
I
~18
I
I
~20
!
I
~22
I
I
0.24
I
I
I
~26
"r 3/8 ( F 3/8 ) Fig. 3.
Temperature dependence of the hopping time. The value of t~IT 3/4 is plotted as a function of T -3/~. The solid line shows the fit to the theory.
energy w h calculated with equations (7) and (8) for values of ~-I=12A and T=I00K are 29A and 14 meV, respectively. The small value of v 0 compared with the phonon frequency (1012sec -I) is considered to be responsible for the discrepancy between the effective Bohr radius of the localized state and the wave length of the phonon which assists the variable range hopping [14]. Up to this point, the discussion of the photoluminescence decay was developed assuming that the hole was trapped at the deep state and the electron diffused by the hopping. However, it is the same even if we assume that the electron is trapped at the deep state and the hole diffuses by the hopping. It cannot be identified from this method whether an electron diffuses or a hole does. The diffusion constant and the drift mobility calculated from the obtained B and v0 in zero electric field at room temperature are 7.2 ×10-9cm2/sec and 2.8×10-7cm2/Vsec, respectively. It is observed by Owen and Robertson [15] from the measurements of the photoconductivity that the zero-field hole mobility is 5×10-7cm2/V sec in a-As2Se 3. The drift mobility obtained from the photoluminescence decay shows the value close to that in a-As2Se 3. We estimates the quantity B using some reasonable parameters. For AE = 0.2eV, I/~ = 12A and N(E) = 3×1021cm-3eV-1, the value at 100K is 37K 3/8. The estimated B is close to that obtained from the experiments. The excitation energy dependence of the effective decay time T and the hopping time t~ at 77K are shown in Fig.4. The photoluminescence decay observed with the excitation energy higher than 2.35eV could be fitted with the decay function given by equation (4). The h o p -
PHOTOLUMINESCENCE
]28
DECAY AFFECTED
' 16 4
i
o-As2S 3 77K EL= 1.16eV
(0.s)
I/)
"-.(0.4) "~. --~
. . . . .
4,-
--~-_
_.
16s
Fig. 4.
I
I
2.1
2.2
I
I
I
2.3 2.4 2.5 Exc. E n e r g y ( W )
I
I
2.6
2.7
Excitation energy dependence of the effective decay time and the hopping time. The numbers noted in parentheses show the values of B of the W-W decay function.
ping time t~ obtained is 1.9x10-5sec. For the photoluminescence observed with the excitation energy lower than 2.35eV, the decay curve could be well fitted with the W-W decay function rather than the equation (4). The effective decay time obtained is about 3x10-5sec. These show that the excitation with the energy higher than 2.35eV can create carriers in the shallow localized states in the band tail where the electron or hole diffuses by the hopping until the geminate recombination occurs. The excitation energy lower than 2.35eV is not enough to overwhelm the Coulomb interaction between an electron (hole) is bounded at a localized hole (electron) by the Coulomb interaction and an localized spin triplet exciton is formed. The explanation for the excitation energy dependence of the photoluminescence decay is supported by the optical band gap of 2.45eV at 77K [16] which is close to 2.35eV.
BY VARIABLE RANGE HOPPING
Vol. 53, No. 2
We showed in this paper that the effect of the variable range hopping in the band tail was observed in the photoluminescence decay at the higher temperature than 36K in a-As2S 3. The drift mobility estimated from the photoluminescence decay has a reasonable value in comparison with that obtained from the measurements of the photoconductivity. The photoluminescence due to the recombination of localized spin triplet excitons is observed only at the temperature lower than 2OK. The photoluminescence decay affected by the variable range hopping is observed at 36K. It is considered that the energy level of a triplet exciton is close to that of an electron-hole pair because of the potential fluctuation produced by random structure so that the exciton is thermally ionized even at the low temperature of 36K. -3/2 Finally, the t long time behavior was also observed for the fast decay luminescence. The hopping time of carriers inducing the fast decay luminescence is estimated to be shorter than 10-8sec from the decay curve. This suggests that another path in which carriers diffuse 1000 times faster than those inducing the slow decay luminescence exists in a-As2S 3. The technique of pico-second measurements will be required to understand the fast path. In summary, we obtained the form of the photoluminescence decay by solving the equation of the motion of the photo-excited carriers which diffused by the variable range hopping in the band tail and recombined by the tunneling process. The equation of the motion has essentially the same form as that used for the photoconduction. Development of this method will lead us to the unified understanding of the photoconduction and photoluminescence in amorphous semiconductors. We will discuss elsewhere the photoluminescence decay in a-Si:H and other chalcogenide glasses.
Acknowledgement We thank K. Osawa and A. Yamaguchi their help in the experiments.
for
References [I] [2] [3]
[4]
[5] [6] [7] [8] [9]
G.S. Higashi & M.A. Kastner, J. Phys. C12, L821 (1979). K. Murayama & T. Ninomiya, Jpn. J. Appl. Phys. 21, L512 (1982). S. Depinna & B.C. Cavenett, Phys. Rev. Lett. 48, 556 (1982); Phil. Mag. B46, 71 (1982) T. Tada, H. Suzuki, K. Murayama & T. Ninomiya, to be published in AIP conference proceeding of Optical Effects in Amorphous Semiconductors, Utah, 1984. K.L. Ngai & K. Murayama, Physica B+C 117/ 118, 980 (1983). G. Williams & D.C. Watts, Trans. Faraday Soc. 66, 80 (1970). K.L. Ngai, Comments Solid State Physics 9, 127 (1979). R.A. Street, Adv. Phys. 30, 593 (1981). K.M. Hong, J. Noolandi & R.A. Street, Phys. Rev. B23, 2967 (1981).
[10] [11]
[12] [13] [14]
[15] [16]
R.A. Street, Adv. Phys. 25, 397 (1976). The decay curve of the fast decay luminescence was fitted with the W-W decay function in Ref.[2]. However, the present detailed experiments show that the curve has the form of the t-3/2 behavior rather than that of the W-W decay function. R.A. Street & N.F. Mott, Phys. Rev. Lett. 35, 1293 (1975). L. Onsager, J. Chem. Phys. 2, 599 (1934); Phys. Rev. 54, 554 (1938). N.F. Mott & E.A. Davis, Electronic Processes in Non-crystalline Materials, 2nd Ed., Clarendon Press, Oxford, 1979. A.E. Owen & J.M. Robertson, J. Non-cryst. Solids 2, 40 (1970). F. Kosek & J. Tauc: Czech. J. Phys. B20, 94 (1970).