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Conduction mechanism in amorphous Si films
Journal of Non-Crystalline Solids 43 (1981) 345-351 North-Holland Publishing Company
345
CONDUCTION MECHANISM IN AMORPHOUS Si FILMS R.M. MEHRA, S.C. AGARWAL, Saurabh RANI, Radhey SHYAM and P.C. MATHUR Department o f Physics and Astrophysics, University of Delhi, Delhi.110007, India Received 26 December 1979 Revised manuscript received 7 October 1980
dc conductivity as a function of temperature has been measured for as-evaporated and annealed films of amorphous Si, grown by the vacuum evaporation technique. The experimental data suggest that conduction in the higher temperature range (-175-300 K) is by the thermally activated holes in the localized states near the valence band edge while conduction in the lower temperature range ( - 7 7 - 1 7 5 K) is found to be thermally assisted tunnelling in the localized states near the Fermi level. The activation energy for both the processes is found to increase with an increase in the annealing temperature. The average hopping distance, calculated for conduction near the Fermi level, is also found to increase with an increase in the annealing temperature.
1. Introduction The properties o f tetrahedraUy coordinated amorphous films o f Si and Ge have been extensively studied because it is attractive to compare the characteristics o f the films with their crystalline counterpart. Besides this, the conductivity and mobility o f a-Si and a-Ge are very much greater than those o f chalcogenides as a result of which it is easier to interpret the experimental data in terms o f various theoretical models for the conduction mechanism. Despite the rich literature dealing with a-Si films, the experimental data from various laboratories for as-evaporated and annealed films are often significantly at variance and it is difficult to obtain a coherent picture about the conduction mechanism in this material. Walley [ 1] and Morgan and Walley [2] have shown that conduction in a-Si films, grown by the sputtering technique, in the temperature range 6 0 - 3 0 0 K is by variable range hopping following Mott's [3] relation P = Po exp(To/T) 1/4. On the other hand Lecomber et al. [4] have shown that the conduction in a-Si films, prepared by the glow discharge technique, is thermally activated in the temperature range 2 0 0 - 3 0 0 K and b y variable range hopping in the lower temperature 0 0 2 2 - 3 0 9 3 / 8 1 / 0 0 0 0 - 0 0 0 0 / $ 0 2 . 5 0 © North-Holland Publishing Company
346
R.M. Mehra et at / Conduction mechanism in amorphous Si films
region. Although Brodsky et al. [5] observed variable range hopping conduction in a-Si films, grown by the vacuum evaporation technique, in the temperature range 100-300 K, yet no physical interpretation of To and Oo could be obtained simultaneously by his data. Pollak et al. [6] have shown that even in the high temperature region the variable range hopping conduction mechanism is applicable to a-Si films grown by the sputtering technique. Paul and Mitra [7] have shown that for sputtered a-Si films the conduction mechanism is variable range hopping in the temperature range 7 7 - 3 0 0 K and is thermally activated in the temperature range 3 0 0 500 K. Hauser [8] also observed variable range hopping in the temperature range 2 5 - 3 0 0 K in similar films. Beyer [9] found that the conduction mechanism in a-Si films, grown by electron beam gun evaporation, is thermally activated in the temperature range 3 0 0 - 5 0 0 K. It is therefore interesting to study the conduction in vacuum evaporated a-Si trims as well as the effect of annealing on the conduction mechanism. In the present work we report dc conductivity measurements and the effects of annealing in a-Si films, grown by the vacuum evaporation technique, in the temperature range 7 7 - 3 0 0 K. It is found that the conduction mechanism in low temperature region is not due to variable range hopping but is rather due to tunnelling conduction near the Fermi level. At higher temperatures it is found from the experimental data that the conduction is due to phonon assisted tunnelling in the localized states near the band edge. 2. Experimental The a-Si films, 1 /~m thick, were deposited on fused quartz substrate by the vacuum evaporation technique in a vacuum of ~ 1 0 6 Torr (To = 300 K). A1-A1 electrodes were evaporated on the Film so as to provide a coplanar configuration. After taking observations on the as-evaporated trims, the Films were annealed for 4 h at 350,400 and 450 K. dc conductivity measurements were made in the temperature range 7 7 - 3 0 0 K. The sample was mounted on a copper block (with electrical insulation) in a cryostat in which cooling of the sample was achieved by conduction through the copper block. The temperature of the sample was increased with the help of a heater coil wrapped around the extended part of the copper block.
3. Result and discussion The temperature variation of the dc conductivity (log e versus l/T) for the as-evaporated and annealed a-Si films are shown in fig. 1. It is found from the figure that the conductivity in the temperature range 300-T1 K follows the relation O----" O 1
exp(-&E/kT)
(1)
R.M. Mehra et al. / Conduction mechanism in amorphous Si films
347
o As Evaporated A 350 K a 400K • 450K
16s
~E t.)
167
74
1(5~
g ¢.~
109
I~ c
J
101 ----ll
3
I
4
I
5
l
I
6
I
7
I0001T
8
r
9
J
I0
I
II
(K-')
Fig. 1. Semilog plots of the conductivity versus reciprocal temperature for as-evaporated and annealed a-Si films. The curve A represents as evaporated, while curves B, C and D represent films annealed at 350,400 and 450 K, respectively.
Table 1 Annealing history, pre-exponential factor and activation energy for higher temperature region of a-Si films Annealing history
o1
AE 1
(S2 -1 cm -1)
(eV)
As-evaporated 350 K, 4 h " 400 K, 4 h 450 K, 4 h
1160.0 143.7 90.5 47.0
0.31 0.33 0.42 0.44
348
R.M. Mehra et al. / Conduction mechanism in amorphous Si films
where Ol is the pre-exponential factor, AE l is the conductivity activation energy and T1 is the temperature below which a knee is observed in the conductivity temperature variation. The calculated values of o~ and AE] for the as-evaporated and annealed films are given in table 1. It is found that the activation energy AEi increases from 0.31 eV to 0.44 eV and the pre-exponential factor ol decreases from 1160 f2 -l cm -1 to 47 ~2-1 cm -1 with the increase of annealing temperature from 300 to 450 K. The magnitudes of the activation energy and pre-exponential factor indicates that the conduction is due to the carriers excited into the localized states at the band edge. For conduction in the localized states near the Fermi level, the values of o~ and AE] should be very much smaller and for conduction in the extended states the values of these parameters are expected to be very much greater [4,10] than the observed values. It has been shown by Spear [10] that annealing decreases the density of localized states near the Fermi level and that the predominant conduction mechanism is by holes for films annealed upto 450 K. It was further shown by him that the Fermi level shifts towards the conduction band as a result of annealing. If the transport is by holes, such a shift in the Fermi level would result in an increase in the activation energy. Since in the present experiment the highest annealing temperature is 450 K and the activation energy increases with the -6 I0 o As Evaporated .350 K n 400 K
o o
• 450 K 167
/x
"TE
L~
ic
I .25
.27
I
I
.28
.29
I
I
I
.30
.31
.32
-33
-r-~ ¢~ol ____,. Fig. 2. Semilog plots of (rT 112 versus (reciprocal temperature) 1/4 for as-evaporated and annealed a-Si films. The lines B, C and D represent f'rims annealed at 350, 400 and 450 K, respectively.
R.M. Mehra et al. / Conduction mechanism in amorphous Si films
349
increase o f annealing temperature, it can be assumed that conduction is mainly by holes in the valence band edge. It may be mentioned here that the observed change in the activation energy with annealing temperature can also be explained b y considering the shifts o f E e and Ev, the band edges o f the conduction and valence band respectively. However the possibility o f the shift o f band edges is denied because it is highly improbable that annealing will increase randomness in the sample [11]. Decrease in al with annealing can be attributed either to the decrease in the density o f localized states near the band edge or to the decrease in the mobility of the carriers in the localized states. It can also be seen in fig. 1 that the temperature range in which the conduction occurs in the localized states near the valence band edge goes on decreasing with increasing annealing temperature. This observation is quite understandable because due to the increase of the activation energy with the increase in annealing temperature the temperature required to activate the carriers to the localized states near the valence band edge is higher for films annealed at higher temperatures. In the lower temperature range T 1 - 7 7 K as can be seen in fig. 1 the log o - l I T variation is almost linear with a very low activation energy. Several workers [ 1,6,8, 10] have obtained variable range hopping conduction in this temperature range in a - S i films. The low temperature data have therefore been replotted in fig. 2 on a log o T 1/2 versus T -1/4 in accordance with Mott's relation for variable range hopping
[31
where (3)
To = 16a3/kN(EF)
with wave been table
k as Boltzman's constant and a -I a measure o f the spatial extension of the function e x p ( - ~ T ) associated with the localized states. The value a -1 has taken [12] to be 10 A. The values o f To obtained from fig. 2 are shown iv 2. Although the condition for the variable range hopping process, T o / T > >
Table 2 Annealing history, temperature To and the density of localized states at the Fermi level calculated from eqs. (2) and (3) for a-Si films Annealing history
TO (K)
N(EF) from eq. (3) (cm-3 eV -1 )
N(EF) from eq. (4) (cm -3 eV-I )
As-evaporated 350K, 4 h 400K, 4 h 450 K, 4 h
4.5 X 104 4.7X 104 8.6× 10 s 2.4 X 106
4.18 3.9 2.1 7.46
3.2 X 103 3.98 3.2X 20 -6 2.43 X 10 -7
X 10~1 X 1021 X 1020 X 1019
350
R.M. Mehra et at / Conduction mechanism in amorphous Si films
is satisfied for the as-evaporated as well as for the annealed films, the values of To are about two orders of magnitude lower than the reported values of To for a-Si films [5,78] grown by glow discharge and sputtering techniques. The magnitude of To/T represents the ratio of the characteristic disorder energy to the thermal energy [ 13] and it appears that in the films grown by the vacuum evaporation technique, there is less disorder energy as compared with films grown by the glow discharge and sputtering techniques. The values of N(EF) calculated from eqs. (2) and (3) using the calculated values of To, are also listed in table 2. Though the results obtained using eq. (3) are of the correct order, the values of N(EF) obtained from eq. (2) are absurdly low. The variable range hopping process, therefore, may be discarded in the present case. In the low temperature region the log o - l i T variation (fig. 1) can therefore be represented by the phonon assisted thermal tunnelling near the Fermi level and can be expressed as [14] o = o2 exp(-AE2/kT)
(4)
o2 = e2 R 2 VphN(EF) e -2aR
(5)
N(EF) = 1/AE2R 3
(6)
where
and
with e as the electronic charge, R the average hopping distance, Vph the phonon frequency (1012 Hz). The values of R and N(EF) have been calculated using eqs. (5) and (6) for the as-evaporated and annealed films and these values are shown in table 3. These values are in good agreement with the results obtained by various workers [2,6,7,8]. It can be seen in the results shown in table 3 that the value of R increase with increasing annealing temperature. Paul and Mitra [7] also reported an increase in the values of R in the variable range hopping conduction regime in a-Si films. Thus in case of a-Si films the conduction process in the higher temperature region is the thermally activated type in the localized states at the band edge and is
Table 3 Annealing history, pre-exponential factor and activation energy for the low temperature region of a-Si Films Annealing history
o2 (s2-1 cm-1 )
AE2 (eV)
R (A)
N(E F) (cm-3 eV-1 )
As-evaporated 350 K, 4 h 400 K, 4 h 450 K, 4h
1.8 × 10 -8 9.0 X 10 -9 1.6 X 10 -9 1.5 × 10 -9
0.0047 0.00479 0.0148 0.0237
117.0 120.8 123.0 133.0
1.3 × 1.1 × 3.6 × 1.8 ×
1020 1020 1019
1019
R.M. Mehra et al. / Conduction mechanism in amorphous Si films
351
phonon assisted tunnelling in the localized states near the Fermi level in the low temperature range.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]
P.A. Walley, Thin Solid Films 2 (1968) 327. M. Morgan and P.A. WaUey,Phil. Mag. 23 (1971) 661. N.F. Mott, Phil. Mag. 19 (1969) 835. A. Madan, P.G. Lecomber and W.E. Spear, J. Non-CrystaUineSolids 20 (1976) 239. M.H. Brodsky and R.J. Gambino, J. Non-Crystalline Solids 8-10 (1972) 739. M. PoUak, M.L. Knotek, H. Kurtman and H. Glick, Phys. Rev. Lett. 18 (1973) 856. D.K. Paul and S.S. Mitra, Phys. Rev. Lett. 31 (1973) 1000. J.J. Hauser, Phys. Rev. B8 (1973) 3817. W. Beyer, Solid St. Commun. 29 (1979) 291. W.E. Spear, Proc. of the 5th Int. Conf. on Amorphous and Liquid Semiconductors, Garmisch Partenkirchen (1973). [ 11 ] S. Hasegawa, S. Yazaki and T. Shimuzu, Solid St. Commun. 26 (1978) 407. [12] V. Ambegaokar, B.I. Halperin and J.S. Langer, Phys. Rev. B4 (1972) 2612. [13] R.M. Mehra, S.C. Agarwal, Saurabh Rani, Radhey Shyam, S.K. Agarwal and P.C. Mathur, Thin Sofid Films 71 (1980) 71. [14] R.M. Hill, Phil. Mag. 24 (1971) 1307.
345
CONDUCTION MECHANISM IN AMORPHOUS Si FILMS R.M. MEHRA, S.C. AGARWAL, Saurabh RANI, Radhey SHYAM and P.C. MATHUR Department o f Physics and Astrophysics, University of Delhi, Delhi.110007, India Received 26 December 1979 Revised manuscript received 7 October 1980
dc conductivity as a function of temperature has been measured for as-evaporated and annealed films of amorphous Si, grown by the vacuum evaporation technique. The experimental data suggest that conduction in the higher temperature range (-175-300 K) is by the thermally activated holes in the localized states near the valence band edge while conduction in the lower temperature range ( - 7 7 - 1 7 5 K) is found to be thermally assisted tunnelling in the localized states near the Fermi level. The activation energy for both the processes is found to increase with an increase in the annealing temperature. The average hopping distance, calculated for conduction near the Fermi level, is also found to increase with an increase in the annealing temperature.
1. Introduction The properties o f tetrahedraUy coordinated amorphous films o f Si and Ge have been extensively studied because it is attractive to compare the characteristics o f the films with their crystalline counterpart. Besides this, the conductivity and mobility o f a-Si and a-Ge are very much greater than those o f chalcogenides as a result of which it is easier to interpret the experimental data in terms o f various theoretical models for the conduction mechanism. Despite the rich literature dealing with a-Si films, the experimental data from various laboratories for as-evaporated and annealed films are often significantly at variance and it is difficult to obtain a coherent picture about the conduction mechanism in this material. Walley [ 1] and Morgan and Walley [2] have shown that conduction in a-Si films, grown by the sputtering technique, in the temperature range 6 0 - 3 0 0 K is by variable range hopping following Mott's [3] relation P = Po exp(To/T) 1/4. On the other hand Lecomber et al. [4] have shown that the conduction in a-Si films, prepared by the glow discharge technique, is thermally activated in the temperature range 2 0 0 - 3 0 0 K and b y variable range hopping in the lower temperature 0 0 2 2 - 3 0 9 3 / 8 1 / 0 0 0 0 - 0 0 0 0 / $ 0 2 . 5 0 © North-Holland Publishing Company
346
R.M. Mehra et at / Conduction mechanism in amorphous Si films
region. Although Brodsky et al. [5] observed variable range hopping conduction in a-Si films, grown by the vacuum evaporation technique, in the temperature range 100-300 K, yet no physical interpretation of To and Oo could be obtained simultaneously by his data. Pollak et al. [6] have shown that even in the high temperature region the variable range hopping conduction mechanism is applicable to a-Si films grown by the sputtering technique. Paul and Mitra [7] have shown that for sputtered a-Si films the conduction mechanism is variable range hopping in the temperature range 7 7 - 3 0 0 K and is thermally activated in the temperature range 3 0 0 500 K. Hauser [8] also observed variable range hopping in the temperature range 2 5 - 3 0 0 K in similar films. Beyer [9] found that the conduction mechanism in a-Si films, grown by electron beam gun evaporation, is thermally activated in the temperature range 3 0 0 - 5 0 0 K. It is therefore interesting to study the conduction in vacuum evaporated a-Si trims as well as the effect of annealing on the conduction mechanism. In the present work we report dc conductivity measurements and the effects of annealing in a-Si films, grown by the vacuum evaporation technique, in the temperature range 7 7 - 3 0 0 K. It is found that the conduction mechanism in low temperature region is not due to variable range hopping but is rather due to tunnelling conduction near the Fermi level. At higher temperatures it is found from the experimental data that the conduction is due to phonon assisted tunnelling in the localized states near the band edge. 2. Experimental The a-Si films, 1 /~m thick, were deposited on fused quartz substrate by the vacuum evaporation technique in a vacuum of ~ 1 0 6 Torr (To = 300 K). A1-A1 electrodes were evaporated on the Film so as to provide a coplanar configuration. After taking observations on the as-evaporated trims, the Films were annealed for 4 h at 350,400 and 450 K. dc conductivity measurements were made in the temperature range 7 7 - 3 0 0 K. The sample was mounted on a copper block (with electrical insulation) in a cryostat in which cooling of the sample was achieved by conduction through the copper block. The temperature of the sample was increased with the help of a heater coil wrapped around the extended part of the copper block.
3. Result and discussion The temperature variation of the dc conductivity (log e versus l/T) for the as-evaporated and annealed a-Si films are shown in fig. 1. It is found from the figure that the conductivity in the temperature range 300-T1 K follows the relation O----" O 1
exp(-&E/kT)
(1)
R.M. Mehra et al. / Conduction mechanism in amorphous Si films
347
o As Evaporated A 350 K a 400K • 450K
16s
~E t.)
167
74
1(5~
g ¢.~
109
I~ c
J
101 ----ll
3
I
4
I
5
l
I
6
I
7
I0001T
8
r
9
J
I0
I
II
(K-')
Fig. 1. Semilog plots of the conductivity versus reciprocal temperature for as-evaporated and annealed a-Si films. The curve A represents as evaporated, while curves B, C and D represent films annealed at 350,400 and 450 K, respectively.
Table 1 Annealing history, pre-exponential factor and activation energy for higher temperature region of a-Si films Annealing history
o1
AE 1
(S2 -1 cm -1)
(eV)
As-evaporated 350 K, 4 h " 400 K, 4 h 450 K, 4 h
1160.0 143.7 90.5 47.0
0.31 0.33 0.42 0.44
348
R.M. Mehra et al. / Conduction mechanism in amorphous Si films
where Ol is the pre-exponential factor, AE l is the conductivity activation energy and T1 is the temperature below which a knee is observed in the conductivity temperature variation. The calculated values of o~ and AE] for the as-evaporated and annealed films are given in table 1. It is found that the activation energy AEi increases from 0.31 eV to 0.44 eV and the pre-exponential factor ol decreases from 1160 f2 -l cm -1 to 47 ~2-1 cm -1 with the increase of annealing temperature from 300 to 450 K. The magnitudes of the activation energy and pre-exponential factor indicates that the conduction is due to the carriers excited into the localized states at the band edge. For conduction in the localized states near the Fermi level, the values of o~ and AE] should be very much smaller and for conduction in the extended states the values of these parameters are expected to be very much greater [4,10] than the observed values. It has been shown by Spear [10] that annealing decreases the density of localized states near the Fermi level and that the predominant conduction mechanism is by holes for films annealed upto 450 K. It was further shown by him that the Fermi level shifts towards the conduction band as a result of annealing. If the transport is by holes, such a shift in the Fermi level would result in an increase in the activation energy. Since in the present experiment the highest annealing temperature is 450 K and the activation energy increases with the -6 I0 o As Evaporated .350 K n 400 K
o o
• 450 K 167
/x
"TE
L~
ic
I .25
.27
I
I
.28
.29
I
I
I
.30
.31
.32
-33
-r-~ ¢~ol ____,. Fig. 2. Semilog plots of (rT 112 versus (reciprocal temperature) 1/4 for as-evaporated and annealed a-Si films. The lines B, C and D represent f'rims annealed at 350, 400 and 450 K, respectively.
R.M. Mehra et al. / Conduction mechanism in amorphous Si films
349
increase o f annealing temperature, it can be assumed that conduction is mainly by holes in the valence band edge. It may be mentioned here that the observed change in the activation energy with annealing temperature can also be explained b y considering the shifts o f E e and Ev, the band edges o f the conduction and valence band respectively. However the possibility o f the shift o f band edges is denied because it is highly improbable that annealing will increase randomness in the sample [11]. Decrease in al with annealing can be attributed either to the decrease in the density o f localized states near the band edge or to the decrease in the mobility of the carriers in the localized states. It can also be seen in fig. 1 that the temperature range in which the conduction occurs in the localized states near the valence band edge goes on decreasing with increasing annealing temperature. This observation is quite understandable because due to the increase of the activation energy with the increase in annealing temperature the temperature required to activate the carriers to the localized states near the valence band edge is higher for films annealed at higher temperatures. In the lower temperature range T 1 - 7 7 K as can be seen in fig. 1 the log o - l I T variation is almost linear with a very low activation energy. Several workers [ 1,6,8, 10] have obtained variable range hopping conduction in this temperature range in a - S i films. The low temperature data have therefore been replotted in fig. 2 on a log o T 1/2 versus T -1/4 in accordance with Mott's relation for variable range hopping
[31
where (3)
To = 16a3/kN(EF)
with wave been table
k as Boltzman's constant and a -I a measure o f the spatial extension of the function e x p ( - ~ T ) associated with the localized states. The value a -1 has taken [12] to be 10 A. The values o f To obtained from fig. 2 are shown iv 2. Although the condition for the variable range hopping process, T o / T > >
Table 2 Annealing history, temperature To and the density of localized states at the Fermi level calculated from eqs. (2) and (3) for a-Si films Annealing history
TO (K)
N(EF) from eq. (3) (cm-3 eV -1 )
N(EF) from eq. (4) (cm -3 eV-I )
As-evaporated 350K, 4 h 400K, 4 h 450 K, 4 h
4.5 X 104 4.7X 104 8.6× 10 s 2.4 X 106
4.18 3.9 2.1 7.46
3.2 X 103 3.98 3.2X 20 -6 2.43 X 10 -7
X 10~1 X 1021 X 1020 X 1019
350
R.M. Mehra et at / Conduction mechanism in amorphous Si films
is satisfied for the as-evaporated as well as for the annealed films, the values of To are about two orders of magnitude lower than the reported values of To for a-Si films [5,78] grown by glow discharge and sputtering techniques. The magnitude of To/T represents the ratio of the characteristic disorder energy to the thermal energy [ 13] and it appears that in the films grown by the vacuum evaporation technique, there is less disorder energy as compared with films grown by the glow discharge and sputtering techniques. The values of N(EF) calculated from eqs. (2) and (3) using the calculated values of To, are also listed in table 2. Though the results obtained using eq. (3) are of the correct order, the values of N(EF) obtained from eq. (2) are absurdly low. The variable range hopping process, therefore, may be discarded in the present case. In the low temperature region the log o - l i T variation (fig. 1) can therefore be represented by the phonon assisted thermal tunnelling near the Fermi level and can be expressed as [14] o = o2 exp(-AE2/kT)
(4)
o2 = e2 R 2 VphN(EF) e -2aR
(5)
N(EF) = 1/AE2R 3
(6)
where
and
with e as the electronic charge, R the average hopping distance, Vph the phonon frequency (1012 Hz). The values of R and N(EF) have been calculated using eqs. (5) and (6) for the as-evaporated and annealed films and these values are shown in table 3. These values are in good agreement with the results obtained by various workers [2,6,7,8]. It can be seen in the results shown in table 3 that the value of R increase with increasing annealing temperature. Paul and Mitra [7] also reported an increase in the values of R in the variable range hopping conduction regime in a-Si films. Thus in case of a-Si films the conduction process in the higher temperature region is the thermally activated type in the localized states at the band edge and is
Table 3 Annealing history, pre-exponential factor and activation energy for the low temperature region of a-Si Films Annealing history
o2 (s2-1 cm-1 )
AE2 (eV)
R (A)
N(E F) (cm-3 eV-1 )
As-evaporated 350 K, 4 h 400 K, 4 h 450 K, 4h
1.8 × 10 -8 9.0 X 10 -9 1.6 X 10 -9 1.5 × 10 -9
0.0047 0.00479 0.0148 0.0237
117.0 120.8 123.0 133.0
1.3 × 1.1 × 3.6 × 1.8 ×
1020 1020 1019
1019
R.M. Mehra et al. / Conduction mechanism in amorphous Si films
351
phonon assisted tunnelling in the localized states near the Fermi level in the low temperature range.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]
P.A. Walley, Thin Solid Films 2 (1968) 327. M. Morgan and P.A. WaUey,Phil. Mag. 23 (1971) 661. N.F. Mott, Phil. Mag. 19 (1969) 835. A. Madan, P.G. Lecomber and W.E. Spear, J. Non-CrystaUineSolids 20 (1976) 239. M.H. Brodsky and R.J. Gambino, J. Non-Crystalline Solids 8-10 (1972) 739. M. PoUak, M.L. Knotek, H. Kurtman and H. Glick, Phys. Rev. Lett. 18 (1973) 856. D.K. Paul and S.S. Mitra, Phys. Rev. Lett. 31 (1973) 1000. J.J. Hauser, Phys. Rev. B8 (1973) 3817. W. Beyer, Solid St. Commun. 29 (1979) 291. W.E. Spear, Proc. of the 5th Int. Conf. on Amorphous and Liquid Semiconductors, Garmisch Partenkirchen (1973). [ 11 ] S. Hasegawa, S. Yazaki and T. Shimuzu, Solid St. Commun. 26 (1978) 407. [12] V. Ambegaokar, B.I. Halperin and J.S. Langer, Phys. Rev. B4 (1972) 2612. [13] R.M. Mehra, S.C. Agarwal, Saurabh Rani, Radhey Shyam, S.K. Agarwal and P.C. Mathur, Thin Sofid Films 71 (1980) 71. [14] R.M. Hill, Phil. Mag. 24 (1971) 1307.