- Email: [email protected]
Electrical transport in inhomogeneous composites
Volume
10, number
9,lO
MATERIALS
LETTERS
Electrical transport in inhomogeneous Zolili
February
199
I
composites
Ndlela
Department
ofphysics,
Stanford
University,
Stanford,
CA 94305,
USA
and Clayton
W. Bates Jr.
Departments
of Materials
Stanford,
C.4 94305,
Received
24 October
Science and Engineering
and Electrical
Engmeering.
Stanford
Universrty.
USA
I990
Resistivity as a function of temperature from 300 K down to 80 K was measured for inhomogeneous composites consisting of Ag-a-Si, Ag-poly-Si, Ag-CuInSez and Au-CuInSez. The volume fraction of metal was varied to produce composites in the dielectric, transition and metal regimes corresponding to resistivities varying over 9 orders of magnitudes ( lo-’ to IO6 R cm). Linear regression analysis of the experimental data revealed a T - ‘I4 temperature dependence for the resistivity. Based on these results we interpret the transport in these systems to be due to a hopping mechanism.
1. Introduction This work investigated a two-component inhomogeneous thin-film composite consisting of metal particles dispersed in a semiconductor matrix. The systems were silver or gold dispersed in CuInSez and silver dispersed in amorphous silicon or polycrystalline silicon and were measured from 80 to 300 K. Inhomogeneous composites classified as disordered solids broadly include lightly and heavily doped semiconductors, compensated semiconductors, and amorphous, glassy and other non-crystalline materials. Disordered solids may include any material in which the normal long-range crystalline structure has been disrupted. These composites are also similar to granular metals (often referred to as cermets or ceramic metals) formed by cosputtering a metal and an insulator. Although the composite systems are not actually granular metals, the similarities are very strong. The microstructural features of these films also classify them as disordered three-dimensional quantum-well structures or random heterostructures whose heterojunctions are formed between the metal 0 167-577x/9
I /$ 03.50 0 199
I - Elsevier Science Publishers
particles and semiconductor material. Study of these inhomogeneous composites was strongly motivated for their potential use as photovoltaics, energy converters, and photoconductors [ 11. They may also be used for photodetection in the optical to the infrared region. These materials are also possible candidates to be used as contacts [ 2 1, and gates [ 3,4]. In addition, these systems are useful in studying physical phenomena as electron localization and size quantization [ 5 ] that can occur in thin metallic films and inhomogeneous materials. As noted above, these materials are interesting physical systems to study. First, the metal-semiconductor mixture does not form compounds but is a true composite, at least within the temperature ranges noted. The amorphous silver-silicon composites in this study were prepared at room temperature, whereas the polycrystalline samples were deposited on sapphire substrates at 600” C. The silver and gold CuInSe, composites were prepared on Schott glass substrates at 275 ‘C, a temperature at which silver or gold compounds are not formed with Cu, In or Se.
B.V. ( North-Holland
)
465
Volume 10, number 9,lO
MATERIALS LETTERS
February I99 1
2. Experimental Thin films of (Ag, Au)-CuInSe* were prepared by sputter deposition of silver or gold followed by chemical spray pyrolysis (CSP) of CuInSe, [ 6-91; Ag-a-Si and Ag-polycrystalline silicon were prepared solely by cosputtering deposition. These methods produced thin films having particle sizes of 50 to 200 8, and an overall thickness ranging from 0.1 to 2 pm, Van der Pauw, Hall, and thermoelectric power measurements were made to determine the electronic transport properties of these composites while X-ray diffraction and a SEM (scanning electron microscope) were used to characterize their crystal structure, stoichiometry, and morphology. X-ray diffractometry scans also confirmed that no silver-silicon compounds were formed. Similarly, no evidence was observed that silver or gold formed compounds with Cu, In, or Se. Transport measurements were performed on an inhouse system with an Andonian dewar, with the commercially available Keithley S110, and with the MMR (Micro Miniature Refrigerator Technologies, Inc.) Hall/van der Pauw system. The Ag-polycrystalline silicon exhibited resistivities as high as 1O8 and 1O9 R cm and as low as lo-’ Q cm while amorphous Ag-Si had values on the order 10e2 to 10e3 0 cm. The resistivities of AuCuInSe, were approximately 10e3 to lo6 R cm and those of Ag-CuInSez were on the order of l-10’ Q cm, values consistent with crystalline and polycrystalline CuInSe, obtained by other researchers [ 11. In all of these composite systems, it was observed that the optical properties could be altered by adjusting the volume fraction of metal to semiconductor, and by changing the metal-particle size distribution [ 10,111.
3. Fractional temperature behavior 3.1. Silver and gold CuInSe, The CWBAg series, films 1 to 4, are displayed in fig. 1 for resistivity versus T - ‘. The difference between samples in the plot is the volume fraction of silver that is incorporated into each film. A general progression from slightly metallic-like behavior is 466
&_*__*
_.&__
._..
. . .._.____
*
. . . . ..__.
A
-4 2
4
6
8
10
12
T-‘(K) x103 Fig. 1. Resistivity Ag-CuInSe* composites in the range 80 to 300 K. The samples differ by the amount of silver incorporated into each composite.
shown progressing upwards from sample CWBAgZB toward strong semiconducting or dielectric-type behavior in sample CWBAg4B. Although quantitative values are not given, the metal volume fraction of all composites ranged from 20 to 50%. The microstructural features are all quite similar to those shown in the inset which shows a Ag-amorphous-silicon composite of 50% Ag. The volume percentages appears larger because it was a layered structured formed by alternate evaporation of silver and silicon. These silver composite films also showed evidence of fractional temperature behavior of n= l/4; representative samples in that series are shown in fig. 2 that display fractional temperature dependence. Fig. 3 compares the linear least-squares fit for sample CWBAg4 with the T-’ and T-II4 curve. Notably, the T -‘/4 dependence also holds for low- and highresistivity Au-CuInSe* in figs. 4 and 5. This fractional temperature dependence is similar to that observed by other researchers on granular metallic films, and also fits the T-II4 dependence proposed by Mott for variable-range hopping conduction in disordered and amorphous semiconductors. At low temperatures, around 1 K in granular alu-
Volume
IO, number
9, IO
MATERIALS
LETTERS
February
I99
I
9
8 8
x
.5 ‘= 6 .” L [r
C
5
4
(4 4
6
8
10
12
I
I
4
6
I
I 10
8
I
12
T-’ (K) x103
T-’ (K) xl O3 9
8
-,C+ InRho4B
-.A-, lnRho3HT 4
3 0.24 0.24
0.26
0.28
o.io
0.26
T-‘14 (K)
0.28
0.30
0.32
T-1’4 (K)
0.32
Fig. 3. Linear least-squares
tit of silver composites.
Fig. 2. Temperature dependence of Ag-CuInSe* composites indicating fractional temperature behavior for (a) In p versus T - ’ (b) lnpversus T-1/4.
cles. Their observations alytical expression
minum, McLean et al. [ 121 also observed a T ~ "4 dependence and suggested hopping conduction in parallel with a percolation path due to metal parti-
The first term represents conduction due to metallic percolation, and the second term represents hopping conduction. Although this expression could not be
n=q,,+o,
led them to propose the an-
exp[ - ( To/T)1/4] .
(1)
467
Volume 10, number 9,lO
MATERIALS LETTERS
o.o-
February 199 1
0 0
1.435
-0.20 g
-0.4-
6 21 -0.6.= .? ,$ -o.a-
0
E 1.425
8
0
p 1.420 tj ._ g r 1.415 -t
0
B 5
0
-l.O0 -1.2-
0
Experimental Data
1.430
0
Least Square Fit
1.410
0
-1.4-45
1.405
2.4
2.6
2.8
3.4 3.0 3.2 IOxT-1’4 (K)
3.6
3.8 I
2.4
Fig. 4. Low-resistivity Au-CuInSe, with fractional temperature behavior.
Fig. 6. Silver-amorphous
00
8 g 14.5.
t
2.8 10 x T-“4 (K)
I
3.0
silicon, In resistivity versus T-“4.
composite-like behavior; in addition linear leastsquares fits for both types of samples indicated the T - ‘I4 dependence illustrated by amorphous silicon in fig. 6.
0
8
5 .z “. .z ‘j .g 13.5. .G
4. Summary
Q 8 Q
13.
Q 0
12.5.
0 0
12 0. . 2.36 2.36
. 2.4
1 2.42
2.44
2.46
2.46
2.5
1Ox T-“4 (K) Fig. 5. High-resistivity Au-CuInSez with fractional temperature behavior.
verified in the 80 to 300 K range, the T -‘I4 behavior was still observed. 3.2. Amorphous/polycrystalline silicon The silver-amorphous-silicon crystalline-silicon composites 468
I
2.6
and silver-polyalso exhibited typical
Transport measurements were performed from 80 to 300 K for silver and gold copper indium diselenide composites, and for silver-amorphous silicon and silver-polycrystalline silicon. Results also indicated that the composite materials measured in this study exhibit fractional temperature dependence of resistivity with T- ‘I4 behavior which is indicative of hopping conduction described by the Mott-Davis model [ 13 1. This behavior is also similar to observations made on granular metals for which a T - ‘I4 dependence has also been recently reported [ 14- 17 1. In addition, in some materials a T -“* dependence is thought to appear below some critical temperature T while T --‘I4 is dominant above this temperature [ f 15 ] ; however, others have reported the opposite behavior [ 18 1. other essential features of the physics may be needed to explain transport in granular metals [ 19,201 that may very well depend upon the existence of localized states.
4,
Volume
10, number
9,lO
MATERIALS
This study has applied a linear least-squares analysis of the data which indicated that the resistivities of our composites have a fractional temperature dependence with n= l/ 4 consistent with measurements obtained from other disordered solids. Again, such transport confirms hopping conduction as predicted by the Mott-Davis model and hopping and tunneling observed in granular metals.
Acknowledgement This study was supported by the National Science Foundation under Grant No. ECS 85-20580 and through ONR under contract NOOO14-86-K-0509 6. Facilities at the Center for Materials Research at Stanford University were used in this investigation.
References [ 1] T.J. Coutts, L.L. Kazmerski
and S. Wagner, Copper indium diselenide for photovoltaic applications (Elsevier, New York 1986). [ 2) C.J. Kircher, Solid State Electron. 14 ( 1971) 507. [ 3) B.L. Crowder and S. Zirpinsky, IEEE Trans. Electron Devices 26 (1979) 369. [4] S.P. Murarka and D.B. Fraser, J. Appl. Phys. 51 ( 1980) 342.
LETTERS
February
199
I
[ 51 E. Abrahams, P.W. Anderson, D.C. Licciardello and T.V. Ramakrishnan, Phys. Rev. Letters 42 ( 1979) 673. [6] C.W. Bates Jr., K.F. Nelson, S.A. Raza, J.B. Mooney, J.M. Recktenwald, L. Macintosh and R. Lamoreaux, Thin Solid Films 88 (1982) 279. [ 7 ] C.W. Bates Jr., M. Uekita, K.F. Nelson, CR. Abernathy and J.B. Mooeny, Appl. Phys. Letters 43 (1983) 851. [8] C.R. Abernathy, C.W. Bates Jr., A.A. Anani and B. Haba, Thin Solid Films I 15 ( 1984) L4 1. [9] C.R. Abernathy, C.W. Bates Jr., A.A. Anani, B. Haba and G. Smestad, Appl. Phys. Letters 45 (1984) 890. [ 10 J N. Alexander and C.W. Bates Jr., Solid State Commun. 5 1 (1984) 331. 111] Q.Y. Chen and C.W. Bates Jr., Materials Research Society Symposium Proceedings, Vol. 90. Infrared materials for detectors and sources (MRS, Pittsburgh, 1987). [12 ] W.L. McLean, P. Lindenfeld and T. Worthington, in: AIP Conference Proceedings, Vol. 40. Electrical transport and optical properties of inhomogeneous media, eds. J.C. Garland and D.B. Tanner (AIP, 1978). [ 13 ] N.F. Mott, Conduction in noncrystalline materials (Oxford Univ. Press, Oxford, 1987). [ 141 R. Ntmeth and R. Mlihlschlegel, Z. Physik B 70 ( 1988) 159. [ IS] C. van Haesendonck and Y. Bruynseraede, Phys. Rev. B 33 (1986) 1684. [ 16 ] G. Deutscher, Phys. Rev. Letters 44 ( 1980) 1150. [ I7 ] 0. Entin-Wohlman, Y. Gefen and Y. Shapira, J. Phys. C 16 (1983) 1161. [ 181 P. Sheng and .J. Klafter, Phys. Rev. B 27 (1983) 2583. [19]C.J.Adkins,J.Phys.C15(1982)7143. [ 201 C.J. Adkins, Institute of Amorphous Studies Series, Vol. 107. Disordered semiconductors, eds. M. Kastner, G. Thomas and S. Ovshinsky (1987).
469
10, number
9,lO
MATERIALS
LETTERS
Electrical transport in inhomogeneous Zolili
February
199
I
composites
Ndlela
Department
ofphysics,
Stanford
University,
Stanford,
CA 94305,
USA
and Clayton
W. Bates Jr.
Departments
of Materials
Stanford,
C.4 94305,
Received
24 October
Science and Engineering
and Electrical
Engmeering.
Stanford
Universrty.
USA
I990
Resistivity as a function of temperature from 300 K down to 80 K was measured for inhomogeneous composites consisting of Ag-a-Si, Ag-poly-Si, Ag-CuInSez and Au-CuInSez. The volume fraction of metal was varied to produce composites in the dielectric, transition and metal regimes corresponding to resistivities varying over 9 orders of magnitudes ( lo-’ to IO6 R cm). Linear regression analysis of the experimental data revealed a T - ‘I4 temperature dependence for the resistivity. Based on these results we interpret the transport in these systems to be due to a hopping mechanism.
1. Introduction This work investigated a two-component inhomogeneous thin-film composite consisting of metal particles dispersed in a semiconductor matrix. The systems were silver or gold dispersed in CuInSez and silver dispersed in amorphous silicon or polycrystalline silicon and were measured from 80 to 300 K. Inhomogeneous composites classified as disordered solids broadly include lightly and heavily doped semiconductors, compensated semiconductors, and amorphous, glassy and other non-crystalline materials. Disordered solids may include any material in which the normal long-range crystalline structure has been disrupted. These composites are also similar to granular metals (often referred to as cermets or ceramic metals) formed by cosputtering a metal and an insulator. Although the composite systems are not actually granular metals, the similarities are very strong. The microstructural features of these films also classify them as disordered three-dimensional quantum-well structures or random heterostructures whose heterojunctions are formed between the metal 0 167-577x/9
I /$ 03.50 0 199
I - Elsevier Science Publishers
particles and semiconductor material. Study of these inhomogeneous composites was strongly motivated for their potential use as photovoltaics, energy converters, and photoconductors [ 11. They may also be used for photodetection in the optical to the infrared region. These materials are also possible candidates to be used as contacts [ 2 1, and gates [ 3,4]. In addition, these systems are useful in studying physical phenomena as electron localization and size quantization [ 5 ] that can occur in thin metallic films and inhomogeneous materials. As noted above, these materials are interesting physical systems to study. First, the metal-semiconductor mixture does not form compounds but is a true composite, at least within the temperature ranges noted. The amorphous silver-silicon composites in this study were prepared at room temperature, whereas the polycrystalline samples were deposited on sapphire substrates at 600” C. The silver and gold CuInSe, composites were prepared on Schott glass substrates at 275 ‘C, a temperature at which silver or gold compounds are not formed with Cu, In or Se.
B.V. ( North-Holland
)
465
Volume 10, number 9,lO
MATERIALS LETTERS
February I99 1
2. Experimental Thin films of (Ag, Au)-CuInSe* were prepared by sputter deposition of silver or gold followed by chemical spray pyrolysis (CSP) of CuInSe, [ 6-91; Ag-a-Si and Ag-polycrystalline silicon were prepared solely by cosputtering deposition. These methods produced thin films having particle sizes of 50 to 200 8, and an overall thickness ranging from 0.1 to 2 pm, Van der Pauw, Hall, and thermoelectric power measurements were made to determine the electronic transport properties of these composites while X-ray diffraction and a SEM (scanning electron microscope) were used to characterize their crystal structure, stoichiometry, and morphology. X-ray diffractometry scans also confirmed that no silver-silicon compounds were formed. Similarly, no evidence was observed that silver or gold formed compounds with Cu, In, or Se. Transport measurements were performed on an inhouse system with an Andonian dewar, with the commercially available Keithley S110, and with the MMR (Micro Miniature Refrigerator Technologies, Inc.) Hall/van der Pauw system. The Ag-polycrystalline silicon exhibited resistivities as high as 1O8 and 1O9 R cm and as low as lo-’ Q cm while amorphous Ag-Si had values on the order 10e2 to 10e3 0 cm. The resistivities of AuCuInSe, were approximately 10e3 to lo6 R cm and those of Ag-CuInSez were on the order of l-10’ Q cm, values consistent with crystalline and polycrystalline CuInSe, obtained by other researchers [ 11. In all of these composite systems, it was observed that the optical properties could be altered by adjusting the volume fraction of metal to semiconductor, and by changing the metal-particle size distribution [ 10,111.
3. Fractional temperature behavior 3.1. Silver and gold CuInSe, The CWBAg series, films 1 to 4, are displayed in fig. 1 for resistivity versus T - ‘. The difference between samples in the plot is the volume fraction of silver that is incorporated into each film. A general progression from slightly metallic-like behavior is 466
&_*__*
_.&__
._..
. . .._.____
*
. . . . ..__.
A
-4 2
4
6
8
10
12
T-‘(K) x103 Fig. 1. Resistivity Ag-CuInSe* composites in the range 80 to 300 K. The samples differ by the amount of silver incorporated into each composite.
shown progressing upwards from sample CWBAgZB toward strong semiconducting or dielectric-type behavior in sample CWBAg4B. Although quantitative values are not given, the metal volume fraction of all composites ranged from 20 to 50%. The microstructural features are all quite similar to those shown in the inset which shows a Ag-amorphous-silicon composite of 50% Ag. The volume percentages appears larger because it was a layered structured formed by alternate evaporation of silver and silicon. These silver composite films also showed evidence of fractional temperature behavior of n= l/4; representative samples in that series are shown in fig. 2 that display fractional temperature dependence. Fig. 3 compares the linear least-squares fit for sample CWBAg4 with the T-’ and T-II4 curve. Notably, the T -‘/4 dependence also holds for low- and highresistivity Au-CuInSe* in figs. 4 and 5. This fractional temperature dependence is similar to that observed by other researchers on granular metallic films, and also fits the T-II4 dependence proposed by Mott for variable-range hopping conduction in disordered and amorphous semiconductors. At low temperatures, around 1 K in granular alu-
Volume
IO, number
9, IO
MATERIALS
LETTERS
February
I99
I
9
8 8
x
.5 ‘= 6 .” L [r
C
5
4
(4 4
6
8
10
12
I
I
4
6
I
I 10
8
I
12
T-’ (K) x103
T-’ (K) xl O3 9
8
-,C+ InRho4B
-.A-, lnRho3HT 4
3 0.24 0.24
0.26
0.28
o.io
0.26
T-‘14 (K)
0.28
0.30
0.32
T-1’4 (K)
0.32
Fig. 3. Linear least-squares
tit of silver composites.
Fig. 2. Temperature dependence of Ag-CuInSe* composites indicating fractional temperature behavior for (a) In p versus T - ’ (b) lnpversus T-1/4.
cles. Their observations alytical expression
minum, McLean et al. [ 121 also observed a T ~ "4 dependence and suggested hopping conduction in parallel with a percolation path due to metal parti-
The first term represents conduction due to metallic percolation, and the second term represents hopping conduction. Although this expression could not be
n=q,,+o,
led them to propose the an-
exp[ - ( To/T)1/4] .
(1)
467
Volume 10, number 9,lO
MATERIALS LETTERS
o.o-
February 199 1
0 0
1.435
-0.20 g
-0.4-
6 21 -0.6.= .? ,$ -o.a-
0
E 1.425
8
0
p 1.420 tj ._ g r 1.415 -t
0
B 5
0
-l.O0 -1.2-
0
Experimental Data
1.430
0
Least Square Fit
1.410
0
-1.4-45
1.405
2.4
2.6
2.8
3.4 3.0 3.2 IOxT-1’4 (K)
3.6
3.8 I
2.4
Fig. 4. Low-resistivity Au-CuInSe, with fractional temperature behavior.
Fig. 6. Silver-amorphous
00
8 g 14.5.
t
2.8 10 x T-“4 (K)
I
3.0
silicon, In resistivity versus T-“4.
composite-like behavior; in addition linear leastsquares fits for both types of samples indicated the T - ‘I4 dependence illustrated by amorphous silicon in fig. 6.
0
8
5 .z “. .z ‘j .g 13.5. .G
4. Summary
Q 8 Q
13.
Q 0
12.5.
0 0
12 0. . 2.36 2.36
. 2.4
1 2.42
2.44
2.46
2.46
2.5
1Ox T-“4 (K) Fig. 5. High-resistivity Au-CuInSez with fractional temperature behavior.
verified in the 80 to 300 K range, the T -‘I4 behavior was still observed. 3.2. Amorphous/polycrystalline silicon The silver-amorphous-silicon crystalline-silicon composites 468
I
2.6
and silver-polyalso exhibited typical
Transport measurements were performed from 80 to 300 K for silver and gold copper indium diselenide composites, and for silver-amorphous silicon and silver-polycrystalline silicon. Results also indicated that the composite materials measured in this study exhibit fractional temperature dependence of resistivity with T- ‘I4 behavior which is indicative of hopping conduction described by the Mott-Davis model [ 13 1. This behavior is also similar to observations made on granular metals for which a T - ‘I4 dependence has also been recently reported [ 14- 17 1. In addition, in some materials a T -“* dependence is thought to appear below some critical temperature T while T --‘I4 is dominant above this temperature [ f 15 ] ; however, others have reported the opposite behavior [ 18 1. other essential features of the physics may be needed to explain transport in granular metals [ 19,201 that may very well depend upon the existence of localized states.
4,
Volume
10, number
9,lO
MATERIALS
This study has applied a linear least-squares analysis of the data which indicated that the resistivities of our composites have a fractional temperature dependence with n= l/ 4 consistent with measurements obtained from other disordered solids. Again, such transport confirms hopping conduction as predicted by the Mott-Davis model and hopping and tunneling observed in granular metals.
Acknowledgement This study was supported by the National Science Foundation under Grant No. ECS 85-20580 and through ONR under contract NOOO14-86-K-0509 6. Facilities at the Center for Materials Research at Stanford University were used in this investigation.
References [ 1] T.J. Coutts, L.L. Kazmerski
and S. Wagner, Copper indium diselenide for photovoltaic applications (Elsevier, New York 1986). [ 2) C.J. Kircher, Solid State Electron. 14 ( 1971) 507. [ 3) B.L. Crowder and S. Zirpinsky, IEEE Trans. Electron Devices 26 (1979) 369. [4] S.P. Murarka and D.B. Fraser, J. Appl. Phys. 51 ( 1980) 342.
LETTERS
February
199
I
[ 51 E. Abrahams, P.W. Anderson, D.C. Licciardello and T.V. Ramakrishnan, Phys. Rev. Letters 42 ( 1979) 673. [6] C.W. Bates Jr., K.F. Nelson, S.A. Raza, J.B. Mooney, J.M. Recktenwald, L. Macintosh and R. Lamoreaux, Thin Solid Films 88 (1982) 279. [ 7 ] C.W. Bates Jr., M. Uekita, K.F. Nelson, CR. Abernathy and J.B. Mooeny, Appl. Phys. Letters 43 (1983) 851. [8] C.R. Abernathy, C.W. Bates Jr., A.A. Anani and B. Haba, Thin Solid Films I 15 ( 1984) L4 1. [9] C.R. Abernathy, C.W. Bates Jr., A.A. Anani, B. Haba and G. Smestad, Appl. Phys. Letters 45 (1984) 890. [ 10 J N. Alexander and C.W. Bates Jr., Solid State Commun. 5 1 (1984) 331. 111] Q.Y. Chen and C.W. Bates Jr., Materials Research Society Symposium Proceedings, Vol. 90. Infrared materials for detectors and sources (MRS, Pittsburgh, 1987). [12 ] W.L. McLean, P. Lindenfeld and T. Worthington, in: AIP Conference Proceedings, Vol. 40. Electrical transport and optical properties of inhomogeneous media, eds. J.C. Garland and D.B. Tanner (AIP, 1978). [ 13 ] N.F. Mott, Conduction in noncrystalline materials (Oxford Univ. Press, Oxford, 1987). [ 141 R. Ntmeth and R. Mlihlschlegel, Z. Physik B 70 ( 1988) 159. [ IS] C. van Haesendonck and Y. Bruynseraede, Phys. Rev. B 33 (1986) 1684. [ 16 ] G. Deutscher, Phys. Rev. Letters 44 ( 1980) 1150. [ I7 ] 0. Entin-Wohlman, Y. Gefen and Y. Shapira, J. Phys. C 16 (1983) 1161. [ 181 P. Sheng and .J. Klafter, Phys. Rev. B 27 (1983) 2583. [19]C.J.Adkins,J.Phys.C15(1982)7143. [ 201 C.J. Adkins, Institute of Amorphous Studies Series, Vol. 107. Disordered semiconductors, eds. M. Kastner, G. Thomas and S. Ovshinsky (1987).
469