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Theory of steady state photoconductivity in amorphous semiconductors
JOURNALOFNON-CRYSTALLINE SOLIDS8--10(1972)947-953© North-HollandPublishingCo.
T H E O R Y O F STEADY S T A T E P H O T O C O N D U C T I V I T Y I N AMORPHOUS SEMICONDUCTORS J. G. SIMMONS and G. W. TAYLOR Electrical Engineering Department, University of Toronto, Toronto, Canada
The theory of photoconductivity described in the adjoining paper is applied to amorphous semiconductors. It is shown that for constant light intensity the photocurrent, Ip, initially increases with decreasing temperature, reaches a maximum value at some temperature, say, Tin, and thereafter decreases monotonically with further decrease in temperature (T< Tin). The photocurrent at the maximum is a few percent of the dark current, I0, corresponding to Tin. At constant temperatures and low intensities, when lp< Io, the photoconductivity is a linear function of the light intensity. At high light intensity, when Ip > Io, Ip is a sub-linear function of the light intensity. The theory is shown to be in good agreement with experimental observation. 1. Introduction Recently, a good deal of data has been accumulated on the photoconductive properties of amorphous solids, in particular the semiconducting chalcogenide glasses 1-6). No adequate self-consistent explanation, based on the currently accepted model of amorphous semiconductors, has been forthcoming. The object of this paper is to present a general treatment of photoconductivity in amorphous semiconductors based on the previous paper and on the Mott 7) and Cohen, Fritszche and Ovshinsky 8) ( M - C F O ) model for amorphous solids. 2. Model and assumptions The model for the amorphous semiconductor that we will consider here is shown in fig. 1. The trapping levels are distributed throughout the band gap of the semiconductor, such that their density decreases exponentially with energy from the band edges towards the middle of the energy gap, in the spirit of the M - C F O models. For the purposes of calculation, the energy decay length, Eo, of the trap distribution has been chosen to be 0.083 eV, which yields a ratio of 100:1 for the trap densities at the band edges to that at the centre of the gap. The total number of traps in the band gap is 2 x 1019 c m - 3 which is of the order usually reported in the literature. 947
948
J. G. SIMMONS AND G. W. TAYLOR
In the spirit of the CFO model, the "tail states" of the conduction band are assumed to be acceptor-type; that is, neutral when empty and negatively charged when filled. The tail states of the valence band are assumed to be donor-type; that is, neutral when filled and positively charged when empty. The concept of overlapping tail states s) is not essential to the theory presented herein and will not be pursued further. For convenience, we assume here that the equilibrium Fermi level Evo separates the two sets of extended states.
IE-Ei[/Eo ;eV
n-3 (eV)-~
IE = 2 X 1 0 1 9 c m -3
N(EJ
Fig. 1. Energy diagram illustrating the trap distribution used in the computations throughout the text.
Evo has been offset below the centre (El) of the energy gap in accordance with the fact that most amorphous semiconductors are apparently p-type. The incident radiation is assumed to be completely and uniformly absorbed by the semiconductor, with unit quantum efficiency. The products, va. and yap, of the thermal velocity (v) of the carriers at their respective mobility edges and the capture cross sections of the electrons (a,) and the holes (ap) are, in the absence of evidence to the contrary, assumed to be temperature independent. For the purposes of computation, a, and ap are assumed to be 10 -15 c m 2, and v is taken to be 10 7 cm s - L
3. Theoretical equations Here we invoke the results of the previous paper. The equations, derived in the previous paperg), that we make use of here are those numbered (17), (18) and (19).
THEORYOFSTEADYSTATEPHOTOCONDUCTIVITY
949
4. Calculations
4.1. EXCESSCARRIERCONCENTRATIONS Fig. 2 illustrates the excess hole (Ap) and electron (An) concentrations as a function of temperature T for various constant electron-hole pair generation rates (G). The particularly interesting aspects of the A p - T curves is the pronounced maxima exhibited by Ap at, say, Tp. Note that Tp increases with increasing light intensity. It can be shown that this maximum occurs when Ap ,,~ Po
2kT EFO -- Ev
Thus for typical values of k T ( =0.025 eV) and Ev0-Ev( =0.6 eV), Ap is approximately a few percent, typically 10~, of the dark hole concentration, Po. It will be noted that An has a similar dependence on temperature as does Ap, but it is always less than Ap. It is noted here that the disparity between
....... Z~n 10~
"°"*°°*°°°°°°°°°°°o.°°°°°° 10;
°° .
~
2
X 10 6
....
Ap, A n
( cm -3) 106
10~
" ° ~ °
10'
~o
I
3
I
4
I
5
I
6
I
7
I
8
I
9
•
Io3/T ("K-;)
Fig. 2. Theoretical Ap, An versus T characteristics for various G
950
J. G. SIMMONS AND G . W . TAYLOR
Ap and An increases the further LEo is offset from the centre of the energy gap, that is, as E F o - El increases. 4.2. MOBILITY The effect of illuminating the sample with light is to redistribute charge in the energy gap such that electrons are trapped in levels above EFo, giving rise to negatively charged centers, and holes are trapped in levels below EFo, giving rise to positively charged centres (see fig. 2 of the previous paper). Thus, if the photo-excited carriers have mean free paths of several tens of angstroms, then at high temperatures the mobility, /~, is determined by phonon scattering, which is theoretically proportional to T -~. However, at low temperatures the mobility will be determined by scattering by the charged centres, that is, by impurity scattering. Thus, in this case T~ /~ o c - - ,
ss
where Ns is the number of charged centres. Thus the mobility versus temperature characteristic will exhibit a maximum, as shown in fig. 3. 10 2
10
Io3/T
F i g . 3.
{'K -~)
Mobility as a function of temperature when charged center scattering is important.
951
THEORY OF STEADY STATE PHOTOCONDUCTIVITY
On the other hand if the uncertainty, Ak/k, in the wave vector k after a scattering event, is given by Ak/k ~ 1, then the mobility will presumably be relatively temperature insensitive. Whichever is the ease, the mobility will not affect the general features of the photo current versus temperature characteristic, as discussed below. 4.3. PHOTOCURRENT VERSUS TEMPERATURE In calculating the Ip versus T characteristic shown in fig. 4, the effect of charge-centre scattering was included. If the mobility were assumed to be constant in temperature, the resulting I p - T characteristic is qualitatively similar to that shown in fig. 4. The main difference between the two curves is that the maximum in the curve is moved to slightly higher temperatures, and the rate of decrease of photocurrent for T < Tmaxis not as pronounced. It is noted that the theoretical curves shown in fig. 4. manifest all the basic features of the experimental curves 1-6). In particular it is seen that: (a) the maximum in the curves moves to higher temperatures, Tm, with increasing light intensity; (b) the magnitude of the maximum is essentially linearly
X l O 15 X l O 14
E
10~
101
31
41
51 103/T
Fig. 4.
~ ( ' K -~)
71
~
9r
:_
Theoretical lp versus T characteristic for various constant G.
952
J. G. SIMMONS AND G. W . TAYLOR
dependent on G; and (c) it can be shown that the ratio lplI o, where I o is the dark current corresponding to Tm, is given by
lp ,,~ Io
2 k Tm Ero - Ev"
Thus, as a result of (c), for typical values of kTm and E r o - Ev, Ip is usually a few percent ( ~ 10%) of I o at the maximum. Also Ip < Io for T> Tin, and Io < lp for T < Tin. By using a value of 10-15 cm 2 for the electron and hole capture cross sections we can obtain relatively good quantitive agreement with the experimental results 1,2,% We hasten to add, however, that there are essential differences in detail between the experimental and theoretical data, which are due to the idealistic trap distribution (fig. 1) used in the theoretical analysis. A detailed treatment of more realistic trap distributions is given elsewhere10). 4.4.
Ip VERSUS
G
Fig. 5 illustrates the Ip-G characteristic for various constant T. The interesting features of these characteristics are as follows. In the low intensity region, when Ip is linear in G, the magnitude of the photocurrent is higher the lower the temperature. However, at higher light levels, when lp is sublinear in G, Ip is greater the higher the temperature. Also the region in which I, is linear in G corresponds to the condition
I.
10"
220 230 250 300
Ip ( A-cm"
lO-~
lo"
10t'
I
10"
10'~
It
10'~
G F i g . 5.
T h e o r e t i c a l i p versus G f o r v a r i o u s c o n s t a n t T.
THEORYOF STEADYSTATEPHOTOCONDUCTIVITY
953
a n d in the region where Ip is s u b l i n e a r G c o r r e s p o n d s to the c o n d i t i o n lo > Io. T h e similarity b e t w e e n the theoretical characteristics in fig. 4 a n d the experim e n t a l characteristics 1,6) is very g o o d . The r e a s o n for the c o m p r e s s e d n a t u r e o f the theoretical curves c o m p a r e d to those o f the e x p e r i m e n t a l curves is a g a i n a result o f the idealized n a t u r e o f the m o d e l shown in fig. 1.
5. Conclusions The p h o t o c o n d u c t i o n characteristics reflect the t r a p p i n g p a r a m e t e r s in a very sensitive m a n n e r , a n d p o s s i b l y p r o v i d e the best means o f d e t e r m i n i n g the p a r a m e t e r s o f these systems.
References 1) B. T. Kolomiets and V. M. Lyubin, Fiz. Tverd. Tela 2 (1960) 52; English Transl. Soviet Phys.-Solid State 2 (1960) 46. 2) A. E. Fagen and H. Fritzsche, J. Non-Crystalline Solids 4 (1970) 480. 3) J. P. deNeufville, Advan. Res. Proj. Agency Rept., Contract DAH 15-70.C-0187. The authors are indebted to Dr. A. E. Fagen for a copy of this report. 4) W. E. Howard and R. Tsu, Phys. Rev. B 1 (1970) 4709. 5) B. T. Kolomiets, Yu. V. Rukhlyadev and V. P. Shilo, J. Non-Crystalline Solids 5 (1971) 389, 402. 6) K. Weiser, J. Non-Crystalline Solids 8-10 (1972) 922. 7) N. F. Mott, Advan. Phys. 16 (1967) 49. 8) M. Cohen, H. Fritzsche and S. R. Ovshinsky, Phys. Rev. Letters 22 (1969) 1065. 9) G. W. Taylor and J. G. Simmons, J. Non-Crystalline Solids 8-10 (1972) 940. 10) J. G. Simmons and G. W. Taylor, to be published.
T H E O R Y O F STEADY S T A T E P H O T O C O N D U C T I V I T Y I N AMORPHOUS SEMICONDUCTORS J. G. SIMMONS and G. W. TAYLOR Electrical Engineering Department, University of Toronto, Toronto, Canada
The theory of photoconductivity described in the adjoining paper is applied to amorphous semiconductors. It is shown that for constant light intensity the photocurrent, Ip, initially increases with decreasing temperature, reaches a maximum value at some temperature, say, Tin, and thereafter decreases monotonically with further decrease in temperature (T< Tin). The photocurrent at the maximum is a few percent of the dark current, I0, corresponding to Tin. At constant temperatures and low intensities, when lp< Io, the photoconductivity is a linear function of the light intensity. At high light intensity, when Ip > Io, Ip is a sub-linear function of the light intensity. The theory is shown to be in good agreement with experimental observation. 1. Introduction Recently, a good deal of data has been accumulated on the photoconductive properties of amorphous solids, in particular the semiconducting chalcogenide glasses 1-6). No adequate self-consistent explanation, based on the currently accepted model of amorphous semiconductors, has been forthcoming. The object of this paper is to present a general treatment of photoconductivity in amorphous semiconductors based on the previous paper and on the Mott 7) and Cohen, Fritszche and Ovshinsky 8) ( M - C F O ) model for amorphous solids. 2. Model and assumptions The model for the amorphous semiconductor that we will consider here is shown in fig. 1. The trapping levels are distributed throughout the band gap of the semiconductor, such that their density decreases exponentially with energy from the band edges towards the middle of the energy gap, in the spirit of the M - C F O models. For the purposes of calculation, the energy decay length, Eo, of the trap distribution has been chosen to be 0.083 eV, which yields a ratio of 100:1 for the trap densities at the band edges to that at the centre of the gap. The total number of traps in the band gap is 2 x 1019 c m - 3 which is of the order usually reported in the literature. 947
948
J. G. SIMMONS AND G. W. TAYLOR
In the spirit of the CFO model, the "tail states" of the conduction band are assumed to be acceptor-type; that is, neutral when empty and negatively charged when filled. The tail states of the valence band are assumed to be donor-type; that is, neutral when filled and positively charged when empty. The concept of overlapping tail states s) is not essential to the theory presented herein and will not be pursued further. For convenience, we assume here that the equilibrium Fermi level Evo separates the two sets of extended states.
IE-Ei[/Eo ;eV
n-3 (eV)-~
IE = 2 X 1 0 1 9 c m -3
N(EJ
Fig. 1. Energy diagram illustrating the trap distribution used in the computations throughout the text.
Evo has been offset below the centre (El) of the energy gap in accordance with the fact that most amorphous semiconductors are apparently p-type. The incident radiation is assumed to be completely and uniformly absorbed by the semiconductor, with unit quantum efficiency. The products, va. and yap, of the thermal velocity (v) of the carriers at their respective mobility edges and the capture cross sections of the electrons (a,) and the holes (ap) are, in the absence of evidence to the contrary, assumed to be temperature independent. For the purposes of computation, a, and ap are assumed to be 10 -15 c m 2, and v is taken to be 10 7 cm s - L
3. Theoretical equations Here we invoke the results of the previous paper. The equations, derived in the previous paperg), that we make use of here are those numbered (17), (18) and (19).
THEORYOFSTEADYSTATEPHOTOCONDUCTIVITY
949
4. Calculations
4.1. EXCESSCARRIERCONCENTRATIONS Fig. 2 illustrates the excess hole (Ap) and electron (An) concentrations as a function of temperature T for various constant electron-hole pair generation rates (G). The particularly interesting aspects of the A p - T curves is the pronounced maxima exhibited by Ap at, say, Tp. Note that Tp increases with increasing light intensity. It can be shown that this maximum occurs when Ap ,,~ Po
2kT EFO -- Ev
Thus for typical values of k T ( =0.025 eV) and Ev0-Ev( =0.6 eV), Ap is approximately a few percent, typically 10~, of the dark hole concentration, Po. It will be noted that An has a similar dependence on temperature as does Ap, but it is always less than Ap. It is noted here that the disparity between
....... Z~n 10~
"°"*°°*°°°°°°°°°°°o.°°°°°° 10;
°° .
~
2
X 10 6
....
Ap, A n
( cm -3) 106
10~
" ° ~ °
10'
~o
I
3
I
4
I
5
I
6
I
7
I
8
I
9
•
Io3/T ("K-;)
Fig. 2. Theoretical Ap, An versus T characteristics for various G
950
J. G. SIMMONS AND G . W . TAYLOR
Ap and An increases the further LEo is offset from the centre of the energy gap, that is, as E F o - El increases. 4.2. MOBILITY The effect of illuminating the sample with light is to redistribute charge in the energy gap such that electrons are trapped in levels above EFo, giving rise to negatively charged centers, and holes are trapped in levels below EFo, giving rise to positively charged centres (see fig. 2 of the previous paper). Thus, if the photo-excited carriers have mean free paths of several tens of angstroms, then at high temperatures the mobility, /~, is determined by phonon scattering, which is theoretically proportional to T -~. However, at low temperatures the mobility will be determined by scattering by the charged centres, that is, by impurity scattering. Thus, in this case T~ /~ o c - - ,
ss
where Ns is the number of charged centres. Thus the mobility versus temperature characteristic will exhibit a maximum, as shown in fig. 3. 10 2
10
Io3/T
F i g . 3.
{'K -~)
Mobility as a function of temperature when charged center scattering is important.
951
THEORY OF STEADY STATE PHOTOCONDUCTIVITY
On the other hand if the uncertainty, Ak/k, in the wave vector k after a scattering event, is given by Ak/k ~ 1, then the mobility will presumably be relatively temperature insensitive. Whichever is the ease, the mobility will not affect the general features of the photo current versus temperature characteristic, as discussed below. 4.3. PHOTOCURRENT VERSUS TEMPERATURE In calculating the Ip versus T characteristic shown in fig. 4, the effect of charge-centre scattering was included. If the mobility were assumed to be constant in temperature, the resulting I p - T characteristic is qualitatively similar to that shown in fig. 4. The main difference between the two curves is that the maximum in the curve is moved to slightly higher temperatures, and the rate of decrease of photocurrent for T < Tmaxis not as pronounced. It is noted that the theoretical curves shown in fig. 4. manifest all the basic features of the experimental curves 1-6). In particular it is seen that: (a) the maximum in the curves moves to higher temperatures, Tm, with increasing light intensity; (b) the magnitude of the maximum is essentially linearly
X l O 15 X l O 14
E
10~
101
31
41
51 103/T
Fig. 4.
~ ( ' K -~)
71
~
9r
:_
Theoretical lp versus T characteristic for various constant G.
952
J. G. SIMMONS AND G. W . TAYLOR
dependent on G; and (c) it can be shown that the ratio lplI o, where I o is the dark current corresponding to Tm, is given by
lp ,,~ Io
2 k Tm Ero - Ev"
Thus, as a result of (c), for typical values of kTm and E r o - Ev, Ip is usually a few percent ( ~ 10%) of I o at the maximum. Also Ip < Io for T> Tin, and Io < lp for T < Tin. By using a value of 10-15 cm 2 for the electron and hole capture cross sections we can obtain relatively good quantitive agreement with the experimental results 1,2,% We hasten to add, however, that there are essential differences in detail between the experimental and theoretical data, which are due to the idealistic trap distribution (fig. 1) used in the theoretical analysis. A detailed treatment of more realistic trap distributions is given elsewhere10). 4.4.
Ip VERSUS
G
Fig. 5 illustrates the Ip-G characteristic for various constant T. The interesting features of these characteristics are as follows. In the low intensity region, when Ip is linear in G, the magnitude of the photocurrent is higher the lower the temperature. However, at higher light levels, when lp is sublinear in G, Ip is greater the higher the temperature. Also the region in which I, is linear in G corresponds to the condition
I.
10"
220 230 250 300
Ip ( A-cm"
lO-~
lo"
10t'
I
10"
10'~
It
10'~
G F i g . 5.
T h e o r e t i c a l i p versus G f o r v a r i o u s c o n s t a n t T.
THEORYOF STEADYSTATEPHOTOCONDUCTIVITY
953
a n d in the region where Ip is s u b l i n e a r G c o r r e s p o n d s to the c o n d i t i o n lo > Io. T h e similarity b e t w e e n the theoretical characteristics in fig. 4 a n d the experim e n t a l characteristics 1,6) is very g o o d . The r e a s o n for the c o m p r e s s e d n a t u r e o f the theoretical curves c o m p a r e d to those o f the e x p e r i m e n t a l curves is a g a i n a result o f the idealized n a t u r e o f the m o d e l shown in fig. 1.
5. Conclusions The p h o t o c o n d u c t i o n characteristics reflect the t r a p p i n g p a r a m e t e r s in a very sensitive m a n n e r , a n d p o s s i b l y p r o v i d e the best means o f d e t e r m i n i n g the p a r a m e t e r s o f these systems.
References 1) B. T. Kolomiets and V. M. Lyubin, Fiz. Tverd. Tela 2 (1960) 52; English Transl. Soviet Phys.-Solid State 2 (1960) 46. 2) A. E. Fagen and H. Fritzsche, J. Non-Crystalline Solids 4 (1970) 480. 3) J. P. deNeufville, Advan. Res. Proj. Agency Rept., Contract DAH 15-70.C-0187. The authors are indebted to Dr. A. E. Fagen for a copy of this report. 4) W. E. Howard and R. Tsu, Phys. Rev. B 1 (1970) 4709. 5) B. T. Kolomiets, Yu. V. Rukhlyadev and V. P. Shilo, J. Non-Crystalline Solids 5 (1971) 389, 402. 6) K. Weiser, J. Non-Crystalline Solids 8-10 (1972) 922. 7) N. F. Mott, Advan. Phys. 16 (1967) 49. 8) M. Cohen, H. Fritzsche and S. R. Ovshinsky, Phys. Rev. Letters 22 (1969) 1065. 9) G. W. Taylor and J. G. Simmons, J. Non-Crystalline Solids 8-10 (1972) 940. 10) J. G. Simmons and G. W. Taylor, to be published.