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Dispersive transport from negatively correlated defect centers
Journal of Non-Crystalline Solids 57 (1983) 81--89 North-Holland Publishing Company
81
D I S P E R S I V E T R A N S P O R T F R O M NEGATIVELY C O R R E L A T E D DEFECT CENTERS B a b a r A. K H A N a n d David A D L E R Department of Electrical Engineering and Computer Science. and Center for Materials Science and Engineering. Massachusetts Institute of Technology, Cambridge. MA 02139. USA
Received 5 October 1982 Revised manuscript received 14 December 1982
The transient current response arising from time-of-flight experiments on a material whose trapping levels are predominantly associated with a well-defined negatively correlated defect is calculated. The resulting behavior is controlled by the relative magnitude of two characteristic times, the transit time, "rt, and the time necessary for interconversion of a neutral acceptor to a neutral donor, r. If rt < r, nondispersive transport results. In contrast, if rt > r, two kinks characterize l(t). The current is initially independent of time, but eventually becomes dispersive, decaying as t k,2 prior to rt and as t -3/2 afterwards. The dispersion results from the spread of times necessary to interconvert the neutral defects.
I. Introduction For m a n y years, it has been clear that the results of time-of-flight ( T O F ) experiments on a large class of chalcogenide glasses c a n n o t be u n d e r s t o o d with the a s s u m p t i o n that free carriers propagate as G a u s s i a n wave packets [1]. Scher a n d Montroll [2] applied a t i m e - d e p e n d e n t r a n d o m - w a l k model in which carriers moved via a m e c a h n i s m with a dispersion of rates, e.g. h o p p i n g through localized states with an array of spatial separations. This model successfully explained m a n y features of the observed transport. The time-dep e n d e n t r a n d o m - w a l k approach was also explicitly applied to transport via multiple t r a p p i n g a n d release involving a d i s t r i b u t i o n of traps [3-5], and the results were used to explain the transport data in a m o r p h o u s selenium [4,5]. N o o l a n d i [6] showed that, in fact, the h o p p i n g and multiple transport models are m a t h e m a t i c a l l y equivalent. I n both models, the current transients have a well-defined transit time, ~'t, which can be identified from the fact that the current decreases as l ( t ) ~ t -I~-~ for t < r,, but c o n s i d e r a b l y faster, as I ( t ) - t -~1 +~), for t > r t. The p a r a m e t e r c~ can take on values from 0 to 1, d e p e n d i n g o n the m e c h a n i s m a n d the physical conditions. Recently, Tiedje a n d Rose [7] a n d i n d e p e n d e n t l y Orenstein a n d K a s t n e r [8] d e m o n s t r a t e d that multiple t r a p p i n g involving an e x p o n e n t i a l d i s t r i b u t i o n of traps can account for b o t h T O F a n d transient p h o t o c o n d u c t i v 0 0 2 2 - 3 0 9 3 / 8 3 / 0 0 0 0 - 0 0 0 0 / $ 0 3 . 0 0 © 1983 N o r t h - H o l l a n d
82
B.A. Khan, D. Adler / Dispersive transport
ity (TP) measurements in a simple manner. In both cases, c~ -- T / T o, where To is the parameter measuring the fall-off in the trap distribution, but in TP experiments the critical time at which the behavior of I ( t ) changes is the recombination time rather than the transit time [9]. In chalcogenide glasses, it is likely that the predominant trapping levels result from positively and negatively charged defects called valence alternation pairs (VAPs) [10]. A spatial distribution of such pairs can be expected in many amorphous chalcogenides, and these can provide the experimental distribution of traps necessary for dispersive transport [11]. A major characteristic of VAPs is the negative effective correlation energy, Ueff, which arises because of the interconvertibility of neutral centers by either a bond breaking or formation. These neutral defects form when photogenerated carriers are trapped by the positively and negatively charged centers in either T O F or TP experiments. Frye and Adler [12] were able to explain their field-effect results on glasses in the T e - A s system by a detailed analysis of the kinetics of neutral-defect interconversion. They found that a potential barrier retards the interconversion at low temperatures and leads to an effective unpinning of the quasi-Fermi energies at relatively short times. It is this unpinning which results in the transient field effect observed in several chalcogenide glasses. When a potential barrier exists between two interconvertible defects characterized by a negative Ueff, the kinetics of carrier trapping become complex even if the charged defects are not spatially distributed. Each interconversion changes a neutral donor to a neutral acceptor or vice versa: consequently, it is possible e.g. for a trapped electron to be released as a free hole. It is thus clear that defect interconversion can introduce a dispersion into the kinetics even for well-defined trap levels. In this paper, we calculate the current response in a T O F experiment for the case of a material whose predominant defect is not spatially correlated but has a negative U~ff with a barrier retarding defect interconversion. We find that such a situation, expected to apply to glasses in the T e - A s system, provides a third mechanism for dispersive transport, in addition to hopping and multiple trapping by a distribution of traps. This new mechanism, however, has some unique characteristics which should enable an experimental determination of its applicability.
2. Analysis of time-of-flight measurements We assume that a single well-defined defect with a negative Uefr predominates. The band structure is sketched in fig. 1, where N D and NA are the respective concentrations of donor and acceptor levels, U is the magnitude of Ueff, and W is the difference in energy between the neutral acceptor and donor centers. We take ~- as the relaxation time governing the interconversion of an acceptor to a donor. It is useful to note that the problem can be quickly analyzed in two limits, namely when ~- is either very small or very large. For small ~-, neutral defect
B.A. Khan, D. Adler / Dispersive transport
IEo
83 Fc
ND
I E D = U/2
-
-
-
Ev
-fE-
Fig. 1. Band d i a g r a m for a s e m i c o n d u c t o r c o n t a i n i n g a total c o n c e n t r a t i o n of N D + N A interconvertible defects c h a r a c t e r i z e d by a n e g a t i v e Uef f.
interconversion is essentially instantaneous. In this case, Uda and Yamada [13] have shown that nondispersive transport results. In the opposite limit, when ~is very large, the electron and hole systems are essentially decoupled and the problem reduces to that of a single, well-defined trap level, which is also nondispersive. In general, ~" cannot be assumed to be either very large or very small compared with the time scale of the measurements [12]. However, it is reasonable to take ~" large compared with trapping time, which is ordinarily of the order of a inverse phonon frequency, - 10-~2s [9]. This enables us to infer that the free carriers reach a quasi-equilibrium with the charged defects, and we can then write the free-electron concentration, n and the free-hole concentration, p, as n = a,,ND°
(1)
p = aeU ° ,
(2)
and
where N ° and N° are the respective concentrations of neutral donors and acceptors, a,,- exp(-E,,/kT),
(3)
ap - exp( - E p / k T ).
(4)
and
E,, and Ep are defined in fig. 1. The rate equations relating the neutral and charged defect concentrations then can be written [12]: ddt ( U ~ + U ° ) = - ~ d (NA_+NO) =--Y1( N ° -
) a---O-AN° D ,
0¢D
(5)
where Nt~ and N~ are the concentrations of charged donors and acceptors,
B.A. Khan, D. Adler / Dispersive transport
84
respectively, a A - exp( - E a / k V ),
(6)
a D -= exp( - E D / k T ) .
(7)
and
In eqs. (6) and (7), E A = U / 2 and E o = U / 2 + W (see fig. 1). Substitution of eqs. (1) and (2) into eq. (5) yields: dN~
1 p
an n
1 dn
dt
r Otp
aDr a,,
Or,, d t
(8)
and dN£
lot A n
1 p
dt
r OtD Otn
"r O%,
1 dp
(9)
Ote d t "
But carrier transport is controlled by the relations: dP_ dt
dNA dt
_.~
dn dt
dNi~ dn d--~ +/*eF~xx '
#hF
(10)
and (11)
where F is field applied in the x direction and/.t h and #~ are the free-hole and free-electron mobilities, respectively, assumed for simplicity to be approximately equal. Restricting our discussion to hole transport, we can take the Laplace transform with respect to t of eqs. (8)-(11), obtaining: d2/gq
dx 2
OtA [- - IffDotnT F [ 1 2 "~ [P'hF OtA k ap #hF-aoa.r
I't2F2otDotn'r t
-(
OtA
O/DOtnT
1 +1 Otp
OtA
) '( +1
1 a,,
~
1+--
OtpT
(
s + p.h E aDa" aAa p
S- -~
1
dx
aA
S /~=0.
(12)
OLn
Since eqs. (3), (4), (6), and (7) show OtDa" = e x p [ - ( E D + E , -
OtAotp
E a-
(13)
Ep)/kT],
then when the Fermi energy, E v, is near mid-gap,
(aDan/otAa
p -- 1)
is small.
85
B.A. Khan, D. Adler / Dispersive transport
and eq. (12) reduces to: -
d2/~+ 1 ( 1 dx 2 ~ c~p
1 ) d/~ a,, dx
-
S
~F2OtDa,,r
ooo{oA( 1) ,(,
+
O/A
- -
O/DO/nT
1+--
O~p
+--
O~pT
aA
}
1+--
ap
s p=0.
O/n
a,, }
(14)
If we take the local photogeneration rate, g(x,t), to be:
then the solution of eq. (14) is just:
(15)
p(x) = n exp(-Ax), where A - as + (bs 2 + cs) 1/2, with:
,)
a~
b
~
O~p O/n
m
,
,(,,244)
2/~2F:
--S - 2+ -O~n~p + - + O~p - + O/n 4 O/p + O~
,
20~A ( 10~DO~n O~D ) la2F2aD~,, r 1 + -Otp - + OLAOIp + OCAOlp . Since the Laplace transform of the current is given by:
I(s) = e#hF foLdX where L is the length of the sample, eq. (15) yields:
I(s)
e ~ h F ( 1 -- e x p ( - A L ) AL )"
(16)
Eq. (16) can be numerically inverted to obtain the transient photocurrent. However, some physical insight can be obtained from an evaluation of the behavior at long times. In the limit of small s, eq. (16) can be approximated as:
,(s)--(eBt~hF/L)(l-exp[-(cs)'/2L])/[as+(cs)'/2
I.
(17)
Inverting eq. (12), we obtain: I ( t ) = ( e ~ # h F / a L ) e x p ( c t / a : ) { e r f c [ ( c t )'/2/a] - e x p ( c t / a z )erfc[(ct )l/2/a
+
c'/ZL/2t I/z ]).
(18)
86
B.A. Khan, D. Adler / Dispersive transport
'° F.,
4
=24x
I(t}
eI -6 I0
I -5 I0
I -4 I0
I -3 I0
I -2 I0
I -I I0
I I
t(secs) Fig. 2. C u r r e n t as a function of time for a s a m p l e with r << r t at several different t e m p e r a t u r e s ,
Since in the limit of large z, exp(z 2) erfc z cc z ~, eq. (18) indicates:
I(T) oc
{ tt~ '/2 3/2
( t < cL2/4), (t > cL2/4).
Such a variation in current is consistent with dispersive transport for a = and the transit time is then given by [2]
¢, = eL2~ 4.
(19)
1/2, (20)
Thus, the experimental value of rt can be used to determine c. In addition, for the case E A > ED, E, > Ep, we can obtain a and b from the relations: b = (~/,~°)
c
and a = -
(b/2)
'/2.
In order to analyze the general solution, eq. (16) was also inverted numerically. Fig. 2 shows the resulting behavior of I(t) for r t = cL2/4 = 24 ms and L = 0.0075 cm, using several different values for %r/a,,. The results are straightforward to interpret. I(t) is initially nondispersive, since the problem is essentially that of transport controlled by a single trap level. For t > r, however, the possibility of trap interconversion leads to dispersive transport with a = 0.5. 3. Discussion Since we did not consider the possibility of carrier recombination, the time dependence of the current is due only to free-carrier extraction at the contacts
B.A. Khan, D. Adler / Dispersive transport
87
2~
I0
I0I F-
z
w oE o£
,62
L~
I
-5
10 1(~6 id5 i64 tG3 i0-2 io-~ I Time
I (secs)
IO ~
F i g . 3. Current as a function of time for a sample with "r >> "rt at two different temperatures.
and defect interconversion. If ~'t > ~', we should thus expect I(t) to be constant for t < ~', and drop quickly near ~-,, as is characteristic of nondispersive transport. A s is shown in fig. 3, this is indeed the case. However, for t >> ~-, the trap interconversion transforms the hole current to electron current and leads to a dispersive behavior, in agreement with the results of Uda and Yamada [13]. In contrast, for 1"< ~t, the sudden drop at ~t does not occur, as is clear from fig. 2. In this case, the temperature dependence manifests itself primarily through the magnitude of the initial (constant) current and the length of time
I(t)
2 IO --
~ ~ ~
t T =O.17ms
0~ ~_
~-- 0.17 ms ,o°
~ 8.9 ms
z W n~ i(~I nr"
L~
,02
~
,64 l
I
I
l
24
ms
I
I
,~6 165 ,6" ,d3 1# io-' ~o ,o'
{secs)
Time F i g . 4. Current response as a function of time for a sample with "rap/a n << 10 6 s for several different values of ~',.
88
B.A. Khan, D. Adler / Dispersive transport
b e f o r e the onset of dispersion. However, a is not t e m p e r a t u r e - d e p e n d e n t as in m u l t i p l e - t r a p p i n g m o d e l s [7-9]. N o t e that b o t h fig. 2 and fig. 3 yield current transients with two kinks. If b o t h of these kinks were to be i n t e r p r e t e d as transient times, it might a p p e a r that two types of drift m o b i l i t y exist, a fast a n d a slow transport. In fact, j u s t such an o b s e r v a t i o n has been recently m a d e by T a k a h a s h i [14] on Se:Bi a n d S e - A s - T e - B i glasses. It should be b o r n e in m i n d that the timescale of the e x p e r i m e n t can p r e c l u d e o b s e r v a t i o n of some of the range of p r e d i c t e d behavior. F o r example, if ~" is sufficiently short that the first kink in the curves of fig. 2 are too fast to be observable, c u r r e n t - t r a n s i e n t s such as those sketched in fig. 4 w o u l d result. Such b e h a v i o r is very similar to the dispersive t r a n s p o r t o r d i n a r i l y o b s e r v e d in a large class of m a t e r i a l s [1].
4. Conclusion W e have c a l c u l a t e d the transient currents arising from time-of-flight experim e n t s in a system for which the traps are p r e d o m i n a n t l y well-defined valence a l t e r n a t i o n pairs. T w o characteristic times then control the resulting behavior, the transit time, ~'t a n d the time, ~-, for the interconversion of a neutral a c c e p t o r to a donor. If ~-< ~'t, the response is initially c o n s t a n t in time, but b e c o m e s dispersive with a = 0.5 once the interconversion times are exceeded. Two kinks in l ( t ) plots should be evident. If ~" > ~'t, the current is nondispersive over a wide range of times, a n d typical G a u s s i a n behavior, i.e. a c o n s t a n t current followed b y a steep decline at ~'t, results. Some d i s p e r s i o n m a y be o b s e r v a b l e at long times. In a n y event, such a material can exhibit b o t h dispersive a n d n o n d i s p e r s i v e t r a n s p o r t , d e p e n d i n g on s a m p l e length a n d t e m p e r a t u r e .
This research was s u p p o r t e d b y the N a t i o n a l Science F o u n d a t i o n M a t e r i a l s R e s e a r c h L a b o r a t o r y G r a n t No. 81-19295.
References [1] [2] [3] [4] [5] [6] [7] [8]
G. Pfister and H. Scher, Adv. Phys. 27 (1978) 747. H. Scher and E.W. Montroll, Phys. Rev. BI2 (1975) 2455. F.W. Schmidlin, Phys. Rev. BI6 (1977) 2362. M. Silver and L. Cohen, Phys. Rev. B I5 (1976) 3276. J. Noolandi, Phys. Rev. BI6 (1977) 4466. J. Noolandi, Phys. Rev. B16 (1977) 4474. T. Tiedje and A. Rose, Solid State Commun. 37 (1981) 49. J. Orenstein and M. Kastner, Phys. Rev. Lett. 46 (1981) 1421.
B.A. Khan, D. Adler / Dispersive transport [9] J. Orenstein, M. Kastner and V. Vanivov, Phil. Mag. B46 (1982) 23. [10] M. Kastner, D. Adler, and H. Fritzsche, Phys. Rev. Lett. 37 (1976) 1504; see also R.A. Street and N.F. Mort, Phys. Rev. Lett. 35 (1975) 1293. [11] D. Adler, J. Physique 42 (1981) C4-3. [12] R.C. Frye and D. Adler, Phys. Rev. B24 (1981) 5812. [13] T. Uda and E. Yamada, J. Non-Crystalline Solids 35-36 (1980) 105. [14] T. Takahashi, J. Non-Crystalline Solids 44 (1981) 239.
89
81
D I S P E R S I V E T R A N S P O R T F R O M NEGATIVELY C O R R E L A T E D DEFECT CENTERS B a b a r A. K H A N a n d David A D L E R Department of Electrical Engineering and Computer Science. and Center for Materials Science and Engineering. Massachusetts Institute of Technology, Cambridge. MA 02139. USA
Received 5 October 1982 Revised manuscript received 14 December 1982
The transient current response arising from time-of-flight experiments on a material whose trapping levels are predominantly associated with a well-defined negatively correlated defect is calculated. The resulting behavior is controlled by the relative magnitude of two characteristic times, the transit time, "rt, and the time necessary for interconversion of a neutral acceptor to a neutral donor, r. If rt < r, nondispersive transport results. In contrast, if rt > r, two kinks characterize l(t). The current is initially independent of time, but eventually becomes dispersive, decaying as t k,2 prior to rt and as t -3/2 afterwards. The dispersion results from the spread of times necessary to interconvert the neutral defects.
I. Introduction For m a n y years, it has been clear that the results of time-of-flight ( T O F ) experiments on a large class of chalcogenide glasses c a n n o t be u n d e r s t o o d with the a s s u m p t i o n that free carriers propagate as G a u s s i a n wave packets [1]. Scher a n d Montroll [2] applied a t i m e - d e p e n d e n t r a n d o m - w a l k model in which carriers moved via a m e c a h n i s m with a dispersion of rates, e.g. h o p p i n g through localized states with an array of spatial separations. This model successfully explained m a n y features of the observed transport. The time-dep e n d e n t r a n d o m - w a l k approach was also explicitly applied to transport via multiple t r a p p i n g a n d release involving a d i s t r i b u t i o n of traps [3-5], and the results were used to explain the transport data in a m o r p h o u s selenium [4,5]. N o o l a n d i [6] showed that, in fact, the h o p p i n g and multiple transport models are m a t h e m a t i c a l l y equivalent. I n both models, the current transients have a well-defined transit time, ~'t, which can be identified from the fact that the current decreases as l ( t ) ~ t -I~-~ for t < r,, but c o n s i d e r a b l y faster, as I ( t ) - t -~1 +~), for t > r t. The p a r a m e t e r c~ can take on values from 0 to 1, d e p e n d i n g o n the m e c h a n i s m a n d the physical conditions. Recently, Tiedje a n d Rose [7] a n d i n d e p e n d e n t l y Orenstein a n d K a s t n e r [8] d e m o n s t r a t e d that multiple t r a p p i n g involving an e x p o n e n t i a l d i s t r i b u t i o n of traps can account for b o t h T O F a n d transient p h o t o c o n d u c t i v 0 0 2 2 - 3 0 9 3 / 8 3 / 0 0 0 0 - 0 0 0 0 / $ 0 3 . 0 0 © 1983 N o r t h - H o l l a n d
82
B.A. Khan, D. Adler / Dispersive transport
ity (TP) measurements in a simple manner. In both cases, c~ -- T / T o, where To is the parameter measuring the fall-off in the trap distribution, but in TP experiments the critical time at which the behavior of I ( t ) changes is the recombination time rather than the transit time [9]. In chalcogenide glasses, it is likely that the predominant trapping levels result from positively and negatively charged defects called valence alternation pairs (VAPs) [10]. A spatial distribution of such pairs can be expected in many amorphous chalcogenides, and these can provide the experimental distribution of traps necessary for dispersive transport [11]. A major characteristic of VAPs is the negative effective correlation energy, Ueff, which arises because of the interconvertibility of neutral centers by either a bond breaking or formation. These neutral defects form when photogenerated carriers are trapped by the positively and negatively charged centers in either T O F or TP experiments. Frye and Adler [12] were able to explain their field-effect results on glasses in the T e - A s system by a detailed analysis of the kinetics of neutral-defect interconversion. They found that a potential barrier retards the interconversion at low temperatures and leads to an effective unpinning of the quasi-Fermi energies at relatively short times. It is this unpinning which results in the transient field effect observed in several chalcogenide glasses. When a potential barrier exists between two interconvertible defects characterized by a negative Ueff, the kinetics of carrier trapping become complex even if the charged defects are not spatially distributed. Each interconversion changes a neutral donor to a neutral acceptor or vice versa: consequently, it is possible e.g. for a trapped electron to be released as a free hole. It is thus clear that defect interconversion can introduce a dispersion into the kinetics even for well-defined trap levels. In this paper, we calculate the current response in a T O F experiment for the case of a material whose predominant defect is not spatially correlated but has a negative U~ff with a barrier retarding defect interconversion. We find that such a situation, expected to apply to glasses in the T e - A s system, provides a third mechanism for dispersive transport, in addition to hopping and multiple trapping by a distribution of traps. This new mechanism, however, has some unique characteristics which should enable an experimental determination of its applicability.
2. Analysis of time-of-flight measurements We assume that a single well-defined defect with a negative Uefr predominates. The band structure is sketched in fig. 1, where N D and NA are the respective concentrations of donor and acceptor levels, U is the magnitude of Ueff, and W is the difference in energy between the neutral acceptor and donor centers. We take ~- as the relaxation time governing the interconversion of an acceptor to a donor. It is useful to note that the problem can be quickly analyzed in two limits, namely when ~- is either very small or very large. For small ~-, neutral defect
B.A. Khan, D. Adler / Dispersive transport
IEo
83 Fc
ND
I E D = U/2
-
-
-
Ev
-fE-
Fig. 1. Band d i a g r a m for a s e m i c o n d u c t o r c o n t a i n i n g a total c o n c e n t r a t i o n of N D + N A interconvertible defects c h a r a c t e r i z e d by a n e g a t i v e Uef f.
interconversion is essentially instantaneous. In this case, Uda and Yamada [13] have shown that nondispersive transport results. In the opposite limit, when ~is very large, the electron and hole systems are essentially decoupled and the problem reduces to that of a single, well-defined trap level, which is also nondispersive. In general, ~" cannot be assumed to be either very large or very small compared with the time scale of the measurements [12]. However, it is reasonable to take ~" large compared with trapping time, which is ordinarily of the order of a inverse phonon frequency, - 10-~2s [9]. This enables us to infer that the free carriers reach a quasi-equilibrium with the charged defects, and we can then write the free-electron concentration, n and the free-hole concentration, p, as n = a,,ND°
(1)
p = aeU ° ,
(2)
and
where N ° and N° are the respective concentrations of neutral donors and acceptors, a,,- exp(-E,,/kT),
(3)
ap - exp( - E p / k T ).
(4)
and
E,, and Ep are defined in fig. 1. The rate equations relating the neutral and charged defect concentrations then can be written [12]: ddt ( U ~ + U ° ) = - ~ d (NA_+NO) =--Y1( N ° -
) a---O-AN° D ,
0¢D
(5)
where Nt~ and N~ are the concentrations of charged donors and acceptors,
B.A. Khan, D. Adler / Dispersive transport
84
respectively, a A - exp( - E a / k V ),
(6)
a D -= exp( - E D / k T ) .
(7)
and
In eqs. (6) and (7), E A = U / 2 and E o = U / 2 + W (see fig. 1). Substitution of eqs. (1) and (2) into eq. (5) yields: dN~
1 p
an n
1 dn
dt
r Otp
aDr a,,
Or,, d t
(8)
and dN£
lot A n
1 p
dt
r OtD Otn
"r O%,
1 dp
(9)
Ote d t "
But carrier transport is controlled by the relations: dP_ dt
dNA dt
_.~
dn dt
dNi~ dn d--~ +/*eF~xx '
#hF
(10)
and (11)
where F is field applied in the x direction and/.t h and #~ are the free-hole and free-electron mobilities, respectively, assumed for simplicity to be approximately equal. Restricting our discussion to hole transport, we can take the Laplace transform with respect to t of eqs. (8)-(11), obtaining: d2/gq
dx 2
OtA [- - IffDotnT F [ 1 2 "~ [P'hF OtA k ap #hF-aoa.r
I't2F2otDotn'r t
-(
OtA
O/DOtnT
1 +1 Otp
OtA
) '( +1
1 a,,
~
1+--
OtpT
(
s + p.h E aDa" aAa p
S- -~
1
dx
aA
S /~=0.
(12)
OLn
Since eqs. (3), (4), (6), and (7) show OtDa" = e x p [ - ( E D + E , -
OtAotp
E a-
(13)
Ep)/kT],
then when the Fermi energy, E v, is near mid-gap,
(aDan/otAa
p -- 1)
is small.
85
B.A. Khan, D. Adler / Dispersive transport
and eq. (12) reduces to: -
d2/~+ 1 ( 1 dx 2 ~ c~p
1 ) d/~ a,, dx
-
S
~F2OtDa,,r
ooo{oA( 1) ,(,
+
O/A
- -
O/DO/nT
1+--
O~p
+--
O~pT
aA
}
1+--
ap
s p=0.
O/n
a,, }
(14)
If we take the local photogeneration rate, g(x,t), to be:
then the solution of eq. (14) is just:
(15)
p(x) = n exp(-Ax), where A - as + (bs 2 + cs) 1/2, with:
,)
a~
b
~
O~p O/n
m
,
,(,,244)
2/~2F:
--S - 2+ -O~n~p + - + O~p - + O/n 4 O/p + O~
,
20~A ( 10~DO~n O~D ) la2F2aD~,, r 1 + -Otp - + OLAOIp + OCAOlp . Since the Laplace transform of the current is given by:
I(s) = e#hF foLdX where L is the length of the sample, eq. (15) yields:
I(s)
e ~ h F ( 1 -- e x p ( - A L ) AL )"
(16)
Eq. (16) can be numerically inverted to obtain the transient photocurrent. However, some physical insight can be obtained from an evaluation of the behavior at long times. In the limit of small s, eq. (16) can be approximated as:
,(s)--(eBt~hF/L)(l-exp[-(cs)'/2L])/[as+(cs)'/2
I.
(17)
Inverting eq. (12), we obtain: I ( t ) = ( e ~ # h F / a L ) e x p ( c t / a : ) { e r f c [ ( c t )'/2/a] - e x p ( c t / a z )erfc[(ct )l/2/a
+
c'/ZL/2t I/z ]).
(18)
86
B.A. Khan, D. Adler / Dispersive transport
'° F.,
4
=24x
I(t}
eI -6 I0
I -5 I0
I -4 I0
I -3 I0
I -2 I0
I -I I0
I I
t(secs) Fig. 2. C u r r e n t as a function of time for a s a m p l e with r << r t at several different t e m p e r a t u r e s ,
Since in the limit of large z, exp(z 2) erfc z cc z ~, eq. (18) indicates:
I(T) oc
{ tt~ '/2 3/2
( t < cL2/4), (t > cL2/4).
Such a variation in current is consistent with dispersive transport for a = and the transit time is then given by [2]
¢, = eL2~ 4.
(19)
1/2, (20)
Thus, the experimental value of rt can be used to determine c. In addition, for the case E A > ED, E, > Ep, we can obtain a and b from the relations: b = (~/,~°)
c
and a = -
(b/2)
'/2.
In order to analyze the general solution, eq. (16) was also inverted numerically. Fig. 2 shows the resulting behavior of I(t) for r t = cL2/4 = 24 ms and L = 0.0075 cm, using several different values for %r/a,,. The results are straightforward to interpret. I(t) is initially nondispersive, since the problem is essentially that of transport controlled by a single trap level. For t > r, however, the possibility of trap interconversion leads to dispersive transport with a = 0.5. 3. Discussion Since we did not consider the possibility of carrier recombination, the time dependence of the current is due only to free-carrier extraction at the contacts
B.A. Khan, D. Adler / Dispersive transport
87
2~
I0
I0I F-
z
w oE o£
,62
L~
I
-5
10 1(~6 id5 i64 tG3 i0-2 io-~ I Time
I (secs)
IO ~
F i g . 3. Current as a function of time for a sample with "r >> "rt at two different temperatures.
and defect interconversion. If ~'t > ~', we should thus expect I(t) to be constant for t < ~', and drop quickly near ~-,, as is characteristic of nondispersive transport. A s is shown in fig. 3, this is indeed the case. However, for t >> ~-, the trap interconversion transforms the hole current to electron current and leads to a dispersive behavior, in agreement with the results of Uda and Yamada [13]. In contrast, for 1"< ~t, the sudden drop at ~t does not occur, as is clear from fig. 2. In this case, the temperature dependence manifests itself primarily through the magnitude of the initial (constant) current and the length of time
I(t)
2 IO --
~ ~ ~
t T =O.17ms
0~ ~_
~-- 0.17 ms ,o°
~ 8.9 ms
z W n~ i(~I nr"
L~
,02
~
,64 l
I
I
l
24
ms
I
I
,~6 165 ,6" ,d3 1# io-' ~o ,o'
{secs)
Time F i g . 4. Current response as a function of time for a sample with "rap/a n << 10 6 s for several different values of ~',.
88
B.A. Khan, D. Adler / Dispersive transport
b e f o r e the onset of dispersion. However, a is not t e m p e r a t u r e - d e p e n d e n t as in m u l t i p l e - t r a p p i n g m o d e l s [7-9]. N o t e that b o t h fig. 2 and fig. 3 yield current transients with two kinks. If b o t h of these kinks were to be i n t e r p r e t e d as transient times, it might a p p e a r that two types of drift m o b i l i t y exist, a fast a n d a slow transport. In fact, j u s t such an o b s e r v a t i o n has been recently m a d e by T a k a h a s h i [14] on Se:Bi a n d S e - A s - T e - B i glasses. It should be b o r n e in m i n d that the timescale of the e x p e r i m e n t can p r e c l u d e o b s e r v a t i o n of some of the range of p r e d i c t e d behavior. F o r example, if ~" is sufficiently short that the first kink in the curves of fig. 2 are too fast to be observable, c u r r e n t - t r a n s i e n t s such as those sketched in fig. 4 w o u l d result. Such b e h a v i o r is very similar to the dispersive t r a n s p o r t o r d i n a r i l y o b s e r v e d in a large class of m a t e r i a l s [1].
4. Conclusion W e have c a l c u l a t e d the transient currents arising from time-of-flight experim e n t s in a system for which the traps are p r e d o m i n a n t l y well-defined valence a l t e r n a t i o n pairs. T w o characteristic times then control the resulting behavior, the transit time, ~'t a n d the time, ~-, for the interconversion of a neutral a c c e p t o r to a donor. If ~-< ~'t, the response is initially c o n s t a n t in time, but b e c o m e s dispersive with a = 0.5 once the interconversion times are exceeded. Two kinks in l ( t ) plots should be evident. If ~" > ~'t, the current is nondispersive over a wide range of times, a n d typical G a u s s i a n behavior, i.e. a c o n s t a n t current followed b y a steep decline at ~'t, results. Some d i s p e r s i o n m a y be o b s e r v a b l e at long times. In a n y event, such a material can exhibit b o t h dispersive a n d n o n d i s p e r s i v e t r a n s p o r t , d e p e n d i n g on s a m p l e length a n d t e m p e r a t u r e .
This research was s u p p o r t e d b y the N a t i o n a l Science F o u n d a t i o n M a t e r i a l s R e s e a r c h L a b o r a t o r y G r a n t No. 81-19295.
References [1] [2] [3] [4] [5] [6] [7] [8]
G. Pfister and H. Scher, Adv. Phys. 27 (1978) 747. H. Scher and E.W. Montroll, Phys. Rev. BI2 (1975) 2455. F.W. Schmidlin, Phys. Rev. BI6 (1977) 2362. M. Silver and L. Cohen, Phys. Rev. B I5 (1976) 3276. J. Noolandi, Phys. Rev. BI6 (1977) 4466. J. Noolandi, Phys. Rev. B16 (1977) 4474. T. Tiedje and A. Rose, Solid State Commun. 37 (1981) 49. J. Orenstein and M. Kastner, Phys. Rev. Lett. 46 (1981) 1421.
B.A. Khan, D. Adler / Dispersive transport [9] J. Orenstein, M. Kastner and V. Vanivov, Phil. Mag. B46 (1982) 23. [10] M. Kastner, D. Adler, and H. Fritzsche, Phys. Rev. Lett. 37 (1976) 1504; see also R.A. Street and N.F. Mort, Phys. Rev. Lett. 35 (1975) 1293. [11] D. Adler, J. Physique 42 (1981) C4-3. [12] R.C. Frye and D. Adler, Phys. Rev. B24 (1981) 5812. [13] T. Uda and E. Yamada, J. Non-Crystalline Solids 35-36 (1980) 105. [14] T. Takahashi, J. Non-Crystalline Solids 44 (1981) 239.
89