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Dispersive band tail relaxation in amorphous semiconductors
Journal of Non-CrystaUineSolids 77 & 78 (1985) 163-166 North-Holland, Amsterdam
163
DISPERSIVE BAND TAIL RELAXATION IN AMORPHOUSSEMICONDUCTORS M. GRONEWALD, B. MOVAGHAR*, B. POHLMANN**, D. WORTZ*** Fachbereich Physik, University of Marburg, F.R. Germany * GEC H i r s t Research Centre, London, Great B r i t a i n * * Fachbereich Physik, Gesamthochschule Essen, F.R. Germany * * * I n s t i t u t f~r Theoretische Physik, University of Heidelberg, F.R. Germany Within the novel non-equilibrium theory of incoherent motion we study the current and energy r e l a x a t i o n of excited charge c a r r i e r s as observed in t r a n s i e n t photoconductivity experiments. The theory is applied to an exponential band t a i l model commonly used f o r a-As2Se 3 and a-Si:H. The t y p i c a l temperature dependence of the decay parameter can be explained in the whole temperature regime and the mean energy loss as a function of i n i t i a l energy is calculated. MOTIVATION AND INTRODUCTION The nature and shape of the band t a i l states in amorphous semiconductors like a-Si:H and a-As2Se3 are of great interest in both theoretical and experimental work.
Here the measurementof current transients is very useful and has
been studied by Kastner and coworkers as well as others I.
In these time of
f l i g h t (TOF) or transient photocurrent (TPC)-measurements, one records the time evolution of an i n i t i a l l y nonequilibrium carrier d i s t r i b u t i o n , which has been generated through excitation by l i g h t or electric f i e l d .
Then the dispersive,
e.g. non-Gaussian time dependenceof the transient signal reflects the d i s t r i bution of the t a i l states. The (until now) commonexplanation has been given in the framework of the Multiple Trapping (MT) model2. shortcomings.
Nevertheless this successful model has some
One ]imitation is that only energetical disorder is considered.
But at low density, positional disorder is also important and gives a broad distribution of relaxation times due to the exponential dependenceof the overlap integral.
Additionally there is direct evidence for the importance of
diffusion in localized band t a i l states (at least in low defect material) which has been inferred by LESR and very recently by IR-excitation of trapped carriers in a-Si:H. To describe and explain these facts properly, we propose a non-equilibrium theory of incoherent diffusive motion within an ensembleof localized states, This work has been supported in part (M. GrOnewald) by the Deutsche Forschungsgemeinschaft. 0022-3093/85/$03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
M. Grfinewald et al. / Dispersive band tail relaxation
164
characterized by energetical and p o s i t i o n disorder.
The formalism has been
developed elsewhere3 and the aim of the present paper is to apply t h i s theory to the c a l c u l a t i o n of current and energy decay in exponential band t a i l s ,
which
are t y p i c a l f o r amorphous semiconductors. MODEL AND RESULTS The t h e o r e t i c a l d e s c r i p t i o n is based on the e q u i l i b r i u m hopping theory of Movaghar and coworkers 4, which t r e a t s the random walk of an e x c i t a t i o n in a disordered system of localized states with asymmetric energy-dependent hopping rates.
For a charged c a r r i e r we assume the f o l l o w i n g rate f o r a t r a n s i -
t i o n from s i t e i to j : Wii~ = v o exp(-2~ r i j
- BMax(O,Ej-Ei)) with B : I/kT the
inverse temperature, ~ inverse l o c a l i z a t i o n radius and ( r i , E i ) the p o s i t i o n and energy of the p a r t i c l e at s i t e i . The theory has been extended to the non-equilibrium s i t u a t i o n a f t e r pulsed initial
e x c i t a t i o n 3.
The c a l c u l a t i o n of the time dependence of both the d i f -
fusion c o e f f i c i e n t D(t) and the energy r e l a x a t i o n E(t) are possible. initial
s t a r t i n g configuration may be f i x e d a r b i t r a r i l y
luation of these q u a n t i t i e s f o r d i f f e r e n t i n i t i a l complished.
As the
in the theory, the eva-
energies may be e a s i l y ac-
This is of great importance f o r exciton d i f f u s i o n in various o r -
ganic m a t e r i a l s .
A second "input" of the theory is the density of states (DOS).
In order to compare our results with the commonly used MT-model, we adopt the widespread exponential band t a i l model p(¢) : exp(¢/¢o), ¢ < O.
But the cur-
rent and energy decay can be calculated f o r any model function.
Results f o r
a full
Gaussian-DOS (as r e a l i z e d in organic glasses) are presented elsewhere3.
In Fig. I D(t) is p l o t t e d against time t f o r various temperatures T (with Eo = kTo slope of the band t a i l ) .
The algebraic power law decay over a large
time regime (more than 10 decades), which has been observed experimentally in amorphous chalcogenides and a-Si:H I is evident.
The dispersive nature of the
t r a n s i e n t s f o r T < To is an u l t i m a t i v e consequence of the inherent disorder present in amorphous semiconductors and therefore can be used to estimate the magnitude of t h i s d i s o r d e r . Of special i n t e r e s t in t h i s context is the temperature dependence of the dispersion parameter ~ (D(t) - t -s, s = I - ~). r e l a t i o n is predicted: ~ = T/T o .
Within the MT-model a simple
A b r i e f summary of the corresponding results
of our hopping theory are given in Fig. 2 together with a comparison to the experimental material a v a i l a b l e from the l i t e r a t u r e I .
The f u l l points are the
results from the theory, the open ones are experimental values f o r hole t r a n sients in a-As2Se 3, a-Si:H and electrons in a-Si:H.
The theory is in good
agreement with the experiments in the e n t i r e temperature regime.
At high T,
M. Grfinewald et al. / Dispersive band tail relaxation
165
Iogvot
)o
5 i
15
h
i
l.O
s=l-ot SK -5-
0.5
£3
_
-10
~
~
T°=52LK
3
3
K
"xx~xx ~ \b
-15
0.1 155K
0.I
116K
0.5 T/To
1.0
FIGURE 2 Photocond. decay parameter s = I vs. normalized temperature T/T o .
" 58K
exp. values FIGURE I Time dependence o f d i f f u s i o n c o e f f i c i e n t D(t) f o r various temperatures T (T O slope of the e x p o n e n t i a l ban ~ t a i l ) . S i t e d e n s i t y n = 6.4.101~cm~ a, wav~ f u n c t i o n decay parameter ~-I = 10-~cm.
theory
To = 550 K
o
holes in a-As2Se3 and a-Si:H
i
To = 310 K
[]
e l e c t r o n s in a-Si:H (from r e f . I)
•
a = T/T o and the hopping t h e o r y e x h i b i t s the same temperature dependence as one o b t a i n s from the MT-model ( s o l i d l i n e ) .
At low temperatures (T/T o ~ 0 . 3 ) ,
the decay is steeper than p r e d i c t e d from MT and the l i m i t i n g ready reached at f i n i t e
T = 0.2 To .
n o n - e q u i l i b r i u m hopping t r a n s p o r t ( i . e . p o r t path.
slope s = I is a l -
We t h e r e f o r e can conclude t h a t at low T r e l a x a t i o n ) becomes the dominant t r a n s -
But even at higher T < To our t h e o r y f u l l y
accounts f o r the e x p e r i -
mental m a t e r i a l and is in agreement with the MT-model, Further i n s i g h t i n t o the r e l a x a t i o n process can be gained by i n s p e c t i n g the energy d i s s i p a t i o n process.
We t h e r e f o r e have c a l c u l a t e d E ( c , t ) ,
energy of a c a r r i e r at time t when s t a r t e d at energy ~ at t = O.
the mean In Fig. 3
E(t) = f de p(~)E(E,t) is plotted against log t for various T, assuming uniform excitation at t = 0 (the same applies to Fig. I ) .
The curves can nearly be
parametrised by E(t) = - Esln t / t o with Es = kT in the considered temperature regime.
We therefore again can conclude that MT-model and hopping theory agree
at high T whereas at lower T, the hopping model predicts a faster relaxation than MT. Fig. 4 exhibits the mean energy for different times as a function of the i n i t i a l (starting) energy and therefore shows the demarcation energy between m o b i l i z e d and immobilized states at a given time. l i n e E(t) = c are s t i l l
A l l s t a t e s on the diagonal
immobilized, the others e x h i b i t e n e r g e t i c a l r e l a x a t i o n
by a given amount ( h o r i z o n t a l
l i n e s ) at d i f f e r e n t
time and are in e q u i l i b r i u m
M. Gr~newald et al. / Dispersive band tail relaxation
166
0
--Iogvot 5i I0 i
0-
15 i
!-
-0.1-
Eo: 0,045 eV
log Vot
-0.I>
-o.3.
-0.3-0.5.
II..U
-0.5 -0.7-
-0.7 -
291K 233K
FIGURE 3 Time e v o l u t i o n of the mean energy f o r d i f f e r e n t temperatures. Same DOSmodel as in Fig. I . (Go : kTo)
with each other.
-07
-o'5
-0'.3
- - E/eV-----
-o'i
FIGURE 4 Energy dependence of the mean energy E(G,t) f o r d i f f e r e n t times t . Same DOS-model as in Fig. I . (Go = kTo)
There seems to be direct relevance of these calculations to
the exciton diffusion problem in GaAsquantum well structures 5.
The excita-
tions are "frozen out", simulating a kind of mobility edge. Further applications and extensions of the theory are: Inclusion of deep traps, e.g. recombination, an exact analysis of the T = 0 situation, resulting in new, even slower decay laws, a detailed investigation of the energy relaxat i o n in experiment, computer simulation and theory, the analysis of sJnglet exciton motion with dipolar (FOrster) interaction. REFERENCES I) D. Monroe, J. Orenstein, and M. Kastner, J. de Physique 42, Suppl. C 559 (198]); D. Monroe, Phys.Rev.Lett. 54 (1985) 146; J.M. Hvamand M.H. Brodsky: J. de Physique 42, Suppl. (1981) C551; T. Tiedje, in: Semiconductors and Semimetals VoI. 21C (Ed. J.I. Pankove), Springer, Berlin (1984) 2) J. Orenstein and M. Kastner, Phys.Rev.Lett. 46 (1981) 1421; T. Tiedje and A. Rose, Solid State Comm. 37 (1980) 49 3) M. GrOnewald, B. Movaghar, B. Pohlmann, and D. W~rtz, Phi1.Mag. B 49 (1984) 341; J. of Non-Cryst.Solids 59/60 (1983) 49; to be published in Phys.Rev. B (1985)
4) B. Movaghar, M. Gr~newald, B. Pohlmann, W. Schirmacher, and D. WOrtz, J. Stat.Phys. 30 (1983) 315 5) J. Hegarty, L. Goldner, and M.D. Sturge, Phys.Rev. B 30 (1984) 7346
163
DISPERSIVE BAND TAIL RELAXATION IN AMORPHOUSSEMICONDUCTORS M. GRONEWALD, B. MOVAGHAR*, B. POHLMANN**, D. WORTZ*** Fachbereich Physik, University of Marburg, F.R. Germany * GEC H i r s t Research Centre, London, Great B r i t a i n * * Fachbereich Physik, Gesamthochschule Essen, F.R. Germany * * * I n s t i t u t f~r Theoretische Physik, University of Heidelberg, F.R. Germany Within the novel non-equilibrium theory of incoherent motion we study the current and energy r e l a x a t i o n of excited charge c a r r i e r s as observed in t r a n s i e n t photoconductivity experiments. The theory is applied to an exponential band t a i l model commonly used f o r a-As2Se 3 and a-Si:H. The t y p i c a l temperature dependence of the decay parameter can be explained in the whole temperature regime and the mean energy loss as a function of i n i t i a l energy is calculated. MOTIVATION AND INTRODUCTION The nature and shape of the band t a i l states in amorphous semiconductors like a-Si:H and a-As2Se3 are of great interest in both theoretical and experimental work.
Here the measurementof current transients is very useful and has
been studied by Kastner and coworkers as well as others I.
In these time of
f l i g h t (TOF) or transient photocurrent (TPC)-measurements, one records the time evolution of an i n i t i a l l y nonequilibrium carrier d i s t r i b u t i o n , which has been generated through excitation by l i g h t or electric f i e l d .
Then the dispersive,
e.g. non-Gaussian time dependenceof the transient signal reflects the d i s t r i bution of the t a i l states. The (until now) commonexplanation has been given in the framework of the Multiple Trapping (MT) model2. shortcomings.
Nevertheless this successful model has some
One ]imitation is that only energetical disorder is considered.
But at low density, positional disorder is also important and gives a broad distribution of relaxation times due to the exponential dependenceof the overlap integral.
Additionally there is direct evidence for the importance of
diffusion in localized band t a i l states (at least in low defect material) which has been inferred by LESR and very recently by IR-excitation of trapped carriers in a-Si:H. To describe and explain these facts properly, we propose a non-equilibrium theory of incoherent diffusive motion within an ensembleof localized states, This work has been supported in part (M. GrOnewald) by the Deutsche Forschungsgemeinschaft. 0022-3093/85/$03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
M. Grfinewald et al. / Dispersive band tail relaxation
164
characterized by energetical and p o s i t i o n disorder.
The formalism has been
developed elsewhere3 and the aim of the present paper is to apply t h i s theory to the c a l c u l a t i o n of current and energy decay in exponential band t a i l s ,
which
are t y p i c a l f o r amorphous semiconductors. MODEL AND RESULTS The t h e o r e t i c a l d e s c r i p t i o n is based on the e q u i l i b r i u m hopping theory of Movaghar and coworkers 4, which t r e a t s the random walk of an e x c i t a t i o n in a disordered system of localized states with asymmetric energy-dependent hopping rates.
For a charged c a r r i e r we assume the f o l l o w i n g rate f o r a t r a n s i -
t i o n from s i t e i to j : Wii~ = v o exp(-2~ r i j
- BMax(O,Ej-Ei)) with B : I/kT the
inverse temperature, ~ inverse l o c a l i z a t i o n radius and ( r i , E i ) the p o s i t i o n and energy of the p a r t i c l e at s i t e i . The theory has been extended to the non-equilibrium s i t u a t i o n a f t e r pulsed initial
e x c i t a t i o n 3.
The c a l c u l a t i o n of the time dependence of both the d i f -
fusion c o e f f i c i e n t D(t) and the energy r e l a x a t i o n E(t) are possible. initial
s t a r t i n g configuration may be f i x e d a r b i t r a r i l y
luation of these q u a n t i t i e s f o r d i f f e r e n t i n i t i a l complished.
As the
in the theory, the eva-
energies may be e a s i l y ac-
This is of great importance f o r exciton d i f f u s i o n in various o r -
ganic m a t e r i a l s .
A second "input" of the theory is the density of states (DOS).
In order to compare our results with the commonly used MT-model, we adopt the widespread exponential band t a i l model p(¢) : exp(¢/¢o), ¢ < O.
But the cur-
rent and energy decay can be calculated f o r any model function.
Results f o r
a full
Gaussian-DOS (as r e a l i z e d in organic glasses) are presented elsewhere3.
In Fig. I D(t) is p l o t t e d against time t f o r various temperatures T (with Eo = kTo slope of the band t a i l ) .
The algebraic power law decay over a large
time regime (more than 10 decades), which has been observed experimentally in amorphous chalcogenides and a-Si:H I is evident.
The dispersive nature of the
t r a n s i e n t s f o r T < To is an u l t i m a t i v e consequence of the inherent disorder present in amorphous semiconductors and therefore can be used to estimate the magnitude of t h i s d i s o r d e r . Of special i n t e r e s t in t h i s context is the temperature dependence of the dispersion parameter ~ (D(t) - t -s, s = I - ~). r e l a t i o n is predicted: ~ = T/T o .
Within the MT-model a simple
A b r i e f summary of the corresponding results
of our hopping theory are given in Fig. 2 together with a comparison to the experimental material a v a i l a b l e from the l i t e r a t u r e I .
The f u l l points are the
results from the theory, the open ones are experimental values f o r hole t r a n sients in a-As2Se 3, a-Si:H and electrons in a-Si:H.
The theory is in good
agreement with the experiments in the e n t i r e temperature regime.
At high T,
M. Grfinewald et al. / Dispersive band tail relaxation
165
Iogvot
)o
5 i
15
h
i
l.O
s=l-ot SK -5-
0.5
£3
_
-10
~
~
T°=52LK
3
3
K
"xx~xx ~ \b
-15
0.1 155K
0.I
116K
0.5 T/To
1.0
FIGURE 2 Photocond. decay parameter s = I vs. normalized temperature T/T o .
" 58K
exp. values FIGURE I Time dependence o f d i f f u s i o n c o e f f i c i e n t D(t) f o r various temperatures T (T O slope of the e x p o n e n t i a l ban ~ t a i l ) . S i t e d e n s i t y n = 6.4.101~cm~ a, wav~ f u n c t i o n decay parameter ~-I = 10-~cm.
theory
To = 550 K
o
holes in a-As2Se3 and a-Si:H
i
To = 310 K
[]
e l e c t r o n s in a-Si:H (from r e f . I)
•
a = T/T o and the hopping t h e o r y e x h i b i t s the same temperature dependence as one o b t a i n s from the MT-model ( s o l i d l i n e ) .
At low temperatures (T/T o ~ 0 . 3 ) ,
the decay is steeper than p r e d i c t e d from MT and the l i m i t i n g ready reached at f i n i t e
T = 0.2 To .
n o n - e q u i l i b r i u m hopping t r a n s p o r t ( i . e . p o r t path.
slope s = I is a l -
We t h e r e f o r e can conclude t h a t at low T r e l a x a t i o n ) becomes the dominant t r a n s -
But even at higher T < To our t h e o r y f u l l y
accounts f o r the e x p e r i -
mental m a t e r i a l and is in agreement with the MT-model, Further i n s i g h t i n t o the r e l a x a t i o n process can be gained by i n s p e c t i n g the energy d i s s i p a t i o n process.
We t h e r e f o r e have c a l c u l a t e d E ( c , t ) ,
energy of a c a r r i e r at time t when s t a r t e d at energy ~ at t = O.
the mean In Fig. 3
E(t) = f de p(~)E(E,t) is plotted against log t for various T, assuming uniform excitation at t = 0 (the same applies to Fig. I ) .
The curves can nearly be
parametrised by E(t) = - Esln t / t o with Es = kT in the considered temperature regime.
We therefore again can conclude that MT-model and hopping theory agree
at high T whereas at lower T, the hopping model predicts a faster relaxation than MT. Fig. 4 exhibits the mean energy for different times as a function of the i n i t i a l (starting) energy and therefore shows the demarcation energy between m o b i l i z e d and immobilized states at a given time. l i n e E(t) = c are s t i l l
A l l s t a t e s on the diagonal
immobilized, the others e x h i b i t e n e r g e t i c a l r e l a x a t i o n
by a given amount ( h o r i z o n t a l
l i n e s ) at d i f f e r e n t
time and are in e q u i l i b r i u m
M. Gr~newald et al. / Dispersive band tail relaxation
166
0
--Iogvot 5i I0 i
0-
15 i
!-
-0.1-
Eo: 0,045 eV
log Vot
-0.I>
-o.3.
-0.3-0.5.
II..U
-0.5 -0.7-
-0.7 -
291K 233K
FIGURE 3 Time e v o l u t i o n of the mean energy f o r d i f f e r e n t temperatures. Same DOSmodel as in Fig. I . (Go : kTo)
with each other.
-07
-o'5
-0'.3
- - E/eV-----
-o'i
FIGURE 4 Energy dependence of the mean energy E(G,t) f o r d i f f e r e n t times t . Same DOS-model as in Fig. I . (Go = kTo)
There seems to be direct relevance of these calculations to
the exciton diffusion problem in GaAsquantum well structures 5.
The excita-
tions are "frozen out", simulating a kind of mobility edge. Further applications and extensions of the theory are: Inclusion of deep traps, e.g. recombination, an exact analysis of the T = 0 situation, resulting in new, even slower decay laws, a detailed investigation of the energy relaxat i o n in experiment, computer simulation and theory, the analysis of sJnglet exciton motion with dipolar (FOrster) interaction. REFERENCES I) D. Monroe, J. Orenstein, and M. Kastner, J. de Physique 42, Suppl. C 559 (198]); D. Monroe, Phys.Rev.Lett. 54 (1985) 146; J.M. Hvamand M.H. Brodsky: J. de Physique 42, Suppl. (1981) C551; T. Tiedje, in: Semiconductors and Semimetals VoI. 21C (Ed. J.I. Pankove), Springer, Berlin (1984) 2) J. Orenstein and M. Kastner, Phys.Rev.Lett. 46 (1981) 1421; T. Tiedje and A. Rose, Solid State Comm. 37 (1980) 49 3) M. GrOnewald, B. Movaghar, B. Pohlmann, and D. W~rtz, Phi1.Mag. B 49 (1984) 341; J. of Non-Cryst.Solids 59/60 (1983) 49; to be published in Phys.Rev. B (1985)
4) B. Movaghar, M. Gr~newald, B. Pohlmann, W. Schirmacher, and D. WOrtz, J. Stat.Phys. 30 (1983) 315 5) J. Hegarty, L. Goldner, and M.D. Sturge, Phys.Rev. B 30 (1984) 7346