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Einstein's relationship for hopping electrons
ELSEVIER
Journal of Non-Crystalline Solids 198-200 (1996) 214-217
Einstein’s relationship for hopping electrons S.D. Baranovskii
a3* , T. Faber a, F. Hensel a, P. Thomas ‘, G.J. Adriaenssens
’ FB Physikalische Chemie und Zentrum ftir Materialwissensrhaften h FB Physik und Zentrum ftir Materialwissenschaften ’ Laboratorium
l’oor Halfgeleiderfwica,
Katholieke
der Philipps-Unil:ersitiit
der Philipps-Urlir:ersitiit lJnil:ersiteit Leuoen,
Marburg, D-35032 Marburg, Germany
Marburg,
Celestijnenlaan
’
D-35032
Marburg,
2OOD. B-3001
Germany
Lewen,
Belgium
Abstract Photoconductivity in amorphous semiconductors at very low temperatures is temperature-independent, being determined by the energy-loss hopping of carriers through localized band-tail states. In such a hopping relaxation, neither diffusion coefficient, D, nor mobility of carriers, k, depend on temperature and the conventional form of the Einstein’s relationship p = eD/kT is not valid. The relationship between p and D for the hopping relaxation of electrons in the exponential band tail is calculated and it is shown that it has the form p a eD/&,,, where E(, is the energy scale of the exponential band tail.
1. Introduction Study of transient and steady-state photocurrents has become one of the most powerful tools in the investigation of transport properties of disordered materials such as amorphous semiconductors, polycrystalline materials, polymers, etc. At low temperatures transport in these materials is dominated by hopping of electrons between localized states. In order to analyze the hopping transport theoretically, one usually calculates first a diffusion coefficient, D, of the hopping electrons and then expresses the carrier mobility via Einstein’s relationship p=&D,
(1)
where e is the elementary charge, k is the Boltzmann constant, and T is the temperature. This procedure [1,2] has become already routine. Moreover,
* Corresponding author. Tel.: +49-6421 285 582; fax: +496421 288 916; e-mail: [email protected]. 0022-3093/96/$15.00 Copyright 0022-3093(95)00685-O
SD1
this procedure is often used not only for transport in the quasi thermal equilibrium, but for non-equilibrium conditions as well, e.g., to calculate photocurrent. It has even been used for transport at very low temperatures [3,4] at which photoconductivity is governed by the energy-loss hopping of electrons in their relaxation through the band-tail states [5,6]. However, in the energy-loss hopping neither mobility, p, nor diffusion coefficient, D, depend on temperature and the relationship in Eq. (1) cannot be valid. Moreover, using Eq. (1) for the mobility with temperature-independent diffusion coefficient, D, leads to the result [3,4] that p increases infinitely with decreasing T. This result cannot be correct, of course. At low T in the non-equilibrium conditions mobility and diffusion coefficient, D, are not related via Eq. (1). It is the aim of this paper to derive the relationship between JL and D for the energy-loss hopping transport of electrons through localized band-tail states. We carry out such a derivation below for the linear transport regime with respect to the applied electric field, i.e., for the case in which field-independent mobility can be introduced.
0 1996 Elsevier Science B.V. All rights reserved.
S.D. Baranouskii et al. / Journal oj’hion-Crystalline Solids198-200 (19961 214-217
2. Diffusion
coefficient
for energy-loss
hopping
at
T=O Let us consider non-equilibrium electrons in the localized tail states of the conduction band. Being interested here in the relationship between p and D only, we do not take recombination into account, so the only process which can happen with an electron is its hop downward in energy (upward hops are not possible at T = 0) to a nearest localized state in the tail. If the distance to this state is R, the rate of such a hop is V= vC,exp( -2R/a),
(2)
where (Y is the localization length of localized states and prefactor, v~), is of the order of the phonon frequency. We can assume that CYdoes not depend on the energy of a state in the tail without loosing generality of the final result as will be seen below. If the spatial distribution of localized tail states is isotropic, the probability to find the nearest neighbor is also isotropic in the absence of the external field and the process of the hopping energy relaxation of electrons is somewhat similar to the diffusion in space. However. the median length of a hop (the distance R to the nearest neighbor available), as well as the median time, r = vP ‘( R), of a hop (see Eq. (2)) increase in the course of relaxation, because the hopping process brings electrons deeper into the tail. Nevertheless one can ascribe a diffusion coefficient to such a process [7] D(R)=+v(R)R’.
(3)
Here V( R)R’ replaces the product of the ‘mean free path’ R and ‘velocity’ 11= Ru( R) and the coefficient l/6 can be found, e.g., in Refs. [2,4]. According to Eqs. (3) and (2), this diffusion coefficient decreases exponentially with increasing R.
energy shape of the density of localized states, g( 6 1, in the tail should be taken into account [51. Let us assume for simplicity that g(c) decreases exponentially into the mobility gap
where energy, F, is measured positive from the mobility edge (E = 0) towards the gap center; N,, is the total concentration of localized tail states and F() is the tailing parameter. Let us consider an electron in a localized tail state at energy, E. If the external electric field with a strength, F, is applied along direction x, the concentration of tail states available for a hop of this electron at T= 0 (i.e., those, which have energies deeper in the tail than E) is [S] N(x)
=N(&)(l
mobility
for the energy-loss
+eFx/s,),
(5)
where N(F)
=lXg(E)dE=Ni,exp tl
-5 i
1
(6)
It was assumed in the derivation of Eq. (5) that eFx> a, which we assume to be valid. Let us calculate the average projection (x) on the field direction of the vector r from the initial state at energy E to the available nearest neighbor among sites with concentration N(x) determined by Eq. (5). Introducing spherical coordinates with the angle 0 between r and x-axis, we obtain
X
3. Electron ,
115
J
Id rr3
cos
@N( r cos 0)
hopping Xelp[--/llgoi~d@sin@
In order to calculate the mobility of electrons in their energy-loss hopping relaxation under the influence of the external field, one should take into account the spatial asymmetry of the hopping process due to the field [5,6]. To estimate this, the
X
rdr’ r”N( r’cos 0)
/0
Substituting
1 Eq. (5) for N(r cos /3>, calculating
(7) the
216
SD. Baranovskii
integrals in Eq. (7) and omitting (eN-‘/3(&)F/&,,>2, we obtain
(x>=
eFN( E)-2’3 3E 0
et al. /Journal
second-order
of Non-Crystalline
terms
r(5/3) (4rr/3)2’3
(8)
:
where r(5/3) = 0.903 is the value of the gammafunction T(x) at x = 5/3 and N(E) is determined by Eq. (6). Eq. (8) gives the average displacement in the field direction of an electron that hops from a state at energy E to the nearest available neighbor in the exponential tail. The average length, ( r), of such a hop is (r)
=~Xr4nN(a)r2exp(-tnN(gjr’)dr = ($rP”3N-‘/3(
E)T($).
(9)
One can ascribe to the hopping process the mobility U P=-= =
(X>V(<~>)
F
eN-*I”( 8) v( (r))
and the diffusion
r(5/3) (10)
(4*/3)2’3 coefficient
(see Eq. (3))
D=i(r)2v((r)) =~~-2/3(~)~((~))
r2(4i3)
(4rr/3)2’3
.
(11)
Expressions (10) and (11) lead to the relationship between p and D of the form 2r(5/3) CL= r2(4,3)
e :,D
e = 2.3
(1996) 214-217
[5] for the density of the steady-state photocurrent and relates it to the diffusion coefficient introduced by Shklovskii et al. [7], one obtains the expression for p, which differs from Eq. (12) by a numerical coefficient only. Relationship between p and D, almost identical to that in Eq. (12), can be found in the paper of Ritter et al. [8]. We would like to emphasize, however, the difference between their result and our Eq. (12). Our relationship between p and D is derived for the energy-loss hopping of electrons at T = 0, i.e., when temperature does not influence the relaxation at all. Ritter et al. [S] obtained their result for the case of thermal quasi-equilibrium, i.e., when the energy distribution of electrons is described by the Fermi-Dirac distribution function f(s) with some quasi-Fermi-level. They derive their result on the basis of the Einstein’s relationship from the textbook of Smith [9] -pn=eD
F
3%
Solids 198-200
= dE)p / -CC
df( &) d&
dE’
(13)
where n is the total concentration of electrons. We cannot agree with their result due to the following reason. Eq. (13) is valid for the case in which electrons at different energy levels posses equal kinetic coefficients p and D [9]. This condition is obviously not valid for hopping transport or relaxation in the exponential band tails. Indeed, for the energy-loss hopping considered above, R in Eq. (2) is of the order of NP “3(~>. Then, using Eqs. (2), (3) and (6), one obtains the diffusion coefficient for electrons at energy, 6:
(12)
This formula replaces the Einstein’s relationship (Eq. (1)) in the case of the energy-loss hopping of electrons in the exponential band tail.
4. Discussion The derivation of (x) in Eqs. (7) and (8) has been carried out here in the spirit of Shklovskii et al. [5], who did not discuss the mobility and its relation to the diffusion coefficient, though. However, if one extracts p from the expression of Shklovskii et al.
(14) This expression shows that D(s) has a very sharp double-exponential dependence on E and cannot be taken out of the integrand in Eq. (13). It is worth noting that we have derived Eq. (12) under the assumption that eF( x ) -X E,,. According to Eq. (81, the quantity (x) is proportional to N-2/3(~) = N;2/3 exp(2&/3s,), i.e., it increases exponentially in the course of the relaxation toward larger localization energies E. It means that the application of Eq. (12) is restricted by shallow states in the tail. The energy border for our theory depends
S.D. Baranouskii et al./Journal
of Non-Cystalline Solids198-200 (1996) 214-217
the magnitude of F, of course. The smaller is F, the deeper in the tail is this border. For very deep states, where the condition e(x) F << co is not fulfilled, it is not possible to introduce a field-independent mobility and non-linear effects with respect to the applied electric field play the decisive role [7].
on
Acknowledgements S.D.B. is thankful to B.I. Shklovskii, who first brought attention [lo] to the Einstein’s relationship in the form close to Eq. (12). Financial support by the Deutsche Forschungsgemeinschaft through the Sonderforschungsbereich 383 is gratefully acknowledged.
117
References 111M.H. Cohen, J. Non-Cryst. Solids 2 (1970) 432. 121N.F. Mott and E.A. Davis, Electronic Processes
in NonCrystalline Materials (Clarendon, Oxford, 197 1) p. 44. [31 N.V. Zykov, Sov. Phys. Semicond. 22 (1988) 1328. [41 C.E. Nehel and R.A. Street, Philos. Mag. B67 (1993) 407. [51 B.I. Shklovskii, H. Fritzsche and S.D. Baranovskii, Phys. Rev. Lett. 62 (1989) 2989. [61 S.D. Baranovskii, H. Fritzsche, E.I. Levin, I.M. Ruzin and B.I. Shklovskii, Sov. Phys. JETP 69 (1989) 773. [71 B.I. Shklovskii, E.I. Levin, H. Fritzsche and S.D. Baranovskii, Advances in Disordered Semiconductors 3, ed. H. Fritzsche (World Scientific, Singapore, 1990) p. 161. [81D. Ritter, E. Zeldov and K. Weiser, Phys. Rev. B38 (1988) 8296. (Cambridge University, 1978) [91 R.A. Smith, Semiconductors p. 172. [lOI B.I. Shklovskii, private communication (1989).
Journal of Non-Crystalline Solids 198-200 (1996) 214-217
Einstein’s relationship for hopping electrons S.D. Baranovskii
a3* , T. Faber a, F. Hensel a, P. Thomas ‘, G.J. Adriaenssens
’ FB Physikalische Chemie und Zentrum ftir Materialwissensrhaften h FB Physik und Zentrum ftir Materialwissenschaften ’ Laboratorium
l’oor Halfgeleiderfwica,
Katholieke
der Philipps-Unil:ersitiit
der Philipps-Urlir:ersitiit lJnil:ersiteit Leuoen,
Marburg, D-35032 Marburg, Germany
Marburg,
Celestijnenlaan
’
D-35032
Marburg,
2OOD. B-3001
Germany
Lewen,
Belgium
Abstract Photoconductivity in amorphous semiconductors at very low temperatures is temperature-independent, being determined by the energy-loss hopping of carriers through localized band-tail states. In such a hopping relaxation, neither diffusion coefficient, D, nor mobility of carriers, k, depend on temperature and the conventional form of the Einstein’s relationship p = eD/kT is not valid. The relationship between p and D for the hopping relaxation of electrons in the exponential band tail is calculated and it is shown that it has the form p a eD/&,,, where E(, is the energy scale of the exponential band tail.
1. Introduction Study of transient and steady-state photocurrents has become one of the most powerful tools in the investigation of transport properties of disordered materials such as amorphous semiconductors, polycrystalline materials, polymers, etc. At low temperatures transport in these materials is dominated by hopping of electrons between localized states. In order to analyze the hopping transport theoretically, one usually calculates first a diffusion coefficient, D, of the hopping electrons and then expresses the carrier mobility via Einstein’s relationship p=&D,
(1)
where e is the elementary charge, k is the Boltzmann constant, and T is the temperature. This procedure [1,2] has become already routine. Moreover,
* Corresponding author. Tel.: +49-6421 285 582; fax: +496421 288 916; e-mail: [email protected]. 0022-3093/96/$15.00 Copyright 0022-3093(95)00685-O
SD1
this procedure is often used not only for transport in the quasi thermal equilibrium, but for non-equilibrium conditions as well, e.g., to calculate photocurrent. It has even been used for transport at very low temperatures [3,4] at which photoconductivity is governed by the energy-loss hopping of electrons in their relaxation through the band-tail states [5,6]. However, in the energy-loss hopping neither mobility, p, nor diffusion coefficient, D, depend on temperature and the relationship in Eq. (1) cannot be valid. Moreover, using Eq. (1) for the mobility with temperature-independent diffusion coefficient, D, leads to the result [3,4] that p increases infinitely with decreasing T. This result cannot be correct, of course. At low T in the non-equilibrium conditions mobility and diffusion coefficient, D, are not related via Eq. (1). It is the aim of this paper to derive the relationship between JL and D for the energy-loss hopping transport of electrons through localized band-tail states. We carry out such a derivation below for the linear transport regime with respect to the applied electric field, i.e., for the case in which field-independent mobility can be introduced.
0 1996 Elsevier Science B.V. All rights reserved.
S.D. Baranouskii et al. / Journal oj’hion-Crystalline Solids198-200 (19961 214-217
2. Diffusion
coefficient
for energy-loss
hopping
at
T=O Let us consider non-equilibrium electrons in the localized tail states of the conduction band. Being interested here in the relationship between p and D only, we do not take recombination into account, so the only process which can happen with an electron is its hop downward in energy (upward hops are not possible at T = 0) to a nearest localized state in the tail. If the distance to this state is R, the rate of such a hop is V= vC,exp( -2R/a),
(2)
where (Y is the localization length of localized states and prefactor, v~), is of the order of the phonon frequency. We can assume that CYdoes not depend on the energy of a state in the tail without loosing generality of the final result as will be seen below. If the spatial distribution of localized tail states is isotropic, the probability to find the nearest neighbor is also isotropic in the absence of the external field and the process of the hopping energy relaxation of electrons is somewhat similar to the diffusion in space. However. the median length of a hop (the distance R to the nearest neighbor available), as well as the median time, r = vP ‘( R), of a hop (see Eq. (2)) increase in the course of relaxation, because the hopping process brings electrons deeper into the tail. Nevertheless one can ascribe a diffusion coefficient to such a process [7] D(R)=+v(R)R’.
(3)
Here V( R)R’ replaces the product of the ‘mean free path’ R and ‘velocity’ 11= Ru( R) and the coefficient l/6 can be found, e.g., in Refs. [2,4]. According to Eqs. (3) and (2), this diffusion coefficient decreases exponentially with increasing R.
energy shape of the density of localized states, g( 6 1, in the tail should be taken into account [51. Let us assume for simplicity that g(c) decreases exponentially into the mobility gap
where energy, F, is measured positive from the mobility edge (E = 0) towards the gap center; N,, is the total concentration of localized tail states and F() is the tailing parameter. Let us consider an electron in a localized tail state at energy, E. If the external electric field with a strength, F, is applied along direction x, the concentration of tail states available for a hop of this electron at T= 0 (i.e., those, which have energies deeper in the tail than E) is [S] N(x)
=N(&)(l
mobility
for the energy-loss
+eFx/s,),
(5)
where N(F)
=lXg(E)dE=Ni,exp tl
-5 i
1
(6)
It was assumed in the derivation of Eq. (5) that eFx
X
3. Electron ,
115
J
Id rr3
cos
@N( r cos 0)
hopping Xelp[--/llgoi~d@sin@
In order to calculate the mobility of electrons in their energy-loss hopping relaxation under the influence of the external field, one should take into account the spatial asymmetry of the hopping process due to the field [5,6]. To estimate this, the
X
rdr’ r”N( r’cos 0)
/0
Substituting
1 Eq. (5) for N(r cos /3>, calculating
(7) the
216
SD. Baranovskii
integrals in Eq. (7) and omitting (eN-‘/3(&)F/&,,>2, we obtain
(x>=
eFN( E)-2’3 3E 0
et al. /Journal
second-order
of Non-Crystalline
terms
r(5/3) (4rr/3)2’3
(8)
:
where r(5/3) = 0.903 is the value of the gammafunction T(x) at x = 5/3 and N(E) is determined by Eq. (6). Eq. (8) gives the average displacement in the field direction of an electron that hops from a state at energy E to the nearest available neighbor in the exponential tail. The average length, ( r), of such a hop is (r)
=~Xr4nN(a)r2exp(-tnN(gjr’)dr = ($rP”3N-‘/3(
E)T($).
(9)
One can ascribe to the hopping process the mobility U P=-= =
(X>V(<~>)
F
eN-*I”( 8) v( (r))
and the diffusion
r(5/3) (10)
(4*/3)2’3 coefficient
(see Eq. (3))
D=i(r)2v((r)) =~~-2/3(~)~((~))
r2(4i3)
(4rr/3)2’3
.
(11)
Expressions (10) and (11) lead to the relationship between p and D of the form 2r(5/3) CL= r2(4,3)
e :,D
e = 2.3
(1996) 214-217
[5] for the density of the steady-state photocurrent and relates it to the diffusion coefficient introduced by Shklovskii et al. [7], one obtains the expression for p, which differs from Eq. (12) by a numerical coefficient only. Relationship between p and D, almost identical to that in Eq. (12), can be found in the paper of Ritter et al. [8]. We would like to emphasize, however, the difference between their result and our Eq. (12). Our relationship between p and D is derived for the energy-loss hopping of electrons at T = 0, i.e., when temperature does not influence the relaxation at all. Ritter et al. [S] obtained their result for the case of thermal quasi-equilibrium, i.e., when the energy distribution of electrons is described by the Fermi-Dirac distribution function f(s) with some quasi-Fermi-level. They derive their result on the basis of the Einstein’s relationship from the textbook of Smith [9] -pn=eD
F
3%
Solids 198-200
= dE)p / -CC
df( &) d&
dE’
(13)
where n is the total concentration of electrons. We cannot agree with their result due to the following reason. Eq. (13) is valid for the case in which electrons at different energy levels posses equal kinetic coefficients p and D [9]. This condition is obviously not valid for hopping transport or relaxation in the exponential band tails. Indeed, for the energy-loss hopping considered above, R in Eq. (2) is of the order of NP “3(~>. Then, using Eqs. (2), (3) and (6), one obtains the diffusion coefficient for electrons at energy, 6:
(12)
This formula replaces the Einstein’s relationship (Eq. (1)) in the case of the energy-loss hopping of electrons in the exponential band tail.
4. Discussion The derivation of (x) in Eqs. (7) and (8) has been carried out here in the spirit of Shklovskii et al. [5], who did not discuss the mobility and its relation to the diffusion coefficient, though. However, if one extracts p from the expression of Shklovskii et al.
(14) This expression shows that D(s) has a very sharp double-exponential dependence on E and cannot be taken out of the integrand in Eq. (13). It is worth noting that we have derived Eq. (12) under the assumption that eF( x ) -X E,,. According to Eq. (81, the quantity (x) is proportional to N-2/3(~) = N;2/3 exp(2&/3s,), i.e., it increases exponentially in the course of the relaxation toward larger localization energies E. It means that the application of Eq. (12) is restricted by shallow states in the tail. The energy border for our theory depends
S.D. Baranouskii et al./Journal
of Non-Cystalline Solids198-200 (1996) 214-217
the magnitude of F, of course. The smaller is F, the deeper in the tail is this border. For very deep states, where the condition e(x) F << co is not fulfilled, it is not possible to introduce a field-independent mobility and non-linear effects with respect to the applied electric field play the decisive role [7].
on
Acknowledgements S.D.B. is thankful to B.I. Shklovskii, who first brought attention [lo] to the Einstein’s relationship in the form close to Eq. (12). Financial support by the Deutsche Forschungsgemeinschaft through the Sonderforschungsbereich 383 is gratefully acknowledged.
117
References 111M.H. Cohen, J. Non-Cryst. Solids 2 (1970) 432. 121N.F. Mott and E.A. Davis, Electronic Processes
in NonCrystalline Materials (Clarendon, Oxford, 197 1) p. 44. [31 N.V. Zykov, Sov. Phys. Semicond. 22 (1988) 1328. [41 C.E. Nehel and R.A. Street, Philos. Mag. B67 (1993) 407. [51 B.I. Shklovskii, H. Fritzsche and S.D. Baranovskii, Phys. Rev. Lett. 62 (1989) 2989. [61 S.D. Baranovskii, H. Fritzsche, E.I. Levin, I.M. Ruzin and B.I. Shklovskii, Sov. Phys. JETP 69 (1989) 773. [71 B.I. Shklovskii, E.I. Levin, H. Fritzsche and S.D. Baranovskii, Advances in Disordered Semiconductors 3, ed. H. Fritzsche (World Scientific, Singapore, 1990) p. 161. [81D. Ritter, E. Zeldov and K. Weiser, Phys. Rev. B38 (1988) 8296. (Cambridge University, 1978) [91 R.A. Smith, Semiconductors p. 172. [lOI B.I. Shklovskii, private communication (1989).