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Onsager's ion recombination model in one dimension
‘.
Volume
23, number
1
CHEMICAL
1,November 1973
PHYSICS LETTERS
‘,’
ONSAGER’S
ION RECOMBINATION
MODEL IN ONE DIMENSION .,’ R. HABERKORN
M.E. MICHEL-BEYERLE
Received 29 August
1973
In order IO cheek the applicability of Onsager's model for ion recombination lo the photoinjection of charge arriers at the surface of molcculx crystals. the random walk of charge carriers under the influence of an cxternnl eleclric field nnd an imngc force h;ls been solved analytically for the onedimensional geomerry. Current-voltage relationships for the three- 2nd the onc-dimensionnl case differ markedly.
The electric field and temperature intrinsic
photocurrents
in anthracene
dependence of crystals [ 1, 11
were found to be well described in terms of a theory developed fcr ion recombination by Onsager [3,4]. This theory assumes that after each initial act of photoionization, the positive and the negative charge carriers are thermalized rapidly, reaching during this process a certain initial separation. From there they start a random walk under the influence of the mutual Coulomb field and the applied electric field, until they 1inally either recombine
or escape the Coulotnb
attraction, thus contributing to a photocurrent. Recently (51, the identical formalism has been applied to the photoinjection of holes into anthracene crystals via exciton decay at metal electrodes. In this case the Coulomb force is exerted by the image charge. The only difference with the intrinsic photoconductivity seems to be the dimensionality of the problem.
In this paper we will show, however, that the solution of Onsager’s model changes drastically with dimensionality, going from the three-dimensional to the one-dimensional case. The main result of the three-dimensional solution is a curreni-voltage relationship (expanded through the linear term in the applied field &) given by
I= gnC 1 + [q3/?e(kT)“] E} ) .I?8
(1)
where I denotes the current, 9 the electric charge of the carriers, E the dielectric constant and kT the thermal energy. The initial separation of the carriers is included in the factor IO(T)_ According to eq. (I), a sensitive test for the applicability of Onsager’s model is the linear Z-E relationship at not too high electric field strengths, with a ratio of the slope/intercept depending on temperature as T-2 and being independent of any other adjustable parameters. These features are in fact verified in the experiments on intrinsic photoconductlvlty in anthracene crystals, [ 1, 21 The essential difference between this three-dimensional and the analogous one-dimensional problem is obvious: In the absence of an external field E in the three-dimensional
case, there is still a finite current
going our to infinity, which corresponds to a radial density distribution ~3:r-l of charge carriers (r being the mutual distance), a diffusion current density i (I. 1-2 and hence a total current I independent of r. In one dimension, on the other hand, a constant diffusion current requires a particIe density a x (x being the distance from the injecting surface), which cannot of course be maintained to infinity. So we expect a current-voltage curve different from eq. (I) in that there should be no intercept at all.
?3, number 1
V&me
CHEMICAL
!n order to solve the onedimensional problem we coilsider the equation i = -D [d.n/dx + (q/U)
dlr/d.x] ,
PHYSICS LElTE8.S
Onsager
]
(2)
[u(x) - 4xg)l
1973
For .the potential (4), this may be expressed as,
with R = particle density.,j = density of particle current, D = diffusion coefficient, 4 = charge of carriers, x = distance irom the surface. For the steady state the equation of continuity is fulfilled by j = const. The charge carrier distribution H(X) is given explicitly by Nx) = e.upC-(4/Q
1 November
:
i=i
0
.I’
m
j-exp (-rc-sjt)dr/
$ exp (-rr-s/t)dl
0
0
,
(7)
.’
where dimensionless variables r = qEa/kT and’s = for applied field ar.d image force, respectively. The integral in the denominator for r> 0 (i.e., E causes a drift in positive x direction) is given by [7]
q2/4ekTa w&e introduced
rt s exp(-rr-s/r)dr
= Z(s~r)“‘Kt(Z(r~)“~)
,
(8)
0 I
X0
where-y0 is an arbitrary distance from the surface. This equation proved to be convenient in various kinds of one-dimensional problems [6]. The electric potential energy is determined by a superposition of image forces and externally applied field E: q”(x) = -&46X
(4)
carriers are dragged away to the counter electrode. In the case of vanishing image force (s = 0), the i = i. [ 1 - exp (-qEa/kT)]
.
(9)
For small E(r < 1) an expansion through linear terms in E yields
n(0) = r@) = 0 , j=i-io,
O
=I
a
i = io(qEa/kT)
; (9
expressing the fact, that the carriers start their random walk at a distance x = a from the surface (i. particles per area and time units are injected) and either recombine with the injecting electrode (X = 0) or are collected by the counter electrode at x =m, contributing to the external current density i. Choosing x0 =a and applying eq. (3) to the two regions 0 < x
(6) [(4/W
dx>l
0
h)-’
.
E,(gZ/4&Ta)
EZ(x) being a function tegral given by [8] : E,(x) = s f-’ I
,
(10)
related to the exponential
in-
exp (-xr)dt
In the relevant region, s S 1 (image force potential greater than thermal energy) one obtains
.!?(4/4~0*)-’ exp (-q2/4ckTa) (11) 0 As argued above, there is no current in the absence of an externally applied electric field in the case.of one-
i=
i
dimensional
4x11dr
0
X (Sew
whereas the integral in the numerator could only be evaluated numerically. _ Nevertheless, the limiting cases interesting in experiments can easily be discussed. At very large fields E(r 9 1) i approaches io, indicating rhat all injected
result is
- qE.K
The boundary conditions are
i=i o Pexp
K, being the modified Bessel function of first order,
geometry.
The temperature
dependence
of the term linear in E is given by an activation term with a temperature independent preexponential factor. For the image force problem occurring in photoinjection processes at the phase boundary molecular crystal/metal, ihe photocurrent-voltage curve does
129
: .
:
; .I.
:
:
,;,
:,.. y.::
V&me
23, number
CHEMICAL
i
PHYSICS
LETTERS
:
1 N&& :.
:, not allow to infer on the applicability of Onsager’s on the basis of a temperature dependent slope/intercept ratio as in the three-dtiensional case. !n the one-dimensional geometry charge separation is model
determined
by an activation
term
which
depends
on
the initial distance 0 between’charge cirriers and the surface. The pre-exponential factor is also dependent on this distance 0. but not explicitly on the temperature. Thus, the experimentally measured slope/intercept ratios in the case of charge carrier injection via exciton drcay 3t metal electrodes [5] do not reflect the Onsager model for charge separation. Furthermore, a priori one would not expect this mechanism to be dominant in photoinjection of holes since the narrow valence band of anthracene crystals [91 should certainly not favour the injection of holes with kinetic energy exceeding kT. Increasing width of lower valence bands seems to be unlikely. This argument would also imply that in intrinsic photoconductivity [ I, 21 it is the electron which predominantly carries kinetic energy. To the case of aqueous redox electrolytes as electrodes Onsager’s model does not apply on the basis of
experimental findings. With aqueous electrodes image forces can be neglected due to the slow orientation polarization of the water molecules as compared to the hopping frequency of charge carriers in anthracene crystals [lo] _Consequently, the saturation current can be reached at field strengths as low as lo3 V/cm. If charge separation would be described in terms of the Onsager model, this saturation field strength would yield an initial distance
(accordtig & eq. (9))
a = ATfqEpt
1973 ::
‘:.
ofcharge,carrikrs from the electrode in the range ofa.., = 2000 #. The mean free path of charge &riers in such tiolecular crystals, however, is of the order ofa lattice parameter. ‘Ttius the boundary condiG+ n(0) =O [eq. (5)], which is essential for the Onsager model, cannot
be valid
in this case.
An interesting field of applicability of the onedimensional Onsager model might be the intrinsic photogeneration of charge carrier pairs in molecular crystals with highly anisotropic, approximately onedimensional electronic conductivity.-
References C.L. Braun and J.F. Hornis, J. Chem. Phgs. 49 (1968) 1967. [2] R.H. Batt, C.L. Braun and J.F. Horn&, Appl. Opt. Suppl. 3 (1968) 20. [3] L. Onsager, J. Chem. Phys. 2 (1934) 599. ] I] R.H. Batt,
[4] L. Onsnger, Phys. Rev. 54 (1938) 554. [S] H. Killcsreitcr and H. Bnessler, Phys. Stal.
Sol. 53b
(1971) 193. [6] H. R~llmnnn and hf. Pope, to be published. ]7] LS. Gradshteyn and I.M. Ryzhik, Tables ofintcgals,
translated by A. Jeffrey (Academic Press, NW York. 1966). [S] hf. Abmmowirz and IA Stegun, Handbook of mathe. marital functions (Dover, New York, 1965). [9] R. Silbey, J. Jortner, S.A. Rice and ht.T. Vala, J. Chem. sums, series and products,
Phys.42
(196.5)
733.
[IO] h1.E. Xfichel-Beyerle
25~1(1972) 1496.
and R. Haberkorn. 2. Naturforsch.
Volume
23, number
1
CHEMICAL
1,November 1973
PHYSICS LETTERS
‘,’
ONSAGER’S
ION RECOMBINATION
MODEL IN ONE DIMENSION .,’ R. HABERKORN
M.E. MICHEL-BEYERLE
Received 29 August
1973
In order IO cheek the applicability of Onsager's model for ion recombination lo the photoinjection of charge arriers at the surface of molcculx crystals. the random walk of charge carriers under the influence of an cxternnl eleclric field nnd an imngc force h;ls been solved analytically for the onedimensional geomerry. Current-voltage relationships for the three- 2nd the onc-dimensionnl case differ markedly.
The electric field and temperature intrinsic
photocurrents
in anthracene
dependence of crystals [ 1, 11
were found to be well described in terms of a theory developed fcr ion recombination by Onsager [3,4]. This theory assumes that after each initial act of photoionization, the positive and the negative charge carriers are thermalized rapidly, reaching during this process a certain initial separation. From there they start a random walk under the influence of the mutual Coulomb field and the applied electric field, until they 1inally either recombine
or escape the Coulotnb
attraction, thus contributing to a photocurrent. Recently (51, the identical formalism has been applied to the photoinjection of holes into anthracene crystals via exciton decay at metal electrodes. In this case the Coulomb force is exerted by the image charge. The only difference with the intrinsic photoconductivity seems to be the dimensionality of the problem.
In this paper we will show, however, that the solution of Onsager’s model changes drastically with dimensionality, going from the three-dimensional to the one-dimensional case. The main result of the three-dimensional solution is a curreni-voltage relationship (expanded through the linear term in the applied field &) given by
I= gnC 1 + [q3/?e(kT)“] E} ) .I?8
(1)
where I denotes the current, 9 the electric charge of the carriers, E the dielectric constant and kT the thermal energy. The initial separation of the carriers is included in the factor IO(T)_ According to eq. (I), a sensitive test for the applicability of Onsager’s model is the linear Z-E relationship at not too high electric field strengths, with a ratio of the slope/intercept depending on temperature as T-2 and being independent of any other adjustable parameters. These features are in fact verified in the experiments on intrinsic photoconductlvlty in anthracene crystals, [ 1, 21 The essential difference between this three-dimensional and the analogous one-dimensional problem is obvious: In the absence of an external field E in the three-dimensional
case, there is still a finite current
going our to infinity, which corresponds to a radial density distribution ~3:r-l of charge carriers (r being the mutual distance), a diffusion current density i (I. 1-2 and hence a total current I independent of r. In one dimension, on the other hand, a constant diffusion current requires a particIe density a x (x being the distance from the injecting surface), which cannot of course be maintained to infinity. So we expect a current-voltage curve different from eq. (I) in that there should be no intercept at all.
?3, number 1
V&me
CHEMICAL
!n order to solve the onedimensional problem we coilsider the equation i = -D [d.n/dx + (q/U)
dlr/d.x] ,
PHYSICS LElTE8.S
Onsager
]
(2)
[u(x) - 4xg)l
1973
For .the potential (4), this may be expressed as,
with R = particle density.,j = density of particle current, D = diffusion coefficient, 4 = charge of carriers, x = distance irom the surface. For the steady state the equation of continuity is fulfilled by j = const. The charge carrier distribution H(X) is given explicitly by Nx) = e.upC-(4/Q
1 November
:
i=i
0
.I’
m
j-exp (-rc-sjt)dr/
$ exp (-rr-s/t)dl
0
0
,
(7)
.’
where dimensionless variables r = qEa/kT and’s = for applied field ar.d image force, respectively. The integral in the denominator for r> 0 (i.e., E causes a drift in positive x direction) is given by [7]
q2/4ekTa w&e introduced
rt s exp(-rr-s/r)dr
= Z(s~r)“‘Kt(Z(r~)“~)
,
(8)
0 I
X0
where-y0 is an arbitrary distance from the surface. This equation proved to be convenient in various kinds of one-dimensional problems [6]. The electric potential energy is determined by a superposition of image forces and externally applied field E: q”(x) = -&46X
(4)
carriers are dragged away to the counter electrode. In the case of vanishing image force (s = 0), the i = i. [ 1 - exp (-qEa/kT)]
.
(9)
For small E(r < 1) an expansion through linear terms in E yields
n(0) = r@) = 0 , j=i-io,
O
=I
a
i = io(qEa/kT)
; (9
expressing the fact, that the carriers start their random walk at a distance x = a from the surface (i. particles per area and time units are injected) and either recombine with the injecting electrode (X = 0) or are collected by the counter electrode at x =m, contributing to the external current density i. Choosing x0 =a and applying eq. (3) to the two regions 0 < x
(6) [(4/W
dx>l
0
h)-’
.
E,(gZ/4&Ta)
EZ(x) being a function tegral given by [8] : E,(x) = s f-’ I
,
(10)
related to the exponential
in-
exp (-xr)dt
In the relevant region, s S 1 (image force potential greater than thermal energy) one obtains
.!?(4/4~0*)-’ exp (-q2/4ckTa) (11) 0 As argued above, there is no current in the absence of an externally applied electric field in the case.of one-
i=
i
dimensional
4x11dr
0
X (Sew
whereas the integral in the numerator could only be evaluated numerically. _ Nevertheless, the limiting cases interesting in experiments can easily be discussed. At very large fields E(r 9 1) i approaches io, indicating rhat all injected
result is
- qE.K
The boundary conditions are
i=i o Pexp
K, being the modified Bessel function of first order,
geometry.
The temperature
dependence
of the term linear in E is given by an activation term with a temperature independent preexponential factor. For the image force problem occurring in photoinjection processes at the phase boundary molecular crystal/metal, ihe photocurrent-voltage curve does
129
: .
:
; .I.
:
:
,;,
:,.. y.::
V&me
23, number
CHEMICAL
i
PHYSICS
LETTERS
:
1 N&& :.
:, not allow to infer on the applicability of Onsager’s on the basis of a temperature dependent slope/intercept ratio as in the three-dtiensional case. !n the one-dimensional geometry charge separation is model
determined
by an activation
term
which
depends
on
the initial distance 0 between’charge cirriers and the surface. The pre-exponential factor is also dependent on this distance 0. but not explicitly on the temperature. Thus, the experimentally measured slope/intercept ratios in the case of charge carrier injection via exciton drcay 3t metal electrodes [5] do not reflect the Onsager model for charge separation. Furthermore, a priori one would not expect this mechanism to be dominant in photoinjection of holes since the narrow valence band of anthracene crystals [91 should certainly not favour the injection of holes with kinetic energy exceeding kT. Increasing width of lower valence bands seems to be unlikely. This argument would also imply that in intrinsic photoconductivity [ I, 21 it is the electron which predominantly carries kinetic energy. To the case of aqueous redox electrolytes as electrodes Onsager’s model does not apply on the basis of
experimental findings. With aqueous electrodes image forces can be neglected due to the slow orientation polarization of the water molecules as compared to the hopping frequency of charge carriers in anthracene crystals [lo] _Consequently, the saturation current can be reached at field strengths as low as lo3 V/cm. If charge separation would be described in terms of the Onsager model, this saturation field strength would yield an initial distance
(accordtig & eq. (9))
a = ATfqEpt
1973 ::
‘:.
ofcharge,carrikrs from the electrode in the range ofa.., = 2000 #. The mean free path of charge &riers in such tiolecular crystals, however, is of the order ofa lattice parameter. ‘Ttius the boundary condiG+ n(0) =O [eq. (5)], which is essential for the Onsager model, cannot
be valid
in this case.
An interesting field of applicability of the onedimensional Onsager model might be the intrinsic photogeneration of charge carrier pairs in molecular crystals with highly anisotropic, approximately onedimensional electronic conductivity.-
References C.L. Braun and J.F. Hornis, J. Chem. Phgs. 49 (1968) 1967. [2] R.H. Batt, C.L. Braun and J.F. Horn&, Appl. Opt. Suppl. 3 (1968) 20. [3] L. Onsager, J. Chem. Phys. 2 (1934) 599. ] I] R.H. Batt,
[4] L. Onsnger, Phys. Rev. 54 (1938) 554. [S] H. Killcsreitcr and H. Bnessler, Phys. Stal.
Sol. 53b
(1971) 193. [6] H. R~llmnnn and hf. Pope, to be published. ]7] LS. Gradshteyn and I.M. Ryzhik, Tables ofintcgals,
translated by A. Jeffrey (Academic Press, NW York. 1966). [S] hf. Abmmowirz and IA Stegun, Handbook of mathe. marital functions (Dover, New York, 1965). [9] R. Silbey, J. Jortner, S.A. Rice and ht.T. Vala, J. Chem. sums, series and products,
Phys.42
(196.5)
733.
[IO] h1.E. Xfichel-Beyerle
25~1(1972) 1496.
and R. Haberkorn. 2. Naturforsch.