Non-Onsager mechanism of long-wave photogeneration in amorphous
selenium at high electric fields
A. Reznik, K. Jandieri, F. Gebhard, and S. D. Baranovskii
Citation: Appl. Phys. Lett. 100, 132101 (2012); doi: 10.1063/1.3697643
View online: http://dx.doi.org/10.1063/1.3697643
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Published by the American Institute of Physics.
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APPLIED PHYSICS LETTERS 100, 132101 (2012)
Non-Onsager mechanism of long-wave photogeneration in amorphous
selenium at high electric fields
A. Reznik,1,a) K. Jandieri,2 F. Gebhard,2 and S. D. Baranovskii2
1
Imaging Research, Sunnybrook Health Sciences Centre, 2075 Bayview Avenue, Toronto M4N 3M5, Canada
Department of Physics and Material Sciences Center, Philipps University Marburg, D-35032 Marburg,
Germany
2
(Received 9 February 2012; accepted 6 March 2012; published online 26 March 2012)
The quantum efficiency of the free-carrier-photogeneration in amorphous selenium avalanche
blocking structures is studied experimentally in a wide range of wavelengths (380–600 nm) at high
electric fields (10 112:5 Vlm1 ). While at comparatively small excitation wavelengths (up to
’ 540 nm), our experimental results are consistent with the Onsager theory of electron-hole pair
dissociation [L. Onsager, Phys. Rev. 54, 554 (1938)], at larger wavelengths (540–600 nm) and high
electric fields Onsager theory fails to explain our results. The reason for the failure of the Onsager
approach is discussed and an alternative theoretical tool is adopted to account for the experimental
C 2012 American Institute of Physics. [http://dx.doi.org/10.1063/1.3697643]
observations. V
To date amorphous selenium (a-Se) remains the only
amorphous photoconductor, which combines high optical
quantum efficiency (g ’ 95% for blue light) with avalanche
multiplication of charge carriers (holes) at high electric fields
(F > 80 Vlm1 ).1,2 Although a high g is exploited in practical a-Se photosensors,3–5 the mechanism of the free-charge
photogeneration is not yet clarified despite intensive studies
in 1970s, which showed that g in a-Se depends on electric
field with strong variation with photon energy.6 A comprehensive analysis of all collected experimental results at that
time was performed by Knights and Davis (KD) (Ref. 6)
who considered photogeneration process as Poole-Frenkel
ionization and thermalization of electron-hole pairs (EHPs)
bound by their Coulomb attraction. An alternative model
was suggested by Pai and Enck (PE) (Ref. 7) who adopted
Onsager dissociation theory8,9 in order to interpret the experimental data on the photogeneration process. Onsager calculated the probability that a pair of oppositely charged ions
separated by some distance r0 in a weak electrolyte would
escape recombination undergoing diffusive/drift motion
under the combined influence of the mutual Coulomb attraction and external electric field F. In the PE model, geminate
carriers are initially created at a certain distance r0 , which
depends on the wavelength of the excitation, k. At large
wavelengths, the distance r0 is short resulting in a high
recombination probability. As the photon energy increases,
so does r0 , and there is a larger probability that the charges
overcome their mutual Coulomb attraction yielding two free
mobile carriers. While only a single fitting parameter r0 is
used in the PE model, the KD approach requires a number of
fitting parameters, such as the recombination lifetime, the
diffusion constant and the phonon frequency. Although for
small r0 (large k) the assumption of the diffusive motion in a
continuous medium is to be taken with caution, the PE model
has been widely accepted for the explanation of the fieldand wavelength-dependencies of the photogeneration yield g
in a-Se. Pai and Enck7 have proven that in the range of excitation wavelengths (380–600 nm) studied in our experiments,
a)
Electronic address: reznika@tbh.net.
0003-6951/2012/100(13)/132101/4/$30.00
as described below, light absorption is dominated by singlephoton processes. Two-photon processes discussed by
Enck10 become relevant at wavelengths above 1000 nm.
Note that most studies of the free-carrier photogeneration in a-Se had been carried out in the times of a-Se applications in xerography, i.e., long before the “new era” of the
avalanche a-Se. This did not allow to check the validity of
their models at high electric fields at which the avalanche
multiplication occurs. Below we show that at strong F above
the avalanche multiplication threshold Fav , the dependencies
of the free-carrier photogeneration quantum efficiency
gðF; kÞ on the electric field F and on the wavelength k cannot
be explained by the Onsager theory. We suggest that the reason for the failure of the Onsager’s approach is the invalidity
of its crucial assumption on the continuous-medium diffusive
motion of the dissociating charge carriers. This assumption
is valid only if the mean-free path of charge carriers between
two scattering events is much smaller than the crucial spatial
scale of the Onsager problem—the initial separation r0
between electron and hole in the dissociating EHP. At large
k and concomitantly at small r0 this condition fails, and the
Onsager’s approach loses its validity. Therefore, in order to
explain the experimental gðF; kÞ at small k, we use an alternative approach that was originally suggested by Tachiya11
for molecular liquids and molecular solids. We adopt this
theoretical tool for amorphous materials and perform MonteCarlo simulations in its framework.
The effective quantum efficiency g incorporates both
the primary mechanism by which EHPs are generated and
also the secondary mechanism by which these free carriers
avalanche. Its field dependence is extracted from the measured photocurrent Iph at known light intensity W for the 8 lm
thick a-Se layer. The details of experimental setup for the
measurements of photocurrent can be found in Ref. 2. A grating monochromator was used to expose the a-Se high-gain
avalanche rushing photoconductor (HARP) targets to light of
a specific wavelength k in the range from 380 nm to 600 nm.
The resulting photocurrent was measured as a function of the
electric field between 10 V=lm and 112:5 V=lm, the upper
limit was dictated by the Geiger breakdown threshold. The
100, 132101-1
C 2012 American Institute of Physics
V
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132101-2
Reznik et al.
Appl. Phys. Lett. 100, 132101 (2012)
effective quantum efficiency g was calculated using the following expression:
g ¼
1
Iph
Wk
;
e
hc
(1)
where e is the elementary charge and hc=k is the incident
photon energy. The field dependences of the effective quantum efficiency g ðFÞ obtained from Eq. (1) for different k
are shown in Fig. 1(a). The effective quantum efficiency
g ðk; FÞ is the product of the dissociation efficiency of photogenerated electron-hole pairs gdis , and the avalanche multiplication gain of the dissociated charge carriers gav ,
g ðF; kÞ ¼ gdis ðk; FÞ gav ðFÞ:
(2)
The data for the short wavelength k ¼ 430 nm in Fig. 1(a)
allows one to determine the field dependence of the avalanche gain gav ðFÞ. Indeed, g reaches unity at electric fields
F 50 V=lm that are significantly lower than the avalanche
multiplication threshold Fav 80 V=lm. The effective quantum efficiency g for k ¼ 430 nm does not change significantly in the range 50 V=lm F 80 V=lm. This allows
us to identify two distinct domains of the applied electric
field with different physical mechanisms responsible for the
field dependence of the effective quantum efficiency. Below
the avalanche threshold Fav , the effective quantum efficiency
g ðFÞ equals the dissociation efficiency gdis ðFÞ because gav
equals unity at F < Fav (no avalanche). Above the avalanche
threshold Fav , the effective quantum efficiency g ðFÞ equals
gav ðFÞ because gdis is saturated at unity already above the
field F 50 V=lm. As a result, the avalanche multiplication
FIG. 2. Field dependences of the dissociation efficiency gdis for different
wavelengths k. Open symbols: experimental results; solid lines: Onsager
approach; filled symbols: simulations in the framework of the Tachiya
model.
gain can be defined for the entire range of electric fields as
shown in Fig. 1(b).
The data in Fig. 1(b) are key to determine the field dependence of the dissociation efficiency, gdis ðFÞ at wavelengths k > 430 nm and electric fields F > Fav , at which the
avalanche multiplication superimposes the dissociation process and the direct measurement of the dissociation efficiency becomes impossible. In accord with Eq. (2), in order
to determine gdis ðFÞ, the measured effective efficiency g
shown in Fig. 1(a) should be divided by the avalanche gain
gav presented in Fig. 1(b). The obtained results for gdis ðFÞ
are shown in Fig. 2 by open symbols for all studied values of
the excitation wavelength k in the entire range of electric
fields. While at electric fields F 50 V=lm our data for
gdis ðFÞ agree with those already known from the literature,6,7
the results obtained above the avalanche threshold F > Fav ,
are currently unique, and we now turn to their analysis.
Let us try to describe theoretically the experimental data
obtained for dissociation efficiency of EHPs gdis ðFÞ as presented in Fig. 2. Since PE (Ref. 7) obtained a good agreement with the experimental data on dissociation of EHPs in
a-Se using Onsager theory, we also start our analysis using
Onsager’s approach.8,9 According to this approach, the dissociation efficiency can be calculated as
gOns ¼ U0
FIG. 1. Effective photogeneration efficiency as a function of electric field
for different values of the excitation wavelength (a); field dependence of the
avalanche multiplication gain in a-Se (b).
1
1
1
X
exp½ðA þ BÞ X
Am X
Bl
;
m! n¼0 l¼mþnþ1 l!
B
m¼0
(3)
where A e2 =er r0 k0 T; B eFr 0 =k0 T; k0 is the Boltzmann
constant, T is the temperature, er is the dielectric constant,
U0 is the photo-excitation efficiency, and r0 is the initial separation between geminate electron and hole. The latter parameter plays the decisive role in all applications of Onsager
theory to interpret experimental data. Since the magnitude of
r0 cannot be determined experimentally, the value of this parameter is usually chosen by fitting experimental data.7 In
order to compare our results with those of Pai and Enck,7 we
start our analysis with the data at low electric fields. We
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132101-3
Reznik et al.
FIG. 3. Initial EHP separation r0 as a function of wavelength k. Open
circles: our estimations, filled circles: data from Ref. 7.
estimate the values of r0 by fitting the expression in Eq. (3)
to the measurements of g at the lowest considered electric
field, F ¼ 10 V=lm. As in Ref. 7, the parameter U0 is
assumed to be independent of the electric field and equal to
unity. The obtained dependence of r0 on k is shown in Fig. 3
by open circles and perfectly agrees with that reported in
Ref. 7 and shown in Fig. 3 by filled circles. This agreement
evidences that our experimental technique is in concord with
the experimental techniques used in Ref. 7 at electric fields
below the avalanche threshold, F < Fav .
Let us insert the obtained values of r0 into Eq. (3) in
order to describe the dependence gdis ðFÞ for the entire range
of electric fields F. The results are shown by solid lines in
Fig. 2. While at small k values the Onsager theory agrees
with experimental data in the entire range of electric fields, it
fails to describe experimental results for the dissociation efficiency at large wavelengths (k 550 nm) and high electric
fields (F 60 V=lm). Since the Onsager approach contains
only one fitting parameter r0 and the magnitude of r0 has
been already established for each wavelength k as given in
Fig. 3, there is no option to account for the remarkable
disagreement between this theory and experimental data at
k 550nm and F 60 V=lm within the frame of Onsager’s
approach. The intriguing question arises then on possible
reasons for the failure of the Onsager description at large values of the wavelength k.
In order to understand the reason for the failure of the
Onsager description at large k, one should recognize in Fig.
3 that large values of k correspond to extremely small values
of the r0 . It is, however, worth noting that Onsager results
are essentially based on the solution of the Smoluchowski
equation, which is valid only if the mean-free path of the diffusing species is the smallest spatial scale in the problem.
This assumption is usually fulfilled in electrolytes considered
initially by Onsager,8,9 although it can easily be broken in
other systems if the mean-free path is comparable to r0 , as
has been already claimed in the literature.12–14
We suggest that the failure of the Onsager theory to
describe our experimental data at large k and high F is
caused by similar reasons. Indeed, the mean-free path of
holes in a-Se at electric fields close to Fav is usually estimated as l ffi 0:6 nm.15 This value is comparable to
r0 ffi 1:0 nm, in Fig. 3 for long wavelengths.
A theoretical description of the dissociation problem in
the case of a carrier mean-free path comparable to r0 has
Appl. Phys. Lett. 100, 132101 (2012)
been suggested by Tachiya.11 This approach is based on simulating the motion of a charge carrier undergoing scattering
events with some characteristic time s being affected by the
Coulomb attraction to the opposite charge carrier and by the
external electric field F. The scattering time s is the key
input parameter of the model.
We performed computer simulations in the framework
of the Tachiya model based on the following principles.11
Electron and hole are initially separated by the distance r0 .
The hole is fixed at the origin of the coordinate system while
the electron moves in the configurational space of the relative coordinates r with acceleration
2
1
e rðtÞ
þ eF ;
(4)
aðtÞ ¼
er jr3 ðtÞj
m
determined by the Coulomb potential and the external electric
field. The electron undergoes a scattering process with time
separation tn between the nth and (n þ 1)th sequential scattering events. The magnitudes of tn are chosen randomly from the
exponential distribution gðtn Þ with the characteristic time s,
gðtn Þdtn ¼
expðtn =sÞ
dtn :
s
(5)
After each scattering event, the velocity components are
randomized according to Maxwell-type probability distribution. The trajectory of the electron between the scattering
events is followed by the standard Beeman method16 and the
total energy of the electron is calculated as
E¼
mv2
e
:
er jrj
2
(6)
It is assumed that the electron has recombined with the hole if
this energy becomes less than a certain value Ecr . Electron is
claimed to escape the recombination if the distance to the hole
becomes larger than a certain distance rcr . Following Tachiya,11
we used the value rcr ¼ 10r with r ¼ e2 =er k0 T. For the critical energy, we assume the value Ecr ¼ 15k0 T. It has been
checked that the simulation results do not change with increasing the absolute values of the parameters Ecr and rcr .
The dissociation probability gdis ¼ n=N was calculated
after performing N runs of the algorithm with the number of
dissociation events n. Depending on the initial separation and
the applied electric field, we performed N ¼ 103 104 runs in
order to obtain well-converged results. It was assumed that
the initial separation r0 depends on the excitation wavelength
as shown Fig. 3. There is only one fitting parameter in the
simulation: the characteristic scattering time s. This parameter
was chosen so that the simulation results for the dissociation
efficiency shown in Fig. 2 by filled symbols exactly reproduce
the experimental data shown in this figure by open symbols.
In order to obtain this perfect agreement, one has to assume
that the scattering time s increases with increasing electric
field as p
shown
ffiffiffiffiffiffiffiffiffiffiffiffiffiin Fig. 4, where s is plotted in units of
s ¼ r = k0 T=m. Taking m equal to the free electron mass
and k0 T ¼ 0:025 eV, one obtains for the characteristic value
of the scattering time s ffi 0:01s the value of the microscopic
mobility at room temperature l ¼ es=m ffi 2 cm2 =ðsVÞ,
which seems reasonable for a-Se. þ-
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132101-4
Reznik et al.
Appl. Phys. Lett. 100, 132101 (2012)
wavelengths, becomes comparable to the diffusion meanfree path of charge carriers. In order to account for the experimental data at k > 540 nm, we adopt a theoretical
approach11 that takes into account the finite size of the carrier mean-free path. The application of this approach to our
results indicates a substantial increase of the scattering time
of charge carriers with increasing electric field.
FIG. 4. Field dependence of scattering time s necessary for the best agreement between the simulated and experimental data in Fig. 2.
Authors are indebted to Dr. A. Nenashev and Dr. F.
Jansson for clarifying discussions with respect to the theoretical model and Dr. Tanioka and his group for the photoconductivity measurements.
1
Remarkably, the same field dependence sðFÞ shown in
Fig. 4 is necessary in order to fit experimental data at all values of k used in our experiments. This universality with
respect to the excitation wavelength and concomitantly with
respect to the initial separation r0 indicates that the dependence sðFÞ should be characteristic for the motion of charge
carriers in a-Se. In crystalline semiconductors, the increase
of the characteristic scattering time with increasing electric
field has been predicted theoretically and evidenced experimentally for the case that scattering on charged defects dominates the motion of carriers.17 For a-Se, more study is
necessary to clarify this issue. Remarkably, the increase of
the scattering time with increasing field, i.e. the increase of
the relaxation distance for holes in a-Se was predicted by
Kasap et al.18 in their analysis of the avalanche multiplication in a-Se. This was also confirmed by experimental study
of Juska and Arlauskas.19,20
From our study, one can conclude that while Onsager
theory is capable to account for the dissociation efficiency of
photogenerated electron-hole pairs in a-Se at short excitation
wavelength (k 540 nm), it fails to describe experimental
data at longer wavelength (k > 540 nm). We suggest that the
reason for the failure of the Onsager approach is the small
size of the photogenerated electron–hole pair that, at long
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2
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