A photoconductor intrinsically has no gain
Yaping Dan1*, Xingyan Zhao1, Abdelmadjid Mesli2
1
University of Michigan – Shanghai Jiao Tong University Joint Institute, Shanghai Jiao Tong
University, Shanghai, China
2
Institut Matériaux Microélectronique Nanosciences de Provence, UMR 6242 CNRS, Université
Aix-Marseille, 13397 Marseille Cedex 20, France
*Correspondence should be addressed to: yaping.dan@sjtu.edu.cn
Abstract
Semiconducting photoconductors have been widely reported to have an extraordinarily high gain (up to 108). In the
past 50 years, the high gain is often explained by a widely accepted theory that the gain is equal to
is the minority recombination lifetime and
the photogenerated excess carriers (
where
the carrier transit time. The theory is derived on the assumption that
are spatially uniform and independent of external voltage bias. In
this Letter, we find that this assumption is not valid for a photoconductive semiconductor in contact with two metal
electrodes. By solving the continuity equation and performing numerical simulations using commercial device
simulators, we conclude that a photoconductor intrinsically has no gain, meaning that the gain will be no more than
1 no matter how short the transit time is. The high gain observed in experiments must come from other extrinsic
effects on which we have offered a brief perspective.
It is well known that avalanche photodiodes and bipolar phototransistors have gain. A
photoconductive semiconductor having gain is surprising, but it is clearly written in the classical
semiconductor physics textbooks[1-3] and widely accepted by the research community for decades[4-8].
The gain theory was derived in 1950s[9], which concluded that the gain of a photoconductor is equal to
the recombination lifetime of minority carriers divided by the transit time that the carriers take to
transport between the two contacts of the device. Therefore, a photoconductor will intrinsically have a
large gain if the transit time is much shorter than the recombination lifetime (by applying a large electric
field on a short device, for instance). The physical explanation for the gain is that the short transit time
allows the photogenerated carriers to circulate in the circuit multiple times before recombination,
equivalent to generating many times more photoexcited carriers[1]. We call this gain theory as “recycling
gain mechanism” for convenience.
Conceptually, according to the theory, the recycling of charge carriers increases the number of
collected carriers but not the concentration of excess carriers in the device. The theory will inevitably lead
to the conclusion of no gain in photoconductivity, which however is in contradiction with most of the
experimental observations[4, 8]. Quantitatively, there is a huge disparity between the gains predicted by
the theory and those measured in experiments. For instance, Matsuo et al [10] observed in 1984 that the
gain of GaAs photoconductive detectors predicted by the recycling gain theory is 3 to 4 orders of
magnitude smaller than the gain measured in the experiments. Similar observations have been made
persistently by other researchers in the past several decades[4, 11, 12]. Some argued that this disparity is
due to the carrier trapping by surface trap states or charge separation by built-in electric fields that
prolongs the recombination lifetime of minority carriers[6, 13, 14]. Others even conceptually mixed up
the trap (capture and emission) and minority recombination lifetime[4], using the long trap lifetime to
replace the short minority recombination lifetime to explain away the disparity. Up to date, this gain
theory is still being widely used to explain the observed photoconductive gain in photoconductors based
on quantum dots [15], nanowires [7, 16] and more recently 2-dimensional materials[17, 18].
In this Letter, we find that this well-known recycling gain theory is highly questionable because
the theory derivation is based on the assumption that the concentration of photogenerated excess carriers
(PEC) in the photoconductor is independent of applied electric field. However, for a semiconductor in
contact with metal (as electrodes), the PECs in the semiconductor are always spatially non-uniform and
therefore electric field dependent. By solving the continuity equation and performing simulations using
the commercial device simulator, we conclude that a photoconductor intrinsically has no gain. It means
that, for a photoconductive semiconductor in contact with metal electrodes, the theoretical gain will never
be greater than 1 no matter how short the transit time is. The photoconductive gain observed in the
experiments must come from other extrinsic effects, on which we will offer a brief perspective at the end
of this Letter.
Let us first go through the theoretical derivation for the recycling gain mechanism in the classical
semiconductor physics textbooks[1]. The gain G of a photoconductor following the definition of internal
quantum efficiency is defined as the number of photogenerated electrons or holes collected by electrodes
divided by the number of photons absorbed in the semiconducting photoconductor.
eq.(1)
where Ac is the cross-sectional area of the device, e the charge unit, and
the photon energy. The
equation can be further written as a ratio of photocurrent density Jph and optical power absorbed by the
device Pabs. The photocurrent density equals to
intensity,
in which
is the electric field
and
the electron and hole mobility, and
and
(= ) the photogenerated electron
P /  is the total number of photons absorbed
and hole concentration, respectively. The denominator abs
per second in the device. If we assume one absorbed photon generating one electron-hole pair, then the
g  Pabs /  / V where V is the device volume given by
carrier generation rate is equal to
with L the length between the two electrodes of the photoconductor.
In general, the photogenerated minority carrier concentration can be written as
n  g  
eq.(2)
where g is the generation rate and τ the recombination lifetime of minority carriers.
By plugging eq.(2) into eq.(1), we will have:
eq.(3)
in which
is the transit time for the minority electrons transporting between the two
contact electrodes of the photoconductor. Note that the transit time has a low limit due to the velocity
saturation. Nevertheless the gain according to eq.(3) can still be very high if the recombination lifetime is
much longer than the transit time by applying a large electric voltage on a short device.
eq.(3) is the theoretical foundation of the recycling gain mechanism. The problem of the theory
derivation is originated from eq.(2) which is assumed to be independent of the electric field intensity. But
for a semiconductor in contact with metal, the distribution of photogenerated excess carriers is always
non-uniform and therefore readily skewed by the electric field, resulting in the voltage-dependent excess
carrier concentration. This can be seen clearly from the continuity equation. For a uniformly doped p-type
semiconductor under small injection condition, the continuity equation at steady state for minority carriers
is expressed as:
eq.(4)
where
and
are the diffusion constant, mobility, recombination lifetime and generation rate of
minority electrons, respectively. With voltage bias, the electric field inside the device is often uniform
(Ohmic contact) but always nonzero. The uniform electric field will zero out the third term. The second
and consequently the first term are zero only if the excess carriers are uniformly distributed. In this case,
eq.(2) is valid.
Figure 1. Energy band diagram of a photoconductor (a) in dark, (b) under light illumination with zero voltage bias
and (c) under light illumination with nonzero voltage bias. The semiconductor is assumed to have the same work
function with the metal.
However, the distribution of photogenerated excess carriers in a semiconductor in contact with
metal is always non-uniform, as shown in Fig.1. For simplicity, we assume that the semiconductor and
the contact metal have the same work function. There is no energy band bending when they are in contact
(Fig.1a). Light is uniformly illuminated on the device from the vertical direction, as shown in the sketch
of Fig.1b and c. Excess minority electrons are excited in the conduction band in the semiconductor and no
excess electrons will be generated in the metal. To maintain continuity, the concentration of excess
minority electrons has to be zero at the semiconductor-metal interface, resulting in excess electrons in the
semiconductor diffusing towards the metal, as shown in the bottom sketch of Fig.1b. At zero voltage bias,
the electron diffusion is anti-symmetric with no net photocurrent flow in the circuit. At non-zero bias, the
electric field will skew the anti-symmetric transport of excess electrons (Fig.1c), creating net photocurrent.
Clearly, the concentration of photogenerated excess carriers is spatially voltage-dependent instead of
following the simple expression of eq.(2). If eq.(2) cannot hold, then the gain expression eq.(3) derived on
the basis of eq.(2) is highly questionable.
To derive the correct expression for the gain, we need to first find the minority carrier distribution
by solving from the continuity equation eq.(4) on the assumption of uniform electric field (the third term
is zero). This assumption is valid for a uniformly doped semiconductor with Ohmic contact at small
injection condition. By applying the boundary conditions Δn = 0 at both x = 0 and x = L, we find:
eq.(5)
√
(“+” for λ1 and “-” for λ2) with the drift length
where
diffusion length
√
and the
.
Figure 2. Spatial distribution of photogenerated excess minority carriers in a photoconductor. Dotted lines are the
solutions of the continuity equation and solid lines are the simulation results. Velocity saturation is excluded from
the simulation.
To validate the solution of the continuity equation given by eq.(5), we performed numerical
simulations on a silicon device using the DEVICE module of the commercial software Lumerical. The
software module numerically solves the Poisson’s equation and the continuity equations for minority and
majority carriers. It can catch the transport behavior of both types of carriers, providing more realistic
results. We suppose that the device under simulation is 10 μm long and 1 μm × 1 μm in cross-section. The
p-type doping concentration is 1017 cm-3 and the generation rate is spatially uniform at 1022 cm-3/s. The
recombination lifetime of minority carriers is set at 1 ns due to, for instance, surface recombination. The
mobility for electrons and holes is 30 cm2/Vs and 30 cm2/Vs, respectively. Different mobilities for
electrons and holes will not change the conclusion. As we show later, the photocurrent will saturate at
high voltage bias. To illustrate that the saturation is not caused by the velocity saturation, the velocity
saturation effect is excluded from the simulation. The spatial distributions of photogenerated excess
carriers are plotted in Fig.2a. The solid and dotted lines denote the concentrations of excess minority
carriers
given by the device simulator and the equation eq.(5), respectively. It is clear that the
simulation results and the solutions of the continuity equation are almost identical.
(
)(
)
(
)(
)
{
eq.(6)
As expected, the anti-symmetric distribution of the excess carriers is skewed by the electric field
(Fig.2), which will create net photocurrent in the circuit. The equation for the minority photocurrent is
given by eq.(6). The expression is rather complicated but can be regressed to the forms that we are more
familiar with at two extreme cases. The first case is when the electric field intensity is close to zero. The
drift length is then nearly zero, much smaller than the diffusion length. Logically, the transit time will be
significantly longer than the recombination lifetime of minority carriers, i.e.
. In this case,
the spatial distribution of photogenerated excess minority carriers remains almost anti-symmetric. If the
diffusion length
is much smaller than the device length L, then the excess minority carriers are
uniformly distributed almost in the entire semiconductor. The uniform distribution of photogenerated
carriers will zero out the first three terms in eq.(4), resulting in
minority excess carriers will then be given by
The photocurrent density of
, consistent with the regression in eq.(6) for
a small electric field. For the case that the electric field intensity E approaches to very large values, the
excess carrier distribution is strongly skewed (like the curve at 20V bias in Fig. 2). The equation
will never satisfy. In this case, the transit time will be much shorter than the recombination lifetime,
i.e.
. The minority electron photocurrent density saturates to
(see more intuitive
explanations in SI) instead of linearly going up, as shown in eq.(6). This is not surprising if we take into
account the fact that the concentration of excess minority carriers decreases as the bias increases, as
shown in Fig.2.
The photogenerated excess majority carriers
also contribute to the photocurrent. Note that the
semiconductor is doped. There is a large background dark current contributed by the majority carriers.
The continuity equation for majority carriers is a nonlinear differential equation, from which it is difficult
to analytically solve the spatial distribution of the excess majority carriers. Nevertheless it is known (also
see SI) that the spatial distributions of excess majority and minority carriers are nearly identical if the
external electric field is not too high, regardless of the difference in mobility of minority and majority
carriers. This phenomenon is called ambipolar transport[1]. As stated above, the excess minority electrons
are nearly uniformly distributed and the electron photocurrent density is given by
on the condition that the electric field intensity E is not strong and the diffusion length is much
smaller than the device length L (Fig.1b). Due to the ambipolar transport phenomenon, the same
conclusion can be reached for the excess majority holes, i.e.
and
. Therefore
the total photocurrent density is governed by
, which is consistent with the
common knowledge and the simulation results shown in Fig.3a at small voltages. In this case, the gain
expression given by eq.(3) still holds except that the gain is much smaller than 1, because
at small electric field intensity as previously analyzed for the minority carriers.
Figure 3. Photocurrent vs voltage for different recombination lifetimes of minority carriers. The device has the same
parameters with the one in Fig.2 except the minority recombination lifetime increasing from 1 ns to 20 ns. At small
voltage, the photocurrent is governed by
(
) while at high voltage, the photocurrent density
saturates to
.
In Fig.3, the total photocurrent density saturates to
at high voltage bias. The
saturation happens more rapidly for a longer minority recombination lifetime due to the fact that the
distribution of excess minority carriers is more readily skewed by the electric field if the recombination
lifetime is longer (eq.(5)). Note that the minority electron photocurrent also saturates to
at high
voltage bias, as indicated by eq.(6). Since the photogenerated majority holes contribute to the
photocurrent as well, it is surprising that the total photocurrent density does not saturates to
.
This is because the electron-hole pairs are separated by the high electric field (Fig.S2 in the supporting
information), similar to the photogenerated carriers in a reverse biased PN junction where the carrier
transport is charge limited, except that the photoconductor has a large background current (dark current).
Indeed, the concentration of photogenerated excess carriers decreases as the electric field intensity
increases, which makes it easier to separate electron-hole pairs at high voltage bias without inducing a
large internal electric field. For photoconductors with a low doping concentration, the photogenerated
electrons and holes are readily separated, resulting in a saturated photocurrent density
voltage bias (Fig.S3 in SI). Since the photocurrent density always saturates to
at high
, we can
conclude from eq.(1) and (3) that the gain of a photoconductor approaches to but never exceeds 1,
meaning that a photoconductor intrinsically has no gain.
Figure 4. Carrier trapping process by trap states of a p-type semiconductor. At small injection condition,
photoexcited excess carriers
and
(
) will shift up the electron quasi Fermi level
, allowing for
states below
to be filled with
. The number of photogenerated hole counterparts
is left in the valence
band to contribute to photoconductivity.
If a photoconductor intrinsically has no gain, then where are the gains observed in the
experiments originated from? The gain must come from the photogenerated charge carrier accumulation
in physical space instead of time domain. Otherwise, the gain (internal quantum efficiency) will not be
greater than 1 as we show above, which also means that there will be no gain in photoconductivity, and
that the responsivity of the photoconductor will not be higher than that of a PN junction photodiode (1
A/W at λ = 1.2 μm). We believe that the gain essentially is originated from the trapping of photogenerated
charge carriers by trap states. The process is briefly described below[19].
Given a p-type semiconductor under light illumination, the minority electron quasi Fermi level
will shift up from the Fermi level
while the majority hole quasi Fermi level
remain nearly intact
at small injection condition, as illustrated schematically in Fig.4. The trap states below
(above
be filled with photogenerated electrons
(=
. The photogenerated hole counterparts
) will
) will be
left in the valence band to contribute to the gain in photoconductivity. The gain is independent of the trap
lifetime and only determined by the density of trap states and the quasi Fermi level shift. As an example,
let us assume that the doping concentration of a semiconductor is
excess carriers are
states are
and the photogenerated
. Suppose that the photogenerated electrons captured by trap
. Accordingly, the same number of photogenerated holes
is
left in the valence band. Then the gain in photoconductivity will be approximately 500, assuming the
same mobility for both electrons and holes. It may look surprising that
and
are significantly
higher than
and
. This is because the generation and recombination from band to band is a dynamic
process, from which the trap states below
can continuously “steal” photogenerated electrons, leaving
the hole counterparts un-trapped in the valence band. The un-trapped holes do circulate in the circuit
under voltage bias. A higher voltage bias will result in a larger photocurrent and photoresponsivity, but
not a higher gain in photoconductivity.
If the trap states are located on the device surfaces, the trapping of photogenerated charge carriers
may alter the net charges on the surfaces, inducing the band bending in the device. This is the so-called
gating effect. But it is a secondary effect introduced by the aforementioned carrier trapping process and
highly dependent on the initial net charges on the surfaces. The gating effect, if exists, may further
complicates the gain mechanism[20].
In conclusion, the widely accepted “recycling gain theory” is based on an assumption that is not
valid for the photoconductor consisting of a semiconductor in contact with metal electrodes. We have
proved that a semiconducting photoconductor intrinsically has no gain in terms of either internal quantum
efficiency or photoconductivity. The widely accepted recycling gain mechanism that the photoconductor
gain is equal to the ratio of minority carrier recombination lifetime and carrier transit time shall be
discarded. The high photoconductive gain observed in experiments must be originated from other
extrinsic effects such as trap states that result in the accumulation of photogenerated excess carriers in
physical space.
Author Contribution
Y. D. conceived the concept, derived the theory and wrote the manuscript. Under the guidance of
Y. D., X. Z. performed the device simulations. A. M. commented on the manuscript.
Acknowledgement
The authors thank Mr. Hongwei Guo at Zhejiang University for useful discussions. The work is
supported by the national “1000 Young Scholars” program of the Chinese central government, the
National Science Foundation of China (61376001), the SJTU-UM Collaborative Research Program and
the “Innovative Research Plan” of the Shanghai Bureau of Education.
Conflict of Interests
The authors declare no conflict of interests.
References
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]
[13]
[14]
[15]
[16]
[17]
D. A. Neamen, Semiconductor Physics and Devices: Basic Principles, Fourth Edition: Publishing
House of Electronics Industry, P. 634, 2011.
Z. B.-S. Liu En-Ke, Luo Jin-Sheng, Semiconductor Physics (in Chinese), Seventh Edition: Publishing
House of Electronics Industry, P. 328.
S. M. SZE and K. N. Kwok, Physics of Semiconductor Devices, Third Edition: John Wiley & Sons, Inc.
G. Konstantatos, M. Badioli, L. Gaudreau, J. Osmond, M. Bernechea, F. P. G. de Arquer, et al.,
"Hybrid graphene-quantum dot phototransistors with ultrahigh gain," Nature Nanotechnology,
vol. 7, pp. 363-368, Jun 2012.
C. H. Liu, Y. C. Chang, T. B. Norris, and Z. H. Zhong, "Graphene photodetectors with ultrabroadband and high responsivity at room temperature," Nature Nanotechnology, vol. 9, pp.
273-278, Apr 2014.
C. Soci, A. Zhang, X. Y. Bao, H. Kim, Y. Lo, and D. L. Wang, "Nanowire Photodetectors," Journal of
Nanoscience and Nanotechnology, vol. 10, pp. 1430-1449, Mar 2010.
Z. X. Wang, M. Safdar, C. Jiang, and J. He, "High-Performance UV-Visible-NIR Broad Spectral
Photodetectors Based on One-Dimensional In2Te3 Nanostructures," Nano Letters, vol. 12, pp.
4715-4721, Sep 2012.
A. Zhang, H. Kim, J. Cheng, and Y. H. Lo, "Ultrahigh Responsivity Visible and Infrared Detection
Using Silicon Nanowire Phototransistors," Nano Letters, vol. 10, pp. 2117-2120, Jun 2010.
R. L. Petritz, "THEORY OF PHOTOCONDUCTIVITY IN SEMICONDUCTOR FILMS," Physical Review,
vol. 104, pp. 1508-1516, 1956.
N. Matsuo, H. Ohno, and H. Hasegawa, "Mechanism of high gain in GaAs photoconductive
detectors under low excitation," Japanese Journal of Applied Physics, Part 2 (Letters), vol. 23, pp.
L299-L301, May 1984.
E. Munoz, E. Monroy, J. A. Garrido, I. Izpura, F. J. Sanchez, M. A. SanchezGarcia, et al.,
"Photoconductor gain mechanisms in GaN ultraviolet detectors," Applied Physics Letters, vol. 71,
pp. 870-872, Aug 18 1997.
C. Soci, A. Zhang, B. Xiang, S. A. Dayeh, D. P. R. Aplin, J. Park, et al., "ZnO nanowire UV
photodetectors with high internal gain," Nano Letters, vol. 7, pp. 1003-1009, Apr 2007.
V. J. Logeeswaran, J. Oh, A. P. Nayak, A. M. Katzenmeyer, K. H. Gilchrist, S. Grego, et al., "A
Perspective on Nanowire Photodetectors: Current Status, Future Challenges, and
Opportunities," Ieee Journal of Selected Topics in Quantum Electronics, vol. 17, pp. 1002-1032,
Jul-Aug 2011.
G. E. Zardas, C. J. Aidinis, E. A. Anagnostakis, and C. I. Symeonides, "On a Predictive Scheme of
Slow Photoconductive Gain Evolution in Expitaxial Layer/Substrate Optoelectronic
Nanodevices," Open Journal of Microphysics, vol. 1, pp. 32-34, 2011.
G. Konstantatos, I. Howard, A. Fischer, S. Hoogland, J. Clifford, E. Klem, et al., "Ultrasensitive
solution-cast quantum dot photodetectors," Nature, vol. 442, pp. 180-183, Jul 2006.
J. S. Jie, W. J. Zhang, Y. Jiang, X. M. Meng, Y. Q. Li, and S. T. Lee, "Photoconductive characteristics
of single-crystal CdS nanoribbons," Nano Letters, vol. 6, pp. 1887-1892, Sep 2006.
O. Lopez-Sanchez, D. Lembke, M. Kayci, A. Radenovic, and A. Kis, "Ultrasensitive photodetectors
based on monolayer MoS2," Nature Nanotechnology, vol. 8, pp. 497-501, Jul 2013.
[18]
[19]
[20]
Z. Y. Yin, H. Li, L. Jiang, Y. M. Shi, Y. H. Sun, G. Lu, et al., "Single-Layer MoS2 Phototransistors,"
Acs Nano, vol. 6, pp. 74-80, Jan 2012.
Y. Dan, "Optoelectronically probing the density of nanowire surface trap states to the single
state limit," vol. 106, ed: Applied Physics Letters, 2015, p. 053117.
M. M. Furchi, D. K. Polyushkin, A. Pospischil, and T. Mueller, "Mechanisms of Photoconductivity
in Atomically Thin MoS2," Nano Letters, vol. 14, pp. 6165-6170, Nov 2014.