Derivation of photocurrent and decay time in the hopping
photoconductive regime
Following Ref. [1], it is possible to draw a parallel between hopping transport (HT) in an
exponential density of states (DOS) tail and multiple trapping and release (MTR) model, with the
transport level in the former playing the role of the mobility edge in the latter. Synthetically
speaking, in HT (MTR) only carriers excited to the transport level (mobility edge) contribute to
current flow.
To derive an expression for the photocurrent in HT regime we assume that holes have a negligible
mobility thus giving a negligible contribution to the current, we disregard diffusion terms and
assume the electric field to be uniform, hence:
J ph  q n
V
.
L
(A1)
Our target is to express the electron density n and the mobility  (which is charge density
dependent) as a function of the volume photogeneration rate G . To this extent we observe that
since holes are practically immobile, an electron has a chance to encounter a hole and finally to
recombine only during its site-to-site hopping, which means only when excited to the transport
energy. Therefore we write the recombination rate:
R
ntr

,
(A2)
where we have introduced the density of carriers excited to the transport energy ntr ( n indicates as
usual the total electron density), whose lifetime  is

1
,
 v th p
(A3)
where  is the recombination center cross section, v th is the thermal velocity of carriers at the
Supplemental material for “Hopping photoconductivity in an exponential density of states”
1
transport level (vide infra), p is the density of recombination centers (since immobile holes are the
ones and only recombination centers). Combining Eqs. A2 and A3 and reminding that in the steady
state R  G and that n  p because ohmic contacts ensure charge neutrality, we obtain a relation
between n , ntr and G :
n
G
.
 v th ntr
(A4)
Then, by expressing both n and ntr as a function of the quasi Fermi level Ef , it is possible to solve
for Ef and to obtain Ef as a function of G . Once Ef G  is known, n G  and  G  are known
and can be substituted in Eq. A1.
The relation between the electron density n and the Fermi energy is readily obtained:
n
0
Ef


 g E  f E  dE   g E  dE  Nt e
Ef
kTc
,
(A5)
where g(E )  Nt kTc  exp E kTc  (for E  0 ), Nt stands for the total number of electron states,
Tc is the DOS characteristic temperature and the Fermi-Dirac statistics has been approximated with
a step function centered at Ef .
The key point is to obtain an expression for ntr  Ef  , a quantity which is rarely dealt with in HT
models. To this extent, we observe that electrical conductivity  can be expressed as the product of
the total carrier concentration with the mobility, viz.   n (where  is the classical bulk
mobility), or equivalently as the product of carrier density at the transport energy with the effective
mobility which pertains to electrons at transport energy, viz.   tr ntr . Since  n  tr ntr , we
define
ntr  n

.
tr
(A6)
Starting from the expression for chemical diffusion coefficient D given in Ref [1,2] and exploiting
Supplemental material for “Hopping photoconductivity in an exponential density of states”
2
the generalized Einstein relation (which reads D   kTc q in an exponential DOS) we calculate
 [3], thus obtaining:
 1
rtr
1 
qrtr2 0 2   Etr Ef  kT  kTc 

e

kT
 

where  is the localization radius, 0 is the attempt to escape frequency and rtr is the effective jump
distance:
rtr 
3Tc

2T
A8)
Regarding tr , we firstly derive an expression for the jump diffusion coefficient Dtr , j of carriers at
2
the transport level and then exploit the Einstein relation [4]. In general D j   rh , viz. it is the
product of a mean effective jump frequency with a mean square of jump distance. In our specific
case, once that an occupied tail state becomes isoenergetic with the transport level (thanks to the
absorption of a suitable amount of thermal energy), the electron residing there can tunnel to the
transport level over a mean distance given by rtr and with an effective jump frequency  tr which
takes into account the tunneling nature of the process [1]:
 tr   0e
2
rtr

  0e
T
3 c
T

 A9)
The jump diffusion coefficient is then given by
Dtr , j 
rtr2 tr

rtr2 0 e
T
3 c
T
,
(A10)
and by means of Einstein relation we finally obtain
qr 2 3 c
tr  tr 0 e T .
kT
T
(A11)
By substitution of Eq. A11, A7 and A5 in Eq. A6, we finally obtain an expression for ntr :
Supplemental material for “Hopping photoconductivity in an exponential density of states”
3
ntr  Nt e
Etr
kTc
e

Etr Ef
kT
.
(A12)
From Eq. A12, ntr can be thought as due to the excitation from Ef to an energy level located at Etr
having an effective density of states given by Nt exp Etr kTc  . The effective density of transport
energy states turns out to be equal to the number of states contained in a kTc -wide DOS slice
centered at Etr .
By means of Eqs. A8 and A9 we can also calculate the thermal velocity which can be defined as the
product of the effective jump distance with the effective jump frequency:
3Tc 0 3 Tc
.
e
2T
T
v th 
(A13)
Then combining Eqs. A4, A5 and A12 we obtain an expression for the quasi Fermi level as a
function of G :



kT
G
Ef  Etr 
ln 

T
Etr

1

k
Tc   v th  Nt e Tc






.
2
 

 
 
(A14)
Substituting Eq. A14 in Eqs. A5 and A7, and combining with Eq. A1 we finally obtain
J ph
Etr
V 
kTc
 qtr
Nt e
L

9  q  0  Tc 

4 kT  T 
2 2
T Tc
T Tc



2
T
3 c
e T
 Tc 


 G  T Tc 


  v th 
Etr
V 
kTc
Nt e
L

T Tc
T Tc



Tc
 G T Tc
.



v
 th 
(A15)
Now we derive an expression for the photocurrent fall time, defined as the time needed to reduce
the photocurrent by a factor e of its steady state value upon switching the light stimulus off. For
this to occur, the concentration of carriers at the transport energy ntr has to drop by the same
Supplemental material for “Hopping photoconductivity in an exponential density of states”
4
fraction and this is accomplished by lowering the quasi Fermi level by kT . But this implies that a
kT -wide slice of states below the steady state quasi Fermi level Ef gets emptied of carriers. To
this extent, carriers have to be excited to the transport energy and then have to recombine.
Assuming that the latter process is the dominant one, the fall time is expressed by:
E f ,0
 dec 
E
f ,0  kT
g( E )dE
R
T

Tc

 Nt e


Etr
kTc
Tc T
T
 Tc



T
 1 T Tc


  v th 
Tc
 1 T Tc
.
 
G 
Supplemental material for “Hopping photoconductivity in an exponential density of states”
(A16)
5
Experimental: device realization and electrical characterization
Devices realization follows the procedure outlined in Ref. [5]. Squaraine (SQ) was synthesized
according to Ref. [6]; [6,6]phenyl-C61-butyric acid methyl ester (PCBM) was 99 % pure (SigmaAldrich). SQ and PCBM were blended in chloroform at a concentration of 19.2mg ml in a 1:3 by
weight ratio (respectively). Prolonged stirring (1 night) was applied before deposition by spincoating, according to a two step scheme ( 100 rpm for 60s followed by 1000 rpm for 60s ).
Photodetectors were realized on fused silica substrates with pre-litographed interdigitated gold
electrodes (thickness 40 nm ) defining interelectrode spacings of 3, 6 and 12 μm and equivalent
width of 5 mm . Thin film transistors (TFT) were realized on substrates made of p+-doped silicon
acting as gate, covered by a 130 nm thick silicon dioxide layer, onto the which gold source and
drain electrodes were lithographed to give a bottom-gate bottom-contact configuration (channel
length (L) and width (W) 6 μm and 15 mm respectively).
Prior to active blend deposition, both substrates for photodetectors and TFTs were functionalized
with a self-assembled monolayer of hexamethyldisilazane (HMDS, Aldrich): after heating up to
110o C for moisture desorption, during cool down substrates were exposed for 15 minutes to
HMDS vapor in a closed Petri dish. The rationale for introducing this step is to improve electron
transport by reducing electron trapping phenomena due to silica hydroxyl groups [7].
All the electrical characterizations were performed in vacuum at 10-6 mbar . Photoconductors were
irradiated by a set of light emitting diodes; current signals were measured with a Keithley SCS2400
in case of continuous wave acquisitions, or by means of a low-noise transimpedance amplifier
connected to an oscilloscope for time resolved measurements. Light intensity was changed by
modulating the driving current of the light emitting diodes and/or by interposition of neutral density
optical filters, and every measuring condition was calibrated thanks to the use of a silicon pin
photodiode.
Supplemental material for “Hopping photoconductivity in an exponential density of states”
6
Transistors were tested in the saturation regime by biasing the source to ground and sweeping gate
and drain (tied together) from 0 V to + 30 V ( -30 V ) to test electrons (holes) mobility [8].
According to Ref. [9], to extract the mobility current measurements were fitted using the following
expression:
ITFT  Cox
W
1
0
VG  VTH 2 ,
L
2
(A17)
where Cox is the dielectric specific capacitance and the mobility is given by a power law as

  0 VG  VTH  . In order to estimate the charge carrier concentration in channel, we
approximate the channel thickness by its Debye length Ld  2 kT  qCox Vg  Vth  [9] and we
calculate the carrier concentration as n  Cox Vg  Vth  qLd (in the gate voltage range where Ld is
larger than 1nm ). Both n and p type TFT transfer characteristic curves are reported in Fig. 1.
10
-8
10
-9
0 = 5.3310 cm /Vs
-7
2
-6
10
0
= 2.6110 cm /Vs
-4
2
= 0.45
= 0.47
10
ITFT [A]
ITFT [A]
-7
10
-8
10
-10
(a)
-5
-10
-15
VG,VD [V]
-20 -25 -30
(b)
-9
10
10
15
20
25
30
35
VG, VD [V]
Figure 1. TFT transfer characteristic curves ( 12μm channel length) obtained by applying at both
drain and gate contacts the same voltage, sweeping from 0 V to 30 V . Holes (a) and electrons (b)
conduction were tested by exploiting negative and positive voltages respectively. Fittings of these
curves with Eq. A17 are shown by red lines, and fitted parameters are reported in the insets.
Supplemental material for “Hopping photoconductivity in an exponential density of states”
7
Reported below are additional measurements which constitute fingerprints of photoconductive
behavior. Figure 2 shows that the reciprocal of the photocurrent scales linearly with the device
length and the linear fitting extrapolation crosses the axes origin. Figure 3 shows that the product of
photocurrent and decay time is independent on the incident optical power.
Finally, Fig. 4 depicts the electron quasi Fermi level as a function of the incident power density
according to Eq. A14, showing that throughout the explored range it lies below the transport level,
which is a mandatory condition for the validity of Eq. A7.
Figure 2: Plot of the reciprocal of the photocurrent versus the device length, ( 30 V bias voltage and
6 mW cm 2 incident power density). The substantial agreement with a linear fitting (red line)
crossing the axes origin is a clear mark of ohmic behavior of contacts.
Supplemental material for “Hopping photoconductivity in an exponential density of states”
8
Figure 3. Product of photocurrent and decay time as a function of the incident power density: from
the measurement an average value of 8.44×10-13 A s has been extracted (red line).
Figure 4. Quasi Fermi level as a function of the incident power density. In the explored range, Ef
lies below Etr (represented by the red line) by more than 200 meV .
Supplemental material for “Hopping photoconductivity in an exponential density of states”
9
[1]
J. Bisquert, Phys. Chem. Chem. Phys. 10, 3175 (2008).
[2]
J. Bisquert, J. Phys. Chem. C 111, 17163 (2007).
[3]
While in Ref [1] and Ref. [2] the chemical diffusion coefficient is given by
D  9 4 Tc T   2 0 1  T Tc  exp 3Tc T   Etr  Ef 1 kT  1 kTc  , here we used a
3
modified expression without the multiplicative factor given by 1  T Tc  . This latter factor
comes from approximating the Fermi-Dirac distribution as a unit function for energies up to
the Fermi level and as a Boltzmann factor for higher energies in Refs. [1,2]. But in the
specific case of our squaraine-based photoconductor, which has a Tc in large excess of T ,
we verified that a step-like approximation of the Fermi-Dirac distribution is more accurate
for the calculation of n in Eq. A6, hence we write the chemical diffusion coefficient in the
form D  9 4 Tc T   2 0 exp  3Tc T   Etr  Ef 1 kT  1 kTc  .
3
[4]
In the case of exponential DOS (and in general when carrier statistics cannot be
approximated by a Boltzmann distribution, which is usually the case for organic
semiconductors), the jump diffusion coefficient D j differs from the chemical diffusion
coefficient D appearing in the expression of diffusion current by the so-called
thermodynamic factor  , which in an exponential DOS reads Tc T . The ratio between the
jump diffusion coefficient and the mobility obeys the Einstein relation, viz. D j   kT q ,
which we actually exploit to derive tr in Eq. A11. The ratio between the chemical diffusion
coefficient and the mobility obeys the generalized Einstein relation, viz.
D    kT q  kTc q , which we actually used to derive  in Eq. A7.
[5]
M. Binda, T. Agostinelli, M. Caironi, D. Natali, M. Sampietro, L. Beverina, R. Ruffo, and F.
Silvestri, Org. Electron. 10, 1314 (2009).
Supplemental material for “Hopping photoconductivity in an exponential density of states”
10
[6]
L. Beverina, A. Abbotto, M. Landenna, M. Cerminara, R. Tubino, F. Meinardi, S.
Bradamante, and G. A. Pagani, Org. Lett. 7, 4257 (2005).
[7]
L. L. Chua, J. Zaumseil, J. F. Chang, E. C. W. Ou, P. K. H. Ho, H. Sirringhaus and R. H.
Friend, Nature 434, 194 (2005).
[8]
Working at VGD  0V is advantageous because when testing electrons (holes) mobility,
accumulation of holes (electrons) is suppressed at drain side thus avoiding ambipolar
behavior and simplifying curve fitting.
[9]
L. Fumagalli, M. Binda, D. Natali, M. Sampietro, E. Salmoiraghi, and P. Di Gianvincenzo, J.
Appl. Phys. 104, 084513 (2008).
Supplemental material for “Hopping photoconductivity in an exponential density of states”
11