Download How a structured vibrational environment controls the performance

How a structured vibrational environment controls the performance of a
photosystem II reaction centre-based photocell
Richard Stones,1 Hoda Hossein-Nejad,1 Rienk van Grondelle,2 and Alexandra Olaya-Castro1, a)
1)
arXiv:1601.05260v1 [physics.bio-ph] 20 Jan 2016
Department of Physics and Astronomy, University College London, Gower Street, London,
WC1E 6BT United Kingdom.
2)
Department of Physics and Astronomy, VU University, 1081 HV Amsterdam,
The Netherlands
Photosynthetic reaction centres are pigment-protein complexes that can transform photo-induced electronic
excitations into stable charge separated states with near-unit quantum efficiency. Here we consider a theoretical photovoltaic device that places a single photosystem II reaction centre between electrodes to investigate
how the mean photo-current and its fluctuations depend on the vibrational environment that assists energy
and electron transfer. Our results indicate that selective coupling to well resolved vibrational modes does
not necessarily offer an advantage in terms of power output but does lead to photo-currents with suppressed
noise levels. The exciton manifold and the structured vibrations assisting electron transfer can also support
the emergence of a phenomenon akin to dynamical channel blockade, whereby excitonic traps can impose
competing routes for population transfer under steady state operation. Our results help characterizing the
device-like functionality of these complexes for their potential integration into molecular-scale photovoltaics.
INTRODUCTION
Life on Earth is fueled by photosynthesis, the process by which plants, algae and certain bacteria convert solar energy into stable chemical energy1 . The initial electron transfer steps during solar energy conversion by these organisms take place in photosynthetic
reaction centres (PRCs), sophisticated trans-membrane
supramolecular pigment-protein complexes that exhibit
a dual device-like functionality. Under illumination, a
PRC complex effectively operates as Nature’s solar cell1
where electronic excitations of chromophores are transformed into stable charge-separated states, with electron
donor and electron acceptor separated by a few nanometres. This sub-picosecond charge separation process occurs with near unit quantum efficiency implying that almost every quanta of energy absorbed results in charge
separated across the PRC2,3 . The same chromophoreprotein structure and energetic landscape promoting this
quantum yield also favours a diode-like behaviour of all
PRCs such that under an appropriate applied bias, electric current flows mostly in one direction4 . Their functional versatility, nanometre size, and near-unit quantum efficiency makes PRCs promising components of the
next-generation of photovoltaic and photoelectrochemical cells5,6 as well as of biomolecular electronics4,7,8 .
A step further in this field is the recent development
of single-molecule techniques that allow measurement of
the photocurrent through a single PRC complex9 . Using
cysteine group mutations, it has been possible to bind
a Photosystem I unit to a scanning probe gold-tip that
acts as both an electrode and a localized light source to
excite and measure the photocurrent of a fully functional
PRC9 . These experiments open up a new platform to
a) Electronic
mail: [email protected]
carry out further investigations on the microscopic principles underlying charge separation in PRCs. In particular, it is foreseeable that besides measuring the currentvoltage features these techniques may allow characterisation of current fluctuations and the associated full counting statistics of electron transport in PRCs. In quantum
transport setups it has been shown that such fluctuations can reveal unknown intrinsic dynamical features of
the quantum system through which electron transport
occurs, including the influence of electron-phonon interactions and coherence10 . A theoretical study along the
lines of full counting statistics for light-harvesting complexes has been carried out11 but so far there has been
no investigation of full counting statistics of charge transport in PRCs.
Charge separation in reaction centres has been subject to extensive spectroscopic studies such that there
is a wealth of knowledge about electron transfer kinetics and charge separation pathways in PRCs isolated
from different organisms. Despite these efforts, however,
the detailed quantum mechanical features underpinning
this process are still unclear and under scrutiny. Both
steady-state and multi-dimensional optical spectroscopy
have revealed that the formation of stable charge separated states in bacterial and plant reaction centres is
strongly affected by the coupling of electronic degrees of
freedom to a wide range of vibrational motions12–14 . Recent works, reporting picosecond lasting quantum beats
among exciton-charge transfer states in the photosystem
II reaction centre (PSIIRC) argue that such phenomena
is supported by the coupling of the excited state to underdamped intramolecular vibrations15 . The possible implications of such vibrations for the electric current output
of a single PRC are unknown.
In this work we envision a photocell device where a
single PSIIRC is placed between two electrodes and investigate the effects of structured vibrational motions in
the photo-induced current and current noise under con-
2
tinuous illumination. Our results show that the noise
strength, as quantified by the Fano Factor, effectively
probes both the structure of exciton manifold and the
structured phonon-environment assisting charge separation. The exciton manifold guarantees a multi-step population transfer process leading to sub-Poissonian statistics. By comparing the situations of a smooth vibrational
environment as opposite to one structured with well resolved vibrational motions, we show that selective coupling to underdamped vibrational modes does not necessarily offer an advantage in terms of power output but
they lead to output currents with suppressed noise levels. This indicates that underdamped vibrations coupled
to charge separation promote ’ordered’ electron transport. Furthermore we show that under these steady state
operation, excitons localized in the non-active branch
together with the structured environment can support
competing dynamical processes during charge separation,
resulting in a transition from sub- to super-Poissonian
electron current statistics. This phenomenon is akin
to dynamical channel blockade observed in solid-state
systems10 . Overall our work enables the formulation of
several design principles for the optimisation of PRCs to
be integrated in nano-electronic or photovoltaic applications.
RESULTS
A photocell based on the photosystem II reaction centre
The prototype complex we consider is the PSIIRC
found in higher plants and algae1 . Crystallography has
provided the arrangement of the chromophores involved
in primary charge transfer16 . As illustrated in Fig. 1 (a),
four chlorophylls (Chls) and two pheophytins (Phe) are
arranged in two branches (D1 and D2 ), where D1 and D2
label the chlorophyll binding proteins in the core of the
reaction centre. The two central chlorophylls are known
as the special pair (PD1 and PD2 ) and they are flanked
by the accessory chlorophylls ChlD1 and ChlD2 , and the
pheophytins PheD1 and PheD2 . Charge separation only
occurs down the D1 branch17,18 . Nonlinear spectroscopy
has revealed that at least two different excited states,
(PD1 PD2 ChlD1 )∗ and (ChlD1 PheD1 )∗ , give rise to two
different pathways for charge separation along the D1
branch13,19,20 where the likelihood of each depends on
the specific protein configuration and the corresponding
disorder of pigment excitation energies19 .
Although the relative contribution of these pathways in
ensemble measurements is not conclusively known, spectroscopy at low temperature and its corresponding theoretical fit indicate that electron transfer is predominantly
initiated from the state (ChlD1 PheD1 )∗20 . Our analysis therefore focuses on the ChlD1 pathway. ChlD1 is
the lowest energy pigment and sufficiently strong interpigment electronic couplings lead to formation of delocalized exciton states upon photo-excitation. This
pathway can therefore be identified as the population
of an exciton state with the largest amplitude in the
pair (ChlD1 PheD1 )∗ which channels energy to an initial
−
charge transfer (CT) state |Chl+
D1 PheD1 i and finally to−
wards a stable or secondary CT state |P+
D1 PheD1 i, with
electron and hole residing on different pigments as illustrated in Fig. 1(b). We consider the exciton and CT state
space as well as the quantum dynamical evolution that
has provided a good description for both steady-state
and transient spectroscopy while reproducing the experimentally measured time scales during primary charge
separation13,15,20 . This model describes charge separation dynamics via rate equations of population transfer combining modified Redfield theory and generalized
Förster theory20 as we illustrate it in Fig. 2 (a). Within
this picture formation of the final charge separated state
is strongly influenced by the competing rates of relaxation among exciton states and rates of transfer from
such excitons to the intermediate CT state. Population
transfer rates reflect the interplay between the electronic
interactions, among excited pigments and between these
and CT states, and the interaction between electronic
states and a wide range of vibrational motions assisting
charge separation. Indeed, fluorescence line narrowing
experiments have given evidence of a highly structured
spectral density of fluctuations associated to electronically excited chromophores21 as shown in Fig. 2 (b).
Thus in this work we examine two key questions: (i)
how does the statistics of electrons flowing through this
PRC witness the exciton manifold leading to charge separation? and (ii) how is such statistics affected by the
structured vibrational environment assisting transitions
among electronic states.
To address the above questions, we consider a theoretical photocell device in which a single PSIIRC unit
is placed between two leads that can supply or take
away electrons from the system, as illustrated in Fig.
1 (a). The charge transfer cycle represented in Fig.
1(b) assumes that electrons are pumped from the left
lead at rate ΓL and leave the system from the right
lead at rate ΓR while the sample is incoherently photoexcited at rate γex . We use values of ΓL = 201cm−1
and γex = γn = 75cm−1 for the upwards rate, where
γ = 0.00125cm−1 and n = 60000, to simulate excitation by concentrated solar radiation22 . ΓR will be varied
to change the voltage between the system and the drain
lead. The state space of the PSIIRC-based photocell is
shown in Figure 2 (a). In this model we include all six
core reaction centre chromophores distributed in the D1
and D2 branches. Although the D2 branch is not directly
involved in charge separation excitons localised on this
branch can act as electronic traps20 thereby affecting the
statistics of electrons flowing through the system, as we
will discuss later. Therefore, the Hilbert space spans the
following states: the ground state |gi, six exciton states
−
|e1 i to |e6 i, the initial CT state |Chl+
D1 PheD1 i ≡ |Ii,
−
+
the secondary CT state |PD1 PheD1 i ≡ |αi and the positively charged state |P+
D1 PheD1 i ≡ |βi which represents
3
FIG. 1: Photosystem II reaction centre schematic model. (a) Schematic diagram of the photocell device
setup with isolated PSIIRC unit. (b) Charge transfer cycle for our model. (i) Neutral ground state of the system
after an electron has been replenished at the special pair. (ii) Excited state manifold after excitation by a photon.
(iii) Primary charge transfer state of the ChlD1 charge separation pathway. (iv) Secondary charge transfer state. (v)
Positively charged state after an electron has been removed from the system.
FIG. 2: Photosystem II reaction centre microscopic model. (a) Energy level diagram showing state space of
the model. Modified Redfield rates are used to describe relaxation between exciton states (labelled |en i).
Generalised Förster theory (GFT) / Marcus theory is used to model charge transfer rates. Phenomenological rates
are indicated with red arrows. The transition |αi → |βi represents hopping of an electron from the system to the
drain lead. |βi → |gi represents an electron hopping from the source lead into the system. The ground state is
coupled to a radiation field exciting the system to the lowest energy exciton state |e1 i. (b) The components of the
spectral density used in the model. The Drude part is scaled relative to the high energy part for clarity. Mode
parameters are shown in Supplementary Table S1.
the ’empty’ state of the system for counting statistics. In
the above we have assumed low enough excitation rates
to guarantee that only single excitation states are populated.
The system dynamics is described by a Lindblad master equation of the form
ρ˙s (t) = (Lex + Lrelax + LCT + Lleads )ρs (t),
(1)
where Lex defines the coupling to an incoherent radi-
ation field which excites the system from the ground
state to the exciton manifold, Lrelax describes relaxation
rates between the exciton states using modified Redfield
theory23,24 , LCT defines initial and secondary CT processes using Marcus/Förster theory24,25 and Lleads describes coupling of the system to the leads. As explained
in the methods section, from this Liouvillian the full
counting statistics can be calculated using Eq. 15 and
enforcing both the Coulomb blockade regime such that
probability of two electron occupancy is negligible26 , and
4
the infinite bias limit enforcing unidirectional electron
flow27 .
The current counting statistics depends on the induced
voltage across a load connecting the states |αi and |βi
with associated energy gap Eαβ = Eα − Eβ . As usually
done in photocells28 , such a voltage is estimated via a
detailed balance condition where the ratio between the
steady state populations of the secondary CT state (ραα )
and the ‘empty’ state (ρββ ) is equilibrated at a temperature T :
ραα
eV = Eαβ + kB T ln
,
(2)
ρββ
with kB is the Boltzmann constant. The population ratio, and associated voltage across the load, can therefore be varied by changing the rate at which electrons
leave the system ΓR . When ΓR → 0 there is a large load
across the PSIIRC-based photocell, defining the open circuit regime. In the opposite limit, when ΓR → ∞, the
voltage tends to zero and defines the short circuit regime.
These limits describe the operation extremes in which the
photocell delivers no power.
To investigate the effect of the structured vibrational
environment on the statistics of the transition from |αi
to |βi, we compare the situations in which (i) the full
structured spectral density J(ω) = JD (ω) + JM (ω) is
considered as opposite to the situation where (ii) only
the smooth low energy component JD (ω) is accounted
for. We denote these cases as structured and smooth
environments. The low and high energy components of
the spectral density are depicted in Fig. 2 (b) and are
given by13 :
JD (ω) =
JM (ω) =
2λωΩc
,
ω 2 + Ω2c
X
2λj ωj2 γj ω
j
(ω 2 − ωj2 )2 + γj2 ω 2
(3)
.
(4)
JD (ω) is the Drude form of a spectral density describing an overdamped Brownian oscillator where λ and Ωc
are the reorganisation energy and cut off frequency, respectively. JM (ω) has been measured experimentally21
and describes a number of discrete high energy modes
coupled to excited pigments, with λj , ωj and γj being
the reorganisation energy, frequency and damping associated with mode j respectively. All parameters used in
the model are detailed in the supplementary information.
the low-energy thermal component of the spectral density
is considered. As expected in both situations, the current
is zero in the open circuit regime and it exhibits a sharp
increase to a constant value as the voltage decreases. The
power delivered by the photocell in both situations is
maximum for a voltage close to Eαβ , that is, when the
steady state populations ραα and ρββ are similar.
The most important feature of Fig. 3 is the fact that
the constant current and maximum power are significantly lower for a photocell with a structured environment. This contrasts with the power enhancement predicted for a simple light-harvesting device that includes
a single resonant mode coherently coupled to electronic
transitions29 .
In our system a wide range of vibrational motions influence the transfer rates among excitons and between
excitons and CT states. Although the energy of some of
the high-energy modes may be close to the excitonic energy gaps, this feature is not satisfied for the majority of
the transitions in the PRC, unlike the case of some lightharvesting complexes30 . Specifically, there is no quasiresonance between vibrational motions and the energy
mismatch between excitons and the primary CT state.
More importantly, CT states are strongly coupled to the
vibrational background with much larger reorganisation
energies than exciton states. This results in a larger shift
between the absorption lineshape of a CT state and the
emission lineshape of an exciton state yielding a reduced
overlap. Consequently transfer rates from excitons to
CT states are decreased. One can see this as a Zeno-like
effect whereby the strongly coupled environment ”measures” the population of a CT state at a very high rate
thereby slowing transfer.
The stronger coupling to the environment of CT states
compared to exciton states, favours forward electron
transfer. This is illustrated for a donor-acceptor dimer
system with a PSIIRC structured spectral density in Fig.
4 (a). The ratio of forward and backward transfer rates
between the donor and acceptor is shown as a function
of the scaling constant for the coupling between the acceptor CT state and the environment. Starting from a
coupling scaling of 3, which we see for the secondary CT
state in the ChlD1 pathway, the transfer rate ratio increases at all temperatures. At lower temperatures this
improvement in the forward transfer is enhanced. Fig. 4
(b) shows the temperature dependence of the current for
our PSIIRC-based photocell. The increase in the current
without coupling to a structured environment is seen at
all temperatures, though interestingly there is an optimal
operational temperature for the photocell.
Photocell current-voltage and power performance
The characteristic current-voltage (I-V) and power (P)
curves depicted in Fig. 3 for our PSIIRC-based photocell
are obtained with simple expressions hIi = eΓR ραα and
P = hIiV . Figure 3 presents the characteristic curves
for the two situations under consideration: (i) when the
spectral density is fully accounted for and (ii) when only
Zero-frequency noise
Full performance characterization of a photocell cannot be limited to the mean current (and associated power
curve). The current fluctuations quantified by second
and higher order cumulants (hI n ic with n > 1) can
5
2.0
4.0
2.4
3.5
J( )
3.0
1.7
2.5
1.0
2.2
2.0
2.0
1.5
1.5
0.5
k f or war d / k backwar d
1.6
JD ( )
power(P / e)
power
current (I / e)
1.5
1.0
0.2
0.4
0.6
0.8
1.0
volt age (V)
1.2
1.4
1.6
1.8
1.6
T = 77.0K
T = 200.0K
T = 300.0K
1.4
0.5
0.0
0.0
2.0
1.2
3.0
0.0
3.2
3.4
3.6
3.8
accept or environm ent coupling scaling
4.0
2.
give information on the correlations between elementary
charge transfer events. In the context of our question
of interest, these fluctuations can then reveal how the
structured vibrational environment may promote or hinder such correlations and their associated noise features.
Here we focus on the long-time limit or zero-frequency
regime of the relative noise strength which is quantified
by the second order Fano factor31 :
F (2) =
hI 2 ic
.
hIi
(5)
This ratio between the zero-frequency second order current cumulant and the mean current asserts the deviation of the underlying statistical process from a Poissonian distribution. A Fano factor of 1 indicates a Poissonian process without correlations among charge transfer events. While deviations from 1 are interpreted as
either super-Poissonian (F (2) > 1) or sub-Poissonian
(F (2) < 1), regimes that can be associated with highly
fluctuating or more stable currents, respectively.
In mesoscopic and quantum systems the Fano factor
has proven to be very sensitive to the structure of the
state space [ref] as well as on the electron-phonon interactions [ref]. This is precisely what is indicated by the
results for the Fano factor of the current through our
photocell device shown in Fig. 5. The analytic form
of the Fano factor for our photocell model is too cumbersome. To gain insight into the behaviour presented
in Fig. 5 it is therefore useful to consider the case of a
single resonant level (SRL)27 . The dynamics of this sys-
2.0
/
h
1.
current
FIG. 3: Mean current (black) and power (red)
versus voltage. For both plots the dashed lines
indicate the results for a Drude spectral density while
solid lines indicate the inclusion of high energy modes.
Calculations carried out at 300K with excitation rate
γ = 0.00125cm−1 and solar photon number n = 60000.
See Supplementary Information for all other parameters.
1.0
0.
0.0
100
1 0
200
t em perat ure K
2 0
300
FIG. 4: Favouring forward electron transfer. (a)
Ratio of forward and backward transfer rates versus
acceptor environmental coupling scaling for a
donor/acceptor dimer system with a PSIIRC spectral
density. The donor-acceptor energy gap is 200cm−1 and
the electronic coupling is 40cm−1 . (b) Current versus
temperature for a PSIIRC photocell model at a fixed
voltage of 1.28V, excitation rate γ = 0.00125cm−1 and
solar photon number n = 60000. See Supplementary
Information for all other parameters.
tem in the basis {|occupiedi, |emptyi} is governed by a
Liouvillian with matrix elements L11 = −L21 = ΓR and
L22 = −L12 = ΓL such that the Fano factor as a function
of voltage takes the form
F (2) (V ) =
1 + exp[−2(eV − E0 )/kB T ]
,
(1 + exp[−(eV − E0 )/kB T ])2
(6)
where E0 is the energy gap between the occupied and
empty states. This function (see Supplementary Fig.
S2) is symmetric about its minimum and approaches
1 at large and small voltages, where electron transfer
events are rare and essentially uncorrelated. The mini-
6
mum occurs for the voltage associated to electrons entering and leaving the system at the same rate (ΓR /ΓL =
ρoccupied /ρempty = 1). This sub-Poissonian behaviour
is a manifestation of the Coulomb blockade regime where
the presence of an electron in the system prevents another
one entering until the system is empty, thereby reducing
the noise.
Figure 5 shows that the noise profiles in both cases,
structured and smooth environment,are not symmetric
around their minima with values lower than 1 for small
voltages. This is due to the fact that transfer rates are
such that the non-equilibrium steady state operation of
the photocell samples the manifold of exciton states assisting charge separation. Specifically, electron transfer
events into the drain lead witness multi-step transfer nature of the process which is ensured by the manifold of
excitons coupled to the primary CT state.
Nevertheless, noise levels are overall lower with the
structured environment as a consequence of a rich interplay between phonon-assisted transfer among excitons as
well as from excitons to primary CT state and the transfer between CT states. To illustrate this interplay we
consider the situation where the rate of secondary CT
transfer is very slow compared to relaxation rates within
the exciton manifold and the rates between the excitons
and primary CT state. In this case, population of this
stable CT state is so slow that all the internal transfers
from the exciton manifold to the primary CT can be described a single step process i.e. there is no sampling
of the multi-level exciton structure, and the Fano factor
tends to 1 for small and larger voltages as shown by the
dotted line in Fig. 5 (top). This means that the system behaves effectively as a SRL (Cf. Fig. 5 (top) with
Supplementary Figure S2). The time scale of secondary
charge transfer is therefore a limiting time-scale which
can lead to a variety of phenomena as it will be further
explored in the next section.
All the above results are computed at T = 300K.
We can therefore conclude that at room temperature the
PSIIRC-based photocell with the structured vibrational
environment delivers less power than a photocell with the
unstructured environment, yet still this is accompanied
by a suppression of current fluctuations and associated
noise close to a minimum of 1/2. This is a consequence of
the nontrivial interplay between vibration-assisted exciton and charge transfer processes underlying the function
of the PSIIRC.
Competing dynamical processes
As discussed above, the transfer rate from primary to
secondary CT state has a dramatic effect on the noise
properties of the PSIIRC-based photocell: it is such that
the non-equilibrium steady state operation of the photocell samples the manifold of exciton states assisting
charge separation. We now investigate the Fano factor
considering an increased secondary charge transfer rate
FIG. 5: Fano factor versus voltage. (a) Top panel:
shows Fano factor for the PSIIRC-based photocell with
the structured environment for a modified (slow)
(dotted line) and the measured (solid line) secondary
charge transfer rate. (b) Bottom panel shows the Fano
factor for the PSIIRC photocell with the smooth
low-energy vibrational environment. Calculations
carried out at 300K with excitation rate
γ = 0.00125cm−1 and solar photon number n = 60000.
See Supplementary Information for all other parameters.
of 2ps−1 , which is faster than the primary charge transfer
rates.
Figure 6 (a) shows the Fano factor for a fixed voltage and as a function of the excitation rate. Strikingly,
the structured phonon-environment supports a transition
from sub- to super-Poissonian statistics for high enough
excitation rates. This is not observed for the unstructured environment and therefore the Fano factor clearly
discriminates the two cases. Most striking is the fact
that such transition does not happen at all, even with
a structured environment, if excitons on D2 branch are
discarded, as shown in Fig. 6 (a) (inset). To understand
the origin of these observations it is convenient to investigate the second-order correlation function g (2) (t), which
is related to the Fano factor as follows32 :
Z ∞
(2)
F = 1 + 2hIi
dt[g (2) (t) − 1].
(7)
0
The function g (2) (t) gives the probability that an electron
tunnelling through the drain lead is detected at time t,
having detected one at an earlier time t = 0 and its formal expression for calculations is given in Eq. (16). The
interpretation of g (2) (t) is the same as in quantum optics where g (2) (0) < g (2) (t) indicates anti-bunching, and
g (2) (0) > g (2) (t) indicates bunching. Figure 6 (b) shows
g (2) (t) for structured and unstructured vibrational envi-
7
1.4
1.2
1.0
0.8
)
g2 t
ronments at an excitation rate for which F (2) > 1. Electron transport in both cases in Fig. 6 (b) is anti-bunched
in the picosecond time scale, that is, g (2) (t < 1ps) < 1.
However,for the structured environment there is a prominent raise of g (2) (t > 1ps) to values larger than one with
a clear maximum. For this curve the positive portion of
g (2) (t) − 1 dominates the integral of Eq. 7 leading to a
Fano factor larger than one. The non-monotonic increase
of g(2)(t) indicates that competing dynamical processes
occur at different time scales and is reminiscent of the
phenomenon of dynamical channel blockade observed in
transport through quantum dots10,32,33 . At short times,
Coulomb blockade and fast pathways through the directly populated lowest exciton state dominate, as such
the probability of an electron tunnelling into the drain
lead at time t (given one was detected at t = 0) increases
with time. At longer times, however, all excitons are
populated including excitons in the D2 branch. For a
sufficiently high transfer to secondary CT state, excitons
in the D2 branch, although not directly coupled to CT
states, constitute an indirect slow pathway of population
transfer. This path thus dominates at intermediate time
scales. Differences in the relative time scales of these two
competing transfer pathways can cause the time electrons
spend in the system to be irregular, which manifests as a
large Fano factor and associated super-Poissonian statistics. Hence, while excitons localized in the D2 branch
do not participate in transient ultrafast charge separation, they can have a profound effect under steady sate
operation of a PSIIRC-based photocell.
(
0.6
0.4
0.2
D r ude
DISCUSSION
0.0
0
Learning how Nature’s molecular machinery manages to harvest and storage energy efficiently promises
to provide lessons for man-made devices with similar
capabilities34 thereby encouraging the development of
nature-inspired molecular scale photovoltaic devices. Research has shown that charge separation and electron
transfer in PRCs are coupled to both broad band and
well resolved vibrational motions. It is however unknown
what the effect of this structured vibrational environment
is in the operation of such complexes as components in
a photovoltaic cell. Present single-molecule technologies,
which have demonstrated the ability to manipulate and
measure the photocurrent of single PRCs, have opened
up the door to investigate this question at the singlecomplex level. Here we have addressed this question by
considering a PSIIRC-based photocell where the ChlD1
charge separation pathway is dominant.
Our results show that the structured environment assisting electron transfer in the PSIIRC-based photocell
device acts to reduce the current and power output.
This is a manifestation of a Zeno-like effect whereby
strong coupling of CT states to well-resolved high energy
modes, and the associated multi-phonon processes, prevent transfer. These observations suggest PSIIRC com-
2
4
6
8
t im e ps
10
12
14
16
FIG. 6: Counting statistics for a photocell device
with a secondary CT rate of 2 ps−1 . (a) Fano
factor versus excitation rate calculated at 77K with a
fixed voltage of 1.42V. Inset: Same parameters but
using a four site PSIIRC system, neglecting the D2
branch. (b) g (2) (t) with a fixed excitation rate of 12
cm−1 . See Supplementary Information for all other
parameters.
plexes where the ChlD1 is dominant are not necessarily
well adapted for photovoltaics. The predicted reduction
in the average photocurrent exhibited upon the inclusion
of coupling to high-energy vibrational modes is not detrimental for photosynthetic electron transfer processes in
vivo. For the operation of PRCs in the biological scenario
it is more important that charge recombination is inhibited so that the energy captured is not wasted. Strong
coupling of CT states to high energy modes may indeed
help to prevent charge recombination by guaranteeing a
slight asymmetry between forward and backward transfers from CT states. This behaviour contrasts with what
8
has been predicted for electronic excitation energy transfer in light-harvesting complexes as for those complexes,
where only exciton states are involved, a structured vibrational environment can indeed enhance excitation energy transfer30,35 . This should not come as a surprise
as light-harvesting complexes and reaction centres have
very different biological functions.
While no advantage is obtained in terms of mean current and power output, strong coupling to well resolved
vibrational modes does result in a reduction of current
fluctuations. This means that the PSIIRC-based photocell delivers currents with lower relative noise strengths
obeying a sub-Poissonian statistics. This lower noise and
associated ordered electron transport is further promoted
by the exciton manifold ensuring the multi-step nature
of the transport process.
Our analysis also show that the presence of exciton
trap states i.e. exciton states weakly coupled to CT
states can support a transition to super-Poissonian statistics. This is a manifestation of the fact that under steady
state operation, trap exciton states can become an indirect pathway for population transport with an markedly
different time scale.
The above discussion allows us to put forward three
principles for the development of functional PRCphotocells devices with different functionalities: (i) If the
purpose of the device is to deliver currents with lower
relative noise strengths as possible then assure the electron transfer process is assisted by a manifold of exciton states and a strongly coupled structured vibrational
environment. The donor exciton states together with a
structured environment guarantee a multi-step, ordered,
population transfer process in the picosecond time scale.
This feature relies on enforcing the Coulomb blockade
limit, where only one electron is present in the system
at a time. To further minimise current fluctuations,
the time scales for population transfer involving different exciton states must be comparable. This amounts to
eliminating trap exciton states, for instance, by carefully
sculpting the energy landscape of the pigment-protein
complexes through genetic engineering. (ii) Take advantage of the selective, strong coupling to specific vibrational modes to enhance forward transfer thereby helping to prevent charge re-combination. (iii) If the desired
function is to achieve a larger mean current irrespective
of how large its fluctuations are, then decouple specific
vibrational motions from electronic states. Modifications
of the electron-phonon interaction can be achieved by
strong coupling of pigment-protein complex to a confined optical cavity mode36,37 . Indeed, the energy exchange of electronic transitions with a strongly coupled
optical mode could help suppressing reorganisation energy of the nuclei thereby increasing the rate of electron
transfer reactions37 . Moreover, the parameters varied in
our calculations, such as voltage and temperature, allow a finer degree of control and optimization of different
current characteristics depending on the operational requirements.
The above principles have been proposed based on the
analysis of PSIIRC where the ChlD1 charge separation
pathway is dominant. Other PSIIRCs can have a dominant PD1 charge separation pathway where exciton and
CT states are coherently coupled19 . In this case it has
been argued that specific modes may indeed support coherent transfer between these states15 . The analysis presented here can be extended to consider photocell devices
where such coherent coupling is at work38 . The implications of coherent interactions for photocell performance
are under way and will be reported in a separate study.
Experimental implementation of our proposal involves
attachment of electrodes to individual PSIIRC units
forming metal-protein junctions. Similar experiments
have been carried out in a scanning tunnelling microscopy
setup with photosystem I (PSI) units9 and fragments
of reaction centre-enriched purple bacterial membranes4 .
The PSI units were genetically engineered with cysteine residues on both the oxidizing and reducing sides
which allowed covalent attachment of the protein to
two electrodes across which the photo-current could be
measured9 . The photo-current generated by PSII units
(that is the protein structure consisting of the PSIIRC
and surrounding light-harvesting chromophores) has also
been measured by depositing the protein on a metal oxide electrode39 . However this was an ensemble measurement with a soluble mediator on the oxidizing side rather
than looking at individual units. Despite the fact that
whole photosystem complex have been used in these experiments, only the reaction centre is functional in charge
transport between the electrodes, especially if the light
used is tuned to excite the reaction centre chromophores.
Advances in semiconductor-protein interfaces offer a potentially more flexible approach, for example depositing a
film of PSI on a p-doped silicon surface40 . This technique
has the advantage that uni-directional electron flow can
be achieved by adjusting the doping type and the doping density of the semiconductor. In addition, band gap
matching of doped silicon to that of the photosynthetic
protein can the enhance the magnitude of the photocurrent.
Current statistics measurements also potentially offer a
non-invasive, single system level probe of charge transfer
phenomena in a wide range of biological41,42 and chemical systems43–45 . Specifically, this technique may be applicable to charge transfer along molecular wires made
from DNA strands41 , to general donor-bridge-acceptor
systems43–45 or to unveil vibrational mechanisms for
odour receptors42.
METHODS
Theory of full counting statistics
Consider a mesoscopic system placed between a source
and drain leads which supply or take away electrons from
the system respectively. We consider the Coulomb block-
9
ade regime where only a single excess electron may occupy the system at any given time, an infinite bias voltage between the leads, guarantees uni-directional electron
flow. Assuming weak coupling between the system and
the leads we can write a Markovian master equation for
the reduced system where we have traced over the lead
degrees of freedom
ρ˙s (t) = Lρs (t).
(8)
Unravelling ρs (t) into components describing n electrons
counted in the drain lead at time t we get the master
equation27
ρ˙s (n, t) = L0 ρs (n, t) + LJ ρs (n − 1, t),
(9)
where L0 describes the evolution of the system between
electron counting events and LJ describes jumps where
an electron tunnels into the drain lead. This equation is
more conveniently solved by Fourier transforming with
respect to n to give
ρ˙s (χ, t) = L(χ)ρs (χ, t),
(10)
P
where ρs (χ, t) = n einχ ρs (n, t) and L(χ) = L0 + eiχ LJ .
χ is a counting field coupling to the lead where counting
takes place. The solution to Eq. 10 is then
ρ(χ, t) = Ω(χ, t − t0 )ρ(χ, t0 ),
(11)
where Ω(χ, t−t0 ) = eL(χ)(t−t0 ) is a time-propagator. The
full counting statistics is encoded in the probability distribution of the number of electrons, n, transmitted to
the drain lead27 , which can be expressed
P (n, t) = tr{ρs (n, t)}.
(12)
Transforming to χ-space gives the characteristic function
of the distribution
X
G(χ, t) =
P (n, t)einχ ,
(13)
n
which can also be expressed26 as
G(χ, t) = tr{Ω(χ, t − t0 )ρs (χ, t0 )}.
(14)
We are interested in the cumulants of the current distribution rather than the characteristic function of the
distribution of the number of particles transferred to the
drain lead. We therefore use the following expression for
the current cumulants
hI N ic =
d ∂ N F (χ, t)
|χ→0,t→∞ ,
dt ∂(iχ)N
(15)
where F (χ, t) = lnG(χ, t) is the cumulant generating
function and the limit t → ∞ ensures the average is carried out in the steady state. Additionally we assume the
system has reached its steady state ρstat by time t = t0
so we include this instead of the initial state in Eq. 14.
We also calculate the second order correlation function
g (2) (t) for electron transport using the equation32
g (2) (t) =
tr{Lj Ω(0, t)Lj ρstat }
.
tr{Lj ρstat }2
(16)
ACKNOWLEDGEMENTS
The authors would like to thank Clive Emary for helpful discussions. Financial support from the Engineering
and Physical Sciences Research Council (EPSRC UK)
Grant EP/G005222/1 and from the EU FP7 Project PAPETS (GA 323901)is gratefully acknowledged.
AUTHOR CONTRIBUTIONS STATEMENT
A.O-C designed the research, R.S and H.H-N carried
out the calculations. RS, H.H-N, RvG and A.O-C analysed the results and wrote the manuscript.
ADDITIONAL INFORMATION
The authors declare no competing financial interests.
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