Maximum efficiency of state-space models of molecular scale engines
The performance of nano-scale energy conversion devices is studied in the framework of state-space models where a device is described by a graph comprising states and transitions between them represented by nodes and links, respectively. Particular segments of this network represent input (driving) and output processes whose properly chosen flux ratio provides the energy conversion efficiency. Simple cyclical graphs yield Carnot efficiency for the maximum conversion yield. We give general proof that opening a link that separate between the two driving segments always leads to reduced efficiency. We illustrate this general result with a simple model of an organic photovoltaic cell, where such an intersecting link corresponds to non-radiative carriers recombination and where the reduced maximum efficiency is manifested as a smaller open-circuit voltage.
Modeling of molecular scale engines such as photovoltaic cells, thermoelectric devices and light emitting diods is often done in terms of a state space description Seifert (2012). In such state-space models the processes underlying the device operation are described as transitions between microscopic system states, leading to the engine description in terms of a graph comprising nodes (=states) connected by bonds that represent transitions between them. On this network, the time evolution is determined by the rates associated with each bonds and fluxes associated with the non-equilibrium dynamics flow along interconnected linear and cyclical paths. Particular segments of this network represent the input (driving) and output processes, where the corresponding rates deviate from restrictions imposed by equilibrium thermodynamics, thus maintaining the system in a non-equilibrium state. Similar models are often used to describe and analyze biochemical networks Hill (1966, 2009); Gerritsma and Gaspard (2010).
An example is shown in Fig. 1 that depicts a single cycle model (henceforth referred to as model SC) for a photovoltaic cell comprising states that represent a charge carrier occupying different sites in the system and one state (denoted ) in which the charge carrier is absent. Transition rates between system states are determined by state energies , the bias voltage and the local temperatures. The latter are taken equal to ambient temperature everywhere except at the transition that is driven by the sun temperature . The different rates satisfy detailed balance conditions:
for all , except
where is the rate coefficient for the transition and , . The cycle affinity is defined by Schnakenberg (1976)
where is the ratio between products of forward and backward rates
The open-circuit (OC) voltage is determined by the condition that the drivings associated with the voltage and with the difference between and balance each other so that the steady state current through the system vanishes. In this case, the cycle affinity vanishes Einax and Nitzan (2014), namely . Denoting the corresponding voltage by and using Eqs. (1)-(3) this leads to
The l.h.s is where is the thermodynamic (“internal”) efficiency of the device and is the steady state current (same through all links). Equation (6) thus yields the Carnot expression for the efficiency in the zero power limit. It also highlights the OC voltage as an important attribute in the quest to understand cell performance Vandewal et al. (2009); Kirchartz et al. (2009); Potscavage et al. (2009); Nayak et al. (2011). In Ref. Einax et al., 2011 we have derived two important generalizations of this result: First, if the transition results from the combinations of radiative and non-radiative processes whose forward/backward ratios are determined by temperatures and , respectively, Eq. (6) is replaced by a similar expression in which is substituted by an effective temperaturenot (a)
where is the ratio between the non-radiative and radiative relaxation rates. Second, if one of the cycle processes involves charge separation that comes at an energy cost (exciton binding energy) , Eq. (6) is replaced by Einax and Nitzan (2014)
Model SC and Eqs. (4)-(7) constitute a convenient framework for analyzing the OC voltage of photovoltaic cells, and, as shown in Ref. Einax and Nitzan, 2014, can be used as a starting point for studying optimal performance in finite power operations, provided that information on the system electronic structure and dynamics can be incorporated into such state-space model. However, this single cycle model is oversimplified; realistic state-space models contain intersecting pathways that makes the analysis that leads to Eqs. (4)-(6) more involved. Indeed, in state-space models of photovoltaic cells such intersecting pathways represent non-radiative losses, e.g., electron-hole recombination processes that are expected to reduce efficiency.
As graphical structures, such intersecting pathways divide the cycle of model SC into separate cycles (see Fig. 2), making it possible to have steady states in which the current in the power-extracting segment vanishes while internal currents exist elsewhere in the system. While intuitively we expect lower efficiencies in such situations, no general proof for this is available.
In this Letter we consider two generalizations of model SC shown in Fig. 2. In one [Fig. 2(a)], an intersecting pathway divides the system cycle into two subcycles without separating between the two driving segments. In the second [Fig. 2(b)], the driving segments are positioned on different cycles. We show that (a) Intersecting pathways of the type shown in Fig. 2(a) do not affect the OC voltage, hence their zero power operation is characterized by the Carnot efficiency. (b) Intersecting pathways of the type shown in Fig. 2(b) lead to zero power efficiency smaller that the Carnot value, manifested for photovoltaic cells be a lower OC voltage.
We start with Model SC and take note of the two sites, and , between which a new pathway will be enabled. The rate ratio, Eq. (5), can obviously be written as where and are ratios of rate products that correspond to the arcs and in Fig. 2. Now enable the pathway between sites and , that is, make and different from zero. The graph is now divided into two subcycles, and in addition to the original cycle. The corresponding cycle rate-product ratios are
so that .
Consider next the situation shown in Fig. 2(a), where the two driving segments are on the same subcycle . Starting from the model SC in the OC situation so that , enabling the link does not lead to current on this bond. To show this note that since all rates on arc satisfy Eq. (1), we have . Equation (8) then yields and therefore . Consequently, connecting the sites and will not create any extra current and subsequently does not affect the open-circuit voltage.
In the situation of Fig. 2(b), each of the subcycles created by enabling the pathway contains a non-equilibrium element: driving by the sun in subcycle and the driving by the voltage in subcycle . Focusing on the latter and using Eqs. (1) and (3) yields
where . Equation (8) then yields . This indicates that with the link enabled the system cannot establish a state where currents through all links in subcycle vanish unless .
To understand the consequence for the open-circuit voltage, let us set and and evaluate the derivative of the OC voltage, , with respect to at . is the value of for which the current between the electrodes, (the blue segment in Figs. 1 and 2), therefore across all links in arc , vanishes; namely for all this links. From Eqs. (1) and (3) it follows that under OC conditions, the probabilities to find the system in states and are related to by
It follows that
where is a positive number and . The derivative on the r.h.s is negative if (, i.e. , and negative in the opposite case, (; (this can be inferred from the current expression, . Hence
Enabling the link in the scheme of Fig. 2(b) thus results in reduction of the OC voltage, namelt of the maximum thermodynamic efficiency.
To illustrate this general observation for a particular system, we consider a simple -level model for an organic photovoltaic cell comprising a donor acceptor (D-A) complex placed between two electrodes (see Fig. 3 and Supplementary Material not, b). The and species are characterized by their HOMO levels, , , and LUMO levels, , that translate into three relevant energy differences (the optical gap), (interfacial LUMO-LUMO gap), and (effective band gap). Charge carriers can enter and leave the system only by transitions between the donor HOMO level and the electrode , and between the acceptor LUMO level and the electrode . These electrodes are represented by free-electron reservoirs at chemical potentials (). An important feature of this model is that a thermal fluctuation is needed to overcome the exciton binding energy of an interfacial charge transfer exciton to generate a free charge carrier. With some simplifying assumptions, the system can be described by a kinetic scheme involving four states. A detailed description of this model and the underlying kinetic scheme was given in Ref. Einax et al., 2011.
An additional link between states and correspond to non-radiative recombination that returns a free electron on the acceptor back to the donor ground state. In the absence of this process, the open-circuit voltage is found to be given by Eq. (7). The maximum thermodynamic efficiency is correspondingly , highlighting the loss associated with the exciton binding energy.
Repeating the cycle analysis when the rates , are finite not (b) yields the following expression for the open-circuit voltage
setting and . and taking the derivative with respect to we find, using also Eqs.(S5-S8)
In conclusion, we have presented a state-space network representation scheme of molecular scale engines and applied it to models of photovoltaic cells. A general argument based on cycle analysis yields conditions under which the opening of intersecting pathways in an otherwise cyclical graph leads to a decrease in the maximum efficiency. In the application to the operation of a photovoltaic cell such intersecting link often corresponds to a carrier recombination process, and its consequence is manifested in a reduced open-circuit voltage. It should be emphasized that similar analysis can be carried out for more complex models that include, for example, polaron formation and hot exciton dissociation, as well as for other types of nano-scale engines, provided that they can be modeled by kinetic transitions in the system state-space. As described elsewhere Einax and Nitzan (2014), such an approach can be used for analysis of optimal performance under finite power operation, although with the unavoidable loss of the generality associated with a thermodynamic description.
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