Non-equilibrium thermodynamics of charge separation in organic solar cells

TL;DR

This paper develops a theoretical framework using stochastic thermodynamics to analyze charge separation in organic solar cells, highlighting the roles of disorder, delocalization, and entropy in efficiency.

Contribution

It introduces a novel non-equilibrium thermodynamic model for charge separation in OSCs, linking microscopic dynamics with thermodynamic quantities and transient behaviors.

Findings

High disorder leads to equilibrium-like site populations.

Delocalized pairs show non-equilibrium behavior at low disorder.

Large Gibbs entropy and initial separation are key for efficiency.

Abstract

This work presents a novel theoretical description of the non-equilibrium thermodynamics of charge separation process in organic solar cells (OSCs). Using the theory of stochastic thermodynamics, we connect the phonon-assisted dynamics and recombination of electron-hole pairs within a photo-excited organic bilayer with the thermodynamic free energy. We analyze the impact of energetic disorder and delocalization on the free energy, average energy and entropy. For high energetic disorder, the site population is well described by equilibrium. We observe significant deviations from equilibrium for delocalized electron-hole pairs at small energetic disorder, representing efficient OSCs. Our results emphasize that both a large Gibbs entropy and large initial separation are required to achieve efficient charge separation. A decrease in free energy barrier with increased distances between charges does not necessarily correlate with the separation yield. Our presented framework can further shed light on the transient thermodynamic free energy, allowing previously inaccessible insight into the individual thermodynamic contributions from energy and entropy on sub-ns timescales. Transient simulations reveal large Gibbs entropy on ps-timescales for even highest disorder, which may explain the efficient separation of "hot" CT states.

Tables

Table 1. Table 1: Activation energies E A subscript 𝐸 A E_{\text{A}} and rate constants k 𝑘 k for the ion migration in MAPbI 3 . The transport times t transport subscript 𝑡 transport t_{\text{transport}} represent an estimated time for the respective ions to cross a 500   nm times 500 nanometer 500\text{\,}\mathrm{nm} thick perovskite layer. shi2018from

Defect	$E_{\text{A}}\left(\mathrm{eV}\right)$	$k\left({\text{s}}^{-1}\right)$	$t_{\text{transport}}\left(\text{s}\right)$
V${}_{\text{Pb}}$	0.8 azpiroz2015defect	$1.2$	${10}^5$
V${}_{\text{MA}}$	0.57 haruyama2015first	$8.97\text{×}{10}^3$	$25$
V${}_{\text{I}}$	0.32 haruyama2015first	$1.5\text{×}{10}^8$	$5\text{×}{10}^{-3}$

Equations

$$k_{ij}=k\times \{\begin{array}{ll}\exp \left(-\frac{\Delta E_{ij}}{k_{\mathrm{B}}T}\right)& \text{if}\Delta E_{ij}>0\\ 1& \text{else}\end{array}.$$

Full text

Perovskite solar cells, ion migration, grain boundaries

I Morphology generation

The generation of grain boundaries within the active perovskite layer is accomplished by the Voronoi tessellation kaiser2018charge. Figure 1 depicts the methodology for the morphology generation in 2D, for the sake of clarity. Centroids of grains, depicted by red spheres, are distributed to match the experimentally observed average grain size. The implemented algorithm connects all centroids within the active layer and computes the Voronoi cells, which comprise the active material closest to the respective centroids. The boundaries of the Voronoi cell are highlighted by the red polygons. Now, the sites of the active material within the simulation box, depicted by black dots, are placed on top of the Voronoi tessellation. We determine all sites which are within a predefined distances to any of the boundaries of the Voronoi cells. Our algorithm projects each site to the cell boundaries and observes the distance to the corresponding boundary. If the distance is below the desired grain boundary width, we determine if the projected site lies within the interface. If this is the case, the site belongs to the grain boundary and the algorithm moves on to the next site. The resulting active perovskite morphology is depicted on the right-hand side in Fig. 1. The blue domains represents the grain interior, while the red domains are grain boundaries.

II Ion dynamics

The migration of ions is crucial for hysteresis effects, stability and reproducibility of the power conversion efficiency within perovskite solar cells. Within the hybrid perovskite methylammonium lead iodide, several movable ions are present in the active material: MA+, Pb2+ and I-. Table 1 summarizes the activation energies, rate constants and transport times for all given ions. For many applications, the migration of lead-ions is negligible due to the high activation energy. Iodide and methylammonium ions migrate on much faster timescales and are thus considered to be responsible for the experimentally observed hysteretic $jV$-curves. Ions can be considered as strongly localized charged particles. Thus, we account for the ion migration by using the Miller-Abrahams hopping rate miller1960impurity

[TABLE]

Within the modeled kinetic Monte Carlo method, MA+ and I- ions can be distributed initially with any desirable densities.

III How to handle different timescales

Compared to the photo-generated charge carriers, ion migration occurs on a much slower timescale. While charge carriers have transport times of $\approx$10\text{,}\mathrm{ns}$$, ions migrate orders of magnitude slower through the perovskite layer. The different timescales are handled by a multi-timescale approach, separating the dynamics of ions from the dynamics of the photo-generated charge carriers. For a specific time, the migration of ions is simulated while the generation of charges is deactivated. Then, the ions are frozen to their current positions and charge generation is activated. The charge carriers now move within the electrostatic potential generated by the applied electric field and the Coulomb interaction with the ions. Hereby, the current density at a certain time and ion distribution is sampled. Iterative repetition of this method allows to simulate both the migration of ions and the charge carrier dynamics on long time scales.

IV Recombination Processes

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bimolecular recombination

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trap assisted recombination