Photon recycling in perovskite solar cells assessed by a detailed-balance compatible dipole emission model
Metal halide perovskites have become one of the most promising classes of materials for optoelectronic applications. This is due to their high efficiencies of conversion between electrical and optical power, the low cost to manufacture, and their suitability for large-area applications.
Among the salient features of this material are:
long non-radiative carrier lifetimes
strong optical absorption
steep absorption edge (Fig. 1a)
Together these properties lead to a sizable internal emission of photons under optical and electrical excitation, and to the re-absorption of a substantial part of this radiation. The process of photon re-absorption and internal re-emission is called photon recycling and can have a beneficial impact on device performance.
In solar cells, photon recycling leads to an increase of the open-circuit voltage (Voc) of the device, a key performance parameter, and a measure for the maximum voltage that can be sustained by the device under solar illumination. In light-emitting devices, it may improve the luminescent external quantum efficiency by fostering the extraction of internally emitted photons via their re-distribution from guided to leaky and out-coupled modes.
As the internal radiative processes are not accessible to experiment, simulations are instrumental to assess the potential benefit of photon recycling to the performance of an optoelectronic device of given architecture and material properties. However, due to the sub-micron thickness of the perovskite active films, the ray optics approach established to quantify photon recycling bulk solar cell devices are not suitable. Furthermore, the standard detailed balance picture for the local rate of radiative recombination based on the van Roosbroeck-Shockley theory [VRS54] assumes a uniform density of photon states in the absorber, neglecting any cavity effects. On the other side, the dipole radiation models employed for the simulation of light emission in thin-film organic LEDs under consideration of cavity effects are not compatible with the detailed balance framework conventionally used to assess emission in solar cell devices and are plagued with non-physical divergencies in the presence of absorbing media.
In order to reconcile the two pictures for a unified description of radiative processes in thin-film perovskite solar cells and LEDs – including photon recycling – we derived a detailed balance compatible parametrization of the dipole radiation model [Aeb21] in terms of the local values of the optical constants and the quasi-Fermi level splitting. The approach is based on the evaluation of the transverse dyadic Green’s function for photons in arbitrary multilayer stacks, which enables a numerically stable simulation of emission in absorbing media and provides the photon flux as well as the rate for emission and re-absorption with spatial and spectral resolution. Based on this information, any optical quantity relevant for the assessment of photon recycling can be obtained directly, from the enhancement of the open-circuit voltage to the evaluation of radiative loss channels and the quantification of parasitic absorption at metallic reflectors and in contact layers.
Figure 1 – (a) Absorption coefficient of MAPI from [Liu19] and corresponding local emission rate from detailed balance. (b) Photon flux at photon energy 1.6 eV in a 280 nm MAPI slab subject to a constant quasi-Fermi level splitting of 1.2 eV. (c) Comparison of the external flux from Poynting vector and generalized Kirchhoff law.
Application of the approach to single methylammonium lead iodide (MAPI) perovskite films (using optical data from [Liu19]) demonstrates the agreement of the global flux from the Green’s function (Fig. 1b) with the predictions from the generalized Kirchhoff law [Wue82] (Fig. 1c) but also reveals the overestimation of the local emission rate by the van Roosbroeck-Shockley relation (VRS) in the case of low film thickness (Fig. 2a).
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In the absence of additional layers, an inspection of the modes contributing to the internal and external emission (Fig. 2b) shows that internal emission is dominated by guided modes, and there is substantial reabsorption even in outcoupled modes, which, however, does not have a parasitic component, as the net internal emission equals the external emission. The enhancement of the open-circuit voltage as a function of film thickness (Fig. 2c) reflects the common absorptance resonances, the larger emission rate of the VRS for thin films, and the convergence of the two pictures in the ray-optics limit.
Figure 2 – (a) Local emission rate for MAPI absorber layers of different thicknesses. (a) Spatial integration of internal rates compared to external flux. (c) Optical enhancement of the open circuit voltage evaluated using the van Roosbroeck-Shockley and Green’s function rates.
In contrast to the common assessment of photon recycling effects based on detailed balance and thermodynamics, the explicit evaluation of parasitic absorption is not necessary. However, parasitic absorption in the individual layers of a complex device stack can be quantified easily using our approach, as the local re-absorption rate is directly accessible (Fig. 3b). In analogy to the quantity relevant for the utilization of external radiation, normalization of the re-absorption rate in a given layer to the total internal emission rate provides a layer-resolved re-absorptance (Fig. 3c).
Figure 3 – (a) Local photon flux in the full device stack of [Liu19] at an external voltage of 1.2 V and photon energy 1.6 eV. (b) Local rate of reabsorption for the situation of (a). (c) Layer resolved spectral absorptance of internally emitted light.
While our approach reaches beyond the state of the art in terms of generality and consistency even on the level of an assessment that is restricted to optical considerations, the explicit expressions that it provides for the local rates of radiative recombination and secondary photogeneration due to re-absorption of internally emitted photons lend themselves to an integration with a full opto-electronic device simulation framework, including charge transport on drift-diffusion level. The dependence on the local quasi-Fermi level splitting is thereby replaced by the relation to the charge carrier density product and the intrinsic carrier density, and steady-state device characteristics are obtained after iterating the evaluation of carrier densities and optical rates (Fig 4a). Such a fully coupled opto-electronic simulation allows for the assessment of the effects of photon recycling in the presence of charge transport and non-radiative recombination in realistic device architectures.
As an example, the current-voltage characteristics are evaluated for the solar cell device structure from [Liu19], under consideration of charge transport in the central device region comprising contact layers, charge transport layers, and non-radiative recombination via defects located at the interfaces of the MAPI absorber (Fig. 4b). To this end, the optical rates inferred from the Green’s function model are used in Fluxim’s optoelectronic device simulation tool Setfos. For comparison, the optical limit of the current-voltage characteristics as determined from the full and net (i.e., with reabsorption deducted) radiative dark saturation current and under the assumption of uniform quasi-Fermi level splitting equaling the external voltage are shown as well. They reflect the impact of parasitic absorption at the metallic reflector and in contact layers by the open-circuit voltage enhancement that is strongly reduced as compared to a free-standing single MAPI layer of identical thickness. The characteristics provided by the full optoelectronic simulation at the radiative limit exhibit similar Voc enhancement, but a slightly lower fill factor originating in the non-trivial dependence of the quasi-Fermi level splitting on the external voltage. Finally, the impact of non-radiative recombination at interface defects is assessed by evaluating the current-voltage characteristics using defect parameters inferred from a rough fit of the experimental characteristics, revealing another drastic reduction of the Voc enhancement (Fig. 4c).
Figure 4 – (a) Iterative computation of charge carrier densities and optical rates. (b) Device stack and band alignment used in the electrical simulation. (c) Current-voltage characteristics for the optical model, the radiative limit of the coupled model and for the full optoelectronic model with defect-mediated recombination (parameters from fit to exp. data).
Conclusions
In conclusion, we propose a novel simulation approach that provides a numerical treatment of emission in absorbing media without non-physical divergencies, unifies the theory of dipole emission with that of radiative rates from detailed balance, and which enables a consistent evaluation of internal and external emission under the consideration of the actual photon modes in the full device stack. Secondary photogeneration due to reabsorption of internally emitted photons is determined in a full-wave picture and is coupled - together with the local emission rate – to a full optoelectronic device simulation framework that allows for the quantification of the effects of photon recycling in the presence of charge transport and non-radiative losses in realistic solar cell device architectures.
This study is going to be presented at the PVSEC 2021 by our colleague Dr. Urs Aeberhard.
