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Theoretical limit of power conversion efficiency for organic and hybrid halide perovskite photovoltaics

We calculated the maximum power conversion efficiency as a function of the optical band gap for organic photovoltaic (PV) cells by assuming that charge separation is accompanied by the energy loss required to dissociate strongly bound charge pairs. The dissociation energy can be estimated from the relationship between the open circuit voltage (VOC) and the optical band gap (Eg). By analyzing the published data on VOC and Eg, the dissociation energy can be estimated. The result could be used as a guide for selecting donor and acceptor materials. We also studied the theoretical limit of power conversion efficiency of hybrid halide perovskite by taking into account the energy loss involved in the carrier transfer from the perovskite phase to the metal oxide charge transport layer.


Introduction
Organic photovoltaic (OPV) cells have many advantages, such as being thin, soft, and light, over conventional inorganic photovoltaic cells owing to characteristics of organic materials. Although the photoelectric power conversion efficiency of organic photovoltaics is lower than that of inorganic photovoltaic cells, it has improved rapidly and now exceeds 10 %. The rapid increase in the power conversion efficiency aroused fundamental interest in the theoretical limit of the conversion efficiency for organic solar cells.
Since the pioneering work of Shockley and Queisser in 1961, 1 the theoretical limit of the power conversion efficiency has been known for inorganic solar cells and is approximately 30%. The Shockley-Queisser (SQ) limit was calculated for PN-junction solar cells and is not applicable for excitonic solar cells. The exciton-binding energy can be at least as large as 0.3−0.5 eV and cannot be ignored in organic photovoltaic cells. Organic photovoltaic cells are composed of donor and acceptor materials, and charge separation against the large exciton-biding energy occurs at the donor-acceptor interface. The application of the SQ limit has been extended to organic photovoltaics. [2][3][4][5][6][7][8][9] Some other approaches have also been developed to study the efficiency, in particular, on the basis of a thermodynamic detailed balance. [10][11][12][13][14] As a practical approach, limits for solar cells were assessed using criteria based on the short circuit currents, open circuit voltage and other quantities. 15 When the SQ limit has been applied to organic photovoltaics, the difference between the optical energy gap and the electronic energy gap has been taken into account. [2][3][4][5][6][7][8][9] The energy difference between these two gaps results in voltage loss by energy dissipation. Recently, we have calculated the limit of power conversion efficiency by assuming irreversible exciton dissociation, 8 while other studies assumed reversible exciton dissociation. [2][3][4][5] We took into account the excess energy required for irreversible charge separation at donor/acceptor (D/A) interfaces. 8 In this work, we briefly introduce our approach and show its consequences with respect to the open circuit voltage and the short circuit currents.
Recently, the power conversion efficiency of hybrid halide perovskite solar cells has been rapidly increased. [16][17][18][19][20][21][22] The metal oxide TiO 2 is used as a charge transport layer in the hybrid halide perovskite photovoltaic cells. Although hybrid halide perovskite solar cells are not classified as excitonic solar cells, the carrier transfer from the perovskite phase to the metal oxide TiO 2 involves intrinsic energy losses. We study the power conversion efficiency of hybrid halide perovskite photovoltaics by taking into account the energy loss involved in the carrier transfer from the perovskite phase to the metal oxide charge transport layer.

Theoretical limit of organic photovoltaic cells
In this section, we summarize the results of our previous work.. 8   calculated using the AM 1.5 spectrum that we denote by j in (E). For simplicity, we assume that all photons of energy higher than band gap energy are absorbed by the cell and converted to carriers, J in (E g ) = ∞ Eg dEj in (E). As in the SQ theory of inorganic PV cells, we assume inevitable loss of carriers by radiative recombination. The loss of carriers by recombination per unit area per unit time can be expressed by 8 where h, k B , and c denote the Planck constant, the Boltzmann constant, and the speed of light. The maximum power conversion efficiency can be obtained from 8 When ∆E DA = 0, the above equation reduces to the SQ limit. The maximum power conversion efficiency is shown as a function of the band gap (nm) in Fig. 2. In Fig. 2, E g (eV) is expressed by wave length using E gap (nm) = 1240(eV· nm)/E g (eV). By increasing ∆E DA , the maximum power conversion efficiency is decreased and the band gap at the peak is shifted toward shorter wavelengths. The blue shift of the peak results from the excess energy required to dissociate excitons.

Dissociation energy
Plausible values of ∆E DA can be estimated from the relationship between the open circuit voltage and the optical band gap. eV OC can be regarded as the electronic band The recombination current density J R (E g − ∆E DA , 0) can be expressed by the carrier densities at the interface and the recombination life time. Essentially the same equation as Eq. (3) can be derived in this case, and V OC can be expressed using the recombination lifetime. 23 The results indicate that the recombination lifetime is related directly to In Fig as shown in Fig. 4. ∆E DA can be estimated from eV OC by this relation. It should be remembered that the relationship given by Eq. (4), could be affected by changes in donor/acceptor ratio, layer thickness, and morphology.

Short-circuit current density
As shown in Fig. 3, the voltage loss to dissociate excitons into free carriers can be 0.3 − 0.4 eV for some combinations of donors and acceptors. The corresponding maximum power conversion efficiency shown in Fig. 2 is much higher than the current maximum power conversion efficiency, which is close to 11%. To quantify the additional loss, we show the short-circuit current density calculated using Eq. (1) and compare it with the experimental data in Fig. 5. The short-circuit current density was obtained from Experimental values are much lower than the theoretical curve. The difference in-  dicates the presence of current loss other than that caused by the radiative recombination characterized by black-body radiation. The short-circuit current is mainly given by J in (E g ), and the contribution of J R (E g − ∆E DA , 0) is negligible for the the optical band gap (eV) shown in Fig. 5. J R (E g − ∆E DA , 0) is significant when E g is close to ∆E DA . It should also be considered that J R (E g − ∆E DA , V ) may not be negligible under certain applied voltages. A large current loss is seen in Fig. 5 when ∆E DA ≤ 1.8 eV, which is probably caused by nonradiative recombination.

Hybrid halide perovskite
Hybrid halide perovskite has been initially developed as a sensitizer of solar cells using electrolytes. 16,17 Recently, the power conversion efficiency of hybrid halide perovskite solar cells has rapidly been increased by avoiding electrolytes, and it has exceeded that of organic photovoltaics. [18][19][20][21][22] A frequently studied material of hybrid perovskite solar cells is methylammonium lead iodide (CH 3 NH 3 PbI 3 ). The energy band diagram of CH 3 NH 3 PbI 3 is shown in Fig. 6. [27][28][29] The optical band gap of CH 3 NH 3 PbI 3 is 1.5 eV.
The dielectric constant of CH 3 NH 3 PbI 3 is around 25, and the binding energy between an electron and a hole is small. 29 The carriers dissociate in CH 3 NH 3 PbI 3 , and the electron and hole lifetimes inside CH 3 NH 3 PbI 3 have been known to be large and are on the order  of µs. [30][31][32][33][34] The electron transfer from CH 3 NH 3 PbI 3 to TiO 2 occurs on a time scale of picoseconds, which is much faster than the time scale of charge recombination inside CH 3 NH 3 PbI 3 . 33,35 For simplicity, we assume that all electrons generated in CH 3 NH 3 PbI 3 transfer to TiO 2 . This assumption may be reasonable for thin layers. The photonabsorbing layers of hybrid halide perovskite can be as thin as sub-µm because the optical absorption of CH 3 NH 3 PbI 3 is much higher than that of conventional inorganic semiconductors. 34 As seen in Fig. 6, the transfer of electrons to TiO 2 is accompanied by an energy loss of 0.2 eV. According to recent reports, interface recombination between electrons in TiO 2 and holes in CH 3 NH 3 PbI 3 occurs on the nanosecond time scale. 33 In hybrid halide perovskites, Wannier excitons are formed and they dissociate inside hybrid halide perovskites. 29 When electrons transfer from hybrid halide perovskites to TiO 2 with a time scale much shorter than the carrier lifetime and a thin layer of hybrid halide perovskites is used, charge recombination inside hybrid halide perovskites can be ignored. Charge recombination at the interface between hybrid halide perovskites and TiO 2 occurs on the nanosecond time scale and is taken into account. Although charge dissociation processes in hybrid halide perovskites are different from those in organic photovoltaic cells, The result for ∆E = 0.2 eV, however, overlaps with the thick solid line in Fig. 7 (not shown).

Conclusions
The