Efficient luminescent solar cells based on tailored mixed-cation perovskites.

We report on a new metal halide perovskite photovoltaic cell that exhibits both very high solar-to-electric power-conversion efficiency and intense electroluminescence. We produce the perovskite films in a single step from a solution containing a mixture of FAI, PbI2, MABr, and PbBr2 (where FA stands for formamidinium cations and MA stands for methylammonium cations). Using mesoporous TiO2 and Spiro-OMeTAD as electron- and hole-specific contacts, respectively, we fabricate perovskite solar cells that achieve a maximum power-conversion efficiency of 20.8% for a PbI2/FAI molar ratio of 1.05 in the precursor solution. Rietveld analysis of x-ray diffraction data reveals that the excess PbI2 content incorporated into such a film is about 3 weight percent. Time-resolved photoluminescence decay measurements show that the small excess of PbI2 suppresses nonradiative charge carrier recombination. This in turn augments the external electroluminescence quantum efficiency to values of about 0.5%, a record for perovskite photovoltaics approaching that of the best silicon solar cells. Correspondingly, the open-circuit photovoltage reaches 1.18 V under AM 1.5 sunlight.

INTRODUCTION

Metal halide perovskites of the composition ABX3 [A = Cs+, CH3NH3+ (MA), or NH=CHNH3+ (FA); B = Pb or Sn; X = Br, I] have recently attracted strong research interest because of their outstanding photovoltaic properties (). The development was triggered by the reports of Kojima et al. () and Im et al. () on liquid electrolyte-based quantum dot solar cells. Adoption of solid-state hole conductor and new deposition techniques (–) boosted the power-conversion efficiency (PCE) from 3% () to the present record of 20.1% (). The latter was reached with perovskites using mixtures of A and X ions with a general formula of FA1−MAPb(I1−Br)3, pioneered by Pellet et al. () and Jeon et al. ().

Although the current PCE of 20.1% is impressive, the open-circuit voltage (VOC) of these solar cells remains below 1.1 V, which is lower than their theoretical VOC limit of 1.32 V (for MAPbI3) in AM 1.5 G sunlight (). The loss of more than 200 mV arises from nonradiative recombination of photogenerated charge carriers and manifests itself in a low external quantum yield (⪅0.01%) for electroluminescence measured at a forward bias potential corresponding to VOC (, ). Reducing nonradiative recombination would pave the way for higher VOC and PCE values. Recent attempts toward this end used additives such as phenyl-C61-butyric acid methyl ester (PCBM) (), oxygen (), and PbI2 (–). Cao et al. () maintained unreacted PbI2 during sequential deposition () of MAPbI3 on a mesoporous TiO2 scaffold, causing an increase in VOC of up to 1.036 V (PCE = 9.7%) at a residual PbI2 level of 1.7%. This was ascribed to elimination of surface states by PbI2, in agreement with photovoltage spectroscopy measurements that also point to a decrease in MAPbI3 defect states by unreacted PbI2 (). Similar beneficial effects have been ascribed by Chen et al. () and Pathak et al. () to the surface enrichment of PbI2 formed by thermal decomposition of MAPbI3. In contrast, Shao et al. () found that such PbI2-enriched perovskite layers are n-doped and contain a high level of defects that were eliminated by the deposition of a PCBM overlayer. In most of these cases, the influence of excess PbI2 was studied using devices with poor or moderate performance (PCE ⪅ 12%). Hence, it appears that it is difficult to attribute these sometimes contradictory effects to the mere presence of excess PbI2.

RESULTS

Here, we report for the first time on a perovskite solar cell (PSC) using a new PbI2-enriched composition that exhibits both very high solar-to-electric PCE and intense electroluminescence. We produce the mixed-cation mixed-halide perovskite films in a single step from a solution of FAI, PbI2, MABr, and PbBr2 in a mixed solvent containing dimethyl formamide (DMF) and dimethyl sulfoxide (DMSO). We vary the molar ratio for PbI2/FAI () from 0.85 to 1.54 while maintaining a fixed molar ratio of 5.67 for PbI2/PbBr2 in the precursor solutions. By using this technique, we fabricate PSC with the structure Au/Spiro-OMeTAD/perovskite/mesoporous TiO2/compact TiO2/FTO. We achieve the highest PCE (20.8%) with , corresponding to a 3% weight excess of PbI2 in the perovskite. The photovoltaic metrics of the device are as follows: short-circuit current density (JSC) = 24.6 mA cm−2, open-circuit voltage (VOC) = 1.16 V, and fill factor (FF) = 0.73 (Fig. 1A). By integrating the incident photon-to-electron conversion efficiency (IPCE) spectrum over the AM 1.5 photon flux, we obtain JSC values that agree within 4% of the measured values. One of the devices was sent for certification to Newport Corporation, an accredited photovoltaic testing laboratory, confirming PCEs of 19.90% (backward scan) and 19.73% (forward scan) with a JSC of 23.2 mA cm−2 and a VOC of 1.13 V (fig. S1). The normalized IPCE spectrum is shown in fig. S1 as well. The commonly observed hysteresis (, ) is not pronounced in our devices, as proven by J-V curves shown in Fig. 1C (table S1), where the voltage sweep rate was varied from 10 to 5000 mV s−1. A cross-sectional scanning electron microscopy (SEM) image of a champion PSC is shown in Fig. 1B, visualizing a thick perovskite capping layer of around 500 nm. A histogram of 40 devices (figs. S2 and S3) indicates good performance reproducibility, with an average PCE of 19.5%. A preliminary stability investigation shows that the devices stored in the dark at room temperature are relatively stable, with a PCE drop of only 0.3% for 1 month (fig. S4 and table S2).

Fig. 1

Basic characteristics of fabricated perovskite solar cells.

(A) J-V curves for the champion solar cell under AM 1.5 G illumination, measured from VOC to JSC. (B) Cross-sectional SEM image of the champion cell. (C) Hysteresis measurements of one PSC at different scanning speeds under AM 1.5 G illumination.

The photovoltaic metrics of the PSCs made by varying in the precursor solution are summarized in Fig. 2 (A to D). For or , the device performance deteriorates considerably. However, in the range of , JSC and FF show a plateau, whereas VOC increases substantially, reaching a maximum of 1.17 V at (1.18 V without aperture; see discussion in Materials and Methods). This is the highest VOC observed so far for lead iodide–based PSCs on mesoporous TiO2, resulting in a PCE larger than 19%. During the rise of VOC, JSC maintains a large value (more than 23 mA cm−2) because the band gap does not alter strongly for the different perovskite compositions, as shown by the absorbance spectra in fig. S5. The FF is around 0.72, indicating that a moderate excess of PbI2 does not retard charge collection, which is efficient in these mixed perovskites, allowing for the use of 500-nm capping layers to obtain a sharp IPCE onset and extraordinarily high photocurrents. Thus, overstoichiometric PbI2 in this range does not strongly influence the composition and optical properties of the perovskite films but improves their electronic quality. This allows for simultaneously high JSC and VOC but avoids the commonly observed “tradeoff” between VOC and JSC when tuning the band gap (, ). In the following sections, we quantify and characterize the PbI2 content remaining in the film. Subsequently, we investigate the role of PbI2 in reducing nonradiative recombination.

Fig. 2

Influence and characterization of remnant PbI2 in the fabricated solar cells and films as a function of the ratio between PbI2 and FAI in the precursor solution.

(A to D) Photovoltaic parameters JSC (A), VOC (B), FF (C), and PCE (D) versus , measured under AM 1.5 G illumination (100 mW cm−2). (E) Fraction of remnant PbI2 (left axis, orange line) and relative perovskite absorbance (right axis, blue line). (F) Mean crystallite sizes of FAPbI3 (left axis, green line) and PbI2 (left axis, orange line) phases determined by Rietveld refinement of thin-film XRD patterns and mean crystallite size ratio of PbI2/FAI (right axis).

SEM images of perovskite films deposited on a mesoporous TiO2/compact TiO2/FTO substrate with varied are shown in fig. S6. The film morphology changes when and , indicating a modification of the perovskite morphology or the appearance of new phases. To further explore the film composition, we conducted thin-film x-ray diffraction (XRD) measurements for perovskite films deposited again on mesoporous TiO2/compact TiO2/FTO substrates (table S3 and figs. S7 and S8). Except for the sample with , a peak at 12.5° is observed, which is attributed to the (001) lattice planes of hexagonal (2H polytype) PbI2. The PbI2 content (Fig. 2E) increases with , showing a 3.8% excess in weight for the most efficient device () and a 7.5% excess in weight for the highest VOC device () (table S1). These values are close to those expected from the molar ratios of the precursors in the spin-coating solution, which correspond to approximately 3 and 10 weight percent, respectively. The relative amount of perovskite (Fig. 2E) deduced from absorption measurements shows a reverse trend as compared with the remnant PbI2 but follows the JSC values (Fig. 2A), indicating that losses in JSC are due to reduced light harvesting rather than reduced charge collection. Because the thicknesses of all perovskite films are similar (fig. S9), the changes in absorbance indicate a significant ratio of nonperovskite material. The opposite trends in the mean crystallite sizes of PbI2 and perovskite (Fig. 2F) could be due to the competitive expansion of two crystal domains. Coincidentally, the mean crystallite size of the perovskite phase follows the same trend as VOC (shown in Fig. 2B). The trend in the crystallite size ratio of PbI2 and perovskite (Fig. 2F) mirrors that of VOC and reaches 0.35 for the film with the highest VOC. Thus, excess PbI2 not only exists as a crystalline phase but also influences the morphology and crystallite size of the perovskite phase. The observed correlation of the open-circuit voltage with the mean crystallite size is likely due to the fact that larger perovskite crystallites exhibit a reduced area of grain boundaries. Consequently, the overall density of defect states is lower.

DISCUSSION

To study the electronic quality of the device and to identify the recombination mechanisms limiting VOC, we performed electroluminescence measurements in which we applied a forward bias to the solar cell in the dark and operated it as a light-emitting diode (LED). Figure 3A contains data for the device (), with a VOC of 1.18 V. The blue curve shows injection current versus voltage, where the signature of a shunt for low voltages is followed by an exponential region dominated by recombination in the device. For voltages larger than 1.2 V, limitations by the series resistance become apparent. The red curve shows the emitted photon flux, which increases exponentially with voltage. The diode ideality factor derived from the “slope” is approximately 2 for the current and 1 for the emitted photon flux, indicating that much of the recombination happens through defects [Shockley-Read-Hall (SRH)], whereas emission originates from band-to-band recombination. The external electroluminescence quantum ?efficiency (EQEEL) increases linearly with injection current (fig. S10), as expected from a device with an ideality factor of 2 (), and approaches 0.5% for currents in the range of JSC. This value translates into a voltage loss of kT/e ln(1/EQEEL) = 0.14 V, confirming the measured VOC = 1.32 V − 0.14 V = 1.18 V [where 1.32 V is the theoretical maximum VOC (radiative limit)], which we determined for this solar cell using its IPCE action spectrum and following the approach of Tress et al. ().

Fig. 3

Characterization of recombination mechanisms and rates.

(A) Current-voltage curve in the dark (blue), emitted photon flux (red), and external electroluminescence quantum efficiency (EQEEL) (green) of a device with . Lines are a guide to the eye, indicating the slopes for an ideality factor of 2 and 1, respectively, assuming a temperature of 320 K. au, arbitrary units. (B and C) PL decay for a film with and . Lines are calculated according to the rate equation in the text. (D) VOC as a function of short-circuit current ISC proportional to the light intensity, which was varied for blue and red illuminations.

An EQEEL of 0.5% is a record for solution-processed solar cells such as organics, where it is commonly <10−6 (), and approaches that of the best silicon solar cells (). Even when compared to perovskite-based LEDs, our EQEEL is among the highest. Our solar cell LED delivers an overall electric power to light conversion efficiency of as high as 0.5% at a voltage of 1.5 V (which is still below Eg/e), whereas reported LEDs show the same efficiency at voltages higher than 3 to 4 V (, ). An LED efficiency of 3.5% has been obtained recently, but under much larger charge carrier densities [that is, not under solar cell operating conditions (2.2 V, 160 mA cm−2, and 50-nm layer thickness)] (). Our high EQEEL indicates a high-quality perovskite film and a very good charge selectivity of the contacts, accompanied by a high built-in potential and/or very balanced injection and transport of electrons and holes.

To analyze the dynamics of recombination, we performed time-correlated single photon counting (TCSPC) measurements at different excitation fluences. The photoluminescence (PL) decays for perovskite films with and deposited on mesoporous TiO2/compact TiO2/FTO substrate are shown in Fig. 3 (B and C) (for all samples in figs. S11 and S12).

For the highest light intensity generating approximately n = 2 × 1017 charges per cubic centimeter, the overall decay becomes faster as a result of direct electron-hole recombination, where the recombination coefficient β ≈ 2…4 × 10−11 cm3 s−1 in dn/dt = −βn2 − kn describes radiative recombination resulting in the expected VOC of ≈1.3 V (derivation in Materials and Methods). For long time scales or decreased excitation intensity, the PL shows monoexponential decay characteristic of SRH recombination. We deduced 220 and 350 ns, respectively, as nonradiative lifetimes 1/k for the two perovskite films with and , consistent with the trend in VOC. In addition, the signal drops strongly in the first few nanoseconds as a result of trap filling. The loss of PL is more significant for the stoichiometric device, in particular when illuminated from the TiO2 side. This indicates that the trap density is lower for as a result of either a better passivation of traps at the perovskite/TiO2 interface or a better quality of the perovskite crystallites.

To further elucidate this, we investigate VOC as a function of illumination intensity, represented by JSC in Fig. 3D (JSC ∝ intensity). The dashed lines visualizing the theoretical slope for SRH recombination indicate that VOC is mainly limited by SRH recombination. The device with shows a reduced slope toward higher intensities. This is consistent with the bending seen in EQEEL as a function of injection current (fig. S10). Because VOC is still below the radiative limit, this reduced slope is most likely due to losses of charges at a nonperfectly selective contact.

We use illumination with different wavelengths to tune the absorption profile in the perovskite film. Because of the strong dependence of the absorption coefficient on wavelength (fig. S5), red light (630 nm) and blue light (460 nm) penetrate the film to varied extent: 90% of the blue light is absorbed over a distance <200 nm away from FTO, whereas red light penetrates much farther. Figure 3D shows a significant difference only for the stoichiometric device, where VOC is lower for blue illumination despite the fact that the generated charge carrier density is higher. This is in accordance with the PL decay and proves that this device shows its highest defect density close to TiO2, the impact of which is reduced in the presence of overstoichiometric PbI2.

From the PL decay and the monochromatic VOC study, we contend that the role of excess PbI2 is to reduce recombination close to the perovskite/TiO2 interface. Thus, PbI2 crystallites, which are located within the mesoporous TiO2 scaffold, prevent recombination of holes photogenerated in this region of the absorber. Furthermore, PbI2 crystallites, which are present within the mesoporous TiO2 layer, favor the growth of bigger perovskite crystallites, which exhibit a reduced area of grain boundaries and, therefore, fewer defects.

To conclude, we have shown that an excess of PbI2 of up to ≈8 wt % can enhance the electronic quality of the perovskite/TiO2 film. With the adoption of this approach, devices based on mixed perovskite enabled a PCE of 20.8% and an open-circuit voltage of 1.18 V, accompanied by a high external electroluminescence quantum efficiency of 0.5%, which is attributed to reduced recombination via defects due to a moderate excess of PbI2. Further optimization along this path will lead to solution-processed solar cells approaching the theoretical limits for open-circuit voltage and efficiency.

MATERIALS AND METHODS

Materials

All materials were purchased from Sigma-Aldrich and used as received, unless stated otherwise.

Synthesis of inorganic/organic halide materials

NH=CHNH3I was synthesized by slowly dropping 15 ml of hydroiodic acid (45 wt % in water) (Applichem) to a solution of 5 g of formamidine acetate in methanol cooled at 0°C. The solution was further stirred for 5 hours at room temperature. The light-yellow solution was concentrated by rotary evaporation at 80°C until no obvious liquid remained. The crude solid was then dissolved by a minimum amount of methanol, reprecipitated in diethyl ether, and filtered. The procedure was repeated three times, and the resulting white solid was collected and dried at 80°C under vacuum for 2 days: 1H nuclear magnetic resonance (DMSO-d6) δ 8.76 (s, 4H), 7.85 (s, 1H) ppm; 13C nuclear magnetic resonance (DMSO-d6) δ 157.05 ppm.

CH3NH3Br was synthesized by slowly dropping 31.1 ml of hydrobromide acid (48 wt % in water) (Fluka) to a solution of 27.86 ml of CH3NH2 (40 wt % solution in absolute methanol; TCI) cooled at 0°C. The solution was further stirred for 5 hours at room temperature. The colorless solution was concentrated by rotary evaporation at 50°C until no obvious liquid remained. The crude solid was then dissolved by a minimum amount of ethanol, reprecipitated in diethyl ether, and filtered. The procedure was repeated three times, and the resulting white solid was collected and dried at room temperature under vacuum for 2 days.

Solar cell preparation

The fluorine-doped tin oxide–coated glass (NSG) was sequentially cleaned using detergent, acetone, and ethanol. A 20- to 30-nm TiO2 blocking layer was deposited on the cleaned FTO by spray pyrolysis, using O2 as carrier gas, at 450°C from a precursor solution of 0.6 ml of titanium diisopropoxide and 0.4 ml of bis(acetylacetonate) in 7 ml of anhydrous isopropanol. A 200-nm mesoporous TiO2 was coated on the substrate by spin coating at a speed of 4500 rpm for 15 s with a ramp-up of 2000 rpm s−1 from a diluted 30-nm particle paste (Dyesol) in ethanol; the weight ratio of TiO2 (Dyesol paste) to ethanol is 5.5:1. After spin coating, the substrate was immediately dried on a hotplate at 80°C, and the substrates were then sintered at 500°C for 20 min. The perovskite film was deposited by spin coating onto the TiO2 substrate. The precursor solution was prepared in a glovebox of 1.35 M Pb2+ (PbI2 and PbBr2) in a mixed solvent of DMF and DMSO; the volume ratio of DMF to DMSO was 4:1. The molar ratio for PbI2 (specified purity >98%; TCI)/PbBr2(purity, 99.999%; Alfa Aesar) was fixed at 0.85:0.15, and the molar ratio for MABr/PbBr2 was fixed at 1:1. A comprehensive analysis carried out by an analytical institution and covering more than 70 elements showed that the purity of the TCI product (PbI2) exceeded 99.99%. Thus, it was much purer than specified, implying that changes in molar ratio within the range of some percentages in the precursor solutions are not superimposed by the effects of impurities.

By changing the amount of FAI, we obtained different molar ratios for PbI2/FAI varying from 1.54 to 1.37, 1.23, 1.16, 1.10, 1.05, 1.00, and 0.85. The spin-coating procedure was performed in an argon flowing glovebox: first, 2000 rpm for 10 s with a ramp-up of 200 rpm s−1; second, 6000 rpm for 30 s with a ramp-up of 2000 rpm s−1. Chlorobenzene (110 μl) was dropped on the spinning substrate during the second spin-coating step 20 s before the end of the procedure. The substrate was then heated at 100°C for 90 min on a hotplate. After cooling down to room temperature, Spiro-OMeTAD (Merck) was subsequently deposited on top of the perovskite layer by spin coating at 3000 rpm for 20 s. The Spiro-OMeTAD solution was prepared by dissolving Spiro-OMeTAD in chlorobenzene (60 mM), with the addition of 30 mM bis(trifluoromethanesulfonyl)imide (from a stock solution in acetonitrile) and 200 mM tert-butylpyridine. Finally, FK209 [tris(2-(1H-pyrazol-1-yl)-4-tert-butylpyridine)-cobalt(III) tris(bis(trifluoromethylsulfonyl)imide); Dyenamo AB] was added to the Spiro-OMeTAD solution (from a stock solution in acetonitrile); the molar ratio for FK209 and Spiro-OMeTAD was 0.03. Finally, 80 nm of gold was deposited by thermal evaporation using a shadow mask to pattern the electrodes.

Characterization

Current-voltage characteristics were recorded by applying an external potential bias to the cell while recording the generated photocurrent with a digital source meter (Keithley model 2400). The light source was a 450-W xenon lamp (Oriel) equipped with a Schott K113 Tempax sunlight filter (Praezisions Glas & Optik GmbH) to match the emission spectrum of the lamp with the AM 1.5 G standard. Before each measurement, the exact light intensity was determined using a calibrated Si reference diode equipped with an infrared cutoff filter (Schott KG-3).

To specify the illuminated area, we used an aperture (shadow mask) with an area of 0.16 cm2, whereas the total device area defined by the overlap of the electrodes was approximately 0.36 cm2. This approach allows for an accurate determination of the short-circuit current density but has drawbacks when determining the open-circuit voltage (VOC) as VOC depends on the dark current flowing through the device according to the equation: VOC = nIDkT/e ln(ISC/I0 + 1). If there is a mismatch between the illuminated area and the dark area, the ratio between the photocurrent ISC and the dark saturation current I0 is artificially changed by this ratio. Therefore, the open-circuit voltage was additionally measured without aperture.

XRD spectra were recorded on an X’Pert MPD PRO (PANalytical) equipped with a ceramic tube providing Ni-filtered (CuKα, λ = 1.54060 Å) radiation and on an RTMS X’Celerator (PANalytical). The measurements were performed in Bragg-Brentano geometry 2θ = 8° to 88°. The samples were mounted without further modification, and the automatic divergence slit (10 mm) and beam mask (10 mm) were adjusted to the dimension of the films. A step size of 0.008° was chosen for an acquisition time of 270.57 s deg−1. A baseline correction was applied to all x-ray thin-film diffractograms to compensate for the broad feature arising from the FTO glass and anatase substrate. The presence of strong thermal diffuse scattering and turbostratic disorder, mainly in the PbI2 films, thwarted a successful Rietveld refinement of up to six phases. Finally, the areas of the (001) maximum of PbI2 and the (011)/(101) peaks of FAPbI3, calculated by means of a pseudo-Voigt function, were used to estimate the weight fractions. These pseudo-Voigt functions also furnished the full widths at half maximum, which were subsequently used to compute the sizes of the coherent domains along the diffraction vectors by means of Scherrer’s equation, setting K = 1. SEM images were recorded using a high-resolution scanning electron microscope (Zeiss Merlin).

Electroluminescence yield

The emitted photon flux was detected with a large-area (1 cm2) Si-photodiode (Hamamatsu S1227-1010BQ) positioned close to the sample. Because of the nonconsidered angular dependence of emission and detector sensitivity, EQEEL was expected to be slightly underestimated (on the order of 10%). The driving voltage was applied using a Bio-Logic SP300 potentiostat, which was also used to measure the short-circuit current of the detector at a second channel.

Absorption spectra were measured on a PerkinElmer ultraviolet (UV)–vis spectrophotometer. Absorbance was determined from a transmittance measurement using an integrating sphere. We used the “PerkinElmer Lambda 950 nm” setup with the integrating sphere system “60 nm InGaAs integrating sphere.” The sources were deuterium and tungsten halogen lamps, and the signal was detected by a gridless photomultiplier with Peltier-controlled PbS detector. The UV WinLab software was used to process the data.

PL and TCSPC experiments

PL spectra were recorded by exciting the perovskite films deposited onto mesoporous TiO2 at 460 nm with a standard 450-W Xenon CW lamp. The signal was recorded with a spectrofluorometer (Fluorolog; Horiba Jobin Yvon Technology FL1065) and analyzed with the software FluorEssence.

The PL decay experiments were performed on the same samples using the same Fluorolog with a pulsed source at 406 nm (Horiba NanoLED 402-LH; pulse width <200 ps, 11 pJ per pulse, approximately 1 mm2 in spot size), and the signal was recorded using TCSPC. The samples were excited from the perovskite and glass side under ambient conditions.

Analysis of the PL decay

From the pump fluence, we estimated an initial photogenerated charge carrier density on the order of 2 × 1017 cm−3 upon excitation at the highest intensity. For a filter with a transmittance of 2.5%, we expected 5 × 1015 cm−3. Assuming that most of the charge carriers in the perovskite are photogenerated (that is, the intrinsic charge carrier density is low), we can set the electron density equal to the hole density and write the continuity equation for photogenerated electrons: dn/dt = −βn2 − kn, assuming that changes in the charge density n are due to either a bimolecular process or monomolecular recombination. Solving this equation for n(t) and assuming that the PL signal is proportional to n2(t), we can calculate the PL decay (solid lines in the main paper).

The ideal solar cell without nonradiative recombination should have k = 0. Then, VOC can be calculated, assuming that the charge carrier generation rate G equals the recombination rate R: G = R = βn2. For a semiconductor using Boltzmann statistics and effective mass approximation, VOC can be written as eVOC = Eg − kT ln(NcNv/np), with the effective densities of states in conduction and valence band: Nc,v = 2 × (2πme,h*kBT/h2)3/2. Thus, knowing the effective mass me,h, temperature T, β, band gap Eg, and light intensity, VOC can be calculated as VOC = Eg/e − kT/e ln(NcNv β/G) ≈ 1.3 V, where we approximated G = JSC/(e × thickness) = 3 × 1027 m−3 s−1. This is a rough estimation where the order of magnitude of the input parameters—but not their exact values—is known. Effective masses have been taken from Giorgi et al. ().