Optical determination of Shockley-Read-Hall and interface recombination currents in hybrid perovskites

Metal-halide perovskite solar cells rival the best inorganic solar cells in power conversion efficiency, providing the outlook for efficient, cheap devices. In order for the technology to mature and approach the ideal Shockley-Queissier efficiency, experimental tools are needed to diagnose what processes limit performances, beyond simply measuring electrical characteristics often affected by parasitic effects and difficult to interpret. Here we study the microscopic origin of recombination currents causing photoconversion losses with an all-optical technique, measuring the electron-hole free energy as a function of the exciting light intensity. Our method allows assessing the ideality factor and breaks down the electron-hole recombination current into bulk defect and interface contributions, providing an estimate of the limit photoconversion efficiency, without any real charge current flowing through the device. We identify Shockley-Read-Hall recombination as the main decay process in insulated perovskite layers and quantify the additional performance degradation due to interface recombination in heterojunctions.

Six decades after the demonstration of the first silicon photovoltaic cell 1 , only a limited number of semiconductors enable single junction photovoltaic devices with power conversion efficiencies (PCEs) exceeding 20% 2 . Among these, hybrid organic-inorganic perovskites (HPs) represent an emerging class of solution-processed semiconductors [3][4][5][6][7][8][9][10] with the potential to approach the Shockley-Queisser limit of the PCE 11 and the prospect to be integrated with established commercial technologies to fabricate cheap, multi-junction solar cells 12,13 , with even higher efficiencies.
The rapid increase in photovoltaic performance has been accompanied by intense research on the photophysics of these materials [14][15][16][17][18][19][20][21][22] . A key aspect of photoconversion in HPs is the recombination current, a ubiquitous mechanism of energy loss in solar cells. Photovoltaic devices operate as non-ideal current generators in which a fraction of photogenerated electrons and holes recombines inside the cell, feeding the internal diode current instead of being injected into the external load 23 . Minimal recombination energy losses are achieved in the ideal case in which electron-hole pairs decay only radiatively; in this regime, the recombination resistance is maximized, and so is the cell voltage. The external electroluminescence quantum yield (EQY EL ) quantifies the amount of radiative recombination with respect to non-radiative losses; the best solar cells show the highest EQY EL (10 −3 -10 −1 ) and therefore operate at a voltage as close as possible to the semiconductor band gap-up to 0.78 E g in GaAs 24 and very recently 0.76 E g in HP 8,20,[25][26][27] .
The simplest architecture of heterojunction solar cells consists of a light absorber sandwiched between two charge-selective semiconductor layers, one of which allows the flow of photoexcited electrons (ETL) but blocks the transmission of holes, while the other conveys only photoexcited holes (HTL) to the opposite electrode (Fig. 1). Nonradiative decay channels are typically associated with two possible mechanisms: one is related to the presence (a) Stand-alone intrinsic layer. The free energy μ oc of photogenerated electron-hole pairs is equal to the energy splitting of the quasi-Fermi levels of electrons in the conduction (F cb ) and valence (F vb ) bands. No electric voltage is present between the two sides due to the absence of electron-and hole-selective contacts. An empty trap level E t in the mid-gap is assumed. R SRH,e(h) is the Shockley-Read-Hall recombination rate of electrons (holes) per unit of volume. Correspondingly, R rad refers to radiative recombinations. (b) HTL-i-ETL double heterojunction. The difference between F cb in the ETL and F vb in the HTL is given by μ = eV, where V is the circuit voltage of the solar cell in absence of electrical losses. As shown in Supplementary Fig. 2, equilibrium conditions of the solar cell in the dark impose that the concentration of trapped electrons varies across the intrinsic layer: trap levels at the centre of the i-semiconductor traps are half-filled so that the recombination rate R SRH,e(h) of the excess carriers is quite large. Conversely, trap levels close to the ETL (HTL) are filled (empty), no trapping of electrons (holes) by mid-gap states is thereby possible, leading to negligible R SRH,e(h) values.

Results
All-optical determination of the electron-hole free energy. According to the theory of non-equilibrium semiconductors, the free energy (or chemical energy) of photogenerated electron-hole pairs, μ, is equal to the energy splitting of the quasi-Fermi levels of electrons in the conduction (F cb ) and valence (F vb ) bands, respectively. To provide an expression for μ in terms of measurable optical quantities, we consider Kirchhoff 's law of radiation, which represents the detailed balance between emission and absorption, generalized by Würfel to account for non-equilibrium electron and hole populations 37 : is the emitted photon current density, proportional to the external photoluminescence intensity, while a(ω) denotes the absorptivity, which depends both on the absorption coefficient and film thickness. Ω is the effective external emission angle. The right-hand side of equation (1) is valid for excitation levels typical of solar illumination, for which the Bose function can be substituted by the Boltzmann distribution. Equation (1) provides the analytical relation between J PL , μ, and the photon current density J 0,rad emitted by the semiconductor in thermal equilibrium with the environment (μ = 0) 38 . To our purposes, J PL can be opportunely expressed in terms of the external photoluminescence quantum yield (EQY PL ), defined as the ratio between J PL and the absorbed excitation photon flux, that is = EQY J J PL PL ex . With this substitution, equation (1) can be reformulated to explicitly link μ to EQY PL : The second term of the right-hand side of equation (2) states a simple but very useful rule: nonradiative decay channels affect the free energy only through the EQY PL , leading to a free-energy loss of In(10 −1 ) kT = 60 meV for a factor of 10 drop of EQY PL at 300 K. Furthermore, equation (2) provides a reformulation of Kirchhoff 's law that allows us to highlight similarities between the optical and the optoelectronic reciprocity relations 39 . The latter ones, proposed ten years ago by Rau, connect the electroluminescent emission and photovoltaic voltage through the equations eV oc = eV oc,rad + kT ln(EQY EL ) and = eV k T ln J J oc,rad sc 0,rad , where e J0,rad represents the radiative charge current in the dark, and eJ sc the short-circuit charge current (note that here the symbol J is used for particle currents, not charge currents, hence the need to multiply currents by the electron charge e). Figure 1 visualizes μ for a stand-alone intrinsic semiconductor layer and for a double heterojunction (HTL-i-ETL; 'i' stands for the intrinsic absorber semiconductor), providing the link between μ and the circuit voltage in solar cells. In the stand-alone layer (Fig. 1a), μ = μ oc is constant throughout the film but no external electric voltage is produced due to the absence of electron-and hole-selective contacts. In the HTL-i-ETL solar cell (Fig. 1b), the free energy is provided by the difference between the energies of the quasi-Fermi levels of electrons in the ETL and in the HTL at the opposite side of the device. µ e in the intrinsic layer equates the circuit voltage V of the solar cell if F cb and F vb do not vary from the transport materials up to the external contacts.
All-optical determination of the ideality factor. The optical reciprocity relation in eq. (2) allows measuring μ without having any information on the specific microscopic mechanisms of electron-hole recombination. Nonetheless, as we are interested in uncovering the decay channels, we need to find out how these processes influence the μ-J ex characteristics. We therefore need to specify a microscopic model for recombinations. Let us consider the simplest case, in which decays of electrons and holes are driven by elementary processes such as monomolecular, bimolecular, or higher order multi-particle interactions. If the corresponding recombination rates for electrons and holes follow a power law α n ( ) ( ) the electron (hole) extraction rate per unit of volume, and α denotes the number of carriers involved in the recombination process (α = 1,2,3 for monomolecular, bimolecular, and trimolecular recombinations respectively). To connect μ to the free electron (n e ) and hole (n h ) concentrations, we use the mass action law, generalized to account for non-equilibrium carriers:  , where m is the ideality factor of the μ-J ex curves, with J 0 being the reverse bias saturation current (note again that charge currents are usually considered instead of particle currents as we are doing here). When J e(h) = 0 (μ = μ oc ), eq. (4) provides an all-optical route to determine the ideality factor m and consequently to identify the recombination mechanisms. Elementary electron-hole annihilation processes (α = 1,2,3,… ) are associated with rational values of m. Band-to-band electron-hole recombinations (n e~nh ,α e = α h = 2,) yield m = 1; nonradiative monomolecular decays of minority carriers in doped semiconductors also lead to m = 1. Auger recombinations are trimolecular processes, so Free energy and ideality factor in perovskite films. The experimental data from single HP layers allow to understand the bulk carrier recombination processes and to determine the maximum attainable free energy μ oc . Figure 2a reports μ oc as a function of ln(I ex ), where I ex is the excitation light intensity delivered by a green CW laser (λ = 532 nm). μ oc was determined from the EQY PL of MAPbI 3 as a function of I ex . We analysed perovskite layers grown with several deposition methods, namely single-step and double-step techniques. According to eq. (2), it is sufficient to assess μ oc,rad at just one excitation intensity, which we chose so that the absorbed photon current density of the green light and solar light at AM1.5G (J sun , ħω > E OG ) were equal (I ex = 50 mW/cm 2 ). In the Supplementary Note 2, we show that μ oc,rad can be usefully written as:  Shockley-Read-Hall recombinations are expected to lead to a rational ideality factor m = 3/2 at low illumination levels. The experimental slope decreases to approximately 2/3 for I ex > 100 W/cm 2 , as foreseen for Auger recombination. At I ex = 50 mW/cm 2 , the rate of photons absorbed by the film matches that one obtained at an illumination level of one sun (AM 1.5G). At this excitation intensity, μ rad = 1.33 eV while µ = .
1 16 SRH eV, with a free energy loss due to Shockley-Read-Hall recombinations µ µ µ . by the semiconductor 37 ; a sun and a T represent the spectrally averaged film absorptivity, weighted by the black-body radiation spectra at T sun = 5541 K and T = 300 ± 20 K, respectively. The temperature T of the electron-hole gas was determined by fitting a Boltzmann distribution to the high-energy tail of the photoluminescence spectrum. f(T, T sun , E gap ) is an analytical function, whose value can be readily calculated as equal to 1.718. In the Shockley-Queisser limit, = 1 a a T sun , hence the radiative free energy only depends on T sun and properties of the material, namely E gap and the effective working temperature T. We then obtained μ oc,rad (J sun ) = 1.33 ± 0.02 eV, where the experimental error was mainly due to the uncertainty in the measurement of T. The effect of the dependence of the absorptivity on the layer thickness is within 10 meV (see Supplementary Note). Our all-optical value is in agreement with estimations of the radiative electrical voltage in HP solar cells, V oc,rad = (1.32-1.34) ± 0.02 V, as independently assessed from electrical characteristics 20 .
The free energy μ oc was estimated through eq. (2) by simply adding to the radiative limit μ oc,rad the term kTln(EQY PL ) due to nonradiative recombinations. Experimental data show that μ is proportional to ln(I ex ), with the slope coefficient giving the ideality factor. For 1 mW/cm 2 < I ex < 100 W/cm 2 , the slope was close to 3/2, independently of the method used to fabricate the MAPbI 3 films, with an average ideality factor = . ± . m 1 48 0 03. In the light of the previous discussion on the m-rational values, experimental findings indicate that electrons and holes follow different power laws, and more specifically one carrier decays obeying a first order process (α = 1) while the other follows a second order process (α = 2). This asymmetry between electrons and holes is consistent with the presence of traps preferentially capturing a single carrier type. Figure 1 shows the recombination processes expected in a stand-alone, undoped HP layer, assuming empty traps 40,41 ; fully occupied trap states would lead to similar conclusions, just swapping the roles of electrons and holes. The capture rate of electrons by intragap levels can be described as a bimolecular process, R SRH,e ∝ n h,t n e , where n h,t is the density of trapped holes. However, in the intrinsic semiconductor all traps are nearly empty, so n h,t is almost equal to the density N t of recombination centres. Consequently, the recombination process is monomolecular (α e = 1). Similarly to the case of electrons, the hole capture rate is R SRH,h ∝ n h n e,t , but in this case the densities of both trapped electrons n e,t and free holes n h increase with the excitation intensity and, additionally, n e,t~nh . Hole recombination is therefore an effective bimolecular process with α h = 2; far from trap saturation, we obtain = + = . We thus conclude that nonradiative recombinations in HPs can be described in the framework of the SRH model. In our samples, SRH recombination limits the available free energy to µ = . For I ex > 100 W/cm 2 , m decreased to ~2/3, as expected for trimolecular annihilations via Auger decay channels. Over a wide range of excitation intensities, as in the μ oc -J ex characteristics presented in this work, two (or more) elementary decay channels contribute to the recombination dynamics. Competition between them is such that the lowest-order electron-hole decays, identified by the lowest power index α and thus the highest m (nonradiative recombinations with m = 3/2 in perovskite films), dominates at low excitation intensities, while the highest order decay (nonradiative Auger recombinations with m = 2/3 in perovskite films) becomes the most important one at high excitation 14,27,42,43 . Similar behaviour is also observed in the I-V characteristics of a Si solar cell, where m decreases from 2 (SRH recombinations) to 1 for increasing voltage 28 . We rule out trap saturation because it would cause hole recombination to become monomolecular, as the population of trapped electrons becomes constant, and consequently m should increase from 3/2 to 2 (α e = α h = 1).
Free energy and ideality factor in perovskite heterojunctions. According to the optical reciprocity relation in eq. (2), the larger EQY PL , the larger the resulting free energy. Therefore, a purely optical analysis of the EQY PL ought to establish if additional interface recombination is setting stricter limits to μ oc in single and double HP heterojunctions with respect to the bulk. Figure 3 shows the EQY PL and corresponding μ oc in these structures at an absorbed monochromatic photon density current corresponding to one sun. The standalone layer has the largest EQY PL , while significantly lower values are measured in the presence of interfaces, both in the i-ETL (perovskite-TiO 2 ) and HTL-i (spiro-MeOTAD-perovskite) heterojunctions. Our interpretation of the evidence is that additional non-radiative recombination channels appear at both interfaces, providing faster non-radiative recombination than in the standalone layer. The full HTL-i-ETL structure, representing a contactless solar cell, shows an even lower EQY PL .
The ideality factor in the heterojunctions was investigated by studying the μ-I ex characteristics (Fig. 4), in the same way as for single HP layers (Fig. 2). Its value deviates from the 3/2 value measured in the intrinsic materials for I ex between 0.01 and 100 suns. Furthermore, the μ oc -I ex characteristics show that m increases with the excitation density, contrary to what expected from the competition between elementary recombination processes. m varies from ≈ 3/2 to ≈ 2 in single heterojunctions, and from ≈ 1 to ≈ 2 in double heterojunctions.
Possible interface recombination processes at the heterojunction interfaces are sketched in Fig. 3. In a steady-state experiment, electron (hole) transfer from the i-layer to the ETL (HTL) is compensated by electron (hole) back-transfer, preventing charge build-up at both sides of the interface. We attribute the lower EQY PL observed in our perovskite heterojunctions with respect to the HP single layers to the activation of such interface decay channels 44,45 . Quenching of optical excitations at interfaces has even been exploited to measure the diffusion length in HPs 46,47 . Interface decay processes are not elementary decays and, as in the case of SRH decays, the ideality factor is expected to increase with the excitation intensity. Excitation-dependent band bending and modification of the energy level alignment close to the junctions, e.g. due to built-in electric fields, could drive non-linear phenomena for the carrier dynamics at the two interfaces. Anyway, the fact that at one sun m ≈ 2 suggests monomolecular recombinations for both electrons and holes (α e ≈ α h ≈ 1).    Figure 5 shows eJ e(h) (μ) and the condition for maximum chemical energy conversion, with efficiency η = µ × × µ e J FF I sc oc sun (FF μ being the filling factor of the μ-J e(h) characteristics, and eJ sc = eJ sun ). The points in Fig. 5 are the experimental data eJ ex = eJ rec (μ oc ) from Figs 2 and 4 with the substitution µ µ → oc 29 . In the Shockley-Queissier limit, when the solar spectrum is perfectly absorbed down to the bandgap energy, with μ rad = 1.33 eV and m = 1, the ultimate limit η SQ = 30.5% is obtained (eJ sc = 25.4 mA/cm 2 , FF μ = 0.91). In the single HP layer, SRH limits the chemical energy to μ oc = 1.16 eV and m = 1.5; furthermore, considering the actual absorption of a 250-nm thick HP film (as typical for solar cells reported in literature), the limiting photoconversion efficiency due to SRH recombination reduces to η SRH = 23.4% (eJ sc = 23.6 mA/cm 2 , FF μ = 0.86). The strictest limit in the investigated double heterojunctions is set by interface recombinations to μ = 0.97 eV (m ≈ 2), resulting in η = 18.2% (eJ sc = 23.6 mA/cm 2 , FF μ = 0.80). Recently, solar cells have been reported based on HP materials with composite cations, including both organic molecules and the inorganic elements Rb and Cs, with very high external photoluminescence efficiency, up 3.6% 8 . Assuming no electrical losses, our analysis predicts for such cells a limit efficiency η = 26.2% (d = 400 nm, eJ sc = 24.9 mA/cm 2 , μ oc = 1.26, FF μ = 0.83).

Conclusions
The Shockley-Queissier model provides the reference, ultimate limit performance of single junction photovoltaic devices, assuming only radiative recombinations and neglecting all losses due to charge transport and extraction. We have developed an all-optical experimental method to assess the deterioration of photoconversion performances with respect to the Shockley-Queissier limit due to nonradiative recombinations. To this aim, the proposed approach allows measuring the upper limit to the open-circuit voltage of a solar cell set by nonradiative electron-hole decays in the intrinsic materials and assessing to what extent each interface in the device introduces additional recombination currents. We are able to identify the nature of decay processes, showing that recombinations in single perovskite layers can be described by the Shockley-Read-Hall model in the presence of defects that preferentially trap either electrons or holes. We estimate the ideality factor and the limit of the solar cell efficiency, without any current flowing through the device, therefore avoiding hysteresis, non-ideal contacts and charge transport losses. Methods Materials. All materials were purchased from Sigma-Aldrich or Alpha Aesar and used as received. Spiro-MeOTAD was purchased from Lumtec. CH 3 NH 3 I was synthesized according to a reported procedure 48 . CH 3 NH 3 (27.86 ml, 40% in methanol, TCI) and hydroiodic acid (30 ml, 57 wt% in water, Aldrich) were mixed at 0 °C and stirred for 2 h. The precipitate was recovered byevaporation at 50 °C for 1 h. The product was washed with diethyl ether three times and finally dried at 60 °C in a vacuum oven for 24 h.

Sample preparation. Glass substrates (Visiontech) and FTO-coated glass substrates (Solaronix)
were cleaned by ultrasonication in a deionized water, 2-propanol and acetone. Substrates were treated to the TL1-washing procedure (washed in double distilled water (Milli-Q water), hydrogen-peroxide (H 2 O 2 ) and ammonia (NH 3 ) 5:1:1 v/v at 80 °C for 10 minutes), then rinsed in double distilled water prior next depositions. A 80 nm-thick TiO 2 dense hole-blocking layer (ETL) was deposited on glass/FTO by spin coating a commercial titaniumdiisopropoxidebis(acetylacetonate) solution (75% in 2-propanol, Sigma-Aldrich) diluted in butanol (0,15 M) twice at 3,000 rpm for 60 sec and annealed at 125 °C. As last step a 0.3 M solution is spin-coated, and annealed at 520 °C. The prepared CH 3 NH 3 I and commercial PbI 2 (99.99% ultradry, Alfa Aesar) were stirred in a mixture of γ -butyrolactone (GBL) and DMSO (2:1vol/vol; GBL, > 99%; DMSO, 99.8%; Sigma-Aldrich) at 60 °C for 12 h, to prepare a solution 1 M. The perovskite precursor solution was coated onto either glass or TiO 2 /FTO substrate by a consecutive two-step spin-coating process at 1,000 and 4,000 r.p.m for 10 and 60 s, respectively with a dripping of dichloromethane at 10 s to the end. After spin-coating, the films were annealed on a hotplate at 100 °C for 10 min. After cooling to room temperature, either a PMMA solution (80 mg/1 mL chloroform) or the spiro-MeOTAD solution was spin-coated on the perovskite layer at 2,500 r.p.m. for 45 s. A spiro-MeOTAD solution was prepared by dissolving 90 mg of spiro-MeOTAD in 1 ml chlorobenzene (99.8%, Sigma Aldrich), to which were added 28.8 μ l of 4-tert-butylpyridine (96%, Sigma-Aldrich), 17.5 μl lithium bis (trifluoromethanesulfonyl) imide (LiTFSI) solution (520 mg LI-TSFI in 1 ml acetonitrile, 99.8%, Sigma-Aldrich). This fabrication process was carried out under controlled atmospheric conditions with a humidity of < 1% and a temperature between 20 and 25 °C. From AFM measurements, we determined a perovksite film thickness of 160 nm.

FTO/TiO 2 /perovskite/spiro-MeOTAD device fabrication. FTO-coated glass substrates (Solaronix)
were cleaned by ultrasonication in deionized water, 2-propanol and acetone. A 80 nm-thick TiO 2 dense hole-blocking layer (bl-TiO 2 ) was then deposited on the substrates by spin coating at 3,000 rpm for 60 sec and annealed at 520 °C using a commercial titanium diisopropoxide bis(acetylacetonate) solution (75% in 2-propanol, Sigma-Aldrich) diluted in butanol (0,3 M). The prepared CH 3 NH 3 I and commercial PbI 2 (99%, Alpha Aesar) for the 1 M CH 3 NH 3 PbI 3 solution were stirred in a mixture of γ -butyrolactone (GBL) and DMSO (2:1 vol/vol; GBL, ≥ 99%; DMSO, 99.8%; Sigma-Aldrich) at 60 °C for 12 h. Before use, these precursor solutions were filtrated using a hydrophilic PTFE syringe filter (pore size of 22 μ m). The filtrated perovskite precursor solution was coated onto bl-TiO 2 /FTO substrate by a consecutive two-step spin-coating process at 1,000 and 4,000 r.p.m for 10 and 60 s, respectively with a dipping of dicloromethan at 10 sec to the end. After spin coating, the films were annealed on a hotplate at 100 °C for 10 min. After cooling to room temperature, the spiro-MeOTAD solution was spin-coated on the perovskite layer at 2,500 r.p.m. for 45 s. A spiro-MeOTAD solution was prepared by dissolving 90 mg of spiro-MeOTAD in 1 ml chlorobenzene (99.8%, Sigma Aldrich), to which were added 28.8 μ l of 4-tert-butylpyridine (96%, Sigma-Aldrich), 17.5 μl lithium bis (trifluoromethanesulfonyl)imide (LiTFSI) solution (520 mg LI-TSFI in 1 ml acetonitrile, 99.8%, Sigma-Aldrich). This fabrication process was carried out under controlled atmospheric conditions with a humidity of < 1% and a temperature between 20 and 25 °C. Finally, 80 nm gold was thermally evaporated on top of the device at a pressure of 5 × 10 −6 mbar for 30 min to form the back contact. The active area of the complete device was 0.09 cm −2 .
Absolute photoluminescence quantum yield. EQY PL was measured according to the method described by de Mello et al. 49 using the beam from a diode-pumped Nd:YVO 4 CW laser (Millennia V) at 532 nm as excitation. The irradiance of the beam was calculated as E exc = P/A, where P is the optical power of the beam, measured by a bolometer, and A is beam area, measured using the knife-edge method. The samples were placed into an integrating sphere (Newport 819C-IS-5.3) and both the scattered laser light and the photoluminescence light were collected through a fibre-optic cable (Avantes FC-V200-1-SR) coupled to a grating spectrometer (Princeton Instruments Acton SpectraPro 2500i equipped with a 150 gr/mm, 600 nm blaze grating) and detected by a LN-cooled CCD camera. The laser beam was angled so that its reflection and/or the PL emission did not strike the output port directly. The spectral response of the detection system has been calibrated so that the number of counts is proportional to the number of photons collected. EQY PL was calculated as: As required by the theoretical framework, the measurements were not corrected for self-absorption.

Time-integrated photoluminescence (TIPL). Samples were placed in a vacuum chamber and excited
with a diode-pumped Nd:YVO 4 CW laser (Millennia V) at 532 nm. The photoluminescence was dispersed with a grating spectrometer (Princeton Instruments Acton SpectraPro 2500i equipped with a 150 gr/mm, 600 nm blaze grating) and detected by a LN-cooled CCD camera.