Maximum Efficiency and Open-Circuit Voltage of Perovskite Solar Cells

Abstract

This chapter serves as an introduction to the general working principles of solar cells. It starts from the thermodynamics of terrestrial solar cells and fundamentals of semiconductor-based photovoltaics, where the theoretical limits of efficiency and open-circuit voltage as a function of the bandgap are discussed. The chapter describes the prediction of the open-circuit voltage when the photovoltaic action spectra and the electroluminescence quantum efficiency are known. The role of subgap states and several sources of nonradiative recombination, including interfaces to the charge-transport layers, are investigated at open-circuit voltage and fill factor of state-of-the-art perovskite solar cells. Based on these factors, organic–inorganic perovskite solar cells with different architectures and compositions are compared with other solar cell technologies. Low disorder and weak nonradiative recombination are shown to be responsible for the superior performance of mixed cation mixed halide perovskite solar cells, allowing for open-circuit voltages of 1.2 V to be achieved at a bandgap of 1.6 eV.

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1 Maximum Power-Conversion Efficiency of a Terrestrial Solar Cell

1.1 Thermodynamics and Black Body Radiation

A solar cell converts energy of light emitted from the sun into electrical energy. The energy flux from the sun is primarily thermal radiation and can be approximated by a black body spectrum at a temperature T _S of ≈5800 K outside the earth atmosphere. Prior to reaching the earth’s surface, narrow spectral bands have been filtered out by gases in the atmosphere, such as ozone, oxygen, and water vapor (Fig. 1). The spectrum is then called AM X, where X denotes the distance that the light has traveled through the atmosphere in multiples of its thickness. The standard spectrum for solar cell characterization is AM 1.5 global corresponding to an angle of incidence of $48^{\text{o}} = \cos^{ - 1} (1.5^{ - 1} )$.

Fig. 1

The standard solar spectrum, AM1.5 g, used for solar cell characterization. The irradiance integrated over wavelength results in 100 mW cm⁻². The spectral shape can be approximated with a black body radiator at 5800 K. The gaps are due to absorption of compounds in the atmosphere (O₃, O₂, H₂O, …). Dash-dotted is the corresponding photon flux, obtained when dividing the irradiance spectrum (red) by the photon energy at each wavelength

Upon absorption of sun light, the absorber’s temperature increases to T _A meaning that the energy transferred from the sun heats the absorber “medium”. Thus, the solar cell can be treated as a heat engine transforming thermal energy into work in the form of electrical energy. The maximum conversion efficiency of a heat engine is the Carnot efficiency, which considers the conservation of entropy contained in thermal radiation. This entropy ${\Delta }S$ has to be released with a heat flux ${\Delta }Q = T_{0} {\Delta }S$, with T ₀ being the temperature of the surroundings, because electrical energy is entropy-free [1]. Thus, the Carnot efficiency is

$$\eta_{\text{C}} = \frac{{T_{\text{A}} - T_{0} }}{{T_{\text{A}} }} = 1 - \frac{{T_{0} }}{{T_{\text{A}} }}.$$

(1)

In order to obtain the maximum Carnot efficiency, the temperature difference between the medium and the surroundings should be maximized to avoid a large heat flux associated with the conservation of entropy. As the temperature of the surroundings is fixed, the medium should be as hot as possible.

Coming back to the energy source, we notice that the heat is transferred from the sun to the medium by radiation. In the optimum case, the medium itself is capable of absorbing all the incident radiation. Thus, it is a black body as well, which according to Kirchhoff’s law reemits part of the absorbed photon flux. Spectrum $B(\lambda , T)$ and emitted power per surface area (I) are described by Planck’s and the Stefan–Boltzmann law, respectively:

$$B\left( {\lambda , T} \right) = \frac{{2hc^{2} }}{{\lambda^{5} }}\frac{1}{{\exp \frac{hc}{{\lambda k_{\text{B}} T}} - 1}},$$

(2a)

$$I = \sigma T^{4} ,$$

(2b)

with wavelength $\lambda$, Planck’s constant h, speed of light c, Boltzmann constant k _B, and Stefan–Boltzmann constant $\sigma .$ Thus, the higher T _A, the more intense the emission. If work is not extracted from the absorber medium, it reaches equilibrium with the sun and the surroundings. For a maximum concentration of the sun light, i.e., a situation where all the emitted flux from the absorber medium is redirected toward the sun, the medium would reach the temperature of the sun (surface) T _S and emit the solar spectrum back to the sun. However, in this situation work cannot be extracted. This requires a lower temperature of the absorber medium, and the fraction of extractable energy can be written as the difference between absorbed and emitted energy flux according to the Stefan–Boltzmann law:

$$\eta_{\text{rad}} = \frac{{\sigma \left( {T_{\text{S}}^{4} - T_{\text{A}}^{4} } \right)}}{{\sigma T_{\text{S}}^{4} }} = 1 - \frac{{T_{\text{A}}^{4} }}{{T_{\text{S}}^{4} }}.$$

(3)

This equation demands for a low T _A, which is in contrast to the requirements for obtaining a high Carnot efficiency. Consequently, the highest solar to work conversion efficiency is reached at a T _A such that $T_{0}\, < \,T_{\text{A}} \,<\,T_{\text{S}}$. The optimum T _A is 2478 K with a corresponding $\eta_{ \hbox{max} } = \eta_{\text{rad}} \eta_{\text{C}}$ of 85 %. For a conversion of, e.g., mechanical work of the heat engine into electricity, we can imagine a generator with almost 100 % power-conversion efficiency. Consequently, the highest power-conversion efficiency of a terrestrial solar cell might approach but cannot exceed 85 %.

In case of a nonconcentrated system, the thermodynamic limit is reduced as the medium will emit into a larger solid angle and thus increase the entropy. The solid angle of the sun seen from the earth is 6.8 × 10⁻⁵ (resulting from the long distance between the earth and the sun and the sun’s diameter), which is a factor 46,200 smaller than the solid angle of emission into a hemisphere [1, 2]. This factor should be considered in the emitted intensity in Eq. 3 and reduces $\eta_{ \hbox{max} }$ to only a few percent. Thus, efficient solar–thermal energy conversion using a heat engine requires concentration of sun light. This is not necessarily the case for photovoltaic systems using bandedge absorbers as we will see in the following sections. They can operate efficiently at low concentrations of the sun light and at ambient temperatures. Nevertheless, every solar-based energy converter, independent of its implementation, is tied to the 85 % limit.

1.2 Semiconductor-Based Photovoltaics

A semiconductor is characterized by a bandgap E _g making its optical properties differ from those of a black body. In an ideal semiconductor, only photons with energy higher than E _g are absorbed by promoting an electron from the valence to the conduction band and leaving a hole in the valence band (Fig. 2). Thus, the onset of the absorption spectrum is at the energy of the bandgap. Incident photons with lower energies are transmitted. The peak energy of the emitted photon spectrum is located close to the bandgap energy as well and almost independent of temperature in contrast to the black body radiation. Mainly the spectral width increases with temperature due to an increased thermal energy of electrons and holes in their respective bands. To be distinguished from thermal (i.e., black body) radiation, this kind of narrow radiation is called luminescence.

Fig. 2

Optoelectronic basics of semiconductors a Sketch of band diagram with bandgap $E_{\text{g}}$, conduction and valence band edge $E_{\text{C}}$ and $E_{\text{V}}$, respectively, and quasi-Fermi levels for electrons $E_{\text{F}}^{n}$ and holes $E_{\text{F}}^{p}$, which are split due to illumination. b Schematics of ideal absorptance and emission

To harvest as many solar photons as possible, the bandgap of the semiconductor should be minimized (Fig. 1), which allows for the highest electron-hole generation rate, and we can write the below equation for the photocurrent:

$$J_{{{\text{ph}},{\text{ideal}}}} = e \mathop \int \limits_{{E_{\text{g}} }}^{\infty } \varPhi_{{{\text{AM}}1,5{\text{g}}}} \left( E \right)dE$$

(4)

with $\varPhi_{{{\text{AM}}1,5{\text{g}}}}$ describing the spectral photon flux, i.e., the number of photons per area, time, and energy (or wavelength) interval. To generate a net electrical current in an external circuit, electrons and holes have to be extracted at two different contacts. For the moment, we only introduce the concept of so-called charge-selective contacts: an electron-selective contact would only allow electrons to pass and reject holes, and vice versa for a hole-selective contact. We will come to implementations of such selective contacts later.

However, a large photocurrent is not sufficient: the relevant parameter is the electrical power of the extracted charge flux. This power is the product of flux and the (average) free energy of one extracted electron-hole pair, which is proportional to the voltage measurable externally. Thus, we need an expression for this energy, which is mainly electrochemical in nature and depends on the electrical potential energy as well as the concentration of charges (chemical energy). We expect the potential energy of the electron-hole pair to scale with the bandgap, consistent with the fact that the peak of luminescence radiation is located at the bandgap energy. This holds even for absorbed photons with much higher energy, which create an electron (hole) that is located much higher (deeper) in the conduction (valence) band compared to the band edges. However, this electron relaxes quickly (ps) to the band edge due to energy transfer to phonons (Fig. 2a), analogously to Kasha’s rule in the photochemistry of electronically excited molecules. The excess energy is lost as heat and the average energy of the electron remains at $E_{\text{C}} + \frac{3}{2}k_{\text{B}} T$. This fast process assisted by phonons is called thermalization. Consequently, the higher the bandgap, the lower are the overall thermalization losses, when absorbing a broad spectrum such as AM1.5 g. Therefore, when tuning the bandgap to reach high efficiency, there is a tradeoff between harvesting as many photons as possible and maximizing the energy of extracted charges.

To determine the optimum bandgap, we need to find an expression for the electrical energy of an extracted electron-hole pair. Coming back to the previous section on thermodynamics, we know that electrical energy is an entropy-free part of the energy. Thus, we need to know the entropy of an electron (hole) in the electron (hole) gas in the conduction (valence) band. After longer derivation [1] it is found that the entropy-free part of the energy, i.e., the electrochemical potential $\eta_{\text{e}}$ of an electron in the conduction band, is:

$$\eta_{\text{e}} = E_{\text{C}} - k_{\text{B}} T\,\ln \frac{{N_{\text{C}} }}{n},$$

(5)

where n is the electron density (number of electrons per unit volume) and N _C is the effective density of states, i.e., a parameter that represents the amount of available states close to the band edge. The second term increases linearly with temperature as expected for an energy flux related to entropy (${\Delta }Q = T{\Delta }S$). ${\Delta }Q$ becomes 0 either at T → 0 K or when the density of states is completely filled ($n = N_{\text{C}}$), i.e., there is only one microscopic state possible without any permutations (although in this case, this equation does not hold any more as it contains approximations valid for low n only) [2]. Analogously, we obtain for (positively charged) holes p:

$$- \eta_{\text{h}} = E_{\text{V}} + k_{\text{B}} T\,\ln \frac{{N_{\text{V}} }}{p}.$$

(6)

In steady state, one electron is extracted together with one hole to conserve charge and obey a continuity equation for electrons. The total energy is then the sum of the electrochemical potentials

$$eV = \eta_{\text{e}} + \eta_{\text{h}} = E_{\text{C}} - E_{\text{V}} - k_{\text{B}} T\,\ln \frac{{N_{\text{C}} N_{\text{V}} }}{np} = E_{\text{g}} - k_{\text{B}} T\,\ln \frac{{N_{\text{C}} N_{\text{V}} }}{np}.$$

(7)

Here, we introduce the externally measurable voltage V, which is expressed as the energy of one extractable electron-hole pair in the electron gas divided by the elementary charge e. This equation shows that the idea of a large bandgap required to maximize the energy of an extracted electron-hole pair is correct.

From Eqs. 5 and 6, we recognize that the electrochemical potentials are the quasi-Fermi energies:

$$n = N_{\text{C}} \exp \left( { - \frac{{E_{\text{C}} - E_{\text{F}}^{n} }}{{k_{\text{B}} T}} } \right) \to E_{\text{F}}^{n} = \eta_{\text{e}}$$

(8a)

$$p = N_{\text{V}} \exp \left( { - \frac{{E_{\text{F}}^{p} - E_{\text{V}} }}{{k_{\text{B}} T}} } \right) \to E_{\text{F}}^{p} = - \eta_{\text{h}}$$

(8b)

As electrons are fermions, their concentration is determined by Fermi Dirac statistics. The Fermi level E _F is defined as the energy at which the probability of finding the fermion is ½. In a metal E _F describes the highest energy for electrons in the bands at $T \to 0\,{\text{K}}$.

To obtain the electron density in general, the statistics is multiplied by the energetic distribution of available states (so-called density of states, DOS). For electrons in the conduction band (CB) we get

$$n = \mathop \int \limits_{\text{CB}} f\left( E \right) \times {\text{DOS}}_{\text{CB}} \left( E \right)dE = \mathop \int \limits_{\text{CB}} \frac{1}{{1 + {\text{exp}}\frac{{E - E_{\text{F}} }}{{k_{\text{B}} T}}}} \times {\text{DOS}}_{\text{CB}} \left( E \right)dE \approx \mathop \int \limits_{\text{CB}} \exp \left( { - \frac{{E - E_{\text{F}} }}{{k_{\text{B}} T}}} \right) \times {\text{DOS}}_{\text{CB}} \left( E \right)dE$$

(9)

This approach results in the previous equations considering a certain DOS and the Boltzmann approximation. Under equilibrium, i.e., no voltage applied or generated by light, we obtain

$$E_{\text{F}}^{n} = E_{\text{F}}^{p}$$

(10)

and

$$np = n_{\text{i}}^{2} = N_{\text{C}} N_{\text{V}} \exp \left( { - \frac{{E_{\text{g}} }}{{k_{\text{B}} T}}} \right),$$

(11)

with the intrinsic (i.e., thermally generated) charge carrier density n _i. The definition of quasi-Fermi levels under illumination or applied voltage (nonequilibrium) is possible as thermalization within a band is commonly much faster than a spontaneous transition of an electron from conduction to valence band (i.e., electron-hole recombination). Thus, electrons and holes are in equilibrium within their respective band but not amongst each other. This situation can be called quasiequilibrium, described by two separate Fermi distributions for electrons and holes.

In Eq. 7 the electrochemical energy of the extracted electron-hole pair is not solely determined by the bandgap, but also contains a term that depends on electron and hole concentrations (chemical energy). In a solar cell under illumination, the concentration of charges is built-up by the absorption of light (the so-called charge carrier generation G), but it cannot reach infinitely large numbers as the competing process of recombination sets in. The rate of band-to-band recombination of an electron with a hole can be expressed as

$$R = \beta np.$$

(12)

Intuitively we can say that the recombination rate R scales with the probability of an electron finding a hole. In steady state at open circuit under a given light intensity $\propto G$, all charges recombine, i.e., R = G and therefore the open-circuit voltage is (Eq. 12 in Eq. 7):

$$eV_{\text{oc}} = E_{\text{g}} - k_{\text{B}} T\,\ln \frac{{N_{\text{C}} N_{\text{V}} \beta }}{G}.$$

(13)

This equation describes $V_{\text{oc}}$ as a function of temperature, light intensity (which is $\propto G$), and bandgap. An expression for the radiative recombination constant $\beta$ will be derived subsequently.

1.3 Radiative Limit of the Open-Circuit Voltage

We want to start from the equilibrium situation and extend it to the operation points of the solar cell. Here, equilibrium means that the semiconductor is in the dark without any applied bias. In this case, “dark” describes a surrounding at a temperature $T_{0} = 300\,{\text{K}}$. This temperature gives rise to a background radiation with a black body spectrum at T ₀ according to Planck’s law

$$\varPhi_{\text{BB}} \left( E \right) = \frac{1}{{4\pi^{2} \hbar^{3} c^{2} }}\frac{{E^{2} }}{{\exp \left( {\frac{E}{{k_{\text{B}} T_{0} }}} \right) - 1}}.$$

(14)

This equation is identical to Eq. 2a, but expressed in terms of a photon flux as a function of energy (instead of intensity as a function of wavelength). The semiconductor absorbs part of this spectrum and has to emit the same part because it is in equilibrium with the surrounding

$${\Phi }_{{{\text{abs}},0}} = \mathop \int \nolimits a\left( E \right) \varPhi_{\text{BB}} \left( E \right)dE = {\Phi }_{{{\text{em}},0}} .$$

(15)

Here, a(E) is the absorptance (0…1), where we could use a Heaviside step function switching from 0 to 1 at E _g for an ideal semiconductor as we did in Eq. 4.

This equality is true not only for the absolute flux, but also for each energy (or wavelength) according to the principle of “detailed balance”:

$$\Phi_{{{\text{em}},0}} \left( E \right) = a\left( E \right) \varPhi_{\text{BB}} \left( E \right).$$

(16)

It means that the emission spectrum $\varPhi_{\text{em}} \left( E \right)$ can be predicted when absorption coefficient and temperature are known [3].

To experimentally determine the emission spectrum, we consider the nonequilibrium case: Additional electrons and holes need to be generated in the semiconductor, either by light or by applying a voltage and injecting charges. The former option results in photoluminescence, and the latter in electroluminescence. We verify the applicability of the detailed balance theory for a CH₃NH₃PbI₃ perovskite solar cell. Rather than using the absorption spectrum, we use the photocurrent action spectrum (external quantum efficiency EQE_PV), which ideally is proportional to the absorption spectrum and easier to obtain experimentally. Based on an extended version of the reciprocity between absorption, emission, and electronic charge transport, a(E) can be interchanged with EQE_PV(E) [4]. A measured EQE_PV onset is shown in Fig. 3a. By multiplying the EQE_PV(E) with $\varPhi_{\text{BB}} \left( E \right)$, shown as dashed line for $T_{0} = 320\,{\text{K}}$, we calculate an emission spectrum (Fig. 3b), which fits the measured electroluminescence spectrum very well. This indicates that the assumption of an unmodified spectrum even under illumination or applied voltage is justified. Coming back to Eq. 12, we can write in equilibrium:

Fig. 3

Bandedge of CH₃NH₃PbI₃ a Spectral photocurrent onset in terms of the external quantum efficiency of a solar cell (solid line) and black body photon flux at 320 K (dashed). b Measured and calculated electroluminescence emission. Data originally published in Ref. [6]

$$G_{0} = \mathop \int \nolimits \alpha \left( E \right) \varPhi_{\text{BB}} \left( E \right)dE = \beta n_{0} p_{0} = \beta n_{\text{i}}^{2} .$$

(17)

Thus, the radiative recombination constant $\beta$ can be calculated knowing the intrinsic charge carrier density n _i (Eq. 11) and the absorption coefficient $\alpha \left( E \right)$. Assuming parabolic band minima (DOS) with an effective electron (hole) mass $m^{*} = 0.228\,\left({0.293} \right)m_{0}$ (m ₀ is the electron rest mass) [5], we can derive N _C,V using Eq. 9 ($\to N_{{{\text{C}},{\text{V}}}} = 2\left( {\frac{{2\pi m^{*} k_{\text{B}} T}}{{h^{2} }} } \right)^{{\frac{3}{2}}}$, derivation not shown), and in turn n _i (Eq. 11). With Eq. 17 we obtain the value of $\beta \approx 10^{ - 11} {\text{cm}}^{3} {\text{s}}^{ - 1}$.

Under bias (or illumination), n and p become larger than n _i and the common Fermi level (Eq. 10) splits into separate (quasi) Fermi levels for electrons and holes according to Eq. 8a, 8b, leading to $np = n_{\text{i}}^{2} \exp \frac{{E_{\text{F}}^{n} - E_{\text{F}}^{p} }}{{k_{\text{B}} T}}$.

Thus, recognizing that eV is the difference between the quasi-Fermi levels at the contacts, the enhanced recombination gives rise to an emitted photon flux ${\Phi }_{\text{em}}$ exponentially increasing with the applied voltage:

$${\Phi }_{\text{em}} = {\Phi }_{{{\text{em}},0}} \exp \frac{eV}{kT}.$$

(18)

If we think of the device operated as a light-emitting diode, ${\Phi }_{\text{em}}$ is achieved at a particular electric current injected into the device:

$$J_{\text{rad}} = e\left( {{\Phi }_{{{\text{em}},0}} \exp \frac{eV}{{k_{\text{B}} T}} - {\Phi }_{{{\text{em}},0}} } \right) = e{\Phi }_{{{\text{em}},0}} \left( {\exp \frac{eV}{{k_{\text{B}} T}} - 1} \right).$$

(19)

We recognize that this equation is identical to the Shockley diode equation when the saturation current is set as $J_{0} = e{\Phi }_{{{\text{em}},0}} .$

If the device is operated as a solar cell under illumination, the photocurrent J _ph should be added to Eq. 19:

$$J_{\text{ideal}} = e{\Phi }_{{{\text{em}},0}} \left( {\exp \frac{eV}{{k_{\text{B}} T}} - 1} \right) - J_{\text{ph}} .$$

(20)

Now, we can write an equation for the open-circuit voltage, setting J _ideal to 0:

$$V_{{{\text{oc}},{\text{rad}}}} = \frac{{k_{\text{B}} T}}{e}\,\ln \left( {\frac{{J_{\text{ph}} }}{{e{\Phi }_{{{\text{em}},0}} }} + 1} \right).$$

(21)

This is the ideal $V_{\text{oc}}$ of a solar cell if only radiative recombination is present. For CH₃NH₃PbI₃ perovskite solar cells, we calculate a value of 1.33 V [6, 7]. Variations in bandgap as a function of film morphology can slightly (by ${\approx}\pm 0.02 {\text{V}}$) alter this value. As we will discuss later, (partially) substituting atoms/ions on the perovskite lattice is a strategy to modify the bandgap.

1.4 Shockley–Queisser Limit

Since current is not extracted at $V_{\text{oc}}$, and the power is the product of current and voltage, the power at open circuit is 0. Instead, it is largest at the so-called maximum power point (MPP), which is found by searching for the maximum of $J \times V$. For CH₃NH₃PbI₃ perovskite with a bandgap of ≈1.55–1.6 eV, the maximum efficiency would be 31–30 %. The ideal current-voltage curve according to Eq. 20 is visualized in Fig. 4a, including the extractable power. The fill factor is defined as ${\text{FF}} = \frac{{{\text{J}}_{\text{MMP}} V_{\text{MPP}} }}{{J_{\text{sc}} V_{\text{oc}} }} = \frac{{\eta I_{\text{light}} }}{{J_{\text{sc}} V_{\text{oc}} }}$ with the short-circuit current density $J_{\text{sc}} = J_{\text{ph}}$ and the incident light intensity I _light. Figure 4b shows the maximum J _SC as a function of the bandgap calculated using Eq. 4 with varied E _g. For an E _gof 1.6 eV, we expect a theoretical photocurrent of 25 mA cm⁻², which does not include any optical or electrical losses that reduce this value by 1–3 mA cm⁻² for the best perovskite solar cells [8, 9]. The resulting power-conversion efficiency (Shockley–Queisser, SQ limit [10]) as a function of the bandgap is shown in Fig. 4c, where an E _g between 1.1 and 1.4 eV allows an efficiency of ≈33 %. The decrease of $\eta$ for larger E _g is due to transmission losses and thus low photon harvesting (low J _SC, cf. Fig. 4b). For reduced E _g, the voltage is low and most of the energy is lost due to thermalization.

Fig. 4

Shockley–Queisser limit a Theoretical J-V curve and power output for an ideal semiconductor with a bandgap of 1.6 eV. b Harvested photon flux and short-circuit current density as a function of the bandgap. c Maximum power-conversion efficiency as a function of bandgap and record values [9, 11] experimentally achieved so far

The losses discussed so far are unavoidable. However, in implementations of solar cells, additional losses occur, which will be discussed in the following sections of this chapter. The circles in Fig. 4c show how closely solar cells based on different semiconductor material with different E _g approach their theoretical radiative limit. Solar cells based on the direct semiconductor GaAs reach an efficiency >28 % coming very close to their theoretical limit. Photovoltaic devices based on silicon, the most common commercial material, show additional losses, but have an efficiency still exceeding 25 %. Perovskite and CIGS solar cells exhibit similar performance, where today’s best perovskite solar cell is already closer to the Shockley–Queisser limit than the best CIGS cell.

2 The Bandgap

2.1 Absorption Onset and Subgap Urbach Tail

The SQ limit was calculated for a step function in absorption. However, as shown in Fig. 3a, the absorption is not a Heaviside function but contains a contribution below E _g. This is detrimental, because these subgap states hardly contribute to photocurrent. However, they do participate in recombination entering Eq. 15 enhanced by a multiplication with $\varPhi_{\text{BB}} (E)$. Thus, a steep absorption onset is preferred. It should follow at least $\exp \left( {\frac{E}{{k_{\text{B}} T_{0} }}} \right)$ to compensate for the exponential increase of $\varPhi_{\text{BB}} (E)$ with decreasing energy. Figure 5 shows the absorption onset of CH₃NH₃PbI₃ perovskite in comparison with other solar cell materials [12]. Due to its direct bandgap the absorption coefficient above bandgap is similar to the one of GaAs and much steeper than in crystalline or amorphous Si. Additionally, the distribution of subgap states is narrow.

Fig. 5

Absorption coefficient of different solar cell materials. The black lines denote an exponential fit to the Urbach tail. Reprinted with permission from [12]. Copyright 2014 American Chemical Society

The absorption coefficient below the bandgap shows a so-called Urbach tail. Its origin is from various sources of disorder that generate exponentially decaying densities of states below the conduction and above the valence band. Those could be impurities, ionic positional disorder, vibrational fluctuations of atoms, etc. Their common feature is that they locally perturb the periodic crystal potential giving rise to fluctuations of the potential that electrons close to the band edges see. The steep rise and resulting low Urbach energy indicates that the disorder in the CH₃NH₃PbI₃ perovskite is low due to its high (nano-) crystallinity. In general, the absorption onset of nano-crystalline perovskite depends on the morphology of the film, mainly on the crystallite size. It shifts toward larger energies with smaller crystallites, possibly due to strain in the material [13].

2.2 Tuning of the Bandgap and Tandem Devices

Until now we have discussed the CH₃NH₃PbI₃ perovskite with a bandgap of 1.55 to 1.6 eV. Similar to other compound semiconductors, a partial replacement of components with others of the same ionic charge and fitting into the lattice (Goldschmidt tolerance factor) facilitates tuning of the bandgap of the perovskite. Here, the organic cation has only minor influence, as it is not directly contributing to the valence and conduction band but changing lattice parameters. Replacing methylammonium (CH₃NH₃) by the bigger formamidinium (HC(NH₂)₂) results in a decrease of E _g (by ≈50 meV), slightly shifting the maximum $\eta$ toward its optimum [14–17]. Using the smaller Cs instead increases the bandgap by ≈100 meV [14, 17]. This trend follows the tight-binding model of more closely spaced atoms having more interaction, and thus stronger splitting of their states into broader bands which in turn results in a more narrow bandgap. In case of the perovskite, size effects controlling the metal–halide–metal bond angles [18] are accompanied with further effects such as modified hydrogen bonding and spin orbit coupling when interchanging the monovalent (organic) cation [19]. Note that due to the antibonding nature of the valence states, the bandgap shows an anomalous decrease with temperature [20].

More influential are replacements of the halide itself using, e.g., Br [17, 21]. Figure 6 shows photocurrent response (EQE_PV) and PL spectra for perovskites with different I:Br molar ratios in the precursor solutions. (These compounds contain a mixture for the organic cation as well because formamidinium–iodide is added to the precursor solution). The data shows that E _g increases monotonically upon increasing substitution of I with Br (similar to highly efficient Cs HC(NH₂)₂ mixed systems [22]). At the same time, the Urbach tail is maintained at low energies of approx. 15 meV. Calculating the maximum $V_{\text{oc}}$ using Eq. 21, we get an increase of $V_{{{\text{oc}},{\text{rad}}}}$ from 1.33 to 1.47 V, when the Br content reaches 40 %. However, the maximum $\eta$ would decrease from 30 to 27 %. Thus, for a single junction solar cell, there is no reason for increasing E _g except that a higher voltage is preferred to a higher current due to practical reasons such as reduction of resistive losses.

Fig. 6

Mixed MA-FA perovskites with mixed halides a The photocurrent onset as a function of the molar Br content, not showing any considerable changes in the Urbach energy. b Photoluminescence spectra confirming a shift of the bandgap. The signal at 1.5–1.6 eV might be due to some iodine rich phases in the samples with larger bromine content

However, tuning the bandgap becomes important when making multijunction solar cells, where several solar cells are stacked on top of each other with the solar cell with the largest E _g employed as the front cell. This approach is a strategy to overcome the Shockley–Queisser limit, as thermalization losses are reduced by collecting the high energy photons at higher voltage. The ideal combination for a two-junction (tandem) solar cell would yield a maximum $\eta$ of > 42 % with E _g1 = 1.0 eV and E _g2 = 1.9 eV [23]. Technologically interesting is a combination of perovskite with silicon [24–26]. For a monolithic integration the photocurrent should be matched, meaning that Si, which delivers a maximum current of almost 42 mA cm⁻², should be combined with a perovskite solar cell providing 21 mA cm⁻². Ideally, this current could be achieved using a mixed halide perovskite with a bandgap of 1.75 eV. Then, half of the incident photon flux is converted into charges that are collected at a larger voltage compared to the voltage obtainable with silicon. This boosts the theoretical efficiency from 33 to >40 %. Using realistic values including optical and electrical losses for Si and perovskite solar cell, efficiencies of around 30 % are expected to be feasible in a tandem configuration [27, 28].

3 Nonradiative Recombination

3.1 Quantum Efficiency of Electroluminescence

So far, the solar cell was treated as ideal, which means that only theoretically unavoidable loss processes have been considered. Regarding recombination, this is radiative recombination only. In that case, the solar cell operates at its radiative limit. However, in reality nonradiative losses are present in addition, reducing charge carrier densities in Eq. 7 and consequently the $V_{\text{oc}}$.

Coming back to the derivation of $V_{\text{oc}}$ based on detailed balance (Eq. 19), we consider a quantum efficiency of emission for the injection current:

$$J_{\text{rad}} \left( V \right) = {\text{EQE}}_{\text{EL}} J_{\text{inj}} \left( V \right).$$

(22)

EQE_EL is the external quantum efficiency of electroluminescence. We can expand the equation for $V_{\text{oc}}$ (Eq. 21) to

$$V_{\text{oc}} = \frac{{k_{\text{B}} T}}{e}\,{ \ln }\left( {{\text{EQE}}_{\text{EL}} \frac{{J_{\text{ph}} }}{{e{\Phi }_{{{\text{em}},0}} }} + 1} \right) \approx V_{{{\text{oc}}, {\text{rad}}}} - \frac{{k_{\text{B}} T}}{e}\,\ln {\text{EQE}}_{\text{EL}} .$$

(23)

Thus, a reduction of the EQE_EL by a factor of 10 is equivalent to a decrease of $V_{\text{oc}}$ by $\frac{{k_{\text{B}} T}}{e}\, \ln 10$ = 60 mV at room temperature. The EQE_EL should be determined at $V_{\text{oc}}$. Practically, it is measured in the dark by driving the solar cell as an LED, detecting the emitted photon flux and dividing it by the injected electron flux.

Knowing EQE_EL and EQE_PV, which is required to calculate $J_{\text{ph}}$ and $e{\Phi }_{{{\text{em}},0}}$ (Eqs. 4 and 15):

$$J_{\text{ph}} = e \mathop \int \nolimits {\text{EQE}}_{\text{PV}} \left( E \right)\varPhi_{{{\text{AM}}1,5{\text{g}}}} \left( E \right)dE; e{\Phi }_{{{\text{em}},0}} = e \mathop \int \nolimits {\text{EQE}}_{\text{PV}} \left( E \right)\varPhi_{\text{BB}} \left( E \right)dE,$$

(24)

we can predict the real $V_{\text{oc}}$ of a given solar cell [4, 29]. Differences in $V_{\text{oc}}$ for different experimental implementations of a solar cell with the same semiconductor are mainly caused by variations of the nonradiative loss that decrease EQE_EL. These changes explain the differences observed in $V_{\text{oc}}$ for perovskite solar cells, which are reported between 0.7 and 1.2 V, whereas $V_{{{\text{oc}},{\text{rad}}}}$ ≈ 1.33 V is a material property. Table 1 shows examples for perovskite fabricated by different methods and employed in different device architectures [6, 9]. Avoiding a hole transport layer decreases EQE_EL due to surface recombination. On the other hand, employing hole transport layer and tuning the morphology increases EQE_EL and thus the measured $V_{\text{oc}}$.

Table 1 Measured $J_{\text{sc}}$ and $V_{\text{oc}}$ compared to calculated values based on measured EQE_PV and EQE_EL

Figure 7 compares the record value of EQE_EL for perovskite solar cells [9] (marked with a green cross) with other technologies [31]. The direct semiconductor GaAs yields the highest values due to a high internal luminescence yield. In silicon, EQE_EL is limited by Auger and surface recombination [32, 33]. The trend line (dashed) visualizes that in experiment the factor determining the efficiency is EQE_EL defining how closely $V_{\text{oc}}$ approaches $V_{{{\text{oc}},{\text{rad}}}}$.

Fig. 7

Power-conversion efficiency for different solar cell technologies normalized to the Shockley-Queisser limits as a function of EQE_EL (here denoted as Q _LED), equivalent to the nonradiative $V_{\text{oc}}$ loss of ${\Delta}V_{{{\text{oc}}, {\text{nr}}}} = - 26\,{\text{mV}} \times \, \ln {\text{EQE}}_{\text{EL}}$ (top axis). The dotted lines define the theoretical limits of various EQE_EL at the denoted bandgaps. Different CH₃NH₃PbI₃ perovskite fabrication technologies are shown in green. The green cross marks the most efficient perovskite solar cell. Different organic solar cells are shown in blue. The dashed line is a guide to the eye representing the approximate experimental trend. Reprinted and modified with permission from [31]. Copyright 2015 by the American Physical Society

3.2 Identifying Recombination Mechanisms

The exact source of nonradiative recombination in perovskites has not yet been identified. In addition to recombination at the contacts, defect recombination in the material itself plays a role. Defects might originate from impurities, grain boundaries, or additional crystalline/amorphous phases, e.g., resulting from unreacted precursors. Furthermore, intrinsic defects due to dislocations, interstitials, vacancies, or antisites might give rise to electronically active states. Theoretical studies on whether these defects are active as recombination centers yielded rather contradictory results [34–37] .

Shockley-Read-Hall (SRH) theory [38] describes recombination through defects states:

$$R = \frac{{np - n_{\text{i}}^{2} }}{{\tau_{p} \left( {n + n_{1} } \right) + \tau_{n} \left( {p + p_{1} } \right)}}$$

(25)

with $n_{1} = N_{\text{C}} \exp \left( { - \frac{{E_{\text{C}} - E_{\text{T}} }}{{k_{\text{B}} T}}} \right)$, $p_{1} = N_{\text{V}} \exp \left( { - \frac{{E_{\text{T}} - E_{\text{V}} }}{{k_{\text{B}} T}}} \right)$, and lifetimes for electrons and holes $\tau_{n,p}$. E _T is the energy of the trap. For multiple traps or for a distribution in energy, the right-hand side of Eq. 25 is a sum over all E _T. Investigating Eq. 25, we find that midgap traps ($E_{\text{C}} - E_{\text{T}} \approx E_{\text{T}} - E_{\text{V}}$) are most active as recombination centers. This is in line with the intuitive explanation that midgap traps allow for equally efficient electron and hole capture. Assuming that most of the present charge carriers are photogenerated, thus $n \gg n_{1}$ and $n \gg n_{\text{i}}$, and considering one dominating SRH lifetime $\tau$, we can approximate R:

$$R \propto \frac{n}{\tau }.$$

(26)

Following the same procedure as for direct recombination in Eqs. 11 and 12, we express $V_{\text{oc}}$ as a function of light intensity in case of SRH recombination

$$eV_{\text{oc}} = E_{\text{g}} - 2 k_{\text{B}} T\,\ln \frac{{\sqrt {N_{\text{C}} N_{\text{V}} } \tau }}{G}.$$

(27)

In general, we can write for the slope of $V_{\text{oc}}$ versus log₁₀ $G/(1{\text{m}}^{ - 3} {\text{s}}^{ - 1} )$: $n_{\text{ID}} k_{\text{B}} T\,\ln 10$, where $n_{\text{ID}} = 1$ for radiative recombination and $n_{\text{ID}} = 2$ for SRH recombination. The parameter n _ID is called the ideality factor and lies between 1 and 2, if both recombination processes are present. It is introduced for the dark J-V curve (cf. Eq. 9) as well:

$$J_{\text{inj}} = J_{0} \left( {\exp \frac{eV}{{n_{\text{ID}} k_{\text{B}} T}} - 1} \right).$$

(28)

Figure 8a shows the J-V curve of a perovskite solar cell in the dark (blue). It is governed by shunt and series resistances for voltages <0.8 and >1.2 V, respectively. The exponential increase of J between the resistance limited regions is described by Eq. 28, where $n_{\text{ID}} = 2$. Consequently, the dominant recombination mechanism is of SRH type. The red curve shows the emitted photon flux with an exponential rise in the same voltage range as the dark curve, but with $n_{{{\text{ID}},{\text{rad}}}} = 1$ proving that radiative recombination results from the recombination of an electron in the conduction band with a hole in the valence band. The greater increase of emission current with voltage, compared to overall current, gives rise to an EQE_EL(green) that increases with voltage. When plotted as a function of the injection current (Fig. 8b), the EQE_EL increases linearly as expected when comparing Eq. 27 with 23 for SRH recombination being dominant.

Fig. 8

Highly luminescent perovskite solar cells: a Injection current (blue), emitted photon flux (red) and EQE_EL as a function of the applied voltage. b EQE_EL as a function of the injection current. The black line describes a linear relation between EQE_EL and the injection current. From [9]. Reprinted with permission from AAAS

Having both recombination processes (Eq. 12 and 26) running in parallel, we can write a rate equation assuming that spatial variations in recombination rates and charge carrier densities are negligible, and the majority of the charge carriers are photogenerated:

$$\frac{dn}{dt} = - \beta n^{2} (t) - \frac{1}{\tau }n(t).$$

(29)

This equation can be solved analytically for n as a function of time t, which can be monitored by the emission as a function of t doing a photoluminescence (PL) decay measurement. A varied light intensity of the laser pulse can be used to tune the initial charge carrier density $n(0)$, which allows a more reliable determination of $\tau$ and $\beta$. Figure 9a shows PL transients including a fit according to the solution of Eq. 29 ($n\left( t \right) = \frac{1}{\tau \beta }\frac{1}{{e^{{\frac{t}{\tau }}} \left( {\frac{1}{\tau \beta n\left( 0 \right)} + 1} \right) - 1}}$), with $\beta \approx 10^{ - 11} {\text{cm}}^{3} {\text{s}}^{ - 1}$ and $\tau =$ 350 ns. The value for $\beta$ coincides with what is expected from the radiative limit, confirming that direct electron-hole recombination is radiative. More sophisticated models including the dynamics of trapping and exciton formation can be applied to fit the PL decay, or other techniques such as transient THz spectroscopy, yielding $\beta \approx 10^{ - 11} {-}10^{ - 10} {\text{cm}}^{3} {\text{s}}^{ - 1}$ [39–41].

Fig. 9

Optoelectronic characterization of mixed perovskite devices a PL decay for illumination from the surface or through the substrate and with high and with 40 times lower light intensity. b $V_{\text{oc}}$ as a function of illumination intensity under red and blue light for two different mixed perovskite devices. From [9]. Reprinted with permission from AAAS

The SRH lifetime $\tau \propto \frac{1}{{N_{\text{T}} \sigma_{\text{T}} }}$ depends on the density $(N_{\text{T}} )$ and capture cross section $(\sigma_{\text{T}} )$ of electronically active recombination centers. For a charged defect, the latter is inversely proportional to the dielectric constant, which characterizes the ability of the material to screen charge [42]. Thus, the high dielectric constant of CH₃NH₃PbI₃ perovskite reported to up to 70 [43] combined with a low N _T for states close to midgap allows for a $\tau$ of several 100 s of ns. Here, the role of crystal size, grain boundaries, and passivation agents is not well understood yet [9].

To understand the effect of $\tau$ on the solar cell’s performance, we calculate $V_{\text{oc}}$ as a function of light intensity for varied $\tau$ (Fig. 10). In the radiative limit, $V_{\text{oc}}$ increases with light intensity showing a slope of 60 mV/decade (dashed), whereas it shows 120 mV/decade in the case of SRH recombination, which dominates at low light intensity. Investigating $V_{\text{oc}}$ at 1 sun (Fig. 10b), we find that a nonradiative lifetime $\tau$ of 10 µs would be sufficient to practically reach the radiative limit. The experimental data (symbols) for two perovskite solar cells based on mesoporous TiO₂ are consistent with a $\tau$ reaching more than 100 ns.

Fig. 10

Calculated influence of the SRH lifetime $\tau$ on $V_{\text{oc}}$ for a 300 nm thick CH₃NH₃PbI₃ perovskite with a bandgap of 1.6 eV and a radiative recombination constant of 10⁻¹¹ cm³s⁻¹. a $V_{\text{oc}}$ as a function of illumination intensity and b $V_{\text{oc}}$ as a function of $\tau$ at 1 sun. When SRH recombination is dominant, $V_{\text{oc}}$ decreases with 120 mV/decade with illumination intensity and with $1/\tau$. Symbols in (a) show experimental data for two-step CH₃NH₃PbI₃ solar cells from Ref. [44] (circles) and for one-step mixed perovskite [45] obtained by a measurement with pulsed light (crosses). Most of the changes in $V_{\text{oc}}$ are due to an increased lifetime, as the bandgap changes only by a value <50 meV. A $\tau > 100\,{\text{ns}}$ is consistent with the PL decay in Fig. 9

So far, we have assumed that charge carrier densities and recombination rates in the perovskite film do not depend on the position within the film. However, the direction of incident illumination and vertical inhomogeneities in the morphology of the perovskite layer and different properties at the contacts give rise to lateral inhomogeneities in recombination parameters and charge carrier densities. A treatment of inhomogeneities requires numerical simulations. However, first conclusions can be drawn from qualitative investigations exploiting different inhomogeneities introduced in G(x) by monochromatic illumination. Blue light penetrates the film less than red light, giving rise to a steeper G(x). A modified $V_{\text{oc}}$ indicates different charge carrier lifetimes close to the surface and in the bulk. The experimental data in Fig. 9b indicates that recombination is stronger at the TiO₂ contact for the device with the lower $V_{\text{oc}}$ [9].

Nonradiative recombination influences the fill factor (FF) as well. This can be seen in Eq. 20, where the ideality factor needs to be introduced in front of $k_{\text{B}} T$. In case of negligible series or shunt resistance losses, the FF only depends on recombination. It can be expressed as a function of recombination rate, and thus of $V_{\text{oc}}$. In the radiative limit at 1 sun, it exceeds 90 % (cf. Fig. 4a) and decreases toward 83 % at 1.2 V and 80 % at 1.0 V (Fig. 11). Thus, a perovskite solar cell with a bandgap of 1.6 eV with optimized contacts and a $J_{\text{sc}}$ of 24 mA cm⁻², but operating at only 1.0 V cannot deliver a FF greater than 80 %. However, a (slightly) larger FF is possible if recombination does not result from the perovskite itself but from surfaces. This seems to be the case for C₆₀-based electron transport layers [46] in inverted perovskite solar cells, where a FF as high as 85 % is reported for a $V_{\text{oc}}$ of 1.03 V [47]. This value requires an $n_{\text{ID}} < 2$, when derived from $V_{\text{oc}}$ as a function of light intensity. This would be an additional indication for surface recombination limiting $V_{\text{oc}}$ [48]. Note that an excessively high FF might result from transient phenomena during measurement, giving rise to hysteresis in the J-V curve [49, 50]. On the other hand, surface recombination can decrease the FF as well. In real solar cells, a low FF is caused by resistive effects due to inhibited charge transport in the absorber or parasitic shunt and series resistances.

Fig. 11

Calculated fill factor for CH₃NH₃PbI₃ perovskite with a bandgap of 1.6 eV and a radiative recombination constant of 10⁻¹¹ cm³s⁻¹. a The fill factor decreases with lower $V_{\text{oc}}$ due to its definition (cf. Fig. 4a). Additionally, a change from the radiation limit to SRH recombination gives rise for the rapid decrease. b Fill factor as a function of SRH lifetime $\tau$

The proximity of the FF to the recombination limit (in case of negligible shunting) can be experimentally verified applying the Suns-$V_{\text{oc}}$ method [44, 51, 52]. Employing this method, a pseudo-J-V curve is constructed from light intensity-dependent $V_{\text{oc}}$ data. Series resistance effects are excluded because when measuring $V_{\text{oc}}$, current does not flow. Assuming that the light intensity is proportional to current, the pseudo-J-V curve for a particular intensity $I_{0}$ reads:

$$J\left( {V_{\text{oc}} \left( I \right)} \right) = J_{sc} \left( {I_{0} } \right)\left( {1 - \frac{I}{{I_{0} }}} \right).$$

(30)

Figure 12 shows such curves (dashed) for a CH₃NH₃PbI₃ perovskite solar cell at different illumination intensities $I_{0}$ compared to the measured J-V curves (solid lines) [44]. Both curves coincide at 0.25 suns indicating that charge transport losses in the perovskite are negligible. At higher light intensities and the consequently higher photocurrents, the series resistance limits the FF, mainly due to the resistance of the transparent conductive oxide.

Fig. 12

Current-voltage curves (solid lines) at different illumination intensities. Pseudo-J-V curves (dashed) obtained using the Suns-$V_{\text{oc}}$ method. Reprinted with permission from [44]. Copyright 2015 American Chemical Society

3.3 Role of the Charge Transport Layers

In the theoretical discussions, we assumed that contacts were perfect. Through experiments, we have already seen that they can change EQE_EL and thus be a source of nonradiative recombination, e.g., if a hole transport layer (HTL) is not used (cf. Table 1).

Thus, an essential role of the charge transport layers is to provide a selective contact. The electron transport layer (ETL) should only allow electrons to pass and block holes, and the HTL vice versa. This property can be put into practice by a large energy barrier for electron (hole) transfer due to offsets in conduction (valence) band. Additionally, recombination at the interface between the absorber and the charge transport layer should be avoided. This type of recombination, where charge recombines at the surface of the absorber, is called surface recombination and it is characterized by a surface recombination velocity. For metals, the surface recombination velocity is usually very high for both electrons and holes. Hence, metals are not a good choice for selective contacts.

In general, interfaces between two materials (heterojunctions) might fulfill the role of separation of charge or excited states, as in the case of organic donor–acceptor heterojunctions [2]. However, from the previous discussions, there are no indications that interfaces are essential for the photovoltaic working principle of perovskite solar cells. Thus, charges are generated and dissociated in the perovskite itself, creating a splitting of the quasi-Fermi levels, which gives rise to the photovoltage. The charge transport layers just serve as contacts. However, this raises the question on the operational principle of perovskite solar cells, where no intentional pn junction is introduced, in contrast to conventional solar cells. From the discussions above, we know that a pn junction itself is not a requirement for a solar cell. Instead, selective contacts are sufficient. The solar cell architecture comprises of a metal–insulator (semiconductor)-metal structure. Charges can be driven toward their respective interface simply by diffusion. Usually, a difference in work function of the two metals (or transparent conductive oxides) is beneficial, though not required, as it introduces a built-in potential similar to the different types of doping in a pn junction. To date it is not clear whether the assumption of a constant electric field in the perovskite is justified. Space charge regions could result from unintentional doping, the existence of mobile ionic species, grain boundaries, or other inhomogeneities in the film.

A lot of research focuses on interchanging ETL or HTL to increase performance and in particular $V_{\text{oc}}$. Regarding the ETL, high $V_{\text{oc}} > 1 {\text{V}}$ has been obtained with several materials such as TiO₂ with TiO₂ or Al₂O₃ scaffold [6], PCBM [53], ZnO [54], or Ti(Nb)O_x [55]. Record values of 1.2 V have been achieved with TiO₂ [9] and SnO₂ [56], where the high $V_{\text{oc}}$ is mainly attributed to an improved morphology of the perovskite. One approach commonly taken up is the tuning of the HOMO level of the organic HTL. It is anticipated that a lower lying HOMO increases $V_{\text{oc}}$. The experimental effort on testing HTLs to increase $V_{\text{oc}}$ has only been partly successful [57–59] but clear trends are mostly missing [60–64]. Instead, improvements of $V_{\text{oc}}$ were obtained by modifying the perovskite [9]. This is expected from the discussion in this chapter. Modifying the HTL or ETL to increase $V_{\text{oc}}$ is only important if recombination at the HTL(Au)/perovskite interface reduces $V_{\text{oc}}$ (possibly independent of HOMO position), as it is the case for, e.g., too thin HTL [44] or PEDOT:PSS as HTL [65]. A deeper lying HOMO accompanied by a higher work function of a doped HTL could increase the built-in potential and thus reduce surface recombination as less charges are reaching the wrong electrode. However, if the surface is well passivated, i.e., recombination at the electrode is not an issue, changes in the HOMO of the HTL will not affect $V_{\text{oc}}$. In addition, the formation of interface dipoles can change band alignment compared to the case predicted by assuming a constant vacuum level at the interface.

Theoretically, $V_{\text{oc}}$ can exceed the difference between the work functions of the contacts or the difference between the conduction and the valence band of electron and hole transport layer, respectively. This is thought to be counter intuitive, but with restrictions this is as normal as the fact that it is possible to apply a voltage larger than those offsets. The results for Br-based perovskite on TiO₂ and HTL spiro-OMeTAD with $V_{\text{oc}}$ approaching 1.5 V indicate that this is possible in experiment as well [66]. Thus, it is more important to tune the HTL and ETL to reduce recombination centers on their surfaces, which might result from additives, and use the charge transport layers to passivate the perovskite surface instead of solely focusing on the energetics.