A phenomenological figure of merit for photovoltaic materials

An FOM for PV absorbers should be a function of the bulk material properties that have a significant influence on the realizable PV efficiency. I propose eight such properties: band gap $E_\mathrm{g}$, non-radiative lifetime τ, carrier mobility µ, doping density n, static dielectric constant ε, DOS effective mass m, and two quantities ($\overline{\alpha}$ and σ) related to the spectral average and spectral dispersion of the absorption coefficient spectrum $\alpha(\lambda)$, respectively (λ is the wavelength of light). These eight bulk properties are the free parameters of the FOM (figure 2). I define the set of the eight properties as $\mathbb{P} = \{\overline{\alpha},\sigma,\tau,\mu,n,\epsilon,m,E_\mathrm{g} \}$. Hence, $\Gamma_{\mathrm{PV}} = f(\mathbb{P})$. Defining these quantities is not trivial, so in the next paragraphs I will clarify the definitions used throughout this article.

$\overline{\alpha}$ is defined as the weighted integral average of $\alpha(\lambda)$ in the spectral region from $\lambda_\mathrm{l} = 300$ nm to $\lambda_\mathrm{g}~ = ~hc/E_\mathrm{g}$, where h is Planck's constant and c is the speed of light. $\lambda_\mathrm{g}$ is the wavelength corresponding to the band gap energy. $\lambda_\mathrm{l}$ is a cut-off wavelength, chosen to disregard the very short wavelengths with negligible photon flux in the AM 1.5G spectrum, and for which $\alpha(\lambda)$ data is often unavailable. The weights are given by the spectral density $\phi(\lambda)$ of the AM 1.5G photon flux. Thus:

$$\begin{equation} \overline{\alpha} = \cfrac{\int_{\lambda_\mathrm{l}}^{\lambda_\mathrm{g}} \alpha\left(\lambda\right)\phi\left(\lambda\right) \,\mathrm{d}\lambda}{\int_{\lambda_\mathrm{l}}^{\lambda_\mathrm{g}} \phi\left(\lambda\right) \,\mathrm{d}\lambda}. \end{equation} \tag{ 1 }$$

The dispersion parameter σ is the weighted standard deviation of the logarithm of $\alpha(\lambda)$ in the same spectral region.

$$\begin{equation} \sigma = \sqrt{\cfrac{\int_{\lambda_\mathrm{l}}^{\lambda_\mathrm{g}} \phi\left(\lambda\right) \left[\text{log}\left(\alpha\left(\lambda\right)\right)-{\text{log}\left(\overline{\alpha}\right)}\right]^2 \,\mathrm{d}\lambda}{\int_{\lambda_\mathrm{l}}^{\lambda_\mathrm{g}} \phi\left(\lambda\right) \,\mathrm{d}\lambda}}. \end{equation} \tag{ 2 }$$

For a lighter notation, I will refer to the average absorption coefficient $\overline{\alpha}$ simply as α in the rest of the article.

n is defined as the equilibrium concentration of electrons in the conduction band of the PV absorber in the dark without an applied voltage. When n is used as a parameter in a FOM, it is simply intended as the concentration of majority carriers regardless of their type (electrons or holes). Assuming homogeneous concentrations of fully ionized single acceptors and single donors ($N_\mathrm{A}$ and $N_\mathrm{D}$, respectively):

$$\begin{equation} n = | N_\mathrm{D} - N_\mathrm{A} |. \end{equation} \tag{ 3 }$$

This definition can easily be extended to the case of multiple types of fractionally ionized dopants that can donate/accept more than one electron.

The carrier lifetime τ is intended as the lifetime associated with non-radiative, Shockley–Read–Hall (SRH) recombination in the bulk [18, 19]. It is defined as:

$$\begin{equation} \tau = \left(v_\mathrm{t}N_\mathrm{t}\Sigma\right)^{-1} \end{equation} \tag{ 4 }$$

Here, $v_\mathrm{t}$ is the thermal velocity of carriers in the material, $N_\mathrm{t}$ is the concentration of the defects responsible for SRH recombination, and Σ is their carrier capture cross section. In general, electrons and holes may have different lifetimes $\tau_\mathrm{n}$ and $\tau_\mathrm{p}$, due to differences in their thermal velocities and (especially) capture cross sections by the dominant defect. To limit the number of free parameters in the FOM expression, I assume that $\tau_\mathrm{n} = \tau_\mathrm{p} \equiv \tau$. For a real PV absorber with significant differences between $\tau_\mathrm{n}$ and $\tau_\mathrm{p}$, it may still be possible to reduce the two separate lifetimes into an effective lifetime that can be used in the FOM, depending on the carrier injection level under illumination. This topic is discussed in the supplementary material. SRH recombination is assumed to be the only active recombination mechanism besides radiative recombination (Auger recombination is neglected).

The carrier mobility µ is defined from the conductivity effective mass M and the mean intra-band carrier scattering, or relaxation, time $\tau_\mathrm{s}$ as:

$$\begin{equation} \mu = \cfrac{q \tau_\mathrm{s}}{M} \end{equation} \tag{ 5 }$$

where q is the elementary charge. Drift- and diffusion mobilities are assumed to be equal to each other. Analogous to the case of the SRH lifetime, different mobilities can generally be expected for electrons ($\mu_\mathrm{n}$) and holes ($\mu_\mathrm{p}$) due to different values of M and $\tau_\mathrm{s}$ in the valence- and conduction bands. Again, I assume $\mu_\mathrm{n} = \mu_\mathrm{p} \equiv \mu$ to limit the number of free parameters in the FOM expression, but it may be possible to reduce carrier-specific mobilities into a single effective µ. A qualitative discussion is given in the supplementary material.

m is the DOS effective mass [20–23]. The DOS effective mass for electrons in the conduction band ($m_\mathrm{e}$) can be determined by considering the absorber's calculated electronic DOS, and fitting it with the following function near the energy of the conduction band minimum ($E_\mathrm{c}$) [24]

$$\begin{equation} \mathrm{DOS}\left(E\right) = \cfrac{1}{2 \pi^2} \left(\cfrac{2 m_\mathrm{e}}{\hbar^2}\right)^{3/2} \left(E - E_\mathrm{c}\right)^{1/2} \end{equation} \tag{ 6 }$$

where $\hbar = h/2\pi$ (h is Planck's constant). An analogous expression is used to determine the DOS hole effective mass $m_\mathrm{h}$ near the energy of the valence band maximum $E_\mathrm{v}$. Similar to the cases of τ and µ, electrons and holes generally have separate values for their DOS effective masses ($m_\mathrm{e}$ and $m_\mathrm{h}$, respectively). To avoid an exceedingly large number of free parameters in the FOM, I assume $m_\mathrm{e} = m_\mathrm{h} \equiv m$ throughout this paper. However, it is straightforward to reduce $m_\mathrm{e}$ and $m_\mathrm{h}$ to a single effective value $m = \sqrt{m_\mathrm{e} m_\mathrm{h}}$ without a significant loss of generality, as explained in the supplementary material. Note that m and M are not equivalent, since they are derived differently and have different physical meanings [21, 22].

The static dielectric constant ε is intended as the value of the dielectric constant in the limit of zero frequency (indicatively, below ${10^{9}}\,\mathrm{Hz}$). Because both ions and electrons can respond to these low-frequency electric fields, ε is a measure of the polarizability of the PV absorber by both ionic and electronic displacements.

The band gap $E_\mathrm{g}$ is intended as the electronic band gap in the bulk of the absorber:

$$\begin{equation} E_\mathrm{g} = E_\mathrm{c} - E_\mathrm{v} \end{equation} \tag{ 7 }$$

where $E_\mathrm{c}$ and $E_\mathrm{v}$ are the edges of extended bulk electronic states making up the conduction- and valence band, respectively. The SQ limit in this article is always calculated from this definition of $E_\mathrm{g}$. The electronic gap defined in equation (7) is always assumed to be equal to the optical band gap, i.e. the photon energy of absorption onset. This equality is enforced by only considering expressions for $\alpha(E)$ in which $\alpha(E) = 0$ when $E \lt E_\mathrm{g}$, and $\alpha(E) \gt 0$ when $E \gt E_\mathrm{g}$ (E is the photon energy). Sub band gap absorption due to, e.g. excitonic absorption, defect absorption, or potential fluctuations is not considered. In the presence of sub band gap absorption, $E_\mathrm{g}$ in its present definition may be difficult to measure and may not represent the most PV-relevant band gap value [25, 26]. In such cases, the PV-relevant band gap definition proposed in [26] may be more appropriate.

Throughout this paper, I assume all properties to be isotropic for simplicity. The implications of direction-dependent properties [27] are briefly discussed in the supplementary material.

As we will see, the definition of the $\Gamma_{\mathrm{PV}}$ FOM includes polynomial functions with fractional exponents, as well as transcendental functions (exponential and logarithmic). To avoid complications and inconsistencies with units, it is necessary to convert the eight properties in the $\mathbb{P}$ set to a unitless form [28] when they are used in an FOM. Throughout this article, the unitless versions of the eight properties are $\alpha/(\mathrm{cm}^{-1})$, $\tau/(\mathrm{s})$, $\mu/({\mathrm{cm}^{2}\mathrm{V}^{-1}\mathrm{s}^{-1}})$, $n/(\mathrm{cm}^{-3})$, $m/(m_0)$, and $E_\mathrm{g}/(\mathrm{eV})$. m₀ is the rest mass of the electron. σ and ε are already unitless.

Once defined, an FOM for PV absorbers should enable a quantitative estimate ($\eta_{\Gamma}$) of the maximum efficiency $\eta_\mathrm{max}$ achievable by a candidate PV absorber with any (realistic) specific set of properties $\hat{\Bbb{P}} = \{\hat{\alpha},\hat{\sigma},\hat{\tau},\hat{\mu},\hat{n},\hat{\epsilon},\hat{m},\hat{E_\mathrm{g}}\}$. Before attempting the definition of an FOM, it is therefore necessary to collect a dataset of $\eta_\mathrm{max}$ versus $\mathbb{P}$ to be used as a benchmark when evaluating the quality of a proposed FOM. This is step A in the overall workflow of this paper, shown in figure 2.

A reliable source for $\eta_\mathrm{max}$ for an absorber with a property set $\hat{\mathbb{P}}$ is a finite-element-based, one-dimensional drift-diffusion simulation of an ideal solar cell incorporating such an absorber. In this work, I have conducted these simulations with the SCAPS program [29]. In the ideal solar cell there are no efficiency losses due to contact layers, interfaces, or a non-optimal device structure. Instead, all efficiency losses with respect to the SQ limit $\eta_\mathrm{sq}$ can be attributed to non-optimal values in some of the bulk properties in the $\mathbb{P}$ set. In practice, the simulated solar cell structure simply consists of the PV absorber and two contacts that are fully carrier-selective. Relying on one-dimensional simulations automatically implies another ideal feature, i.e. that the absorber's properties are homogeneous in the plane of the solar cell, which is often not the case in real materials [30]. All simulation parameters used to model this ideal device structure are shown in table S1, supplementary material.

The requirement for an optimal device structure implies that the drift-diffusion simulation used to estimate $\eta_\mathrm{max}$ must be carried out at the optimal thickness $d_\mathrm{opt}$, i.e. the absorber thickness at which the highest efficiency is reached [10]. The computational procedure to obtain $d_\mathrm{opt}$ is visualized in figure 1(a). Briefly, a batch of simulations is executed at different absorber thicknesses d for each unique set of material properties $\hat{\mathbb{P}}$ considered in this work. The thickness at which η is maximized is taken as the optimal thickness $d_\mathrm{opt}$ for that particular $\hat{\mathbb{P}}$ property set, and the corresponding value of η is defined as $\eta_\mathrm{sim}$ for that particular $\hat{\mathbb{P}}$ property set (figure 1(a)). The consistency of the drift-diffusion-simulated values of $J_\mathrm{sc}$, $V_\mathrm{oc}$, FF, and $\eta_\mathrm{sim}$ with their maximum values allowed by the SQ limit is shown in figure S1, supplementary material.

I assume that $\eta_\mathrm{sim}$ is a good estimator of $\eta_\mathrm{max}$ ($\eta_\mathrm{sim} \approx \eta_\mathrm{max}$) due to the ideality of the simulated solar cell structure. Thus, a dataset of $\eta_\mathrm{sim}$ values for various combinations of properties can be used as a training set to assess the appropriateness of various possible expressions of $\Gamma_{\mathrm{PV}}$, in a similar spirit to machine learning approaches. When collecting the training dataset, it is important to sample different combinations of the eight properties within physical ranges. The property ranges sampled in this work are shown in table 1. In total, I employ a dataset of 2573 $\eta_\mathrm{sim}$ values to derive the $\Gamma_{\mathrm{PV}}$ FOM. The device simulation workflow used to collect these values is summarized in step A of figure 2. All eight properties are changed independently of each other. For example, µ is not automatically changed when changing m, even though µ is likely to depend on m, everything else being equal. Likewise, α and σ are not automatically modified when changing $E_\mathrm{g}$, even though the actual $\alpha(E)$ spectrum is adjusted to ensure that the energy at which $\alpha(E)$ becomes greater than zero always corresponds to $E_\mathrm{g}$ (see also previous section). Additional assumptions, approximation, definitions, and neglected physical phenomena are given in the supplementary material.

Table 1. The eight bulk properties of the absorber ($\mathbb{P}$ set) entering the definition of the $\Gamma_{\mathrm{PV}}$ FOM. The 'Sampled range' column gives the minimum and the maximum value of each property in the $\eta_\mathrm{sim}$ versus $\mathbb{P}$ dataset. Since the $\Gamma_{\mathrm{PV}}$ FOM has been developed from data within these ranges, the efficiency prediction ($\eta_{\Gamma}$) for a PV absorber with properties falling outside of the sampled range may be grossly incorrect. The 'Default value' column gives the value used for each property when that property is kept fixed. In some figures in this article, data is shown for 'high' and 'low' values of a certain property. The 'high' and 'low' values are also given in the same column. Unitless versions of these properties (i.e. the property divided by its unit) are used in all figures of merit in this article.

Property	Symbol	Unit	Sampled range	Default value
Band gap	$E_\mathrm{g}$	eV	(0.7, 2.0)	1.2
Average of absorption coefficient	α	$\mathrm{cm}^{-1}$	$(5 \times 10^{3},5 \times 10^{5})$	$2.7 \times 10^{5}$ ('high': $2.7 \times 10^{5}$; 'low': $1.9 \times 10^{4}$)
Dispersion of absorption coefficient	σ		(0.2, 1.8)	0.33 (with 'high α': 0.33; with 'low α': 1.42)
SRH recombination lifetime	τ	$\mathrm{s}$	$(1 \times 10^{-15},1 \times 10^{3})$	Never fixed
Carrier mobility	µ	$\mathrm{cm}^2\,\mathrm{V}^{-1}\mathrm{s}^{-1}$	$(1 \times 10^{-2},1 \times 10^{9})$	$1 \times 10^{8}$ ('high': $1 \times 10^{8}$; 'low': 1)
Doping density	n	$\mathrm{cm^{-3}}$	$(1 \times 10^{10}, 1 \times 10^{18})$	$1 \times 10^{10}$ ('high': $1 \times 10^{18}$; 'low': $1 \times 10^{10}$)
Static dielectric constant	ε		(1, 100)	10 ('high': 100; 'low': 1)
DOS effective mass	m	m0	(0.12, 2.5)	0.54 ('high': 2.5; 'low': 0.12)

Throughout this paper, PV efficiencies will be expressed either as absolute power conversion efficiencies ($\eta_{\Gamma}$ and $\eta_\mathrm{sim}$), or as fraction of the maximum efficiency $\eta_\mathrm{sq}$ allowed by the SQ limit [5, 31] for an absorber of band gap $E_\mathrm{g}$. Such SQ-normalized efficiencies are labeled $\eta_\mathrm{sim,sq} \equiv \eta_\mathrm{sim}/\eta_\mathrm{sq}$ and $\eta_{\Gamma,\mathrm{sq}} \equiv \eta_{\Gamma}/\eta_\mathrm{sq}$.

I start from a baseline case aimed at reproducing the ατ FOM proposed by Kaienburg et al [10]. In the baseline case τ and α are varied but µ is kept infinite. σ, n, ε, m, and $E_\mathrm{g}$ are all fixed. n is set to ${10^{10}}\,\mathrm{cm}^{-3}$ to reproduce a nearly intrinsic, fully depleted absorber. τ is varied over 15 orders of magnitude (${10^{-15}}\,\mathrm{s} \lt \alpha \lt {1}\,\mathrm{s}$). α is varied over three orders of magnitude (${10^{3}}\,\mathrm{cm}^{-1} \lt \alpha \lt {10^{6}}\,\mathrm{cm}^{-1}$) by tuning the b prefactor in the generic expression $\alpha(E) = b(E-E_\mathrm{g})^{1/2}$ inspired by direct-gap semiconductors.

A plot of $\eta_\mathrm{sim}$ versus ατ (figure 3(a)) reveals that the ατ FOM is reasonably (but not perfectly) appropriate under these conditions, since the $\eta_\mathrm{sim}$ values line up rather close to a unique logarithmic function of ατ. Outside the logarithmic region, $\eta_\mathrm{sim}$ approaches 0 and $\eta_\mathrm{SQ}$ in the limits of low and high ατ product, respectively. The small spread between the curves produced by different values of α (figure 3(a)) is likely due to the fact that α has been varied by a factor 1000 in figure 3(a), compared to the factor 1.5 employed by Kaienburg et al [10].

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Simply setting n to a higher fixed value to simulate heavy doping (${10^{18}}\,\mathrm{cm}^{-3}$) radically changes this picture. As shown in figure 3(a), the different curves at constant α no longer line up. In particular, changing α from ${10^{5}}\,\mathrm{cm}^{-1}$ to ${10^{6}}\,\mathrm{cm}^{-1}$ hardly has any effect on the efficiency at constant τ, as demonstrated by the one-decade horizontal shift between these two curves in most of the ατ range of interest. Hence, the ατ FOM appears to be approximately valid under the conditions simulated by Kaienburg et al but it may not be generalizable to other conditions.

To develop a new figure of merit step by step, I temporarily return to the initial assumption of $n = {10^{10}}\,\mathrm{cm}^{-3}$ and let the dispersion of the absorption coefficient σ vary on top of α and τ. The different σ values are obtained by using the eight hypothetical $\alpha(E)$ spectra shown in figure 3(b) in the drift-diffusion simulations. These spectra have $4.8 \times 10^{3}\,\mathrm{cm}^{-1} \lt \alpha \lt 4.7 \times 10^5\,\mathrm{cm}^{-1}$ and $0.29 \lt \sigma \lt 1.42$. See the supplementary Material for more details. Letting σ vary is a way to account for the diversity of absorption processes in the semiconductors that may be considered for the role of PV absorbers. Some features that influence the value of σ are direct versus indirect band gaps, different joint densities of states, different nature of optical transitions away from the band edges etc.

When σ is allowed to vary, the ατ FOM is no longer adequate (figure 4(a)) even in the limit of low doping density. Specifically, materials with a high σ like c-Si can only reach a substantially lower efficiency at constant ατ than materials with a low σ like chalcogenide perovskites. The problem with a high σ is that the carrier generation profile inside the absorber will be the sum of many exponentially-decaying profiles (corresponding to the different photon energies) with very different extinction depths. Thus, even if α remains constant, a thicker film is necessary to absorb the same fraction of the AM1.5G spectrum, resulting in a larger available volume for recombination.

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To incorporate the σ dependence, one can define a $\Gamma_\mathrm{A}$ FOM (A for 'absorption') as

$$\begin{equation} \Gamma_\mathrm{A}\left(\alpha,\tau,\sigma\right) = A_1 A_2 \end{equation} \tag{ 8 }$$

where

$$\begin{equation} \begin{aligned} A_1 & = \tau \alpha^{\sigma^{-0.27}} \\ A_2 & = 1+\left(\cfrac{a_1\sigma^{10}}{\alpha\tau }\right)^{0.4} \end{aligned} \end{equation} \tag{ 9 }$$

and $a_1 = 1 \times 10^{-7}$. α and σ are in unitless form (see Methods section). The A₁ factor is similar to the ατ FOM, with the addition of the σ^−0.27 exponent to α, which extends its validity to different absorption coefficient dispersions σ. The A₂ factor is an additional correction for the low ατ region, and it only 'turns on' when $\alpha\tau \lt {10^{-7}}\,\mathrm{}\sigma^{10}$. If the same $\eta_\mathrm{sim}$ data points in figure 4(a) are plotted against the $\Gamma_A$ FOM (figure 4(b)), the points line up close to a single function. This demonstrates the appropriateness of $\Gamma_\mathrm{A}$ when α, σ, and τ are allowed to vary and the other bulk properties are fixed to their default values in table 1.

As we have already seen in figure 3, changing n can have a large impact on $\eta_\mathrm{sim}$. Figure 5(a) shows the effect of varying n between ${10^{10}}\,\mathrm{cm}^{-3}$ and ${10^{18}}\,\mathrm{cm}^{-3}$ for two combinations of α and σ, while µ is still kept infinite. Clearly, modifications to the $\Gamma_\mathrm{A}$ FOM are necessary to account for doping effects.

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To understand some of the trends, it is useful to consider the splitting of the quasi-Fermi levels (quasi-Fermi level splitting ($\mathrm{QFLS}$)) in an n-type absorber at open circuit under illumination. Under some assumptions [32]:

$$\begin{equation} \mathrm{QFLS} = E_\mathrm{g} + k_\mathrm{B}T \text{ln}{\left[\frac{h^6}{4 \left(2\pi k_\mathrm{B}T\right)^3} \frac{\left(n + \Delta n\right) \left(n_\mathrm{i}^2/n +\Delta p\right)}{\left(m_\mathrm{e} m_\mathrm{h}\right)^{\frac{3}{2}}}\right]} \end{equation} \tag{ 10 }$$

where $k_\mathrm{B}$ is Boltzmann's constant, T is the temperature, $n_\mathrm{i} = \sqrt{np}$ is the absorber's intrinsic carrier concentration and Δn and Δp are the excess carrier densities with respect to their equilibrium values in the dark (n and $n_\mathrm{i}^2/n$ respectively). In many realistic scenarios, the QFLS equals the $V_\mathrm{oc}$ limit of a PV absorber [8, 33].

From equation (10), $V_\mathrm{oc}$ (and therefore also $\eta_\mathrm{sim}$) are expected to rise logarithmically with increasing doping density when $n \gg \Delta n$ ('low injection' conditions) and be independent of doping density when $n \ll \Delta n$ ('high injection' conditions). This general trend is visible in figure 5(a). The transition between low- and high-injection conditions occurs at lower values of n as α is decreased, because Δn and Δp are lower due to the increased $d_\mathrm{opt}$. In the low-α case, the transition shifts to higher n as τ is increased, because Δn and Δp become higher due to the lower SRH recombination rate. The slope change in the low-α curves at high n when $\alpha \tau \simeq {10^{-6}}\,\mathrm{}$ is due to $J_\mathrm{sc}$ losses from incomplete light absorption (due to a low $d_\mathrm{opt}$) below this threshold.

It seems as if we could keep increasing the efficiency by increasing n to even higher values than $1 \times 10^{18}\,\mathrm{cm}^{-3}$. However, several side effects can occur at very high doping density, such as a decrease of τ due to a higher density of recombination centers, a decrease of µ due to ionized impurity scattering, and an increasing weight of Auger recombination which is otherwise neglected in this study. In addition, the drift-diffusion simulation is no longer accurate because it relies on the Boltzmann approximation of the Fermi–Dirac distribution. This approximation breaks down at very high doping densities (see details in the supplementary material). Therefore, I do not consider doping densities above $1 \times 10^{18}\,\mathrm{cm}^{-3}$.

To account for the effects of different doping densities, the $\Gamma_\mathrm{A}$ FOM can be modified into the $\Gamma_\mathrm{AD}$ FOM (D for 'doping') as follows:

$$\begin{equation} \Gamma_\mathrm{AD}\left(\alpha,\tau,\sigma,n\right) = \cfrac{A_1 A_2 D_1}{D_2 D_3} \end{equation} \tag{ 11 }$$

where

$$\begin{equation} \begin{aligned} D_1 & = \left(1+d_1\cfrac{n}{\alpha^{\;2}}\right)^{0.25\text{log}{\left(\alpha\right)}-0.3} \\ D_2 & = 1+{\left(d_2 \tau \alpha\right)}^{0.02\text{log}{\left(n\right)}} \\ D_3 & = 1+\left(\cfrac{d_3}{\alpha\tau }\right)^{0.6} \end{aligned} \end{equation} \tag{ 12 }$$

and $d_1 = 6 \times 10^{-7}$, d₂ = 10, $d_3 = 1 \times 10^{-8}$. α, σ and n are in unitless form (see Methods section).

According to the D₁ factor, inclusion of doping effects in the FOM is important when $n \gt \alpha^2/d_1$. For example, when $\alpha = 1 \times 10^5\,\mathrm{cm}^{-1}$ low-injection conditions are realized when $n \gt {10^{16}}\,\mathrm{cm}^{-3}$. When $\alpha = 1 \times 10^4\,\mathrm{cm}^{-1}$, low-injection conditions are realized when $n \gt {10^{14}}\,\mathrm{cm}^{-3}$. The D₂ factor is a correction for the high ατ region, as it only turns on when $\alpha \tau \gt 0.1$. The D₃ factor is a additional correction for the low ατ region, as it only turns on when $\alpha \tau \lt 1 \times 10^{-8}$. The predictive quality of the $\Gamma_\mathrm{AD}$ FOM is shown in figure 5(b). The $\eta_\mathrm{sim}$ values line up near a single function when plotted against $\Gamma_\mathrm{AD}$ (figure 5(b)). This confirms the appropriateness of $\Gamma_\mathrm{AD}$ as a descriptor of $\eta_\mathrm{sim}$ when α, σ, τ, and n are allowed to vary and the other bulk properties are fixed to their default values in table 1.

Allowing for finite values of the carrier mobility µ is a particularly important step, because the current generation of efficiency-prediction models based on the principle of detailed balance [5–9] still rely on the assumption of infinite mobilities. Thus, a method to estimate the maximum efficiency of candidate PV absorbers with finite mobilities would be highly desirable. Efficiency losses related to µ are mainly manifested in $J_\mathrm{sc}$ and FF, because finite mobilities may lead to less-than-ideal carrier collection efficiency at voltages up to open circuit (figure 1).

In the study by Kaienburg et al [10] the relevant FOM changed from ατ to $\alpha \tau \mu$ when µ fell below the $0.001\,\mathrm{cm}^2\,\,\mathrm{Vs}^{-1}$ threshold. This threshold is, however, not a universal feature of a generic PV absorber, as demonstrated by varying µ between $0.1\,\mathrm{cm}^{2}\,\,\mathrm{Vs}^{-1}$ and $100\,\mathrm{cm}^{2}\,\,\mathrm{Vs}^{-1}$ in figure 6(a). According to the figure, a $0.001\,\mathrm{cm}^2\,\,\mathrm{Vs}^{-1}$ threshold might be realistic only if n is low and the ατ product is so high that $\eta_\mathrm{sim}$ is close to the SQ limit for all but the lowest mobilities. In all other cases, the mobility threshold for a µ-dependent efficiency is higher, often by many orders of magnitude. Low mobilities are generally more tolerable if ατ is high and if n is low. When $n = 1 \times 10^{18}\,\mathrm{cm}^{-3}$, even a very high mobility of $10\,000\,\mathrm{cm}^2\,\,\mathrm{Vs}^{-1}$ is still below the threshold at low ατ products, and it has a detrimental effect on the efficiency (figure 6(a)). On the other hand, a mobility of $100\,\mathrm{cm}^2\,\,\mathrm{Vs}^{-1}$ is close to the threshold in the same ατ range when $n = 1 \times 10^{10}\,\mathrm{cm}^{-3}$. These features can be understood on a qualitative level. As α increases, the optimal thickness of the absorber decreases, so carrier collection is facilitated. Hence, the requirements on µ are less stringent. As τ increases, the diffusion length ($\propto \sqrt{\mu\tau}$) and the drift length ($\propto \mu\tau$) remain unchanged if µ decreases by the same amount [8].

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To account for the effects of different mobilities, the $\Gamma_\mathrm{AD}$ FOM can be modified to the $\Gamma_\mathrm{ADT}$ FOM (T for 'transport') as follows:

$$\begin{equation} \Gamma_\mathrm{ADT}\left(\alpha,\tau,\sigma,n,\mu\right) = \cfrac{A_1 A_2 D_1}{D_2 D_3 \left(1 + T_1 T_2\right)} \end{equation} \tag{ 13 }$$

where

$$\begin{equation} \begin{aligned} T_1 & = \cfrac{10}{\mu} \left( 1+t_1\text{exp}\left(-0.1\alpha\right) n^{1-0.74 \, \text{exp}{\left(-\alpha /t_2\right)}} \right)^{0.6} \\ T_2 & = \left(\cfrac{1+ t_3 n}{t_4 \tau 10^{\frac{0.5}{\sigma}}}\right)^{0.6} \end{aligned} \end{equation} \tag{ 14 }$$

and $t_1 = 2 \times 10^{4}$, $t_2 = 1.5 \times 10^{5}$, $t_3 = 3 \times 10^{-17}$, $t_4 = 7.8 \times 10^{5}$. All properties are in unitless form (see Methods section). Since µ only appears at the denominator of the T₁ factor with an exponent of one, the $\Gamma_\mathrm{ADT}$ FOM is directly proportional to µ when µ is well below a threshold defined by $T_1 T_2 \gt 1$. This mobility threshold corresponds to

$$\begin{equation} \mu \lt 10 \left(\cfrac{\left(1+t_3 n\right) \left(1+t_1\text{exp}\left(-0.1\alpha\right) n^{1-0.74 \, \text{exp}{\left(-\alpha /t_2\right)}}\right)}{t_4 \tau 10^{\frac{0.5}{\sigma}}} \right)^{0.6} \end{equation} \tag{ 15 }$$

where all properties are again in unitless form. This analysis confirms that a linear dependence of the FOM on mobility (as in the simple $\alpha \tau \mu$ FOM by Kaienburg et al) is justified below a certain mobility threshold. Rather than being a constant universal value, the mobility threshold is a function of the other bulk material properties (equation (15)) and can vary by many orders of magnitude. The predictive quality of the $\Gamma_\mathrm{ADT}$ FOM is shown in figure 6(b).

When a low µ and a low τ are combined with a low α (low efficiency region in figure 6(c)) a low doping density becomes more favorable than a high doping density. Although a higher n is still preferable to achieve a higher $V_\mathrm{oc}$ (equation (10)), this effect is overcompensated by a $J_\mathrm{sc}$ improvement at lower n. The reason is that, when n is sufficiently low, an electric field is present throughout the absorber at short circuit due to full carrier depletion. The electric field adds a drift component to carrier transport, which improves the carrier collection efficiency and results in a higher $J_\mathrm{sc}$ with respect to the high-n case where transport mainly occurs by diffusion [34].

The µ-dependent simulations at low α (figure 6(c)) also reveal that $\eta_\mathrm{sim}$ may saturate to a value lower than the SQ limit, even as τ keeps increasing (figure 6(c)). The reason why the SQ limit is never reached is that the diffusion length is ultimately limited by the radiative lifetime. Therefore, once the SRH lifetime has surpassed the radiative lifetime, the diffusion length can no longer be increased and $\eta_\mathrm{sim}$ remains constant even if τ is increased further [12]. If the diffusion length is smaller than the absorption depth due to an unfavorable combination of a low µ and a low α, the SQ limit is never reached. To take this effect into account in the FOM, the $\Gamma_\mathrm{ADT}$ FOM can be modified to the $\Gamma_\mathrm{ADTS}$ FOM (S for 'saturation') as follows:

$$\begin{equation} \Gamma_\mathrm{ADTS}\left(\alpha,\tau,\sigma,n,\mu\right) = \cfrac{A_1 A_2 D_1}{D_2 D_3 \left(1 + T_1 T_2\right)\left(1+S_1 S_2\right)} \end{equation} \tag{ 16 }$$

where

$$\begin{equation} \begin{aligned} S_1 & = \cfrac{0.1 \alpha^{0.75}\tau}{\mu} \\ S_2 & = 1+ \left(\cfrac{s_1}{\alpha}\right)^{20}\mu^{0.5}\text{log}{\left(n\right)} \end{aligned} \end{equation} \tag{ 17 }$$

and $s_1 = 4.8 \times 10^{3}$. All properties are in unitless form (see Methods section). The predictive quality of the $\Gamma_\mathrm{ADTS}$ FOM is shown in figure 6(d).

Varying the static dielectric constant ε of the PV absorber within two orders of magnitude ($1 \unicode{x2A7D} \epsilon \unicode{x2A7D} 100$) has a weaker effect on $\eta_\mathrm{sim}$ compared to the other bulk properties examined so far (figure 7(a)). Neglecting possible indirect effects of ε on other properties (discussed later below), the main role of ε in a PV absorber is to modulate the width of depletion regions adjacent to the contacts, since ε enters Poisson's equation [35, 36]. The width of depletion regions may influence the diode ideality factor and the carrier collection efficiency [8]. Thus, changes in $\eta_\mathrm{sim}$ due to a varying ε are mainly due to $J_\mathrm{sc}$ and FF changes. Depletion region widths are proportional to $\sqrt{\epsilon/n}$, [36] so we may observe parallels between the effects of ε and the effects of n on $J_\mathrm{sc}$ and FF. As we have seen in the previous section, high values of n generally lead to a higher $\eta_\mathrm{sim}$, with a low n being preferable only when µ, α, and τ are all low (figure 6(c)). Accordingly, $\eta_\mathrm{sim}$ increases with decreasing ε when µ is high, independent of the ατ product (figure 7(a)). The opposite trend is observed when µ is low and the ατ product is not too high. These results confirm again that narrow depletion regions (small ε) are generally preferable in 'good' PV absorbers (high $\alpha\tau\mu$ product), whereas having wide depletion regions is more convenient in 'bad' PV absorbers (low $\alpha\tau\mu$ product). The dependence of $\eta_\mathrm{sim}$ on ε weakens with decreasing n, because at sufficiently low doping densities the PV absorber may be fully depleted regardless of the value of ε. For an example, compare the curves with $n = {10^{18}}\,\mathrm{}$ and $n = {10^{16}}\,\mathrm{}$ in figure 7(a).

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The dependence of $\eta_\mathrm{sim}$ on the DOS effective mass m of the PV absorber (figure 7(b)) is significant. This is expected, because m enters the expression for the QFLS (equation (10)). Thus, we expect that $V_\mathrm{oc}$ (the PV parameter directly related to the QFLS) should increase with decreasing m. Note that $n_\mathrm{i} \propto m^{3/2}$ in equation (10), but the $n_\mathrm{i}^2/n$ term is small compared to Δp under 1-Sun illumination so this additional dependence is negligible. The m-dependence of $V_\mathrm{oc}$ is stronger than its n-dependence, because the expression inside the logarithm in the QFLS expression (equation (10)) depends on $n/m^3$ under low injection conditions, and on $1/m^3$ under high injection conditions. Thus, the m-dependence is not affected by the carrier injection level. In figure 7(b), m is varied between 0.12 m₀ and 2.5 m₀. As expected, $\eta_\mathrm{sim}$ decreases with increasing m. For the high-mobility case—where equation (10) is valid—the efficiency indeed scales as $n/m^3$, because the overlapping curves (solid and dashed red line) have roughly the same $n/m^3$ ratio (both n and m³ are changed by a factor 100). Note that the range of doping densities employed in figure 7(b) is sufficiently high to ensure low injection conditions (see also figure 5(a)).

The effect of both ε and m can be captured reasonably well by adjusting some of the factors in the $\Gamma_\mathrm{ADTS}$ FOM expression in equation (16). However, since no new factors are needed when introducing these two new properties, the inclusion of variable ε and m in a FOM will only be shown in the next section as part of the final $\Gamma_{\mathrm{PV}}$ FOM including a variable band gap.

There is an important caveat regarding the effects of ε and m on PV efficiency. m and ε influence solar cell efficiency both directly (i.e. independent of the other bulk properties) and indirectly (i.e. by affecting the values of the other bulk properties, some of which explicitly depend on m and/or ε). As we have seen, the main direct effect of ε is on the width of depletion regions, [35] whereas the main direct effect of m is on QFLS (equation (10)). I emphasize that only direct effects are simulated in figure 7, because all the variable properties are independently changed in this study (see Methods section). However, both ε and m have substantial indirect effects [11, 32]. Some examples are given in the supplementary material. Importantly, many of these indirect phenomena affect the efficiency in the opposite direction of the direct effects modeled in figure 7. Hence, it is incorrect to draw conclusions such as 'increasing ε generally leads to lower efficiencies in the high-mobility limit' and 'increasing m generally leads to lower efficiencies' based on the trends in figure 7. This conclusion would only be true if changing ε and m did not affect the other properties.

A very well-known effect of changing the band gap $E_\mathrm{g}$ of a PV absorber is modification of its SQ limit $\eta_\mathrm{sq}$. Since calculation of $\eta_\mathrm{sq}(E_\mathrm{g})$ is straightforward, this effect is taken into account by normalizing the simulated efficiency by $\eta_\mathrm{sq}$, as done in most plots in this paper.

Plotting $\eta_\mathrm{sim,sq}$ data for absorbers with three different band gaps (figure 8(a)) demonstrates that normalization by $\eta_\mathrm{sq}$ is not sufficient to predict the efficiency of PV absorbers with variable band gaps via the $\Gamma_\mathrm{ADTS}$ FOM. This is not unexpected because $E_\mathrm{g}$ enters the definition of many quantities that are relevant for PV efficiency. Examples are the QFLS (equation (10)), the intrinsic carrier density $n_\mathrm{i}$, the absorption coefficient spectrum $\alpha(E)$, the generation rate, and the SRH recombination rate due to a mid-gap defect [8, 36, 37]. Interestingly, the $\Gamma_\mathrm{ADTS}$ FOM generally overestimates the efficiency potential of narrow band-gap absorbers and underestimates the efficiency potential of of wide band-gap absorbers (figure 8(a)), especially when $\Gamma_\mathrm{ADTS}$ is low. Due to the complex dependency between $E_\mathrm{g}$ and $\eta_\mathrm{sim,sq}$ and the usage of a complex FOM in figure 8(a), this trend is not straighforward to rationalize. From further analysis of the simulated data, the trend seems to mainly be driven by the $E_\mathrm{g}$ term in the QFLS expression in equation (10). If the excess carrier densities Δn and Δp only decrease weakly with increasing $E_\mathrm{g}$, then the QFLS is roughly equal to $E_\mathrm{g}$ minus a positive constant. Hence, the QFLS becomes a larger fraction of $E_\mathrm{g}$ as $E_\mathrm{g}$ increases, thus increasing $\eta_\mathrm{sim,sq}$ for wide-gap absorbers.

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Clearly, the $\Gamma_\mathrm{ADTS}$ FOM must be extended to address the complex efficiency-band gap dependency. The outcome is the $\Gamma_{\mathrm{PV}}$ FOM as a function of all eight bulk properties in the $\mathbb{P}$ set. This is the final FOM extension worked out in this article, corresponding to the end of Step B in figure 2. To develop the final $\Gamma_{\mathrm{PV}}$ FOM, I employ simulation data on absorbers with band gaps of 0.7 eV and 2.0 eV, while the other properties in the $\mathbb{P}$ set are also varied. The rationale is to test a band gap range appropriate for PV absorbers that may be considered for single-, double- and triple-junction solar cells [38–40]. Note that many new simulated data points are employed for $E_\mathrm{g} = 1.2\,\text{eV}$ absorbers when developing the final $\Gamma_{\mathrm{PV}}$, including other absorption coefficient spectra besides the ones shown in figure 3(b). This is done to check the robustness of the $\Gamma_{\mathrm{PV}}$ FOM and of the previously defined temporary FOMs.

I propose the following $\Gamma_{\mathrm{PV}}$ FOM to include the effects of variable ε, m, and $E_\mathrm{g}$:

$$\begin{equation} \begin{aligned} \Gamma_{\mathrm{PV}}\left(\mathbb{P}\right) &\equiv \Gamma_{\mathrm{PV}}\left(\alpha,\tau,\sigma,n,\mu,\epsilon,m,E_\mathrm{g}\right) = {E_\mathrm{g}}^{2.5}\left(\cfrac{\mathcal{A}_1 \mathcal{A}_2 \mathcal{D}_1}{\mathcal{D}_2 \mathcal{D}_3 \mathcal{D}_4 \, \left(1+\mathcal{T}_1 \mathcal{T}_2 \mathcal{T}_3\right) \, \left(1+\mathcal{S}_1 \mathcal{S}_2\right)}\right)^{{E_\mathrm{g}}^{-0.8}}. \end{aligned} \end{equation} \tag{ 18 }$$

The various factors in equation (18) are defined as:

$$\begin{align} \mathcal{A}_1 & = \cfrac{\overline{a}_1 \tau \alpha^{\sigma^{-\overline{a}_2 {E_\mathrm{g}}^{0.5}}}}{m^2} \end{align} \tag{ 19 }$$

$$\begin{align} \mathcal{A}_2 & = 1+\left(\overline{a}_3\cfrac{\sigma^{10}}{\alpha\tau }\right)^{0.4} \end{align} \tag{ 20 }$$

$$\begin{align} \mathcal{D}_1 & = \left(1+\overline{d}_1 \cfrac{n}{\epsilon^{0.8} \alpha^2}\right)^{0.22\text{log}{\left(\alpha/\overline{d}_2\right)}} \end{align} \tag{ 21 }$$

$$\begin{align} \mathcal{D}_2 & = \left(1+\overline{d}_3 \cfrac{n}{\alpha^2 \tau }\right)^{0.05 {E_\mathrm{g}}^4} \end{align} \tag{ 22 }$$

$$\begin{align} \mathcal{D}_3 & = 1+\cfrac{{\left(\overline{d}_4 {E_\mathrm{g}}^{8.5} \tau \alpha^{0.68{E_\mathrm{g}}^{-1.5}}\right)}^{\frac{\text{log}{\left(10n/\epsilon\right)}}{\overline{d}_5}}}{1+10^{\frac{E_\mathrm{g}-1.5}{0.1}}} \end{align} \tag{ 23 }$$

$$\begin{align} \mathcal{D}_4 & = 1+\left(\cfrac{\overline{d}_6}{{E_\mathrm{g}}^{17}\alpha\tau }\right)^{0.6} \end{align} \tag{ 24 }$$

$$\begin{align} \mathcal{T}_1 & = \cfrac{\overline{t}_1\left(E_\mathrm{g}+0.5\right)^{11}}{m \epsilon^{0.5}\mu^{\left(\overline{t}_2{E_\mathrm{g}}^{4.3}+0.9\right)}} \end{align} \tag{ 25 }$$

$$\begin{align} \mathcal{T}_2 & = \left(\cfrac{1+ \overline{t}_8 n\left(1+{E_\mathrm{g}}^{5.1}\right)}{\overline{t}_9 \tau 10^{\frac{0.5}{\sigma}}}\right)^{0.47{E_\mathrm{g}}^{1.25}} \end{align} \tag{ 26 }$$

$$\begin{align} \mathcal{T}_3 & = \left( 1 + \mathcal{T}_3^{\prime} \mathcal{T}_3^{^{\prime\prime}} \right)^{\overline{t}_7 {E_\mathrm{g}}^8 +0.6} \end{align} \tag{ 27 }$$

$$\begin{align} \mathcal{T}_3^{\prime} & = \overline{t}_3\left(1+ \overline{t}_4 {E_\mathrm{g}}^{10}\right)\text{exp}{\left(-0.1\alpha^{\frac{0.5}{1+\overline{t}_5 {E_\mathrm{g}}^{10}}} \right)} \end{align} \tag{ 28 }$$

$$\begin{align} \mathcal{T}_3^{^{\prime\prime}} & = 1+ \cfrac{\cfrac{0.16}{m^3} \left(\cfrac{n m^3}{0.16} \right)^{1-0.74 \, \text{exp}{\left(-\alpha /\overline{t}_6\right)}}}{1+10^{\frac{E_\mathrm{g}-1.5}{0.01}}} \end{align} \tag{ 29 }$$

$$\begin{align} \mathcal{S}_1 & = \cfrac{10^{2.4 E_\mathrm{g}} \alpha^{0.75}\tau}{\overline{s}_1 m^2 \mu^{\frac{1}{1+\overline{s}_2 {E_\mathrm{g}}^{10}}}} \end{align} \tag{ 30 }$$

$$\begin{align} \mathcal{S}_2 & = 1+ \left(\cfrac{\overline{s}_3}{\alpha}\right)^{20} \mu^{0.5}\text{log}{\left(n\right)} \end{align} \tag{ 31 }$$

where all properties are in unitless form (see Methods section). The quantities $\overline{a}_i$ ($i \in \{1,2,3\}$), $\overline{d}_i$ ($i \in \{1,2,\ldots,6\}$), $\overline{t}_i$ ($i \in \{1,2,\ldots,9\}$), and $\overline{t}_i$ ($i \in \{1,2,3\}$) are fixed parameters. Their values are given in the supplementary material. Note that $\Gamma_{\mathrm{PV}}$ contains new factors that were not present in the $\Gamma_\mathrm{ADTS}$ FOM. Also, most of the factors that were present in $\Gamma_\mathrm{ADTS}$ have been modified to generalize it to realistic ε, m, and $E_\mathrm{g}$ ranges. To avoid confusion, I have then defined the factors of the $\Gamma_{\mathrm{PV}}$ FOM as $\mathcal{A}_i$, $\mathcal{D}_j$, $\mathcal{T}_k$, $\mathcal{S}_l$ etc instead of as A_i , D_j , T_k , S_l etc. Since some of the fixed parameters have been adjusted as well, they are represented by barred lowercase letters in the $\Gamma_{\mathrm{PV}}$ FOM, instead of as simple lowercase letters.

We are now one step away from being able to estimate the maximum efficiency of a generic PV material from its $\Gamma_{\mathrm{PV}}$. The final step (step C in figure 2) is to find an appropriate function that gives a good fit to the $\eta_\mathrm{sim,sq}$ versus $\Gamma_{\mathrm{PV}}$ data in figure 8(b). A temporary function that gives a fair fit (figure 9(a)) is the following:

$$\begin{equation} \eta^{^{\prime}}_{\Gamma} = \begin{cases} 0 & \text{if } \Gamma_{\mathrm{PV}} < {10^{-6}}\,\mathrm{}\\ \left[\frac{1}{8} \text{log}\left(\Gamma_{\mathrm{PV}}\right)+\frac{3}{4}\right]\eta_\mathrm{sq} & \text{if } {10^{-6}}\,\mathrm{} \unicode{x2A7D} \Gamma_{\mathrm{PV}} \unicode{x2A7D} {10^{2}}\,\mathrm{}\\ \eta_\mathrm{sq} & \text{if } \Gamma_{\mathrm{PV}} > {10^{2}}\,\mathrm{}\\ \end{cases}. \end{equation} \tag{ 32 }$$

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This piecewise expression is instructive, because it shows that the maximum efficiency approaches zero for very low values of $\Gamma_{\mathrm{PV}}$, and it approaches the SQ limit $\eta_\mathrm{sq}$ for very high values of $\Gamma_{\mathrm{PV}}$, as expected from solar cell physics. For intermediate values of $\Gamma_{\mathrm{PV}}$ spanning eight orders of magnitude, the maximum achievable efficiency is proportional to the logarithm of $\Gamma_{\mathrm{PV}}$ minus a constant. A similar conclusion was drawn for the particular case of the ατ FOM [10].

The piecewise expression in equation (32) does not give a good fit in the $10^{-8} \lt \Gamma_{\mathrm{PV}} \lt 10^{-5}$ region (figure 9(a)). To predict efficiency limits as accurately as possible, I propose the following function instead:

$$\begin{equation} \eta_{\Gamma} = \cfrac{\eta_\mathrm{sq}}{\left(1+\cfrac{k_1 \, {\Gamma_{\mathrm{PV}}}^{-0.235}}{1+k_2 \, {\Gamma_{\mathrm{PV}}}^{0.869}}\right)\left(1+k_3 \, {\Gamma_{\mathrm{PV}}}^{-0.362}\right) }. \end{equation} \tag{ 33 }$$

The three k_i terms are fixed parameters, with their values given in the supplementary material. $\eta_{\Gamma,\mathrm{sq}}$ is simply equal to $\eta_{\Gamma} / \eta_\mathrm{sq}$. The expression proposed in equation (33) gives a good fit to the $\eta_\mathrm{sim,sq}$ versus $\Gamma_{\mathrm{PV}}$ data (figure 9(a)), with $R^2 = 0.995$ and a mean squared error of 11.7% absolute (figure 9(b)). Assuming that $\eta_\mathrm{sim} \approx \eta_\mathrm{max}$, we can conclude that the $\Gamma_{\mathrm{PV}}$ FOM enables immediate prediction of the maximum efficiency of a generic PV absorber with an absolute error of $\pm \sqrt{11.7\%} = \pm 3.4\%$ on the fraction of the SQ limit. As an example, we can estimate the maximum efficiency achievable by the hybrid perovskite absorber CH₃NH₃PbI₃ (MAPI) with a band gap of 1.55 eV and a set of bulk properties typical of its current state of the art, as given in table 2. MAPI is a convenient material for this analysis because it is structurally isotropic, so its properties do not have a strong direction dependence. Furthermore, τ, µ and m in MAPI have similar values for electrons and holes, [21, 43, 47] so the definitions of the $\mathbb{P}$ set proposed in the methods section is rather unambiguous.

Table 2. Best guesses for the values of the eight properties in the $\Bbb{P}$ set exhibited by CH₃NH₃PbI₃ (MAPI) films used to fabricate state-of-the-art MAPI solar cells with power conversion efficiency up to 21.3$\%$. The SQ limit $\eta_\mathrm{sq}$ of MAPI is given, together with the maximum efficiency predicted for MAPI films with these specific property values as a fraction of the SQ limit ($\eta_{\Gamma,\mathrm{sq}}$), and as absolute power conversion efficiency ($\eta_{\Gamma}$).

	$E_\mathrm{g}$	α		τ	µ	n		m	$\eta_\mathrm{sq}$	$\eta_{\Gamma,\mathrm{sq}}$	$\eta_{\Gamma}$
	(eV)	(cm−1)	σ	(s)	(cm2 V−1s−1)	(cm−3)	ε	(m0)	(%)	(% of $\eta_\mathrm{sq}$)	(%)
MAPI	1.55 [41]	$9.9 \times 10^{4}$ [42]	0.63 [42]	$6.9 \times 10^{-7}$ [43]	10[44]	$1 \times 10^{12}$ [45]	33.5 [46]	0.15 [21]	31.4	$(85.4 \pm 3.4)$	$(26.8 \pm 1.1)$

The efficiency limit derived by substituting MAPI's specific $\hat{\mathbb{P}}$ set (table 2) into equation (33) is $(85.4 \pm 3.4){\%}$ of its SQ limit of $\eta_\mathrm{sq}\,(1.55\,\text{eV}) = 31.4\%$. This corresponds to a maximum power conversion efficiency of $(26.8 \pm 1.1)\,{\%}$, a few percent better than the highest efficiency I am aware of (21.3%) for a pure MAPI absorber without alloying [48]. The main message is that device-level improvement may help improve efficiencies in MAPI solar cells up to the $(26.8 \pm 1.1){\%}$ limit, but bulk property improvement in MAPI films is likely necessary to exceed this limit. Importantly, we can also estimate the efficiency potential of a hypothetical 'bad' MAPI sample with the same properties as state-of-the-art samples, except for a lower carrier lifetime τ = 10 ns. The corresponding efficiency limit is $(20.4 \pm 1.1){\%}$. If, in addition, the mobility of the 'bad' sample is also lower than its state-of-the-art value (say $\mu = 0.1\,\mathrm{cm}^2\,\mathrm{Vs}^{-1}$), the efficiency limit further drops to $(13.5 \pm 1.1)\,{\%}$.