# A figure-of-merit-based framework to evaluate photovoltaic materials

Andrea Crovetto  E-mail<EMAIL_ADDRESS>National Centre for Nano Fabrication and
Characterization (DTU Nanolab), Technical University of Denmark, 2800 Kongens
Lyngby, Denmark

###### Abstract

I propose a general quantitative framework to evaluate the quality, track the
historical development, and guide future optimization of photovoltaic (PV)
absorbers at any development level, including both experimentally synthesized
and computer-simulated materials. The framework is based on a PV figure of
merit designed to include efficiency limitations due to imperfect photocarrier
collection that are not included in classic detailed balance methods. Figure-
of-merit-driven efficiency limits including collection losses are calculated
for 28 experimentally synthesized PV absorbers and 9 PV absorbers simulated by
electronic structure methods. Among emerging absorbers for which no working
solar cells have yet been reported, there are very large differences in their
likelihood to achieve high PV efficiencies.

Figure 1: Predictive ability of the $\alpha\tau$ (a), $\alpha\tau\mu$ (b), and
$\Gamma_{\mathrm{PV}}$ figures of merit (c). The maximum efficiency of over
2500 hypothetical absorbers with different combinations of the $\alpha$,
$\sigma$, $\tau$, $\mu$, $n$, $\epsilon$, $m$, and $E_{\mathrm{g}}$ properties
(as obtained by drift-diffusion simulation) is plotted against the three FOMs.
These data points constituted a training set in the original development of
the $\Gamma_{\mathrm{PV}}$ FOM. [1] Efficiencies are expressed as fraction of
the SQ limit, indicated by a red dashed line. The $\Gamma_{\mathrm{PV}}$ FOM
splits the efficiency versus FOM space into an accessible and an inaccessible
region.

The pace of exploration of new photovoltaic (PV) absorber materials has
accelerated in recent years, owing in part to the popularization of high-
throughput materials research methods and the desire to find new absorbers
with similar performance and process tolerance to halide perovskites but
without their stability and toxicity drawbacks. Various problems in emerging
PV materials research are still largely unresolved, specifically: (i)
quantifying the efficiency potential of new absorbers during their early
development stage, (ii) guiding their optimization, and (iii) helping
researchers make rational decisions on which materials are worth
investigating.

These problems are fundamentally linked to the methods available to estimate
the maximum PV efficiency of semiconducting absorbers. The physical principle
of detailed balance between photon absorption and emission is the cornerstone
of the most influential efficiency limits, which consist of the classic
Shockley-Queisser (SQ) limit [2, 3] and subsequent extensions to include the
effect of non-radiative recombination. [4, 5, 6, 7] Even though these methods
are powerful predictors of open-circuit voltage limits in solar cells, they do
not consider efficiency limitations arising from imperfect carrier collection,
which often occurs under various combinations of carrier mobilities, doping
density, and dielectric constant. [8, 9] Carrier collection losses affect
short circuit current densities and fill factors but are difficult to model in
a unified manner. [5, 8, 9] Thus, detailed-balance methods provide upper
bounds to the maximum efficiency achievable by a real PV absorber.

To address this problem, here I propose an alternative route to the definition
of efficiency limits. Rather than deducing them from fundamental physics (as
in detailed balance methods), I suggest a phenomenological efficiency limit by
an inductive methodology based on the recently developed
$\Gamma_{\mathrm{PV}}$ photovoltaic figure of merit (FOM). [1] The
$\Gamma_{\mathrm{PV}}$ FOM provides a general quantitative framework to
evaluate the quality, track the historical development, and guide future
optimization of a generic PV absorber at any development level.

The $\Gamma_{\mathrm{PV}}$ FOM is a unitless function of eight properties of
the PV absorber’s bulk:

$\Gamma_{\mathrm{PV}}=f(\alpha,\sigma,\tau,\mu,n,\epsilon,m,E_{\mathrm{g}})$
(1)

Here, $\alpha$ and $\sigma$ are parameters describing the spectral average and
spectral dispersion of the absorption coefficient spectrum, $\tau$ is the non-
radiative recombination lifetime, $\mu$ is the carrier mobility, $n$ is the
doping density, $\epsilon$ is the static dielectric constant, $m$ is the
density-of-states (DOS) effective mass, and $E_{\mathrm{g}}$ is the band gap.
Definitions of these properties are given in Ref. [1], including potential
strategies to deal with direction- and carrier-type dependence.

The $\Gamma_{\mathrm{PV}}$ FOM is defined as:

$\Gamma_{\mathrm{PV}}={E_{\mathrm{g}}}^{2.5}\left(\cfrac{\mathcal{A}_{1}\mathcal{A}_{2}\mathcal{D}_{1}}{\mathcal{D}_{2}\mathcal{D}_{3}\mathcal{D}_{4}\,(1+\mathcal{T}_{1}\mathcal{T}_{2}\mathcal{T}_{3})\,(1+\mathcal{S}_{1}\mathcal{S}_{2})}\right)^{{E_{\mathrm{g}}}^{-0.8}}$
(2)

The various factors
$\mathcal{A}_{1},\mathcal{A}_{2},\mathcal{D}_{1}...\,\mathcal{S}_{2}$ in Eq. 2
are expressions [1] containing the eight properties listed in Eq. 1.

The value of $\Gamma_{\mathrm{PV}}$ can be calculated for any candidate PV
absorber at any development level, as long as the eight bulk properties in Eq.
1 have been measured, calculated, or estimated. Once $\Gamma_{\mathrm{PV}}$
has been determined for a certain absorber, the maximum PV efficiency
achievable by that absorber in a planar single-junction solar cell under the
the standard 1-Sun, AM1.5G spectrum can be estimated as
$\eta_{\mathrm{\Gamma}}$, defined as:

$\eta_{\mathrm{\Gamma}}(\Gamma_{\mathrm{PV}},\eta_{\mathrm{sq}})=\cfrac{\eta_{\mathrm{sq}}}{\left(1+\cfrac{k_{1}\,{\Gamma_{\mathrm{PV}}}^{-0.235}}{1+k_{2}\,{\Gamma_{\mathrm{PV}}}^{0.869}}\right)\left(1+k_{3}\,{\Gamma_{\mathrm{PV}}}^{-0.362}\right)}$
(3)

Here, $\eta_{\mathrm{sq}}$ is the Shockley-Queisser limiting efficiency of the
absorber, which is readily calculated from $E_{\mathrm{g}}$, [2] and
$k_{1},k_{2},k_{3}$ are fixed parameters. [1] For easier comparison between
different PV absorbers with different band gaps, $\eta_{\mathrm{\Gamma}}$ will
often be expressed as a fraction of the Shockley-Queisser limit. This SQ-
normalized efficiency is defined as
$\eta_{\mathrm{\Gamma,sq}}=\eta_{\mathrm{\Gamma}}/\eta_{\mathrm{sq}}$.

The predictive capability of $\Gamma_{\mathrm{PV}}$ is shown in Fig. 1 and
compared to the case of the simpler $\alpha\tau$ and $\alpha\tau\mu$ FOMs.
[10] The three plots in Fig. 1 show the maximum SQ-normalized efficiency of
over 2500 hypothetical absorbers (as obtained by explicit drift-diffusion
simulation) versus the three FOMs calculated for the same absorbers. In the
simulation [1] the only PV losses besides radiative recombination are due to
non-optimal values of the eight bulk properties used to define
$\Gamma_{\mathrm{PV}}$ (Eq. 1). It is apparent that the $\alpha\tau$ and
$\alpha\tau\mu$ FOMs are generally not predictive of efficiency, although they
are useful indicators under certain conditions. [10, 1] On the other hand, the
$\Gamma_{\mathrm{PV}}$ FOM seems to have higher predictive power because the
simulated data roughly follows a single function that approaches zero for low
$\Gamma_{\mathrm{PV}}$ values and $\eta_{\mathrm{sq}}$ for high
$\Gamma_{\mathrm{PV}}$ values (Fig. 1(c)). This function is just the SQ-
normalized version of the $\eta_{\mathrm{\Gamma}}$ function defined in Eq. 3
and it is shown in Fig. 2 as a black line.

The $\Gamma_{\mathrm{PV}}$ FOM naturally leads to the definition of an
”accessible region” (below the $\eta_{\mathrm{\Gamma,sq}}$ curve in Fig. 2)
which is the efficiency range that PV absorbers can realistically access as a
function of their figure of merit. The ”inaccessible region” above the
$\eta_{\mathrm{\Gamma,sq}}$ curve would be allowed by the SQ limit alone, but
is not allowed by the additional losses introduced by non-optimal values of
the eight bulk properties. The error bar of the $\eta_{\mathrm{\Gamma,sq}}$
efficiency limit is $\pm 3.4\%$ absolute, [1] under the assumption that the
values of $\alpha$, $\sigma$, $\tau$, $\mu$, $n$, $\epsilon$, $m$, and
$E_{\mathrm{g}}$ used to calculate $\Gamma_{\mathrm{PV}}$ are known exactly.
This corresponds to an absolute error around $\pm 1.1\%$ on the actual
$\eta_{\mathrm{\Gamma}}$ efficiency (not SQ-normalized) for PV absorbers in
the $1.1\text{\,}\mathrm{e}\mathrm{V}1.5\text{\,}\mathrm{e}\mathrm{V}$ band
gap range, and less for absorbers outside this range.

Figure 2: Performance status of 28 experimentally synthesized PV absorbers,
expressed as record efficiency (y-axis, normalized to their SQ limit) versus
$\Gamma_{\mathrm{PV}}$ FOM (x-axis). Record solar cells for all materials
except SbSI have a planar architecture. The thick black line is the
$\eta_{\mathrm{\Gamma,sq}}$ efficiency limit (Eq. 3). The dotted line
indicates the Auger limit for c-Si and mc-Si (more stringent than their SQ
limit). In this representation, the SQ limit is always 100%. The size of the
data points is proportional to the logarithm of the number of publications
referring to each absorber in a PV context. The color of the data points
indicates the band gap of each absorber. The small light grey data points are
the training set used to derive the $\Gamma_{\mathrm{PV}}$ FOM (identical to
the data points in Fig. 1(c)). They give a visual impression of the $\pm
3.4\%$ absolute error bar of $\eta_{\mathrm{\Gamma,sq}}$ as an estimator of
efficiency limits. The material properties used to calculate
$\Gamma_{\mathrm{PV}}$ for each absorber are listed in Table I, Supporting
Information. The maximum experimental efficiencies used for each material are
listed in Table III, Supporting Information.

In Fig. 2, the maximum (SQ-normalized) experimental efficiency
$\eta_{\mathrm{exp,sq}}$ demonstrated by 28 emerging and established PV
absorbers in a planar solar cell configuration is plotted as a function of
their corresponding $\Gamma_{\mathrm{PV}}$ FOM. As expected, all absorbers lie
in the accessible region defined by the $\Gamma_{\mathrm{PV}}$ FOM. Several
absorbers lie within error bar of the $\eta_{\mathrm{\Gamma,sq}}$ limit,
indicating that solar cell architectures, contact layers, and interfaces are
sufficiently optimized and do not introduce significant losses at the current
level of bulk material quality in these absorbers. GaAs is the absorber with
both the highest FOM (20) and the maximum SQ-normalized experimental
efficiency (88%). Alloyed perovskites based on CH5N2PbI3 (triple-cation and
double-anion) have the second highest FOM, followed by c-Si and CH3NH3PbI3
perovskites (MAPbI3). Since the efficiency of c-Si cells is limited by Auger
recombination due to particularly weak absorption in Si, [11] there is very
limited scope for efficiency increase by improvement of bulk material
properties in c-Si (Fig. 2), except for increase in $\alpha$, which would make
c-Si cells less limited by Auger recombination. Multicrystalline silicon (mc-
Si), Cu(In,Ga)Se2 (CIGSe), and CdSeTe have similar FOMs
($0.5\text{\,}1\text{\,}$) and SQ-normalized record efficiencies (around 70%).

According to the FOM analysis, further efficiency increases in GaAs, c-Si, mc-
Si, and CIGSe cells are unlikely unless their bulk properties are improved. On
the other hand, higher efficiencies in CdSeTe and perovskite solar cells are
not prohibited by their current material quality, so device and interface
optimization may be a promising path towards improved solar cell performance.
In fact, the $\Gamma_{\mathrm{PV}}$ FOM suggests that it may already be
possible for state-of-the-art perovskite alloys to surpass the efficiency of
GaAs solar cells without further work on their bulk properties.

Many PV absorbers fall in the intermediate $0.003\text{\,}0.3\text{\,}$
$\Gamma_{\mathrm{PV}}$ range (Fig. 2), but none of them lie within error bar
of the $\eta_{\mathrm{\Gamma,sq}}$ limit, indicating that they all suffer from
interface or device-level losses in state-of-the art cells. Absorbers with
higher cumulative development efforts such as CdTe, amorphous silicon (a-Si)
and kesterites (CZTS, CZTSe, and ACZTSSe) tend to lie closer to their FOM-
driven efficiency limit compared to less developed absorbers such as BiOI,
Zn3P2, and Cu2BaSn(S,Se) (CBTSSe). The record efficiency of kesterite
absorbers has recently climbed to 14.9% after years of stagnation [12] and the
$\Gamma_{\mathrm{PV}}$ value of the current state-of-the-art kesterite
absorbers (0.15) implies an efficiency limit of about 22% without further bulk
property improvement. Nevertheless, this FOM value is still lower than in the
best Sn-based perovskites. These low-band gap perovskites have also
experienced significant progress in both record efficiency and bulk material
quality, with CH5N2SnI3 (FASnI3) now exhibiting a FOM of 0.25 and an
efficiency limit of about 23%. Interestingly, two novel PV absorbers (BaZrS3
and BaCd2P2) have already reached this intermediate FOM range despite their
very short history and lack of functioning solar cells.

Some of the considered absorbers fall in the low FOM region
($\Gamma_{\mathrm{PV}}<0.001$). Among them, the record efficiency of AgBiS2
seems difficult to increase without improving the bulk properties of the
absorber, a factor that may slow down the rapid progress AgBiS2 solar cells
have experienced in recent years. [13] The chemically related NaBiS2 shows
significantly less promise as a PV absorber due to its lower FOM. SnS has
about the same FOM value as NaBiS2 but it seems to already have reached its
(low) efficiency limit, possibly due to its longer development history. The
figure of merit of the recently proposed antiperovskite Ag3SI is still so low
($<${10}^{-7}\text{\,}$$) that even a perfect device structure and perfect
interfaces would not help this absorber reach meaningful efficiencies. SbSI
lies in the inaccessible region (although within error bar of
$\eta_{\mathrm{\Gamma,sq}}$). This may partially be due to its mesoporous cell
architecture, for which the $\eta_{\mathrm{\Gamma,sq}}$ efficiency limit does
not strictly apply. [1]

Figure 3: Quantification of the current quality of 28 PV absorbers at the
device/interface level (y-axis) and at the bulk absorber property level
(x-axis). Along the dashed line the device quality factor is equal to the
absorber quality factor. For materials located above this line, improving the
bulk properties of the absorber is likely more urgent than improving the
device structure, interfaces, contact/transport layers in their solar cells.
For materials located below the dashed line, absorber development is likely
less urgent. The white contour lines indicate device/absorber quality factor
combinations with a constant $\eta_{\mathrm{exp,sq}}$ efficiency limit
(labeled as fraction of the SQ limit). Red data points indicate materials that
would have a $>10\%$ (relative) higher efficiency limit if their carrier
mobility could be increased from their current experimental value to infinity.
The other materials (black points) would benefit less from a mobility
increase.

The combination of experimental record efficiencies $\eta_{\mathrm{exp,sq}}$
and the $\Gamma_{\mathrm{PV}}$ figure of merit enables researchers to quantify
the relative urgency of device-level optimization and bulk absorber-level
optimization for a given single-junction PV technology. In Fig. 3, a device
quality factor defined as $\eta_{\mathrm{exp,sq}}/\eta_{\mathrm{\Gamma,sq}}$
is plotted against an absorber quality factor simply defined as
$\eta_{\mathrm{\Gamma,sq}}$. For some PV technologies (e.g., BiOI, Zn3P2,
CBTSSe, BaZrS3, BaCd2P2) the absorber quality factor is substantially higher
than the device quality factor. In these cases, optimization of device
structure, interfaces, contact and/or transport layers is arguably more urgent
than improvement of the bulk properties of the absorber. For other
technologies (e.g., CIGSe, SnS, AgBiS2, SbSI) the device quality factor is
close to 100% and improving the absorber quality seems to be the only
realistic way to significantly improve their current record efficiency. By
changing the $\mu$ values of the different PV materials (with everything else
fixed) and applying Eqs. 2, 3, it is possible to determine which PV materials
are limited by low carrier mobilities. It turns out that over half of the
materials shown in Figs. 2, 3 would have an efficiency limit higher by at
least 10% relative if their carrier mobilities could be increased to infinity.
These are examples of PV materials for which the classic detailed balance
analysis would overestimate efficiency limits.

Figure 4: Maximum efficiencies achievable by 9 PV absorbers, according to the
$\Gamma_{\mathrm{PV}}$ FOM using DFT-calculated properties (pale colors) and
experimentally measured properties on state-of-the-art specimens (bright
colors). These efficiency limits are compared to classic detailed-balance
efficiency limits including non-radiative recombination (but not considering
finite values of $\mu$, $n$, and $\epsilon$). The top of the green bars
corresponds to the $\eta_{\mathrm{\Gamma,sq}}$ limit. The top of the orange
bars is the current record efficiency. The purple bars represent the gap in
efficiency between the $\eta_{\mathrm{\Gamma,sq}}$ limit and the SQ limit. The
material properties used to obtain $\Gamma_{\mathrm{PV}}$ from DFT modeling
are listed in Table II, Supporting Information.

A key question that naturally arises at this point is whether it is actually
possible to improve the bulk properties of a given absorber beyond its current
state of the art. For example, is there any hope of developing SnS beyond its
modest 18% absorber quality factor? Could we have foreseen the significant
recent FOM increase of kesterite absorbers? Can perovskite absorbers become
even better? To search for answers, we can turn to first-principles
calculations of material properties. Until recently, only six out of the eight
bulk properties necessary to calculate the FOM ($\alpha$, $\sigma$, $n$,
$\epsilon$, $m$, $E_{\mathrm{g}}$) were within reasonable reach of materials
modeling methods based on density functional theory (DFT). However, recent
methodological advances [14, 15] now allow computational scientists to also
estimate $\tau$ by calculating carrier capture coefficients of point defects,
[16] as well as $\mu$ by more efficient calculation of carrier scattering
rates. [17]

Hence, it is also possible to calculate the $\Gamma_{\mathrm{PV}}$ FOM and the
corresponding $\eta_{\mathrm{\Gamma,sq}}$ efficiency limit using
computationally-determined properties, even for the case of materials that
have never been synthesized. Some examples are shown in Fig. 4. The DFT-
derived $\eta_{\mathrm{\Gamma,sq}}$ limits (pale colors) in materials with
significant accumulated experience (MAPbI3, CdTe, CZTS, CZTSe, SnS) are quite
close to the corresponding limits derived from experimental properties (bright
colors). This is a remarkable finding, implying that the ultimate efficiency
potential of these absorbers could have been quantitatively predicted with
reasonable accuracy before commencing any experimental development work.
Still, the experiment/computation agreement can sometimes be partially
attributed to error cancellation. Furthermore, comparison of PV material
properties from experiment and computation should always be done with an
awareness of possible limitations. A discussion is given in the Supporting
Information.

Two other results in Fig. 4 are worth noting. First, the very recently
proposed BaCdP2 [18] is the only absorber in Fig. 4 that may be able to reach
the SQ limit according to the DFT-based $\Gamma_{\mathrm{PV}}$ FOM. The above-
discussed good agreement between $\eta_{\mathrm{\Gamma,sq}}$ values obtained
on a computer and in the lab for more established absorbers gives credibility
to this exciting finding. Second, efficiency limits calculated through the
$\Gamma_{\mathrm{PV}}$ FOM can be compared to the corresponding limits
calculated with current state-of-the-art detailed-balance methods including
non-radiative recombination (finite $\tau$) but no collection losses (infinite
$\mu$). [4, 5, 6, 7] For DFT-modeled absorbers, these methods are often
referred to as the spectroscopically limited maximum efficiency (SLME)
following the treatment in Ref. [4]. As expected, the inductive
$\eta_{\mathrm{\Gamma,sq}}$ efficiency limit is tighter than the deductive
SLME limit due to the explicit inclusion of $\mu$, $n$, and $\epsilon$.

Another interesting application of the $\Gamma_{\mathrm{PV}}$ FOM is tracking
the progress of different PV absorbers and their record solar cells through
history. These timelines are plotted in Fig. 5 for four PV absorber types and
can be related to technological breakthroughs in absorber processing and to
device structure/interface innovations. As an example, let us follow the
history of CIGSe. Record CIGSe cells between 1985 and 1996 were operating at
the maximum efficiency allowed by the absorber’s bulk quality at that time.
During this period, the FOM of CIGSe improved due to increases in $\tau$ and
$n$ following the introduction of a Cu-rich growth phase, as well as Na and Ga
incorporation. [19] Steady improvements to the three-stage evaporation process
in CIGSe over the next decade [19] resulted in significantly longer carrier
lifetimes and a large increase in the $\Gamma_{\mathrm{PV}}$ FOM. However, the
device quality as of 2009 was no longer sufficient for CIGSe cells to achieve
their full potential. The main breakthrough allowing CIGSe solar cells to
again approach their $\eta_{\mathrm{\Gamma,sq}}$ efficiency limit in the
2020’s was the introduction of an alkali post-deposition treatment (2013) to
improve the charge-separating heterointerface. [20]

Figure 5: Development history of four classes of PV absorbers. Kesterites
include absorbers with the general (Ag,Cu)2ZnSn(S,Se)4 formula; CIGSe stands
for Cu(In,Ga)Se2; MAPbI3 perovskites only include materials without alloying.
Perovskite alloys are absorbers with the general (MA,FA,Cs)Pb(I,Br,Cl)3
formula. Numbers next to each data point refer to the year of publication. The
dashed line is the $\eta_{\mathrm{\Gamma,sq}}$ efficiency limit (Eq. 3, same
as in Fig. 2). The small grey data points are the training set used to derive
the $\Gamma_{\mathrm{PV}}$ FOM and giving an impression of the error bar of
the $\eta_{\mathrm{\Gamma,sq}}$. Raw data is listed in Tables IV–VI,
Supporting Information.

In conclusion, the $\Gamma_{\mathrm{PV}}$ FOM can be considered a
generalization of previously proposed $\alpha\tau$ and $\alpha\tau\mu$ FOMs.
It seems to be applicable across a wide range of realistic combinations of
bulk properties of possible PV absorber materials (Fig. 1). The efficiency
limit $\eta_{\mathrm{\Gamma}}$ set by the $\Gamma_{\mathrm{PV}}$ FOM is more
stringent than the corresponding limit set by deductive methods based on the
principle of detailed balance. The main reason is that efficiency losses by
imperfect carrier collection (primarily influencing $J_{\mathrm{sc}}$ and FF)
are taken into account in $\Gamma_{\mathrm{PV}}$ by including the carrier
mobility, doping density, and dielectric constant of the PV absorber as
relevant properties. Applying the $\Gamma_{\mathrm{PV}}$ FOM to 28 established
and emerging PV materials (Fig. 2) allows us to quantitatively visualize their
current development status, to decide whether to prioritize bulk absorber
improvement or device-level improvement (Fig. 3), and to track the impact of
different innovations in absorber processing versus device/interface
processing and design (Fig. 5). With some caveats, the
$\eta_{\mathrm{\Gamma}}$ efficiency limit obtained using DFT-calculated
properties seems to be an acceptable descriptor of the ultimate performance
level that may be expected from candidate PV absorbers (Fig. 4). This gives
credibility to computational screening workflows that include calculation of
the eight properties necessary to evaluate the $\Gamma_{\mathrm{PV}}$ FOM.
Intriguingly, some exotic phosphide compounds recently identified via such
methods exhibit higher efficiency potential then many established PV
absorbers.

## Acknowledgements

This work was co-funded by the European Union (ERC, IDOL, 101040153). Views
and opinions expressed are however those of the author only and do not
necessarily reflect those of the European Union or the European Research
Council. Neither the European Union nor the granting authority can be held
responsible for them. This work was supported in part by a research grant
(42140) from VILLUM FONDEN.

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