# Light Management in Perovskite Photovoltaic Solar Cells: a perspective

Florian Berry1 Raphaël Mermet-Lyaudoz1 Jose Maria Cuevas Davila1 Djihad Amina
Djemmah1 Hai Son Nguyen1,2<EMAIL_ADDRESS>Christian Seassal1 Erwann
Fourmond1 Céline Chevalier1 Mohamed Amara1 Emmanuel Drouard 1
<EMAIL_ADDRESS>1Univ Lyon, Ecole Centrale de Lyon, CNRS, INSA
Lyon, Université Claude Bernard Lyon 1, CPE Lyon, INL, UMR5270, 69130 Ecully,
France  2Institut Universitaire de France (IUF)

###### Abstract

Light Management (LM) is essential for metal-halide perovskite solar cells in
their race for record performance. In this review, criteria on materials,
processes and photonic engineering are established such as to enhance mainly
the short circuit current density, towards high energy yields. These criteria
are used to analyse a large panel of solutions envisaged in the literature for
single junction cells. Moreover, a perspective based on rigorous
electromagnetic simulations performed on various comparable structures is
proposed in order to clarify the conclusions, and to pave the way to further
performance enhancement in the case of all-perovskite, two-terminal tandem
cells.

KYEWORDS: light trapping, single junction perovskite cell, all perovskite
tandem cells

##  I Introduction

Perovskite photovoltaic solar cells (PPSC) have reached efficiency comparable
to those of the best crystalline silicon (c-Si) devices within only a few
years nre , thanks to the tremendous electrical and optical properties of the
metal-halide hybrid perovskites materials Snaith (2013); Jeon  _et al._
(2015a); Tress (2017). For single junction PPSC, the best performance yet
demonstrated reach or even overpass a short-circuit current density $J_{sc}$
of 25.8 $mA.cm^{-2}$, an open circuit voltage $V_{oc}$ of 1.179 $V$, and a
fill factor $FF$ of 0.846, leading to power conversion efficiency $PCE$ of
25.7% nre . These achievements are already quite close to the theoretically
expected maximum values, especially for the $J_{sc}$, which is typically of
27.83 $mA.cm^{-2}$ under AM1.5G illumination, given that the band gap energy
is estimated in their studies to 1.53 $eV$. Furthermore, tandem PPSC made of
at least two different perovskite materials appears promising bust also
challenging and less mature. In the case of the most common 2 terminal (2T)
configuration, a net performance increase could have been recently
demonstrated compared to single junction PPSC Lin  _et al._ (2022).

As for every optoelectronic device, light management (LM) in PPSC is of
primary importance to reach the best performance. As schematized in Figure 1,
the LM is expected to impact three key performance parameters. First, thanks
to the LM, impinging sunlight can be efficiently collected and then absorbed
leading to a high $J_{sc}$. Then the LM can help to enhance the $V_{oc}$
thanks to the so-called photon recycling (PR)Miller  _et al._ (2012), provided
a strong external luminescence Ross (1967) and considering the semiconductor
exhibits weak non radiative recombinations. Finally, as recently proposed, the
energy yield (EY) tends to substitute to the two previous criteria usually
considered under standard test conditions (STC). For the latter, requested
studies, however, remain complex since they intend to couple illumination
models and angular-dependant optical, thermal and electrical models of the
cell for one year Peters and Buonassisi (2018). However, simply considering
the incidence angle in the evaluation of absorption and thus $J_{sc}$ is a
first step towards EY improvement.

Figure 1: The three main correlated aspects that governed Light Management.

As illustrated in Figure 1, the LM in PPSC results from three interdependent
considerations: the photonic engineering (PE) at the wavelength scale, the
material choices, and the nanofabrication processes. Practically, the last two
directly define the PPSC architecture.

In brief, still regardless of the envisaged semiconductor, the LM can invoke
several kinds of PE. Most frequently, it results in anti-reflection (AR) over
the whole absorbed spectral range, leading to a larger $J_{sc}$ as well as
possibly larger EY if this the AR effect also occurs over a wide angular
domain. Moreover, beyond the AR effects, light trapping (LT) mechanisms can be
involved at the band edge of the perovskite material, also leading to a larger
$J_{sc}$ Bermel  _et al._ (2007); Zhou and Biswas (2008), whereas enhanced PR
thanks to absorption suppression just above the band gap could enhance
$V_{oc}$ Sandhu  _et al._ (2013), still provided previously mentioned
conditions. This well established PE is reminded in more details in section
IV.2. the LM could even be extended to the infrared domain to possibly limit
the parasitic sub band gap absorption and enhance the thermal radiation, so as
to decrease the operating temperature Dumoulin  _et al._ (2021). Finally, for
the specific case of 2T tandem PPSC, the LM is required to ensure at the same
time absorption optimization and current matching between the two cells.

In the specific case of metal-halide perovskites, intrinsic properties make
these materials particularly relevant for LT, compared to the standard case of
c-Si solar cells. First, metal-halide perovskites are direct band gap
semiconductors, the light is then more easily absorbed in a submicron thick
layer, where LT and possibly PR can occur at their band edge, as detailed
later. Second, their refractive indices are lower than that of c-Si, thus, AR
effect is easier to obtain over the spectral range of interest. Lastly, the
ability to recycle photons has been observed/shown in the specific case of
lead iodide perovskite material Pazos-Outón  _et al._ (2016).

Other key properties of metal-halide perovskites, mainly electrical and with
regards to aging, have direct consequences on the typical architecture of the
PPSC. The diffusion lengths of the carriers in the perovskites imply both
contacts should be in the vicinity of the photocarrier generation, thereby
covering both sides of the cell Yang  _et al._ (2019). Additionally, one of
these contacts should obviously be transparent to the sunlight. The double
heterojunction architecture is widespread, leading to the use of additional
intermediate layers, the electron/hole transport layers (ETL/HTL). Then, the
back metallic contact, ideally made of noble metal, can thus act as a back
mirror. In any case, a careful optical design is required to keep low
parasitic absorption in these surrounding layers. Moreover, an effective
encapsulation should be added to prevent from the degradation of the
perovskite. It should be carefully chosen considering the impact of its
optical properties.

Let us finally envisage the wide panel of low-cost elaboration techniques and
strategies enabling the structuration of the perovskite thin film at various
scales, from the sub-micron to the over-micron one. First PPSC mainly relied
on a mesoporous architecture, in which the perovskite infiltrated a scaffold.
Then, progress in the elaboration techniques of the perovskite in liquid or
vapor phases basically results in a better microcrystallization, and thus
larger diffusion lengths. Simple planar architectures have therefore replaced
mesoporous ones Liu  _et al._ (2013); Liu and Kelly (2014); Zhao  _et al._
(2019); thanks to these far larger diffusion lengths, current planar PPSC
overpass the performances of the best mesoporous PPSC.

Following these huge modifications, the LM has evolved: if the mesoporous
architecture could have been optimized for efficient LM thanks to the scaffold
(typically made of TiO2) to enhance the $J_{sc}$, most of the planar cells
demonstrated in the last few years (see e.g. supporting information of ref Xu
_et al._ (2020)) exhibit performances approaching the limits, up to about 90%
of the maximum achievable $J_{sc}$, mostly by simply taking care of AR.

However, refined the LM still enables additional improvements of the
performances towards the limits. For example, reduced optical losses thanks to
LT especially at the band edge helps to gain the last missing $mA/cm^{2}$, as
detailed in the following. In addition, the $V_{oc}$ could be improved thanks
to an accurate treatment of the photocarrier recombination mechanisms and the
associated rates.

A few review papers discussed on various demonstrations of LM for PPSC at the
time of their writing Zhang and Toudert (2018); Zhang  _et al._ (2019). In
this paper, we focus on PE mainly in corrugated dielectric, without any
additional metallic material inside or at the vicinity of the perovskite
material. Indeed, if using metal can induce LT due to plasmonics effect, other
phenomena such parasitic absorption can frequently balance this effect Siavash
Moakhar  _et al._ (2020).

After a brief presentation of the main PPSC architectures and material choices
that impact LM, and summarizing the PE, we propose to further analyze various
published structures relying on a wave optics approach. Thus, by unraveling
the various mechanisms involved, we are able to compare the various proposed
strategies to deduce guidelines for further optimization. These guidelines are
finally illustrated thanks to simulations of the optical properties of a few
case studies.

##  II Key PPSC materials and architectures

The LM in the PPSC directly derive from their architecture, i.e. a stack of
layers of various materials and thus optical indices, each layer having a
thickness of the order of a few tenth of wavelengths, and with possible
structurations from small to large scales compared to the wavelength. The main
properties of the perovskite materials impacting LM have already been
discussed. However, comparisons will appear difficult and need to be treated
carefully, due to discrepancies between the structures investigated, as
detailed hereafter.

###  II.1 Patterning of the PPSC

Figure 2: Various typical architectures of complete PPSC obtained using
nanofabrication. a. Almost flat (possible nanoscale roughness) b. patterned
perovskite using "bottom-up" approach, i.e. structuration of the ETL/HTL, then
overcoating with perovskite; c. patterned perovskite using "top down"
approach, i.e. perovskite coating then structuration and overcoating. d. Flat
perovskite but patterned surrounding layers and e. Patterned substrate coated
with conformal stack. All envisaged patterned can be either periodic or
aperiodic.

Let us present an overview of the main patterning processes and deriving
architectures that have been proposed in the literature. Due to the low
diffusion lengths in the first fabricated metal-halide hybrid perovskites,
first PPSC used mesoporous architectures to limit recombination. Indeed, TiO2
scaffolds were infiltrated with the perovskite material, enabling a more
efficient collection of the carriers, and, in some cases, LM Lee  _et al._
(2015); Lin  _et al._ (2016); Ramos  _et al._ (2016). Nevertheless, as already
mentioned in the introduction, most promising approaches relies nowadays in
planar architectures.

In the following, layers will be considered as almost flat provided their
roughness is at a sub-wavelength, nanometre scale, especially thanks to a good
crystallinity in the case of the perovskite.

However, a 2D structuration can be introduced in the PPSC stack, notably for
single junction PPSC. In this frame, the perovskite layer itself can be one of
the patterned layers, or only surrounding layers are structured, and the
perovskite can be considered as flat. The various envisaged technical
solutions within each of the two strategies are sketched in Figure 2 b-e.
Patterning of the perovskite layer can be achieved by various approaches. Such
a structuration can first be generated by depositing this active layer on top
of patterned layers, for example the ETL/HTL standing underneath ("bottom-up"
like approach, Figure 2 b.); the pattern can be either random Xu  _et al._
(2020); Huang  _et al._ (2016); Pascoe  _et al._ (2016), or periodic Paetzold
_et al._ (2015); Abdelraouf and Allam (2016); Wang  _et al._ (2018a); Kim  _et
al._ (2019). Alternatively, the perovskite layer itself can be directly
patterned after its deposition ("top-down" like approach, Figure 2 c.), again
either in a random Wang  _et al._ (2018b, c) or periodic way Schmager  _et
al._ (2019a, b).

Another approach is to realize flat perovskite layers and to corrugate other
adjacent layers (Figure 2 d.), such as the encapsulation layer Nguyen  _et
al._ (2016); Li  _et al._ (2019); Kim  _et al._ (2021), transport layers Kang
_et al._ (2016); Qarony  _et al._ (2018); Wang  _et al._ (2021a), possibly
associated with the metal back contact Wei  _et al._ (2017); Zhang  _et al._
(2018), or only the front contacts Hossain  _et al._ (2020a); Tockhorn  _et
al._ (2020), and finally a glass substrate covering layer Tavakoli  _et al._
(2015); Dudem  _et al._ (2016); Jošt  _et al._ (2017); Peer  _et al._ (2017);
Martinez-Denegri  _et al._ (2019); Thangavel  _et al._ (2020).

Finally, the front substrate or the back contact layer can be patterned within
a broad range of sizes, from the hundreds of nanometers to almost the
millimeter; then the other layers of the stack are conformally deposited,
leading to a fully structured cell Qarony  _et al._ (2018); Wang  _et al._
(2016); Du  _et al._ (2016); Shi  _et al._ (2017); Soldera and Taretto (2018);
Tooghi  _et al._ (2020); Wang  _et al._ (2019); Soldera  _et al._ (2020);
Haque  _et al._ (2020) (Figure 2 e.). It is noticeable that among these
numerous papers, only two Tockhorn  _et al._ (2020); Soldera  _et al._ (2020)
are dealing with experimental results. Indeed, spin coating of perovskite on a
patterned substrate remains challenging. The former mesoporous architecture
has recently been revisited under the form of opal like structuring Lobet  _et
al._ (2020).

###  II.2 Influence of the choice of Materials

####  II.2.1 Perovskite medium

The choice of the absorbing material for PPSC results from multiple
constraints. It should first exhibit an optimal band gap energy, in the range
1.1 - 1.4 $eV$Shockley and Queisser (1961), as well as optimal electrical
properties and stability. Then, various deposition processes can be envisaged
for a given metal halide perovskite. It may mainly lead to various grain sizes
of the material during the crystallization process. For sizes of the order of
the wavelength, this can affect LM.

Methylammonium lead iodide perovskite material (CH3NH3PbI3, MAPI) has been
widely considered in the early stages of PPSC development, thanks to its
rather simple chemical composition, and despite $E_{g}$ slightly above the
optimal range. It is even noticeable that different $E_{g}$ are reported for
MAPI typically from 1.55 to 1.6 $eV$ (see e.g. Jiang  _et al._ (2015)). This
implies a more than 7% uncertainty on the maximum achievable $J_{sc}$, that is
already of the order of the achievable gain using LM as reported in the
following. Thus, the quantitative comparison between different studies, even
implying the same material within the meaning of the chemical composition, is
rather delicate. Differences of aging of the perovskite could also impact the
comparison. Finally, various dispersion models can be used to fit the
dielectric function Li  _et al._ (2020a), leading to differences that also
impact the optical indices that are used for simulations.

For all these reasons, it has to be emphasized that, in the following, only
the relative impact of LM could be put into perspective, but the various given
performances cannot be accurately compared. Finally, if MAPI can still be used
as a typical case study, other materials have been developed in the past few
years. They are more stable and exhibit a slightly lower $E_{g}$, mainly
thanks to the substitution of MA by FA (Formamidinium, (NH2)2CH), possibly
with Cs, as can be seen later.

####  II.2.2 Transport and contact layer materials

For the transport layers, mainly polymer materials are used, as well as thin
layers of inorganic materials such as TiO2. It is noticeable than materials
such as those based on C60 fullerene, acting as ETL, exhibit significant
absorption at the shortest wavelengths of the solar spectrum, whereas polymers
used as HTL mainly absorb at larger wavelengths, close to and above the band
gap of the perovskite. Then, there is often a compromise to be found between
suitable band levels, high electrical conductivity, and barrier to possible
migrations, and low parasitic absorption, even if the thicknesses of these
transport layers remain limited.

##  III Photonic engineering

Figure 3: Main Light Management strategies a. Without peculiar LM, reflectance
can be important. b. Using a flat structure, or a patterned that can be
considered as homogeneous (e.g. given its subwavelength dimensions), LM can
lead to a rather broadband AR effect as well as to Fabry Perot modes that can
enhance the absorption. c. Using a random texturation, LM can results from
scattering, possibly Lambertian, that can also enhance the absorption. d.
Using a periodic or strongly correlated patterned, in addition to AR and Fabry
Perot effects, some Light Trapping can occur, drastically enhancing the
absorption especially at band edge of perovskite.

Let us now summarize the various kinds of PE, mostly at the wavelength scale,
that can be envisaged to improve LM in both single-junction and 2T multi-
junction PPSC, and their impact on geometrical requirements in the PPSC. They
are sketched in Figure 3. Most of the strategies derive from those already
developed for other kinds of direct or even indirect band gap material used in
thin film Massiot  _et al._ (2020).

In this frame, LM results from various light-structure interaction mechanisms.
These mechanisms occur mainly either in the vertical direction (i.e.
orthogonal to the stack) or in the directions of the patterns. Depending on
the scale of the structure with respect to the wavelength, three main models
can be used: first, at sub-wavelength scales, an effective medium; then, at
wavelength scale, wave optics, or, at larger scales, geometrical optics.

###  III.1 Photonic engineering in multilayer unpatterned stack

It is well known that geometrical optics are mainly suitable when the
dimensions are far larger than the wavelength. Thus, it cannot accurately
describe the interference effects that occur within the thickness of the
numerous thin films that compose the cell. In contrast, the optical indices of
medium textured at subwavelength scales can be homogenized in a approximated
way using Effective Medium Theories such as the Bruggeman model. Then,
descriptions involving wave optics are the most rigorous, and wave optics
mechanisms might also be the most promising ones.

Let us first briefly recall the two basic mechanisms able to enhance light
harvesting in a flat stack, i.e. without the assistance of any patterning. In
a rather high index, highly absorbing material, a properly designed stack
enables to drastically reduce the impedance mismatch with the surrounding
environment, whereas in a finite thickness absorbing layer, especially with a
low absorption, it is rather a Fabry Perot like approach that can be used,
leading to highly enhanced but narrow band absorption.

Within a typical PPSC stack, having a metallic contact that can act has a back
mirror, these two approaches can be invoked. Schematically, at short
wavelength, the large extinction coefficient of the perovskite enables the
single pass absorption in the perovskite film, but AR effect is required. At
larger wavelengths close to band gap, where the absorption drops, multiple
passes are required, the second approach is well suited (Figure 3 b.).

However, the exact nature of the layers, and thus their optical indices and
thicknesses are restricted by other constraints such as energy band levels,
electrical resistance, risk of shunting, or even diffusion barriers for some
species. Consequently, parasitic optical absorption has to be carefully
studied. Moreover, broadband enhancement of the absorption under normal
incidence is unlikely to be obtained using such simple architectures.

###  III.2 Photonic engineering in multilayer, patterned stack

In addition to the continuum of propagative waves, light can be confined into
the discrete set of in-plane, transverse guided modes existing in the stack.
It is noticeable that the in-plane wave vector of any of these guided modes is
larger than those of the free space modes. Thus these modes cannot be simply
coupled from the free space in a perfectly flat stack, without any periodic or
aperiodic corrugation. Then, the various kinds of in-plane structurations -
having dimensions from the wavelength scale up to two orders larger - all
induce diffraction. Whereas the impinging light lies around the normal
incidence and has thus negligible in-plane wave vector $k_{in\parallel}$, the
light inside the device is diffracted. Indeed, the electromagnetic field can
be expanded over a set of plane waves thanks to the in-plane spatial
frequencies induced in the medium by the structuration. The reflected waves
also experience diffraction accordingly. Considering the various kinds of
structurations and thus of the resulting spatial frequencies, three main cases
can be envisaged:

  1. i)

A random structuration tends to induce isotropic diffraction, so Lambertian
scattering. Moreover, this scattering does not depend on the wavelength within
the absorbed spectral range. It can typically lead to broadband AR effect
(Figure 3 c.). However the absorption enhancement remains limited. Indeed,
given the already large absorption (except at the band edge) of the
perovskite, and the back reflection induced by the metallic contact, it will
be at most of the order of $4n^{2}$ with an index lower than other materials,
in particular inorganic ones Yablonovitch and Cody (1982).

  2. ii)

A correlated disordered structuration that enhances a specific set of spatial
frequencies, leads to a spectral dependent absorption enhancement. A careful
choice of the sizes of the patterns is required to enhance the absorption in
the desired spectral range Vynck  _et al._ (2012).

  3. iii)

As a particular case from the previous case, periodic, so-called photonic
crystal (PC) structurations can be envisaged. Most of them consists in a
square or triangular lattice a simple pattern such as pillars or holes with a
vertical profile, or smoother pyramidal, or even parabolic profiles. They
mainly lead to discrete modal properties. A few complex patterns arranged in a
periodic way have been explored Martins  _et al._ (2013); Oskooi  _et al._
(2014); Bozzola  _et al._ (2014); Ding  _et al._ (2016); Li  _et al._ (2020b),
enabling an absorption enhancement in a targeted spectral range thanks to a
larger local density of modes.

In any case, the diffraction efficiency, i.e. the amount of light that is
effectively diffracted, and that is thus intended to be absorbed, strongly
depends on the pattern of the structuration, including the optical index
contrast, as well as on the diffraction order $p$. Whereas, as already
mentioned, for scattering, the diffraction efficiency hardly depends on the
direction, for diffraction by periodic structures, it is generally larger for
the first diffraction order.

In this frame, it is noticeable that a LT phenomenon rigorously occurs when
the impinging light is coupled thanks to, at least, strongly correlated
structure, ideally periodic patterns, having respectively a correlation length
or a period $\Lambda$, within at least one guided mode of the stack (Figure 3
d.). According to a perturbation approach Camarillo Abad  _et al._ (2020), in
a 1D case for sake of illustration without losing generality, a phase matching
condition can be written under the form:

$\beta_{m}=k_{in\parallel}+p\frac{2\pi}{\Lambda}$ (1)

Where $\beta_{m}$ is the in-plane wave number of the $m^{th}$ guided mode of
the stack. Due to the time reversal symmetry, the light coupled into the
guided mode, if not fully absorbed, can be decoupled after a certain length.
These modes are thus so-called "quasi-guided mode" (QGM).

Such modes can drastically enhance the light path inside the waveguide layer.
Thus, if the in-coupled QGM is mainly confined in the perovskite layer rather
than in the surrounding layers, the useful absorption is enhanced. Moreover,
it also depends on the absorption; more precisely, the absorption is optimized
at critical coupling, i.e. when an equilibrium is reached between diffraction
efficiency and absorption. Consequently, the spectral bandwidth of the LT is
set since the quality factor of the mode at the optimal absorption is of the
order of $n/2\kappa$, $n,\kappa$ being respectively the real and imaginary
parts of the complex optical index Park  _et al._ (2009).

Such a resonant LT appears promising to enhance the low absorption close to
the band gap of the perovskite, provided it is not reduced elsewhere at
shorter wavelengths. More precisely, within the width of the resonance, the
absorption can be significantly larger than the previously mentioned broadband
limit Yu  _et al._ (2010). It could also be used in tandem PPSC mainly to
optimize the $J_{sc}$, especially provided LT occurs in a guided mode
specifically confined in one of the perovskite layers. As already discussed,
for the other spatial frequencies that do not lead to LT, patterning can still
result in a rather broadband AR effect, in addition to the one resulting from
the design of the stack. Further analysis of all these effects can be found
elsewhere Zanotto  _et al._ (2010); Brongersma  _et al._ (2014).

In addition to these PE approaches that contribute at first order to enhance
the absorption and so the $J_{sc}$, other strategies could be used to enhance
the $V_{oc}$, using a PR mechanism. In a simplified way, PR is supposed to
induce at the $V_{oc}$ operating point a high density of photons into the
absorbing materials, and so to give a chance to electrons-holes pairs to
recombine radiatively rather than in a non radiative way. Typically, this
requires at first order to inhibit the radiation, and consequently the
absorption, in the luminescence spectral domain of the semiconductor. Thus,
enhancing simultaneously the $J_{sc}$ using LT and the $V_{oc}$ using PR is a
compromise Sandhu  _et al._ (2013).

In any case, it has to be emphasized that all these effects should ideally
target the perovskite layer only and not enhance the parasitic absorption in
the other layers.

###  III.3 Main geometrical requirements on patterns for Light Trapping

As mentioned previously, 2D in-plane structurations are able to diffract the
impinging sunlight under a quasi-normal incidence into the large number of
modes of the PPSC, possibly including the guided modes. Indeed, the
architectures of both single and multi-junction PPSC reviewed in the section
II.1 can be schematized as one or several sub-micron thick high index
perovskite layers lying between two lower indices ETL and HTL and possibly
also lower index layers in the multi-junction case. The whole is lying on
medium index TCO on a low index glass substrate and coated on top with metal.
Such structures exhibit several guided modes in both TE and TM polarization
states. Let us first make the assumption of weak corrugation that doesn’t
change significantly the effective indices of the stack of active layers lying
on a substrate. The largest effective index modes - typically 2.2 - are mainly
confined in the perovskite, whereas the other modes, whose effective indices
are below 1.9, can be confined in the whole stack of layers. In this frame,
for normal incident light ($k_{in//}=0$) and for the highest efficient
coupling at order $p=1$, equation (1) can be rewritten under the form:

$\beta_{m}=\frac{2\pi N_{eff,m}}{\lambda}=\frac{2\pi}{\Lambda}$ (2)

where $\Lambda$ is the period, and could even be a characteristic length in
correlated patterns. This 1D case can easily be extended to 2D patterns.

Thus, to be able to efficiently couple into the fundamental guided mode, which
is mainly confined into the perovskite, at wavelengths corresponding to its
band edge, e.g. 750 $nm$, $\Lambda$ should be around 340 $nm$, whereas it
should be larger than 400 $nm$ for higher order guided modes. This results in
a pseudo-guided mode, as discussed at the end of section III.2. At shorter
wavelengths, such structurations can also couple light into the various low
effective index modes (including Fabry Perot like modes) of the stack, that
may also result in a broadband AR effect. This typically implies that all the
micron scale or even larger patterns in any of the layers of the stack are
rather unlikely to couple the impinging light into guided modes in the low
absorption domain, where such a coupling is of main interest. Such patterns
only induce coupling into the Fabry Perot like modes of the stack and/or
diffraction at high orders with a limited efficiency.

##  IV Examples of reported light management strategies in PPSC

We report here on some studies to illustrate concrete LM strategies. For any
case, the chosen perovskite material and its thickness are of primary
importance. First of all, planar PPSC as envisaged in the section II.1 for
their architecture and in section III.1 for the corresponding PE are
discussed, putting mainly into evidence the importance of the thicknesses on
LM. Then, results on in-plane patterned PPSC are reviewed, with focuses on the
involved PE.

###  IV.1 Optimization of the thicknesses in planar PPSC

Most studies related to thickness optimization only focus on that of the
perovskite layer, and report mainly on the derived useful absorption, as
detailed just below. However, the whole stack has been considered in a few
papers, with a view to investigate the parasitic absorption, as discussed
later.

####  IV.1.1 Choice of the perovskite thickness in single junction PPSC

As already mentioned, the optimal thickness of the sole perovskite layer is
chosen as a compromise between maximizing optical absorption and keeping low
bulk recombination of the carriers. In addition to the analysis of a few
seminal cases of single junction PPSC, we present a synthesis of several
studies, mainly experimental, that report on the effect of the perovskite
layer thickness of planar single junction PPSC, first and mainly made of MAPI,
then of other lead-halide perovskites. The detailed performances of the
representative cells are summarized in Table 1 and, when available, the EQE
are plotted in Figure 4 in a spectral range limited from 400 to 800 $nm$ for
sake of simple comparison.

##### MAPI

One of the first high efficiency planar PPSC Liu  _et al._ (2013) which is
made of an about 300 $nm$ thick MAPICl layer exhibited a $J_{sc}$ of 21.5
$mA.cm^{-2}$. In this seminal paper, M. Liu et al. already mentioned the
possibility of an optimum thickness due to the balance between absorption and
recombination. It was shown later by Y. Da et al. Da  _et al._ (2018) that
this cell suffered from optical losses that represented 11.3% of the energy
losses, and especially more than 20% of the achievable $J_{sc}$. This was
mainly related to reflectance and parasitic absorption in the FTO. This thus
emphasizes the importance of minimizing these effects by a careful design of
the stack.

Figure 4: EQE of various PPSC also reported in Table 1 Liu and Kelly (2014);
Momblona  _et al._ (2014); Liu  _et al._ (2014, 2019); Correa-Baena  _et al._
(2016); Jeong  _et al._ (2021); Ball  _et al._ (2015); Lin  _et al._ (2015).

Then, the first high-efficiency PPSC, solution processed at room temperature,
proposed by D. Liu et al. Liu and Kelly (2014) was made of an about 300 $nm$
thick MAPI layer. A $J_{sc}$ of 20.4 $mA.cm^{-2}$ was obtained, so about 5
$mA.cm^{-2}$ below the maximum achievable $J_{sc}$ given a typical $E_{g}$ of
MAPI. The EQE (see Figure 4) was rather flat and remained lower than 0.8. The
deficit could be due to both optical losses (too low absorption in the
perovskite) and electrical losses (too high recombination).

C. Momblona et al. Momblona  _et al._ (2014) conducted an experimental study
on PPSC made of MAPI deposited by thermal evaporation, especially by
increasing its thickness from 160 to 900 $nm$. If the $J_{sc}$ increased
rather monotonously with the perovskite layer thickness, it never exceeded
20.4 $mA.cm^{-2}$. The main parameter that decreased with increasing
perovskite layer thickness was the fill factor. As a result, the cell owning
optimal performance was obtained for a perovskite layer thickness around 300
$nm$. More precisely, whereas the $J_{sc}$ slightly increases, both the
$V_{oc}$ and the $FF$ significantly decreased at thicknesses larger than 300
$nm$. At these early ages of PPSC, the corresponding optimal PCE was 12.7%,
with a $J_{sc}$ of 18.8 $mA.cm^{-2}$. Then, leading further investigations on
the $J_{sc}$ deficit, it could be derived from the EQE (see Figure 4) that
optical losses in almost 300 $nm$ thick perovskite were important at both
short wavelengths and from 600 to 700 $nm$, compared to the 900 $nm$ thick
perovskite layer that fully absorbs. The enhanced absorption in the range 700
to 750 $nm$ is likely due to a cavity effect. Thus, even if such a thickness
of about 300 $nm$ appears to be optimal under the PCE point of view, it
remains too thin to avoid optical losses. In addition, even with such thin
absorbing layers, such cells still suffers from electrical losses.

In a similar way, D. Liu et al. Liu  _et al._ (2014) also then studied PPSC
consisting in a ITO/ZnO/MAPI/P3HT/Ag stack, in which MAPI was obtained using
thermal evaporation or solution processing. Again, in spite of a larger
absorption and limited short-circuit for thicker films, a limited diffusion
length led to an optimum thickness of the MAPI layer of about 330 $nm$ for the
solution processed PPSC. It is noticeable that the corresponding EQE (see
Figure 4) was rather low and flat. This could be related to scattering due to
sub-micron crystallinity that mitigates cavity effects at larger wavelengths.

More recently, Y. Liu et al. Liu  _et al._ (2019), still using MAPI, similarly
concluded that there were more power loss in PPSC when the thickness of the
MAPI layer was either less or more than about 300 $nm$. They noticed that the
grain size increased with the thickness, in favor of a larger thickness.
However, it was noticeable that, at wavelengths larger than 650 $nm$, the EQE
of the 303 $nm$ thick MAPI PPSC (see Figure 4) was lower than the one of the
564 $nm$ thick MAPI PPSC. This is related to too low absorption in this
spectral range.

##### Other lead-halide perovskites

As an alternative, J.P. Correa-Baena et al. Correa-Baena  _et al._ (2016) used
multi-cations perovskites materials; these were recently under the spotlight,
especially considering their better stability Jeon  _et al._ (2015b). Using
FA0.83MA0.17Pb(I0.83Br0.17)3 together with mesoporous TiO2, the thickness that
maximises the PCE was found to be at least 480 $nm$. The corresponding
$J_{sc}$ was almost 24.4 $mA.cm^{-2}$. Considering the band gap of this
material, reported to be about 1.63 eV Jacobsson  _et al._ (2016), this is
already the maximum achievable value. The EQE (see Figure 4) is noticeably
high (also thanks to the an IQE of almost 100%) and flat at large wavelength.
This could be related to the mesoporous TiO2 that induces some scattering
effect.

Unlike the previous papers, M. Rai et al. Rai  _et al._ (2020) focused on the
$V_{oc}$ deficit related to non-radiative recombination as a function of the
thickness, expressed in terms of molar concentration of the precursor
solution. This time, the studied perovskite was Cs0.2FA0.8Pb(I0.85Br0.15)3,
whose band gap has been found to be 1.62 $eV$, so close to the one of MAPI.
They noticed as well that the $V_{oc}$ decreases with the thickness, whereas
the $J_{sc}$ increases. As for the previously mentioned paper, the grain size
increases with the thickness; the molar concentration of the precursor
solution that maximizes PCE correspond to thickness of the perovskite layer of
about 400 $nm$, so 30% larger than the one usually obtained using MAPI. This
might be due to a larger diffusion length of the photocarriers for this
perovskite.

K.B. Nine et al. Nine  _et al._ (2020) simulated optically and electrically
the effect of the thickness of several FACsPI layers, leading to a larger
optimal thickness, of almost 600 $nm$, so twice the thickness usually found
using MAPI. Indeed, better electrical properties of this perovskite material,
especially carrier mobility, allow such a thicker layer.

##### Synthesis

Material | Thickness ($nm$) | $J_{sc}(mA.cm^{-2})$ | $V_{oc}(V)$ | $FF$ | PCE (%) | Ref
---|---|---|---|---|---|---
MAPICl | 330 | 21.5 | 1.07 | 0.67 | 15.4 | Liu  _et al._ (2013)
MAPI | 300 | 20.4 | 1.03 | 0.749 | 15.7 | Liu and Kelly (2014)
MAPI | 285 | 18.8 | 1.07 | 0.63 | 12.7 | Momblona  _et al._ (2014)
MAPI (solution process) | 330 | 17 | 0.94 | 0.62 | 11.8 | Liu  _et al._ (2014)
MAPI | 303 | 21.3 | 1.07 | 0.715 | 18.4 | Liu  _et al._ (2019)
FAMAPBrI | 480 | 24 | 1.14 | 0.75 | 20.8 | Correa-Baena  _et al._ (2016)
FAPI | 600 | 26.35 | 1.189 | 0.817 | 25.59 | Jeong  _et al._ (2021)
MAPI | 492 | 21.56 | nr | nr | nr | Ball  _et al._ (2015)
MAPI | 350 | 21.9 | 1.05 | 0.72 | 16.5 | Lin  _et al._ (2015)
Table 1: Experimental performances of planar PPSC with optimized perovskite
thickness (nr: not reported)

It is noticeable that these studies tend to an optimal thickness of about 300
$nm$ when using MAPI. This thickness could lead to state-of-the-art PCE of J.
Li et al. Li  _et al._ (2018), with noticeably high $J_{sc}$ of 24.1
$mA.cm^{-2}$, mainly thanks to an ETL including graphdiyne, which improves
electrical properties. Then, FAPI is the perovskite used for the best up-to-
date single junction PPSC Jeong  _et al._ (2021) which performance were
mentioned in the introduction, with a thickness of about 600 $nm$, even if,
among numerous refinements of the architecture, the rough, about 50 $nm$ thick
TiO2 layer already may help to trap the light.

It appears that choosing a thickness smaller than the one maximizing the
$J_{sc}$ could help to keep low resistance, so high FF and PCE. Then, with
such a $J_{sc}$ around 90% of the maximum achievable $J_{sc}$, LM can optimize
the absorption in the perovskite, especially at photon energies close to the
band gap, where the absorption starts to decrease.

In any case, it is of high interest to understand the reasons why large
variations of the internal perovskite absorption spectrum have been obtained
for similar thicknesses of the same perovskite material (MAPI), as can be
noticed through the cases 2 to 5 of Table 1 and the corresponding EQE plotted
in Figure 4. They might be due to differences in the surroundings layers
(thicknesses, indices) or in the perovskite itself, like microcrystallinity
that can induce scattering. These phenomena will be described in the
following, starting with the analysis of the effect of the surroundings layers
in a unpatterned stack.

####  IV.1.2 Single junction unpatterned PPSC stack analysis

If there can be an optimal thickness for the perovskite layer, the effect of
the other layers, especially on the absorption and reflectance, should also be
analyzed. Here, we review a selected set of publications focusing on this
issue.

J.M. Ball et al. Ball  _et al._ (2015) studied planar PPSC using an optical
model based on the transfer-matrix formalism with experimentally determined
complex refractive index data. They focused on a typical stack made of
FTO/TiO2/MAPI/SPIRO-OMeTAD/Au. Under the assumption of $E_{g}=1.56eV$, the
detailed analysis revealed that for a calculated $J_{sc}$ of 21.56
$mA.cm^{-2}$, parasitic absorption induces a lack of about 1.72 $mA.cm^{-2}$,
and reflection losses corresponding to a $J_{sc}$ decrease of 1.36
$mA.cm^{-2}$, so a total of more than 10% of the collected current. On the
other hand, the IQE losses can be estimated to another 10%. Moreover, most of
the optical losses take place in the 420 $nm$ thick FTO. This underlines the
importance of taking care of all the layers of the stack.

(a) Optical-field distribution for four wavelengths: for $\lambda<500nm$ the
optical-field distribution follows the Beer–Lambert law and no optical field
reaches to the back electrode as a result of the high absorption coefficient.
In such cases the absorption is saturated and no optical interference occurs.
For $\lambda>500nm$ the optical field is governed by low-finesse cavity
interference.

(b) EQE spectra of devices with different MAPI-layer thicknesses. For
$\lambda<500nm$ there is minimal influence of the film thickness, but for
$\lambda<500nm$ the EQE is strongly thickness dependent because of the optical
interference (Fabry Perot like modes)

Figure 5: PPSC properties as a function of the MAPI layer thickness Lin  _et
al._ (2015). Reproduced with permission

Q. Lin et al. Lin  _et al._ (2015) deeply investigated the properties of PPSC
made of MAPI. Thanks to their measurement of the optical indices of the MAPI
and of the other materials, they simulated the properties of PPSC, assuming an
IQE of 100%. They identified the two absorption regimes of a flat PPSC,
previously discussed in section III.1, namely the single pass and the cavity
regime. Thus, thanks to optical cavity effects at wavelengths larger than 500
$nm$, the noticeably high EQE of their cell (see Figure 4) appeared to be
optimized for a thickness of about 370 $nm$ (see Figure 5). The experimental
data noticeably confirmed the simulation study, since a cell having a 350 $nm$
thick MAPI layer led to the best properties, $J_{sc}=21.9mA.cm^{-2}$,
$V_{oc}=1.05V$, $FF=0.72$ and $PCE=16.5\%$. Given the resulting limited
thickness of the MAPI, the obtained $J_{sc}$ was 87% of the maximum achievable
value. These remarkable results were also obtained thanks to the optimization
of the p and n type interlayers that were used to optimize the
heterojunctions. It was especially shown that the PCDTBT p type interlayer has
to be as thin as possible ( $\sim 5nm$), because of its absorption, even if it
enabled a nice crystallinity of the MAPI. Otherwise, given the almost 100%
IQE, most of the 4 to 5 $mA.cm^{-2}$ decrease of the photocurrent (depending
on the exact value of the band gap energy of the considered MAPI) corresponds
to optical losses, which could be either parasitic absorption or reflectance.

M. Van Eerden et al. van Eerden  _et al._ (2017) analyzed the optical losses
of 350 $nm$ thick MAPI layers PPSC. Within the considered cell, most of the
losses are related to reflectance (equivalent to 4.2 $mA.cm^{-2}$), whereas
the parasitic absorption remains limited to the half, and despite a
subwavelength roughness (typically from 10 to 50 nm RMS) at both ITO/TiO2 and
MAPI/SPIRO-OMeTAD interfaces.

####  IV.1.3 Analysis of unpatterned, up-to-date record 2T tandem PPSC

(a) Stack of materials

(b) Simulated $J_{sc}$ as function of wide-band gap and narrow-band gap
perovskite layer thicknesses.

(c) EQE of both subcells.

Figure 6: Best up-to-date 2T tandem, all perovskite PPSC Xiao  _et al._
(2020). Reproduced with permission.

When compared to other tandem architectures, the 2T case leads to possibly
limited optical losses, but such cells require an equilibrium of the current
densities delivered by the two subcells. Among recent results Wang  _et al._
(2021b), K. Xiao et al. Xiao  _et al._ (2020) have deeply investigated 2T
tandem PPSC (see Figure 6(a)) both numerically and experimentally. They have
shown using optical simulations that the reachable $J_{sc}$, which is indeed
the smallest $J_{sc}$ delivered by each of the subcell, strongly depends on
the thicknesses of each perovskite layer (see Figure 6(b)), provided that
thicknesses of the other layers have been set taking into account both
electrical and optical properties (low parasitic absorption). Indeed, it
appears that the top perovskite has to be thick enough to absorb enough light,
but that a too thick layer also absorbs too much light, and finally that the
overall $J_{sc}$ is limited by the bottom sub cell. The thickness of the
perovskite bottom sub cell has to be large enough, more than 1.1 $\mu m$. When
fabricated on a small surface, such a cell exhibits even a slightly larger
$J_{sc}$ than simulated, likely due to the too pessimistic predicted
absorption. Even if this result remains a record at the time of its
publication, its EQE (see Figure 6(c)) reveals some non-idealities. Indeed, in
addition to the relatively low values of the EQE plateaus, there is an overlap
of the absorption domains of the two perovskite materials between 500 and 700
$nm$. This means that some undesirable thermalization still occurs below 700
$nm$ in the bottom sub cell.

###  IV.2 In-plane structuration for light management in single junction PPSC

At this stage, it appears that even the best reported cells exhibit an EQE
that can still be improved using PE at the wavelength scale, even if most of
them already benefit from the roughness of the microcrystalline perovskite.

In this section, we will highlight important criteria that should be met to
ensure a fair demonstration of LM. The report on selected studies is organized
as a function of the various architectures as envisaged in section II.1.

####  IV.2.1 General criteria for fair LM demonstration

The impact of LM on the performance of a solar cell can be demonstrated by
comparing a patterned device with an unpatterned reference. In case of
superficial structurations or with the addition of specific LM layers, the
absorption improvement appears generally obvious compared to the same
unpatterned stack, or, even more favorable, to the unpatterned stack without
the additional flat LM layer. Moreover, integrating LM layer could lead to a
better charge collection, by decreasing the carrier path; the specific LM
effect would then be difficult to distinguish. Therefore, great care should be
taken in the definition of a reference structure or device. In particular, the
volume of perovskite materials should be as close as possible in both
structures. Additionally, the net enhancement of EQE, efficiency or even yield
value can be accurately estimated only if the unpatterned reference has been
optimized first. These conditions should be met in order to discuss LM for
PPSC performance optimization.

####  IV.2.2 Periodic patterning for resonant LT

##### Superficial structuration and light management layers

(a) Schematic of the PPSC.

(b) Simulated absorption spectrum for microlens array of period $~{}700nm$ and
height $800nm$. The absorption of flat solar cell is overlaid for comparison.

Figure 7: PPSC with microlens array on air-glass side Peer  _et al._ (2017).
Reproduced with permission.

A. Peer et al. Peer  _et al._ (2017) simulated planar PPSC with a stack
including a 400 $nm$ thick MAPI layer. The cell lay on top of a 700 $\mu m$
thick glass substrate. A micro lens array was then added on the top face of
the glass substrate, by imprinting a polymer such as polystyrene (see Figure
7(a)). Each micro lens of the triangular lattice had a smooth profile close to
truncated pyramids. For an aspect ratio period/height of the micro lenses
close to 1, and a period of 700 $nm$, optimized gain of 6.3% of the $J_{sc}$
was reported compared to the flat reference, whose EQE noticeably shown dips
likely due to cavity effects in the cell (see Figure 7(b)). Then, according to
the EQE of the cell coupled to the micro lens array, the overall resulting
broadband enhancement of the absorption is due at lower wavelength to AR
effect, and to LT at the band edge. The optimal period of the pattern is
rather large, in agreement with a low effective index of the coupled guided
mode. It means that LT occurs in the mode that is hardly guided in the
perovskite, but that has a nonnegligible overlap with the pattern. This
overlap remains anyway limited, as revealed by the high quality factor of the
resonance. Therefore, the absorption reaches almost 1, so is close to critical
coupling conditions.

(a) SEM (left) and AFM (right) images of the spin-coated layer (top) of
grating patterned layer (middle) and moth-eye patterned layer (bottom).

(b) External quantum efficiency (EQE) spectra of PPSC, and relative
enhancement obtained by dividing the spectra of grating and moth-eye patterned
devices by that of flat one.

Figure 8: PPSC with bio inspired PCBM HTL Wei  _et al._ (2017). Reproduced
with permission.

J. Wei et al. Wei  _et al._ (2017) fabricated bioinspired back electrodes by
imprinting the HTL made of PCBM before the conformal deposition of the
Bphen/Ag back contact. Patterns were either a periodic sinusoidal 1D grating
or a uniform 2D moth-eye structure (see Figure 8(a)), with a typical pitch of
about 600 $nm$. It has then been experimentally shown (see Figure 8(b)) that
the EQE of the 240 $nm$ thick MAPICl PPSC in increased by 10% up to 40% at the
band edge, mainly thanks to the 2D moth eye pattern. The overall $J_{sc}$
increased by 14.3%, mainly due to absorption increase, but a lower series
resistance was also measured for patterned cells, which led to a slight
increase of the IQE. As shown by there FDTD simulations of periodic
structures, this is due light diffraction into the guided modes, rather than
plasmonic effects. It can be confirmed by Fourier Analysis of the top view of
the moth eye pattern that a significant part of the spatial frequencies lies
in the optimal range for LT (see Figure SI-S1).

Figure 9: EQE of the patterned PPSC (solid line) and of their planar
references (dashed line) (left) and EQE enhancement ((right) of reported
simulated $J_{sc}$ enhancements of Table 2, mainly resulting from Light
Management, for MAPI single junction PPSC, with any kind of in-plane pattern
(additional layers, structuration of the perovskite or conformal perovskite)
and whatever the main identified photonic concept Peer  _et al._ (2017); Wei
_et al._ (2017); Kim  _et al._ (2021); Schmager  _et al._ (2019a); Du  _et
al._ (2016); Hossain  _et al._ (2020a); Qarony  _et al._ (2018)

H. Kim et al. Kim  _et al._ (2021) simulated nanosphere arrays of TiO2
conformally coated with silica on a standard MAPI PPSC stack, with a MAPI
thickness limited to 100 $nm$. Provided suitable spacing and nanosphere
diameter, the $J_{sc}$ could jump from 14 to 18.7 $mA.cm^{-2}$. Authors
invoked the Mie scattering effect of the array to explain the enhancement, but
it appears that the FEM simulation has likely been done using periodic
boundary conditions, and thus that a LT effect occurs at several wavelengths,
such as 705 or 790 $nm$ for those at the band edge of MAPI. This is in line
with the larger EQE enhancement close to the band edge, as can be seen in
Figure 9. It remains that the overall $J_{sc}$ is anyway limited due to the
unusually low thickness.

##### Structuration of the perovskite

(a) Layer stack of the simulated patterned PPSC, along with associated layer
thicknesses. Three configurations of the active layer are simulated (1) a
planar reference, (2) a cylindrical indention into the perovskite layer of
variable depth; and (3) a hole geometry which corresponds to the maximum
cylindrical indention.

(b) Absorptance in the perovskite layer of the hole pattern and the 120 $nm$
deep indentation compared to the corresponding planar reference and the
theoretical Yablonovitch limit for two different initial perovskite layer
thicknesses of (left) 200 $nm$ and (middle) 300 $nm$. $J_{sc}$ for all data
are compared (right). The displayed nanophotonic patterns employ a geometrical
$ff=0.4$ and a period of 380 $nm$.

Figure 10: Various 2D in plane patterns of the MAPI layer in a PPSC Schmager
_et al._ (2019a). Reproduced with permission

R. Schmager et al. Schmager  _et al._ (2019a) simulated a classical MAPI PPSC
(see Figure 10(a)). Compared to the planar configuration, etching of the MAPI
layer was envisaged, thanks to a square lattice of holes, filled with Spiro-
OMeTAD, with various etching depths. To avoid any short circuit, an optimized
partial etching of 120 $nm$ led to a 5.6% increase of the $J_{sc}$, compared
to the initially flat 300 $nm$ thick MAPI layer, provided an equivalent volume
of perovskite. It results from a LT at the band edge, but without any sharp
resonance, as can be observed on the spectral response of other patterns (see
Figure 10(b)).

Figure 11: External quantum efficiency and reflectance spectra of the planar
and nanoimprinted perovskite solar cells. The nanoimprinted perovskite layer
has a period of 480 $nm$. The nanoimprinted perovskite solar cell shows
enhanced absorption and current generation close to the band gap. For energies
below the band gap (wavelength larger than 790 $nm$) discrete peaks are
visible in the reflectance measurement. Schmager  _et al._ (2019b). Reproduced
with permission. Figure 12: EQE of the patterned PPSC (solid line) and of
their planar references (dashed line) (left) and EQE enhancement ((right) of
Table 3, mainly resulting from Light Management, for single junction PPSC,
with any kind of in-plane pattern (additional layers, structuration of the
perovskite or conformal perovskite) and whatever the main identified photonic
concept Schmager  _et al._ (2019b); Tockhorn  _et al._ (2020); Paetzold  _et
al._ (2015); Dudem  _et al._ (2016); Jošt  _et al._ (2017); Zhang  _et al._
(2018); Pascoe  _et al._ (2016)

The same group Schmager  _et al._ (2019b) also realized nanoimprinted PPSC
made of Cs0.1(FA0.83MA0.17)0.9Pb(I0.83Br0.17)3. Their experimental results
show a relative improvement of 2% of the PCE compared to the planar reference
for the complete, gold coated solar cell. This was obtained thanks to an
increase of the $J_{sc}$ from 19.1 to 19.4 $mA.cm^{-2}$ and noticeably
identical $V_{oc}$ and $FF$.The EQE of the patterned cell was improved at
wavelengths larger than 680 $nm$. A coupling to a quasi-guided mode is
observed in the perovskite band edge (see Figure 11), leading to a significant
EQE enhancement (see Figure 12).

##### Textured substrate

(a) Three-dimensional perspective and two-dimensional side view of the
inverted vertical cone perovskite unit cell.

(b) Simulated absorption and reflection spectra of the optimized (period
$a=600nm$, radius $R=380nm$) inverted cone PC and planar PPSC. The MAPI
thickness of the planar cell is 180 $nm$, corresponding to the “equivalent
bulk thickness” of MAPI in the inverted-cone PC device.

Figure 13: MAPI PPSC patterned with inverted conesDu  _et al._ (2016).
Reproduced with permission.

Q. G. Du et al. Du  _et al._ (2016) simulated a strongly corrugated MAPI
layer. It was obtained thanks to cones realized in ITO coated glass substrate
with an intermediate conformal PEDOT layer lying on ITO (see Figure 13(a)).
Thanks to optimized period and radius, a 15-17% improvement was expected in
the PCE compared to the best flat cells of equivalent volume (see Figure
13(b)). The optimal period of about 400 $nm$ fits well the required period for
LT into the guided modes, given the fact that the effective index of the modes
is lower than our examples, due to the strong corrugation of the MAPI filled
with low index material. AR and resonant light trapping are thus
simultaneously obtained.

####  IV.2.3 Periodic structures for non resonant LM: anti-reflection and
scattering

##### Superficial structuration and light management layers

M. I. Hossain et al. Hossain  _et al._ (2020a) proposed to coat the MAPBI like
perovskite, having a thickness from 100 to 400 $nm$, with zinc oxide. More
precisely, a first thick, lightly doped ZnO ensured the top contact, then it
could be covered by texturations patterned either as pyramids or as non-
resonant metasurfaces. The period was set to 800 $nm$. Accordingly to the
mechanism typically occurring using such a period, no sharp resonance enhances
the absorption at the band-edge, only an AR like effect occurs. Both kinds of
patterns lead to simulated equivalent $J_{sc}$ enhancements of about 3
$mA.cm^{-2}$ whatever the thickness of the perovskite between 100 and 400
$nm$, so a relative enhancement roughly from 25% to 12% within the same range,
but only 10% for the thickness 300 $nm$.

P. Tockhorn et al. Tockhorn  _et al._ (2020) simulated and fabricated a 550
$nm$ thick PPSC made of a mixed cation, mixed halide
Cs0.05(FA0.83MA0.17)0.95Pb(I0.83Br0.17) perovskite material on various
periodically patterned glass substrates, in p-i-n configuration. The top side
of the glass substrates was coated with low index, NaF thin film. Compared to
the planar reference, patterned structures could exhibit a $J_{sc}$ up to 1
$mA/cm^{2}$ larger. This results from a broadband enhancement of the EQE,
including at the band edge, mainly attributed to AR effect, as verified by
simulations where the volume of perovskite has been kept constant. This is in
line with the flat EQE enhancement, as can be seen in Figure 12 The resulting
PCE reaches 19.7%, 1% absolute above the one of the planar reference.

##### Structuration of the perovskite

U. W. Paetzold et al. Paetzold  _et al._ (2015) proposed to pattern the front
ITO electrode with a square lattice of pillars. The ETL was then corrugated,
as well as the about 320 $nm$ thick MAPICl layer that planarized the stack,
before the HTL and Al back contact deposition. They noticed increased
absorption and EQE as the lattice period decreased, up to the smallest value
envisaged, i.e. 500 $nm$. This led to an increase ofthe $J_{sc}$ by 5%.
Broadband enhancements were observed, at wavelengths shorter than the band
edge of the material, as well as a limited LT at the band edge, according to
the EQE enhancement plotted in Figure 12. This is likely due to the lack of
large enough spatial frequencies to couple the quasi-guided modes of the
structure, given a period slightly above the optimal range. In addition, there
might be a slight change in the volume of perovskite between the patterned
cells compared to the flat reference.

##### Textured substrate

W. Qarony et al. Qarony  _et al._ (2018) calculated the EQE for three various
configurations of PPSC using the same volume of perovskite and including moth
eye periodical patterns with a period of about 150 $nm$, typically leading to
scattering. The fist one had only a pattern at the top air / ZnO interface
(thus is more comparable to former structures), but the two others considered
a conformally patterned stack, either on a patterned Al substrate, or on a
patterned NiO/ITO layer. The patterned Al substrate clearly led to a lower EQE
due to additional parasitic absorption. Moreover, a slightly lower $J_{sc}$
was obtained with the conformal stack on the flat Al substrate compared to the
top patterned stack. Even this last case exhibits a low EQE enhancement (see
Figure 12).

####  IV.2.4 Aperiodic patterning for LM

##### Superficial structuration and light management layers

B. Dudem et al. Dudem  _et al._ (2016) proposed a multifunctional inverted
micro-structured pyramidal Polydimethylsiloxane (PDMS) AR layer for enhancing
the device efficiency, through AR and self cleaning. The MAPI layer was 340
$nm$ thick. The sizes of the pyramid were in the range from 1 to 10 $\mu m$,
therefore too large for efficient diffraction. Compared to flat PDMS, the
$J_{sc}$ increases by 0.38 $mA.cm^{-2}$ up to 21.25 $mA.cm^{-2}$,
corresponding to a limited AR effect, as confirmed by the flat EQE enhancement
(see Figure 12).

Rather similarly, M. Jošt et al. Jošt  _et al._ (2017) fabricated a so-called
light management foil on the glass substrate of a planar 270 $nm$ thick MAPI
cell, which led to a limited EQE enhancement (see Figure 12). At first, the
thick glass substrate prevents from a strong overlap between any guided mode
into the perovskite and the foil. Moreover, whatever the glass thickness, the
lack of LT is also related to the far too low spatial frequencies resulting
from the texturation. Indeed, thanks to available top view of the foil, a
Fourier transform analysis reveals (see Figure SI-S2) that most of Fourier’s
components lie below 7 $\mu m^{-1}$, whereas the optimum $\beta_{m}$ should be
in the range 18 - 34 $\mu m^{-1}$ at first order of diffraction, and even
larger for other orders.

On the other side of stack, H. Zhang et al. Zhang  _et al._ (2018) measured
and modelled the impact of the roughness of the back mirror on the back
scattering of MAPICl PPSC. The reference was made of a perovskite layer with a
significant roughness after crystallization, coated with a thick Spiro layer
that planarizes, then a flat gold mirror. With a thinner HTL, the gold mirror
replicated the roughness of the perovskite layer leading to light back
scattering. The PCE increased from 19.3% to 19.8%, mainly thanks to the
$J_{sc}$ increase from 22.7 to 23.6 $mA.cm^{2}$. It is noticeable that the
obtained EQE enhancement is the largest at the band edge. This could be
related to the grain size that appears to be in the range 200 - 500 $nm$, so
including the optimal range for efficient LT.

##### Structuration of the perovskite

A. R. Pascoe et al. Pascoe  _et al._ (2016) proposed textured MAPI at the
scale of several hundreds of nanometers thanks to gas crystallization. It
enhanced the EQE at wavelengths larger than 550 $nm$ (see Figure 12), thanks
to an induced so-called scattering. Accordingly, the averaged $J_{sc}$
increased from 21.3 for planar references having an about 300 $nm$ thick MAPI
layer to 22.1 $mA.cm^{2}$ for patterned samples of comparable volume. As
previously, the noticeably large EQE enhancement close to the band edge can be
related to the typical grain size of the order of 500 $nm$.

####  IV.2.5 Synthesis

In the various previously described studies reporting on $J_{sc}$ enhancements
thanks to LM, it can be noticed that most of the possible architectures
envisaged in the section II.1 have been considered, using all the PE described
in section III. Tables 2 and 3 summarize the $J_{sc}$ enhancements reported in
the various previously described studies.

From Table 2, focusing on simulations of periodic patterning of MAPI based
single junction PPSC, it comes out that LT can indeed significantly increase
the $J_{sc}$ for very thin perovskite layers. The enhancement is more limited
at a thickness of about 300 $nm$. It remains that an accurate comparison of
the $J_{sc}$ enhancements is not possible, since performance enhancement
strongly depends on the choice of the unpatterned reference, and especially
its optimization in terms of LM as discussed previously. However, the EQE
enhancements (Figure 9) can clearly reach larger values thanks to LT than for
AR.

From Table 3, focusing on experimental results, using various perovskite
materials, the $J_{sc}$ enhancements are of the same order as the simulated
ones, again with the same precautions as above. As for the simulated PPSC, the
EQE enhancements (Figure 12) also reach larger values using patterns at the
scale of the wavelength in the material. Moreover, the periodic case lead to
the largest enhancement.

Table 2: Reported simulated $J_{sc}$ enhancements mainly resulting from Light Management, for MAPI single junction PPSC, with any kind of in-plane pattern (additional layers, structuration of the perovskite or conformal perovskite) and whatever the main identified PE Perov.thick. ($nm$ ) |  $J_{sc}$ enhanc. (%) | Pattern. type | Photonic concept | Ref
---|---|---|---|---
400 | 6.3 | periodic add. patterned layer | Broadband AR and LT | Peer  _et al._ (2017)
120 | 31.7 | periodic add. patterned layer | Broadband AR and LT | Kim  _et al._ (2021)
300 | 5.6 | periodic struct. of the perovskite | Broadband AR and LT | Schmager  _et al._ (2019a)
180 | 17 | periodic conformal perovskite | Broadband AR and LT | Du  _et al._ (2016)
300 | 10 | periodic add. patterned layer | Broadband AR | Hossain  _et al._ (2020a)
300 | 10 | periodic conformal perovskite | Broadband AR | Qarony  _et al._ (2018)
Table 3: Reported experimental $J_{sc}$ enhancements mainly resulting from Light Management, for single junction PPSC, with any kind of in-plane pattern, and whatever the main identified PE Material | Perov.thick. ($nm$) |  $J_{sc}$ enhanc. (%) | Pattern. type | Photonic concept | Ref
---|---|---|---|---|---
MAPI | 240 | 14.3 | periodic add. patterned layer | Broadband AR and LT | Wei  _et al._ (2017)
CsFAMAPBI | 370 | 2 | periodic struct. of the perovskite | Broadband AR and LT | Schmager  _et al._ (2019b)
CsFAMAPBI | 550 | 6.3 | periodic, struct. of the ETL | Broadband AR | Tockhorn  _et al._ (2020)
MAPICl | 320 | 5 | periodic struct. of the perovskite | Broadband AR | Paetzold  _et al._ (2015)
MAPI | 340 | 1.8 | aperiodic add. patterned layer | Broadband AR | Dudem  _et al._ (2016)
MAPI | 270 | 4.8 | aperiodic add. patterned layer | Broadband AR | Jošt  _et al._ (2017)
MAPICl | 300 | 4 | aperiodic add. patterned layer | Broadband AR | Zhang  _et al._ (2018)
MAPI | 300 | 3 | aperiodic substrate corrugation | Broadband AR | Pascoe  _et al._ (2016)

###  IV.3 LM for PR

LM for $V_{oc}$ enhancement thanks to PR in PPSC is discussed in a limited
number of publications.

S. Nanz et al. Nanz  _et al._ (2019) investigated mainly theoretically the
effect of various kinds of LM strategies on the PR, for multilayer stacks that
are part of PPSC, mainly without the HTL and metallic contact. They were thus
able to derive an upper limit $\Delta V_{oc}$ for each case, under the
assumption of pure radiative recombination. According to the summarized
principle reminded previously, the $\Delta V_{oc}$ resulting from PR in a
patterned multilayer was in between the values obtained with the Lambertian
multilayer and the rigorously planar multilayer. Indeed, the Lambertian
multilayer, as it consists in a better absorber, also radiates the
luminescence, whereas, for targeted thicknesses, luminescence can be partly
guided, leading to recycling. Then, the quasi-guided mode of patterned
multilayer led to enhanced PR compared to the Lambertian multilayer, and also
an enhanced $J_{sc}$ compared to the flat multilayer. However, A. Bowman et
al. Bowman  _et al._ (2020) showed that using a more realistic model including
recombination, PR was rather unlikely to occur at maximum peak power even if
the cell only interacts with a limited solid angle. In this context, it thus
appears more promising to increase the absorption and thus the extraction, to
the detriment of recycling.

##  V Simulations and Perspective

As shown in the previous section, a limited number of studies, mainly focused
on $J_{sc}$ enhancements, evidenced a LM effect in PPSC. This synthesis also
illustrates that due to a lack of common references, different kinds of PE can
hardly be compared, so as the most promising architectures for AR and light
trapping. Therefore, we propose to simulate some of the promising patterns, to
be able to compare their performances using typical materials and architecture
of a PPSC.

###  V.1 Methodology

The Rigorous Coupled Wave Analysis Moharam and Gaylord (1981) suits well for
the simulation of the stacks periodically patterned under plane wave
illumination. We have used the $S^{4}$ code Liu and Fan (2012) available in
the Solcore package Alonso-Álvarez  _et al._ (2018). The derived $J_{sc}$ are
obtained using AM1.5G spectrum Gueymard (1995) provided an IQE of 1.

Optical indices used are from S. Manzoor et al. Manzoor  _et al._ (2018) for
MAPI, from K.R. McIntosh et al. McIntosh  _et al._ (2009) for PMMA, and from
J.M. Ball et al. Ball  _et al._ (2015) for all other materials: Sodalime Glass
as a substrate, ITO, TiO2, doped Spiro-OMeTAD (as HTL material), and gold.
Thicknesses are set to 100 $nm$ for top contact, 20 $nm$ for ETL, and 300 $nm$
for Au.

The ETL is supposed to be a thin TiO2 layer, dense and flat, to avoid
scattering, and to limit parasitic absorption compared to other envisaged
organic materials.

###  V.2 Optimization of the key layers thicknesses in various planar single-
junction PPSC

Let us first consider a planar PPSC on an infinitely thick substrate,
illuminated through this substrate under normal incidence (see Figure 14). The
thicknesses of the MAPI layer $th_{MAPI}$ and of the HTL $th_{HTL}$ are
supposed to vary in the ranges 300 - 700 $nm$ and 200 - 400 $nm$ respectively.
As can be seen in Figure SI-S3, the $J_{sc}$ of such a cell increases with
$th_{MAPI}$. Moreover, for a given $th_{MAPI}$, the $th_{HTL}$ has a non-
negligible influence in the $J_{sc}$; e.g. for the smallest MAPI thickness of
the considered range, the $J_{sc}$ can be increased by more than 0.5
$mA.cm^{-2}$, up to about 21.74 $mA.cm^{-2}$.

It remains that the low $J_{sc}$ for the thinnest considered MAPI is due to
lower absorption at long wavelengths, as can be seen in Figure SI- S5, for a
given thickness of HTL of 240 $nm$.

Given the possible identified advantages of using a thinner MAPI layer, mainly
for electrical properties, its thickness will be set to 300 $nm$ in the
following. An AR PMMA layer is then coated on the glass substrate, which
thickness is set to 1 $mm$ (see Figure 14), with a negligible roughness. For
the chosen $th_{MAPI}$, the coupled influence of PMMA thickness, $th_{PMMA}$,
and $th_{HTL}$ is studied. According to Figure SI- S6, a significantly
increased $J_{sc}$, up to about 21.92 $mA.cm^{-2}$, can be obtained for
$th_{PMMA}=360\ nm$ together with $th_{HTL}=250\ nm$. This last structure will
be then used as a planar but optimized reference for fair estimation of the
impact of PE in the following. The corresponding spectrum is drawn in Figure
15.

###  V.3 Introduction of various 2D PC to enhance the current density

Figure 14: Various configurations of single junction PPSC simulated in the
perspective. (a) planer PPSC on an infinite glass substrate, to study the
effect of MAPI and Spiro-OmeTAD thicknesses (see results in Figure SI-S3), (b)
Planar reference having a 300 $nm$ thick MAPI layer, and various PMMA and
Spiro-Ometad thicknesses to maximise absorption (see results in Figure SI-S6),
(c) Cross section of a 2D square lattice of cylindrical holes in MAPI, to
study the effect of the corresponding PC parameters (see results in Figures
SI-S7 and 15), (d) Cross section of a 2D square lattice of cylindrical holes
in PMMA, to study the effect of the corresponding PC parameters (see results
in Figures SI-S8 and 15), (e) Cross section of two 2D distinct square lattices
of cylindrical holes in PMMA and MAPI, to study the effect of PC parameters
(see results in Figures SI-S9 and 15)

As already discussed, to further increase the absorption and thus the
$J_{sc}$, the most efficient strategies should be to couple the impinging
light into the guided modes thanks to properly designed patterns. In the
following, we envisage 2D square lattices of cylindrical patterns. Each
resulting 2D PC owns typically three parameters that can be optimized: i) its
period $P$, ii) its filling fraction ($ff$) which is the ratio between the
hole surface and the period surface, and iii) its thickness $t$. These 2D PC
can be located either (see Figure 14):

  * •

in the MAPI layer, made of holes in the MAPI layer filled with HTL material;
for a fair comparison, the volume of MAPI material is the same as the planar
PPSC, so its total thickness changes. Moreover, $t<th_{MAPI}$ to prevent from
short circuits between HTL and ETL. A $t_{HTL}=250nm$ thick slab of HTL is
kept for planarization;

  * •

in the top PMMA layer, patterned in a PC of air holes, with $t=th_{HTL}$, in
favor of the diffraction efficiency, given the low index of the PMMA;

  * •

simultaneously at the two previous locations, but each PC has its own set of
parameters.

In the following studies, all the PC parameters are scanned over realistic
ranges. The step for $P$ and $t$ is 5 $nm$, whereas only 3 $ff$ have been
envisaged: 0.3, 0.4 and 0.5; these appear to be the most realistic values
compatible with a large area patterning at a reasonable cost.

Figure 15: Simulated absorption spectra for various 1 mm thick glass substrate
PPSC with maximized short circuit current density: the planar PPSC (Figure 14
(b)), the PPSC with a 2D PC into the MAPI (Figure 14 (c)), the PPSC with a 2D
PC in the PMMA (Figure 14 (d)), and the PPSC with both 2D PC (Figure 14 (e)).
Small amplitude, high spectral resolution Fabry Perot resonances take place in
the $1mm$ thick glass substrate.

####  V.3.1 Single junction: 2D PC in the perovskite layer

With respect to the best planar PPSC coated with PMMA, an increase of almost 1
$mA.cm^{-2}$, leading to a $J_{sc}$ about 22.88 $mA.cm^{-2}$ is obtained
thanks to a PC into the MAPI (see Figure SI- S7), with $P=400\ nm$, $ff=0.4$
and $t=110\ nm$. It has been checked that $ff$ of 0.3 and 0.5 lead to lower
current densities than the optimal one, but still larger than the reference.
The corresponding spectra in Figure 15 reveal that the improvement is mainly
due to a larger band edge absorption, so LT, as confirmed by the absorption
enhancement in the same Figure.

It is noticeable that other periods, smaller than 550 $nm$, can lead to more
limited $J_{sc}$ enhancements. However, it has been checked that using periods
around 5 and 10 $\mu m\pm 0.5$ lead to a $J_{sc}$ lower than 21.85
$mA.cm^{-2}$. It confirms that such periods, far larger than the sub-micron
optimal one, do not enable an efficient diffraction and thus do not lead to
any $J_{sc}$ enhancement, when compared to an optimized planar reference.

####  V.3.2 Single junction: 2D PC in the PMMA covering layer

With respect to the best planar PPSC coated with PMMA, an increase of almost
0.9 $mA.cm^{-2}$, leading to a $J_{sc}$ about 22.75 $mA.cm^{-2}$, is obtained
thanks to a PC in the PMMA layer. The 2D PC parameters are $P=665\ nm$,
$ff=0.4$ and $t=615\ nm$ (see Figure SI-S8). It has been checked that $ff$ of
0.3 and 0.5 lead to lower current densities. It can be noticed on the
corresponding spectra in Figure 15 that the improvement is due to both a AR
effect and limited LT at band edge absorption, since it implies low effective
index guided modes that are the only able to interact, weakly, with the
pattern on top of the thick substrate. The enhancement remains lower than the
one induced by the 2D PC in the perovskite layer. Again, it has been checked
that periods far larger than the optimal one, typically around around 5 and 10
$\mu m\pm 0.5$ lead to a $J_{sc}$ lower than 22 $mA.cm^{-2}$, so to a limited
enhancement, because of a reduced diffraction efficiency.

####  V.3.3 Single junction: Combination of the two 2D PC

Given the previous enhancements, a structure that combines the two 2D PC, one
in the PMMA and the second at the MAPI/HTL interface, can be envisaged. It is
noticeable that RCWA method implies that the period of such a combined
architecture is a integer $N_{PMMA}$ times the period of the 2D PC in the PMMA
($P_{PMMA}$), and another integer $N_{MAPI}$ times the pitch of the 2D PC in
the MAPI ($P_{MAPI}$); other PC parameters ($ff_{PMMA}$, $t_{PMMA}$ on the one
hand, $ff_{MAPI}$, $t_{MAPI}$ on the other hand) can differ (see Figure 14).
For the sake of illustration of a possible further $J_{sc}$ enhancement, it
has been chosen to set the parameters of the 2D PC at the MAPI/HTL as for the
optimized cell with a flat PMMA layer, i.e. $P_{MAPI}=400\ nm$,
$ff_{MAPI}=0.4$, $t_{MAPI}=110\ nm$, as well as $ff_{PMMA}=0.5$ and
$t_{PMMA}=405\ nm$, among the thinnest most favorable values of the 2D PC in
PMMA associated with planar MAPI. Then, $N_{PMMA}$ is scanned from 5 to 8 and
$N_{MAPI}$ from 9 to 15, given the fact that $P_{PMMA}$ is typically larger
$P_{MAPI}$ according to the previous studies. The simulated $J_{sc}$ displayed
in Figure SI-S9 shows that a limited increase, of about 0.26 $mA.cm^{-2}$, up
to 23.14 $mA.cm^{-2}$, is possible provided $P_{PMMA}=530\ nm$. It can be seen
in Figure 15 that such $J_{sc}$ enhancement results from both LT and AR
effects compared to the planar reference. If the full space of various
possible PC parameters has not been scanned, and thus the previous parameters
not fully optimized, the interest of such a combination is yet demonstrated.

###  V.4 2T tandem

Figure 16: Various configurations of 2T tandem PPSC simulated in the
perspective. (i) Planar ref on an infinite glass substrate with varying
perovskite materials thicknesses, to reach and equilibrium of $J_{sc}$. (ii)
Cross section of a 2D square lattice of ITO pillars in 1.77-eV perovskite, to
enhance $J_{sc,1.77eV-perovskite}$ (iii) Cross section of a 2D square lattice
of ITO pillars in the 1.77-eV perovskite, and of dots of 1.22-eV perovskite in
HTL. Inspired by the configuration proposed by K. Xiao et al. Xiao  _et al._
(2020)

A 2T tandem PPSC generally exhibits thermalization, as the one previously
described in section IV.1.3. Thus, we will consider this example as a case
study for possible improvement. In the following simulations, the optical
indices they provided for the considered materials are used. The planar stack
(see Figure 16), considered as a reference, is close to the one shown in
Figure 6(a). Oppositely to our previous studies, the glass substrate is again
supposed to be infinitely thick to avoid the additional interferences in the
substrate. Within the stack, the layer thicknesses (except perovskites) have
been set to realistic values such as $t_{ITO}=100\ nm$, $t_{NiO}=t_{NVPB}=10\
nm$ (simplified version of a mixed material ETL), $t_{C60}=10\ nm$ for both
HTL, as well as $t_{PEDOT-PSS}=10\ nm$ and $t_{Cu}=100\ nm$. As justified
later, the SnO2 layer, with $t_{SnO_{2}}=100\ nm$, acts as an optical spacer
(the 1 $nm$ thick Au layer has thus been neglect).

Setting the thicknesses of both perovskite layers to $t_{1.77\ eV\ PK}=400\
nm$ (also the maximum in the considered range) and $t_{1.22\ eV\ PK}=880\ nm$
(in the range 800 - 1200 $nm$) leads to the highest $J_{sc}$ of 16.25
$mA.cm^{-2}$, that appears to be limited by the 1.77 $eV$ perovskite subcell.
This value is slightly larger than the one obtained by K. Xiao et al..
Moreover, it is obtained in our case for different perovskite thicknesses, due
to a mismatch between the thicknesses of the charge transport layers we have
chosen and author’s choices. However, our derived absorbance spectra for both
sub cells (see Figure 17) still exhibits a thermalization effect. Simply
increasing $t_{1.77\ eV\ PK}$ could be at the expense of the charges
collection.

In this frame, the possible enhancement of the $J_{sc,1.77\ eV\ PK}$ thanks to
a 2D PC at the ITO / 1.77 $eV$ perovskite layer interface is studied. The 2D
PC consists of a square lattice of ITO pillars, coated with conformal ETL and
then with a 1.77 $eV$ perovskite layer that planarizes the corresponding
subcell, while keeping an equivalent volume of perovskite as in the planar
tandem cell.

(a) Simulated absorption spectra of the various envisaged 2T tandem PPSC.

(b) Zoom on LT in 1.77-eV perovskite material

Figure 17: 2T tandem PPSC simulated in the perspective, Inspired by the
configuration proposed by K. Xiao et al. Xiao  _et al._ (2020).

A significant enhancement of $J_{sc,1.77\ eVPK}$, up to 17.92 $mA.cm^{-2}$,
has been found for $P=325\ nm$, $ff=0.5$ (here defined as the ITO filling
fraction) and $t=160\ nm$. The reason is twofold (see spectra in Figure 17): a
broadband AR as well as a limited LT at the band edge of the 1.77 $eV$
perovskite material that slightly reduces the thermalization. LT specifically
occurs in one of the absorbing materials, to the detriment of the other one.
It results from a guided mode mainly confined in the perovskite of interest,
with a limited overlapping in the other perovskite, especially thanks to the
rather large, 100 $nm$ SnO2 optical spacer. Indeed, this layer can prevent
evanescent coupling between quasi guided modes of the perovskites.
Additionally, it could protect the 1.77 $eV$ perovskite material during the
fabrication, but the junction between the two subcells might then also be less
efficient.

Moreover, this first step leads to a strong $J_{sc}$ disequilibrium between
the two sub cells, since $J_{sc,1.22\ eV\ PK}$ even slightly decreases to 15.7
$mA.cm^{-2}$, compared to the planar reference. To increase in a second step
$J_{sc,1.22\ eV\ PK}$, this low-Eg perovskite layer can also be patterned
rather than simply increasing its already large thickness.

Starting from the last structure, a second 2D PC is introduced at the 1.22
$eV$ perovskite material / HTL interface. To target a LT at larger wavelength,
close to the band edge of this perovskite, its period has to be increased.
However, given the constraint induced by the boundary conditions of the
simulation, we simply chose a supercell having a period twice the one of the
single PC device, with one pattern in the 1.22 $eV$ perovskite material, and
two patterns in the 1.77 $eV$ perovskite material (see Figure 16), while
keeping $ff$ and thicknesses constant. Moreover, again, the volume of this
perovskite is kept constant, HTL acting as a planariser, with a minimum
thickness of 10 $nm$ (with a larger volume as in the planar alternative).

It appears that $J_{sc,1.77\ eVPK}$ remains unchanged, whereas $J_{sc,1.22\
eV\ PK}$ is increased up to 16.2 $mA.cm^{-2}$, reducing, but not cancelling
the disequilibrium. As expected, this results from LT at the band edge in the
1.22 $eV$ perovskite material (see Figure 17). This last result appears as a
proof of concept to integrate two PC in order to induce LT, as well as
possibly AR, in two different absorbing layers of a given stack, provided each
one exhibits guided modes without overlapping. If not yet optimized, it is
noticeable that, providing the first subcell is planarized, the two PC could
practically have independent parameters, offering additional degrees of
freedom to reduce disequilibrium. Given the required spacer, such a concept
could be also applied to other kind of three or four terminal multijunctions
cells.

##  VI Conclusion - Perspectives

Light management for PV solar cells was mainly intended to increase the
$J_{sc}$, simply thanks to a larger absorption. When using metal-halide
perovskite materials, LM is shown to result from an interaction between the
various material choices, the patterning processes and the PE. Moreover,
demonstrating sole LM needs to meet some rigorous criteria to get rid of mixed
electrical, material and photonic effects.

The investigations presented in this review have demonstrated that the easiest
$J_{sc}$ enhancement is indeed the most frequently envisaged effect,
especially for single junction cells, compared to the still highly challenging
$V_{oc}$ enhancement, and the immature studies of the EY due to the lack of
studies at the module scale. As demonstrated by several authors, even
thicknesses optimization within flat PPSC can enhance absorption thanks to
Fabry Perot effects while taking care of the electrical constraints. To
further increase the absorption at the perovskite band edge, an in-plane
pattern, at various scales, can be introduced in one of the layers. However,
patterns at the wavelength scale appeared as the most efficient. In any case,
the $J_{sc}$ enhancement compared to already optimized flat reference remained
limited to a few percent. As regards the photonic regimes, the broadband AR
effect was the most frequently observed, but only LT was able to significantly
enhance the absorption at the band edge, as confirmed by our own simulations.
In addition, we confirmed that direct patterning of the perovskite leads to
more efficient LT than an additional layer on top of the substrate. Finally,
in 2T tandem cells, we showed that one PC in each subcell can enhance each of
the $J_{sc}$ separately.

Beyond, some of the strategies developed for these opaque, single junction
PPSC can be adjusted for other applications of hybrid perovskites for sun
light harvesting. Indeed, accurately tuned spectral absorption is required in
semitransparent single junction. Moreover, the same concepts can be tweaked to
tailor reflectivity spectrum for perovskite-based color printing devices
Gholipour  _et al._ (2017); Gao  _et al._ (2018); Fan  _et al._ (2019); Yoo
_et al._ (2021) and engineering absorption management in full-color perovskite
detectors Hossain  _et al._ (2020b); Qarony  _et al._ (2020).

This work is even part of far larger context. Indeed, other studies are
ongoing concerning materials, which need to be more stable, and processes. In
this frame, it can be noticed that record cells might not share the same
encapsulation strategies as more realistic, large surface cells and modules
Wang  _et al._ (2021c), both for aging and safety reasons Wu  _et al._ (2021a,
b). All perovskite tandem cells are even more challenging on the fabrication
point of view Zheng  _et al._ (2020), but are also very promising since they
combine all the potential of perovskite based cells and modules with high
yields.

Finally, according to the reciprocity relation between absorption and emission
Rau (2007), light extraction strategies in perovskite LED Gholipour  _et al._
(2017); Wang  _et al._ (2017); Tiguntseva  _et al._ (2017), as well as
resonances of high quality factor in perovskite-based laser Chen  _et al._
(2016); Pourdavoud  _et al._ (2017); Qin  _et al._ (2020) can use similar
concepts to those derived in this work. Even LM in 2T tandem cells could be
mimicked in white LEDs obtained by stacking several emitting materials. For
LEDs, a high light extraction efficiency is a even more the key of good
efficiency.

Acknowledgement:

RCWA simulations were performed on the Newton computer cluster facilities
operated by PMCS2I at Ecole Centrale de Lyon and on the CNRS/IN2P3 Computing
Center in Lyon. R. M. L. acknowledges project EMIPERO (ANR-18-CE24-0016).

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— SUPPLEMENTAL MATERIAL —

Figure S1: Fourier space analysis of the moth-eye pattern proposed by Wei et
al Wei  _et al._ (2017). The 615 pixels of the real space view correspond to
$9.26\mu m$; so, the resolution is $0.015\mu m$ per pixel. Thus, the largest
spatial frequency in the Fourier space is $66\mu m^{-1}$. From equation 2, the
spatial frequency range enabling the coupling of a plane wave under normal
incidence into a guided mode having an effective index of two in the
absorption domain of MAPI is $[18-34\mu m^{-1}]$. It thus appears that some
significant spatial frequencies of the pattern lie in the domain of interest,
especially at large wavelengths (low spatial frequencies), partially
justifying LT. Figure S2: Fourier space analysis of the moth-eye pattern
proposed by M. Jǒst et al Jošt  _et al._ (2017). The 828 pixels of the real
space view correspond to $24.55\mu m$; so, the resolution is $0.03\mu m$ per
pixel. Thus, the largest spatial frequency in the Fourier space is $33\mu
m^{-1}$. From equation 2, the spatial frequency range enabling the coupling of
a plane wave under normal incidence into a guided mode having an effective
index of two in the absorption domain of MAPI is $[18-34\mu m^{-1}]$. Most of
the significant spatial frequencies of the pattern lie outside the domain of
interest. LT is thus unlikely in such a device. Figure S3: Simulated short
circuit current density, under AM1.5G spectrum under normal incidence provided
a unitary IQE for the planar MAPI PPSC as a function of HTL and perovskite
thickness. Figure S4: Simulated short circuit current density under AM1.5G
spectrum under normal incidence provided a unitary IQE for the planar MAPI
PPSC as a function of HTL and perovskite thickness. Figure S5: Simulated
absorption spectra of a planar MAPI PPSC as a function of the perovskite
thickness for a 0.24 $\mu m$ thick HTL. Figure S6: Simulated short circuit
current density under AM1.5G spectrum under normal incidence, provided a
unitary IQE of a planar 0.3 $\mu m$ thick MAPI PPSC as a function of HTL and
PMMA thickness. Figure S7: Simulated short circuit current density under
AM1.5G spectrum under normal incidence provided a unitary IQE of a 2D MAPI
PPSC with 0.25 $\mu m$ thick flat HTL, covering 0.32 $\mu m$ thick PMMA layer
as a function of the hole depth and the period for $ff$ of 0.4. For all
configurations, the volume of MAPI remained the same as that of the 0.3 $\mu
m$ thick unpatterned layer previously considered. Figure S8: Simulated short
circuit current density under AM1.5G spectrum under normal incidence provided
a unitary IQE, of a planar 0.3 $\mu m$ thick MAPI PPSC, with a 0.25 $\mu m$
thick flat HTL coated with 2D PC PMMA layer as a function of thickness and
period, for $ff$ of 0.4 Figure S9: Simulated short circuit current density
under AM1.5G spectrum under normal incidence provided a unitary IQE of a PPSC
combining two 2D PCs, one in PMMA and the second at the MAPI/HTL interface.
The 2D PC in MAPI has a pitch of 0.4 $\mu m$, thickness of 0.11 $\mu m$, and
the $ff$ is 0.4. The $ff$ of the 2D PC in PMMA is set to 0.5, and the
thickness of the PMMA layer is set to 0.405. The simulated PPSC thus has a
period which is $N$ times the pitch of the 2D PC with PMMA and $N^{\prime}$
times the pitch of the 2D PC with MAPI with 0.4 $\mu m$ Figure S10: Simulated
short circuit current density, under AM1.5G spectrum under normal incidence
provided a unitary IQE, of a 2T tandem PPSC combined with a 2D PC in the large
bandgap subcell as a function of thickness and period, provided $ff=0.5$.