|  Laser-Annealing and Solid-Phase Epitaxy of Selenium Thin-Film Solar Cells
---|---
|  Rasmus Nielsen,∗a Tobias H. Hemmingsen,a Tobias G. Bonczyk,a Ole Hansen,b
Ib Chorkendorff,a and Peter C. K. Vesborga
|  Selenium has resurged as a promising photovoltaic material in solar cell
research due to its wide direct bandgap of 1.95 eV, making it a suitable
candidate for a top cell in tandem photovoltaic devices. However, the
optoelectronic quality of selenium thin-films has been identified as a key
bottleneck for realizing high-efficiency selenium solar cells. In this study,
we present a novel approach for crystallizing selenium thin-films using laser-
annealing as an alternative to the conventionally used thermal annealing
strategy. By laser-annealing through a semitransparent substrate, a buried
layer of high-quality selenium crystallites is formed and used as a growth
template for solid-phase epitaxy. The resulting selenium thin-films feature
larger and more preferentially oriented grains with a negligible surface
roughness in comparison to thermally annealed selenium thin-films. We
fabricate photovoltaic devices using this strategy, and demonstrate a record
ideality factor of n=1.37, a record fill factor of FF=63.7%, and a power
conversion efficiency of PCE=5.0%. The presented laser-annealing strategy is
universally applicable and is a promising approach for crystallizing a wide
range of photovoltaic materials where high temperatures are needed while
maintaining a low substrate temperature.

##

††footnotetext: a SurfCat, DTU Physics, Technical University of Denmark,
DK-2800 Kgs. Lyngby, Denmark.††footnotetext: b DTU Nanolab, National Center
for Nano Fabrication and Characterization, Technical University of Denmark,
DK-2800, Kgs. Lyngby, Denmark.††footnotetext: ∗ E-mail:
<EMAIL_ADDRESS>† Electronic Supplementary Information (ESI)
available: Additional figures. See DOI: 00.0000/xxxxxxxxxx/

## INTRODUCTION

Photovoltaics (PV) is considered an attractive renewable energy technology of
key importance in the mitigation of climate change1. While silicon solar cells
have been dominating the global PV market for decades, a family of inorganic
thin-film PV materials has emerged in recent years2, 3. Thin-film PV offers a
compelling pathway towards cost-competitive alternatives to silicon solar
cells, as well as complementary materials for use in tandem devices4, 5. Among
these emerging inorganic materials, selenium is a particularly promising
candidate due to its single element composition, low melting point, and direct
bandgap between 1.8 and 2.0 eV making it suitable for integration with lower
bandgap photovoltaic devices6, 7, 8, 9. However, despite recent advances in
the device architecture improving the power conversion efficiency (PCE) of
selenium thin-film solar cells from 5.0% to 6.5%10, the PCE is still too low
for tandem integration to be viable.

The key factors influencing the device performance of state-of-the-art
selenium solar cells have been identified as the crystallinity and the
morphology of the photoabsorber11. Typically, this layer is fabricated by
evaporating an amorphous layer of selenium, followed by a crystallization
process to transform the as-deposited film into the desired trigonal phase.
The crystallization step is almost exclusively carried out by annealing the
structure at temperatures of $\sim$200∘C on a pre-heated hotplate in air, but
it is challenging to precisely control crystal growth using this strategy, and
the optimal temperature is essentially a compromise between the degree of
crystallinity and the surface morphology. Therefore, crystallizing the
selenium thin-films using thermal annealing may be a bottleneck for realizing
higher efficiency PV devices, and alternative approaches should be
investigated.

Fig. 1: Comparison of the as-deposited and the laser-annealed surface. (A)
Schematics illustrating the crystallization of the amorphous film induced by
scanning the laser beam across the surface of the sample. (B) Optical and, (C)
top-view SEM-images of 3 areas on the surface of the selenium thin-film. The
area in the orange square has not been exposed to the laser, the area in the
green square has been exposed during the first half of the scan, and the area
in the blue square has been exposed during the second half of the scan. (D)
The absorbance of a 300 nm selenium thin-film in the (as-deposited) amorphous
and (thermally) crystalline phase.

Laser-annealing is a compelling alternative to thermal annealing, as the
energy induced upon exposure may be accurately localized in time and space12.
This enables excellent control of the crystallization process as well as the
crystal growth direction. The latter is particularly important in the case of
trigonal selenium, as the optoelectronic material properties are
anisotropic13. The crystallographic orientation of the grains in the selenium
thin-film is therefore expected to affect the PV device performance. The
crystallization of amorphous selenium under the influence of light was first
studied half a century ago, where it was observed that illumination increases
the growth rate of crystallites and modifies their external morphology14, 15.
The proposed mechanism suggests that the growth rate is determined by the flux
of holes towards the grain boundary, and that the crystallites formed act as a
sink for holes resulting from the topography of the energy bands. In a more
recent study by Hadar et al.11, the effect of light during the crystallization
of selenium was investigated in the context of photovoltaic devices. By
combining thermal annealing with strong white light illumination, they
observed larger crystal grains and a reduction in the surface roughness.
Furthermore, a significant improvement to the fill-factor (FF) was reported,
indicating that the charge transport properties could be strongly affected by
the non-equilibrium growth conditions induced by light.

In this work, we developed a novel laser-annealing strategy and report the
first laser-annealed selenium thin-film solar cells. Unlike previous reports,
our approach utilizes a monochromatic light source, which enhances the spatial
confinement of the photo-generated carriers. Moreover, our laser-annealing
strategy involves illuminating the photoabsorber through a semitransparent
substrate, which, to the best of our knowledge, has never been reported before
for any other material. As a result, a buried crystalline seed-layer is
formed, serving as a growth template for solid-phase epitaxy. Compared to the
thermally annealed control samples, the laser-annealed selenium thin-films
exhibit negligible surface roughness and the presence of larger, more
preferentially oriented grains. Our photovoltaic devices demonstrate a record
fill factor FF=63.7%, a record ideality factor n=1.37, and a power conversion
efficiency of 5.0%. The two-step crystallization process presented here is
proposed as a universally applicable approach for achieving high-quality
crystalline thin-films of any photovoltaic material synthesized on a substrate
relevant for photovoltaic devices.

## EXPERIMENTAL DETAILS

FTO-coated glass substrates are ultrasonically cleaned in Milli-Q water,
acetone and isopropanol, and dried using a nitrogen gun. The clean substrates
are placed in a shadow mask and introduced into a commercial magnetron sputter
chamber with a base pressure of $\textit{P}_{\text{base}}\sim 10^{-7}$ mbar.
Here, the samples are sputter cleaned in 20 mTorr of Ar for 60 seconds,
followed by the deposition of a $\sim 45$ nm layer of TiO${}_{\text{2}}$. The
TiO${}_{\text{2}}$ thin-film is formed by sputtering metallic Ti in a reactive
atmosphere of Ar/O${}_{\text{2}}$ (60/6 sccm) at 5 mTorr with an elevated
substrate temperature of 400∘C. To further improve the crystallinity and
increase the density of oxygen vacancies, the samples are subsequently
annealed at 500∘C for 1 hour under high vacuum. Oxygen vacancies
$V_{\text{O}}^{+2}$ are known to act as native n-type dopants in
TiO${}_{\text{2}}$, thus improving the transport of electrons through the
selective contact.

Fig. 2: Comparing the superstrate laser-annealing strategy to the standard
thermal annealing strategy. (A) Schematics illustrating how energy is induced
into the selenium thin-film by either lasing or heating through the substrate.
(B) AFM-images and, (C) top-view SEM-images of the selenium thin-film surfaces
after the two respective processes, complemented by (D) cross-sectional SEM-
images. At this processing step the device structure comprise
SLG/FTO/TiO${}_{\text{2}}$/Te/poly-Se. (E) Optical image of the selenium thin-
film surface as a function of increasing laser dose.

The samples are left to cool down to room temperature under high vacuum, and
then briefly exposed to air in the transfer to a custom-built thermal
evaporator with a base pressure of $\textit{P}_{\text{base}}\sim 10^{-8}$
mbar. Here, a thin layer of $\approx$1 nm of Te deposited, followed by the
deposition of $\approx$300 nm of Se. The purpose of the ultra-thin Te layer is
to improve adhesion and uniformity of the selenium thin-film during the
crystallization process16. The batch of reference samples are crystallized by
thermal annealing in air at 190∘C for 4 minutes in a pre-heated aluminium
oven. The other samples are placed in a 3D-printed sample holder facing a 445
nm continuous-wave diode laser. The laser beam is focused to a diameter of
$\textit{d}\approx$1.75 mm, and the optical power density is adjusted to
$\textit{P}\approx$100 W/cm2. The laser beam is scanned in a serpent-like
raster pattern across the surface of the samples using a set of stepper motors
(step size and line separation of 40 $\mu$m), and the total laser dose is
varied by adjusting the raster speed. After the crystallization step, $\sim$15
nm of MoO${}_{\text{x}}$ is sputter deposited from a ceramic
MoO${}_{\text{3}}$ target in a reactive atmosphere of Ar/O${}_{\text{2}}$
(60/0.6 sccm) at 5 mTorr. Finally, Au-contacts are sputter deposited on the
samples to complete a device structure comprising
glass/FTO/TiO${}_{\text{2}}$/Te/Se/MoO${}_{\text{x}}$/Au. The experimental
details have been elaborated in the Supplementary Information, along with
descriptions of the various characterization methods.

## RESULTS AND DISCUSSION

The initial laser-annealing experiments were inspired by the novel approach
developed by You et al. for crystallizing perovskite thin-film
photoabsorbers17, and have been schematically illustrated in Figure 1A. A
laser spot is scanned across the surface of the as-deposited film,
transforming the layer from an amorphous into the desired crystalline phase in
the wake of the beam. Figure 1B shows an optical image of a 12 mm x 10 mm x
300 nm selenium thin-film, where the top half of the sample has been left
unexposed as a reference of the as-deposited film. The bottom half of the
sample has been laser-annealed by scanning the laser beam in a raster pattern
initiated on the left-hand side. From visual inspection, the surface of the
sample is divided into three distinctive areas; the as-deposited film
(orange), the film exposed during the first half of the laser scan (green),
and the film exposed during the second half of the laser scan (blue). The
three regions have been examined using top-view SEM as shown in Figure 1C. In
the first half of the scan, large crystallites are observed in the surface of
the film, and the coexistence of the amorphous and crystalline (trigonal)
phases is confirmed using GIXRD (ESI, Fig S.6). However, as the scan progress,
the morphology changes and the integrity of the surface is compromised by
pinholes. As the optical power density (100 W/cm${}^{\text{2}}$) and the
raster speed (12.5 mm/s) remain constant during the scan, the observed
morphological changes cannot be attributed to a variation in the laser dose.
Another possible explanation is a thermal runaway, resulting from a gradual
rise in the temperature of the sample surface. Considering the influence of
temperature and crystallinity on absorptivity, it is conceivable that the
optical energy is absorbed within a progressively thinner surface layer as the
scan proceeds. Consequently, the surface temperature may rise to a point where
selenium starts to re-evaporate, leading to the formation of voids and
pinholes.

Fig. 3: Photovoltaic device performance as a function of increasing laser
dose. (A) Current-voltage (J-V) curves measured under 1 Sun conditions. (B)
External quantum efficiency (EQE) spectra measured under short-circuit
conditions. (C) Performance metrics as a function of increasing laser dose.
(D) XRD patterns (Cu K$\alpha$) of a selenium thin-film crystallized using
superstrate laser-annealing (0.32 s/mm) compared to a thermally annealed
control sample. Peaks marked with an asterisk are attributed to the FTO-coated
glass substrate.

To gain a deeper understanding of the optical dispersion of the laser within
the film, we measured UV-visible absorbance spectra of 300 nm amorphous and
crystalline selenium thin-films synthesized on quartz substrates, as shown in
Figure 1D. The relative absorbance between the two phases of the 445 nm laser
is 1.35, and the penetration depth $\delta=1/\alpha\left(\lambda\right)$ is 38
nm and 49 nm in the crystalline and amorphous phase, respectively. These
findings suggest that the optical energy of the laser is primarily absorbed in
close proximity to the surface of the film, and becomes even more localized
once large crystallites have been formed. Due to the higher absorbance of the
crystalline phase, the temperature of the crystallites is expected to surpass
that of the surrounding amorphous material. If thermal gradients across the
grain boundaries promote crystal growth, opting for a longer wavelength laser
to further increase the relative absorbance could potentially lead to the
formation of even larger grains. Nevertheless, due to the high vapor pressure
of selenium, the parameter space for achieving a void- and pinhole-free
crystalline thin-film through laser annealing of the surface is exceedingly
narrow and not necessarily consistent throughout the scan.

Seeing that it was not possible to fabricate functional selenium solar cells
using the aforementioned laser-annealing strategy, an alternative approach was
developed, as schematically illustrated in Figure 2A. By taking advantage of
the transparency of the SLG/FTO/TiO${}_{\text{2}}$ substrate to the 445 nm
laser, laser-annealing of the amorphous selenium thin-film was performed
through the substrate. Using this "superstrate" laser-annealing strategy, the
optical energy is predominantly absorbed in the vicinity of the carrier
separating junction instead of the open surface from where selenium can re-
evaporate away.

Figure 2E displays an optical image of the selenium thin-films after laser-
annealing through the substrate, in comparison to the as-deposited and
thermally annealed films. The left-hand sample represents the first
superstrate laser-annealed sample, receiving the same laser dose as the sample
in Figure 1B. However, as only a slight darkening of the film was observed,
the raster speed was successively halved for each subsequent sample until the
dose had increased by a factor of 10. The visual appearance of the superstrate
laser-annealed samples significantly differs from that of both the substrate
laser-annealed and the thermally annealed samples. Instead of a grey
appearance, the color is a dark purple/black shade, which may be attributed to
distinct surface morphology differences observed through AFM and top-view SEM
in Figure 2B and C, respectively. The root-mean-square roughness was
quantified over a $\mu$m${}^{\text{2}}$ area, measuring 4.2 nm and 44.4 nm on
the laser-annealed and thermally annealed samples, respectively. Top-view SEM-
images revealed only slight penetration of the crystallites into the surface
of the laser-annealed sample, in contrast to the thermally annealed sample,
where large, bulky grains and their boundaries are clearly visible. Moreover,
significant differences in the external morphology of the grains in the laser-
annealed and thermally annealed samples are observed, consistent with previous
findings reported by Dresner and Stringfellow14. To gain further insights,
cross-sectional SEM-images are acquired as shown in Figure 2D, revealing the
formation of selenium crystallites buried under a still amorphous layer of
selenium. Additional top-view and cross-sectional SEM-images can be found in
the Supplementary Information.

Fig. 4: Investigating the effect of a final post-annealing processing step.
(A) Process flow and a cross-sectional SEM-image of a laser-annealed device.
(B) Same as (A) for a thermally annealed control device. (C) Current-voltage
(J-V) curves measured under 1 Sun conditions. (D) External quantum efficiency
(EQE) spectra measured under short-circuit conditions. The first derivatives
of the onsets have been included to visualize the shift of the inflection
point. (E) Schematic illustration of solid-phase epitaxial (SPE) growth of
crystalline selenium thin-films.

Following the superstrate laser-annealing step, MoO${}_{\text{x}}$ and Au are
sputter deposited to finalize a device structure comprising
SLG/FTO/TiO${}_{\text{2}}$/Te/poly-Se/a-Se/MoO${}_{\text{x}}$/Au. We examine
the device performance with varying laser doses, quantifying the dose using
the reciprocal raster speed while maintaining a fixed optical power density of
100 W/cm2. Figure 3A and B shows current-voltage (J-V) and external quantum
efficiency (EQE) measurements. We observe a threshold for the laser dose
required for the device to function, with no evident effect on PV performance
upon exceeding this threshold. This observation has been illustrated in Figure
3C. It is noteworthy that the selenium thin-film undergoes a visual
transformation exactly at this threshold, as seen in Figure 2E. This suggests
a significant change in the degree of crystallinity. However, despite
achieving functioning devices with power conversion efficiencies of PCE=4%,
"S-kinks" are observed in the J-V–curves near the open-circuit voltage.
S-kinks are often reported in the context of emerging photovoltaic devices,
arising from bias-dependent charge transport barriers at one of the carrier-
selective contacts18. As previously observed from the cross-sectional SEM-
images in Figure 2D, the large selenium crystallites are buried under a layer
of still amorphous selenium. As amorphous selenium is known to be highly
resistive, this could pose a critical challenge for efficient charge carrier
extraction.

Charge transport in selenium thin-films is rather complex due to the trigonal
symmetry of the crystalline phase, which leads to highly anisotropic carrier
mobilities9. Aligning the (001) plane parallel to the substrate, which
corresponds to an arrangement of the helical chains along the charge transport
direction, should improve charge transport and ultimately enhance device
performance. To investigate the crystallographic orientation of the grains in
the laser-annealed selenium thin-films, the XRD patterns shown in Figure 3D
are recorded using Bragg-Brentano geometry. When comparing the laser-annealed
sample to the thermally annealed control sample, a pronounced difference in
the relative intensities of the (100) and (101) reflections is observed. This
result indicates that a larger fraction of the grains in the laser-annealed
sample exhibit a preferential orientation compared to those in the thermally
annealed sample. As the optical energy of the laser is absorbed in the
amorphous selenium thin-film, a thermal gradient is formed in the superstrate
configuration, extending from the TiO${}_{\text{2}}$/Se-interface to the open
surface. This thermal gradient may potentially facilitate the preferential
growth of the helical chains along its direction.

Fig. 5: Analysis of carrier transport losses. (A) Current-voltage (J-V) curves
reconstructed from SunsVoc measurements. (B) Chart illustrating the loss
mechanisms lowering the device FF below the SQ-limit. (C) Apparent carrier
concentration as a function of profiling depth from C-V measurements.

In continuation of the promising results obtained through superstrate laser-
annealing, an additional processing step is introduced. After the deposition
of the top layers, the entire stack is thermally annealed at 190∘C for 4 min
in air to facilitate the crystallization of the remaining amorphous layer of
selenium, while maintaining the smooth morphology. This transformation is
evident from the well-defined crystal facets observed in the cross-sectional
SEM-image in Figure 4A. The resulting device stands in contrast to the control
device shown in Figure 4B. Schematic diagrams illustrating the final
processing steps have been included alongside the SEM-images to emphasize the
crystallization step as the only difference between the two devices. To avoid
confusion with the thermal annealing step used for our control devices, the
final processing step is referred to as "post-annealing".

Figure 4C and D shows IV and EQE measurements of a superstrate laser-annealed
device before and after post-annealing, in comparison to a thermally annealed
control device. The apparent S-kink is eliminated upon post-crystallization of
the amorphous layer of selenium; however, this improvement comes at the
expense of a reduced short-circuit current density. This reduction is also
observed in the EQE, showing a decrease in carrier collection across all
wavelengths. These findings suggest that the amorphous selenium layer
passivates the interface with the hole-selective MoO${}_{\text{x}}$ contact,
thereby reducing the interfacial recombination velocity. In contrast, the
control device features no layer of amorphous selenium, and the EQE and short-
circuit current density are similar to those of the laser-annealed sample
before post-annealing. This result may be attributed to the formation of a
passivating oxide layer when the control device is thermally annealed in air.
Alternatively, as the transformation of amorphous to trigonal selenium
involves a 10% volume reduction19, the post-annealing process may impose
strain or stress on the already deposited top layers in the case of the laser-
annealed device.

Another notable difference resulting from the post-annealing of the laser-
annealed sample is a shift in the photovoltaic bandgap. The photovoltaic
bandgap is determined from the inflection point of the EQE in Figure 4D. It
has been predicted that strain has a pronounced influence on the bandgap of
selenium20. However, the fact that the inflection points of the control device
and the laser-annealed device after post-annealing are identical suggests that
other factors may be responsible for the observed shift. Another hypothesis
pertains to the alloying of selenium and tellurium, which is known to decrease
the bandgap of selenium21. In our devices, tellurium is present as a thin
adhesion layer of approximately 1 nm. As a result of the more localized
heating during laser-annealing, the bulk inter-diffusion of tellurium is
likely confined to a smaller volume of the selenium thin-film compared to the
thermally annealed control samples. Consequently, the local concentration of
tellurium is anticipated to be higher in the selenium crystallites before the
post-annealing process. During the post-annealing process, tellurium is
expected to diffuse throughout the width of the photoabsorbing layer, leading
to a more diluted local concentration and ultimately an effective increase in
the bandgap. The presence of tellurium at the surface and throughout the
photoabsorbing layer is confirmed using XPS depth profiling (Figure S.3).

The post-annealing process is depicted in Figure 4E, showing a crystalline
seed layer in intimate contact with an amorphous layer of selenium. During the
post-annealing, the c-Se/a-Se interface moves upwards in a process known as
solid-phase epitaxy (SPE). Typically, SPE involves depositing an amorphous
layer onto of a crystalline substrate, followed by a heating step to
crystallize the deposited film using the substrate as a growth-template.
However, our two-step crystallization process differs in that the seed layer
is formed within the as-deposited film, making it applicable to other material
systems synthesized on a substrate transparent to the laser wavelength. SPE
has been extensively studied for the growth of c-Si and c-Ge, with reported
growth temperatures as low as 500∘C and 350∘C for Si and Ge, respectively22,
23. These temperatures are significantly lower than those typically used for
growing crystalline silicon and germanium. This implies that the optimal
crystal growth temperature of selenium could be reduced by growing the
crystalline thin-film from a buried seed layer. Such an approach would
effectively mitigate the morphological challenges encountered in selenium
thin-films crystallized at temperatures exceeding 180∘C11.

The post-annealing treatment effectively eliminated the charge transport
limitations, leading to an impressive fill factor of FF=63.7% and a power
conversion efficiency of PCE=5.0%. It is worth noting that this FF value is
the highest reported among selenium solar cells, and that the FFs of the
laser-annealed samples reproducibly outperform those of the thermally annealed
control samples, regardless of the range of laser doses investigated (Figure
S.9 and S.10). The improved FF could be attributed to the more optimally
aligned crystal grains and the flat surface morphology. Previous studies have
shown that selenium solar cells are highly sensitive to absorber thickness,
with an optimal value of approximately 300 nm8. The flat morphology of the
laser-annealed sample eliminates potential shunting pathways, such as
pinholes, and reduces the local variations in the absorber thickness, leading
to more spatially homogeneous devices.

Despite achieving a record fill factor with great reproducibility regardless
of laser dose, the obtained value still falls significantly below the SQ-limit
for a 1.95 eV bandgap material, which is FF${}_{\text{SQ}}$=92.1%. To
investigate the origin of this significant fill factor deficit, we conducted
SunsVoc measurements to reconstruct IV curves with no carrier transport
losses, shown in Figure 5A. Remarkably, the device ideality factor for the
laser-annealed sample was found to be n=1.37, which is significantly better
than any other previously reported value in literature. However, the ideality
factor worsened after the post-annealing, supporting the hypothesis that the
amorphous selenium layer passivates the poly-Se/MoO${}_{\text{x}}$-interface.
A loss chart has been constructed in Figure 5B by combining the pseudo fill
factors (pFF) obtained from SunsVoc measurements, where the effects of series
resistance has been eliminated, with the actual device fill factors. This
chart illustrates the residual losses that are not accounted for by series
resistance, resulting from non-radiative recombination occurring at interfaces
and within the bulk of the photoabsorber. Comparing the laser-annealed sample
to the thermally annealed control sample, we observed lower non-radiative
recombination losses prior to the post-annealing, with carrier transport
losses being dominant. After the post-annealing process, the laser-annealed
sample demonstrated a higher fill factor compared to the control sample.
However, upon conducting a carrier transport loss analysis, it was determined
that the improvement in fill factor was not attributed to lower non-radiative
recombination losses. Instead, it is primarily due to a reduction in transport
losses. As the highly resistive layer of amorphous selenium has been fully
crystallized after this processing step, the reduced transport losses could be
the result of an improved photoabsorber conductivity. Finally, capacitance-
voltage measurements have been conducted to examine the apparent carrier
concentration of both devices. Despite having a lower apparent carrier
concentration, the laser-annealed device demonstrate fewer transport losses,
which supports the hypothesis that the crystal grains making up the
photoabsorbing layer are in fact more ideally aligned and better electrically
contacted.

An important distinction between laser-annealing and thermal annealing lies in
the thermal temperature of the charge carriers within the material during the
growth process. When photo-generated charge carriers split the Fermi-level
into two quasi-Fermi levels, the electrons and holes are no longer in thermal
equilibrium with their surroundings. This non-equilibrium state of the charge
carriers has implications for the formation energies, and hence
concentrations, of both extrinsic and intrinsic defects in the crystal24.
Therefore, laser-annealing has the potential to serve as a defect-engineering
tool for non-equilibrium growth of high-quality crystalline thin-films,
levering the chemical potential of charge carriers without introducing
extrinsic dopants.

## CONCLUSIONS

In summary, we have introduced a novel laser-annealing strategy and
successfully fabricated the first laser-annealed selenium thin-film solar
cells. Our two-step crystallization approach involves laser-annealing through
a device-relevant semitransparent substrate, resulting in the formation of
buried selenium crystallites, serving as a growth template for solid-phase
epitaxy. In comparison to thermally annealed control devices, our laser-
annealed selenium thin-films exhibit notable improvements, including reduced
surface roughness and the presence of larger, more preferentially oriented
grains. The resulting photovoltaic devices demonstrate promising performance
metrics, with a record fill factor of FF=63.7%, a record ideality factor of
n=1.37, and a power conversion efficiency of PCE=5.0%.

This work underscores the potential of laser-annealing as a viable alternative
to thermal annealing for enhancing the performance of selenium thin-film solar
cells. Furthermore, the two-step crystallization process presented here
showcases its universal applicability for achieving high-quality crystalline
thin-films across various photovoltaic materials on substrates relevant to
photovoltaic devices. With its inherent advantages of being fast, scalable,
and energy-efficient, laser-annealing may pave the way for the development of
more efficient and cost-effective thin-film solar cells based not only on
selenium but also on other inorganic materials.

## Conflicts of interest

There are no conflicts to declare.

## Acknowledgements

This work was supported by the Independent Research Fund Denmark (DFF) grant
0217-00333B.

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# Efficient Measurement of the Bi-photon Spatial Mode Entanglement with
Stimulated Emission Tomography

Yang Xu  Department of Physics and Astronomy, University of Rochester,
Rochester, New York 14627, USA<EMAIL_ADDRESS>Saumya Choudhary  The
Institute of Optics, University of Rochester, Rochester, New York 14627, USA
Robert W. Boyd  Department of Physics and Astronomy, University of Rochester,
Rochester, New York 14627, USA  Department of Physics, University of Ottawa,
Ottawa, Ontario K1N 6N5, Canada  The Institute of Optics, University of
Rochester, Rochester, New York 14627, USA

###### Abstract

The technique of stimulated emission tomography (SET) provides excellent
characterization of SPDC sources of bi-photon states since it increases the
average number of photons detected by several orders of magnitude than the
traditional coincidence counting method. In a SET experiment, the signal
caused by the vacuum fluctuation in SPDC is replaced by a more intense
prepared seed with the same mode properties, resulting in an amplification of
the corresponding idler. Based on this idea, our experiment uses the
difference frequency generation (DFG), a purely classical second-order
nonlinear process, to measure the orbital angular momentum (OAM) spectrum of
an entangled photon pair produced by a Type-II SPDC crystal. We inject the
seed beam at 780 nm with different Laguerre-Gaussian modes together with a
pump beam at 405 nm into a Type-II BBO crystal and measure the Laguerre-
Gaussian mode distribution of the idler at 842 nm. We observe a strong idler
production and good agreement with the theoretical prediction of the OAM
spectrum. We expect that this experiment paves the way for the efficient
measurement of bi-photon wavefunctions produced by ultra-thin SPDC sources and
also the characterization of high-dimensional entangled photon pairs produced
in SPDC.

††preprint: APS/123-QED

Introduction.—Entangled photon pairs have become an essential part of many
quantum imaging [1, 2, 3] and quantum information experiments[4, 5]. Recent
studies have shown that high-dimensional entanglement has unique advantages in
developing secure quantum communication protocols[6], demonstrating the
violation of Bell’s inequality[7, 8], and substantiating the EPR paradox[9].
It was shown that the orbital angular momentum (OAM) of photons is a
convenient and natural choice of a high-dimensional basis[10]. Therefore, the
OAM-entangled photon pairs have become a strong candidate for the
implementation of high-dimensional quantum information protocols[11, 12]. As a
result, there has been a plethora of research efforts, both theoretically[13,
14, 15] and experimentally[16, 17], for the accurate and efficient
characterization of high-dimensional OAM entanglement in spontaneous
parametric down-conversion (SPDC).

In a more general sense, the spatial modes of signal and idler photons
produced by SPDC are entangled. The spatial structure of the down-converted
photons produced in SPDC can be written in terms of a mode decomposition of
their two-photon wave function in a complete set of spatial mode basis.
Usually, the spatial modes of photons are decomposed into Laguerre-Gaussian
(LG) modes, $LG_{p}^{l}$ where $l$ represents the orbital angular momentum
carried by the photon, $l\hbar$, and $p$ corresponds to the radial mode.
Measuring the probability amplitude of each basis mode is the core task in the
characterization of high-dimensional spatial mode entanglement. There have
been extensive research efforts[18, 19, 20, 16] on the measurement of the two-
photon entanglement in OAM modes, also known as the spiral bandwidth. In these
works, the use of image-rotating interferometers[16, 19] and the traditional
coincidence counting method[21] are the two common ways to measure the spiral
bandwidth. Because of the low efficiency of SPDC processes, these two methods
require painstaking alignment, an environment free from perturbations, and
long data acquisition time. Furthermore, the SPDC experiments so far are only
capable of measuring the OAM ($l$) correlation of entangled photon pairs where
the contributions from individual radial modes ($p$) are summed up[20].

However, the radial mode ($p$) is also an important quantum number describing
the radial-momentum-like property of Laguerre-Gaussian modes. The entanglement
in radial modes offers a larger number of qubits that can be used in quantum
communication[22]. In addition, the use of radial modes is essential in
entanglement control with topological knots of phase and polarization
singularities[23]. These knot structures can be generated by specific
superposition of radial and orbital angular momentum modes[24]. It has been
shown by experiments that the knot structures are robust against
turbulence[25]. This feature may become useful in the propagation of
entanglement in turbulent media. A full modal analysis, including the radial
modes, of the two-photon spatial mode entanglement produced by SPDC has been
worked out in theory[13]. In principle, the theoretical results can be
verified by converting to the fundamental mode of single-mode fibers and
measuring the coincidence rate, but the measurement in a real-world laboratory
setting is experimentally challenging due to the extremely low photon pair
production rate if they are projected onto a specific spatial mode.

Fortunately, recent works[26] have shown that one can study the stimulated
parametric down-conversion to extract information about the bi-photon state
produced in its spontaneous counterparts. This technique is known as
stimulated emission tomography (SET). By exploiting the link between
spontaneous and stimulated processes, SET offers an efficient way to measure
the bi-photon state with a signal-to-noise ratio several orders of magnitude
larger than the traditional quantum state tomography method[27, 28]. This
large enhancement in signal-to-noise ratio allows the use of classical
detectors in the measurement of the joint spectral density of entangled photon
pairs[29, 30]. In this letter, we apply the SET theory and demonstrate a new
way to efficiently measure the entanglement of Laguerre-Gaussian (LG) modes of
the photon pair produced in SPDC pumped by a Gaussian beam. By changing the LG
modes of the input seed beam so that the output signal ranges through all the
possible LG modes that a spontaneously generated signal might have in the
absence of the seed, and measuring the relative magnitude of the amplified
idler output in the presence of the seed, we can reconstruct the bi-photon LG
mode spectrum of the photon pair produced in the absence of the seed.

Theory.—Consider a typical SPDC setup that consists of a continuous Gaussian
pump beam with frequency $\omega_{p}$ propagating along the $z$ direction
incident on a thin nonlinear crystal of length $L$. The output bi-photon state
after the nonlinear crystal in the momentum domain is given by[31]

$\displaystyle\ket{\psi}=\iint
d\mathbf{k}_{s}d\mathbf{k}_{i}\Phi(\mathbf{k}_{s},\mathbf{k}_{i})\hat{a}^{\dagger}_{s}(\mathbf{k}_{s})\hat{a}^{\dagger}_{i}(\mathbf{k}_{i})\ket{0}$
(1)

where $\Phi(\mathbf{k}_{s},\mathbf{k}_{i})$ is the bi-photon wave function
describing the pump and the phase matching conditions,
$\hat{a}^{\dagger}_{s}(\mathbf{k}_{s})$,
$\hat{a}^{\dagger}_{i}(\mathbf{k}_{i})$ are creation operators for the signal
and idler modes with wave vectors $\mathbf{k}_{s}$ and $\mathbf{k}_{i}$. The
subscripts $p,s,i$ refers to the pump, signal and idler respectively. To
characterize the spatial mode entanglement of an SPDC source, we need to
calculate the coincidence probability
$P_{p_{s},p_{i}}^{l_{s},l_{i}}=\left|C_{p_{s},p_{i}}^{l_{s},l_{i}}\right|^{2}$
where the coincidence amplitude $C_{p_{s},p_{i}}^{l_{s},l_{i}}$ is given by
the integral

$\displaystyle C_{p_{s},p_{i}}^{l_{s},l_{i}}=\iint
d\mathbf{k}_{s}d\mathbf{k}_{i}\Phi(\mathbf{k}_{s},\mathbf{k}_{i})\left[LG_{p_{s}}^{l_{s}}(\mathbf{k}_{s})\right]^{*}\left[LG_{p_{i}}^{l_{i}}(\mathbf{k}_{i})\right]^{*}$
(2)

The coincidence probability $P_{p_{s},p_{i}}^{l_{s},l_{i}}$ corresponds to the
probability for finding a signal in an LG mode characterized by
$LG_{p_{s}}^{l_{s}}$ and an idler photon in the mode $LG_{p_{i}}^{l_{i}}$.
Assuming a monochromatic Gaussian pump and a thin crystal, the coincidence
amplitude becomes [32, 13]:

$\displaystyle C_{p_{s},p_{i}}^{l,-l}$ $\displaystyle\propto
A^{\absolutevalue{l}}_{p_{s},p_{i}}\frac{\left(1-\gamma_{s}^{2}+\gamma_{i}^{2}\right)^{p_{s}}\left(1+\gamma_{s}^{2}-\gamma_{i}^{2}\right)^{p_{i}}(-2\gamma_{s}\gamma_{i})^{\absolutevalue{l}}}{(1+\gamma_{s}^{2}+\gamma_{i}^{2})^{p_{s}+p_{s}+\absolutevalue{l}}}$
$\displaystyle\times{}_{2}F_{1}\left(-p_{i},-p_{s};-p_{i}-p_{s}-\absolutevalue{l};\frac{1-(\gamma_{s}^{2}+\gamma_{i}^{2})^{2}}{1-(\gamma_{s}^{2}-\gamma_{i}^{2})^{2}}\right)$
(3)

where
$A^{\absolutevalue{l}}_{p_{s},p_{i}}={(p_{s}+p_{i}+\absolutevalue{l})!}/{\sqrt{p_{s}!p_{i}!(p_{s}+\absolutevalue{l})!(p_{i}+\absolutevalue{l})!}}$,
${}_{2}F_{1}$ is the Gaussian hypergeometric function,
$\gamma_{i}=w_{p}/w_{i}$ and $\gamma_{s}=w_{p}/w_{s}$ are the inverses of the
normalized signal and idler beam waist measured with respect to the pump beam.
We noticed that the orbital angular momentum conservation, $l_{s}=-l_{i}=l$,
must be satisfied as a result of the integral in Eq. 2.

Due to the symmetry between the signal and the idler, we have $w_{p}=w_{i}$,
so $\gamma_{i}=\gamma_{s}=\gamma$. For computational and experimental
simplicity, we set $p_{s}=p_{i}=0$ in our experiment. Thus, the coincidence
probability we want to measure in our experiment further simplifies to

$\displaystyle
P_{0,0}^{l,-l}=\absolutevalue{C_{0,0}^{l,-l}}^{2}\propto\left(\frac{2\gamma^{2}}{1+2\gamma^{2}}\right)^{2\absolutevalue{l}}$
(4)

Figure 1: Schematic of the experimental setup. L: lens; HWP: half-wave plate;
BPF: band-pass filter; SMF: single-mode fiber; DM: dichroic mirror. A Type-II
BBO crystal is pumped by a 405 nm CW laser beam with a Gaussian spatial
profile (drawn in blue). The seed OAM is generated by a hologram printed on
SLM. The seed prepared in the first diffraction order is picked out by the
pinhole and injected into the Type-II BBO crystal together with the Gaussian
pump beam. The output idler is passed to the OAM projection measurement setup
where the intensity of the projected beam is measured to construct the OAM
spectrum

Experiment.—Fig. 1 shows the SET setup used to measure the LG mode spectrum
with SET. We pump a 10-mm-long Type-II beta barium borate (BBO) crystal with a
horizontally polarized, 30 mW, c.w. 405 nm collimated Gaussian beam
($\text{LG}_{0}^{0}$). The pump beam is produced by a laser diode and has a
diameter of 2 mm. The 780-nm seed beam carrying a fixed OAM charge, $l_{s}$,
is picked out at the first diffraction order by passing a collimated c.w.
Gaussian beam at 780 nm through an OAM-generating hologram printed on SLM 1
(Meadowlark E-series 1920 $\times$ 1200 SLM). The prepared seed beam in a
Laguerre-Gaussian mode $\text{LG}_{0}^{l_{s}}$ is then injected into the BBO
nonlinear crystal together with the Gaussian pump beam at 405 nm. Both the
pump and the seed beam were spectrally filtered with a narrowband filter (10
nm) centered at 405 nm and 780 nm respectively. A half-wave plate was placed
after each narrowband spectral filter to control the polarization of the pump
and the signal beam so that the phase-matching condition of the difference
frequency generation is satisfied. By aligning the seed beam with the SPDC
signal of the same polarization, the seed beam is amplified by a stimulated
parametric down-conversion process and the idler beam with an opposite OAM
number $l_{i}=-l_{s}$ is generated. The down-converted idler beam is selected
out by a narrowband (10 nm) filter centered at 840 nm. A second SLM (Santec
SLM-200) is used to project the filtered idler onto different OAM
eigenmodes[33, 34]. The hologram we put on SLM2 converts the
$\text{LG}_{0}^{l_{i}}$ into a Gaussian beam. The converted Gaussian beam is
then coupled to a single-mode fiber (SMF). A single photon avalanche diode
(SPAD) is used to measure the total number of photons coupled into the SMF.

Result.—We first verify the validity of the thin-crystal approximation in our
experiment. The 405-nm Gaussian pump beam we use in our experiment has a waist
of 2.7 mm. The thin-crystal approximation for this pump beam is valid when
$L\ll z_{R}$ where $L$ is the length of the crystal and $z_{R}$ is the
Rayleigh range of the pump beam. This condition is equivalent to
$w_{p}/\sqrt{\lambda_{p}L}\gg 1$ [35]. In our experiment, the length of our
Type-I BBO crystal is $L=2.0$ mm, so the quantity
$w_{p}/\sqrt{\lambda_{p}L}\approx 94.8\gg 1$. Thus, the thin-crystal
approximation is suitable for our experiment.

We measure the number of photons collected by the SPAD over ten 5-second time
windows for different seed OAM modes ranging from $l=-6$ to $l=6$. To
calibrate the measurement, we block all beams and collect the dark counts
collected by the SPAD to characterize the background effect over twenty
5-second time intervals. The mean value and the fluctuation of dark counts
will be subtracted in our data analysis.

In Fig. 2, we show the measurements of the two-photon joint spectral amplitude
(JSA) in OAM space. The two-dimensional (2D) JSA is measured in the following
way. First, we fix the static OAM generating hologram on SLM1 so that the seed
beam has a predetermined spatial profile $LG^{l_{s}}_{0}$. The idler produced
in the parametric down-conversion is then passed to another hologram that
generates $LG^{l_{i}}_{0}$ on SLM2. This hologram on SLM2 converts the
$LG^{l_{i}}_{0}$ component in the idler beam into a Gaussian profile in the
first diffraction order. Finally, we scan through multiple $l_{i}$ values by
altering the OAM-generating hologram on SLM2 and record the corresponding
number of photons coupled to the SPAD for each $l_{i}$. We switch to a
different seed mode and repeat the procedure above to measure the full 2D JSA
matrix. We notice that only the blocks where is satisfied are significant
given a Gaussian pump beam $(LG^{0}_{0})$ at 405 nm. The measured two-photon
JSA agrees with theoretical calculations and thus confirms the conservation of
OAM in collinear SPDC processes.

Figure 2: Two-photon JSA by SET compared to theoretical predictions. The
hologram on SLM2 iterates through OAM modes along each column where the seed
OAM is held fixed. The magnitude of JSA is then measured in the number of
photons coupled into the SMF. The width of the pump is kept constant at 2 mm.
(a) OAM spectrum at $w_{i}=w_{s}=0.72$ mm (b) OAM spectrum at
$w_{i}=w_{s}=1.08$ mm (c) OAM spectrum at $w_{i}=w_{s}=1.35$ mm.

In Fig. 3, we plot the LG mode probability distribution for two different seed
waists using our SET experiment and compare them to theoretical calculations.
By comparing the two spectra at two different seed waists, we notice an
increase in the spectrum width as the seed beam waist decreases. This effect
will be discussed in detail later in this section. In Fig. 3(a) where the seed
width is larger($\gamma_{s}=\gamma_{i}=2.03$), one may observe a noticeable
discrepancy between theoretical predictions and the measured data for higher
OAM modes. This is due to the decrease in the coupling efficiency of the fiber
coupler connected to the SMF. The projection phase mask on SLM2 converts the
beam with a higher OAM number into a wider Gaussian profile. Thus, for each
OAM projection phase mask on SLM2, the fiber coupling efficiency varies
slightly. This deviation from the optimal fiber coupling becomes obvious for
smaller $\gamma_{s}$ (larger $w_{s}$) since the width of the converted
Gaussian starts to exceed the capability of the objective we use in our fiber
coupling system. Fortunately, this problem can be easily solved in principle
by using an objective with a suitable effective focal length for the
projection measurement of each OAM mode.

Figure 3: OAM spectrum measured by SET compared to theoretical predictions.
The beam width of the output idler is the same as the seed width due to the
nature of a stimulated process. The spread in the OAM spectrum is clearly
observed as the normalized seed and idler beam widths increase suggesting a
larger spatial overlap between the seed and the pump. (a) OAM spectrum at
$\gamma_{i}=\gamma_{s}=2.03$ (b) OAM spectrum at $\gamma_{i}=\gamma_{s}=3.05$

Now, we study the dependence of the OAM spectrum on seed and pump waists. As
shown in Eq. 2, the weight of each LG mode should depend on the ratio between
the pump and the signal/idler widths, $\gamma_{s,i}=w_{p}/w_{s,i}$. In
stimulated parametric down-conversions, the relation $\gamma_{s}=\gamma_{i}$
is automatically satisfied, so the probability amplitude of each OAM mode only
depends on the normalized beam waist $\gamma=\gamma_{s}=\gamma_{i}$. To better
understand the OAM spectral broadening dependent on the beam waist in a
quantitative way, we plot the normalized coincidence probability of the photon
pair production, $\left|C_{0,0}^{l,-l}\right|^{2}$, as a function of the pump-
normalized beam waist, $\gamma$, for three different values of $l_{i}$ and
$l_{s}$ with $l_{i}+l_{s}=0$ in Fig. 4. We notice an increase in the width of
the spectrum as the waists of the signal and idler decrease
($\gamma=\gamma_{i}=\gamma_{s}$ increases). This broadening in spectral width
is due to the fact that a smaller seed beam waist allows higher OAM modes to
have spatial overlap with a fixed pump beam. This results in a higher
numerical value of the overlap integral between the two electric field’s
spatial profiles.

Figure 4: Dependence of the OAM spectral amplitude on the normalized seed beam
waist for $l_{s}=-l_{i}=1,2,3$. $\gamma=\gamma_{s}=\gamma_{i}=w_{p}/w_{s,i}$
The plot shows that the spatial mode spectral amplitude decreases for higher
OAM numbers and for larger signal/idler beam waists.

Discussion.—In the sections above, we proposed and demonstrated an efficient
method to measure the high-dimensional entanglement of photon pairs produced
by various SPDC processes. In order to show the great efficiency improvement
with our SET method, we examine the SPDC process where the Gaussian pump is
down-converted into a fundamental Gaussian signal and idler
$(p_{p},l_{p})=(p_{s},l_{s})=(p_{i},l_{i})=(0,0)$ as a benchmark case. We fix
our pump power at 30 mW. Then we compare the coincidence detection rate
measured by the traditional coincidence counting method and the idler photon
production rate measured by our SET method. In the coincidence counting setup,
the signal and idler are coupled to single-mode fibers, and the coincidence
detection rate is measured by single photon detectors and a coincidence-
counting circuit with a time window of $\sim 13$ ns. Using this method, we
measure a coincidence rate of $\sim$ 200 $\text{s}^{-1}$. In comparison, we
measure a photon detection rate of $\sim$ 65,000 photons per second with a
Gaussian seed beam of $\sim$ 9 mW using our SET technique. This orders-of-
magnitude improvement suggests that our experiment based on SET offers a time-
saving method to fully characterize the spontaneous nonlinear responses of
ultra-thin bulk crystals or novel nonlinear metasurfaces [36] because the
intensity of the nonlinear response depends on the propagation length of the
beam inside the material. Another main advantage of our experimental scheme is
our ability to efficiently measure the full modal structure of the entangled
photon pairs produced by SPDC, including both radial and angular parts of
Laguerre-Gaussian modes. Current methods based on interferometry[37, 38, 20]
or coincidence-counting[39] only measure the OAM spectrum of the down-
converted photon pair where the contribution from different radial modes are
summed up as long as they share the same angular mode. A direct projection
measurement of the spatial mode on the down-converted photon pairs is
technically challenging due to the extremely low pair production rate. Our
method can be applied to directly measure the full two-photon spatial mode
spectrum, including both angular and radial modes, with modest modification.
To achieve this, we can simply replace the projection measurement unit in our
experiment setup with the recently developed radial mode sorters [40] or the
latest multiplane light conversion method[41] that can sort the spatial mode
of the incident beam in an arbitrary set of spatial mode basis.

Conclusion.—In conclusion, we report the first proof-of-principle experiment
to measure the full spatial mode spectral amplitude of entangled photon pairs
in high-dimensional space produced by an SPDC source with an efficient
classical method. We measure the OAM spectral amplitude for the lowest radial
modes ($p_{i}=p_{s}=0$) and study its dependence on beam waist in a complete
set of Laguerre-Gaussian spatial basis ($\text{LG}_{l}^{p}$). We proved that
our method is also efficient and robust to measure the joint spectral
amplitude for arbitrary signal and idler spatial modes produced in SPDC. Our
experimental demonstration with the SET technique paves the way for the
possible full characterization of many weak down-conversion sources such as
novel nonlinear metasurfaces[36].

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