# Low temperature synthesis of BiFeO3 nanoparticles with enhanced
magnetization and promising photocatalytic performance in dye degradation and
hydrogen evolution

M. A. Basith∗, Nilufar Yesmin, and Rana Hossain  Nanotechnology Research
Laboratory, Department of Physics, Bangladesh University of Engineering and
Technology, Dhaka-1000, Bangladesh.
*Corresponding author<EMAIL_ADDRESS>

DOI: 10.1039/c8ra04599b

###### Abstract

In this investigation, we have synthesized BiFeO3 nanoparticles by varying
hydrothermal reaction temperatures from 200 ∘C to 120 ∘C to assess their
visible-light driven photocatalytic activity along with their applicability
for hydrogen production via water splitting. The rhombohedral perovskite
structure of BiFeO3 is formed for hydrothermal reaction temperature up to 160
∘C, however, for a further decrement of reaction temperature a mixed sillenite
phase is observed. The XRD Rietveld analysis, XPS analysis and FESEM imaging
ensure the formation of single-phase and well crystalline nanoparticles at 160
∘C reaction temperature with 20 nm of average size. The nanoparticles
fabricated at this particular reaction temperature also exhibit improved
magnetization, reduced leakage current density and excellent ferroelectric
behavior. These nanoparticles demonstrate considerably high absorbance in the
visible range with low band gap (2.1 eV). The experimentally observed band gap
is in excellent agreement to the calculated band gap using the first-
principles calculations. The favorable photocatalytic performance of these
nanoparticles has been able to generate more than two times of solar hydrogen
compared to that produced by bulk BiFeO3 as well as commercially available
Degussa P25 titania. Notably, the experimentally observed band gap is almost
equal for both bulk material and nanoparticles prepared at different reaction
temperatures. Therefore, in solar energy applications, the superiority of BFO
nanoparticles prepared at 160 ∘C reaction temperature may be attributed not
only to solely their band gap but also to other factors, such as reduced
particle size, excellent morphology, well crystallinity, large surface to
volume ratio, ferroelectricity and so on.

##  I Introduction

Nanotechnology has made tremendous impact in contribution to the solution of
local as well as global energy crisis by converting sunlight directly into
electric/chemical power. A promising area for the application of
nanotechnology is the hydrogen economy. As a non-polluting source of energy,
hydrogen (H2) can be a good alternative fuel of the future ref1 ; ref2 ; ref3
without polluting the environment. H2 production can be performed from
different domestic feedstock including hydrocarbon fossil fuels and water.
Notably, H2 produced by splitting of water ref11001 ; ref11002 ; ref11003
under solar light illumination are of great promise as a fuel that is
essentially carbon free and inexhaustible in nature with potential for
generating low cost power in power plants as well as in vehicles of next
generation.

Photocatalytic disintegration of water ref7003  into H2 and O2 while
irradiated by light in the visible region has been considered as one of the
promising routes for H2 production as an endless source of clean fuel for many
applications. In 1972, Fujishima and Honda used successfully titania (TiO2)
photoelectrode for the decomposition of water ref3  under illumination with
sunlight and without any applied electric power. Since then many propositions
along with numerous researches have been conducted on hydrogen production via
water splitting ref4 . In the past decade, extensive studies were performed on
a broad range of materials, including modified TiO2 ref5 , SrTiO3 ref6  and
metal oxide-metal hybrids like ZnO/(La,Sr)CoO3 ref7  and CdS/TiO2/Pt ref8  to
improve their photocatalytic water splitting ability under UV-visible light
irradiation. Recently, significant amount of research is being done with a
view to improving the photocatalytic activity of the multiferroic materials
such as bismuth iron oxide (BiFeO3) to investigate its potentiality for solar
hydrogen generation ref9 ; ref1 . BiFeO3 (BFO) intrigues with its multiferroic
properties being potentially applicable in energy-related problems especially
for photocatalytic hydrogen production imputed to its relatively small band-
gap (2.6 eV) ref11 ; ref12 ; ref13 ; ref14 . It is reported that BiFeO3
generates hydrogen at a greater amount than commercially available titania
under exact parametric conditions ref9 . Utilization of a somewhat broader
sunlight spectrum along with a large polarization value of BFO efficiently
generated electron-hole pairs ref101 . Notably, BFO could potentially supplant
the widely investigated photocatalyst TiO2 that has an inherent limitation of
wide band gap (3.2 eV), qualifying its utilization to only 4% of the solar
spectrum ref102 . Despite the great potential it possesses, the somewhat
inferior photocatalytic performance of bulk BiFeO3 restrains its
commercialization ref15 ; ref16 .

Multiferroic BiFeO3 was discovered in 1960, however, it remains difficult to
synthesize single phase BFO nanoparticles with excellent morphology owing to
the fact that the temperature range for stability is in fact quite narrow for
this material. If the synthesis condition, particularly the processing
temperature is not subject to accurate control, other impurity phases such as
Bi2Fe4O9 appears in BFO nanoparticles. Therefore, a number of synthesis
techniques for BFO nanoparticles have been developed at different
temperatures. These include soft chemical routes (e.g. sol-gel method) ref23 ,
ultrasonication technique ref17 , microemulsion technique ref24  and
microwave-hydrothermal process ref25 . Most of these synthesis techniques,
inherently cost inefficient, comprise of higher than 400 oC calcination
temperatures for achieving phase purity for BFO which introduces irregular
morphology and broad particle size distribution. While sol-gel has gained
popularity among different wet chemical techniques, it has a stringent
requirement of an annealing temperature of 600 oC as the concluding step as
well as a post treatment similar to solid-state synthesis techniques for
elimination of impurity phases. Combustion synthesis can yield nanosized BFO
powder at the expense of phase purity ref27  being inferior to the ones
synthesized by the soft chemical route ref23 . Hence, synthesis of phase pure
BFO nanoparticles at a moderate preparation condition has become quite
challenging. Since many years, hydrothermal synthesis of phase pure BFO
powders gained intensive research interest due to its potential in
synthesizing crystalline ceramics at a temperature of 200 ∘C or even lower
without any further step of calcination. This is a cost effective and simple
method, and the primacy of this technique is its significantly lower required
temperature than both solid-state and sol-gel syntheses. During the
preparation of BFO, such a low processing temperature qualifies the reactants
from getting volatilized as well as minimizing the calcination introduced
impurities.

Therefore, it is obvious that enormous effort was paid to conduct extensive
research to synthesize pure BFO nanoparticles using hydrothermal technique.
However, there is still lack of a thorough investigation on synthesis,
characterization and application of pure BFO nanoparticles with desired phase
and crystallinity prepared at very low temperature using a facile, low cost
synthesis route like hydrothermal. It is indeed very essential to shed light
to know different crystallographic phases formed at different reaction
temperatures and finally figure out the appropriate reaction temperature to
obtain phase pure BFO nanoparticles ensuring their quality as required for
practical applications. In this investigation, we delineate the synthesis of
BFO nanoparticles adapting a facile hydrothermal process for different
reaction temperatures and a fixed KOH concentration of 10 M ref75 . With a
view to knowing the crystallographic phases, we scrutinized Rietveld
refinement on powder XRD data of the samples synthesized over a wide range of
temperatures varying from 200 ∘C to 120 ∘C. Notably, in most of the previous
hydrothermal investigations either crystallographic phases of BFO
nanoparticles were not identified by Rietveld or related analysis and/or it
was done at a particular temperature ref75 ; ref200  not like here over a wide
range of temperatures. We observed that well-dispersed, single-phase
perovskite BFO nanoparticles with average particle size 20 nm can be produced
by hydrothermal process at temperature as low as 160 ∘C (from here this
material is referred as BFO 160 unless otherwise specified). A further
reduction of temperature to 140 ∘C did not allow fabrication of single phase
perovskite structure rather produce sillinite type mixed phases nanoparticles.
Interestingly, the single-phase BFO 160 nanoparticles demonstrate excellent
multiferroic, optical and photocatalytic properties. The hydrogen evolution
experiment exhibited that BFO 160 nanoparticles generated more than two times
hydrogen than that of BFO bulk material and commercially available TiO2 under
the same experimental conditions.

##  II Experimental details

###  II.1 Sample Preparation

In the present investigation, the chemical reagents are analytical grade
bismuth nitrate (Bi(NO3)3.5H2O), ferric nitrate (Fe(NO3)3.9H2O), and potassium
hydroxide (KOH). In a typical procedure, 1mmol bismuth nitrate pentahydrate
[Bi(NO3)3.5H2O] and 1mmol ferric nitrate nonahydrate [Fe(NO3)3.9H2O] in
stoichiometric proportions were dissolved in 50mL of 10M KOH solution and
stirred it for 4h ref75 ; ref26 ; ref200 . The mixture was transferred to a
Teflon-lined autoclave (stainless steel, capacity 100 ml) and was heated at
different reaction temperatures between 200 ∘C to 120 ∘C for 6 hours followed
by natural cooling to bring it to room temperature. Centrifugation was applied
to collect the final products and consequent rinsing was performed with
distilled water and ethanol. Prior to further characterization, they were
dried at 90 ∘C for 6 hours. Bulk BiFeO3 sample had also been prepared from
high purity oxides using solid state reaction method with an objective of
comparing its properties to the synthesized nanoparticles ref76 .

###  II.2 Materials Characterization

The crystal structures and phase composition of the synthesized samples were
determined from XRD data using a diffractometer (PANalytical Empyrean) with a
Cu X-ray source (Wavelength, ${\lambda}$: Kα1= 1.540598 Åand Kα2 = 1.544426 Å.
The particle size along with morphology was observed with help of field
emission scanning electron microscopy (FESEM) (XL30SFEG; Philips, Netherlands
and S4300; HITACHI, Japan). X-ray photoelectron spectroscopy (XPS) was used to
investigate the chemical states of synthesized nanoparticles. A vibrating
sample magnetometer (VSM) was used for measuring the magnetic properties of
the BFO powders at room temperature. The leakage current density along with
polarization vs electric field hysteresis loops were measured using a
ferroelectric loop tracer in conjunction with external amplifier (10 kV).
Pellets were prepared by pressing the powders with a hydraulic press followed
by annealing at 750 ∘C with high heating rate (20 ∘C/min) with a view to
measuring the electrical properties ref201 . An Ultraviolet-visible (UV-vis)
spectrophotometer (UV-2600, Shimadzu) was used to obtain diffuse reflectance
spectra (DRS) of the samples for wavelength ranging from 200 to 800 nm. BaSO4
powder, a nonabsorbing standard material, was used as the reference for total
reflectance with the focused beam spot size being approximately of 2 mm2.

###  II.3 Theoretical Calculation of Optical Properties

With a view to understanding the optical properties of BFO bulk and BFO 160
nanoparticles, the optical absorption was calculated based on crystallographic
structural parameters obtained from Rietveld refinement ref701  of this
investigation. The calculations were performed using first-principles density
functional theory (DFT) within the plane wave pseudopotential (PWPP) framework
as implemented in the Cambridge Serial Total Energy Package (CASTEP) ref702 .
The generalized gradient approximation (GGA) with the Perdew-Burke-Ernzerhof
(PBE) exchange-correlation functional within ultrasoft pseudopotentials (USP)
was implemented to describe the electron-ion interaction of valance electrons
of $Bi\left(6s^{2}6p^{3}\right)$ , $Fe\left(3d^{6}4s^{2}\right)$ and
$O\left(1s^{2}2p^{4}\right)$ ref703 . Plane wave cutoff and k-point sampling,
tested with a cutoff of 400 eV and a k-point sampling of 6$\times$6$\times$6,
was found sufficient for the rhombohedral unit cell of the samples. Spin
polarized mode during self-consistent field (SCF) calculations was endorsed
and a SCF tolerance of $10^{-7}$ eV/atoms was used. To assure a precise
agreement with experimental results, the on-site Coulomb interaction was
included in the DFT+ U approach with U = 3 eV to 7 eV with interval 0.5 eV for
Fe 3d orbital ref704 . The optical absorption coefficient was obtained by the
equation $\leavevmode\nobreak\
\alpha=\sqrt[]{2\mu_{0}\omega\leavevmode\nobreak\
}\sqrt[]{\leavevmode\nobreak\
\sqrt[]{\varepsilon_{1}^{2}\left(\omega\right)+\varepsilon_{2}^{2}\left(\omega\right)-\varepsilon_{1}\left(\omega\right)}}$,
where $\varepsilon_{1}\left(\omega\right)$ and
$\varepsilon_{2}\left(\omega\right)$ are frequency dependent real and
imaginary parts of dielectric function, $\omega$ is photon frequency,
$\leavevmode\nobreak\ \mu_{0}$ is the permeability of free space ref705 . The
real part of the dielectric function $\varepsilon_{1}\left(\omega\right)$ can
be evaluated from the imaginary part $\varepsilon_{2}\left(\omega\right)$ by
the famous Kramers-Kronig relationship ref706 . Direct optical band gap of the
samples was obtained from the equation $\leavevmode\nobreak\ \left(\alpha
h\upsilon\right)^{n}=A\left(h\upsilon-E_{opt}\right)$, where A is a constant
and n denotes the transition type as follows: n = 2 for direct allowed, 2/3
for direct forbidden, 1/2 for indirect allowed and 1/3 for indirect forbidden
transitions ref707 . By extrapolating the linear proportion of the
$\left(\alpha h\upsilon\right)^{n}\leavevmode\nobreak\ vs\leavevmode\nobreak\
h\upsilon$ plot to $\left(\alpha h\upsilon\right)^{n}=0$ the optical band gap
of the samples was calculated using Tauc plot ref708 .

###  II.4 Photocatalytic performance

The photocatalytic performance of BFO bulk material and hydrothermally
prepared nanoparticles at different reaction temperatures were subject to
evaluation by photodegradation of Rhodamine B (RhB) ref7004  under
illumination of visible light in aqueous solutions using a 500 W Xe lamp as a
solar simulator accompanied with a visible cutoff filter (${\lambda}$ $\geq$
420 nm). Beginning with a concentration of 15 mg/L of RhB, 80 mg of catalyst
was later added to the solution in a quartz glass reactor for each experiment.
Before illumination, the mixture was magnetically stirred in the dark for 1
hour for achieving the equilibrium between adsorption and desorption
processes. After sufficient agitation, 4 mL of mixture was taken followed by
centrifugation at 5000 rpm for 10 min so that the catalyst powders are
removed. The concentration of RhB was then estimated from the measurement of
the maximum absorbance at 553 nm with a UV-visible spectrophotometer. Testing
for stability, the remaining photocatalyst powders in suspension were
separated by centrifugation after photocatalytic degradation of RhB and
distilled water was used to wash them to remove the residual RhB. Prior to
using them in another photocatalytic reaction, they were further dried. The
same procedure was followed three times. For comparison commercially available
Degussa P25 titania nanoparticles were used to perform dye degradation
experiment under the same experimental conditions.

###  II.5 Hydrogen Generation

A photocatalytic hydrogen generation experiment has been performed in a
slurry-type photochemical reactor. In a typical experiment, 20mg of catalyst
particles is measured and initiated in the reaction container with 30ml water.
Magnetic stirring is applied to mix the solution while purging the system with
argon gas for 30min to ensure an atmosphere, necessarily inert in nature as
required for the splitting process. A 500W Xenon lamp was used for
photoillumination. The gas was collected when each of the experiments ended
and scrutinized in the gas chromatography (GC) device that is equipped with a
thermal conductivity detector (TCD) and a gas analyzer for identifying the gas
components. The GC programming, set up with a reverse polarization, made the
hydrogen peaks occur in an upwards direction so that we can make a comparative
analysis of the peak intensities of the different gases that were produced.
The hydrogen evolution test was also performed for commercially available
Degussa P25 titania nanoparticles under the same experimental conditions.

##  III Results and discussions

###  III.1 Structural characterization

Fig. 1 shows the Rietveld refined XRD patterns of BFO nanoparticles
synthesized by hydrothermal process at reaction temperatures 200 ∘C, 180 ∘C,
160 ∘C, 140 ∘C and 120 ∘C with the peaks indexed on the basis of Rhombohedral
structure (JCPDS File No. 71-2494). For the comparison, XRD pattern of BFO
bulk polycrystalline sample prepared via conventional solid state reaction is
also depicted in Fig. 1 (a). Rietveld refinement was performed with the
FULLPROF package ref301 . The structural variables and constituent phases (in
wt%) calculated from the Rietveld refined XRD spectra of the synthesized
materials have been provided in Supplemental Table 1 ref401 . The small values
of Rwp and Rp shown in Supplemental Table 1 ref401  indicate very good fit for
hydrothermally prepared BFO nanoparticles to the Rhombohedral structure.
Reproduction of all the observed reflections can be performed from this
structural model. Calculated lattice parameters (a and c) have been listed in
Table 1 for all the major phases.

Figure 1: Rietveld plots of XRD patterns of (a) BFO bulk material synthesized
by solid state reaction technique and hydrothermally prepared nanoparticles at
reaction temperatures (b) 200 ∘C (c) 180 ∘C (d) 160 ∘C (e) 140 ∘C (f) 120 ∘C.

As was mentioned, the major phase of the synthesized materials is observed to
be of Rhombohedral crystal structure belonging to space group R3c. However,
some secondary impurity phases are present for both bulk polycrystalline
material and hydrothermally prepared nanoparticles at reaction temperatures
140 ∘C and 120 ∘C. For nanoparticles prepared at hydrothermal reaction
temperatures 200 ∘C to 160 ∘C no obvious impurity phase (like Fe2O3, Fe3O4,
Bi2Fe4O9 etc,) has been determined to the precision limit of XRD. Moreover, it
is seen that for hydrothermal reaction temperature up to 160 ∘C, the
nanoparticles (Fig. LABEL:fig1 (b-d)) present a single-phase structure. For a
further decrease of reaction temperature to 140 ∘C and 120 ∘C a structural
transition to sillenite type Bi25FeO40 (Fig. 1 (e-f)) phase is formed. Thus,
the structural scrutiny clearly indicates that the minimal reaction
temperature for obtaining perovskite type pure BFO phase is 160 ∘C. Therefore,
we have carried out further characterization for nanoparticles prepared
hydrothermally at reaction temperatures of 200 ∘C-160 ∘C.

From the refinements of the XRD patterns, the Fe-O-Fe bond angle and FeO6
octahedral tilt angle $\omega$ are recapitulated in Table 1. The Fe-O-Fe bond
angle of BFO bulk material and nanoparticles prepared at 200 ∘C, 180 ∘C and
160 ∘C reaction temperatures are 160.8753∘, 158.6353∘, 158.3842∘, 152.8377∘,
respectively as shown in Table 1. The average Fe-o-Fe bond angle is maximum
for bulk BFO material, however, a monotonic decrease of the average Fe-O-Fe
angle from 158.6353∘ to 152.8377∘ is observed with decrease of hydrothermal
reaction temperatures. Correspondingly, we observed an increase of the FeO6
octahedral tilt angle ($\omega$) with the decrease of reaction temperatures.
The alterations in the Fe-O-Fe bond angle and FeO6 octahedral tilt angle play
role to modify the magnetic behavior of the synthesized nanoparticles.

Table 1: Structural parameters determined from XRD spectra

Sample | a=b | c | Volume | bond angle |  FeO6 tilt
---|---|---|---|---|---
| (Å) | (Å) |  (Å)3 |  Fe-O-Fe (∘) |  angle ($\omega$ in ∘)
BFO Bulk | 5.5775(1) | 13.8663(1) | 373.577 | 160.8753 | 10.310
BFO (200 ∘C) | 5.5743(1) | 13.8612(3) | 373.003 | 158.6353 | 10.682
BFO (180 ∘C) | 5.5761(1) | 13.8634(3) | 373.314 | 158.3842 | 10.807
BFO (160 ∘C) | 5.5758(1) | 13.8656(4) | 373.327 | 152.8377 | 15.857

###  III.2 Morphology analysis

The morphology of the surface and the size of the fabricated particles were
analyzed by using FESEM imaging. The surface morphology and their
corresponding histograms of bulk BFO sample produced by solid state reaction
technique and BFO materials synthesized by hydrothermal method at 200 ∘C, 180
∘C and 160 ∘C reaction temperatures are shown in Fig. 2. According to Fig. 2
(a) and corresponding histogram Fig. 2 (b), the average grain size of bulk BFO
material is around 1 $\mu$m. The size of the bulk BiFeO3 was also determined
from the XRD pattern (maximum intensity peak) using Scherrer equation and
found to be around 600 nm which is comparable to the value obtained from FESEM
image (Fig. 2 (a) and (b)). FESEM images in figure 2 (c-d) show BFO materials
synthesized for the reaction temperatures 200 ∘C, and 180 ∘C, respectively.
Fig. 2 (e-f) depicts the BFO nanoparticles synthesized at 160 ∘C reaction
temperature and their corresponding histogram for size distribution.

Figure 2: FESEM images of surface morphology of (a) BFO bulk material
synthesized by solid state reaction technique and (b) corresponding histogram.
Hydrothermally prepared nanoparticles at reaction temperatures (c) 200 ∘C (d)
180 ∘C, respectively. The nanoparticles and their corresponding histogram
prepared at hydrothermal reaction temperature 160 ∘C are depicted in (e) and
(f), respectively.

The electronic micrographs obtained by FESEM, Fig. 2 (c-d), show the particles
are prone to get interconnected with one another and aggregating to larger
particles at higher reaction temperatures i.e. at 200 ∘C and 180 ∘C. However,
for a comparatively lower reaction temperature of 160 ∘C, spherical and well
dispersed BFO nanoparticles, Fig. 2 (e-f) with sizes ranging from 20 nm to 30
nm is formed. Our investigation demonstrates that the morphology of the
synthesized BiFeO3 materials changed from agglomerations to regular spherical
shape with a narrow distribution of size when the reaction temperature is
reduced from 200 ∘C to 160 ∘C. Previous investigation also reported such a
change in morphology of BFO nanoparticles by varying the concentration of KOH
for a fixed hydrothermal reaction temperature of 200 ∘C ref881 . We think,
from energy consumption point of view it is much more desirable to control the
surface morphology at a lower hydrothermal reaction temperature of 160 ∘C
under a fixed concentration of KOH. Moreover, Bi is highly volatile and such a
low processing temperature prevents reactants from getting volatilized during
the formation of BFO powders and minimize the amount of impurities as was
mentioned earlier.

During the hydrothermal preparation of BFO powders, the size of the final
products depends on two mutually competitive processes: crystal nucleation and
crystal growth ref882 . Provided the rate of crystal nucleation is greater
than that of growth, then the product particle size becomes smaller. We expect
that in our experiment with the decrease of reaction temperature to 160 ∘C,
the growth rate was decreased, however, nucleation rate was increased and
consequently the average size of the particles was significantly reduced. The
smaller particle size may provide an increased surface area to volume ratio
and substantial adsorption capacity, that is required to obtain a high
photocatalytic performance.

###  III.3 X-ray photoelectron spectroscopic analysis

The XRD analysis and FESEM imaging confirmed that single phase, well
crystalline BFO nanoparticles formed at 160 ∘C reaction temperature may meet
the demand of practical applications. Therefore, X-ray photoelectron
spectroscopy was carried out to further identify the chemical composition of
BFO 160 nanoparticles which were prepared at hydrothermal reaction temperature
160 ∘C. Fig. 3 shows the typical full survey of XPS spectrum of BFO 160
nanoparticles. Observing the survey spectrum, aside from the weak peak of C
1s, only O, Fe, and Bi core levels were detected, indicating that the
resulting product is highly pure. Fig. 3 (b-d) shows the high resolution XPS
spectra of the Bi 4f, Fe 2p and O 1s core levels for the BFO 160
nanoparticles, respectively. The core level XPS spectrum of Bi can be
distinguished by two Gaussian peaks at 4 f7/2 and 4 f5/2 corresponding to the
binding energy of 157.64 eV and 162.93 eV, respectively. The spin-orbit
splitting energy of the Bi 4f is 5.3 eV ref883 ; ref884 . Fig. 3 (c) shows a
high resolution of XPS measurements for Fe contents. Two main peaks: one at
710.9 eV and another at 724.7 eV have been assigned to corresponding states of
Fe 2p3/2 and 2p1/2 with the energy for spin-orbit splitting being 13.35 eV for
Fe 2p ref883 . For a better insight of the chemical composition we have
scrutinized the O 1s peak of BFO 160 nanoparticles (Fig. 3 (d)) and bulk
BiFeO3 materials Fig. 3(e). The uniformly single XPS peak of O 1s (530.29 eV)
demonstrated that the BiFeO3 nanoparticles synthesized at 160 ∘C hydrothermal
reaction temperature prepared in this investigation is indeed a compound of
single-phase. The XPS spectrum of O 1s core level for bulk BiFeO3 materials,
as shown in Fig.3 (e), can be de-convoluted into two symmetric Gaussian peaks.
The lower binding energy peak at 529.8 eV corresponds to the O 1s core
spectrum, while the higher binding energy peak is related to the oxygen
vacancy in the bulk BFO sample.

Figure 3: XPS spectra of the as-synthesized BFO 160 nanopowders (a) a typical
XPS survey spectrum, (b–d) high-resolution XPS core spectra of Bi 4f, Fe 2p,
and O 1s, respectively. (e) XPS spectra of O 1s of BFO bulk materials.

###  III.4 Magnetic characterization

The field dependent magnetic hysteresis loops (M-H) of bulk BFO material and
hydrothermally synthesized nanoparticles are investigated at room temperature
with an applied magnetic field of up to $\pm$15 kOe. In Fig. 4, the obtained
magnetization for bulk BFO prepared by solid state reaction technique changes
linearly with magnetic field which demonstrates its antiferromagnetic nature
ref33 ; ref11113 . However, the magnetic behavior of BFO nanoparticles varies
depending on hydrothermal reaction temperatures. We have calculated the
remanent magnetization (Mr) of the synthesized samples from the M-H hysteresis
loops ref500 ; ref11112 . The maximum magnetization (Ms) at an applied
magnetic field of 13 kOe is also calculated from the hysteresis loops. Both Mr
and Ms are higher for BFO nanoparticles prepared at 160 ∘C hydrothermal
reaction temperature. In particular at 13 kOe applied magnetic field, the Ms
value is 11.3 emu/g which is exceptionally high compared to the values
reported in literature ref501 ; ref502 . In investigation ref501 , BFO
nanoparticles with a particle size of 14 nm were prepared using sol-gel method
for annealing temperature of 400 ∘C and the obtained Ms was 1.55 emu/g at 50
kOe. The observed Ms was 1.4 emu/g at 70 kOe for 18 nm BFO nanoparticles
fabricated by sol-gel method with annealing temperature 425 ∘C ref502 . In our
investigation, such a remarkable high value of magnetization (11.3 emu/g) of
the hydrothermally synthesized nanoparticles at reaction temperature 160 ∘C
demands an extensive investigation of the magnetic behavior of this material.

Figure 4: The room temperature M–H hysteresis loops of BFO bulk material
synthesized by solid state reaction technique and hydrothermally prepared
nanoparticles at reaction temperatures 200 ∘C, 180 ∘C and 160 ∘C.

The rationale behind improved magnetization can be imputed to several factors,
such as: (i) The Rietveld refinement of XRD data (Table I) clearly reveals
that the Fe-O-Fe bond angle is minimum for nanoparticles with average size 20
nm produced at hydrothermal reaction temperature 160 ∘C. Such a decrement in
Fe-O-Fe bond angle is crucial for developing ferromagnetism due to the
strengthening of magnetic interaction caused by decreased Fe-O-Fe bond angle.
This is consistent with previous investigations which reported that
ferromagnetism can be realized in BFO by tuning Fe-O-Fe bong angle with
reduced particle size ref503 ; ref504 . (ii) At room temperature,
magnetization in G-type antiferromagnetic bulk BFO is subdued along with a
spin helical ordering structure showing a periodicity of 62 nm. Compared to
bulk materials, in the case of phase pure nanoparticles, the periodic spiral-
regulated spin structure can be modified with the decrease in size of
particles, and hence magnetization increases.(iii) The enhanced ferromagnetism
in BFO nanoparticles with average particle 20 nm may be due to the reduced
size of the phase pure nanoparticles. With decrease in particle size, the
surface-to-volume ratio is enhanced and hence the surface spins contribute
more to the arrant magnetic moment of the particle ref17 ; ref500 ; ref501 ;
ref502 .

Figure 5: (a) Leakage current density of BFO bulk material and nanoparticles
prepared at 160 ∘C hydrothermal reaction temperature. The polarization vs.
electric field (P–E) hysteresis loops of (b) BFO bulk material and (c)
nanoparticles prepared at 160 ∘C hydrothermal reaction temperature.

To further confirm the multiferroicity of hydrothermally prepared BFO 160
nanoparticles, P–E hysteresis loops were investigated. Prior to conducting P-E
hysteresis, the leakage current density (J) versus electric field (E)
measurements were carried out.

From Fig. 5 (a), it can be implied that the leakage current density of the
hydrothermally synthesized BFO 160 nanoparticles is smaller compared to that
of the bulk. Impurity phases along with oxygen vacancies contribute to the
high leakage current of bulk material ref601 ; ref502 . Previous demonstration
from Fig. 1 (d) for BFO 160 nanoparticles indicate inhibition of impurity
phases. Furthermore, Fig. 3 shows that in BFO 160 nanoparticles, we observed
only O 1s core spectrum, and obviously no oxygen vacancy related defects.
Therefore, we think that the oxygen vacancies, principally induced by
volatilized Bi3+ ion in the bulk material, were suppressed in BFO 160
nanoparticles and consequently the density of leakage current was reduced.

Fig. 5 (b-c) shows the polarization of ferroelectric hysteresis loops of (b)
bulk BFO and (c) BFO 160 nanoparticles measured for varying electric fields.
Driving frequency maintained at 50 Hz, continuously increasing electric field
resulted in the increase of remanent polarization as stronger electric field
with its higher level of driving power contributed to the reversal of
ferroelectric domains ref603 .

Freely movable charges are expected to contribute more to the electrical
hysteresis loop for bulk BFO. Thus, the bulk BFO material shows a round shaped
P-E loop as observed in Fig. 5 (b) owing to high leakage current which is
evident from Fig. 5 (a). For BFO 160 nanoparticles, the contribution of the
dynamic charges to the polarization is reduced due to the decrease in rounded
shapes of the loops. Furthermore, the P-E loops of BFO nanoparticles tend to
become increasingly typical which can be associated with the reduction in
leakage current density ref604  and decreased space charge defect ref605
compared to that of bulk material. Thus, the measurement of electrical
properties shows an improvement in ferroelectric behavior of BFO 160
nanoparticles. While comparing the P-E loops between BFO bulk and BFO 160
nanoparticles, Fig. 5 (b) and (c) also demonstrate that the significantly
improved breakdown voltage up to 100 kV/cm for BFO 160 nanoparticles compared
to 10 kV/cm for bulk material.

###  III.5 Optical characterization

Photocatalytic performance of a material depends on its ability to effectively
absorb visible light. Optical band gap signifies the minimum energy of the
photons that a material can absorb to generate electron-hole pairs via
interband transition. To determine the optical band gap, the diffuse
reflectance spectra of the synthesized materials were obtained from UV-vis
spectrophotometric measurements. Determining F(R), a parameter proportional to
the absorption coefficient, from the diffuse reflectance value using Kubelka-
Munk function, we can have an estimate of the optical band gapref606 ,

$F(R)=\frac{(1-R)^{2}}{2R}.$ (1)

The band gap of the materials can be calculated by forming the Tauc plot using
the following equation ref606 ,

$F(R)^{*}h\nu=A(h\nu-E_{g})^{n},$ (2)

where, h$\nu$, A, and Eg denote the energy of photons, constant of
proportionality, and optical band gap, respectively. n is a number that takes
the value of 1/2 and 2 for direct and indirect transitions, respectively.
Since, BFO is widely regarded as a material with direct band gap, we have
considered n=2 and obtained the Tauc plots accordingly as shown in Fig. 6.

Figure 6: Tauc plots for band gap estimation of BFO bulk material synthesized
by solid state reaction technique and hydrothermally prepared nanoparticles at
reaction temperatures 200 ∘C, 180 ∘C and 160 ∘C.

Fig. 6 displays $[F(R)*h\nu]^{2}$ vs $h\nu$ curves for optical band gap
calculation of BFO bulk materials and nanoparticles hydrothermally synthesized
at different reaction temperatures. Extrapolating the linear region of the
curve to the energy axis, we can determine the band gap. Optical band gap for
bulk BFO material is found to be 2.1 eV. For hydrothermally prepared
nanoparticles at reactions temperatures 200 ∘C, 180 ∘C and 160 ∘C, the direct
band gaps are 2.05 eV, 2.11 eV and 2.1 eV, respectively. This energy band gap
is consistent with value reported in Ref. ref15 ; ref11217 . However, in those
investigations, nanoparticles were produced at considerably higher annealing
temperatures. The small band gap exhibited by synthesized BFO bulk and
nanoparticles indicates their potential as good photocatalysts since they are
capable of absorbing visible range photons effectively. However, all the
synthesized materials have almost equal band gaps and hence, superiority of a
particular sample in photocatalytic applications cannot attributed solely to
its band gap. Rather, there are other factors which will also play important
roles in determining the efficiency of a material as photocatalyst such as
particle size, surface morphology, phase purity, ability to suppress electron-
hole recombination etc.

####  III.5.1 Theoretical calculation of band gap

BFO bulk and BFO 160 nanoparticles were subject to First-principles
calculation with a view to comparing the experimentally obtained values of
band gap to the theoretically calculated ones. While standard DFT methods are
widely in use for such calculations, experimentally obtained outcomes of
BiFeO3 are more closely comparable to the DFT calculated values when Hubbard U
parameter, a measure of effective on-site Coulomb interaction, is chosen
carefully ref2003 ; ref709 ; ref711 ; ref712 ; ref713 . Experimentally
obtained crystallographic parameters were refined by Rietveld method and their
geometry was optimized prior to using them for DFT calculations. PDOS,
electronic band structure and Optical absorption spectra of both BFO bulk and
BFO 160 nanoparticles were obtained via DFT calculations using GGA-PBE
functionals. While performing DFT calculations, U value was varied from 3.0 eV
to 7.0 eV at 0.5 eV interval. DFT calculated absorption spectra closely match
the experimentally obtained ones for U = 4.5 eV and therefore, further
observations were carried out for this value of U. As we can observe from the
PDOS spectra (Fig. 7(a) and 7(b)) of both BFO bulk and BFO 160, conduction
band minima(CBM) is dominated by 3d orbital of Fe atoms and 2p orbital of
oxygen atoms dominate the valence band maxima(VBM). We can infer from this
observation that the electron transitions from VBM to CBM can be attributed to
these two orbitals and there are no major d-d transitions involved. From the
electronic band structures as shown in Fig. 7(c) and 7(d), VBM lies between Z
and L point for BFO bulk and BFO 160 nanoparticles and hence the calculated
direct band gap for both materials found to be 2.1 eV. Since, transition from
2p orbital of oxygen atoms to 3d orbitals of Fe atoms is allowed and direct,
our use of n = 2 in the formation of Tauc plot from experimentally obtained
absorption coefficients is justified. Fig. 7 (e) shows the theoretical optical
absorption coefficients of BFO bulk and BFO 160 obtained from our DFT
calculation. These absorption spectra were transformed into Tauc plots as
shown in Fig. 7 (f) and tangents were drawn to the linear region of the plot.
Extrapolating the tangent to the energy axis, optical band gap of the
corresponding material was found from the energy axis intercept. As observed
from Fig. 7 (f), both BFO bulk and BFO 160 exhibit the same direct optical
band gap of 2.1 eV. These theoretically calculated values are in excellent
agreement with the ones experimentally determined (2.1 eV for both BFO bulk
and BFO 160). With this agreement between theoretical and experimental
calculation, we can conclude that there has been no significant change in the
optical band gap for BFO bulk and nanoparticles.

Figure 7: DFT calculated PDOS spectra of (a) BFO bulk and (b) BFO 160; band
structures of (c) BFO bulk and (d) BFO 160; DFT calculated absorption spectra
of (e) BFO bulk and BFO 160; (f) plot of DFT calculated ($\alpha$$h\nu$)2 vs.
$h\nu$ for BFO bulk and BFO 160.

###  III.6 Photocatalytic activity

The photocatalytic performance of the fabricated materials was studied
extensively through degradation of the typical organic contaminant Rhodamine-B
(RhB) under visible light illumination ref1405  at ${\lambda}$ $\geq$ 420 nm.
Fig. 8 (a) shows how intensity of the absorbance peak of RhB for BFO 160
nanoparticles typically varies under visible-light irradiation. The absorbance
peak corresponding to RhB decreases gradually with increasing time, implying
that RhB has been decomposed by BFO. Due to the well known fact that RhB dye
shows a general resistance to decomposition, the implication of the results is
that the photocatalytic activity of BFO nanoparticles is quite efficient and
hence indicates the promise as visible light photocatalysts for BFO
nanoparticles.

For evaluating the efficiency of degradation of RhB, the maximum intensity
ratio C/C0 is plotted in Fig. 8 (b), in which C0 and C are the maximum initial
intensity (0 h) and maximum intensity at a specific time (1-4 h) respectively
for the absorption spectra of RhB under stimulated sunlight irradiation. A
blank test is evaluated for RhB (Fig. 8 (b)) and it exhibits a negligible
degradation efficiency which clearly shows a nominal self-degradation
potential of RhB. For example, after 4 hours of irradiation without BFO
materials, RhB was degraded by less than 3%, whereas the bulk BFO
photocatalyst could decompose 56% of RhB after 4 h irradiation. Notably, when
hydrothermally prepared nanoparticles were used as photocatalysts, the
degradation efficiency was increased to 59 % and 61 % for nanoparticles
prepared at 200 ∘C and 180 ∘C hydrothermal reaction temperatures.
Interestingly, RhB photodegradation attains a significant improvement in
efficiency to 79 % for BFO 160 nanoparticles which were prepared at 160 ∘C
reaction temperature.

Figure 8: (a) Absorption spectra of rhodamine B (RhB) under visible-light
irradiation in the presence of BFO nanoparticles prepared at 160 ∘C
hydrothermal reaction temperature. (b) Photocatalytic degradation of RhB as a
function of the visible light irradiation time for blank sample, Degussa P25
titania, BFO bulk material and hydrothermally prepared nanoparticles at
reaction temperatures 200 ∘C, 180 ∘C and 160 ∘C. (c) Pseudo-first order
kinetics fitting data for the photodegradation of RhB. (d) Volume of H2
evolution as a function of irradiation time during the photocatalytic
splitting of water. (e) Schematic illustration of RhB degradation and water
splitting mechanism using BFO nanoparticles as catalysts under visible light
irradiation.

In a similar investigation BiFeO3 particles with the size of 100-150 nm were
synthesized at the temperature as low as 120 ∘C via a urea-assisted
hydrothermal process ref9001 . Their photocatalytic activity was evaluated by
the degradation of RhB in aqueous solution under visible-light irradiation
using a 500 W Xe lamp with a cutoff filter (${\lambda}$ $\geq$ 420 nm). The
condition for photocatalytic experiment was same as that of our experiment.
Moreover, the optical band gap of the particles was 2.1 eV which is also
consistent with that of the nanoparticles synthesized in the present
investigation. However, in Ref. [ref9001 ] it was observed that only 40% of
RhB was decomposed after 4 h visible-light irradiation. Though the band gap of
the BFO particles synthesized both in Ref. [ref9001 ] and in our investigation
was same, however, the photoctalytic dye degradation is much higher while BFO
160 nanoparticles are used as photocatalysts. In another recent investigation
ref9002 , the photocatalytic activity of BFO nanoparticles with a particle
size of 40 nm was evaluated by photocatalytic decomposition of RhB in aqueous
solution under visible light irradiation. It is observed that the pure BFO
photocatalyst could decompose 22.3% of RhB after 270 min visible light
irradiation.

The photodegradation of BFO 160 nanoparticles is much higher compared to that
of other materials in particular commercially available Degussa P25 titania
nanoparticles and BFO bulk materials. For comparison, the photodegradation
efficiency of BFO 140 and BFO 120 materials was also evaluted under the same
experimental condition. The degradation efficiency was found to 53 % and 42 %
for using BFO 140 and BFO 120 particles, respectively. Our investigation
clearly demonstrated that the photocatalytic activity of BFO 160 nanoparticles
for photocatalytic decomposition of RhB in aqueous solution under visible
light irradiation is higher compared than that of other materials under
investigation. The enhanced photodegradation efficiency of BFO 160
nanoparticles might have resulted from the reduced particle size with
excellent crystallinity. Looking at the SEM image, Fig. 2 (e-f), significantly
greater surface area of the BFO 160 nanoparticles can be attributed to the
higher photodegradation efficiency. It should also be noted that compared to
commercially available photocatalyst TiO2, that responds only to UV
irradiation, BFO nanoparticles are more advantageous as they make use of the
visible portion of the spectrum.

For quantitative investigation of the rate of degradation, the experimental
data were fitted adapting a first-order model as elucidated by the following
equation, ln(C0/C) = kt ref801 , where C0 and C denote the respective
concentrations of RhB at distinct irradiation times of t0 and t, k is the
first-order rate constant and can be regarded as a basic kinetic parameter for
a number of different photocatalysts. For the photodegradation of RhB, the
pseudo-first order kinetics fitting data has been shown in Fig. 8 (c). The
corresponding reaction rate constant (k) values were calculated to be 7.66 x
10-3, 2.27 x 10-1, 2.47 x 10-1, 2.72 x 10-1, and 3.23 x 10-1 min-1 for the
blank sample, bulk BFO, hydrothermally prepared nanoparticles at reaction
temperatures 200 ∘C, 180 ∘C and 160 ∘C, respectively. Among them, the BFO 160
nanoparticles exhibited the highest k value, which is 1.42 times higher than
that of bulk BFO. Notably, BFO 160 nanoparticles were subject to recyclability
test under the same test conditions. The test result indicates excellent
stability of these nanoparticles after 4 cycles.

Table 2: Calculated band gap energy (Eg), rate constant (K), and degradation efficiency of BFO bulk and hydrothermally prepared nanoparticles. Sample | Eg | Rate constant | Degradation
---|---|---|---
| in eV | (K) x 10 -1 | (%) after 4 h
BFO Bulk | 2.1 | 2.27 | 56
BFO (200 ∘C) | 2.05 | 2.47 | 59
BFO (180 ∘C) | 2.11 | 2.72 | 61
BFO (160 ∘C) | 2.1 | 3.23 | 71

###  III.7 Photocatalytic Hydrogen Production

Now, we have examined the potential of BFO materials for hydrogen generation
via water splitting under visible light irradiation ref11001 . No hydrogen
evolution could be detected when a blank experiment without any photocatalyst
was performed in dark conditions. Fig. 8 (d) shows the hydrogen evolution rate
of BFO bulk material as well as hydrothermally prepared BFO 160 nanoparticles
in mL H2/g catalyst plotted against visible light irradiation time in hours
(h). For comparison, we have inserted in Fig. 8 (d) the hydrogen production
rate of commercially available Degussa P25 titania nanoparticles. The outcome
of our investigation demonstrates that BFO 160 nanoparticles prepared
hydrothermally at reaction temperatures 160 ∘C generate nearly two times
hydrogen compared to that of bulk BFO material after 4 h of illumination.
Compared to commercially available TiO2, the production rate of BFO 160
nanoparticles is more than double after 4 h of illumination. In a separate
investigation ref9 , BFO nanoparticles as photocatalyst were also used for
solar hydrogen generation from water and the production rate after 4 h
illumination is equivalent with that of our investigation. But here the
noteworthy point is that, in Ref. [ref9 ], the nanoparticles of size 50-60 nm
were produced at comparatively high temperature sol-gel process. Particularly
the nanoparticles used for hydrogen production in Ref. [ref9 ] was subject to
annealing at 600 ∘C for 2 hours in N2 atmosphere. Whereas in the present
investigation a facile, low cost hydrothermal process has been used for
fabrication of nanoparticles and the reaction temperature was only 160 ∘C
which ensures the less energy consumption.

###  III.8 Photocatalytic Mechanism

The photocatalytic performance under visible light irradiation depends on a
number of parameters cooperating with each other ultimately ameliorating the
photocatalytic performance. Fig. 8 (e) schematically illustrates degradation
of RhB and water splitting mechanism using BFO nanoparticles under visible
light irradiation ref11002 .

Photocatalysis is an electrochemical process that involves the transfer of the
photogenerated electrons and holes between a semiconductor and an electrolyte.
The semiconductor in this investigation is BFO, and the electrolyte is the RhB
solution. When the BFO nanoparticles are dispersed in the RhB solution under
visible light irradiation, the electrons in their valence band are excited to
the conduction band by absorbing the photons of the visible light. This
transition creates holes in the valence band of BFO. On the other hand, an
electrolyte in equilibrium has one or more redox couples with specific redox
potentials. If the potential of the photogenerated electrons in a
semiconductor is lower than any of these redox couples, the electrons will
flow into the electrolyte to perform reduction of that couple. Similarly, if
the potential of the photogenerated holes of the semiconductor is higher than
a redox couple, the holes will flow into the electrolyte to oxidize it. The
reduced and oxidized species can initiate further reactions that are
responsible for degrading the electrolyte. Besides determining the potentials
of the conduction band minima (CBM) and valence band maxima (VBM) of BFO, we
need to investigate the redox potentials of the redox couples that dictate the
degradation of RhB to learn about the photocatalytic mechanism behind the
degradation of RhB. The potentials of CBM and VBM of BFO have been determined
to be 0.3 V and 2.4 V respectively using the absolute electronegativity theory
by Nethercot and Butl and et al. refNeth ; refBut .

The degradation of RhB dye depends primarily on two critical redox reactions.
First, the photogenerated electron can react with the surface adsorbed O2
(redox potential: -0.16 V vs. NHE) to form O${}_{2}^{-}$, which will further
react with RhB to cause degradation refXXX ; refXXXXX . Second, the
photogenerated holes can react with the OH- ionized from the water molecules
to produce $\cdot OH$ (redox potential: 2.38 V vs. NHE). This $\cdot OH$ can
further oxidize the RhB molecules. Therefore, a photocatalyst needs to possess
a CBM <-0.16 V to drive the first reaction and VBM >2.38 V to drive the second
one efficiently. Therefore, only the photogenerated holes are capable of
performing the second reaction. However, when photons with energy higher than
2.56 eV are incident on BFO nanoparticles, the electrons in their valence band
can be excited to the higher states of the conduction band where their
potential can be more negative than -0.16 V. These photogenerated electrons
will be able to perform the first reaction. Since BFO can absorb visible light
photons with energies higher than 2.56 eV, both these reactions can contribute
to the degradation of RhB ref11109 . In addition, RhB self-photoexcitation may
also play a role in assisting the photodegradation process. The RhB molecules
can absorb the visible range photons to reach an excited state of a potential
of -1.09 V refNANO . Since the potential of electrons in this state is more
negative than the CBM of BFO nanoparticles, the electrons of the photoexcited
state will be transferred to the conduction band of BFO. However, these
electrons cannot participate in the photocatalytic degradation process as they
cannot reduce O2 to O${}_{2}^{-}$. Consequently, these electrons will be
accumulated in the conduction band of BFO and the transfer of electrons from
the photoexcited state of RhB will stop when both states reach the same
potential. Hence, the self-photoexcitation of RhB may not have a significant
effect on the photocatalytic degradation of RhB.

It is well known that the size of the particles and surface area significantly
effect the enhancement of the photocatalytic degradation. In our
investigation, the degradation of RhB with BFO 160 nanoparticles is
considerably greater than that of bulk BFO owing to their smaller size and
higher surface area. The band gap is another important parameter as the low
band gap of photocatalysts might increase the photocatalytic activities
ref2004 . We observed that the band gap is almost same (2.1 eV) for
nanoparticles prepared at all hydrothermal reaction temperatures as well as
the bulk BFO. While sunlight hits the photocatalyst, photons possessing
greater energy than the optical band gap of the photocatalyst cause stronger
light absorption. The estimated band gap 2.1 eV for both BFO bulk and
nanoparticles implied that light with $\lambda$ less than 590 nm could be
absorbed by e– to jump from valence band (VB) to conduction band (CB) which
covers a broad region of the solar energy spectrum. With all the materials
under scrutiny having almost equal band gap, the other factors prove to be
more important for achieving higher photocatalytic efficiency of BFO 160
nanoparticles rather than band gap itself.

It was mentioned already in introduction section that, BFO is an well known
multiferroic materials. It is evident that the ferroelectric property of a
material enhances its ability to decolorize Rhodamine-B, a typical dye
molecule, while irradiated by solar light ref1001 . Ferroelectric materials
have a spontaneous polarization that is caused by the non-centrosymmetry of
the crystal structure. As nanoparticles are more strained than their bulk
counterparts, their crystal structures deviate from the bulk structure and
give rise to even greater non-centrosymmetry for BFO. This essentially
increases the magnitude of spontaneous polarization of BFO nanoparticles,
especially for BFO 160 with smallest particle sizes as evident from our
experimental values. The BFO 160 nanoparticles, assumed to be creating
electrical field in the surrounding medium, improves redox reactions along
with the adsorption of dye molecules, especially polar molecules such as water
on the photocatalyst surface. This adsorption helps the redox reactions to
commence easily as the carrier transport is now relatively easier.
Furthermore, it is anticipated that ferroelectrics can create channels to
transfer the charge carriers to the surface of photocatalyst and significantly
reduce the recombination probabilities as the photogenerated electron-hole
pairs are driven to opposite directions by the electric field they possess.

##  IV Conclusions

We have produced single-phase, well crystalline BiFeO3 nanoparticles with
favorable morphology using a facile, low cost hydrothermal process at reaction
temperature as low as 160 ∘C. The experimentally observed band gap (2.1 eV)
was almost equal for both bulk and all nanoparticles prepared at different
reaction temperatures and was well consistent with value obtained from the
first principles calculation. However, the BFO 160 nanoparticles prepared at
temperature 160 ∘C demonstrated much higher photocatalytic activity compared
to the photocatalytic performance showed by bulk BFO and other nanoparticles.
The enhanced photocatalytic activities of BFO 160 nanoparticles may be
associated with their crystallinity, phase purity, excellent morphology,
higher surface to volume ratio, and the efficient separation and migration of
photogenerated charge carriers. As efficient photocatalyst, the synthesized
BFO 160 nanoparticles generated more than two times of solar hydrogen via
water splitting than that of bulk BiFeO3 as well as commercially available
Degussa P25 titania. The outcome of this investigation also demonstrated an
improved multiferroic properties of technologically important BiFeO3
nanoparticles prepared at temperature as low as 160 ∘C which is cost effective
with less energy consumption. BiFeO3 nano-structured particles prepared
through this investigation have shown a great promise in enhancing the
production of solar H2, a carbon free fuel, using two important renewable
sources: water and solar energy.

##  V Acknowledgements

This work was financially supported by Ministry of Education, Government of
Bangladesh (Grant No. PS 14267) and the Infrastructure Development Company
Limited (IDCOL), Dhaka, Bangladesh.

##  VI Appendix A. Supplementary data

Supplementary data to this article can be found online.

##  VII Data availability

The raw and processed data required to reproduce these findings cannot be
shared at this time due to technical or time limitations.

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[a]Matthew Feickert

# pyhf: pure-Python implementation of HistFactory with tensors and automatic
differentiation

Lukas Heinrich  Giordon Stark

###### Abstract

The HistFactory p.d.f. template is per-se independent of its implementation in
ROOT and it is useful to be able to run statistical analysis outside of the
ROOT, RooFit, RooStats framework. pyhf is a pure-Python implementation of that
statistical model for multi-bin histogram-based analysis and its interval
estimation is based on the asymptotic formulas of “Asymptotic formulae for
likelihood-based tests of new physics”. pyhf supports modern computational
graph libraries such as TensorFlow, PyTorch, and JAX in order to make use of
features such as auto-differentiation and GPU acceleration. In addition,
pyhf’s JSON serialization specification for HistFactory models has been used
to publish 23 full probability models from published ATLAS collaboration
analyses to HEPData.

##  1 Introduction

Measurements in High Energy Physics (HEP) aim to determine the compatibility
of observed events with theoretical predictions. The relationship between them
is often formalised in a statistical model $f(\bm{x}|\bm{\phi})$ describing
the probability of data $\bm{x}$ given model parameters $\bm{\phi}$. Given
observed data, the likelihood $\mathcal{L}(\bm{\phi})$ then serves as the
basis to test hypotheses on the parameters $\bm{\phi}$. For measurements based
on binned data (histograms), the HistFactory [1] family of statistical models
has been widely used for likelihood construction in both Standard Model (SM)
measurements (e.g. Refs. [2, 3]) as well as searches for new physics (e.g.
Ref. [4]) and reinterpretation studies (e.g. Ref. [5]). pyhf [6, 7] is
presented as the first pure-Python implementation of the HistFactory
specification. In addition to providing a Python and command line API for
HistFactory model building and inspection, it leverages modern open source
$n$-dimensional array libraries to take advantage of automatic differentiation
and hardware acceleration to accelerate the statistical inference and reduce
the time to analyst insight.

##  2 HistFactory Formalism

HistFactory statistical models — described in depth in Ref. [8] and Ref. [9] —
center around the simultaneous measurement of disjoint binned distributions
(channels) observed as event counts $\bm{n}$. For each channel, the overall
expected event rate is the sum over a number of physics processes (samples).
The sample rates may be subject to parametrised variations, both to express
the effect of free parameters $\bm{\eta}$ and to account for systematic
uncertainties as a function of constrained parameters $\bm{\chi}$, whose
impact on the expected event rates from the nominal rates is limited by
constraint terms. In a frequentist framework these constraint terms can be
viewed as auxiliary measurements with additional global observable data
$\bm{a}$, which paired with the channel data $\bm{n}$ completes the
observation $\bm{x}=(\bm{n},\bm{a})$. The full parameter set can be
partitioned into free and constrained parameters
$\bm{\phi}=(\bm{\eta},\bm{\chi})$, where a subset of the free parameters are
declared parameters of interest (POI) $\bm{\psi}$ (e.g. the signal strength)
and all remaining parameters as nuisance parameters $\bm{\theta}$.

$f(\bm{x}|\bm{\phi})=f(\bm{x}|\overset{\begin{subarray}{c}\text{free}\\\
\downarrow\end{subarray}}{\bm{\eta}},\underset{\begin{subarray}{c}\uparrow\\\
\text{constrained\hskip
28.45274pt}\end{subarray}}{\bm{\chi}})=f(\bm{x}|\overset{\begin{subarray}{c}\text{\hskip
56.9055ptparameters of interest}\\\
\downarrow\end{subarray}}{\bm{\psi}},\underset{\begin{subarray}{c}\uparrow\\\
\text{\hskip 28.45274ptnuisance parameters}\end{subarray}}{\bm{\theta}})$ (1)

The overall structure of a HistFactory probability model is then a product of
the analysis-specific model term describing the measurements of the channels
and the analysis-independent set of constraint terms:

$f(\bm{n},\bm{a}\,|\,\bm{\eta},\bm{\chi})=\underbrace{\color[rgb]{0,0,1}{\prod_{c\in\mathrm{\,channels}}\prod_{b\in\mathrm{\,bins}_{c}}\textrm{Pois}\left(n_{cb}\,\middle|\,\nu_{cb}\left(\bm{\eta},\bm{\chi}\right)\right)}}_{\begin{subarray}{c}\text{Simultaneous
measurement}\\\ \text{of multiple
channels}\end{subarray}}\underbrace{\color[rgb]{1,0,0}{\prod_{\chi\in\bm{\chi}}c_{\chi}(a_{\chi}|\,\chi)}}_{\begin{subarray}{c}\text{constraint
terms}\\\ \text{for ``auxiliary measurements''}\end{subarray}},$ (2)

where within a certain integrated luminosity one observes $n_{cb}$ events
given the expected rate of events $\nu_{cb}(\bm{\eta},\bm{\chi})$ as a
function of unconstrained parameters $\bm{\eta}$ and constrained parameters
$\bm{\chi}$. The latter has corresponding one-dimensional constraint terms
$c_{\chi}(a_{\chi}|\,\chi)$ with auxiliary data $a_{\chi}$ constraining the
parameter $\chi$. The expected event rates $\nu_{cb}$ are defined as

$\nu_{cb}\left(\bm{\phi}\right)=\sum_{s\in\mathrm{\,samples}}\nu_{scb}\left(\bm{\eta},\bm{\chi}\right)=\sum_{s\in\mathrm{\,samples}}\underbrace{\left(\prod_{\kappa\in\,\bm{\kappa}}\kappa_{scb}\left(\bm{\eta},\bm{\chi}\right)\right)}_{\text{multiplicative
modifiers}}\,\Bigg{(}\nu_{scb}^{0}\left(\bm{\eta},\bm{\chi}\right)+\underbrace{\sum_{\Delta\in\bm{\Delta}}\Delta_{scb}\left(\bm{\eta},\bm{\chi}\right)}_{\text{additive
modifiers}}\Bigg{)}$ (3)

from constant nominal rate $\nu_{scb}^{0}$ and a set of multiplicative and
additive rate modifiers $\bm{\kappa}(\bm{\phi})$ and $\bm{\Delta}(\bm{\phi})$.

##  3 pyhf

Through adoption of open source $n$-dimensional array (“tensor” in the machine
learning world) computational Python libraries, pyhf decreases the
abstractions between a physicist performing an analysis and the statistical
modeling without sacrificing computational speed. By taking advantage of
tensor calculations and hardware acceleration, pyhf can achieve comparable or
better performance than the C++ implementation of HistFactory on data from
real LHC analyses in most situations. pyhf’s default computational backend is
built from NumPy and SciPy, and supports TensorFlow, PyTorch, and JAX as
alternative backend choices. These alternative backends support hardware
acceleration on GPUs, and in the case of JAX JIT compilation, as well as auto-
differentiation allowing for calculating the full gradient of the likelihood
function — all contributing to speeding up fits.

###  3.1 JSON Schema

The structure of the JSON specification of HistFactory models [8] used by pyhf
closely follows the original XML-based specification [1]. The JSON
specification for a HistFactory workspace is a primary focus of Ref. [8], but
a workspace can be summarised as consisting of a set of channels (an analysis
region) that include samples and possible parameterised modifiers, a set of
measurements (including the POI), and observations (the observed data). Figure
1 demonstrates a simple workspace representing the measurement of a single
two-bin channel with two samples: a signal sample and a background sample. The
signal sample has an unconstrained normalisation factor $\mu$, while the
background sample carries an uncorrelated shape systematic. The background
uncertainties for the bins are 10% and 20% respectively. Use of this JSON
specification has allowed for the publication of 23 full statistical models
from ATLAS analyses to HEPData at the time of writing in 2022. This has been a
significant step forward in enabling reinterpretation and recasting of LHC
results by the broader particle physics community [10].

⬇

{

"channels": [

{ "name": "singlechannel",

"samples": [

{ "name": "signal",

"data": [5.0, 10.0],

"modifiers": [ { "name": "mu", "type": "normfactor", "data": null} ]

},

{ "name": "background",

"data": [50.0, 60.0],

"modifiers": [ {"name": "uncorr_bkguncrt", "type": "shapesys", "data":
[5.0,12.0]} ]

}

]

}

],

"observations": [

{"name": "singlechannel", "data": [50, 60]}

],

"measurements": [

{ "name": "Measurement", "config": {"poi": "mu", "parameters": []} }

]

}

Figure 1: A toy example of a 2-bin single channel workspace with two samples.
The signal sample has expected event rates of 5.0 and 10.0 in each bin, while
the background sample has expected event rates of 50.0 and 60.0 in each bin.
An experiment provided the observed event rates of 50.0 and 60.0 for the bins
in that channel. The uncorrelated shape systematic on the background has 10%
and 20% uncertainties in each bin, specified as absolute uncertainties on the
background sample rates. A single measurement is defined which specifies $\mu$
as the POI [8].

###  3.2 Enabling Analysis Ecosystems

In addition to being used in ATLAS analyses, and in the flavor physics
community [11, 12], pyhf has been used as a computational engine for
reinterpretation studies by the particle physics phenomenology community [13,
14] and as the inference engine for Scikit-HEP library cabinetry [15], as well
as other more analysis specific open source projects [16, 17]. The adoption of
pyhf as a library for other projects to build upon has large implications for
establishing standards and providing improvements across ecosystems of
analysis tools. Of particular note, the Institute for Research and Innovation
in Software for High Energy Physics (IRIS-HEP) [18] has adopted pyhf as a core
part of its Analysis Systems pipeline — a demonstrator model for modern
distributed computing for experiments in the high-luminosity LHC (HL-LHC) era
— which has provided rigorous testing of its interoperability with other
tools. Improvements to pyhf directly impact all the areas highlighted in
Figure 2. In addition to its computational abilities, pyhf is highly portable
given its pure-Python nature and use of dependencies, like SciPy, that are
broadly trusted in computational science and have been built for ubiquitous
architectures. This allows for full pyhf runtimes to be natively used in novel
environments, such as the Pyodide port of CPython to WebAssembly/Emscripten.
While Pyodide is not optimal for serious computational use cases, the ability
to use the full pyhf API allows for creation of statistical linting and
visualization tools that use the same tooling as in production while
leveraging interactivity of web native platforms enabled by Pyodide and the
PyScript framework.

Figure 2: Overview of the IRIS-HEP Analysis Systems pipeline for analysis in
the HL-LHC era. The red outline indicates the areas of the pipeline in which
pyhf is used either directly or as an underlying library.

##  4 Conclusions

pyhf is the first pure-Python implementation of the HistFactory specification
that leverages modern open source $n$-dimensional array libraries as
computational backends to exploit automatic differentiation and hardware
acceleration to speed up fits and reduce the time to scientific insight. It
provides a Python and command line API for building, inspection, and to
perform statistical inference for HistFactory models, and its JSON model
serialization has enabled publication of full statistical models from the
ATLAS collaboration and improved reinterpretations. As pyhf is an open source
library that has been built as part of the Scikit-HEP community project it has
been readily adopted by a growing number of other libraries and tools as a
computational and inference engine, allowing for improvements in the library
API and computational backends to propagate to the broader user community.
Growing community support and interaction, adoption across the broader
particle physics community, and rigorous testing from LHC experiments and
IRIS-HEP systems has demonstrated that pyhf has become a key component of the
growing ecosystem of Pythonic open source scientific tools in particle
physics.

## Acknowledgments

Matthew Feickert’s contributions to this work were supported by the U.S.
National Science Foundation (NSF) under Cooperative Agreement OAC-1836650
(IRIS-HEP). Lukas Heinrich is supported by the Excellence Cluster ORIGINS,
which is funded by the Deutsche Forschungsgemeinschaft (DFG, German Research
Foundation) under Germany’s Excellence Strategy – EXC-2094-390783311.

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