and M. A. Basith*

# DyFeO3 electrode material with ultra-wide voltage window for aqueous
symmetric supercapacitors

Mohasin Tarek  Ferdous Yasmeen<EMAIL_ADDRESS>Nanotechnology
Research Laboratory, Department of Physics, Bangladesh University of
Engineering and Technology, Dhaka-1000, Bangladesh

###### Abstract

Aqueous supercapacitors (SCs) encounter limitations in operational voltage and
energy density due to the low decomposition voltage of water. Here, we
fabricate aqueous symmetric supercapacitors (ASSCs) employing DyFeO3 as an
electrode material. This hybrid SC in a 0.5 M Na2SO4 aqueous electrolyte
exhibits a significantly high working voltage of 2.5 V, with an energy density
of 41.81 Wh kg-1 at a power density of 1250 W kg-1, maintaining 94%
capacitance retention after 5000 cycles. By incorporating 20% volume of
acetonitrile with water in the electrolyte, we extend the potential window to
3.1 V, with an energy density of 84.43 Wh kg-1 at a power density of 1550 W
kg-1. The as-fabricated ASSC shows promising stability during a 300-hour float
voltage test with almost intact capacitance retention and Coulombic
efficiency. For the first time, our study unveils the potential of porous
DyFeO3 as an electrode material for advancing ASSCs, featuring an
unprecedented ultra-wide voltage window, along with significantly large energy
and power densities.

###### keywords:

Symmetric hybrid supercapacitor, Ultra-wide electrochemical stability, Aqueous
electrolyte, Dysprosium orthoferrite

##  1 Introduction

Electrochemical supercapacitors (SCs) are emerging as promising alternatives
for sustainable and reliable energy storage devices [1]. Their advantages,
including high power density, long cycle life, fast charge-discharge rates,
wide operating temperature range, and safety, make them particularly
compelling when compared to traditional batteries [2, 3, 4]. Nevertheless, the
limited energy densities, typically around 1-10 Wh kg-1, observed in
commercially available supercapacitors significantly constrain their broad
application [5, 6]. Consequently, there is a pressing need for extensive
research involving various parameters in particular electrode materials and
electrolytes to enhance energy densities and associated factors like cost,
safety, and lifetime to unlock the full potential of SCs [1]. In the present
context, maximizing energy density (E) is achieved through the effective
expansion of the voltage window (V), leveraging the square proportionality
between V and E for a specific capacitance. Notably, the stability window of
aqueous supercapacitors (SCs) is restricted by the narrow stability range of
water, approximately 1.23 V [7, 8]. In pursuit of a significant expansion of
stability windows for constructing SCs, the careful selection of electrode
materials and the judicious choice of electrolytes emerge as key factors [9,
6, 10].

To enhance the electrochemical energy storage capabilities of electrode
materials, various strategies, including doping, nanocomposite formation,
molecular cross-linking, intricate structural design, and chemical
modifications, are employed [11, 7, 12, 13, 6]. While employing these
strategies enhances electrochemical energy storage capabilities, it is crucial
to acknowledge that they can often introduce complexity, reduce reliability,
and increase costs. Particularly, in designing asymmetric SC, a common
strategy to enhance voltage windows involves employing diverse complex
electrode materials for the cathode and anode [14]. This approach, coupled
with intentional imbalances in mass and charge between the electrodes,
amplifies the intricacy and cost associated with both the fabrication and
functionality of the SCs [14]. Therefore, the adoption of a unified,
perovskite-like electrode material for both the cathode and anode, capable of
demonstrating a hybrid charge storage mechanism involving simultaneous
utilization of electrochemical double-layer capacitance and pseudocapacitance,
may prove advantageous [15, 16].

Notably, perovskite oxides, with the formula ABO3 (where A represents a
lanthanide or alkali earth element, and B denotes a transition metal), have
been a focal point in electrode research due to their structural stability,
compositional adaptability, and inherent oxygen vacancies [15, 16]. Among
various perovskite oxide materials, dysprosium orthoferrite (DyFeO3),
crystallizing in a perovskite orthorhombic structure [17], has been
extensively studied for its potential applications in magnetic data storage,
spintronics devices, and as a multiferroic material [18, 19, 20]. Despite
these well-explored areas, DyFeO3’s unique electronic structure—characterized
by a suitable band gap, diverse oxidation states of the constituent elements,
and inherent oxygen vacancies—designates it as a promising candidate for
applications beyond spintronics, particularly in electrochemical energy
storage systems [15]. The diverse oxidation states of the constituent
elements, Dy and Fe, may enable redox reactions, particularly in applications
involving charge transfer within electrochemical systems [15]. Notably, the
inherent oxygen vacancies can promote more efficient redox reactions at the
electrode-electrolyte interface, enhance the material’s surface reactivity,
and serve as defect sites that facilitate ion diffusion within the material,
potentially influencing the specific capacitance of the DyFeO3 electrode
material [15, 21, 22, 23]. Moreover, the porous structure of DyFeO3, achieved
through a controlled fabrication route, offers numerous electroactive sites,
thereby enhancing the charge storage capacity in SCs [15]. This characteristic
not only improves the material’s suitability for SC applications but also
helps mitigate water-splitting reactions at elevated voltages, consequently
expanding the electrochemical stability window (ESW) of the electrolyte
solution. Hence, the distinctive characteristics of DyFeO3 have strongly
motivated us to employ it for the first time as electrode material in high-
performance SC applications, demonstrating its potential for advancing
electrochemical energy storage devices.

Apart from the electrode material, the choice of a suitable electrolyte is a
crucial factor that significantly impacts the stability and overall
effectiveness of SCs [10, 24]. Currently, nonaqueous electrolytes are
extensively used in prominent energy storage systems for portable electronics
and automotive purposes due to their wide operating voltage range and high
energy density; however, challenges such as flammability, significant economic
costs, and environmental concerns restrict their widespread applicability [24,
25]. In contrast, aqueous electrolytes present a promising alternative,
offering cost-effectiveness, high ionic conductivity, intrinsic safety, and
environmental compatibility [11, 10, 7, 8]. Considering the advantageous
features of aqueous electrolyte solutions, a 0.5 M Na2SO4 aqueous solution is
chosen as the electrolyte in this investigation. This choice is based on its
neutral pH, non-corrosive nature, small hydrated ion size, excellent
conductivity, and cost-efficiency [24, 26, 27].

To further expand the ESW of aqueous electrolytes, various approaches have
been explored, such as adjusting the pH, increasing electrolyte concentration,
modifying the potential of zero voltage of electrodes, introducing redox-
active additives, balancing electrode masses, and passivating electrode
surfaces [7, 11, 28]. However, these strategies, whether employed individually
or collectively, have demonstrated only modest success in expanding the ESW.
For instance, employing a super-concentrated electrolyte with 21 M lithium
bis(trifluoromethanesulfonyl)imide (LiTFSI) resulted in a “water-in-salt”
(WIS) system, extending the electrochemical stability window (ESW) to 2.4 V
[29]. Another study utilized a 17 M sodium perchlorate aqueous electrolyte,
achieving an ESW of 2.3 V with a Coulombic efficiency of 66% after 10,000
cycles [7]. Conversely, incorporating organic solvents such as acetonitrile,
ethanol, and ethylene glycol as additives in a WIS electrolyte has shown
promise in enhancing the ESW of electrolyte solutions [8, 30]. Interestingly,
the incorporation of organic solvents to increase the ESW, primarily by
lowering salt concentration and decreasing the activity of free water
molecules, proves to be a simple and cost-effective technique [8].

Therefore, in addition to a pure aqueous 0.5 M Na2SO4 electrolyte, we
developed an aqueous-dominant and dilute salt-containing solid electrolyte
interface (SEI) system. This involved incorporating only 10% and 20%
volumetric percent acetonitrile (AN) with water as the solvent medium for the
0.5 M Na2SO4 electrolyte. Subsequently, we fabricated coin cell
supercapacitors utilizing DyFeO3 nanoparticles as the electrode material for
both the cathode and anode, with electrolyte solutions of 0.5 M Na2SO4(aq.)
and 0.5 M Na2SO4(aq.)/20%AN. Notably, the diverse oxidation states, porous
structure, and inherent oxygen vacancies of the DyFeO3 electrode material, in
combination with the 0.5 M Na2SO4 aqueous electrolyte solution, collectively
mitigate water splitting. This breakthrough extends the ESW to 2.5 V, marking
to the best of our knowledge, the first instance of such an achievement for a
hybrid aqueous symmetric supercapacitor. Furthermore, the electrolyte solution
with a 20% volumetric organic additive, AN, i.e.,0.5 M Na2SO4(aq.)/20%AN,
remarkably expanded the ESW to 3.1 V. Thus, the as-fabricated aqueous
electrolyte-dominant hybrid symmetric SC demonstrates an impressive energy
density of 84.43 Wh kg-1 at a power density of 1550 W kg-1, coupled with
excellent rate capability and cycle performance. These results surpass the
integrated performance of any previously reported aqueous electrolyte-
dominated symmetric SC.

##  2 Results

Figure 1: Structural and morphological characterization of the DyFeO3
nanoparticles. a, The Rietveld-refined powder X-ray diffraction (XRD) pattern
indicates the crystalline nature of DyFeO3 in the orthorhombic phase with the
pnma space group. b, FESEM image reveals the porous structure of DyFeO3
nanoparticles. c, TEM image also confirms the porous structure of DyFeO3, as
observed in FESEM imaging; the inset of (f) provides a closer view of the pore
within a nanoparticle. d-f, Elemental mapping images exhibit the distribution
of Dy, Fe, and O atoms in DyFeO3 nanoparticles.

The as-synthesized DyFeO3 crystallizes in an orthorhombic structure with the
space group pnma[4, 31], as confirmed by the Rietveld refinement of the powder
XRD pattern presented in Fig. 1. The crystallographic parameters and
reliability factors are provided in Supplementary Table S3. The calculated
crystallinity (93%) confirms the formation of highly crystalline DyFeO3
particles. The successful formation of DyFeO3 nanoparticles with a porous
structure is evident in the FESEM and TEM images, as depicted in Fig. 1(b) and
1(f), respectively. Most of the pore sizes estimated from the electron
microscopy images range from approximately 10 to 45 nm (ESI Fig. S5(a)).
Subsequently, EDX analysis, as illustrated in supplementary Fig. S5(b), was
conducted to confirm the presence of each element in the synthesized material.
The experimentally determined mass and atomic percentages of the constituent
elements were found to closely match their theoretical values, as detailed in
supplementary Table S1. Following the EDX analysis, the elemental mapping of
DyFeO3 in Fig. 1(c-e) reveals a uniform distribution of Dy, Fe, and O
throughout the material, confirming its homogeneity.

Figure 2: The chemical state analysis of DyFeO3 nanoparticles through XPS. a,
The full survey spectrum reveals distinct peaks corresponding to the oxidation
states of Dy4+, Dy3+, Fe2+, Fe3+, and O2-. The identified oxidation states may
enhance the electrochemical charge storage capability of DyFeO3 as electrode
material. High-resolution XPS spectra of b, Dy 4p, c, Dy 3d, d, Fe 2p, e, Fe
3p, and f O 1s further detail the specific binding energies associated with
each element. The high-resolution XPS spectrum of the O 1s region f, reveals a
satellite peak at 531.05 eV, indicating the presence of oxygen vacancies in
DyFeO3 nanoparticles. These vacancies have the potential to enhance the
electrochemical performance of DyFeO3 electrode material, as discussed in the
text.

X-ray photoelectron spectroscopy (XPS) was employed to analyze the chemical
state of synthesized DyFeO3 nanoparticles, as illustrated in Fig.2. The survey
spectrum of the DyFeO3 nanoparticles depicted in Fig.2(a) confirmed the
presence of Dy, Fe, and O, consistent with the results from energy-dispersive
X-ray (EDX) analysis. High-resolution XPS spectra of DyFeO3 nanoparticles,
presented in Fig.2(b-f), revealed the coexistence of Dy4+, Dy3+, Fe2+, Fe3+,
and O2- ions [32, 33, 34, 35]. These oxidation states might play the role of
enhancing the charge storage capacity of the nanocomposite through a
pseudocapacitive mechanism involving Faradaic redox reactions. The specific
binding energies of all the XPS peaks for DyFeO3 nanoparticles are documented
in Table S2 in the supplementary information. Note that, in Fig.2(f), the
high-resolution XPS spectra of O-1s were deconvoluted, yielding two distinct
peaks at binding energies of 529.53 and 531 eV. These peaks correspond to
metal-oxygen bonds and oxygen vacancies, respectively, with molar ratios of
61.33% and 38.67% [35]. These vacancies can arise from the charge imbalance
resulting from the mixed oxidation states of rare earth ions (Dy3+ and Dy4+)
and transition metal ions (Fe2+ and Fe3+) within the crystal lattice. The
presence of oxygen vacancies in DyFeO3 is likely to enhance charge carrier
generation and improve charge transfer, potentially playing a promising role
in enhancing the material’s capacitance.

Notably, in the synthesis of DyFeO3 perovskite nanoparticles using the sol-gel
method, we introduced the chelating agent ethylene glycol to the solution to
generate a polymeric metal cationic network that serves as the precursor for
the gel. The thermal decomposition of ethylene glycol released gases including
water vapor, carbon dioxide, and volatile organic compounds [36, 37]. These
evolved gases generated internal pressure within the gel matrix, seeking paths
of least resistance and ultimately leading to the formation of voids or pores.
Concurrently, the decomposition induces thermal stress, initiating microcracks
or voids in regions where internal pressure exceeds the material’s structural
strength [36]. Furthermore, the generated gas bubbles act as nucleation sites
for pore formation, coalescing during the synthesis process to establish
larger voids [38]. It is noteworthy that the calcination process at 750 ∘C
also facilitated the removal of volatile components, such as organic residues
and solvent remnants. This crucial process contributed significantly to the
formation of the porous structure in DyFeO3. Moreover, during calcination, we
exposed the material to high temperatures in a controlled atmosphere using N2
gas. This process leads to the extraction of oxygen atoms from the lattice,
resulting in the generation of oxygen vacancies. The porous morphology and the
presence of these vacancies enhance the surface area of DyFeO3, thereby
influencing its reactivity and functional properties. Collectively, these
factors are essential to enhance the electrochemical performance of the DyFeO3
electrode material [15].

Figure 3: FTIR spectroscopy and solvation structure analysis were conducted on
various electrolyte solutions to gain insights into their molecular
interactions. a, The FTIR spectra reveal a stretching mode in the range of
3337 to 3356 cm-1, associated with hydrogen-bonded water molecules. The
absence of a sharp decline in this stretching mode with changing acetonitrile
(AN) concentration, coupled with a shift towards higher frequencies, suggests
the dominance of hydrogen bonds in hybrid electrolyte solutions. This
prevalent H-bonding diminishes the activity of free water molecules. b, SSIP,
CIP and AGG solvation and coordination structure of ions (Na+ and
$\text{SO}_{4}^{2-}$) in AN incorporated 0.5 M Na2SO4(aq.) electrolyte
solution. c, A schematic illustration of the solvation structure and
interfacial interaction between electrode and electrolyte ions in the AN
incorporated 0.5 M Na2SO4(aq.) electrolyte solution. In this system, the
synergistic effect of electrolyte components and electrode material
significantly reduces the activity of free water molecules, which
substantially enhances the ESW of the electrolyte solution.

Before conducting electrochemical characterization, we examined the influence
of various organic additives on the performance of water-in-salt (WIS)
electrolytes. This was done through the incorporation of additives such as
acetonitrile (AN) and ethylene glycol (EG) into a 0.5 M Na2SO4 aqueous
electrolyte. The physical and chemical properties of the various electrolytes
and solvent media were documented in Table S3 and Table S4. Comparative
analysis revealed that, for enhancing the ESW of the WIS (0.5 M Na2SO4(aq.))
electrolyte, AN emerged as the most favorable organic additive. To elucidate
the mechanism underlying the influence of AN additive on the ESW of the hybrid
electrolytes, FT-IR was employed to verify the O–H bending and H-bending
stretching vibration modes. In Fig. S14, the peak at 1641.64 cm-1 is
attributed to the O–H bending vibration mode of water molecules [8]. An
observable shift of the bending vibration peak from 1641.64 to 1645.64 cm-1
was noted with increasing AN ratio, indicative of enhanced O–H bonding,
particularly in the 0.5 M Na2SO4(aq.)/20%AN electrolyte solution [8]. In Fig.
3(a), the stretching mode at 3337.6 cm-1 corresponds to hydrogen-bonded water
molecules. While a sharp decline in the stretching mode was not observed with
varying AN concentrations but peak shifted from 3337.6 to 3356.1 cm-1,
indicating the dominance of hydrogen bonds in the electrolyte solutions [8].
These findings suggest that strengthened O–H bonding and hydrogen bonds
contribute to the reduction in the activity of free water molecules and the
inhibition of water decomposition. Moreover, AN decreases the viscosity of the
electrolyte, with the bonding between AN and H2O identified as stronger than
the bonds between H2O molecules. This phenomenon aids in the formation of a
robust solid electrolyte interface (SEI) layer in 0.5 M Na2SO4(aq.)/20%AN
electrolyte solutions [11].

The ion association interactions of AN incorporated 0.5 M Na2SO4(aq.)
electrolyte solutions were investigated which are schematically shown in Fig.
3(b-c). According to the number of coordination bonds between anions and
cations of electrolytes, the ion association interaction is generally divided
into three categories [8, 39]. First, solvent-separated ion pair (SSIP)
involves solvation molecules coordinating with either Na+ or
$\text{SO}_{4}^{2-}$ ions, keeping them separated within the solvent. This
structure indicates a weaker ion association, as the ions are surrounded by
solvent molecules, suggesting higher ion mobility. Second, contact ion pair
(CIP) is characterized by a $\text{SO}_{4}^{2-}$ anion being coordinated
directly to a Na+ cation, forming a stable structure with direct ion contact.
This arrangement implies a moderate ion association strength, with the ions in
closer proximity. The prevalence of CIP structures may indicate a more
intimate interaction between Na+ and $\text{SO}_{4}^{2-}$, potentially
influencing the kinetics of electrochemical reactions [8]. Third, anion–cation
aggregate (AGG) refers to the coordination of $\text{SO}_{4}^{2-}$ anions with
two or more Na+ ions or vice versa, forming larger aggregates or clusters.
This structure suggests strong ion association and reduced solvation of ions
within the aggregate, leading to decreased ion mobility. AGG structures may be
more common in concentrated solutions or solvents with lower dielectric
constants [8, 39]. Conversely, in our electrolyte system, the SSIP and CIP
structures were prevalent due to the dilute electrolyte and high dielectric
constant of water, facilitating strong solvation capabilities through hydrogen
bonding and dipole interactions. The addition of AN introduced hydrogen bonds
with water molecules and reduced the activity of free water molecules. The
coordination between solvent molecules and electrolyte ions weakened with
increasing AN content, leading to a reduction in the decomposition of water
molecules on the electrode surface. The low viscosity of AN facilitated ion
mobility towards electrode surfaces, enhancing charge transport, while its
relatively high dielectric constant contributed to the formation of a robust
solvation layer [11]. These observed coordination structures were supported by
FTIR spectra and confirmed the synergistic effect between solvent molecules
and electrolyte ions which might be a key factor to improve ESW.

Figure 4: Electrochemical performances of the fabricated ASSCs (ASSC-1 and
ASSC-2) using DyFeO3 nanoparticles as an electrode material with 0.5 M
Na2SO4(aq.) and 0.5 M Na2SO4(aq.)/20%AN electrolyte solutions, graphite sheet
as a current collector and Whatman filter paper as a separator. a, Schematic
and real illustration of the ASSCs. b, CV curves at a fixed scan rate of 100
mV s-1 show the large integrated CV area with ultra-high (3.1 V) potential
window of ASSC-2 compared to ASSC-1. c, LSV curves demonstrate that the water
oxidation potential of the ASSC-1 and ASSC-2 exceeds the theoretical water
oxidation potential (1.23 V). d, GCD curves at a fixed current density of 1 A
g-1. e, At various current densities, the fabricated ASSCs exhibit high
specific capacitance, $C_{sp}$ upto 253.01 F g-1. f, Nyquist plots show the
very low impedance value i.e. low value of $R_{s}$ and $R_{ct}$, inset shows
the enlarged view of the high-frequency region of Nyquist plots.

###  2.1 Electrochemical performance of as-fabricated aqueous symmetric
supercapacitors (ASSCs)

We aim to develop ASSCs using electrode materials with hybrid charge storage
capability and electrolytes with a wide ESW, predominantly composed of water.
In pursuit of our objectives, we initially conducted cyclic voltammetry (CV),
linear sweep voltammetry (LSV), galvanostatic charge-discharge (GCD), and
electrochemical impedance spectroscopy (EIS) analyses on the DyFeO3 electrode
material. These analyses were performed in various electrolytic solutions,
including 0.5 M Na2SO4(aq.), 0.5 M Na2SO4(aq.)/10%AN, 0.5 M Na2SO4(aq.)/10%EG,
0.5 M Na2SO4(aq.)/20%AN and 0.5 M Na2SO4(aq.)/20%EG, using a three-electrode
configuration. Detailed methodologies, along with corresponding outcomes and
graphical representations, are extensively outlined in the ESI. We observed
that even without additives, DyFeO3 electrode material in a 0.5 M Na2SO4(aq.)
electrolyte demonstrated exceptional electrochemical performance, featuring a
notable ESW of 2.5 V, high specific capacitance (370.9 F g-1), prolonged cycle
life (94%), and low impedance ($R_{s}$ = 5.8 $\Omega$). Furthermore, the
material exhibited elevated EDL capacitance, which is crucial for the
development of high-power-density supercapacitors. Conversely, when subjecting
the DyFeO3 electrode material to electrochemical testing with an electrolyte
solution comprising 0.5 M Na2SO4(aq.)/20%AN, aimed at further increasing the
ESW, the results revealed a widened ESW (3.1 V), increased capacitance (450 F
g-1), relatively prolonged cycle life (92%), and a suitable equilibrium
between the EDL and pseudocapacitance.

In light of this favorable electrochemical performance, we fabricated the
ASSCs using a coin cell arrangement to extend our investigation into the
electrochemical properties of the DyFeO3 electrode material in both 0.5 M
Na2SO4(aq.) and 0.5 M Na2SO4(aq.)/20%AN electrolyte solutions. To enhance
clarity, for the remainder of this manuscript, the aqueous symmetric
supercapacitor featuring DyFeO3 electrode material and 0.5 M Na2SO4(aq.)
electrolyte solution is denoted as ASSC-1, while the supercapacitor utilizing
0.5 M Na2SO4(aq.)/20%AN electrolyte solution as ASSC-2. Initially, we
conducted CV measurements of ASSC-1 at 2.5 V and ASSC-2 at 3.1 V across
different scan rates to assess cyclic reversibility, as illustrated in
supplementary Fig. S14(a-b). A substantial increase in the integrated CV area
of ASSC-2, compared to ASSC-1, was observed, as shown in Fig. 4(b). This
enhancement is attributed to the incorporation of 20% AN in 0.5 M Na2SO4(aq.)
electrolyte solution.

To assess the ESW of the ASSC-1 and ASSC-2, an LSV test was conducted at a
constant scan rate of 0.5 mV s-1, as illustrated in Fig. 4(c). The LSV curves
reveal an impressive ultra-wide ESW of 2.5 V in additive-free ASSC-1 and 3.1 V
in ASSC-2. Remarkably, minimal water splitting is observed within these
potential windows. This broad electrochemical stability window can be ascribed
to the unique characteristics of our synthesized electrode material, including
diverse oxidation states, oxygen vacancies, and porous structure, along with
the selected electrolyte composition featuring high conductivity, high
dielectric constant, low viscosity, and effective solvation [15]. The oxygen
vacancies present in DyFeO3 nanoparticles serve as adsorption sites for Na+
and $\text{SO}_{4}^{2-}$ ions within the electrolyte and influence the
interaction between the electrode and electrolyte ion, which may provide
alternative pathways for charge transfer that do not lead to water-splitting
reactions. Moreover, the nano-size pores present in DyFeO3 electrode
facilitate efficient ion diffusion, as they are larger than the hydrated ion
size of the electrolyte (as detailed in Table S3) [40]. This characteristic
allows Na+ and $\text{SO}_{4}^{2-}$ ions to easily access the interior of the
porous structure, enhancing the transport of charge carriers during the
electrochemical process [41]. The synergistic effects of oxygen vacancies, the
porous structure of DyFeO3 nanoparticles, and the structure of the 0.5 M
Na2SO4(aq.) electrolyte solution collectively contributed to achieving an
ultra-wide ESW of ASSC-1. Conversely, the ESW of the ASSC-2 extends up to 3.1
V, attributed to the synergistic interplay of the DyFeO3 electrode material
and the solvation structure of 0.5 M Na2SO4(aq.)/20%AN electrolyte solution,
as discussed in the earlier section.

In Fig. 4(d), the comparative GCD curves of ASSC-1 and ASSC-2 exhibit that the
charging-discharging duration of the ASSC-2 surpasses that of the ASSC-1. To a
comprehensive understanding of the charging-discharging time under varied
current densities (1, 2, 3, 4, 5, 7, and 10 A g-1), the GCD curves of the
ASSC-1 and ASSC-2 are presented in Fig. S15 (a,b). Notably, GCD curves of
ASSC-1 exhibit a triangular shape, indicative of EDL capacitive-dominant
behavior [42]. On the other hand, the GCD curves of ASSC-2 indicate pseudo-
capacitive-dominant behavior. Moreover, the $C_{sp}$ values were determined at
various current densities using Eq. 1 based on the GCD curves, and the
findings are depicted in Fig. 4(e) [43, 44]. The ASSC-1 demonstrates a
$C_{sp}$ of 192.6 F g-1 at low current density (1 A g-1) and 141.3 F g-1 at
high current density (10 A g-1), whereas for ASSC-2, the corresponding Csp
values are 253 F g-1 and 169.5 F g-1, respectively. In Fig. 4(f), the Nyquist
plots reveal that the series resistance ($R_{s}$) of the ASSC-1 and ASSC-2 is
2.33 and 3.51 $\Omega$, respectively and the charge transfer resistance
($R_{ct}$) is 0.7 and 0.4 $\Omega$, respectively. The reduced impedance
parameters observed can be ascribed to the presence of highly conductive
small-sized ions (Na+ and $\text{SO}_{4}^{2-}$), along with the low viscosity
and high dielectric constant of the solvent medium, which facilitates the easy
penetration of electrolyte ions into the porous DyFeO3 nanoparticles.

Figure 5: Ragone plot, specific capacitance retention and Coulombic efficiency
of the ASSC-1 and ASSC-2. a, Ragone plots exhibit high energy density
comparable with Li-ion battery and high power density comparable with EDLC. b,
At room temperature the fabricated aqueous symmetric supercapacitors exhibit
impressive capacitance retention up to 81% and Coulombic efficiency up to 94%
after 5000 GCD cycles, proving their durability over commercially available
energy storage devices. c, The float voltage test also revealed superior
capacitance retention and Coulombic efficiency of ASSC-2. d, At various
temperatures (5 to 75 oC) excellent capacitance retention and Coulombic
efficiency are observed which is critically important for their practical
application.

The performance of energy storage devices is significantly influenced by their
energy and power density, as these metrics play a crucial role in determining
operational efficiency. To evaluate these key parameters, we calculated the
energy and power density of the ASSCs using GCD data, employing Eq. 2 and Eq.
3 and the results are depicted in Fig. 5(a) through Ragone plot [44].
Specifically, at a power density of 1250 W kg-1, ASSC-1 shows an energy
density of 41.81 W h kg-1, and ASSC-2 exhibits an energy density of 84.43 W h
kg-1 at a power density of 1550 W kg-1. Remarkably, even at higher power
densities, ASSC-1 and ASSC-2 maintain substantial energy density. For
instance, at a power density of 12500 W kg-1, ASSC-1 achieves an energy
density of 30.65 W h kg-1, and similarly, ASSC-2 exhibits an energy density of
56.57 W h kg-1 at 15500 W kg-1. These findings highlight the excellent energy
storage and delivery capabilities of our fabricated ASSCs.

Our fabricated ASSCs exhibit notable rate capability and durability at room
temperature, as shown in Fig. 5(b), retaining 94% and 87% of the initial
capacitance for ASSC-1 and ASSC-2, respectively. This performance is
particularly significant for applications with frequent cycling, such as
electric vehicles and renewable energy systems. Additionally, ASSC-1 and
ASSC-2 exhibit Coulombic efficiencies of 94% and 91% respectively, indicating
efficient charge-discharge processes and minimal energy losses. The observed
high Coulombic efficiency is crucial for the overall performance of the energy
storage devices, ensuring effective retrieval of a substantial proportion of
stored energy during discharge.

The stability of the ASSC-2 was rigorously examined through the float voltage
test, offering a more reliable assessment compared to repeated GCD cycling
[33]. In this experiment, the assembled coin cell supercapacitor underwent a
voltage application of 3.1 V, and a series of five charge and discharge cycles
were carried out at a constant current density of 10 A g-1 with intervals of
12 hours, as illustrated in Fig. 5(c). The ASSC-2 exhibited remarkable
stability over an extended testing period of 300 hours, maintaining
approximately 100 $\pm$ 5 % of its initial capacitance and Coulombic
efficiency. These results demonstrate the robustness and efficiency of the
ASSC-2 within a wide potential window of 3.1 V, which is essential for long-
term stability under challenging operational conditions.

To investigate the practical applicability of our fabricated ASSC-2 across
diverse temperature environments, the capacitance retention and Coulombic
efficiency at high and low temperatures (5 to 75 ∘C) were calculated as shown
in Fig. 5(d). All calculations were conducted relative to results obtained at
25 ∘C. The values of capacitance retention and Coulombic efficiency decreased
at lower temperatures, with the lowest values observed at 5 ∘C, being 75% and
80%, respectively, attributed to sluggish kinetics and diffusion processes
[45, 23]. Conversely, values increased up to 55 °C due to enhanced kinetics
and efficient diffusion, reaching a maximum at this temperature of 135% and
98%, respectively. However, above 55 °C, both capacitance retention and
Coulombic efficiency decreased, reaching their lowest points at 75 ∘C —74% and
82%, respectively. This suggests that at higher temperatures, the advantages
gained from improved kinetics are offset by heightened internal resistance.
Despite these fluctuations, the ASSC-2 demonstrated superior durability and
Coulombic efficiency across the wide temperature range, highlighting its
promising practical applicability.

To assess the stability of DyFeO3 electrode material following 5000 GCD
cycles, once again we conducted XRD spectra, FESEM images, and electrochemical
analyses (CV, GCD, and EIS) on the ASSC-2. The post-cycle characterizations
provided in the Supplementary Information confirm the suitability of our
synthesized DyFeO3 nanoparticles as efficient electrode material for symmetric
supercapacitors with a broad ESW.

The data in supplementary Table S7, including specific capacitance, ESW,
capacitance retention, and energy density, demonstrate the superior
performance of our fabricated ASSCs compared to other assembled
supercapacitors. For instance, when tested in a costly 17 M NaClO4/H2O
electrolyte solution, the AC//AC symmetric supercapacitor retains 66.1%
capacitance after 10,000 GCD cycles within a 2.3 V potential window [7]. In
contrast, ASSC-1, operated in a cost-effective and highly conductive 0.5 M
Na2SO4(aq.) electrolyte solution with an ESW of 2.5 V, exhibits an impressive
94% capacitance retention after 5000 GCD cycles.

Comparatively, a porous carbon symmetric supercapacitor, operating in a high-
density 20 M LiTFSI WIS electrolyte solution with a wide 2.4 V ESW, exhibits a
specific capacitance of 63 F g-1 at a current density of 0.5 A g-1,
accompanied by an energy density of 44 W h kg-1 at 564 W kg-1 [46]. In
contrast, our fabricated ASSC-2, featuring an extended ESW of 3.1 V,
demonstrates a specific capacitance of 253.01 F g-1 and an energy density of
84.43 Wh kg-1 at a power density of 1550 W kg-1, significantly surpassing the
performance of numerous similarly assembled supercapacitors. In addition to
its exceptional electrochemical properties, our fabricated electrolyte system
(refer to Table S6) demonstrates a significantly lower production cost
compared to many other electrolyte systems used previously. Although the cost
of DyFeO3 nanoparticles as an electrode material is slightly higher than that
of commercially available carbon derivatives, the integrated production cost
of our fabricated ASSCs might be more economical, considering their
distinctive energy storage performances and the use of a low-cost,
environmentally friendly electrolyte system.

Figure 6: Schematic illustration of charge storage mechanism of the ASSC-2.
Upon the application of voltage, the electrolyte ionizes, producing Na+ and
$\text{SO}_{4}^{2-}$ ions. These ions migrate toward the electrodes, forming
an Inner Helmholtz Plane (IHP) adjacent to the electrode surface and an outer
Helmholtz plane (OHP) extending into the bulk electrolyte. The interaction
between the IHP and OHP defines the EDL architecture. This setup enables
efficient charge separation, creating a high-capacitance region near the
electrode surface. DyFeO3 nanoparticles, known for their porous structures,
play a crucial role by intercalating electrolyte ions. This intercalation
enhances the system’s pseudocapacitive behavior and charge storage capacity.
The electrolyte’s solvation structure, along with DyFeO3’s porous structure
and oxygen vacancy, collectively contribute to increasing the electrochemical
stability window to 3.1 V. Thus, the intricate charge storage mechanism and
the energy storage potential of the system are demonstrated.

The CV, GCD, and EIS analyses of the ASSCs illustrate the combined influence
of EDL and pseudo-capacitive mechanisms on overall charge storage, as
schematically depicted in Fig. 6 and further explained in the supplementary
information. In this context, the EDL, composed of IHP and OHP, forms through
electrostatic charge separation at the electrode-electrolyte interfaces.
Simultaneously, the pseudo-capacitance results from either electrolyte ion
intercalation into the DyFeO3 electrode surface or redox interactions between
DyFeO3 nanoparticles and electrolyte ions. In Fig. S18, our fabricated ASSC-2
effectively illuminated an LED, showcasing the real-world utility of DyFeO3 as
a hybrid electrode material for high-voltage ASSCs.

In conclusion, we present the first demonstration of a 2.5 V aqueous symmetric
supercapacitor utilizing single-structured DyFeO3 both as cathode and anode.
The porous DyFeO3 nanostructure emerges as a promising supercapacitor
electrode material for practical applications, offering simplicity in
preparation, environmental friendliness, and favorable electrochemical
performance across a wide potential window in pure aqueous and aqueous
dominating electrolytes. Incorporating 20% acetonitrile in a 0.5 M Na2SO4
aqueous electrolyte expanded the electrochemical stability window of the
fabricated symmetric supercapacitor to 3.1 V, surpassing that of common
aqueous-dominating ones. The superior performance of this hybrid
supercapacitor can be attributed to the synergistic interplay of oxygen
vacancies, the porous nanostructure, varied oxidation states of the
constituent elements of DyFeO3, and the solvation structure of the
electrolytes. The as-fabricated hybrid supercapacitor displays a high
capacitance of 253 F g-1 at a current density of 1 A g-1, an impressive energy
density of 84.43 Wh kg-1 at a power density of 1550 W kg-1 in 0.5 M
Na2SO4(aq.)/20%AN electrolyte, maintaining 87% capacitance retention, and 92%
Coulombic efficiency after 5000 cycles. This significant advancement paves the
way for the development of the next generation of high-performance energy
storage devices based on a porous rare-earth orthoferrite electrode material
in aqueous or aqueous-dominating electrolyte solutions.

## References

  *   * Pomerantseva et al. [2019] Pomerantseva, E., Bonaccorso, F., Feng, X., Cui, Y., Gogotsi, Y.: Energy storage: The future enabled by nanomaterials. Science 366(6468), 8285 (2019)
  * Frith et al. [2023] Frith, J.T., Lacey, M.J., Ulissi, U.: A non-academic perspective on the future of lithium-based batteries. Nat. Commun. 14(1), 420 (2023)
  * Albertus et al. [2018] Albertus, P., Babinec, S., Litzelman, S., Newman, A.: Status and challenges in enabling the lithium metal electrode for high-energy and low-cost rechargeable batteries. Nat. Energy 3(1), 16–21 (2018)
  * Du et al. [2010] Du, Y., Cheng, Z.X., Wang, X.L., Dou, S.X.: Lanthanum doped multiferroic DyFeO3: Structural and magnetic properties. J. Appl. Phys. 107(9), 09–908 (2010)
  * Simon and Gogotsi [2008] Simon, P., Gogotsi, Y.: Materials for electrochemical capacitors. Nat. Mater. 7(11), 845–854 (2008)
  * Wang et al. [2012] Wang, G., Zhang, L., Zhang, J.: A review of electrode materials for electrochemical supercapacitors. Chem. Soc. Rev. 41(2), 797–828 (2012)
  * Bu et al. [2019] Bu, X., Su, L., Dou, Q., Lei, S., Yan, X.: A low-cost “water-in-salt” electrolyte for a 2.3 v high-rate carbon-based supercapacitor. J. Mater. Chem. A 7(13), 7541–7547 (2019)
  * Huang et al. [2023] Huang, S., Li, Z., Li, P., Du, X., Ma, M., Liang, Z., Su, Y., Xiong, L.: Ultrahigh-voltage aqueous electrolyte for wide-temperature supercapacitors. J. Mater. Chem. A 11(28), 15532–15539 (2023)
  * Yu et al. [2015] Yu, Z., Tetard, L., Zhai, L., Thomas, J.: Supercapacitor electrode materials: nanostructures from 0 to 3 dimensions. Energy Environ. Sci. 8, 702–730 (2015)
  * Pal et al. [2019] Pal, B., Yang, S., Ramesh, S., Thangadurai, V., Jose, R.: Electrolyte selection for supercapacitive devices: a critical review. Nanoscale Adv. 1(10), 3807–3835 (2019)
  * Xu et al. [2022] Xu, J., Ji, X., Zhang, J., Yang, C., Wang, P., Liu, S., Ludwig, K., Chen, F., Kofinas, P., Wang, C.: Aqueous electrolyte design for super-stable 2.5 V LiMn2O4$||$Li4Ti5O12 pouch cells. Nat. Energy 7(2), 186–193 (2022)
  * Wang et al. [2019] Wang, K., Huang, L., Eedugurala, N., Zhang, S., Sabuj, M.A., Rai, N., Gu, X., Azoulay, J.D., Ng, T.N.: Wide Potential Window Supercapacitors Using Open-Shell Donor-Acceptor Conjugated Polymers with Stable N-Doped States. Adv. Energy Mater. 9(47), 1902806 (2019)
  * Wang et al. [2020] Wang, J., Tian, L., Xie, W., Wang, X., Long, X., Sun, K., Emin, A., Liu, D., Fu, Y., Chen, Q., et al.: A hierarchical interconnected nanosheet structure of porous $\delta$-MnO2 on graphite paper as cathode with a broad potential window for NaNO3 aqueous electrolyte supercapacitors. ACS Appl. Energy Mater. 3(3), 2614–2622 (2020)
  * Huang et al. [2022] Huang, J., Yuan, K., Chen, Y.: Wide voltage aqueous asymmetric supercapacitors: advances, strategies, and challenges. Adv. Funct. Mater. 32(4), 2108107 (2022)
  * Cao et al. [2021] Cao, Y., Liang, J., Li, X., Yue, L., Liu, Q., Lu, S., Asiri, A.M., Hu, J., Luo, Y., Sun, X.: Recent advances in perovskite oxides as electrode materials for supercapacitors. Chem. Commun. 57(19), 2343–2355 (2021)
  * Mefford et al. [2014] Mefford, J.T., Hardin, W.G., Dai, S., Johnston, K.P., Stevenson, K.J.: Anion charge storage through oxygen intercalation in LaMnO3 perovskite pseudocapacitor electrodes. Nat. Mater. 13(7), 726–732 (2014)
  * Afanasiev et al. [2021] Afanasiev, D., Hortensius, J., Ivanov, B., Sasani, A., Bousquet, E., Blanter, Y., Mikhaylovskiy, R., Kimel, A., Caviglia, A.: Ultrafast control of magnetic interactions via light-driven phonons. Nat. mater. 20(5), 607–611 (2021)
  * Cao et al. [2016] Cao, S., Chen, L., Zhao, W., Xu, K., Wang, G., Yang, Y., Kang, B., Zhao, H., Chen, P., Stroppa, A., et al.: Tuning the weak ferromagnetic states in dysprosium orthoferrite. Sci. Rep. 6(1), 37529 (2016)
  * Biswas et al. [2022] Biswas, B., Michel, V.F., Fjellvåg, Ø.S., Bimashofer, G., Döbeli, M., Jambor, M., Keller, L., Müller, E., Ukleev, V., Pomjakushina, E.V., et al.: Role of Dy on the magnetic properties of orthorhombic DyFeO3. Phys. Rev. Mater. 6(7), 074401 (2022)
  * Zeng et al. [2020] Zeng, Z., Xu, Y., Zhang, Z., Gao, Z., Luo, M., Yin, Z., Zhang, C., Xu, J., Huang, B., Luo, F., et al.: Rare-earth-containing perovskite nanomaterials: design, synthesis, properties and applications. Chem. Soc. Rev. 49(4), 1109–1143 (2020)
  * Hussain and Bang [2023] Hussain, S.K., Bang, J.H.: Overview of the oxygen vacancy effect in bimetallic spinel and perovskite oxide electrode materials for high-performance supercapacitors. PCCP 25(17), 11892–11907 (2023)
  * Zhu et al. [2014] Zhu, J., Li, H., Zhong, L., Xiao, P., Xu, X., Yang, X., Zhao, Z., Li, J.: Perovskite oxides: Preparation, Characterizations, and Applications in Heterogeneous Catalysis. ACS Catal. 4(9), 2917–2940 (2014)
  * Shafi et al. [2018] Shafi, P.M., Joseph, N., Thirumurugan, A., Bose, A.C.: Enhanced electrochemical performances of agglomeration-free LaMnO3 perovskite nanoparticles and achieving high energy and power densities with symmetric supercapacitor design. Chem. Eng. J. 338, 147–156 (2018)
  * Zhong et al. [2015] Zhong, C., Deng, Y., Hu, W., Qiao, J., Zhang, L., Zhang, J.: A review of electrolyte materials and compositions for electrochemical supercapacitors. Chem. Soc. Rev. 44(21), 7484–7539 (2015)
  * Park et al. [2019] Park, J., Yoo, Y.-E., Mai, L., Kim, W.: Rational design of a redox-active nonaqueous electrolyte for a high-energy-density supercapacitor based on carbon nanotubes. ACS Sustain. Chem. Eng. 7(8), 7728–7735 (2019)
  * Fic et al. [2012] Fic, K., Lota, G., Meller, M., Frackowiak, E.: Novel insight into neutral medium as electrolyte for high-voltage supercapacitors. Energy Environ. Sci. 5(2), 5842–5850 (2012)
  * Chang et al. [2014] Chang, Z., Yang, Y., Li, M., Wang, X., Wu, Y.: Green energy storage chemistries based on neutral aqueous electrolytes. J. Mater. Chem. A 2(28), 10739–10755 (2014)
  * Chun et al. [2015] Chun, S.-E., Evanko, B., Wang, X., Vonlanthen, D., Ji, X., Stucky, G.D., Boettcher, S.W.: Design of aqueous redox-enhanced electrochemical capacitors with high specific energies and slow self-discharge. Nat. Commun. 6(1), 7818 (2015)
  * Hasegawa et al. [2016] Hasegawa, G., Kanamori, K., Kiyomura, T., Kurata, H., Abe, T., Nakanishi, K.: Hierarchically porous carbon monoliths comprising ordered mesoporous nanorod assemblies for high-voltage aqueous supercapacitors. Chem. Mater. 28(11), 3944–3950 (2016)
  * Dou et al. [2018] Dou, Q., Lei, S., Wang, D.-W., Zhang, Q., Xiao, D., Guo, H., Wang, A., Yang, H., Li, Y., Shi, S., et al.: Safe and high-rate supercapacitors based on an “acetonitrile/water in salt” hybrid electrolyte. Energy Environ. Sci. 11(11), 3212–3219 (2018)
  * Jaiswal et al. [2011] Jaiswal, A., Das, R., Adyanthaya, S., Poddar, P.: Surface effects on morin transition, exchange bias, and enchanced spin reorientation in chemically synthesized DyFeO3 nanoparticles. J. Mater. Chem. C 115(7), 2954–2960 (2011)
  * Liu et al. [2017] Liu, X., You, J., Wang, R., Ni, Z., Han, F., Jin, L., Ye, Z., Fang, Z., Guo, R.: Synthesis and absorption properties of hollow-spherical Dy2Cu2O5 via a coordination compound method with [DyCu(3,4-pdc)2(OAc)(H2O2]•10.5H2O precursor. Sci. Rep. 7(1), 13085 (2017)
  * Owusu et al. [2017] Owusu, K.A., Qu, L., Li, J., Wang, Z., Zhao, K., Yang, C., Hercule, K.M., Lin, C., Shi, C., Wei, Q., et al.: Low-crystalline iron oxide hydroxide nanoparticle anode for high-performance supercapacitors. Nat. Commun. 8(1), 14264 (2017)
  * Yang and Sacher [2009] Yang, D.-Q., Sacher, E.: Characterization and oxidation of Fe nanoparticles deposited onto highly oriented pyrolytic graphite, using X-ray photoelectron spectroscopy. J. Mater. Chem. C 113(16), 6418–6425 (2009)
  * Liu et al. [2020] Liu, R., Wang, Z., Peng, S., Bi, J., Wu, J., Ye, Z.-G.: Oxygen-vacancy-controlled magnetic properties with magnetic pole inversion in BiFeO3-based multiferroics. J. Am. Ceram. Soc. 103(2), 1097–1104 (2020)
  * Kistler [1931] Kistler, S.S.: Coherent expanded aerogels and jellies. Nature 127(3211), 741–741 (1931)
  * Al-Muhtaseb and Ritter [2003] Al-Muhtaseb, S.A., Ritter, J.A.: Preparation and properties of resorcinol-formaldehyde organic and carbon gels. Adv. Mater. 15(2), 101–114 (2003)
  * Feinle et al. [2016] Feinle, A., Elsaesser, M.S., Huesing, N.: Sol-gel synthesis of monolithic materials with hierarchical porosity. Chem. Soc. Rev. 45(12), 3377–3399 (2016)
  * Moon et al. [2022] Moon, J., Kim, D.O., Bekaert, L., Song, M., Chung, J., Lee, D., Hubin, A., Lim, J.: Non-fluorinated non-solvating cosolvent enabling superior performance of lithium metal negative electrode battery. Nat. Commun. 13(1), 4538 (2022)
  * Vu et al. [2012] Vu, A., Qian, Y., Stein, A.: Porous electrode materials for lithium-ion batteries-how to prepare them and what makes them special. Adv. Energy Mater. 2(9), 1056–1085 (2012)
  * Chen et al. [2022] Chen, Z., Danilov, D.L., Eichel, R.-A., Notten, P.H.: Porous electrode modeling and its applications to Li-ion batteries. Adv. Energy Mater. 12(32), 2201506 (2022)
  * Lei et al. [2013] Lei, C., Amini, N., Markoulidis, F., Wilson, P., Tennison, S., Lekakou, C.: Activated carbon from phenolic resin with controlled mesoporosity for an electric double-layer capacitor (EDLC). J. Mater. Chem. A 1(19), 6037–6042 (2013)
  * Melchior et al. [2018] Melchior, S.A., Raju, K., Ike, I.S., Erasmus, R.M., Kabongo, G., Sigalas, I., Iyuke, S.E., Ozoemena, K.I.: High-voltage symmetric supercapacitor based on 2D titanium carbide (MXene, Ti2CTx)/carbon nanosphere composites in a neutral aqueous electrolyte. J. Electrochem. Soc.  165(3), 501–511 (2018)
  * Tarek and Basith [2023] Tarek, M., Basith, M.: MoS2 mediated tuning of CuCo2S4-MoS2 nanocomposites for high-performance symmetric hybrid supercapacitors. J. Mater. Chem. C 11(47), 16605–16622 (2023)
  * Masarapu et al. [2009] Masarapu, C., Zeng, H.F., Hung, K.H., Wei, B.: Effect of temperature on the capacitance of carbon nanotube supercapacitors. ACS nano 3(8), 2199–2206 (2009)
  * Xu et al. [2019] Xu, S.-W., Zhang, M.-C., Zhang, G.-Q., Liu, J.-H., Liu, X.-Z., Zhang, X., Zhao, D.-D., Xu, C.-L., Zhao, Y.-Q.: Temperature-dependent performance of carbon-based supercapacitors with water-in-salt electrolyte. J. Power Sources 441, 227220 (2019)
  * Ansari et al. [2019] Ansari, A.A., Ahmad, N., Alam, M., Adil, S.F., Ramay, S.M., Albadri, A., Ahmad, A., Al-Enizi, A.M., Alrayes, B.F., Assal, M.E., et al.: Physico-chemical properties and catalytic activity of the sol-gel prepared Ce-ion doped LaMnO3 perovskites. Sci. Rep. 9(1), 7747 (2019)

##  3 Acknowledgements

We sincerely acknowledge the University Grant Commission of Bangladesh for
financial support, which has been instrumental in the successful execution of
this research. We also acknowledge Bangladesh University of Engineering and
Technology for providing essential resources crucial for conducting our
experiments.

##  4 Competing financial interests

The authors declare no competing financial interests.

##  5 Methods

###  5.1 Synthesis of DyFeO3 nanoparticles

Nanoparticles of the DyFeO3 perovskite were synthesized using a sol-gel
technique, as illustrated in Fig. S1 [47]. Initially, stoichiometric amounts
of Dy(NO3)3·5H2O (Sigma-Aldrich, 99.80%) and Fe(NO3)3·9H2O (Sigma-Aldrich,
98.00%) were individually dissolved in 100 ml of deionized water and stirred
for 15-20 minutes using a magnetic stirrer. Subsequently, the solutions were
combined, and citric acid (C6H8O7) was introduced into the mixture. citric
acid served as a chelating agent, forming stable complexes with the metal
ions. The solution’s pH was adjusted to 7 by incorporating NH4OH.
Subsequently, the chelating agent, ethylene glycol, was introduced to generate
a polymeric metal cationic network, serving as the precursor for the gel.
After four hours, the temperature was raised to 200 ∘C, leading to the
complete combustion of the gel and the formation of the desired powder. The
combustion process entails the decomposition of the organic components within
the gel, liberating gases and leaving behind metal oxides. The resulting
material was finely grounded using an agate mortar. To achieve the desired
crystallization and porous structure, the material underwent calcination at
750 ∘C for 6 hours, with a heating rate of 5 ∘C per minute in a nitrogen
environment. The controlled heating rate of 5 ∘C per minute minimizes thermal
stress, facilitating controlled decomposition. Nitrogen gas flow, particularly
during the initial stages of the calcination process, ensures an oxygen-free
environment. This controlled atmosphere is crucial for preventing unwanted
oxidation reactions and influencing the evolution of gases during
decomposition. The reproducible formation of porosity in DyFeO3 was confirmed
through the judicious selection of solvent (here water), systematic adjustment
of gel precursor concentration (here ethylene glycol), careful control of
temperature and optimization of the calcination temperature in a N2
environment.

###  5.2 Characterization methods and instrumentation

####  5.2.1 Structural and morphological characterization

The crystal structure of the synthesized materials was determined using a
Rigaku SmartLab X-ray diffractometer with copper K-Alpha radiation. Further
insights were gained through a thorough analysis of the powder X-ray
diffraction (XRD) patterns, employing the Rietveld refinement method with the
Fullprof software program. The surface morphology was analyzed using both
Field Emission Scanning Electron Microscopy (FESEM) and Transmission Electron
Microscopy (TEM). Elemental insights were obtained through energy-dispersive
X-ray (EDX) spectra, meticulously revealing the elemental composition of the
synthesized samples. Fourier Transform Infrared Spectroscopy (FTIR) was
employed for a comprehensive examination of chemical bonds and functional
groups. X-ray Photoelectron Spectroscopy technique revealed the chemical
states and binding energies inherent in the DyFeO3.

####  5.2.2 Electrochemical Characterizations

Cyclic voltammetry (CV), linear sweep voltammetry (LSV), galvanostatic charge-
discharge (GCD), and electrochemical impedance spectroscopy (EIS) analyses
were systematically performed on the DyFeO3 electrode material across diverse
electrolyte solutions. These assessments were conducted using both three-
electrode and symmetric two-electrode configurations, including our fabricated
coin cell supercapacitor. The experiments were carried out utilizing an
electrochemical workstation (Metrohm Autolab PGSTAT302N). The detailed
procedure for preparing the working electrode slurry is outlined in the
Supplementary Information, and Fig. S2 schematically illustrates both the
preparation process of the working electrode slurry and the electrochemical
setup of the three-electrode system.

Assembly of coin cell supercapacitors in symmetric configuration:
Supplementary Fig. S3 depicts the configuration of the CR2032 coin cell and
the corresponding fabrication steps. In the context of the coin cell
supercapacitor, DyFeO3 nanoparticles served as the electrode material for both
the anode and cathode. The electrolyte comprised of 0.5 M Na2SO4(aq.) and 0.5
M Na2SO4(aq.)/20%AN solutions, Whatman filter paper functioned as the
separator, and disc-shaped graphite sheets served as current collectors for
both the anode and cathode. In the coin cell assembly, a slurry of the active
material was uniformly applied to two individual graphite sheets (diameter
1.65 cm) at a consistent loading of 10 mg cm-2. The treated graphite sheets,
designated as the anode and cathode, were then dried in an oven at 80 oC for
12 hours. A circular Whatman filter paper, saturated for 24 hours with 0.5 M
Na2SO4(aq.) and 0.5 M Na2SO4(aq.)/20%AN electrolyte solutions, was positioned
between the modified graphite sheets to finalize the coin cell arrangement.
Subsequently, the coin cell supercapacitor was sealed under a pressure of 800
kg using an automatic coin cell assembling machine (TMAX-160S).

###  5.3 Mathematical analysis

####  5.3.1 Electrochemical Measurement

Specific capacitance ($C_{sp}$)

The specific capacitance is computed using the subsequent equation tailored
for the symmetric two-electrode system (detailed derivation is provided in the
supplementary information). [43, 44].

$C_{sp}=\frac{4J\Delta{t}}{\Delta{V}}$ (1)

Where, J, $\Delta{t}$ and $\Delta{V}$ stand for current density, time to
discharge and potential window, respectively.

Energy density ($E_{sp}$) and Power density ($P_{sp}$)

The specific energy density ($E_{sp}$, measured in W h kg-1), and specific
power density ($P_{sp}$, measured in W kg-1) for the symmetric two-electrode
system (pertaining to a single electrode) were computed based on the GCD using
Equations 4 and 5, respectively [43, 44].

$E_{sp}=\frac{C_{sp}(\Delta{V})^{2}\times 1000}{8\times 3600}$ (2)
$P_{sp}=\frac{E_{sp}\times 3600}{\Delta{t}}$ (3)

Capacitance retention (%)

$\text{Capacitance retention (\%)}=\left(\frac{\text{Capacitance at
}n^{\text{th}}\text{ cycles}}{\text{Initial capacitance}}\right)\times 100\%$
(4)

Coulombic efficiency (%)

$\text{Coulombic efficiency (\%)}=\left(\frac{\text{Discharging
time}}{\text{Charging time}}\right)\times 100\%$ (5)


#  Tight Exponential Strong Converse for Source Coding Problem with Encoded
Side Information

Daisuke Takeuchi and Shun Watanabe  The Department of Computer and Information
Sciences, Tokyo, 184-8588, Japan (email<EMAIL_ADDRESS>The Department
of Computer and Information Sciences, Tokyo, 184-8588, Japan (email:
<EMAIL_ADDRESS>
###### Abstract

The source coding problem with encoded side information is considered. A lower
bound on the strong converse exponent has been derived by Oohama, but its
tightness has not been clarified. In this paper, we derive a tight strong
converse exponent. The achievability part is derived by a careful analysis of
the type argument. The converse part is proved by a judicious use of the
change-of-measure argument, which was introduced by Gu-Effros and further
developed by Tyagi-Watanabe. Interestingly, the soft Markov constraint, which
was introduced by Oohama as a proof technique, is naturally incorporated into
the characterization of the exponent.

Keywords: source coding with side information, network information theory,
strong converse theorem, exponent of correct probability, change of measure
argument.

##  1 Introduction

For source coding problems, the (weak) converse theorems state that at
transmission rates under the theoretical limit, there is no error-free coding
scheme as the block length tends to infinity, i.e. the error probability
cannot go to $0$. On the other hand, the strong converse theorems state that
at transmission rates under the theoretical limit, the error probability
converges to $1$ as the block length tends to infinity.

In this paper, we investigate source coding problem with encoded side
information (a helper) proposed by Wyner [14], Ahlswede and Körner [1], which
is known as one-help-one problem or Wyner-Ahlswede-Körner (WAK) coding
problem. The WAK network is described in Fig. 1. This network consists of two
encoders and one decoder. Both encoders are connected to the decoder by each
channel, and the decoder aims to reproduce the source observed by the first
encoder. As shown in [14] and [1], the information theoretical limit involves
an auxiliary random variable with a Markov chain, which makes this problem
difficult.

We are interested in the strong converse theorem of WAK network. Ahlswede,
Gács and Körner [2] proved that the correct probability goes to $0$ if the
rate pair is outside the rate region. After that, Gu and Effros [7] showed
that the correct probability decreases exponentially. Also, Oohama [10]
derived an explicit lower bound of this exponent function. However, it is
unknown whether that bound is tight or not. In this paper, we derive a tight
characterization of the exponent.

In the proof of the exponent function, we use some techniques such as the type
method and the change of measure argument. In the direct part (achievability),
we apply the type covering argument (cf. [4, Chapter 9]).

We prove the converse part with the change of measure argument. The change of
measure argument was introduced by Gu and Effros [7], [8] as a proof method
for proving strong converse theorems for centralized coding problems, such as
the Gray-Wyner network [6]. More recently, Tyagi and Watanabe [11] further
developed the change of measure argument. Among other things, they enabled to
use the change of measure argument to distributed coding problems including
the WAK network (see also an early attempt in [13]). In this paper, we derive
the strong converse exponent by a judicious use of the recipe provided in
[11].

Even though our derivation of the strong converse exponent is quite different
from that in [10], we use the idea of soft Markov constraint introduced by
Oohama in a series of papers [9], [10], which is also a key component in [11].
Interestingly, the soft Markov constraint is incorporated as a part of the
characterization of the exponent.

In the channel coding, there are two kinds of expressions for the strong
converse exponent, the one by Arimoto [3] and the other by Dueck and Körner
[5]. While the lower bound on the strong converse exponent by Oohama [10] is
closer to the expression by Arimoto, our characterization of the exponent is
along the lines of the expression by Dueck and Körner.

This paper is organized as follows. In Section 2, after introducing notations,
we define the problem that we treat. In Section 3, we state our main result.
In Section 4, we give the proof of our result. In Section 5, we conclude the
paper.

##  2 Preliminaries

Figure 1: WAK network

###  2.1 Notations

Random variables are in capital case (e.g.$\,X$), and their realization are in
lower case (e.g.$\,x$). All random variables take values in some alphabets
that are in calligraphic letters (e.g.$\,\mathcal{X}$). We shall restrict our
attention to finite alphabets only. The cardinality of $\mathcal{X}$ is
denoted by $|\mathcal{X}|$. The probability distribution of random variable
$X$ is denoted by $P_{X}$. Random vectors and their realizations in the $n$th
Cartesian product $\mathcal{X}^{n}$ are denoted by $X^{n}=(X_{1},\dots,X_{n})$
and $x^{n}=(x_{1},\dots,x_{n})$, respectively. Also, define
$\mathcal{P}(\mathcal{X})$ as the set of all of $P_{X}$. For information
quantities, we use the same notations as [4], such as the entropy $H(X)$, the
mutual information $I(X\land Y)$ and the KL divergence $D(P_{X}||Q_{X})$. For
simplicity, we write $\mathcal{T}_{X}$ for the type class of type $P_{X}$.
Also, we define the total variation distance $d_{\textrm{var}}$ between two
discrete distributions $P$ and $Q$ on a finite set $\mathcal{X}$ as

$d_{\textrm{var}}(P,Q)=\frac{1}{2}\sum_{x\in\mathcal{X}}|P(x)-Q(x)|.$

In the following, the base of logarithm and exponentiation is $2$.

###  2.2 WAK Network

We consider the network described in Fig. 1. For a given source $P_{XY}$ on a
finite alphabet $\mathcal{X}\times\mathcal{Y}$, this coding system consists of
two encoders

$\varphi_{1}^{(n)}:\mathcal{X}^{n}\to\mathcal{M}_{1}^{(n)},\
\varphi_{2}^{(n)}:\mathcal{Y}^{n}\to\mathcal{M}_{2}^{(n)},$

and one decoder

$\psi^{(n)}:\mathcal{M}_{1}^{(n)}\times\mathcal{M}_{2}^{(n)}\to\mathcal{X}^{n}.$

For simplicity, we write encoders and a decoder as $\varphi_{1},\varphi_{2}$
and $\psi$, respectively. Input sequences $X^{n}$ and $Y^{n}$ are encoded by
$\varphi_{1}$ and $\varphi_{2}$ separately, and the decoder outputs the
estimation $\hat{X}^{n}=\psi(\varphi_{1}(X^{n}),\varphi_{2}(Y^{n}))$ of
$X^{n}$.

For i.i.d. $(X^{n},Y^{n})\sim P_{X^{n}Y^{n}}$, the error probability of code
$\Phi_{n}=(\varphi_{1},\varphi_{2},\psi)$ is defined as

$P_{\textrm{e}}(\Phi_{n}|P_{X^{n}Y^{n}}):=\Pr(\hat{X^{n}}\neq X^{n}).$

A rate pair $(R_{1},R_{2})$ is called an achievable rate pair if there exists
a sequence of codes such that

$\limsup_{n\to\infty}\frac{1}{n}\log|\mathcal{M}_{1}^{(n)}|\leq R_{1},\
\limsup_{n\to\infty}\frac{1}{n}\log|\mathcal{M}_{2}^{(n)}|\leq R_{2},$

and

$\lim_{n\to\infty}P_{\textrm{e}}(\Phi_{n}|P_{X^{n}Y^{n}})=0.$

Also, the achievable rate region $\mathcal{R}_{\textrm{WAK}}$ is denoted by

$\mathcal{R}_{\textrm{WAK}}=\\{(R_{1},R_{2}):(R_{1},R_{2})\text{ is
achievable}\\}.$

Wyner [14], Ahlswede and Körner [1] determined $\mathcal{R}_{\textrm{WAK}}$ as
below.

######  Theorem 1.

Consider the region

$\displaystyle\mathcal{R}=\bigcup_{\begin{subarray}{c}X-Y-U\\\
|\mathcal{U}|\leq|\mathcal{X}|+1\end{subarray}}\\{(R_{1},R_{2}):$
$\displaystyle R_{1}\geq H(X|U),$ $\displaystyle R_{2}\geq I(Y\land U)\\},$

where the Markov chain $X-Y-U$ means

$P_{UXY}=P_{X}P_{Y|X}P_{U|Y}.$

Then, it holds

$\mathcal{R}_{\textrm{WAK}}=\mathcal{R}.$

Our main purpose is to find an explicit form of the exponent function for the
error probability converging to $1$ as $n\to\infty$ when $(R_{1},R_{2})$ is
not achievable. To achieve this aim, we define the correct probability
$P_{\textrm{c}}$ and the optimal exponent function $G$:

$\displaystyle P_{\textrm{c}}(\Phi_{n}|P_{X^{n}Y^{n}})$
$\displaystyle=1-P_{\textrm{e}}(\Phi_{n}|P_{X^{n}Y^{n}}),$ $\displaystyle
G^{(n)}(R_{1},R_{2}|P_{X^{n}Y^{n}})$
$\displaystyle=\min_{\Phi_{n}}\left(-\frac{1}{n}\right)\log
P_{c}(\Phi_{n}|P_{X^{n}Y^{n}})$ $\displaystyle G(R_{1},R_{2}|P_{XY})$
$\displaystyle=\lim_{n\to\infty}G^{(n)}(R_{1},R_{2}|P_{X^{n}Y^{n}}).$

According to [10], the limit in the definition of $G$ exists, and the exponent
function $G(R_{1},R_{2}|P_{XY})$ is a convex function of $(R_{1},R_{2})$.

Ahlswede, Gács and Körner [2] proved that the correct probability
$P_{\textrm{c}}$ goes to $0$ as $n\to\infty$ . Gu and Effros [7] showed that
the convergence speed of $P_{\mathrm{c}}$ is exponential. Also, Oohama [10]
derived a lower bound on $G$. However, it was not clarified whether the lower
bound in [10] is tight or not. In the next section, we shall show our result.

##  3 Main Result

In this section, we show our main result. Before showing it, we set

$\displaystyle
F(R_{1},R_{2}|P_{XY})=\min_{P_{\tilde{U}\tilde{X}\tilde{Y}}\in\mathcal{P}(\mathcal{U}\times\mathcal{X}\times\mathcal{Y})}\biggl{\\{}D(P_{\tilde{U}\tilde{X}\tilde{Y}}||P_{\tilde{U}|\tilde{Y}}P_{XY})+\left|I(\tilde{U}\land\tilde{Y})-R_{2}\right|^{+}:$
$\displaystyle\hphantom{----------------}R_{1}\geq
H(\tilde{X}|\tilde{U}),|\mathcal{U}|\leq|\mathcal{X}||\mathcal{Y}|+2\biggr{\\}},$
(1)

where $|a|^{+}=\max\\{a,0\\}$.

Now we show our main theorem by using this quantity.

######  Theorem 2.

For any $P_{XY}$, we have

$G(R_{1},R_{2}|P_{XY})=F(R_{1},R_{2}|P_{XY}).$

##  4 Proof

###  4.1 Direct Part

In this section, we prove the direct part. Since we are interested in a lower
bound on the correct probability $P_{\mathrm{c}}$, it suffices to construct a
code such that sequences $(x^{n},y^{n})$ in a fixed type class ${\cal
T}_{\bar{X}\bar{Y}}$ can be decoded correctly. First, we fix
$P_{\tilde{U}\tilde{X}\tilde{Y}}$ that attains the minimum in (1) for
$(R_{1}-2\epsilon,R_{2})$ where $\epsilon$ is a small positive number. Let
$P_{\bar{U}\bar{X}\bar{Y}}$ be a joint type satisfying

$d_{\textrm{var}}(P_{\bar{U}\bar{X}\bar{Y}},P_{\tilde{U}\tilde{X}\tilde{Y}})\leq\frac{|\mathcal{U}||\mathcal{X}||\mathcal{Y}|}{n}.$

Note that, by the continuity of entropy, we see

$\displaystyle H(\bar{X}|\bar{U})$ $\displaystyle\leq
H(\tilde{X}|\tilde{U})+\epsilon$ $\displaystyle\leq R_{1}-\epsilon,$

for sufficiently large $n$. In order to prove the direct part, we use the
following covering lemma.

######  Lemma 3.

For a joint type $P_{\bar{U}\bar{Y}}$, there exists a codebook ${\cal
C}=\\{u_{1}^{n},\ldots,u_{L}^{n}\\}\subseteq{\cal T}_{\bar{U}}$ such that

$L\leq\exp\\{nI(\bar{U}\land\bar{Y})+(|\mathcal{U}||\mathcal{Y}|+4)\log(n+1)\\},$
(2)

and such that for any $y^{n}\in\mathcal{T}_{\bar{Y}}$, there exists
$u^{n}\in\mathcal{C}$ satisfying
$(u^{n},y^{n})\in\mathcal{T}_{\bar{U}\bar{Y}}$.

###### Proof.

Since the strategy of proving this lemma is almost the same as [12, Lemma 4],
we omit it. ∎

We describe an overview of code construction. For this coding scheme, we want
to evaluate the correct probability $P_{\textrm{c}}$. For $\varphi_{1}$, we
use the standard random binning with rate $R_{1}$. For $\varphi_{2}$, we use
the quantization using the test channel $P_{\bar{U}|\bar{Y}}$. However,
$R_{2}$ may be smaller than $I(\bar{U}\land\bar{Y})$ that is needed to
quantize sequences in $\mathcal{T}_{\bar{Y}}$ via the test channel
$P_{\bar{U}|\bar{Y}}$. In that case, $\varphi_{2}$ assigns messages to a part
of the quantization codebook, and assign a prescribed constant message to the
remaining. The detail of the coding scheme is described as follows.

#### Encoding of $\varphi_{1}$

Randomly and independently assign an index
$m_{1}=F(x^{n})\in\\{1,2,\dots,2^{nR_{1}}\\}$ to each sequence
$x^{n}\in\mathcal{X}^{n}$. The set of sequences with the same index $m_{1}$
forms a bin $\mathcal{B}(m_{1})$. Observing $x^{n}\in\mathcal{B}(m_{1})$,
$\varphi_{1}$ sends the bin index $m_{1}$.

#### Encoding of $\varphi_{2}$

Consider a codebook ${\cal C}$ given by Lemma 3 and function $f:{\cal
T}_{\bar{Y}}\to\\{1,\ldots,L\\}$ such that $(u_{f(y^{n})}^{n},y^{n})\in{\cal
T}_{\bar{U}\bar{Y}}$. Without loss of generality, we assume that
$|f^{-1}(1)|\geq|f^{-1}(2)|\geq\cdots\geq|f^{-1}(L)|$. We define a function
$\tilde{f}$ as

$\tilde{f}(y^{n})=\begin{cases}f(y^{n})&f(y^{n})\leq 2^{nR_{2}}\\\
1&f(y^{n})>2^{nR_{2}}\end{cases}.$

Then, upon observing $y^{n}\in{\cal Y}^{n}$, $\varphi_{2}$ sends
$m_{2}=\tilde{f}(y^{n})$ if $y^{n}\in{\cal T}_{\bar{Y}}$ and $m_{2}=1$
otherwise.

#### Decoding.

Observing $(m_{1},m_{2})=(\varphi_{2}(x^{n}),\varphi_{2}(y^{n}))$, $\psi$
declares the estimate
$\hat{x}^{n}\in\mathcal{T}_{\bar{X}|\bar{U}}(u_{m_{2}}^{n})$ if it is the
unique sequence satisfying $F(\hat{x}^{n})=m_{1}$; otherwise it declares a
prescribed sequence.

#### Analysis of the correct probability.

We bound the correct probability $P_{\mathrm{c}}$ averaged over random bin
assignments. We lower bound the correct probability quite generously, and
estimate the probability such that the following three events occur
consecutively.

  1. 1.

The event $E_{1}$ such that $y^{n}\in\mathcal{T}_{\bar{Y}}$ and
$\tilde{f}(y^{n})=f(y^{n})$: we denote the probability of this event by
$P_{\textrm{c},1}=\Pr(E_{1})$.

  2. 2.

The event $E_{2}$ such that $E_{1}$ occurs and $x^{n}$ is included in the
conditional type class ${\cal
T}_{\bar{X}|\bar{U}\bar{Y}}(u^{n}_{f(y^{n})},y^{n})$: we denote the
conditional probability of this event conditioned that $E_{1}$ occurs by
$P_{\textrm{c},2}=\Pr(E_{2}|E_{1})$.

  3. 3.

The event $E_{3}$ such that $E_{1}\wedge E_{2}$ occurs and there exists no
sequence $\hat{x}^{n}\in{\cal T}_{\bar{X}|\bar{U}}(u_{f(y^{n})}^{n})$
satisfying $\hat{x}^{n}\neq x^{n}$ and $F(\hat{x}^{n})=F(x^{n})$: we denote
the conditional probability of this event conditioned on the first and second
events by $P_{\textrm{c},3}=\Pr(E_{3}|E_{1}\wedge E_{2})$.

By noting that ${\cal
T}_{\bar{X}|\bar{U}\bar{Y}}(u_{f(y^{n})}^{n},y^{n})\subseteq{\cal
T}_{\bar{X}|\bar{U}}(u_{f(y^{n})}^{n})$, we can verify that the decoder $\psi$
outputs the correct source sequence $x^{n}$ if the three events occur
$E_{1},E_{2}$, and $E_{3}$ consecutively. Thus, we can lower bound the correct
probability as

$P_{\textrm{c}}\geq\prod_{i=1}^{3}P_{\textrm{c},i}.$

In the following, we shall calculate each probability $P_{\textrm{c},i}$.

When $L\leq 2^{nR_{2}}$, the second condition $\tilde{f}(y^{n})=f(y^{n})$ of
the event $E_{1}$ is always satisfied. On the other hand, by noting that
$|f^{-1}(i)|$ is sorted in descending order, when $L>2^{nR_{2}}$, we have

$\frac{1}{2^{nR_{2}}}\sum_{i=1}^{2^{nR_{2}}}|f^{-1}(i)|\geq\frac{1}{L}\sum_{i=1}^{L}|f^{-1}(i)|=\frac{|{\cal
T}_{\bar{Y}}|}{L},$

which implies

$\sum_{i=1}^{2^{nR_{2}}}\frac{|f^{-1}(i)|}{|{\cal
T}_{\bar{Y}}|}\geq\frac{2^{nR_{2}}}{L}.$

Therefore, by using [4, Lemma 2.6], we have

$\displaystyle P_{\textrm{c},1}$ $\displaystyle=\Pr\\{Y^{n}\in{\cal
T}_{\bar{Y}}\land[f(Y^{n})\leq 2^{nR_{2}}]\\}$ $\displaystyle=P_{Y^{n}}({\cal
T}_{\bar{Y}})\sum_{i=1}^{2^{nR_{2}}}\frac{|f^{-1}(i)|}{|{\cal T}_{\bar{Y}}|}$
$\displaystyle\geq P_{Y^{n}}({\cal
T}_{\bar{Y}})\min\left\\{1,\frac{2^{nR_{2}}}{L}\right\\}$
$\displaystyle\geq(n+1)^{-|\mathcal{Y}|}2^{-n(D(P_{\bar{Y}}||P_{Y})+|(1/n)\log
L-R_{2}|^{+})}$
$\displaystyle\geq(n+1)^{-|\mathcal{Y}|}2^{-n(D(P_{\bar{Y}}||P_{Y})+|I(\bar{U}\land\bar{Y})-R_{2}+\delta_{n}|^{+})},$
(3)

where $\delta_{n}=(1/n)(|\mathcal{U}||\mathcal{Y}|+4)\log(n+1)$.

By using [4, Lemma 2.6], we have

$\displaystyle P_{\textrm{c},2}$
$\displaystyle=\sum_{\begin{subarray}{c}y^{n}\in{\cal T}_{\bar{Y}}\\\
\tilde{f}(y^{n})=f(y^{n})\end{subarray}}\frac{1}{K}P_{X^{n}|Y^{n}}({\cal
T}_{\bar{X}|\bar{U}\bar{Y}}(u_{f(y^{n})}^{n},y^{n}))$
$\displaystyle\geq(n+1)^{-|\mathcal{U}||\mathcal{X}||\mathcal{Y}|}2^{-nD(P_{\bar{X}|\bar{U}\bar{Y}}||P_{X|Y}|P_{\bar{U}\bar{Y}})},$
(4)

where $K=\sum_{i=1}^{2^{nR_{2}}}|f^{-1}(i)|$ is the number of $y^{n}\in{\cal
T}_{\bar{Y}}$ satisfying $\tilde{f}(y^{n})=f(y^{n})$. Also, since the decoding
method searches the unique sequence $\hat{x}^{n}$ among
$\mathcal{T}_{\tilde{X}|\tilde{U}}(u^{n}_{f(y^{n})})$ such that the bin index
coincides with $x^{n}$, $P_{\textrm{c},3}$ can be evaluated as

$\displaystyle P_{\textrm{c},3}$
$\displaystyle=1-\sum_{(x^{n},y^{n})}\Pr\\{(X^{n},Y^{n})=(x^{n},y^{n})|E_{1}\land
E_{2}\\}$ $\displaystyle\hphantom{--}\cdot\Pr\\{\exists\hat{x}^{n}\neq
x^{n}~{}~{}\mathrm{s.t.}~{}~{}F(\hat{x}^{n})=F(x^{n}),\hat{x}^{n}\in\mathcal{T}_{\bar{X}|\bar{U}}(u_{f(y^{n})}^{n})\\}$
$\displaystyle\geq
1-\sum_{(x^{n},y^{n})}\Pr\\{(X^{n},Y^{n})=(x^{n},y^{n})|E_{1}\land E_{2}\\}$
$\displaystyle\hphantom{--}\cdot\sum_{\begin{subarray}{c}\hat{x}^{n}\in\mathcal{T}_{\bar{X}|\bar{U}}(u_{f(y^{n})}^{n})\\\
\hat{x}^{n}\neq x^{n}\end{subarray}}\Pr\\{F(\hat{x}^{n})=F(x^{n})\\}$
$\displaystyle\geq
1-\sum_{\hat{x}^{n}\in\mathcal{T}_{\bar{X}|\bar{U}}(u^{n})}2^{-nR_{1}}$
$\displaystyle\geq 1-2^{-n(R_{1}-H(\bar{X}|\bar{U}))}$ $\displaystyle\geq
1-2^{-n\epsilon},$ (5)

where we use [4, Lemma 2.5], and a condition $R_{1}-\epsilon\geq
H(\bar{X}|\bar{U})$ in the last inequality. Therefore, $P_{\textrm{c}}$ can be
evaluated as

$\displaystyle P_{\textrm{c}}$
$\displaystyle\geq\prod_{i=1}^{3}P_{\textrm{c},i}$
$\displaystyle\geq(n+1)^{-|\mathcal{Y}|}2^{-n(D(P_{\bar{Y}}||P_{Y})+|I(\bar{U}\land\bar{Y})-R_{2}+\delta_{n}|^{+})}$
$\displaystyle\quad\cdot(n+1)^{-|\mathcal{U}||\mathcal{X}||\mathcal{Y}|}2^{-nD(P_{\bar{X}|\bar{U}\bar{Y}}||P_{X|Y}|P_{\bar{U}\bar{Y}})}(1-2^{-n\epsilon})$
$\displaystyle=(n+1)^{-(|\mathcal{U}||\mathcal{X}|+1)|\mathcal{Y}|}(1-2^{-n\epsilon})$
$\displaystyle\quad\cdot
2^{-n(D(P_{\bar{U}\bar{X}\bar{Y}}||P_{\bar{U}|\bar{Y}}P_{XY})+|I(\bar{U}\land\bar{Y})-R_{2}+\delta_{n}|^{+})},$
(6)

where we used the following equality

$\displaystyle D(P_{\bar{U}\bar{X}\bar{Y}}||P_{\bar{U}|\bar{Y}}P_{XY})=$
$\displaystyle
D(P_{\bar{X}|\bar{U}\bar{Y}}||P_{X|Y}|P_{\bar{U}\bar{Y}})+D(P_{\bar{Y}}||P_{Y}),$
(7)

which is derived by the definition of the conditional KL-divergence.

Thus, by rearranging (6), we have

$\displaystyle\frac{1}{n}\log\frac{1}{P_{\textrm{c}}}\leq
D(P_{\bar{U}\bar{X}\bar{Y}}||P_{\bar{U}|\bar{Y}}P_{XY})\\!+\\!|I(\bar{U}\land\bar{Y})-R_{2}+\delta_{n}|^{+}$
$\displaystyle+(|\mathcal{U}||\mathcal{X}|+1)|\mathcal{Y}|\frac{\log(n+1)}{n}\\!+\\!\frac{1}{n}\log\frac{1}{1-2^{-n\epsilon}}.$
(8)

Consequently, by taking the limit of $n$ and by the continuity of the
information quantities, we obtain

$G(R_{1},R_{2}|P_{XY})\leq F(R_{1}-2\epsilon,R_{2}|P_{XY}).$

Finally, by the continuity of $F$ with respect to $R_{1}$, we obtain the
desired result.

###  4.2 Converse Part

In this section, we prove the converse part by using the change of measure
argument. Define
$\mathcal{C}=\\{(x^{n},y^{n}):\psi(\varphi_{1}(x^{n}),\varphi_{2}(y^{n}))=x^{n}\\}$
as the set of sequences without error, and define the changed probability
measure

$\displaystyle P_{\tilde{X}^{n}\tilde{Y}^{n}}(x^{n},y^{n})$
$\displaystyle=\Pr\\{(X^{n},Y^{n})\\!=\\!(x^{n},y^{n}):(X^{n},Y^{n})\\!\in\\!\mathcal{C}\\}$
$\displaystyle=\frac{P_{X^{n}Y^{n}}(x^{n},y^{n})}{P_{X^{n}Y^{n}}(\mathcal{C})}\mathds{1}[(x^{n},y^{n})\in\mathcal{C}],$
(9)

where $\mathds{1}$ is the indicator function. Then, we can verify

$D(P_{\tilde{X}^{n}\tilde{Y}^{n}}||P_{X^{n}Y^{n}})=\log\frac{1}{P_{\textrm{c}}}.$
(10)

First, define random variables $\tilde{M_{1}}=\varphi_{1}(\tilde{X}^{n})$ and
$\tilde{M_{2}}=\varphi_{2}(\tilde{Y}^{n})$. Since $nR_{1}\geq
H(\tilde{M_{1}})$, we can write a chain of inequalities

$\displaystyle nR_{1}$ $\displaystyle\geq H(\tilde{M_{1}})$ $\displaystyle\geq
H(\tilde{M_{1}}|\tilde{M_{2}})$
$\displaystyle=H(\tilde{M_{1}}|\tilde{M_{2}})-H(\tilde{M_{1}}|\tilde{M_{2}},\tilde{X}^{n})$
$\displaystyle=I(\tilde{M_{1}}\land\tilde{X}^{n}|\tilde{M_{2}})$
$\displaystyle=H(\tilde{X}^{n}|\tilde{M_{2}})-H(\tilde{X}^{n}|\tilde{M_{1}},\tilde{M_{2}})$
$\displaystyle=H(\tilde{X}^{n}|\tilde{M_{2}}).$ (11)

Here, note that $H(\tilde{M_{1}}|\tilde{M_{2}},\tilde{X}^{n})=0$ since
$\tilde{M_{1}}$ is generated from $\tilde{X}^{n}$; and
$H(\tilde{X}^{n}|\tilde{M_{1}},\tilde{M_{2}})=0$ since $\psi$ observes
$\tilde{M_{1}}$ and $\tilde{M_{2}}$, then outputs the error-free estimation of
$\tilde{X}^{n}$. Also, we consider $H(\tilde{X}^{n}|\tilde{M_{2}})$. Denoting
$X_{j}^{-}=(X_{1},\dots,X_{j-1})$, we obtain

$\displaystyle H(\tilde{X}^{n}|\tilde{M_{2}})$
$\displaystyle=\sum_{j=1}^{n}H(\tilde{X}_{j}|\tilde{M_{2}},\tilde{X}_{j}^{-})$
$\displaystyle\geq\sum_{j=1}^{n}H(\tilde{X}_{j}|\tilde{M_{2}},\tilde{X}_{j}^{-},\tilde{Y}_{j}^{-})$
$\displaystyle=nH(\tilde{X}_{J}|\tilde{U}_{J},J),$ (12)

where $J\sim\mathtt{unif}(\\{1,\dots,n\\})$ is the time-sharing random
variable and is assumed to be independent of all the other random variables
involved. Also, we put
$\tilde{U}_{J}=(\tilde{M_{2}},\tilde{X}_{J}^{-},\tilde{Y}_{J}^{-})$.

Next, since $I(\tilde{M_{2}}\land\tilde{X}^{n}|\tilde{Y}^{n})=0$, we have

$\displaystyle D(P_{\tilde{X}^{n}\tilde{Y}^{n}}||P_{X^{n}Y^{n}})$
$\displaystyle=D(P_{\tilde{X}^{n}\tilde{Y}^{n}}||P_{X^{n}Y^{n}})+I(\tilde{M_{2}}\land\tilde{X}^{n}|\tilde{Y}^{n})$
$\displaystyle=D(P_{\tilde{X}^{n}\tilde{Y}^{n}}||P_{X^{n}Y^{n}})+H(\tilde{X}^{n}|\tilde{Y}^{n})-H(\tilde{X}^{n}|\tilde{Y}^{n},\tilde{M_{2}}).$
(13)

By applying [11, Proposition 1], we obtain

$\displaystyle
D(P_{\tilde{X}^{n}\tilde{Y}^{n}}||P_{X^{n}Y^{n}})+H(\tilde{X}^{n}|\tilde{Y}^{n})\geq
n(D(P_{\tilde{X}_{J}\tilde{Y}_{J}}||P_{XY})+H(\tilde{X}_{J}|\tilde{Y}_{J})).$
(14)

Also,

$\displaystyle H(\tilde{X}^{n}|\tilde{Y}^{n},\tilde{M_{2}})$
$\displaystyle=\sum_{j=1}^{n}H(\tilde{X}_{j}|\tilde{Y}^{n},\tilde{M_{2}},\tilde{X}_{j}^{-})$
$\displaystyle\leq\sum_{j=1}^{n}H(\tilde{X}_{j}|\tilde{Y}_{j},\tilde{M_{2}},\tilde{X}_{j}^{-},\tilde{Y}_{j}^{-})$
$\displaystyle=nH(\tilde{X}_{J}|\tilde{Y}_{J},\tilde{U}_{J},J).$ (15)

Therefore, we obtain

$\displaystyle D(P_{\tilde{X}^{n}\tilde{Y}^{n}}||P_{X^{n}Y^{n}})\geq
n(D(P_{\tilde{X}_{J}\tilde{Y}_{J}}||P_{XY})+I(\tilde{U}_{J},J\land\tilde{X}_{J}|\tilde{Y}_{J})).$
(16)

Moreover, since $nR_{2}\geq H(\tilde{M_{2}})$, we have

$\displaystyle nR_{2}$ $\displaystyle\geq H(\tilde{M_{2}})$
$\displaystyle=H(\tilde{M_{2}})-H(\tilde{M_{2}}|\tilde{X}^{n},\tilde{Y}^{n})$
$\displaystyle=I(\tilde{M_{2}}\land\tilde{X}^{n},\tilde{Y}^{n})$
$\displaystyle=H(\tilde{X}^{n},\tilde{Y}^{n})-H(\tilde{X}^{n},\tilde{Y}^{n}|\tilde{M_{2}}).$
(17)

Then, we can consider the following inequality

$\displaystyle D(P_{\tilde{X}^{n}\tilde{Y}^{n}}||P_{X^{n}Y^{n}})$
$\displaystyle\geq
D(P_{\tilde{X}^{n}\tilde{Y}^{n}}||P_{X^{n}Y^{n}})+H(\tilde{X}^{n},\tilde{Y}^{n})-H(\tilde{X}^{n},\tilde{Y}^{n}|\tilde{M_{2}})-nR_{2}.$
(18)

A very similar argument in [11, Proposition 1] leads to

$\displaystyle
D(P_{\tilde{X}^{n}\tilde{Y}^{n}}||P_{X^{n}Y^{n}})+H(\tilde{X}^{n},\tilde{Y}^{n})=n(D(P_{\tilde{X}_{J}\tilde{Y}_{J}}||P_{XY})+H(\tilde{X}_{J},\tilde{Y}_{J})).$
(19)

Also,

$\displaystyle H(\tilde{X}^{n},\tilde{Y}^{n}|\tilde{M_{2}})$
$\displaystyle=\sum_{j=1}^{n}H(\tilde{X}_{j},\tilde{Y}_{j}|\tilde{M_{2}},\tilde{X}_{j}^{-},\tilde{Y}_{j}^{-})$
$\displaystyle=nH(\tilde{X}_{J},\tilde{Y}_{J}|\tilde{U}_{J},J).$ (20)

Therefore, we obtain

$\displaystyle\frac{1}{n}D(P_{\tilde{X}^{n}\tilde{Y}^{n}}||P_{X^{n}Y^{n}})$
$\displaystyle\geq
D(P_{\tilde{X}_{J}\tilde{Y}_{J}}||P_{XY})+H(\tilde{X}_{J},\tilde{Y}_{J})-H(\tilde{X}_{J},\tilde{Y}_{J}|\tilde{U}_{J},J)-R_{2}$
$\displaystyle=D(P_{\tilde{X}_{J}\tilde{Y}_{J}}||P_{XY})+I(\tilde{U}_{J},J\land\tilde{X}_{J},\tilde{Y}_{J})-R_{2}$
$\displaystyle=D(P_{\tilde{X}_{J}\tilde{Y}_{J}}||P_{XY})+I(\tilde{U}_{J},J\land\tilde{X}_{J}|\tilde{Y}_{J})+I(\tilde{U}_{J},J\land\tilde{Y}_{J})-R_{2}.$
(21)

Combining (10), (16) and (21), we have

$\displaystyle\frac{1}{n}\log\frac{1}{P_{\textrm{c}}}\geq$ $\displaystyle
D(P_{\tilde{X}_{J}\tilde{Y}_{J}}||P_{XY})+I(\tilde{U}_{J},J\land\tilde{X}_{J}|\tilde{Y}_{J})+\left|I(\tilde{U}_{J},J\land\tilde{Y}_{J})-R_{2}\right|^{+}.$
(22)

Noting (11) and (12), we have a condition that $R_{1}\geq
H(\tilde{X}_{J}|\tilde{U}_{J},J)$. As a consequence, we obtain

$\displaystyle\min_{\psi,\varphi_{1},\varphi_{2}}\frac{1}{n}\log\frac{1}{P_{\textrm{c}}}$
$\displaystyle\geq\min_{P_{\tilde{U}\tilde{X}\tilde{Y}}}\left[D(P_{\tilde{X}\tilde{Y}}||P_{XY})+I(\tilde{U}\land\tilde{X}|\tilde{Y})+\left|I(\tilde{U}\land\tilde{Y})-R_{2}\right|^{+}\right],$
(23)

where the minimization of $P_{\tilde{U}\tilde{X}\tilde{Y}}$ is taken under the
condition $R_{1}\geq H(\tilde{X}|\tilde{U})$. Note that we can verify

$\displaystyle
D(P_{\tilde{X}\tilde{Y}}||P_{XY})+I(\tilde{U}\land\tilde{X}|\tilde{Y})$
$\displaystyle=D(P_{\tilde{X}\tilde{Y}}||P_{XY})+D(P_{\tilde{U}|\tilde{X}\tilde{Y}}||P_{\tilde{U}|\tilde{Y}}|P_{\tilde{X}\tilde{Y}})$
$\displaystyle=D(P_{\tilde{U}\tilde{X}\tilde{Y}}||P_{\tilde{U}|\tilde{Y}}P_{XY}).$

Consequently, we have

$G(R_{1},R_{2}|P_{XY})\geq F(R_{1},R_{2}|P_{\tilde{X}\tilde{Y}}),$

except the cardinality bound, which will be discussed in the next section.

###  4.3 Cardinality Bound

By the support lemma [4, Lemma 15.4], we can restrict the cardinality of $U$
to $|\mathcal{U}|\leq|\mathcal{X}||\mathcal{Y}|+2$ as follows. We set the
following functions on ${\cal P}(\mathcal{X}\times\mathcal{Y})$:

$\displaystyle g_{1}(P_{\tilde{X}\tilde{Y}})$
$\displaystyle=H(\tilde{X}|\tilde{Y}),$ $\displaystyle
g_{2}(P_{\tilde{X}\tilde{Y}})$ $\displaystyle=H(\tilde{X}),$ $\displaystyle
g_{3}(P_{\tilde{X}\tilde{Y}})$ $\displaystyle=H(\tilde{Y}).$

Then, observe that

$\displaystyle P_{\tilde{X}\tilde{Y}}(x,y)$
$\displaystyle=\sum_{u}P_{\tilde{U}}(u)P_{\tilde{X}\tilde{Y}|\tilde{U}}(x,y|u),$
$\displaystyle H(\tilde{X}|\tilde{U},\tilde{Y})$
$\displaystyle=\sum_{u}P_{\tilde{U}}(u)g_{1}(P_{\tilde{X}\tilde{Y}|\tilde{U}}(\cdot,\cdot|u)),$
$\displaystyle H(\tilde{X}|\tilde{U})$
$\displaystyle=\sum_{u}P_{\tilde{U}}(u)g_{2}(P_{\tilde{X}\tilde{Y}|\tilde{U}}(\cdot,\cdot|u)),$
$\displaystyle H(\tilde{Y}|\tilde{U})$
$\displaystyle=\sum_{u}P_{\tilde{U}}(u)g_{3}(P_{\tilde{X}\tilde{Y}|\tilde{U}}(\cdot,\cdot|u)).$

Note that
$I(\tilde{U}\land\tilde{X}|\tilde{Y})=H(\tilde{X}|\tilde{Y})-H(\tilde{X}|\tilde{U},\tilde{Y})$
and $I(\tilde{U}\land\tilde{Y})=H(\tilde{Y})-H(\tilde{Y}|\tilde{U})$.
Therefore, by the support lemma, it suffices to take
$|\mathcal{U}|\leq|\mathcal{X}||\mathcal{Y}|-1+3=|\mathcal{X}||\mathcal{Y}|+2$.

##  5 Conclusion

In this paper, we derived the tight strong converse exponent for the WAK
network. The rate region of the WAK network has two characteristics: an
auxiliary random variable and the Markov chain. Because of these
characteristics, deriving the tight strong converse exponent of the WAK
network has been regarded as a challenging problem. We believe that our proof
methods can be also applied for other multi-terminal network problems.

Even though we derived the tight characterization of the exponent, computation
of the characterization is non-trivial. Thus, it is a future research agenda
to clarify whether our characterization strictly improves on the
characterization of Oohama or not.

## Acknowledgments

The authors would like to thank Prof. Yasutada Oohama and Prof. Jun Chen for a
valuable discussion.

## References

  * [1] R. Ahlswede and J. Körner, “Source coding with side information and a converse for degraded broadcast channels,” IEEE Trans. Inf. Theory, vol. IT-21, pp. 629-637, 1975.
  * [2] R. Ahlswede, P. Gács and J. Körner, “Bounds on conditional probabilities with applications in multi-user communication,” Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete, vol. 34, no. 2, pp. 157-177, 1976.
  * [3] S. Arimoto, “On the converse to the coding theorem for discrete memoryless channels,” IEEE Trans. Inf. Theory, vol. IT-19, pp. 357-359, 1973.
  * [4] I. Csiszár and J. Körner, Information Theory, Coding Theorems for Discrete Memoryless Systems, 2nd ed. Cambridge, U.K.: Cambridge Univ. Press, 2011.
  * [5] G. Dueck and J. Körner, “Reliability function of a discrete memoryless channel at rates above capacity,” IEEE Trans. Inf. Theory, vol. IT-25, pp. 82-85, 1979.
  * [6] R. M. Gray and A. D. Wyner, “Source coding for a simple network,” The Bell System Technical Journal, vol. 53, no. 9, pp. 1681-1721, 1974.
  * [7] W. Gu and M. Effros, “A strong converse for a collection of network source coding problems,” In Proc. of the 2009 IEEE Int. Symp. on Inf. Theory, pp. 2316-2320, 2009.
  * [8] W. Gu and M. Effros, “A strong converse in source coding for supersource networks,” In Proc. of the 2011 IEEE Int. Symp. on Inf. Theory, pp. 395-399, 2011.
  * [9] Y. Oohama, “Exponential strong converse for source coding with side information at the decoder,” Entropy, vol. 20, no. 5, 2018.
  * [10] Y. Oohama, “Exponential strong converse for one helper source coding problem,” Entropy, vol. 21, no. 6, 2019.
  * [11] H. Tyagi, S. Watanabe, “Strong Converse using Change of Measure Arguments,” IEEE Trans. on Inf. Theory, vol. 66, no. 2, pp. 689-703, 2020.
  * [12] S. Watanabe, “Second-Order Region for Gray-Wyner Network,” IEEE Trans. on Inf. Theory, vol. 63, no. 2, pp. 1006-1018, 2017\.
  * [13] S. Watanabe, “A converse bound on Wyner-Ahlswede-Körner network via Gray-Wyner network,” In Proc. of the 2017 IEEE Inf. Theory Workshop, pp. 81-85, 2017.
  * [14] A. D. Wyner, “On source coding with side information at the decoder,” IEEE Trans. Inf. Theory, vol. IT-21, pp. 294-300, 1975\.
  * [15] A. D. Wyner and J. Ziv, “The rate-distortion function for source coding with side information at the decoder,” IEEE Trans. Inf. Theory, vol. IT-22, pp. 1355-1678, 1975.