Experimental and DFT analysis of antisite disorder and Griffiths phase in Dy₂CoMnO₆: structural, electronic, and magnetic insights

Abstract

The intricate interaction of emergent degrees of freedom, ground state degeneracy, and competing magnetic interactions in oxide-based double perovskite frustrated magnets can give rise to exotic excitations and correlated quantum phenomena with promising technological applications. This study investigates the structural, magnetic, and electronic properties of Dy₂CoMnO₆ synthesized via solid-state reaction. Rietveld refinement of XRD data confirms a monoclinic P2₁/n structure, with Raman spectra revealing BO₆ polyhedral dynamics. XPS analysis indicates mixed valence states of Co and Mn. Magnetic measurements show ferromagnetic ordering below T_c = 87 K due to Co²⁺–Mn⁴⁺ superexchange, along with a Griffiths phase above T_c (T_G = 91 K) and canted antiferromagnetic correlations at low temperature. A coercivity of 6.8 kOe at 10 K further supports FM behavior. Density functional theory (DFT) calculations validate the stability of Dy₂CoMnO₆ in the ferromagnetic phase, showing that the ferromagnetic state is energetically more favorable than the antiferromagnetic state. The direct band gap of Dy₂CoMnO₆ was found to be approximately 1.47 eV, while the indirect band gap was around 1.04 eV. Electronic band structure and density of states calculations predict band gaps of 1.60 eV and 3.20 eV for the minority and majority spin orientations, respectively, confirming the semiconductor nature of Dy₂CoMnO₆. These calculated values closely match the experimentally observed band gap.

---

1. Introduction

The quest for materials with novel functionalities for next-generation spintronic devices has driven intensive research into compounds exhibiting concurrent electric and magnetic ordering. This coexistence opens pathways to manipulate one order parameter through control of the other, unlocking multifunctional properties.^1–3 Among these materials, perovskites with the ABO₃ structure stand out for their extraordinary properties, including ferroelectricity, multiferroicity, and giant magnetoresistance (GMR), making them promising candidates for advanced technological applications.^4,5 A further evolution of these materials is achieved by doubling the perovskite unit cell, forming double perovskites (DPs) with the formula A₂B′B′′O₆. This structure allows extensive cationic substitutions at the A and B sites, enabling incorporation of transition metals and rare-earth elements to tailor physical properties for diverse applications.^6–8

Rare-earth-based double perovskites (A₂B′B′′O₆) with rock-salt ordering have garnered significant attention due to their intriguing phenomena, including magnetoelectric coupling, multiferroicity, spin-polarized conductivity, and superconductivity.^9,10 Particularly, systems with B-site cations like Co, Mn, and Ni exhibit insulating behavior and high-temperature ferromagnetism, drawing interest for their potential in multifunctional devices.^11–16 The RE₂CoMnO₆ (RCMO) family, where RE denotes a rare-earth element, offers remarkable tunability in electronic and magnetic properties through modifications of the rare-earth ionic radii (r_R³⁺) and external pressure, making them suitable for spintronic applications.^17–20

The magnetic properties of RE₂CoMnO₆ compounds are dominated by superexchange interactions between Co²⁺ and Mn⁴⁺ ions, governed by the Goodenough–Kanamori rules.^21,22 Structural distortions and variations in the ionic radii significantly influence these interactions, impacting the magnetic transition temperatures.^23,24 Additionally, antisite disorder, where Co and Mn cations interchange positions, introduces complexities such as spin-glass states and secondary magnetic transitions due to competing magnetic interactions (Co²⁺–Co²⁺ or Mn⁴⁺–Mn⁴⁺). Such disorder, along with antiphase boundaries, can profoundly alter the material's magnetic and electronic behaviour.^25–28

Research on analogous rare-earth double perovskites underscores the critical role of antisite disorder in governing structural, electronic, and magnetic properties. Notable studies on La₂CoMnO₆,^29,30 Sm₂CoMnO₆,^11,18,31 Nd₂CoMnO₆^11,32 and Tb₂CoMnO₆³³ reveal a spectrum of fascinating phenomena, including exchange bias effects, Griffiths phases, and spin–phonon coupling, emphasizing the intricate interplay between cationic ordering and physical properties. Nayak et al.³¹ examined Sm₂CoMnO₆ (SCMO) polycrystals and found both ordered (monoclinic P2₁/n) and disordered (orthorhombic Pnma) phases. They observed an exchange bias effect of 10 kOe at 2 K, a spontaneous exchange bias below 20 K, and a ferromagnetic ground state with a 1.3 eV band gap.

Further advancing the field, Aswathi et al.³⁴ investigated Pr₂FeMnO₆, uncovering a negative exchange bias below 200 K due to the Dzyaloshinskii–Moriya (DM) interaction and magnetic exchange anisotropy. Their thermomagnetic analysis revealed a high-temperature magnetic transition at T_c = 576 K, with the presence of ferromagnetic short-range correlations above T_c, suggesting a Griffiths-like phase. Das et al.²⁹ explored Nd₂CoMnO₆, identifying a Griffiths phase, and Bhatti et al.¹⁸ investigated Sm₂CoMnO₆, revealing spin–phonon coupling, a high dielectric constant, and frequency-dependent dielectric response driven by electron hopping.

Dy₂CoMnO₆ (DCMO), in particular, exhibits complex phenomena such as Griffiths phase, mixed valence states, and short-range magnetic ordering. Xin et al.³⁵ reported that DCMO crystallizes in a monoclinic P2₁/n structure and hosts a Griffiths phase, unlike Gd₂CoMnO₆, due to distinct 3d–4f exchange interactions and antisite disorder disrupting long-range ferromagnetic order. Chanda et al.³⁶ demonstrated excellent photocatalytic efficiency of DCMO nanoparticles with a bandgap of 1.93 eV and p-type polaronic conduction. Pedro et al.³⁷ revealed ferrimagnetic ordering accompanied by structural phase coexistence, whereas Das et al.³⁸ observed ferroelectric-paraelectric transitions, polaronic conduction following the OLPT model, and an antiferromagnetic ground state. Muddelwar et al.³⁹ further explored DCMO as a supercapacitor electrode, revealing significant electrochemical performance. Structural studies showed CoO₆ and MnO₆ octahedral tilting and electron density distortions mapped using GFourier analysis. The micron-sized flat particles provided faster electrochemical reactions and shorter ion/electron diffusion paths. Electrochemical characterization indicated a specific capacitance of 100.9 F g⁻¹ at 1 A g⁻¹ with remarkable cyclic stability of 87% over 10 000 cycles, suggesting its potential as a next-generation supercapacitor electrode material. Further advancing this discourse, Valian et al.⁴⁰ synthesized urchin-like DCMO nanostructures via a modified Pechini route, employing novel combinations of chelating and cross-linking agents to regulate particle morphology. The resultant nanostructures displayed superior photocatalytic performance, achieving 88% methylene blue dye degradation under ultraviolet irradiation within 120 minutes, underscoring their efficacy as photocatalytic materials. Essaoud et al.⁴¹ utilized density functional theory (DFT) to probe the electronic, magnetic, and thermoelectric properties of DCMO, revealing its ferrimagnetic semiconducting behavior with spin-dependent indirect bandgaps of 0.31 eV (minority spin) and 1.65 eV (majority spin). The mixed covalent-ionic bonding characteristic of Dy–O, Co–O, and Mn–O was highlighted by topological analysis using quantum theory of atoms in molecules (QTAIM). At higher temperatures (ZT > 0.5 above 500 K), when p-type hole doping outperformed electron doping, the material demonstrated encouraging thermoelectric performance. Notably, DCMO demonstrated exceptional magnetic robustness, with dominant dysprosium contributions, maintaining structural and magnetic stability under hydrostatic pressures up to 35 GPa. Anirban et al.⁴² studied Dy₂CoMnO₆, revealing a monoclinic phase with structural distortions. Charge transport showed activation energies of 0.16 eV (≤75 °C) and 0.37 eV (≥100 °C). Dielectric properties exhibited non-Debye behavior, and the leakage current density followed Ohmic conduction, described by the Schottky barrier model.

In this work, we undertake a comprehensive investigation of the structural, electronic, and magnetic properties of DCMO, with particular emphasis on elucidating the contributions of antisite disorder and 3d–4f exchange interactions to its functional behavior. By bridging the domains of structural modulation and physical phenomena, this study seeks to advance the understanding of structure–property correlations and broaden the technological applicability of DCMO despite significant progress in understanding Dy₂CoMnO₆, a comprehensive understanding of antisite disorder's impact on its structural, electronic, and magnetic properties remains elusive. Current research often separates experimental observations from theoretical models, resulting in fragmented insights into long-range magnetic ordering, charge transport, and electronic interactions. This study combines experimental techniques and DFT calculations to address these gaps, focusing on the effects of antisite disorder. By linking experimental data with theoretical predictions, it aims to deepen the understanding of rare-earth double perovskites, paving the way for their integration into advanced multifunctional devices.

2. Experimental and computational details

• Sample preparation – To synthesize polycrystalline Dy₂CoMnO₆ double perovskite using a solid-state synthesis route, high-purity precursors including Dy₂O₃, CoO, and MnO₂ are initially acquired. These precursors undergo a meticulous grinding process for 4 hours in an agate mortar to ensure thorough mixing. Subsequently, the mixed powders are subjected to calcination at 1200 degrees celsius for 12 hours in a controlled atmosphere furnace to facilitate solid-state reaction and formation of the desired Dy₂CoMnO₆ phase. After calcination, the powder is ground again for 4 hours to achieve a fine, homogeneous mixture. Next, the powder is pelletized using a hydraulic press applying 10 tons of pressure to form uniform pellets. These pellets are then sintered at 1200 degrees celsius for 24 hours to promote densification and enhance crystallinity. Following sintering, the samples are gradually cooled and subjected to various characterization techniques.

• Characterization details – Post-synthesis, an exhaustive characterization of the polycrystalline Dy₂CoMnO₆ double perovskite was undertaken utilizing a suite of sophisticated analytical techniques. A Bruker D8 Advance apparatus was used to perform the powdered X-ray diffraction (XRD) investigation using CuKα radiation (λ = 1.54056 Å). Subsequent Rietveld refinement of the diffraction patterns was meticulously executed using the FullProf Suite, where the pseudo-Voigt function facilitated peak profile fitting, leveraging crystallographic input data sourced from relevant literature. Using the KBr pellet approach, Fourier-transform infrared (FT-IR) absorption spectra were obtained using a PerkinElmer FT-IR spectrometer (Spectrum 1000, Japan). Using a 633 nm He–Ne laser running at full intensity, Raman spectroscopy was carried out using a LaBRAM HR spectrometer (Horiba, France SAS). X-Ray photoelectron spectroscopy (XPS) measurements were carried out at the Indus-2 synchrotron facility, RRCAT Indore, employing photoemission electron spectroscopy (PES) on beamline BL-14. In an ultra-high vacuum environment with a pressure of 5 × 10⁻⁹ mbar, the experimental setup included a bending magnet source of 1.5 T, a Si(111) double-crystal monochromator, and a hemispherical analyzer/detector system (Phoibos 225, Specs manufacture). Magnetization measurements were executed using a Quantum Design SQUID VSM, spanning the temperature range of 10 ≤ T ≤ 350 K and subjected to a magnetic field of H = 500 Oe. For low-field magnetization assessments, the reset magnet mode was employed to negate any residual or stray field effects. Ultraviolet-visible (UV-Vis) absorption spectra were recorded with a CARY Varian-300 spectrophotometer, ensuring a comprehensive elucidation of the material's optical characteristics.

• Computational details: The Vienna Ab initio Simulation Package (VASP) was used to conduct geometrical optimization and electronic calculation in this study within the context of density-functional theory (DFT).⁴³ Both electron–electron interactions and ionic core effects were precisely incorporated using the projector-augmented wave (PAW) approach. The calculations for the exchange–correlation potential were carried out using the Perdew–Burke–Ernzerhof (PBE) functional within the generalized gradient approximation (GGA). To better handle the localization of the d-electron states in the Co and Mn atoms, we applied a Hubbard correction (DFT+U), setting the effective U values at 4.80 eV for Co and 4.50 eV for Mn.⁴⁴ To guarantee accurate geometrical and electrical properties, a kinetic cut-off energy of 500 eV was established. We also conducted band structure calculations that included spin–orbit coupling (SOC), applying the DFT+U method to precisely capture electron interactions and localization. For optimisation and partial density of states analysis, a dense 10 × 8 × 6 k-point mesh created with the Monkhorst–Pack (MP) technique was also used. Structure relaxation is performed using the conjugate gradient algorithm until the total energy converges to within 10⁻⁴ eV and the Hellmann–Feynman force on each atom is less than 0.003 eV A⁻¹.

3. Results and discussion

3.1. Structural analysis

The X-ray diffraction (XRD) pattern obtained at room temperature for the Dy₂CoMnO₆ double perovskite compound is utilized for determining the bulk phase, estimating compositions, and establishing lattice parameters. The results of the X-ray diffraction (XRD) pattern and the Rietveld refinement obtained using the FULL PROF SUITE are presented in Fig. 1. The pattern covers a 2θ range of 10° to 70°. The DCMP powder's crystalline structure has been confirmed to display monoclinic crystal symmetry, which is represented by the space group P2₁/n.³⁵ There are no observable impurity peaks within the resolution limits of the instrument, indicating that the material is highly pure and single-phase. Monoclinic symmetry has been effectively indexed for the predominant reflections that exceed the background threshold. Table 1 provides information on the structural characteristics that were simulated using the XRD pattern. In comparing the results, research utilizing XRD and neutron techniques on the parent compounds DyCoO₃ and DyMnO₃ has confirmed their crystal structures to be orthorhombic (Pnma). Nevertheless, upon introducing a mixture of DyCoO₃⁴⁵ and DyMnO₃⁴⁶ compounds, the crystal structure of DCMO undergoes a shift to a monoclinic configuration with reduced symmetry (space group P2₁/n).³⁸ This change is ascribed to the partial replacement of Dy at the Dy site and Co at the Co site by Mn, or vice versa.

Fig. 1 (a) Rietveld refinement of powder X-ray diffraction pattern of Dy₂CoMnO₆; (b) enlarged view of the selected 2θ range (32.25°–34.50°) highlighting the fitting accuracy in that region.

Table 1 Structural characteristics of Dy₂CoMnO₆ derived from Rietveld analysis of XRD pattern

Dy2CoMnO6
Space group			P21/n (14)
Lattice parameters
a(Å)	b (Å)	c (Å)	α	β	γ
5.2540	5.5755	7.4902	90°	90.0126°	90°
Volume (Å^3)			219.415504

---

Atomic positions
Atom	Site	x	y	z	Occupancy
Dy	4e	−0.01200	0.07276	0.25208	1.0
Co	2c	0.00000	0.50000	0.00000	0.5
Mn	2d	0.50000	0.00000	0.00000	0.5
O1	4e	0.08839	0.47656	0.27729	1.0
O2	4e	0.62803	0.28493	0.06634	1.0
O3	4e	0.72874	0.26667	0.45298	1.0

---

Reliability factors
R p	R wp	R exp	R Bragg	R F	χ ^2
26.2	20.7	17.97	5.694	5.984	1.32

---

Bond distance (Å)
(Co–O1) × 2	(Co–O2) × 2	(Co–O3) × 2	(Mn–O1) × 2	(Mn–O2) × 2	(Mn–O3) × 2
2.13215	2.34620	1.94401	1.73660	1.79527	1.96149

---

Bond angle
Mn–O1–Co	150.793°	Mn–O2–Co	134.888°	Mn–O3–Co	157.506°

---

Fig. 2 illustrates the schematic illustration of the Dy₂CoMnO₆ unit cell, depicting the distribution of ions across crystallographic positions. As per Table 1, these positions correspond to 4e for Dy³⁺ ions, 2c for Co²⁺ ions, 2d for Mn⁴⁺ ions, and 4e for O²⁻ ions. Six O²⁻ ions encircle each Co²⁺ and Mn⁴⁺ ion to create CoO₆ and MnO₆ octahedra, respectively.⁵ The BO₆ octahedra tilting in the direction of the b-axis is described by the Glazer system a⁻b⁺a⁻. The schematic diagram shows tilt in both CoO₆ and MnO₆ octahedra. To determine the cations’ valence states quantitatively, bond valence sums (BVS) were calculated. The final structural parameters are shown in Table 1, which also includes important bond distances and angles related to CoO₆ and MnO₆ octahedra, as well as BVS values for Dy₂CoMnO₆.

Fig. 2 (a) The crystal structure with monoclinic space group of Dy₂CoMnO₆ double perovskite along c-axis, (b) and (c) visual representation of bond angles between CoO₆ and MnO₆ octahedra.

The refined structural parameters presented in Table 1, along with the monoclinic crystal structure depicted in Fig. 2, clearly illustrate that the bond lengths and bond angles, such as Co–O and Mn–O, and Co/Mn–O–Mn/Co, respectively, do not exhibit identical values, with Co–O/Mn–O bond lengths and Co/Mn–O–Mn/Co bond angles deviating from 180°.²⁹ This finding strongly implies that the Dy₂CoMnO₆ (DCMO) system exhibits a distorted crystal structure. Furthermore, the Co–O/Mn–O bond lengths and Co/Mn–O–Mn/Co bond angles in DCMO differ from those observed in an ordered double Perovskite (DP) compound at room temperature. In disordered systems, it is well known that disorder and deformed crystal structures in perovskites frequently affect magnetic contests and cause frustration.^14,47,48 Thus, a similar situation of magnetic competition and frustration is anticipated in our disordered Dy₂CoMnO₆ system.

3.2. Group theory analysis of structural Raman-active modes

Raman spectroscopy, a widely recognized non-destructive and inelastic method, is employed for investigating vibrational modes, functional groups, and determining distortion levels in polycrystalline samples.⁴⁹ The formation of peaks at specific shifts provides quantifiable insights into symmetric/anti-symmetric vibrations, stretching/bending modes, and the excitations necessary for various bonds, phonons, and ions.^50,51 Raman analysis of Dy₂CoMnO₆ DP was conducted at room temperature, with the results displayed in Fig. 3.

Fig. 3 Raman spectra of Dy₂CoMnO₆.

The Raman spectra, stemming from the monoclinic structure identified by space group P2₁/n (C_2h⁵, no. 14), can be interpreted as a combination of in-phase (+) and antiphase (−) tilts of CoO₆ and MnO₆ octahedra, occurring along the [001] direction and within the basal plane of the pseudo-cubic cell.^52–54 This aligns with the a⁻a⁻c⁺ notation according to Glazer's system. These vibrational modes observed in Raman spectroscopy predominantly reflect the dynamic characteristics of Co/MnO₆ octahedra and Co/Mn–O bonds. Additionally, this monoclinic arrangement shows a direct group–subgroup relationship with tetragonal symmetry of I4/m. By employing factor group analysis, we ascertain the allocation of zone-center vibrational modes in relation to the corresponding representations of C_2h⁵ point groups and monoclinic space group.^55,56 The outcomes are outlined in Table 2.

Table 2 The irreducible representations of the normal modes of vibrations of Dy₂CoMnO₆ double perovskite at the zone center for P2₁/n (no. 14) space group and C_2h⁵ point group with corresponding site symmetry

Atom	Site	Symmetry	Distribution
Dy	4e	1	3Ag + 3Au + 3Bg + 3Bu
Co	2c	−1	3Au + 3Bu
Mn	2d	−1	3Au + 3Bu
O1	4e	1	3Ag + 3Au + 3Bg + 3Bu
O2	4e	1	3Ag + 3Au + 3Bg + 3Bu
O3	4e	1	3Ag + 3Au + 3Bg + 3Bu
Γ = 12A g + 18A u + 12B g + 18B u
Γ Raman = 12Ag + 12Bg
Γ Acoustic = Au + 2Bu
Γ IR = 17Au + 16Bu

---

Hence, it is anticipated that the monoclinic Dy₂CoMnO₆ compound will exhibit 24 Raman-active vibrations, i.e.; 12A_g and 12B_g modes. Utilizing the factor-group analyses outlined above, a meticulous fitting of the Raman spectra for the Dy₂CoMnO₆ double perovskite compound was conducted. The outcomes are illustrated in Fig. 4(a)–(c), wherein the spectrum acquired at 300 K is segmented into three wavenumber ranges: 60–300 cm⁻¹ (Fig. 4(a)), 300–650 cm⁻¹ (Fig. 4(b)), and 650–950 cm⁻¹ (Fig. 4(c)) for enhanced clarity. The observed modes and their corresponding proposed assignments, which will be discussed further below, are documented in Table 3. Out of the anticipated 24 phonons for monoclinic compounds, we successfully detected and fitted a sum of 21 Lorentzian peaks [illustrated by the solid lines in Fig. 4(a)–(c)], as only a few peaks with very low intensity were disregarded during the Lorentzian fitting process. The peak positions and full-width at half-maximum (FWHM) values are detailed in Table 3.

Fig. 4 Lorentzian fitted Raman spectra of Dy₂CoMnO₆ into three wavenumber ranges: (a) 60–300 cm⁻¹, (b) 300–650 cm⁻¹, and (c) 650–950 cm⁻¹.

Table 3 Raman fitting parameters for Dy₂CoMnO₆ double Perovskite ceramics

Band	Wavenumber (cm ^−1 )	FWHM (cm ^−1 )
1	96.4568	19.34464
2	105.3659	7.90978
3	118.3337	27.41265
4	134.258	21.6095
5	152.5968	6.34473
6	197.5378	4.90579
8	261.193	14.39484
9	281.8333	1.59602
10	349.6054	12.33295
11	374.234	21.79637
12	432.6127	20.40047
13	482.8098	7.27032
14	495.4944	10.77868
15	522.7165	8.58732
16	565.8443	19.57891
17	620.1975	10.97171
18	619.9716	9.44831
19	689.9308	11.72895
20	728.7796	155.1789
21	927.2323	27.29348

---

The Raman representation corresponding to the monoclinic (P2₁/n) structure of Dy₂CoMnO₆ can be expressed as given in eqn (i):

Γ = T(3A_g + 3B_g) + L(3A_g + 3B_g) + ν₁(A_g + B_g) + ν₂(2A_g + 2B_g) + ν₅(3A_g + 3B_g) (i)

In this context, L and T denote the librational and translational phonon modes, respectively. The notation ν₁ refers to the totally symmetric stretching, ν₂ denotes the asymmetric stretching, and ν₅ represents the symmetric bending. Within the monoclinic lattice framework, the unit cell accommodates two formula units of Dy₂CoMnO₆, giving rise to the manifestation of A_g and B_g symmetry components across all vibrational modes. It is reasonable that these symmetry components, particularly of the internal vibrational modes, converge at nearly identical frequencies, necessitating the employment of polarized Raman spectroscopy on single crystals to achieve their resolution.

In contrast, for polycrystalline sample, as examined in the present study, the inherent averaging of crystallographic orientations results in a simplification of the observed spectral features. Consequently, the Raman spectra are anticipated to display a solitary band corresponding to the symmetric stretch (ν₁), a doublet associated with the asymmetric stretch (ν₂), and a triplet attributed to the bending mode (ν₅). The L and T vibrational modes encompass the frequency range of 90–300 cm⁻¹, predominantly arising from the dynamics of the (Co/Mn)O₆ octahedra and Dy–O bonds.^50,57,58 The translational (T) modes within this interval are attributed to the displacement of the Dy cation, with specific lattice modes identified at 118.33, 134.25, 152.59, 197.53, and 261.19 cm⁻¹ peaks located within the 320–350 cm⁻¹ and 430–440 cm⁻¹ ranges correspond to librational (L) lattice vibrations, which stem from the rotational dynamics of the Dy cation. Notably, librational modes are discernible at 281.83 cm⁻¹, 349.60 cm⁻¹, 374.23 cm⁻¹, and 432.61 cm⁻¹. For polycrystalline samples, a rigorous and exhaustive assignment of all experimentally observed modes via polarization analysis is precluded due to orientation averaging. However, important modes related to the MnO₆ octahedra can be clarified by comparing the vibrational modes of the prototypical Fm3m cubic structure with the P2₁/n monoclinic structure.

The peak observed at 728 cm⁻¹ is assigned to the A_g mode, representing the totally symmetric stretching of the MnO₆ octahedra (ν₁), wherein the Dy atom remains immobile as the oxygen atoms oscillate along the Co–O–Mn axis. Asymmetric oxygen stretching vibrations, with the Dy atom held stationary, are designated as the ν₂ (B_g) mode, occurring at 565.84 cm⁻¹ and 689 cm⁻¹.⁵⁹ The bending dynamics of oxygen within the octahedra, classified under the ν₅ mode, manifest at 482.80 cm⁻¹ and 495.49 cm⁻¹, with their frequencies modulated by the chemical nature of the Co ions.^60,61 Additionally, the mode at 927 cm⁻¹ is identified as the ν₁ mode, signifying the totally symmetric stretching of oxygen atoms along the Co–O–Mn axis within the MnO₆ octahedra.^58,62,63

3.3. FTIR spectroscopy

To identify the chemical groups, present on the surface of the prepared Dy₂CoMnO₆, Fourier-transform infrared (FT-IR) analysis was conducted at room temperature, as illustrated in Fig. 5. We have identified several prominent bands in the spectrum. According to existing literature, the infrared bands within the range of 400–800 cm⁻¹ correspond to metal–oxygen stretching vibrations at the A and B-sites of the A₂BB′O₆ double perovskite. A detailed description of metal–oxygen vibration modes is illustrated in Fig. 5(b), which is inset within Fig. 5(a). The two bands at 438.94 cm⁻¹ and 557.40 cm⁻¹ are attributed to Dy–O bonds. The characteristic peaks at 633 cm⁻¹ and 570 cm⁻¹ are attributed to Co–O bonds,^64,65 while the peak at 620 cm⁻¹ corresponds to Mn–O bonds.^66,67 Due to exposure of the prepared samples to ambient atmosphere, Dy-carbonate species are formed on the surface of the samples. The band observed at 1385 cm⁻¹ is assigned to the asymmetric stretching of carbonate groups. The C=O double bond absorbs infrared light at wavenumbers between approximately 1600–1900 cm⁻¹.⁶⁶ The band at 3433 cm⁻¹ occurs due to the O–H vibrations.⁶⁶

Fig. 5 FTIR spectrum of Dy₂CoMnO₆.

3.4. Electronic properties by X-ray photoelectron spectroscopy

Antisite disorder (ASD) constitutes an intrinsic phenomenon within double perovskites such as Re₂CoMnO₆, exerting a profound and multifaceted influence on their structural, electronic, and magnetic characteristics. The diminution of the ionic radius of the rare-earth ion (Re³⁺) exacerbates deviations in the Co–O–Mn bond angles from the idealized 180°, amplifying lattice distortions and destabilizing the coherent B-site ordering of Co²⁺ and Mn⁴⁺ ions.³² This disruption facilitates stochastic cation interchange, engendering the emergence of trivalent Co³⁺ and Mn³⁺ ions as a compensatory mechanism to uphold charge neutrality. The resultant alteration in the magnetic exchange pathways attenuates the ferromagnetic superexchange interaction Co²⁺–O–Mn⁴⁺, supplanted by feebler interactions such as Co³⁺–O–Mn³⁺, thereby diminishing magnetic moments and inducing phenomena such as spin-glass behavior or intricate magnetic ground states. Concurrently, the augmented lattice distortion intensifies electron-lattice coupling, perturbing electronic transport and potentially engendering polaronic conduction mechanisms. Thus, ASD, synergistically with the contraction of the RE³⁺ ionic radius, intricately modulates the structural framework and functional attributes of RE₂CoMnO₆ double perovskites.^28,48 To validate the aforementioned hypothesis, X-ray photoelectron spectroscopy (XPS) measurements were conducted.

X-Ray photoelectron spectroscopy (XPS) emerges as a particularly effective method for scrutinizing the valence states and ligand coordination of constituent elements within a material. In materials containing open-shell ions, the interaction of a core electron vacancy with the open-shell leads to the development of numerous structures evident in XPS analysis. To deduce oxidation states and ligand coordination accurately, one must consider not only the primary photoelectron peak features but also related satellite peaks, chemical shifts, and their corresponding intensities. Understanding various physical characteristics of a material hinges upon grasping its electronic structure. Thus, we have thoroughly examined the XPS spectra of the current material, DCMO, to glean insights into its electronic structure. All peak positions have been assigned based on the National Institute of Standards and Technology (NIST) database. The deconvolution analysis of core-level XPS peaks for relevant ions was performed using a combination of Lorentzian and Gaussian distribution functions. The survey scan XPS spectra recorded at 300 K, depicted in Fig. 6, confirm the presence of Dy, Co, Mn, O, and C elements within the system.³⁵ The absence of extrinsic elements confirms the sample's purity, while the detection of the C 1s peak, typical in such analyses, is attributed to molecules from the surrounding air absorbed onto the surface.

Fig. 6 Survey scan XPS spectra of Dy₂CoMnO₆ recorded at 300 K.

High-resolution XPS spectra of Dy-3d, Co-2p, Mn-2p, and O-1s core levels for the as-prepared sample are depicted in Fig. 7(a)–(d), respectively. Due to the spin–orbit interaction, each spectrum is split into two components. One component has a lower J (total angular momentum) value, corresponding to the (l − s) coupling, and it appears on the higher energy side. The other component has a higher J value, corresponding to the (l + s) coupling, and it appears on the lower energy side. The XPS spectra of the Dy 3d core level, along with the fitting of peaks, are depicted in Fig. 7(a). It's apparent from the figure that the two distinct spin-orbital split peaks, Dy 3d_5/2 and Dy 3d_3/2, are situated at 1297.77 eV and 1335.5 eV, respectively, with a spin-orbital splitting energy of 37.73 eV. Detailed analysis of the XPS spectrum indicates that the Dy cations are present in the +3 oxidation state, consistent with previous reports in the literature.³⁸Fig. 7(b) displays the Co-2p core level XPS spectrum of the Dy₂CoMnO₆ double perovskite compound at room temperature. The spectrum exhibits two prominent peaks at 782 and 797 eV corresponding to Co-2p_3/2 and Co-2p_1/2, respectively, due spin orbital coupling.²⁸ Deconvolution of the 2p_3/2 peak reveals two distinct peaks corresponding to Co²⁺ 2p_3/2 and Co³⁺ 2p_3/2 at 785.186 and 781.83 eV, respectively. Similarly, the Co 2p_1/2 peak can be deconvoluted into Co²⁺ 2p_1/2 and Co³⁺ 2p_1/2 peaks at 798.744 and 797.019 eV, respectively.^48,55 In cobalt-based systems, a change of approximately 1 electron volt (eV) in binding energy signifies a corresponding alteration in the valence state of cobalt. Consequently, when employing XPSPEAK 4.1 software for fitting, two peaks are typically considered for each spin–orbit coupling, along with their corresponding charge transfer satellites. Multiple electron excitation of Co²⁺ and Co³⁺ in the cobalt species results in the appearance of two shake-up satellite peaks at 790.025 and 804.240 eV. Therefore, the presence of satellite peaks in the Co 2p XPS spectrum suggests the existence of Co²⁺ ions in the DCMO compound. It is worth mentioning here that the asymmetry and broadening observed in the peaks indicate the presence of mixed valence states of the Co ions. The peak positions and line-shape observed in the Co 2p XPS spectrum are similar to earlier reports, indicating the presence of mixed valence Co ions. For the present system, the observed Co 2p doublet separation of 15 eV suggests the coexistence of both Co²⁺ and Co³⁺ ions. The Co 2p XPS spectrum has been deconvoluted to estimate the concentrations of different Co ions (Co²⁺/Co³⁺), as illustrated in Fig. 7(b). While Co³⁺ ions are predominantly present in the system, the presence of Co²⁺ ions are also unavoidable.^5,33 This, in turn, can lead to competing exchange interactions in Dy₂CoMnO₆.

Fig. 7 High-resolution XPS spectra of (a) Dy-3d, (b) Co-2p, (c) Mn-2p, and (d) O-1s core levels Dy₂CoMnO₆.

The core level Mn 2p XPS spectrum of the present material is illustrated in Fig. 7(c). This spectrum is broadly divided into two spin–orbit coupling peaks, namely Mn 2p_3/2 and Mn 2p_1/2, observed at 644 eV and 656 eV, respectively.¹⁸ These peaks result from the spin–orbit splitting of 2p orbitals, with a splitting energy of 12 eV. The peaks observed near ∼643.38 and 654.76 eV suggest the presence of Mn³⁺ ions, whereas peaks near ∼645.75 and 656.79 eV indicate Mn⁴⁺ charge states. As a result, XPS analysis demonstrates the presence of Co²⁺, Co³⁺, Mn³⁺, and Mn⁴⁺ charge states for the B-site cations.^56,63 The observed magnitudes of these charge states are provided in Table 4. Comparing the chemical formula of the investigated compound with the selected formula, Dy₂CoMnO₆, using the content of Co and Mn verifies the formula. Furthermore, the presence of Mn³⁺ and Co³⁺ confirms our hypothesis regarding the Jahn–Teller (JT) distortion in the system. The O 1s core level XPS spectrum is depicted in Fig. 7(d). The peak at 530.467 eV corresponds to the distinctive peak of O²⁻ ions, indicating the bonding between metal and oxygen (M–O–M). Meanwhile, the peak at 531.976 eV is associated with defect sites having low oxygen coordination. The broad signal at 532.625 eV suggests that oxygen ions are ionized and associated with the subsurface of the Dy₂CoMnO₆ ceramic through physisorption or chemisorption of oxygen molecules.³³ This implies that the subsurface is composed of oxygen ions, which may be classified as excess oxygen or O⁻ species, given their lower electron concentration compared to O²⁻ ions. The presence of mixed states of Co and Mn elements suggests the absence of full long-range ordering between Co and Mn elements in the Dy₂CoMnO₆ double perovskite.⁶⁸

Table 4 XPS parameters of Dy₂CoMnO₆ double perovskite

Atom	Orbital	Binding energy	FWHM	Area	Peak specification
Dy	3d	1297.77	8.364	254 824.90	3d5/2
		1305.22	7.927	21 698.55	Satellite
		1335.5	6.574	217 812.6	3d3/2
		1339.58	3.932	10 180.71	Satellite
Co	2p 3/2	781.830	3.974	7991.513	Co3^+
		785.186	5.463	7180.572	Co^2+
		790.025	4.373	2422.829	Satellite
	2p 1/2	797.019	3.678	2944.458	Co^3+
		798.744	4.095	3240.034	Co^2+
		804.240	5.924	4204.529	Satellite
Mn	2p 3/2	643.379	4.052	5074.523	Mn^4+
		645.748	5.624	6495.841	Mn^3+
	2p 1/2	654.759	4.419	2286.352	Mn^4+
		656.791	5.233	3580.431	Mn^3+
O	1s	530.467	2.556	11 910.12	Lattice oxygen
		531.976	3.113	12 448.57	O^2− vacancy
		532.625	6.058	48 062.01	Chemisorbed oxygen

---

Cationic charge states observed in XPS analysis
Cations/valence state	2 ^+	3 ^+	4 ^+
Co (valence state %)	47.33	52.67
Mn (valence state %)		56.14	43.86

---

The presence of Co³⁺ and Mn³⁺ ions confirm our hypothesis regarding the system's Jahn–Teller (JT) distortion. Recent studies have shown that JT distortion can be induced by Co²⁺ ions in the high-spin (HS) state within an octahedral (O_h) unit. Additionally, the presence of Mn ions in the 3+ and 4+ states can also contribute to JT distortion in the ceramic. This disordering of cationic charges results in non-collinear magnetic behavior in the ceramics. Further discussion on field and temperature-dependent measurements of Dy₂CoMnO₆ will be presented in the subsequent section.

3.5. DFT analysis on magnetic-structural stability

Density functional theory (DFT) was employed to optimize the structure and determine the energetically stable magnetic configuration. Twenty atoms make up the typical unit cell of Dy₂CoMnO₆, which crystallises in the monoclinic space group P2₁/n (no. 14). The (4e) Wyckoff locations are occupied by four dysprosium atoms, whereas the (2c) and (2d) positions are occupied by two cobalt and two manganese atoms, respectively. The oxygen atoms are found to occupy twelve (4e) Wyckoff locations from the fitted CIF file using Rietveld refinement.

To identify the most stable magnetic phase between antiferromagnetic and ferromagnetic configurations, energy–volume curves were plotted (Fig. 8). Dy₂CoMnO₆ is energetically more stable in the ferromagnetic phase, as the curves clearly demonstrate that the ferromagnetic state exhibits a lower energy compared to the antiferromagnetic state.⁶⁹

Fig. 8 Energy versus volume curve for Dy₂CoMnO₆ in FM and AFM configuration.

To ascertain the magnetic ground state, we systematically examined the AFM configuration along with multiple spin-alignment variants of ferrimagnetic (FiM) states. The computed relative energies are presented in Table 5. A comprehensive analysis of these calculations revealed that the FM configuration exhibits the lowest energy, in agreement with magnetization measurements. This finding indicates that the predominant exchange interaction between the Co and Mn atoms in DCMO is ferromagnetic in nature.

Table 5 Calculated relative total energies (eV) for various spin configurations. (Co-1, Co-2, Mn-1, and Mn-2 represent distinct atomic sites within the unit cell; refer to Fig. 2 for details)

Magnetic ordering	Spin alignment				Relative energy (eV)
	Co1	Co2	Mn1	Mn2
FM	↑	↑	↑	↑	0.00
AFM	↑	↓	↑	↓	0.10
Ferri-1	↑	↓	↑	↑	1.27
Ferri-2	↓	↓	↑	↓	0.91
Ferri-3	↓	↓	↓	↑	0.90
Ferri-4	↓	↑	↑	↑	1.27
Ferri-5	↑	↓	↓	↓	0.19
Ferri-6	↑	↑	↓	↑	1.23

---

Quantum exchange interactions, which are connected to atoms’ magnetic moments, distances from one another, and any applied external magnetic field, are what cause the microscopic magnetic behaviour in materials. The Heisenberg effective spin Hamiltonian can be used to model these interactions:

where μ_B is Bohr magneton, g_i is gyromagnetic ratio, represents the spin operator, h⃑ is external magnetic field, and J_ij is exchange coupling constant, which depends on the distance amongst atoms.

The total and partial magnetic moments of the Dy₂CoMnO₆ were calculated in its most stable state, as depicted in Fig. 8 and Table 5. The results indicate that the molecule has a significant net magnetic moment of 11.843μ_B. Dysprosium (Dy) delivers 0.076μ_B to the unit cell's magnetic moment, according to an analysis of the contributions from each of the atoms. Manganese (Mn) contributes significantly with 6.305μ_B, followed by cobalt (Co) with 5.324μ_B, playing a key role in the perovskite structure. This large overall magnetic moment likely arises from the overlap between Co-3d and Mn-3d orbitals. Oxygen atoms contribute 0.138μ_B, further supporting the magnetic properties. Additionally, superexchange interactions between Mn⁴⁺ and Co²⁺ ions, mediated by oxygen, involve the half-filled Mn-3d-e_g and empty Co-3d-e_g orbitals, enhancing the magnetic moment.

3.6. Magnetization and evolution of Griffith phase

In Dy₂CoMnO₆, the presence of three magnetically distinct ions-Dy, Co, and Mn-each characterized by disparate interaction energy scales within the spin lattice, engenders a highly intricate network of competing exchange interactions, which inevitably culminates in a disordered magnetic state. Consequently, the system may transition towards an unconventional magnetic phase as a result of these discordant interactions. To elucidate the ground state magnetic properties of DCMO, we performed dc magnetization measurements, M(T), as a function of temperature on a hard pellet of DCMO. These measurements were executed in zero-field cooled (ZFC) and field-cooled (FC) modes under a magnetic field of 500 Oe, utilizing a quantum design vibrating sample magnetometer (VSM). This methodology facilitates the examination of the intricate dynamics of competing exchange interactions and the potential development of an atypical magnetic phase arising from these conflicting interactions.

The dc magnetization of the DCMO ceramic sample was recorded across a temperature range of 10 to 350 K, employing both zero-field-cooled (ZFC) and field-cooled (FC) protocols under an applied field of 500 Oe. As depicted in Fig. 9(a), the magnetization versus temperature (M–T) curves exhibit pronounced irreversibility between the ZFC and FC processes, along with a notable peak in the ZFC curve. The observed peak at approximately 80 K is likely indicative of the localized freezing of magnetic moments, reflecting the characteristics of a conventional spin-glass state, where the magnetic moments freeze randomly at lower temperatures.^9,13,34 A typical transition from paramagnetic (PM) to ferromagnetic (FM) behavior, occurring around the transition temperature (T_c) of 87 K as shown in Fig. 9(b), is attributed to the ferromagnetic superexchange interactions between Co²⁺ (t⁵_2g,e²_g) and Mn⁴⁺ (t³_2g,e⁰_g) ions,^16,32 with T_c being determined from the minimum of the first derivative of the magnetization in the zero-field-cooled (ZFC) mode, as illustrated in the inset of Fig. 9(a).

Fig. 9 (a) Temperature dependence of the magnetization measured under an applied magnetic field of 500 Oe during zero-field-cooled (ZFC) and field-cooled (FC) protocols; (b) corresponding dM/dT versus temperature plot highlighting the transition temperature more precisely.

In DCMO, the magnetic properties are influenced by the contributions from both rare-earth Dy³⁺ (4f¹⁰, ⁵I₈; J = 8) ions and transition metal ions (Co²⁺ and Mn⁴⁺). From the Curie–Weiss (CW) fit, expressed as [eqn (iii)]; where θ is the Curie–Weiss temperature and C is the Curie constant, the values of C and θ are determined to be 33.37 emu K Oe⁻¹ mol⁻¹ and 18.77 K, respectively. The positive value of θ indicates the presence of ferromagnetic (FM) interactions between Dy³⁺ moments and the transition metal ion sublattice (Co²⁺/Mn⁴⁺), which may also be associated with the crystal electric field excitations of the ⁵I₈ multiplet of the Dy³⁺ ion.⁵

Fig. 10(a) exhibits the variation of inverse susceptibility with temperature for DCMO compound. A down turn can be noticed in the magnetic susceptibility below T_c. There is a broaden peak around ∼30 K, which is relevant to the canted – AFM correlations below T_c. The emergence of a decrease in magnetic susceptibility below T_c is a clear indication of the existence of the Griffiths phase (GP). The Griffiths temperature T_G = 91 K is defined as the maximum of the d(1/χ/)/dT vs. temperature plot as shown in the inset of Fig. 10(b).

Fig. 10 (a) Inverse susceptibility (χ⁻¹) as a function of temperature fitted with the Curie–Weiss law in the high-temperature region; (b) temperature derivative of inverse susceptibility (dχ⁻¹/dT) plotted to identify the Griffiths temperature (T_G = 91 K), highlighting the onset of short-range magnetic correlations.

Between T_c and T_G temperatures, it is evident that the 1/χ − (T) curve doesn’t exhibit a linear trend, suggesting a deviation from the Curie–Weiss (C–W) law for the inverse susceptibility within this temperature range. However, the 1/χ data above T_G can be accurately modelled using the C–W law, signifying a characteristic indicative of paramagnetic behavior. The effective magnetic moment of the compound was evaluated experimentally in units of Bohr magneton (μ_B) by using the following equation:⁷⁰

Here N and k_B represent the Avogadro number and Boltzmann constant, respectively. The experimental value of the effective magnetic moment is found to be 16.33μ_B. The magnetic moment was also evaluated theoretically with the free ionic moments of Dy³⁺ (10.48μ_B), Mn⁴⁺ (3.87μ_B), Mn³⁺ (4.89μ_B), and Co³⁺ (4.89μ_B), Co²⁺ (3.87μ_B) by the following formula:⁷¹The obtained theoretical value of the effective magnetic moment is 16.09μ_B. It is evident that the experimental value is higher than the theoretical value. The observed difference can be attributed to the existence of short-range ferromagnetic magnetic ordering within the paramagnetic matrix. This order could potentially arise from magnetic inhomogeneity or a slight oxygen deficiency in the sample during preparation.

As depicted in Fig. 10(a), the inverse susceptibility follows the Curie–Weiss law at higher temperatures; however, it deviates from linear fitting below ∼100 K. The detected deviation is the characteristic feature of the GP, which originated because of the presence of short-range ferromagnetic clusters in the paramagnetic region. The GP is typically described by the following relation:^72,73

χ⁻¹ = (T − T^rd_c)^1−λ (v)

Here, T^rd_c is the random critical temperature, which occurs in a FM state in the PM matrix and λ is an exponent that lies between 0 and 1. To calculate the value of λ, the inverse susceptibility data is fitted by eqn (v) as shown in Fig. 11. The obtained values of λ and T^rd_c are 0.5 and 87.91 K, respectively. The contribution of GP (% GP = ((T_G − T_c)/T_c) × 100) is 4.59%.⁷⁴

Fig. 11 Inverse susceptibility as a function of temperature under 500 Oe applied magnetic field. Solid red line shows the fitting by Griffith model.

Magnetization (M) as a function of applied magnetic field (H) recorded at 10 K and 350 K is presented in Fig. 12(a). As depicted in Fig. 12(a), the magnetization experiences a rapid increase at lower magnetic fields, and at higher fields, saturation is not achieved, even when the field reaches ±50 kOe. The observed hysteresis, along with the absence of saturation at higher fields, may be because of the presence of antiferromagnetic (AFM) correlations and the coexistence of competing magnetic orders in both phases. Hysterias loop at 10 K shows a coercivity of 6.8 kOe also suggests the FM nature of the compound. The hysteresis loop recorded at 350 K exhibits a linear pattern in response to the applied magnetic field, lacking any indication of magnetization saturation. This behavior aligns with expectations for the paramagnetic region of a FM/AFM system.

Fig. 12 (a) M–H hysteresis loop measured at 10 K and 350 K in the range ±50 kOe applied magnetic field. (b) Arrott plot for the DCMO compound at 10 K, solid red line is high field extrapolation to obtain spontaneous magnetization.

In Fig. 12(b), we have presented Arrott plot extracted from virgin curve of M–H data shown in Fig. 12(a). Arrott plot (M²vs. H/M) offers an effective tool to understand the nature of magnetic state. Fig. 12(b) displays a negative intercept, indicating the absence of spontaneous magnetization in this material at 10 K. The absence of spontaneous magnetization confirms that the compound is not pure ferromagnetic.

3.7. Optical properties

UV-Visible absorbance spectroscopy was utilized to investigate the optical properties of the synthesized Dy₂CoMnO₆ double perovskite compound. Fig. 13(a) displays the absorbance spectra recorded at 300 K within the wavelength range of 350–1100 nm. The broad absorption band, centered at 560 nm (2.21 eV), arises from charge transfer between the Mn(3d) and O(2p) electronic states. The energy band gap was calculated from the absorption spectrum using the Tauc relation,⁷⁵ given by:

αhν = A(hν − E_g)ⁿ (vi)

Fig. 13 (a) Absorption spectra and (b) Tauc plot of Dy₂CoMnO₆.

Here, n is either ½ or 2 for direct or indirect transitions, hν is the photon energy, α denotes the absorption coefficient, A represents a constant that is independent of photon energy, and E_g is the optical band gap. As illustrated in in Fig. 13(b), visualisations of (αhν)² and (αhν)^1/2vs. hν were plotted based on this relationship.

The optical band gap values were determined by extrapolating the linear region of the absorption edge to the point where (αhν)² = 0 and (αhν)^1/2 = 0. The direct band gap was found to be approximately 1.47 eV, while the indirect band gap was approximately 1.04 eV. In comparison, Chanda et al. reported a band gap of 1.93 eV for Dy₂CoMnO₆ nanoparticles synthesized via the sol–gel method.³⁶ This value, located within the visible spectrum, highlights the material's suitability for visible-light-driven applications.

The observed direct band gap in this study is slightly lower than previously reported values, potentially due to the particle size and polycrystalline nature of the synthesized Dy₂CoMnO₆. The reduced band gap enhances its potential as a visible-light photocatalyst, emphasizing its applicability in photocatalytic processes.

To further understand the optical properties of Dy₂CoMnO₆, first-principles density functional theory (DFT) calculations were performed to obtain the absorption coefficient as a function of photon energy. The computed absorption spectrum, presented in Fig. S1 of the ESI,† reveals two distinct optical band gaps at approximately 0.9 eV and 2.61 eV. These values were compared with the experimental results obtained from UV-Vis spectroscopy, which indicated an indirect band gap of 1.04 eV and a direct band gap of 1.47 eV, with a broad absorption centered around 2.21 eV due to charge transfer transitions between Mn(3d) and O(2p) states. The lower-energy DFT band gap (∼0.9 eV) aligns well with the experimentally determined indirect band gap (1.04 eV), suggesting the presence of intermediate electronic states or defect-related transitions. However, the higher-energy DFT band gap (2.61 eV) is larger than the experimental direct band gap (1.47 eV) and the reported literature value of 1.93 eV. This discrepancy may arise due to the limitations of the exchange–correlation functional used in the calculations, as generalized gradient approximation (GGA) functionals are known to overestimate or underestimate band gaps depending on the material. Additionally, factors such as the polycrystalline nature, particle size effects, and synthesis conditions of the experimental sample could contribute to the observed deviation.^76,77 Despite these variations, the experimental and theoretical results consistently highlight Dy₂CoMnO₆ as a promising material for visible-light-driven applications.

3.8. Electronic band structure and density of states

The interaction of atoms within solid crystalline structures leads to the formation of energy bands, resulting from the subdivision of discrete electronic energy levels into an immense number of closely spaced sub-levels, ultimately producing what appears as a continuous energy spectrum. This band structure is characterized by the distribution of Kohn–Sham eigenvalues along specific k-point trajectories within the first Brillouin zone. Such bands exert a profound influence on the electronic properties of materials, governing a wide range of phenomena including optical and thermoelectric effects. In the context of the Dy₂CoMnO₆ compound, the transition of electrons from the valence band to the conduction band is contingent upon their ability to acquire energy exceeding the bandgap. Upon absorbing sufficient energy, an electron transitions from its valence band state to a conduction band state. Subsequently, the electron may return to the valence band, releasing energy in the process. If the material exhibits a direct band gap, this release of energy occurs in the form of photon emission (radiative emission). Conversely, in the case of an indirect band gap, the energy is dissipated as a phonon (thermal emission). The nature of the band gap, whether direct or indirect, thus critically determines the mechanisms of energy release and the resulting electronic and thermal behavior of the material. Fig. 14 illustrates the band structure spectra of Dy₂CoMnO₆ at magnetic equilibrium, distinguishing between the minority and majority spin channels. These spectra provide insight into the spin-resolved electronic structure of the material, highlighting the behavior of electrons in both spin orientations. The differentiation between minority and majority spins is crucial for understanding the material's magnetic and electronic properties, as it reflects the spin polarization and the contribution of each spin channel to the overall electronic band structure. This spin-resolved analysis is particularly significant in materials like Dy₂CoMnO₆, where magnetic ordering and electronic interactions are tightly coupled.

Fig. 14 Electronic band structure of Dy₂CoMnO₆ in spin up and spin down configuration.

Dy₂CoMnO₆ exhibits semiconductor behavior, characterized by an indirect band gap. For spin-up electrons, the band gap spans from the Z to D points in the Brillouin zone, with a value of 1.6054 eV. In the spin-down channel, the band gap is larger, measured at 3.2029 eV, corresponding to a transition between the Γ and Z points. The overall band gap, considering the spin-up channel, is 1.6054 eV. This spin-dependent band gap structure underlines the compound's complex electronic behavior, with the differing gaps for spin-up and spin-down electrons contributing to its magnetic and electronic properties. Essaoud et al.⁴¹ meticulously analyzed the electronic properties, focusing on the total and partial density of states, utilizing the advanced full-potential linearized augmented plane wave (FP-LAPW) method within the WIEN2k framework. Their results revealed a conspicuous absence of electronic states near the Fermi level, extending up to 1.65 eV for the minority spin channel and 0.31 eV for the majority spin channel, highlighting distinct spin-dependent electronic characteristics.

The total and projected density of states (TDOS and PDOS) of Dy₂CoMnO₆ have been analyzed to elucidate its electronic and magnetic properties. As shown in Fig. 15(a)–(f), the system exhibits a significant lack of states at the Fermi level in both spin channels, confirming its semiconducting nature. The computed energy gap is approximately 1.6054 eV and 3.2029 eV for the minority and majority spin orientations, respectively.

Fig. 15 Total and partial density of states (DOS) of Dy₂CoMnO₆ in spin up and spin down configuration.

To understand the origin of magnetism in Dy₂CoMnO₆, we further examine the element- and orbital-resolved projected density of states (PDOS) contributions (Fig. 15(c)–(f)). The Co-3d and Mn-3d states exhibit strong spin polarization, with Co contributing 5.324μ_B and Mn contributing 6.305μ_B, indicating their dominant role in the magnetic ordering. The Co-3d orbitals show significant exchange splitting, where the spin-up and spin-down states are highly asymmetric, confirming a high-spin configuration for Co²⁺. Similarly, the Mn-3d states display notable exchange splitting, indicative of Mn⁴⁺ in a high-spin state. This spin asymmetry is a key factor contributing to the overall magnetization of the system.

In addition, the Dy-4f states are located deep in the valence band and do not significantly contribute to the overall magnetic moment, apart from minor hybridization effects, leading to a small Dy moment of 0.076μ_B. The oxygen atoms (O-2p states) display weak spin polarization, contributing 0.138μ_B, reinforcing the role of superexchange interactions in stabilizing the ferromagnetic state. The strong hybridization between O-2p and the transition metal 3d orbitals further supports the observed magnetic ordering.

The computed total magnetic moment of Dy₂CoMnO₆ is 11.843μ_B per formula unit, primarily arising from the cooperative exchange interactions between Mn⁴⁺ (6.305μ_B) and Co²⁺ (5.324μ_B) ions, mediated by oxygen ligands. The half-filled Mn-3d-e_g and empty Co-3d-e_g orbitals participate in a strong superexchange mechanism, stabilizing the ferromagnetic state.

We have employed the GGA+U approach to accurately account for electron–electron correlation in the localized Co-3d and Mn-3d orbitals. The applied Hubbard-U values (U_eff = 4.80 eV for Co and 4.50 eV for Mn) provide an improved description of their electronic and magnetic behavior. The PDOS analysis, along with the computed magnetic moments, confirms that the magnetism in Dy₂CoMnO₆ is predominantly driven by the exchange splitting of Co-3d and Mn-3d states, reinforced by oxygen-mediated superexchange interactions.

Spin–orbit coupling (SOC) in Dy₂CoMnO₆ (DCMO) significantly affects its electronic structure, magnetic anisotropy, and exchange interactions. The strong SOC associated with Dy leads to substantial splitting of the Dy 4f states, which influences their hybridization with the Co and Mn 3d orbitals. Additionally, SOC modifies the Co²⁺–O–Mn⁴⁺ superexchange interactions, altering the stability of different magnetic configurations. This effect induces magnetocrystalline anisotropy, impacting spin alignment and potentially stabilizing non-collinear magnetic structures. Furthermore, SOC enhances magnetoelectric coupling, contributing to complex spin–lattice interactions that can lead to spin-driven ferroelectricity in DCMO.

The electronic band structure of Dy₂CoMnO₆, shown in Fig. S2 of ESI,† is computed with SOC effects and reveals a direct band gap of 1.59 eV. The dispersion curves along high-symmetry points of the Brillouin zone (Γ–Z–D–B–Γ–A) illustrate the electronic transitions and the influence of SOC on band splitting. The separation of the valence band maximum (VBM) and conduction band minimum (CBM) confirms the semiconducting nature of DCMO. The inclusion of SOC modifies band dispersion, impacting the effective mass of charge carriers and the transport properties of the material. The electronic interactions between the Co and Mn 3d states and O 2p orbitals play a critical role in determining charge transfer and hybridization effects in the system.

The total and partial density of states (PDOS) calculations, depicted in Fig. S3 of ESI,† provide further insights into the electronic structure. The valence band is primarily composed of O 2p states hybridized with Mn 3d and Co 3d orbitals, whereas the conduction band is dominated by Dy 4f states. The strong localization of Dy 4f orbitals in the conduction band suggests a minimal contribution to electrical conductivity but plays a crucial role in magnetic exchange interactions. The presence of sharp peaks in the DOS indicates strong electronic correlations, particularly in the Dy 4f states, which are expected due to their localized nature. The hybridization of Dy 4f, Mn 3d, Co 3d, and O 2p states highlights the complex electronic structure of DCMO, where SOC significantly influences both electronic and magnetic properties.

The incorporation of SOC in electronic structure calculations reveals critical modifications in the band structure and density of states of Dy₂CoMnO₆. The SOC-induced band splitting, 3d–4f hybridization, and enhanced magnetic anisotropy play crucial roles in determining the transport and magnetic behavior of the material. These findings provide a deeper understanding of the fundamental electronic interactions in DCMO and suggest its potential application in spintronic and magnetoelectric devices, where SOC-driven effects can be utilized for advanced functionalities.

3.9. Hydrodynamic diameter and zeta potential

The stability of colloidal particles in a specific medium is essential for various applications, such as catalysis, pharmaceuticals, and biomedical fields. This stability can be categorized into two primary types: (i) the prevention of particle agglomeration, known as electrostatic stability, and (ii) the prevention of sedimentation due to gravity, referred to as mechanical stabilization. The stability of colloidal particles, where electrostatic interactions among particles dominate over gravitational forces, can be evaluated using measurements such as zeta potential ζ, hydrodynamic diameter (HD), ionic conductivity, and electrophoretic mobility, often determined through dynamic light scattering (DLS).

Zeta potential is a critical property of particles, reflecting the charges developed at the interface between the particle surface and the surrounding liquid medium.⁷⁸ It is typically measured indirectly through electrophoretic mobility or particle velocity under an applied electric field. The relationship between zeta potential and surface charges on particles is complex and can be described by Henry's equation:

where μ_e, ε, and η correspond to electrophoretic mobility, the dielectric constant of the solvent, and viscosity, respectively. The dimensionless Henry's function f(ka) ranges from 1 to 1.5, depending on the ratio of particle size to the Debye length.

Zeta potential is influenced by factors such as particle size, shape, surfactant type and coverage, as well as the solvent's properties, including chain length, polarity, and dielectric constant. By adjusting the electrostatic interactions between particles, the zeta potential can be increased, leading to enhanced stability. A stable dispersion is generally indicated by a zeta potential value greater than ±30 mV. The size of the primary particles, surface coating, shape, material composition, and additives are some of the factors that affect the stability or aggregation of particle suspensions. Aggregation kinetics are controlled by the mobility of aggregates or the charge on particle surfaces; increased aggregation reduces particle mobility in the dispersion medium, which results in a lower zeta potential and, in turn, a less stable colloidal solution.

To examine the zeta potential (ζ) and hydrodynamic diameter (HD), we first optimized the solvent to achieve good stability, choosing deionized water (DI) as the dispersion medium for Dy₂CoMnO₆. The intensity distribution of HD and ζ, dispersed in DI, is shown in Fig. 16(a). In the first run, two peaks were observed in the HD data at 757 nm and 7288 nm, with the absolute value of ζ recorded at 243 mV and −29.68 mV in DIW (Fig. 16(b)). The HD results indicate that particles exhibit a narrower size distribution in ethanol, suggesting greater stability compared to water. To ensure measurement accuracy, the data were recorded three times. Interestingly, the two peak values of HD at 757 nm and 7288 nm in the first run shifted to three peaks at 352.28 nm, 1851.93 nm, and 10 497.63 nm in the second run. In the third run, the distribution returned to two peaks at 871.523 nm and 6113.69 nm. The average zeta potential was found to be −29.78 mV, with a mobility of −2.325 × 10⁻⁴ cm² V⁻¹ s⁻¹ and an electric field of −16.37 V cm⁻¹.

Fig. 16 (a) Intensity distribution of hydrodynamic diameter and (b) zeta potential in deionized water of Dy₂CoMnO₆.

4. Conclusion

A single-phase polycrystalline Dy₂CoMnO₆ sample was synthesized using the solid-state reaction method and stabilized in a monoclinic crystal structure with the space group P2₁/n. XPS analysis revealed mixed valence states for Co (Co²⁺, Co³⁺) and Mn (Mn³⁺, Mn⁴⁺). Density functional theory (DFT) was employed to optimize the structure and explore the magnetic configuration of Dy₂CoMnO₆. The energy–volume curves confirm that Dy₂CoMnO₆ is more stable in the ferromagnetic state than in the antiferromagnetic state. Its magnetism arises from quantum exchange interactions influenced by atomic moments and interatomic distances. The ferromagnetic phase exhibits a total magnetic moment of 11.843μ_B, primarily due to Co-3d and Mn-3d orbital overlap, with Mn⁴⁺–O–Co²⁺ superexchange interactions enhancing the overall magnetization.

DC magnetization measurements in zero-field-cooled (ZFC) and field-cooled (FC) modes revealed a spin-glass state at ∼80 K and a ferromagnetic transition at 87 K. Curie–Weiss fitting confirmed ferromagnetic interactions between Dy³⁺ and Co²⁺/Mn⁴⁺, with crystal field effects from Dy³⁺. The inverse susceptibility curve indicated a Griffiths phase (GP) between T_c and T_G, deviating from the Curie–Weiss law. The experimentally measured magnetic moment (16.33μ_B) exceeded the theoretical value (16.09μ_B), likely due to short-range ferromagnetic ordering within the paramagnetic matrix, possibly influenced by magnetic inhomogeneity or oxygen deficiencies. Studies on antisite disorder in Dy₂CoMnO₆ reveal that Co/Mn mixing on the B-site induces magnetic inhomogeneity and competing interactions, complicating the system's magnetic behavior. Magnetization measurements showed no saturation at high fields, indicating coexisting magnetic orders. The 10 K hysteresis loop exhibited 6.8 kOe coercivity, confirming ferromagnetism, while the 350 K loop was linear, indicating paramagnetism. These results highlight the role of antisite disorder in tuning the material's magnetic properties, making Dy₂CoMnO₆ promising for spintronic and magnetic applications.

Author contributions

SD contributed to the conceptualization, investigation, writing original-draft and data curation, VK contributed to DFT calculations and reviewing and editing of manuscript, KD contributed to formal analysis and investigation, KS contributed to magnetic analysis, ME contributed to formal analysis, DG contributed to formal analysis, RKS contributed to resources and validation, GP contributed to methodology, supervision and reviewing, NKG contributed to the supervision and reviewing and editing of manuscript.

Data availability

The data supporting the findings of this study are available from the author upon reasonable request.

Conflicts of interest

There are no conflicts to declare.

Acknowledgements

The authors express their sincere gratitude to the Madhya Pradesh Council of Science and Technology (MPCST) for their additional financial support. Special thanks are extended to Dr Vasant Sathe for his assistance with Raman measurements, Dr Mukul Gupta for XRD analysis, and Dr Rajiv Rawat for magnetization studies at UGC-CSR Indore, who provided measurement facilities. The authors also acknowledge Mr Ajay Rathore for his support in Raman measurements, Mr Vinod Savaner for particle size analysis and zeta potential measurements, Dr Piyush Patel for UV-Vis spectroscopy, and Prof. Bharat Modhera of MANIT, Bhopal, for conducting FTIR measurements. SD sincerely thanks Dr Anchit Modi for his initial guidance in the synthesis of the material. A special thanks to Dr Devendra Kumar Pandey for his invaluable help with the Rietveld refinement process. VK thanks to Dr Debesh R. Roy who provided the High-Performance Clusters (HPC) facility of C-DAC Centre for Development of Advanced Computing, Pune India. VK acknowledges the computational facility support provided by the Sackler Center for Computational Molecular and Materials Science at Tel Aviv University, Israel. Some Calculations for this research were conducted on the Lichtenberg high performance computer of the TU Darmstadt, Germany.

References

1. R. C. Sahoo, D. Paladhi, P. Dasgupta, A. Poddar, R. Singh, A. Das and T. K. Nath, Antisite-disorder driven large exchange bias effect in phase separated La_1.5Ca_0.5CoMnO₆ double perovskite, J. Magn. Magn. Mater., 2017, 428, 86–91,  DOI:10.1016/j.jmmm.2016.12.018.
2. K. Aswathi, J. Pezhumkattil Palakkal and M. RaamaVarma, Magnetization sign reversal, exchange bias, and Griffiths-like phase in orthorhombic perovskite Pr₂FeMnO₆, J. Magn. Magn. Mater., 2019, 476, 45–53,  DOI:10.1016/j.jmmm.2018.12.044.
3. T. K. Mandal, M. Croft, J. Hadermann, G. van Tendeloo, P. W. Stephens and M. Greenblatt, La₂MnVO₆ double perovskite: A structural, magnetic and X-ray absorption investigation, J. Mater. Chem., 2009, 19(25), 4382–4390,  10.1039/b823513a.
4. G. R. Haripriya, C. M. N. Kumar, R. Pradheesh, L. M. Martinez, C. L. Saiz, S. R. Singamaneni, T. Chatterji, V. Sankaranarayanan, K. Sethupathi, B. Kiefer and H. S. Nair, Contrasting the magnetism in La_2-xSr_xFeCoO₆ (x = 0,1,2) double perovskites: The role of electronic and cationic disorder, Phys. Rev. B, 2019, 99(18) DOI:10.1103/PhysRevB.99.184411.
5. S. Kundu, A. Pal, A. Chauhan, K. Patro, K. Anand, S. Rana, V. G. Sathe, A. G. Joshi, P. Pal, K. Sethupathi, B. R. K. Nanda and P. Khuntia, Electronic structure and magnetic properties of 3d-4f double perovskite material, Phys. Rev. Mater., 2022, 6(10), 104401,  DOI:10.1103/PhysRevMaterials.6.104401.
6. J. Cedervall, S. A. Ivanov, E. Lewin, P. Beran, M. S. Andersson, T. Faske, G. V. Bazuev, P. Nordblad, M. Sahlberg and R. Mathieu, On the structural and magnetic properties of the double perovskite Nd₂NiMnO₆, J. Mater. Sci.: Mater. Electron., 2019, 30(17), 16571–16578,  DOI:10.1007/s10854-019-02035-z.
7. D. N. Singh, D. K. Mahato and T. P. Sinha, Structural and electrical characterization of La₂ZnMnO₆ double perovskite, Phys. B, 2018, 550, 400–406,  DOI:10.1016/j.physb.2018.08.051.
8. S. A. Ivanov, M. S. Andersson, J. Cedervall, E. Lewin, M. Sahlberg, G. V. Bazuev, P. Nordblad and R. Mathieu, Temperature-dependent structural and magnetic properties of R₂ MMnO₆ double perovskites (R = Dy, Gd; M = Ni, Co), J. Mater. Sci.: Mater. Electron., 2018, 29(21), 18581–18592,  DOI:10.1007/s10854-018-9976-1.
9. H. S. Nair, R. Pradheesh, Y. Xiao, D. Cherian, S. Elizabeth, T. Hansen, T. Chatterji and T. Brückel, Magnetization-steps in Y₂CoMnO₆ double perovskite: The role of antisite disorder, J. Appl. Phys., 2014, 116(12), 123907,  DOI:10.1063/1.4896399.
10. S. Chanda, S. Saha, A. Dutta and T. P. Sinha, Structural and transport properties of double perovskite Dy₂NiMnO₆, Mater. Res. Bull., 2015, 62, 153–160,  DOI:10.1016/j.materresbull.2014.11.021.
11. W. Z. Yang, X. Q. Liu, H. J. Zhao and X. M. Chen, Magnetic, dielectric and transport characteristics of Ln₂CoMnO₆ (Ln = Nd and Sm) double perovskite ceramics, J. Magn. Magn. Mater., 2014, 371, 52–59,  DOI:10.1016/j.jmmm.2014.07.017.
12. M. J. Hosen, M. A. Basith and I. M. Syed, Structural, magnetic and optical properties of disordered double perovskite Gd₂CoCrO₆ nanoparticles, RSC Adv., 2023, 13(26), 17545–17555,  10.1039/d3ra02233a.
13. E. E. Ateia, A. T. Mohamed and H. Elshimy, The impact of antisite disorder on the physical properties of La₂FeB′′O₆ (B′′ = Fe, Ni and Co) double perovskites, Appl. Nanosci., 2020, 10(5), 1489–1499,  DOI:10.1007/s13204-020-01356-4.
14. C. L. Li, L. G. Wang, X. X. Li, C. M. Zhu, R. Zhang, H. W. Wang and S. L. Yuan, Magnetic field-induced metamagnetism and magnetocaloric effect in double perovskites Re₂CoMnO₆ (Re = Sm, Dy), Mater. Chem. Phys., 2017, 202, 76–81,  DOI:10.1016/j.matchemphys.2017.09.009.
15. J. Navarro, J. Nogués, J. S. Muñoz and J. Fontcuberta, Antisites and electron-doping effects on the magnetic transition of Sr₂FeMoO₆ double perovskite, Phys. Rev. B: Condens. Matter Mater. Phys., 2003, 67(17), 174416,  DOI:10.1103/PhysRevB.67.174416.
16. R. C. Sahoo, Y. Takeuchi, A. Ohtomo and Z. Hossain, Exchange bias and spin glass states driven by antisite disorder in the double perovskite compound LaSrCoFeO₆, Phys. Rev. B, 2019, 100(21), 214436,  DOI:10.1103/PhysRevB.100.214436.
17. G. R. Haripriya, R. Pradheesh, K. Sethupathi and V. Sankaranarayanan, The order of magnetic phase transitions in disordered double perovskite oxides Sm₂FeCoO₆ and Dy₂FeCoO₆, AIP Adv., 2018, 8(10), 101340,  DOI:10.1063/1.5042757.
18. I. N. Bhatti, I. N. Bhatti, R. N. Mahato and M. A. H. Ahsan, Physical properties of nano-crystalline Sm₂CoMnO₆: Structure, magnetism, spin–phonon coupling and dielectric study, Phys. B, 2020, 582, 411975,  DOI:10.1016/j.physb.2019.411975.
19. T. D. Rao, S. Marik, D. Singh and R. P. Singh, Observation of exchange bias effect in La₂NiIrO₆, J. Alloys Compd., 2017, 705, 849–852,  DOI:10.1016/j.jallcom.2017.02.115.
20. M. Retuerto, Á. Muñoz, M. J. Martínez-Lope, J. A. Alonso, F. J. Mompeán, M. T. Fernández-Díaz and J. Sánchez-Benítez, Magnetic Interactions in the Double Perovskites R₂NiMnO₆ (R = Tb, Ho, Er, Tm) Investigated by Neutron Diffraction, Inorg. Chem., 2015, 54(22), 10890–10900,  DOI:10.1021/acs.inorgchem.5b01951.
21. C. L. Bull, D. Gleeson and K. S. Knight, Determination of B-site ordering and structural transformations in the mixed transition metal perovskites La₂CoMnO₆ and La₂NiMnO₆, J. Phys.: Condens. Matter, 2003, 15(29), 4927,  DOI:10.1088/0953-8984/15/29/304.
22. M. Alam and S. Chatterjee, B-site order/disorder in A₂BB′O₆ and its correlation with their magnetic property, J. Phys.: Condens. Matter, 2023, 35(22), 223001,  DOI:10.1088/1361-648X/acc295.
23. J. Blasco, J. L. García-Muñoz, J. García, G. Subías, J. Stankiewicz, J. A. Rodríguez-Velamazán and C. Ritter, Magnetic order and magnetoelectric properties of R₂CoMnO₆ perovskites (R = Ho, Tm, Yb, and Lu), Phys. Rev. B, 2017, 96(2), 024409,  DOI:10.1103/PhysRevB.96.024409.
24. A. Hossain, P. Bandyopadhyay and S. Roy, An overview of double perovskites A₂B′B′′O₆ with small ions at A site: Synthesis, structure and magnetic properties, J. Alloys Compd., 2018, 740, 414–427,  DOI:10.1016/j.jallcom.2017.12.282.
25. M. K. Kim, J. Y. Moon, S. H. Oh, D. G. Oh, Y. J. Choi and N. Lee, Strong magnetoelectric coupling in mixed ferrimagnetic-multiferroic phases of a double perovskite, Sci. Rep., 2019, 9(1), 5456,  DOI:10.1038/s41598-019-41990-9.
26. D. Mazumdar and I. Das, Role of 3d-4f exchange interaction and local anti-site defects in the magnetic and magnetocaloric properties of double perovskite Ho₂CoMnO₆ compound, J. Appl. Phys., 2021, 129(6), 063901,  DOI:10.1063/5.0041257.
27. A. Hossain, A. K. M. Atique Ullah, P. Sarathi Guin and S. Roy, An overview of La₂NiMnO₆ double perovskites: synthesis, structure, properties, and applications, J. Sol-Gel Sci. Technol., 2020, 93(3), 479–494,  DOI:10.1007/s10971-019-05054-8.
28. W. Liu, L. Shi, S. Zhou, J. Zhao, Y. Li and Y. Guo, Griffiths phase, spin–phonon coupling, and exchange bias effect in double perovskite Pr₂CoMnO₆, J. Appl. Phys., 2014, 116(19), 193901,  DOI:10.1063/1.4902078.
29. R. R. Das, P. N. Lekshmi, S. C. Das and P. N. Santhosh, Competing short-range magnetic correlations, metamagnetic behavior and spin–phonon coupling in Nd₂CoMnO₆ double perovskite, J. Alloys Compd., 2019, 773, 770–777,  DOI:10.1016/j.jallcom.2018.09.171.
30. P. R. Mandal and T. K. Nath, Evolution of Griffith phase in hole doped double perovskite La_2-xSr_xCoMnQ₆ (x = 0.0,0.5, and 1.0), Mater. Res. Express, 2015, 2(6), 066101,  DOI:10.1088/2053-1591/2/6/066101.
31. A. Nayak, C. H. Prashanth, D. Bala, I. R. Reddy, K. Tarafder, V. Adyam and K. Jyothinagaram, Low field-cooled induced large exchange bias effect and DFT calculations in ferromagnetic Sm₂CoMnO₆, Solid State Commun., 2024, 378, 115408,  DOI:10.1016/j.ssc.2023.115408.
32. A. Rathi, H. Borkar, P. K. Rout, A. Gupta, H. K. Singh, A. Kumar, B. Gahtori, R. P. Pant and G. A. Basheed, Antisite disorder-driven large electric polarization in multiferroic Nd₂CoMnO₆, J. Phys. D: Appl. Phys., 2017, 50(46), 465001,  DOI:10.1088/1361-6463/aa8b57.
33. K. Anand, M. Alam, A. Pal, P. Singh, S. Kumari, A. G. Joshi, A. Das, A. Mohan and S. Chatterjee, Existence of Griffiths phase and unusual spin dynamics in double perovskite Tb₂CoMnO₆, J. Magn. Magn. Mater., 2021, 528, 167697,  DOI:10.1016/j.jmmm.2020.167697.
34. K. Aswathi, J. Pezhumkattil Palakkal and M. RaamaVarma, Magnetization sign reversal, exchange bias, and Griffiths-like phase in orthorhombic perovskite Pr₂FeMnO₆, J. Magn. Magn. Mater., 2019, 476, 45–53,  DOI:10.1016/j.jmmm.2018.12.044.
35. Y. Xin, L. Shi, J. Zhao, X. Yuan, S. Zhou, L. Hou and L. Yang, Nature of Griffiths phase and ferromagnetic 3d-4f interaction in double-perovskite Dy₂CoMnO₆, J. Alloys Compd., 2022, 893, 162222,  DOI:10.1016/j.jallcom.2021.162222.
36. S. Chanda, R. Maity, S. Saha, A. Dutta and T. P. Sinha, Double perovskite nanostructured Dy₂CoMnO₆ an efficient visible-light photocatalysts: synthesis and characterization, J. Sol-Gel Sci. Technol., 2021, 99(3), 600–613,  DOI:10.1007/s10971-021-05605-y.
37. P. L. Pedro, P. Barrozo, D. A. Landinez-Tellez, R. F. Jardim, W. M. Azevedo and J. Albino Aguiar, Structural and magnetic properties of Ln₂CoMnO₆ (Ln = Dy and La) produced by combustion synthesis, J. Supercond. Novel Magn., 2013, 26(7), 2521–2524,  DOI:10.1007/s10948-012-1689-8.
38. R. Das and R. N. P. Choudhary, Studies of electrical, magnetic and leakage-current characteristics of double perovskite: Dy₂CoMnO₆, J. Alloys Compd., 2021, 853, 157240,  DOI:10.1016/j.jallcom.2020.157240.
39. R. K. Muddelwar, J. Pani, A. B. Lad, H. Borkar and V. M. Gaikwad, A novel double perovskite Dy₂CoMnO₆ as supercapacitor electrode with efficient electrochemical performance, J. Solid State Chem., 2024, 329, 124455,  DOI:10.1016/j.jssc.2023.124455.
40. M. Valian, F. Beshkar and M. Salavati-Niasari, Urchin-like Dy₂CoMnO₆ double perovskite nanostructures: new simple fabrication and investigation of their photocatalytic properties, J. Mater. Sci.: Mater. Electron., 2017, 28(17), 12440–12447,  DOI:10.1007/s10854-017-7065-5.
41. S. S. Essaoud, S. M. Azar, A. A. Mousa and A. Y. Al-Reyahi, DFT-based investigation of electronic-structure, magnetic and thermoelectric properties of Dy₂CoMnO₆ double perovskite, Phys. Scr., 2023, 98(7), 075930,  DOI:10.1088/1402-4896/acdd2c.
42. S. Anirban, R. Roy and A. Dutta, Structure, charge carrier conduction, dielectric properties and leakage current density of Dy₂CoMnO₆ double perovskite, J. Alloys Compd., 2022, 928, 167184,  DOI:10.1016/j.jallcom.2022.167184.
43. J. Hafner, Ab-initio simulations of materials using VASP: Density-functional theory and beyond, J. Comput. Chem., 2008, 29(13), 2044–2078,  DOI:10.1002/jcc.21057.
44. A. Floris, I. Timrov, B. Himmetoglu, N. Marzari, S. de Gironcoli and M. Cococcioni, Hubbard-corrected density functional perturbation theory with ultrasoft pseudopotentials, Phys. Rev. B, 2020, 101(6), 64305,  DOI:10.1103/PhysRevB.101.064305.
45. K. Knížek, Z. Jirák, P. Novák and C. de La Cruz, Non-collinear magnetic structures of TbCoO₃ and DyCoO₃, Solid State Sci., 2014, 28, 26–30,  DOI:10.1016/j.solidstatesciences.2013.12.001.
46. C. Lu and J.-M. Liu, DyMnO₃: A model system of type-II multiferroics, J. Materiomics, 2016, 2(3), 213–224,  DOI:10.1016/j.jmat.2016.04.004.
47. M. Retuerto, Á. Muñoz, M. J. Martínez-Lope, J. A. Alonso, F. J. Mompeán, M. T. Fernández-Díaz and J. Sánchez-Benítez, Magnetic Interactions in the Double Perovskites R₂NiMnO₆ (R = Tb, Ho, Er, Tm) Investigated by Neutron Diffraction, Inorg. Chem., 2015, 54(22), 10890–10900,  DOI:10.1021/acs.inorgchem.5b01951.
48. P. R. Mandal, A. Khan and T. K. Nath, Antisite disorder driven magnetodielectric and magnetocaloric effect in double perovskite La_2-xSr_xCoMnO₆ (x = 0.0, 0.5, 1.0), J. Appl. Phys., 2020, 128(2), 024104,  DOI:10.1063/1.5135305.
49. R. R. Jones, D. C. Hooper, L. Zhang, D. Wolverson and V. K. Valev, Raman Techniques: Fundamentals and Frontiers, Nanoscale Res. Lett., 2019, 14(1), 231,  DOI:10.1186/s11671-019-3039-2.
50. A. Dias, L. A. Khalam, M. T. Sebastian and R. L. Moreira, Raman-spectroscopic investigation of Ba₂InTaO₆ and Sr₂InTaO₆ perovskites, J. Solid State Chem., 2007, 180(7), 2143–2148,  DOI:10.1016/j.jssc.2007.05.021.
51. A. Dias, L. A. Khalam, M. T. Sebastian, M. M. Lage, F. M. Matinaga and R. L. Moreira, Raman scattering and infrared spectroscopy of chemically substituted Sr₂LnTaO₆ (Ln = lanthanides, Y, and In) double perovskites, Chem. Mater., 2008, 20(16), 5253–5259,  DOI:10.1021/cm800969m.
52. R. F. Jucá, G. D. Saraiva, A. J. Ramiro de Castro, F. F. de Sousa, F. G. S. Oliveira, I. F. Vasconcelos, G. D. S. Souza, J. M. Soares, C. H. N. Cordeiro and W. Paraguassu, Structural, vibrational and magnetic properties of monoclinic La₂FeMnO₆ double perovskite, Vacuum, 2022, 202, 111140,  DOI:10.1016/j.vacuum.2022.111140.
53. B. Manoun, J. M. Igartua, P. Lazor and A. Ezzahi, High temperature induced phase transitions in Sr₂ZnWO₆ and Sr₂CoWO₆ double perovskite oxides: Raman spectroscopy as a tool, J. Mol. Struct., 2012, 1029, 81–85,  DOI:10.1016/j.molstruc.2012.05.077.
54. A. P. Ayala, I. Guedes, E. N. Silva, M. S. Augsburger, M. Del and J. C. Pedregosa, Raman investigation of A₂CoBO₆ (A = Sr and Ca, B = Te and W) double perovskites, J. Appl. Phys., 2007, 101(12), 123511,  DOI:10.1063/1.2745088.
55. R. X. Silva, M. C. Castro Júnior, S. Yáñez-Vilar, M. S. Andújar, J. Mira, M. A. Señarís-Rodríguez and C. W. A. Paschoal, Spin–phonon coupling in multiferroic Y₂CoMnO₆, J. Alloys Compd., 2017, 690, 909–915,  DOI:10.1016/j.jallcom.2016.07.010.
56. Y. Bai, Y. Xia, H. Li, L. Han, Z. Wang, X. Wu, S. Lv, X. Liu and J. Meng, A-Site-Doping Enhanced B-Site Ordering and Correlated Magnetic Property in La_2–xBi_xCoMnO₆, J. Phys. Chem. C, 2012, 116(32), 16841–16847,  DOI:10.1021/jp302735x.
57. R. L. Moreira, R. P. S. M. Lobo, S. L. L. M. Ramos, M. T. Sebastian, F. M. Matinaga, A. Righi and A. Dias, Raman and infrared spectroscopic investigations of a ferroelastic phase transition in Ba₂ZnTeO₆ double perovskite, Phys. Rev. Mater., 2018, 2(5), 054406,  DOI:10.1103/PhysRevMaterials.2.054406.
58. R. L. Andrews, A. M. Heyns and P. M. Woodward, Raman studies of A₂MWO₆ tungstate double perovskites, Dalton Trans., 2015, 44(23), 10700–10707,  10.1039/c4dt03789h.
59. S. Halder, M. S. Sheikh, B. Ghosh and T. P. Sinha, Octahedral distortion induced phonon vibration and electrical conduction in A₂NdSbO₆ (A = Ba, Sr, Ca), Mater. Chem. Phys., 2017, 199, 508–521,  DOI:10.1016/j.matchemphys.2017.06.049.
60. B. Manoun, J. M. Igartua, P. Lazor and A. Ezzahi, High temperature induced phase transitions in Sr₂ZnWO₆ and Sr₂CoWO₆ double perovskite oxides: Raman spectroscopy as a tool, J. Mol. Struct., 2012, 1029, 81–85,  DOI:10.1016/j.molstruc.2012.05.077.
61. Y. Tamraoui, A. Hachmi, R. el, Haloui, B. Manoun, Y. Tamraoui, S. Louihi, M. Jahid, A. Hachmi, R. el, Abkar, M. A. el Aamrani, O. Ait, S. Ahmed, L. Bih and P. Lazor, Crystal structure and phase transitions in new series of double perovskite oxides Ba_2-xSr_xCaTeO₆ (0 ≤ x ≤ 2): X-ray diffraction and Raman spectroscopy studies, J. Appl. Surf. Interfaces, 2017, 1(3), 35–48.
62. B. Manoun, A. Ezzahi, S. Benmokhtar, A. Ider, P. Lazor, L. Bih and J. M. Igartua, X-ray diffraction and Raman spectroscopy studies of temperature and composition induced phase transitions in Ba_2-xSr_xZnWO₆ (0 ≤ x ≤ 2) double perovskite oxides, J. Alloys Compd., 2012, 533, 43–52,  DOI:10.1016/j.jallcom.2012.03.075.
63. M. N. Iliev, M. v Abrashev, A. P. Litvinchuk, V. G. Hadjiev, H. Guo and A. Gupta, Raman spectroscopy of ordered double perovskite La₂CoMnO₆ thin films, Phys. Rev. B: Condens. Matter Mater. Phys., 2007, 75(10), 104118,  DOI:10.1103/PhysRevB.75.104118.
64. O. A. Fouad, S. A. Makhlouf, G. A. M. Ali and A. Y. El-Sayed, Cobalt/silica nanocomposite via thermal calcination-reduction of gel precursors, Mater. Chem. Phys., 2011, 128(1), 70–76,  DOI:10.1016/j.matchemphys.2011.02.072.
65. C.-W. Tang, C.-B. Wang and S.-H. Chien, Characterization of cobalt oxides studied by FT-IR, Raman, TPR and TG-MS, Thermochim. Acta, 2008, 473(1), 68–73,  DOI:10.1016/j.tca.2008.04.015.
66. P. Shirazi, M. Rahbar, M. Behpour, M. Ashrafi and M. Behpour, La₂MnTiO₆ double perovskite nanostructures as highly efficient visible light photocatalysts, New J. Chem., 2020, 44(1), 231–238,  10.1039/c9nj04932k.
67. A. Harbi, H. Moutaabbid, Y. Li, C. Renero-Lecuna, M. Fialin, Y. le Godec, S. Benmokhtar and M. Moutaabbid, The effect of cation disorder on magnetic properties of new double perovskites La₂NiCo1-MnO₆ (x = 0.2-0.8) The effect of cation disorder on magnetic properties of new double perovskites La₂NiCo1-MnO₆ (x = 0.2-0), J. Alloys Compd., 2018, 778, 105–114,  DOI:10.1016/j.jallcom.2018.10.360.
68. A. Pal, S. Ghosh, A. G. Joshi, S. Kumar, S. Patil, P. K. Gupta, P. Singh, V. K. Gangwar, P. Prakash, R. K. Singh, E. F. Schwier, M. Sawada, K. Shimada, A. K. Ghosh, A. Das and S. Chatterjee, Investigation of multi-mode spin–phonon coupling and local B-site disorder in Pr₂CoFeO₆ by Raman spectroscopy and correlation with its electronic structure by XPS and XAS studies, J. Phys.: Condens. Matter, 2019, 31(27), 275802,  DOI:10.1088/1361-648X/ab144f.
69. Y. Jia, Q. Wang, P. Wang and L. Li, Structural, magnetic and magnetocaloric properties in R₂CoMnO₆ (R = Dy, Ho, and Er), Ceram. Int., 2017, 43(17), 15856–15861,  DOI:10.1016/J.CERAMINT.2017.08.158.
70. N. Panwar, K. Singh, K. Kanwar, Y. Bitla, S. Kumar and V. S. Puli, Low-Temperature Magnetic and Magnetocaloric Properties of Manganese-Substituted Gd_0.5Er_0.5CrO₃ Orthochromites, Crystals, 2022, 12, 263,  DOI:10.3390/cryst12020263.
71. K. Singh and N. Panwar, Observation of Griffiths phase, spin glass behaviour and magnetocaloric effect in frustrated Ga₂Mn_2-xCr_xO₇ (x = 0, 0.1, 0.3, 0.5) pyrochlore compounds, Curr. Appl. Phys., 2023, 53, 86–93,  DOI:10.1016/j.cap.2023.06.014.
72. F. Khachnaoui, N. ben Amor, K. Nouri, M. Bejar and E. Dhahri, Investigation of Griffiths-like phase at low temperature in a new magnetocaloric compound, Al₂Mn₂O₇, J. Phys. Chem. Solids, 2021, 148, 109605,  DOI:10.1016/J.JPCS.2020.109605.
73. M. Feng, Z. Xie, Z. Zou, X. Jiang, X. Zheng, C. Xu, W. Zhang, F. Yu and W. He, Effects of Eu³⁺ doping on the structural, magnetocaloric effect and critical behavior of Ruddlesden-Popper compounds La_1.4-xEu_xSr_1.6Mn₂O₇ (0 ≤ x ≤ 0.1), J. Alloys Compd., 2025, 1010, 177111,  DOI:10.1016/j.jallcom.2024.177111.
74. I. S. Debbebi, H. Omrani, W. Cheikhrouhou-Koubaa and A. Cheikhrouhou, A-site deficiency effects on the critical behavior of La_0.6Ca_0.15·0.05Ba_0.2MnO₃, J. Phys. Chem. Solids, 2018, 113, 67–73,  DOI:10.1016/J.JPCS.2017.10.018.
75. S. Dubey, V. Kumar, K. Dubey, C. Sahu, A. Modi, U. K. Gautam, R. K. Sharma, F. Z. Haque, G. Pagare and N. K. Gaur, Tailoring the structural and electro-optical properties of avisible-light emitting BaZrO₃ photocatalyst: integrating DFT and comprehensive experimental analysis, Nanoscale, 2024, 16(38), 18086–18107,  10.1039/D4NR00517A.
76. V. Kumar and D. R. Roy, Strain-induced band modulation and excellent stability, transport and optical properties of penta-MP2 (M = Ni, Pd, and Pt) monolayers, Nanoscale Adv., 2020, 2(10), 4566–4580,  10.1039/D0NA00503G.
77. V. Kumar, K. Rajput and D. R. Roy, Electric field-induced band modulation of predicted ternary 2D MXC3 [M:X = As:Ge, Sb:Sn and Bi:Pb] with strong stability and optical properties, Carbon, 2021, 172, 791–803,  DOI:10.1016/J.CARBON.2020.10.082.
78. V. Sharma, C. Chotia, Tarachand, V. Ganesan and G. S. Okram, Influence of particle size and dielectric environment on the dispersion behaviour and surface plasmon in nickel nanoparticles, Phys. Chem. Chem. Phys., 2017, 19(21), 14096–14106,  10.1039/c7cp01769c.

---

Footnote

† Electronic supplementary information (ESI) available. See DOI: https://doi.org/10.1039/d5tc00299k

---