[System]
You are a **scientific‐rigor auditor**. You will receive the parsed contents of a research paper. Your job is to identify **only** those errors or flaws that directly undermine the **scientific validity** of the paper’s methods, analyses, or conclusions.

Your sole focus is to identify flaws—such as errors in experimental design, data integrity, calculations, statistical inference, or reproducibility—that directly call into question the validity of a specific claim, paragraph, or the paper as a whole.

**Do not** report issues that are purely presentational, rhetorical, stylistic, or related to citation practices.

---

After you’ve done a **detailed walkthrough** of the paper, output exactly in this format—no extra keys or commentary:

```
<analysis>
{detailed walk‑through of how you checked each section/figure and why you flagged (or did not flag) any flaw}
</analysis>

<response>
{
  "has_error": <true|false>,
  "errors": [
    {
      "location": "Section 2.1",
      "description": "Claim that ‘all X are Y’ is contradicted by citation [5], which shows examples of non‑Y X."
    },
    {
      "location": "Figure 3",
      "description": "XAxis labeled ‘Time (s)’ but units appear to be milliseconds; scale bar mismatches caption."
    }
    // …more entries…
  ]
}
</response>
```

- **Do not** include any other keys or prose outside these two tagged blocks.
- **Do not** report stylistic or citation issues.
- Be as precise as possible about **where** (section ID or figure/table) and **what** the scientific flaw is.
- Each description for the errors must be rooted in a scientific rationale explaining why it is 'wrong' (not how it could be improved).

Begin your analysis now.

[User]
This study focused on investigating cobalt ferrite nanoparticles doped with trivalent Al³⁺ and Gd³⁺ ions across compositions ranging from CoFe₂₋₂ₓAlₓGdₓO₄ (x = 0.00, 0.02, 0.04, 0.06, 0.08). The nanoparticles were synthesized using the sol–gel auto-ignition method with citric acid as a chelating agent. Structural analysis via Rietveld-refined X-ray diffraction confirmed the formation of single-phase nanoparticles with a cubic spinel structure. Morphological examination through scanning electron microscopy revealed spherical-shaped grains. Elemental analysis using energy-dispersive X-ray analysis indicated consistent composition and high purity. Infrared spectra analysis verified the presence of characteristic modes typical of spinel ferrite structures. Magnetic properties assessed by vibrating sample magnetometry demonstrated soft magnetic behavior with lower coercivity. DC electrical resistivity measurements indicated a decrease in resistivity with increasing Al³⁺-Gd³⁺ co-doping, while dielectric studies showed enhanced properties in this regard. Overall, the findings suggest that these co-doped cobalt ferrite nanoparticles hold promise for applications in magneto-electronic devices.

Vaibhav K. Raut¹ · Sandeep B. Somvanshi² · Elmuez A. Dawi³ · Chandrakant T. Birajdar¹

Received: 18 May 2024 / Accepted: 24 September 2024 / Published online: 5 October 2024
© The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature 2024

Keywords spinel ferrite nanoparticle • cobalt ferrites • Al-Gd co-doping • DC Resistivity • dielectric properties

Highlights
• CoFe2O4 nanoparticles co-doped with Al3+ and Gd3+ were synthesized using sol-gel auto-combustion.
• SEM revealed spherical nanoscale grains, and XRD showed increased lattice constants with co-doping.
• Saturation magnetization and magneton number decreased with higher co-doping levels.
• Co-doping improved electrical conductivity and showed typical frequency-dependent dielectric behavior.
• These nanoparticles are promising for magneto-electronic and high-frequency applications.

## 1 Introduction

In the last two decades, nanomaterials and nanotechnology have been revolutionizing across various areas, offering innovative solutions and capabilities [1–3]. These advances demonstrate the wide-range impacts of nanotechnology across the different sectors, promising further breakthroughs in the future [4–6]. Nanomaterials have transformed energy and electronics sectors, offering significant improvements in performances, effectiveness, and functional ability [7–9]. While nanomaterials offer promising advantages, challenges such as scalability, cost-effectiveness, and environmental impact remain [10–12]. Recent developments in magnetic nanomaterials (MNPs) have concentrated on improving their functional abilities, showcasing their versatile nature and potentiality in numerous areas [13–15]. These advancements highlight ongoing endeavors to expand their capabilities for practical applications in areas such as health sectors, environmental issues, and energy-related applications [16–18].

Ferrite nanoparticles (FNPs) are a unique type of MNPs that are composed of iron oxide in combination with other metallic elements [19]. FNPs exhibit distinctive magnetic features due to their smaller size, typically ranging from 1–100 nm [20, 21]. FNPs are utilized across different areas, which consist of biomedical and electronic sectors [22–24]. In the biomedical field, FNPs have applications in drug delivery, magnetic hyperthermia, and as carriers for drugs that can be activated by an inductive field to target and eliminate cancer cells [25]. They are also employed as contrasting agents in magnetic resonance imaging (MRI). In electronic sectors, FNPs are employed in data storage, sensor, and for shielding against 'electromagnetic interference' [26]. Their small size and large surface area make FNPs particularly appropriate for nanoelectronics applications [19, 27].

Spinel ferrites (SFs), characterized by the notation M2+Fe3+2O4, where M2+ represents common divalent metal ions, demonstrate exceptional magnetic and electronic features due to their nanoscale size compared to bulk ferrites [28, 29]. These properties can be tailored for specific applicability through adjustments in particle size, composition, and synthesis methods [30]. Nanoscale spinel ferrites are typically synthesized using techniques such as chemical precipitation, sol-gel processes, hydrothermal reactions, and thermal decomposition of precursors [22, 31, 32]. By manipulating synthesis conditions, sizes, shapes, and magnetic characteristics of SFs nanoparticles can be optimized to create nanomaterials with tailored properties suitable for diverse applications [33].

Cobalt ferrite (CoFe2O4) nanoparticles (NPs) belong to the spinel ferrite family, characterized by a crystal structure comprising cobalt (II) oxide (CoO) and iron (III) oxide (Fe2O3) [34]. These nanoparticles exhibit intriguing magnetic and physical properties that make them highly versatile [23, 35]. Doped cobalt ferrite nanoparticles involve intentionally introducing dopant into the crystal lattice of CoFe2O4 [36–38]. This doping alters various features of the SFs, including 'magnetic, electrical, optical, and catalytic' characteristics [39–41]. The preparation of doped CoFe2O4 NPs follows a similar procedure as those for pristine CoFe2O4, with a dopant precursor being added during the preparation process. The precise controlling of dopant concentrations and allocation within the crystal lattice is crucial for fine-tuning the features of these nanoparticles to suit specific application. This capability to modify and optimize their properties underscores the potential of doped cobalt ferrite nanoparticles across a wide range of technological and industrial uses.

Co-doped cobalt ferrite nanoparticles represent a specialized category where CoFe2O4 NPs are altered by the incorporation of multiple types of dopants in their 'crystal lattice' during synthesis [42, 43]. This co-doping strategy allows for the manipulation and enhancement of various properties, leveraging the synergistic effects of different dopants. Typically, co-doped CoFe2O4 NPs include combinations of transition metal ions like nickel (Ni), manganese (Mn), zinc (Zn), and copper (Cu), as well as rare earth ions. The incorporation of these dopants, whether divalent or trivalent, such as aluminum (Al3+), chromium (Cr3+), and gadolinium (Gd3+), among others, can significantly alter the characteristics of CoFe2O4 NPs. The selection and amount of dopants are crucial parameters that can be precisely controlled to tailor the properties of the nanoparticles.

for specific applications. This fine-tuning capability enables researchers to optimize parameters such as magnetic behavior, electrical conductivity, optical properties, and catalytic activity, thereby expanding the potential applications of co-doped CoFe₂O₄ NPs in various fields of technology and industry.

In the literature, numerous studies have explored the enhanced properties of trivalent ion-doped CoFe₂O₄ nanoparticles across various applications. Kamran et al. [44] focused on trivalent cerium doping in cobalt ferrite nanoparticles, revealing enhanced transport properties relevant to resistive RAM applications. Rather et al. [45] reported on the physical and chemical properties of aluminum-doped mixed cobalt ferrite nanoparticles. Jing et al. [46] explored the structure and magnetic properties of CoFe₂O₄ NPs doped with various rare earth elements, examining the effects of different dopant positions. Despite extensive research on 'trivalent ion' co-doped ferrites, studies specifically on Al³⁺-Gd³⁺ co-doped CoFe₂O₄ NPs are not often reported in current literature. This gap suggests a potential area for future research to explore the combined effects of these particular dopant on the properties and applications of CoFe₂O₄ NPs. Such studies could further advance our understanding and utilization of these materials in diverse technological areas.

In this work, we present a series of CoFe₂O₄ nanoparticles co-doped with Al³⁺ and Gd³⁺ ions, formulated as CoFe₂₋₂ₓAlₓGdₓO₄, synthesized using the sol-gel auto-ignition technique. We comprehensively investigated the different properties of these nanoparticles through rigorous characterization methods. Our findings are expected to offer important insights into the properties and prospective application of these co-doped nanoparticles. By systematically analyzing their structural integrity, infrared absorption spectra, surface characteristics, magnetic behavior, and electrical properties, we aim to establish a solid foundation for their effective utilization in practical applications. It is anticipated that the outcomes of this work will contribute to advancing the understanding and optimizing the performance of CoFe₂O₄ nanoparticles co-doped with Al³⁺ and Gd³⁺. This knowledge could potentially pave the way for their enhanced application in various real-world scenarios, ranging from electronics and magnetic storage to biomedical and environmental technologies.

The metal ion source materials for Co²⁺, Fe³⁺, Al³⁺, and Gd³⁺ in the form of nitrate, citric acid (C₆H₈O₇), ammonia solution (NH₃), and deionized water employed as precursor materials were obtained from Sigma Aldrich and were used without further purification. The sol-gel auto-ignition synthesis technique was employed to synthesize CoFe₂₋₂ₓAlₓGdₓO₄ NPs. All the nitrates in proper proportion were initially dissolved in deionized water and then mixed thoroughly. Citric acid was utilized as the fuel and was mixed with the nitrate precursors solution to achieve a mole ratio of 3:1 (citric acid : metal cations). The pH of the solution was then set to ~7 by adding ammonia solution dropwise. The temperature of the pH-adjusted solution was increased gradually from 60°C to 90°C and then to 115°C. Initially, a sol (solution) was obtained which was gradually converted into a gel. The temperature was further raised to 120°C, causing the gel to auto-ignite and transform into a fine powdered state. The loose powder obtained was further grounded and sintered at 600°C for 6 h to remove any remaining moisture and organic residues from the sample. This synthesis approach ensured the production of CoFe₂₋₂ₓAlₓGdₓO₄ nanoparticles in a controlled manner, suitable for subsequent detailed characterization studies.

The characterization of the synthesized nanoparticles included:

- X-ray Diffraction (XRD): Conducted using a Bruker D8 Advance equipped with a Cu kα source to analyze cubic structure formation and determine nanoscale crystallite size.
- Scanning Electron Microscopy (SEM) with energy dispersive X-ray Analysis (EDAX): Utilized ZEISS, EVO LS15 equipment to examine nanoparticle morphology and composition.
- Transmission Electron Microscopy (TEM): Employed a Tecnai T20 microscope to further analyze the morphology and structure of the nanoparticles at higher resolution [47].
- Fourier Transform Infrared Spectroscopy (FT-IR): Performed using a Thermo Scientific Nicolet iS10 to study chemical bonding and functional groups in the nanoparticles.
- Nitrogen physisorption in conjugation with Brunauer–Emmett–Teller (BET) and Barrett–Joyner–Halenda (BJT) analyses: Utilized Microtrac BELSORP-maxII to measure the surface area of the nanoparticles.
- Vibrating Sample Magnetometer (VSM): Employed a Lakeshore VSM 7407 to measure magnetic properties and characterize M–H loops.
- DC Resistivity Measurement: Used the two-probe technique to study electrical conductivity properties.
- Dielectric Measurements: Conducted using an LCR-Q meter Hioki 3532-50 to analyze the dielectric properties of the NPs.

The structural analysis of the synthesized nanoparticles was conducted using X-ray diffraction (XRD) patterns, and Rietveld refinement was performed using FullProf software, as illustrated in Fig. 1(a–e). The superiority of the Rietveld refinement was validated through various parameters listed in Table 1. The XRD patterns in Fig. 1(a–e) exhibit distinct

cubic spinel structure. Each peak aligns well with previous research on cobalt ferrite nanoparticles indicating high structural integrity and phase purity without impurity peaks [48, 49]. Furthermore, the XRD results show a slight shift in Bragg’s angle, suggesting successful co-doping of Al³⁺ and Gd³⁺ ions in the CoFe₂O₄ lattice (Fig. 1(a–e)). This shifting in the Bragg angles confirms the incorporation of Al³⁺ and Gd³⁺ ions into the crystal structure, validating the effectiveness of the co-doping process. The lattice constant was determined using [50],

\( a = d\sqrt{h^2 + k^2 + l^2} \)                                            (1)

where, d is the interplanar spacing, and h, k, l are the Miller indices.

reflection planes indexed as ‘(220), (311), (222), (400), (422), (511), (440), (620), and (533)’. These reflections validate the occurrence of a single-phase face-centered

The lattice constant of the CoFe₂₋₂ₓAlₓGdₓO₄ nanoparticles ranged from 8.382 Å to 8.396 Å, showing an increasing trend with the co-doping, as detailed in Table 2. This variation in lattice constant is attributed to the substitution of Fe³⁺ ions (with a smaller ionic radius of 0.64 Å) by Gd³⁺ ions having larger ionic radii (0.94 Å). The lattice constant values were also estimated by Rietveld refinement analysis as shown in Table 2. The theoretical lattice constant values are in close agreement with the calculated lattice constant values from XRD data. The crystallite size (t) values were determined by [51],

t = 0.9λ / (β cosθ) nm       (2)

where, λ is the wavelength, β is the full width at half maximum and θ is the angle.

The crystallite size of the CoFe₂₋₂ₓAlₓGdₓO₄ NPs ranged from 19 nm to 23 nm, exhibiting a decreasing trend with the co-doping, as detailed in Table 2. This variation indicates that the incorporation of Al³⁺ and Gd³⁺ ions influences the growth of crystallites in the nanoparticles. Furthermore, the observed crystallite sizes (19 nm to 23 nm) confirm the nanoscale behavior of the synthesized NPs. The dislocation density (δ) was calculated by [52],

δ = 1 / t² lines/m²     (3)

where, t is the crystallite size.

The dislocation density (δ) values varied between 19.39 × 10⁻⁴ lines/m² to 26.41 × 10⁻⁴ lines/m² as given in Table 2. The X-ray density values were determined by [52],

dₓ = (Z × M) / (V × Nₐ) gm/cm³      (4)

where, Z is the coordination number, M is the molecular weight, V is the unit cell volume, and Nₐ is the Avogadro’s number (6.022 × 10²³).

The X-ray density of the CoFe₂₋₂ₓAlₓGdₓO₄ NPs varied from 5.292 g/cm³ to 5.429 g/cm³, showing an increasing trend with the co-doping, as detailed in Table 2. This change in X-ray density reflects the alteration in the overall mass per unit volume due to the incorporation of larger Gd³⁺ ions into the crystal lattice compared to Fe³⁺ ions. Similarly, the bulk density values, determined using the Archimedes principle, ranged from 3.552 g/cm³ to 3.692 g/cm³, as indicated in Table 2. These values typically differ slightly from X-ray density due to factors such as porosity and the presence of air or other voids within the nanoparticle structure. The % porosity values were calculated by [53],

P = (1 - d_B / dₓ) %      (5)

where, d_B is the bulk density and dₓ is the X-ray density.

The % porosity varied between 31% to 33% with co-doping showing decreasing behavior as shown in Table 2. The hopping lengths L_A and L_B were determined by [53],

L_A = a√(3/4)      (6)

L_B = a√(2/4)      (7)

where, a is the lattice constant.

The hopping length values provided in Table 3 demonstrate an increase with higher levels of co-doping. This trend indicates that as the doping concentration of these ions increases, the hopping length within the crystal lattice also increases. The increase in hopping length can

be directly linked to the expansion of the lattice parameter. The bond lengths were estimated using equation [50],

\( d_{AX} = a \sqrt{3\left(u - \frac{1}{4}\right)} \)   (8)

\( d_{BX} = a \sqrt{3u^2 - \frac{11}{4}u + \frac{43}{64}} \)   (9)

\( d_{AXE} = a \sqrt{2\left(2u - \frac{1}{2}\right)} \)   (10)

\( d_{BXE} = a \sqrt{2(1 - 2u)} \)   (11)

\( d_{BEU} = a \sqrt{4u^2 - 3u + \frac{11}{16}} \)   (12)

where u is the oxygen positional parameter.

The bond lengths reported in Table 3 exhibit an increase with the co-doping. This observation is consistent across all values, indicating that as the concentration of Al^{3+} and Gd^{3+} ions increases, the bond lengths within the CoFe_{2-2x}Al_xGd_xO_4 NPs also increase. The ionic radii (r_A and r_B) values were calculated using [52],

\( r_A = \left(u - \frac{1}{4}\right)a\sqrt{3} - r(O^{2-}) \)   (13)

\( r_B = \left(\frac{5}{8} - u\right)a - r(O^{2-}) \)   (14)

The values of ionic radii increase with the co-doping, as presented in Table 3. Considering the experimentally determined lattice constant and oxygen positional parameter (u = 0.381 Å), the interionic distances between cations (Me-Me: b, c, d, e, f) and between cations and anions (Me-O: p, q, r, s) were evaluated. These interionic distances play a crucial role in defining the predominant magnetic characteristics and are instrumental in elucidating the crystallographic structure. The interionic bond distances are calculated by following relations [50] and its values are provided in Table 4.

\( p = a\left(\frac{5}{8} - u\right) \)   (15)

\( q = a\sqrt{3}\left(u - \frac{1}{4}\right) \)   (16)

\( r = a\sqrt{11}\left(u - \frac{1}{4}\right) \)   (17)

\( s = a\sqrt{3}\left(\frac{u}{3} + \frac{1}{8}\right) \)   (18)

\( b = \sqrt{2}\left(\frac{a}{4}\right) \)   (19)

\( c = \sqrt{11}\left(\frac{a}{8}\right) \)   (20)

\( d = \sqrt{3}\left(\frac{a}{4}\right) \)   (21)

\( e = \sqrt{3}\left(\frac{3a}{8}\right) \)   (22)

\( f = \sqrt{6}\left(\frac{a}{4}\right) \)   (23)

From Table 4, it is evident that interionic distances increase with the increase in co-doping. This observation implies that as the concentration of Al^{3+} and Gd^{3+} ions in the CoFe_{2-2x}Al_xGd_xO_4 NPs rises, both the Me-O (distances between cations and oxygen ions) and Me-Me (distances between cations) distances expand. The expansion of these interionic distances suggests a weakening of the interionic bonds within the crystal lattice. The bond angles were estimated using equations [52],

\( \theta_1 = \cos^{-1}\left(\frac{p^2 + q^2 - c^2}{2pq}\right) \)   (24)

\( \theta_2 = \cos^{-1}\left(\frac{p^2 + r^2 - e^2}{2pr}\right) \)   (25)

\( \theta_3 = \cos^{-1}\left(\frac{2p^2 - b^2}{2p^2}\right) \)   (26)

The calculated bond angles presented in Table 5 indicate specific trends i.e. θ₁, θ₂, and θ₅ increase, while θ₃ and θ₄ decrease with increasing co-doping. This variation in bond angles suggests structural modifications and changes in interatomic interactions within the crystal lattice.

SEM images of typical samples of CoFe₂₋₂ₓAlₓGdₓO₄ (x = 0.00 and 0.04) were examined to analyze their morphology, as depicted in Fig. 2. The images clearly show spherical-shaped nanoparticles with nanoscale grains. Additionally, some agglomeration is observed, likely due to interactions between the magnetic nanoparticles. The grain size was estimated using the linear intercept method, yielding values between 34 and 37 nm. These nanoscale grain sizes confirm the nanocrystalline nature of the samples, consistent with the findings from XRD analyses. The observed agglomeration suggests interactions between nanoparticles, which could further influence their collective magnetic behavior. The elemental composition of typical samples of CoFe₂₋₂ₓAlₓGdₓO₄ (x = 0.00 and 0.04) was analyzed using energy dispersive X-ray analysis (EDAX), as illustrated in Fig. 3. The EDAX spectra exhibited excellent agreement between the measured elemental percentages and the expected weight and atomic percentages based on stoichiometry. The spectrum clearly shows the presence of Co, Fe, O, Al, and Gd, confirming the composition of CoFe₂₋₂ₓAlₓGdₓO₄ nanoparticles. Importantly, the absence of any detectable traces of other metals in the EDAX spectrum indicates the high purity of the samples. The surface morphology and microstructure of typical samples of CoFe₂₋₂ₓAlₓGdₓO₄ (x = 0.00 and 0.04) were investigated using transmission electron microscopy (TEM), as depicted in Fig. 4. The TEM images reveal spherical-shaped particles with noticeable agglomeration, likely caused by magnetic interactions among the particles. The average particle size estimated from TEM images are in between 19–22 nm. This nanometer-scale particle size confirms the nanocrystalline nature of the prepared nanoparticles, which is consistent with the findings from XRD analysis. The observed agglomeration highlights the

θ₄ = Cos⁻¹ [ (p² + s² − f²) / (2ps) ]        (27)

θ₅ = Cos⁻¹ [ (r² + q² − d²) / (2rq) ]        (28)

scale particle size confirms the nanocrystalline nature of the prepared nanoparticles, which is consistent with the findings from XRD analysis. The observed agglomeration highlights the

magnetic interactions between nanoparticles, which can influence their collective magnetic behavior. The nanocrystalline behavior observed in TEM aligns well with XRD results, affirming the structural integrity and potential applications of these nanoparticles in various advanced technologies.

The FT-IR spectra were recorded within the range of 380 cm⁻¹ to 1000 cm⁻¹, as illustrated in Fig. 5. The FT-IR spectra exhibit two distinct absorption bands around 600 cm⁻¹ and 400 cm⁻¹, characteristic of the spinel ferrite structure, specifically assigned to the tetrahedral (A) and octahedral (B) sites, respectively, within the spinel structure [54, 55]. The absorption wavenumber ν₁ and ν₂ are given in Table 6. It is observed from Table 6 that ν₁ shifts to higher wavenumber with co-doping. This shift occurs because as more Al³⁺ and Gd³⁺ ions are introduced, Fe³⁺ ions shift towards the oxygen ions at the (A) site, thereby reducing the Fe³⁺-O²⁻ distance. This reduction leads to an increase in the ν₁ wavenumber. Conversely, ν₂ shifts to a lower wavenumber with co-doping. This shift is likely influenced by changes in the lattice dynamics or atomic interactions within the spinel structure as a result of dopant incorporation. The exact mechanism behind this shift requires further detailed analysis, but it reflects the complex interplay between dopants and lattice vibrations. The force constants Kt and KO (Table 6) were estimated by [50],

\[k_t = 7.62 \times M_1 \times \vartheta_1^2 \times 10^{-7} \text{N/m}\]        (29)

\[k_O = 10.62 \times M_2 / 2 \times \vartheta_2^2 \times 10^{-7} \text{N/m}\]   (30)

\[
k_{av} = \frac{k_t + k_O}{2}
\]      (31)

Here, the M₁ and M₂ are molecular weights of (A) and [B] site, respectively. It is observed that the K_O is smaller in comparison with the K_t.

The physisorption of non-reactive gases, specifically nitrogen in this study, was employed to determine surface area and pore parameters. Nitrogen adsorption-desorption isotherms were recorded to analyze the textural characteristics, and the recorded isotherm curves for typical samples of CoFe₂₋₂ₓAlₓGdₓO₄ (x = 0.00, 0.04, and 0.08) NPs are depicted in Fig. 6(a–c). These isotherms exhibit a typical Langmuir type I-V isotherm with H3 hysteresis plots, characteristic of mesoporous materials. From the BET and BJH analyses, various surface parameters were derived and are detailed in Table 7. The BET surface area values ranged from 37 m²/gm to 46 m²/gm. It is notable from Table 7 that the BET surface area increases with the co-doping. This increase suggests enhanced surface accessibility and adsorption capacity due to structural modifications induced by dopant incorporation. Furthermore, the average pore diameter and pore volume values also show an increase with increasing co-doping, as observed in Table 7. This indicates that the introduction of Al³⁺ and Gd³⁺ ions promotes the formation of larger pores within the nanoparticles, potentially enhancing their porosity and surface area. The observed increases in surface parameters with co-doping underscore the tunability of these nanoparticles for applications requiring specific surface properties, such as in catalysis, adsorption, and sensing technologies.

The M-H hysteresis loops displayed in Fig. 7 indicate a soft magnetic nature of all the samples. The magnetic parameters extracted from these loops are summarized in Table 8. It is observed from Table 8 that the saturation magnetization (Mₛ) decreases with co-doping, ranging from 49.18 emu/g to 65.23 emu/g. This decrease in Mₛ at higher concentrations of Al³⁺ and Gd³⁺ ions can be attributed to the substitution of magnetic Fe³⁺ ions by non-magnetic Al³⁺ and Gd³⁺ ions (which are non-magnetic at room temperature) within the crystal lattice [56, 57]. Additionally, the alteration of the exchange interactions between (A) and [B] sites may lead to either strengthening or weakening of the (B)-(B) and (A)-(B) interfaces, further influencing magnetization values [58, 59]. The decrease in Mₛ can also be accredited to the reduction in super-exchange interactions due to the co-doping of non-magnetic ions, thereby lowering the overall magnetization of the nanoparticles. Furthermore, the coercivity (Hc) shows a decreasing trend with co-doping. The lowest Hc value observed was 74.81 Oe for the sample with x = 0.08. The values of magneton number (nB) and magnetocrystalline anisotropy constant (K) were estimated by equations [60],

\[ n_B = \left( \frac{\text{Mol. Weight} \times M_S}{5585} \right) \mu_B \]    (32)

\[ K = \frac{H_C \times M_S}{0.96} \]    (33)

where, Mₛ is the magnetization and Hc is the coercivity.

behavior similar to the saturation magnetization upon co-doping. As indicated in Table 8, the Mr shows a decreasing trend with co-doping. The Mr/MS values are noted to be lower than 0.5, indicating a multi-domain structure in the samples.

The value of the observed magneton number and magnetocrystalline anisotropy constant (K) are given in Table 8. The K values were found to be in the range of 3.83–12.11 K × 10^3 (erg/Oe) and decreased with co-doping. The magnetization characteristics, including the remanence magnetization and the remanence ratio (Mr/MS), exhibit

The electrical properties of CoFe2–2xAlxGdxO4 (x = 0.00, 0.02, 0.04, 0.06, and 0.08) nanoparticles were analyzed by measuring DC electrical resistivity (DCR). The DCR can be obtained by [60],

\[
\rho_{RT} = \frac{RA}{L}
\] (34)

where, R = resistance, A = πr^2 and L = width of the pellet. Figure 8 depicts the changes in electrical resistivity (ρ) with the co-doping. The data clearly show a decrease in resistivity when these ions are introduced. One possible reason for this reduction is the decrease in the abundance of Fe^3+ ions, which diminishes the likelihood of electron hopping between Fe^3+ and Fe^2+ ions [61, 62]. Additionally, nano-crystalline spinel ferrite is known to feature more grain boundaries, which act as barriers to electron movement, thereby reducing electrical resistivity. Factors such as grain boundaries, porosity, and stoichiometry significantly influence the material’s resistivity by introducing additional insulating barriers. These factors collectively influence the material’s electrical properties and are crucial considerations in optimizing its performance for various applications. Apart from grain boundaries, factors such as porosity and stoichiometry (the exact ratio of elements within the material) also play significant roles in determining the electrical resistivity of spinel ferrites. Higher porosity generally increases resistivity by introducing more insulating barriers. According to Verwey’s model, electron conduction in spinel ferrite occurs via hopping among ions of the same element at octahedral [B] sites with different valency. Both Fe^2+ and Fe^3+ ions reside at these sites, facilitating electron conduction as electrons move between them.

The dielectric properties of all the samples were investigated by measuring the dielectric constant (ε′), dielectric loss (ε″), and dielectric loss tangent (tan δ) using an LCR-Q meter at room temperature across varying frequencies. Figure 9 illustrates the variation of dielectric constant with frequency. It reveals a consistent decrease in ε′ as the frequency increases, indicating a frequency-independent trend across all samples. The dielectric parameters can be affected many parameters like particle sizes, structures, stoichiometric composition, and synthesis techniques [63, 64]. The noted decrease in ε′ is related to the microstructure of the material. Ferrite materials typically consist of conducting grains separated by less conducting grain boundaries. At ‘lower frequencies’, the polarization behavior within the material, specifically space-charge polarization around non-ferroelectric regions adjacent to ferroelectric zones, contributes to the decrease in ε′. As the frequency increases, the electrical polarizability stabilizes, resulting in a more consistent ε′ value at higher frequencies. The relationship between ε″, frequency, and tan δ is explored through Figs. 10 and 11 in the present study. Figure 10 illustrates that ε″ decreases exponentially as frequency increases. This decrease is similar to the behavior of ε′, even though ε″ exhibits a more prominent decrease compared to ε′. The tan δ, shown in Fig. 11, also demonstrates an exponential decrease with increasing frequency. The noted trend of tan δ can be attributed to Maxwell-Wagner interfacial polarization. This phenomenon arises due to variations in conductivity and dielectric constant within the material, particularly at interfaces between different phases or grains. Factors influencing tan δ include carrier concentration and structural uniformity. Furthermore, the study reveals that the dielectric parameters are influenced by the composition, specifically the co-doping. It is noted that increasing the concentration of these dopants enhances all dielectric characteristics. This compositional dependence underscores the importance of precise control over material composition in optimizing dielectric properties for various technological applications.

Trivalent Al3+ and Gd3+ ions were successfully co-doped into CoFe2O4 NPs with varying compositions represented as CoFe2-2xAlxGdxO4 (x = 0.00, 0.02, 0.04, 0.06, 0.08) using the sol-gel auto-combustion method. X-ray diffraction (XRD) analysis confirmed the formation of phase pure cubic spinel structures for all compositions. The lattice constant increased with Al3+ and Gd3+ ions co-doping due

to the difference in ionic radii, highlighting the significant impact of co-doping on structural parameters. SEM imaging showed spherical grain shapes with some particle agglomeration, showing nanoscale characteristics. The average grain size corroborated the nanoscale nature of the particles. FTIR spectra revealed peaks corresponding to the cubic spinel ferrite structure. Magnetic properties showed a decrease in both saturation magnetization and magneton number with increasing Al3+ and Gd3+ ions co-doping levels. Conversely, the DC electrical resistivity decreased with higher co-doping levels, indicating improved electrical conductivity. Dielectric measurements exhibited a decrease in dielectric characteristics as frequency increased, a typical behavior in such materials. Overall, the co-doping of Al3+ and Gd3+ ions induced significant alterations in the features

of CoFe2O4 NPs. These modifications enhance their suitability for diverse applicability consisting high-frequency appliances, magnetic storage systems, biomedicine field, and beyond, showcasing their potential in advanced technological domains.

The research data generated and analyzed during the current study are available from the corresponding author upon reasonable request.

Acknowledgements The author Vaibhav K. Raut acknowledges Chhatrapati Shahu Maharaj Research Training and Human Development Institute (SARTHI), Pune (Govt. of Maharashtra, India) for CSMNRF Fellowship.

Author contributions Mr. Vaibhav K. Raut conceived and designed the research study, conducted the experiments, analyzed the data, and drafted the manuscript. Prof. Chandrakant T. Birajdar provided supervision, contributed to experimental design, assisted in data analysis, and participated in manuscript preparation. Prof. Elmuez A. Dawi and Dr. Sandeep B. Somvanshi contributed to data collection, interpretation, and manuscript writing, and critically revised the manuscript.

Conflict of interest The authors declare no competing interests.

Ethics This research study was conducted in compliance with ethical standards.

1. Batoo KM, Hadi M, Chauhan A, Verma R, Singh M, Aldossary OM, Bhargava GK (2022) High-frequency applications of bismuth-doped Co–Zn ferrite nanoparticles for electromagnetic interference filter and multilayer inductor chip fabrication. Appl Phys A 128:283

2. Mokhosi SR, Mdlalose W, Nhlapo A, Singh M (2022) Advances in the synthesis and application of magnetic ferrite nanoparticles for cancer therapy. Pharmaceutics 14:937

3. Rao B, Shanmukhi P, Mammo TW, Kothandan D, Aregai T, Desta T, Kahsay M, Hagos G, Murali N, Batoo KM (2024) Investigation effect of Cr3+ substituted on enhanced dielectric and magnetic properties of co-cu nano ferrites for high-density data storage applications. Appl Phys A 130:1–14

4. Hadi M, Batoo KM, Chauhan A, Aldossary OM, Verma R, Yang Y (2021) Tuning of structural, dielectric, and electronic properties of Cu doped Co–Zn ferrite nanoparticles for multilayer inductor chip applications. Magnetochemistry 7:53

5. Qin H, He Y, Xu P, Huang D, Wang Z, Wang H, Wang Z, Zhao Y, Tian Q, Wang C (2021) Spinel ferrites (MFe2O4): Synthesis, improvement and catalytic application in environment and energy field. Adv Colloid Interface Sci 294:102486

6. Kiran A, Akhtar MN, Yousaf M, Batoo KM, Aldossary OM, Khan SN (2021) Influence of Y3+, Yb3+, Gd3+ cations on structural and electromagnetic properties of CuFe2O4 nanoferrites prepared via one step sol-gel method. J Rare Earths 39:1224–1231

7. Kefeni KK, Mamba BB, Msagati TA (2017) Application of spinel ferrite nanoparticles in water and wastewater treatment: a review. Sep Purif Technol 188:399–422

8. Srivastava R, Yadav BC (2012) Ferrite materials: introduction, synthesis techniques, and applications as sensors. Int J Green Nanotechnol 4:141–154

9. Sutka A, Mezinskis G (2012) Sol-gel auto-combustion synthesis of spinel-type ferrite nanomaterials. Front Mater Sci 6:128–141

10. Wani TA, Suresh G, Masrour R, Batoo KM, Rasool A (2023) A structural, morphological, optical and magnetic study of nickel-substituted zinc (Ni–Zn) ferrite nanoparticles synthesized via glycine assisted gel autocombustion synthesis route. Mater Chem Phys 307:128169

11. Jauhar S, Kaur J, Goyal A, Singhal S (2016) Tuning the properties of cobalt ferrite: a road towards diverse applications. RSC Adv 6:97694–97719

12. Mammo TW, Gebresilassie TA, Shanmukhi P, Teklehaimanot BT, Murali N, Batoo KM, Hussain S (2024) Study of structural, electrical and magnetic properties of co-substituted Co1−2xNixMgxFe2O4 (0≤x≤0.25) nanoferrite materials. Appl Phys A 130:178

13. Mugutkar AB, Gore SK, Mane RS, Batoo KM, Adil SF, Jadhav SS (2018) Magneto-structural behaviour of Gd doped nanocrystalline Co-Zn ferrites governed by domain wall movement and spin rotations. Ceram Int 44:21675–21683

14. Batoo KM, Hadi M, Verma R, Chauhan A, Kumar R, Singh M, Aldossary OM (2022) Improved microwave absorption and EMI shielding properties of Ba-doped Co–Zn ferrite. Ceram Int 48:3328–3343

15. Kumar SR, Priya GV, Aruna B, Raju M, Parajuli D, Murali N, Verma R, Batoo KM, Kumar R, Narayana PL (2022) Influence of Nd3+ substituted Co0.5Ni0.5Fe2O4 ferrite on structural, morphological, dc electrical resistivity and magnetic properties. Inorg Chem Commun 136:109132

16. Razia N, Shakeel K, Asokan K, Hilal A, Imran K (2012) Magnetic and electrical properties of In doped cobalt ferrite nanoparticles. J Appl Phys 112:084321. https://doi.org/10.1063/1.4759436

17. Köseoğlu Y, Alan F, Tan M, Yilgin R, Öztürk M (2012) Low temperature hydrothermal synthesis and characterization of Mn doped cobalt ferrite nanoparticles. Ceram Int 38:3625–3634

18. Karimi Z, Mohammadifar Y, Shokrollahi H, Asl SK, Yousefi G, Karimi L (2014) Magnetic and structural properties of nano sized

Dy-doped cobalt ferrite synthesized by co-precipitation. J Magn Magn Mater 361:150–156

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.