Impact of Antisite Disorder on the Resistivity of Strontium Ferromolybdate Ceramics

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

The manuscript Suchaneck et al., addresses a fundamental and persistent challenge in the field of spintronics. Authors investigated polycrystalline strontium ferromolybdate (Sr2FeMo6-δ, SFMO) synthesized via solid state reaction, focusing on the influence of Fe/Mo antisite disorder (ASD) on electrical conductivity across a cryogenic to ambient temperature regime (5-300K). The core contribution of this work is the application of a composite conductivity model that couples fluctuation induced tunneling (FIT) for the low temperature regime with a metallic power law behavior for the high temperature regime. The authors state that ASD systematically alters the microscopic parameters of intergrain barriers, specifically effective barrier height, width, and area, thereby shifting the conductivity maximum and altering the resistive upturn observed at low temperatures. The experimental approach utilizing Rietveld refinement to quantify the order parameter (P) and correlating it with transport data is methodologically reasonable in principle. The observed trends, particularly the decrease in effective barrier area and increase in barrier width with increasing disorder, offer potential insights into the degradation mechanisms of spin polarized transport in disordered media.

However currently the manuscript suffers from significant theoretical inconsistencies regarding the physical justification of the conductivity model (specifically the circuit topology), interpretation of granular geometry evolution during sintering and disentanglement of antisite effects from secondary phase segregation. The authors must address these major issues before it can be reconsidered for publication.

Major comments:

1. The most critical theoretical issue within the manuscript lies in the formulation of the total conductivity model. In Eq. 11 of the manuscript authors define the total conductivity (σ). This mathematical formulation implies a parallel connection between the intragrain metallic pathway (the first term) and the intergrain tunneling pathway (the second term). Physically this suggests that charge carriers have the choice to bypass the resistive grain boundaries entirely and flow through a continuous metallic network or alternatively flow through a tunneling channel shunting the metal.

This contradicts the fundamental microstructure of a polycrystalline ceramic described by the standard Brick Layer model (BLM). In a dense ceramic, charge carriers traverse a sequential path through grain interiors and across grain boundaries. Consequently total resistance is the sum of the components, not the conductance. The appropriate effective medium approximation for a granular metal-insulator system (where grains are metallic and boundaries are insulating/tunneling barriers) is a series resistor network: ⍴total = ⍴grain + ⍴boundary. 

 

If the authors claim that their system exhibits infiltration of metallic pathways that bypass grain boundaries, this requires microstructural evidence (like conductive AFM) which is not performed. If the parallel formulation is intended to model parallel spin channels within a single grain (as seen in some half-metallic models), authors must explicitly state this in the manuscript.

 

2. The use of a parallel model (sum of conductivities) inherently suppresses the contribution of the high resistance component. At low temperatures where the tunneling term is highly resistive (low conductivity) and the metallic term is low resistive (high conductivity), a parallel model would be dominated by the high conductivity metallic term. This contradicts the observed experimental "upturn" in resistivity at low temperatures, which typically arises because the high resistance tunneling barrier dominates the series circuit [https://doi.org/10.1002/pssb.202200012]. The authors must rigorously justify this circuit topology or reformulate the model using series resistivity addition.
3. The methodology section indicates that different ASD levels were achieved by varying the annealing time from 10-27 hours at 1373 K. It is a well established phenomenon in ceramic processing that extended sintering at high temperatures (>1000 C) leads to significant grain growth. The conductivity model depends explicitly on the geometric factor 2w/d (barrier width over grain size). Authors does not clarify whether the grain size d was measured independently for each sample (e.g. via SEM) or assumed constant. If d increases significantly with annealing time (as expected for grain growth kinetics), then 2w/d would decrease purely due to geometric coarsening. This geometric effect could mathematically be shown as a change in barrier parameters (w or σFIT) during the fitting process. The authors must provide grain size statistics for each sample (SFMO-57 through SFMO-92). If d varies, this variation must be explicitly factored into the extraction of w and A. Justifying modification in transport solely to ASD without correcting for grain coarsening is questionable.
4. The manuscript attributes changes in tunneling parameters to ASD. However the synthesis of SFMO is notoriously sensitive to redox conditions. Antisite disorder is often accompanied by the segregation of secondary phases, particularly insulating SrMoO4 or Fe, depending on the local oxygen partial pressure [https://doi.org/10.1016/j.jssc.2011.10.043]. The FIT model relies on tunneling through a dielectric barrier. In SFMO ceramics, this barrier is often a thin shell of SrMoO4 [https://doi.org/10.3390/electronicmat5010001]. Does increasing ASD (microscopic disorder) cause the barrier width w to increase or do the synthesis conditions that favor ASD (shorter annealing/non-equilibrium) simply leave thicker remnants of the insulating SrMoO4 phase at the grain boundaries? The manuscript suggests ASD increases w but this may be a correlation rather than a direct causality. The discussion needs to address whether "disorder" is effectively an alternate justification for "incomplete synthesis" which naturally results in thicker impurity barriers.
5. The authors have published extensively on this topic in Ceramics (2024) and Physica Status Solidi B (2022). These works utilize the same FIT + metallic conductivity model. The current manuscript must explicitly differentiate its findings. While the focus here appears to be the parametric dependence on ASD, but theoretical derivation is nearly identical. To ensure significant novelty, the analysis should focus more deeply on the quantitative link between the order parameter P and the specific tunneling parameters (V_0, w, A) rather than re-deriving the conductivity minimum model itself.

Minor comments:

• The text in Eq. 3 should clarify the origin of the empirical parameter b. Theoretical predictions suggest b=8μB/f.u. for simple antiferromagnetic coupling of antisites, but experimental values vary [DOI:10.1103/PhysRevB.67.174416]. Comparing their extracted b value with literature would strengthen the magnetic characterization.
• T_1 parameter represents the energy required for an electron to cross the insulating gap. The fitted values should be cross-referenced with the band gap of SrMoO4 (~ 4-5 eV) or the activation energies found in similar double perovskites to validate that the mechanism is indeed tunneling and not variable range hopping (dx.doi.org/10.1021/jz200853v ,J. Phys. Chem. Lett. 2011, 2, 1931–1936)

The fitting parameters in Table 2 (Sample SFMO-57 vs SFMO-92) show distinct trends. The discussion should explicitly link the magnitude of σFIT to the effective barrier area A. If disorder decreases A (as claimed), does this imply that ASD blocks the efficient tunneling channels and forces current through narrower "hot spots"?

Author Response

 

Please see the attachment

Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

The work examines how antisite disorder, Fe_Mo and Mo_Fe, affects the electrical conductivity of Sr₂FeMoO₆ ceramics produced by solid‑state reaction, highlighting a low‑temperature conductivity maximum resulting from the interplay between bulk metallic resistivity and inter-grain tunnelling. By fitting the temperature‑dependent conductivity to combined metallic and fluctuation‑induced tunnelling models, the authors find that increasing antisite disorder raises the effective barrier height and width while decreasing the barrier area, consequently shifting the conductivity maximum to higher temperatures. Using the order parameter (57 % ~ 92 %) to quantify disorder, the study establishes a direct correlation between higher disorder and larger tunnelling‑barrier parameters, linking structural defects to the observed transport behaviour. This work is thorough and engaging and could be potentially recommended for publication, conditional on the following revisions:

1. The XRD results require full refinement and microstructure analysis. Given that the phases are known and the diffraction patterns appear clean, the Rietveld refinement should not be difficult.
2. To further solidify the results, complementary conductivity measurements, such as Hall‑effect or magnetotransport studies, are necessary to separate carrier concentration from mobility and to verify whether the observed shifts in the conductivity maximum arise primarily from changes in barrier height/width or from alterations in the electronic band structure caused by the antisite defects.

Author Response

 

Please see the attachment

Author Response File: Author Response.pdf

Round 2

Reviewer 1 Report

Comments and Suggestions for Authors

The authors have made a significant effort to address several points raised in the previous review, specifically regarding the microstructural stability and the magnetic order parameter. However the response regarding the core theoretical framework (the conductivity model) still remains scientifically questionable and contradictory to standard transport theory.

1. The authors maintained the parallel conductivity model (σ_total = σ_metal + σ_tunnel) and attempted to justify it in their response by claiming that "series intergrain resistance may be neglected" due to bias effects. This response contains a fundamental logical flaw. In a parallel circuit current follows the path of least resistance. At low temperatures metallic grains are highly conductive while the tunneling barriers are resistive. If these were in parallel the current would bypass the barriers entirely flowing through the metallic channel. The total conductivity would essentially mimic the metal (σ_total ~ σ_metal) and no low temperature resistivity growth (semiconducting behavior) would be observed. The authors observe a resistivity growth proves that the tunneling barriers are acting as a bottleneck to charge transport which physically dictates a series description (or a percolation model where no continuous metallic path exists). By neglecting the series resistance in their argument they are mathematically negating the tunneling mechanism they are trying to fit.
2. The authors added text stating that their sintering temperature (1373 K) is too low to cause significant grain growth, citing literature on SrTiO3 and claimed that SEM images showed no changes. While they did not provide the requested quantitative grain size statistics, their argument regarding the thermal stability of the microstructure at these specific synthesis conditions is conceivable. The added text clarifies their assumption that d is constant, which allows the mathematical fitting to proceed even though quantitative proof is missing.

The authors must either: Reformulate the model using the standard series resistor network (ρ_total = ρ_metal + ρ_tunnel) which correctly describes a polycrystalline ceramic where current must cross grain boundaries or

rigorous evidence such as microstructural schematics or percolation studies to prove the existence of a continuous, parallel metallic path that bypasses the grain boundaries. Additionally an explanation is needed for why this path does not negate the tunneling signal at low temperatures.

Author Response

Please see the attachment.

Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

Given the revisions made by the authors the paper can be accepted now.

Author Response

Thank you for your time and efforts reviewing our paper once more.

Round 3

Reviewer 1 Report

Comments and Suggestions for Authors

The authors have corrected the fundamental physical error in their theoretical framework. By switching to the series resistivity model, the interpretation of the transport data is now scientifically justified and consistent with the granular nature of the ceramics. The additional clarifications regarding grain size stability and magnetic parameters are sufficient to support their conclusions. The manuscript meets the standard for publication.