The Origin of the Low-Temperature Minimum of Electrical Resistivity in Strontium Ferromolybdate Ceramics

Round 1

Reviewer 1 Report

referee could not understand \rho_0 = 1/\sigma_{0,FIT}.

\rho_0 is residual resistance, is not it?

That causes the inconsistence of eq 11 when T=0.

T_!/T_0 is 1 to 5, that induces large difference in exp function.

Author should explain around this in the the text.

First, in equation (5), authors assume the sum of the residual resistance and the resistive component of the polynomial.
Then we derive the final result, equation (11), as the sum of the electrical conductivities.
The sum of electrical conductivities must be parallel circuits.
Also, in Equation (5), the sum of the residual resistance and the resistive component of the polynomial means that it is a series circuit. The reciprocal of the sum of the resistivities, which is the average electrical conductivity.
Thus, Equation (11) is a combination of components of different origins, parallel and series circuits.
it should be a reasonable explanation to introduce an assumption as to why σ0,FIT and the residual resistance component are related.

Author Response

referee could not understand \rho_0 = 1/\sigma_{0,FIT}.

\rho_0 is residual resistance, is not it?

We have now denoted rho_0 as the residual metallic resistivity.

That causes the inconsistence of eq 11 when T=0. T_1/T_0 is 1 to 5, that induces large difference in exp function.

In the case of fluctuation induced tunneling, the residual resistivity corresponds to a value for both temperature and barrier height (i.e. T₁) tending to zero.

Author should explain around this in the the text.

We have added the following text to the manuscript: Unfortunately, only a few publications provide values of both w and d. When matching the FIT model and metallic conductivities, the inverse of the metallic residual resistivity corresponds to the FIT model conductivity for temperature and barrier height tending to zero. Thus, we introduce for sake of simplicity an effective model resistivity rho_0 = 1/sigma_0,FIT which is valid for 2w/d ≈ 1.

First, in equation (5), authors assume the sum of the residual resistance and the resistive component of the polynomial.

Equation (5) describes metallic resistivity as its value obtained at zero temperature (the residual resistivity) and a temperature-dependent term attributed to charge carrier scattering.

Then they derive the final result, equation (11), as the sum of the electrical conductivities. The sum of electrical conductivities must be parallel circuits.

The authors thank the referee for indicating this incorrect description. In a granular structure there are intragrain and intergrain contributions to conductance. Conductivity or resistivity measurements yield a value averaged over a large number of series connections of grains and intergrain barriers along a charge carrier trajectory which on their part are connected in parallel. In literature this is described by the so-called brick model. The barrier height for intergrain tunneling is in the order of 10 meV. It decreases with increasing magnetic flux. Due to this low barrier height, the impact of the barriers disappears at elevated temperatures and high magnetic fluxes resulting in a measured intragrain metallic conductivity. Intergrain tunneling becomes visible at lower temperatures.

Let us consider a brick model consisting of cube-shaped grains of metallic conductivity covered at the surface by grain boundaries creating nano-sized intergrain tunneling barriers. Two types grain boundaries appear one with a normal vector parallel to the applied field (perpendicular boundaries) and a second with a normal vector perpendicular to the applied field (parallel boundaries). The corresponding DC equivalent circuit is a parallel connection of intragrain resistance and intergrain resistance of parallel grain boundaries in series with the intergrain resistance of perpendicular grain boundaries [DOI: 10.1557/JMR.1998.0219]. In our case, the series intergrain resistance may be neglected since with regard to a very small barrier height, a bias of already a few mV sufficiently increases tunneling current thus decreasing barrier resistance. On the other hand, the resistance of parallel boundaries remains high since the normal vector of the grain boundary is along an equipotential line. Finally, we arrive at a parallel connection described by equation (14).

Correspondingly, the manuscript text was improved.

Also, in Equation (5), the sum of the residual resistance and the resistive component of the polynomial means that it is a series circuit.

In this work, the granular structure of strontium ferromolybdate ceramics is considered as consisting metallic grains described by equation (5) and intergrain tunneling barriers described by equation (10). Due to the occurrence of a residual resistivity in metals, the intragrain resistivity is a series connection of an initial value at zero temperature and a temperature dependent-term which is proved by a huge number of experimental research. Since inside a grain all spins are in the same direction, the metallic contribution of the spin-down channel is dominating due to the presence of states at the Fermi level, while the semiconductor-like contribution of the spin-up channel may be neglected due to the absence of corresponding states at the Fermi level. Spin-up conductivity comes into play when exp(-0.8eV/kT) becomes a sufficient value.

The reciprocal of the sum of the resistivities, which is the average electrical conductivity.

For the sum of reciprocal resistivities of grains and grain boundaries we have now introduced a model conductivity since geometric parameters of the brick model are commonly unknown. It represents some average conductivity

Thus, Equation (11) is a combination of components of different origins, parallel and series circuits.

This is true and corresponds to the equivalent circuit of a brick model. Equation (11) describes a parallel connection of intragrain metallic conductivity, i.e. the reciprocal of equation (5), and intergrain tunneling conductivity, equation (10). Here, metallic resistivity, equation (5), is a series connection of residual resistivity and a temperature dependent term.

it should be a reasonable explanation to introduce an assumption as to why σ0,FIT and the residual resistance component are related.

Done, see above.

Reviewer 2 Report

This work proposes a new explanation for the low-temperature resistivity minimum in ceramic strontium ferromolybdate (SFMO), attributing it to intergrain fluctuation-induced tunneling rather than a semiconductor-metallic transition. A modified model of total conductivity for SFMO ceramics and thin films is derived, showing validity across various conditions. I would suggest acceptance after minor revision. Following are my questions and concerns about this work

1. The author claims that the low-temperature conductivity mechanism with high probability originates from intragranular tunneling. How can the author exclude other parameters?
2. The equation proposed in the work seems can fit data plots in reported publications. My concern is that even though the fitting looks good, its universality is still uncertain to me since the number of samples is still small.

Author Response

This work proposes a new explanation for the low-temperature resistivity minimum in ceramic strontium ferromolybdate (SFMO), attributing it to intergrain fluctuation-induced tunneling rather than a semiconductor-metallic transition. A modified model of total conductivity for SFMO ceramics and thin films is derived, showing validity across various conditions. I would suggest acceptance after minor revision. Following are my questions and concerns about this work

Thank you for your efforts reviewing the paper.

The author claims that the low-temperature conductivity mechanism with high probability originates from intragranular tunneling. How can the author exclude other parameters?

Spin-dependent intergrain tunneling in strontium ferromolybdate ceramics is a well-established in literature fact. In this work, we attribute intergrain tunneling to the special case of fluctuation-induced tunneling, i.e. we consider ceramics synthesized under certain conditions. Fluctuation-induced tunneling occurs in the presence of conducting grains separated by nano-sized energy barriers where large thermal voltage fluctuations occur when the capacitance of an intergrain junction is in the order of 0.1 fF. This happens in cold-pressed strontium ferromolybdate ceramics and strontium ferromolybdate ceramics subjected to post-synthesis thermal treatment when locally SrMoO₄ barriers are formed. Here, the barrier thickness is in the order 1 to 3 nm, while the barrier area is several hundred square micrometers.

This was already explained in the manuscript:

In the absence of a magnetic field, the temperature dependence of the conductivity of SFMO ceramics was well-described by the fluctuation-induced tunneling (FIT) model, e.g., by the presence of conducting grains separated by nano-sized energy barriers where large thermal voltage fluctuations occur when the capacitance of an intergrain junction is in the order of 0.1 fF.

Now, we have added to the manuscript text:

Nano-sized energy barrier with a barrier width of a few nanometers and a barrier area of several hundred square micrometers usually occur at grain surfaces in cold-pressed or thermally treated under oxidation conditions ceramics.

And in conclusions:

… we propose a novel explanation of the low-temperature resistivity minimum in ceramic strontium ferromolybdate possessing nano-sized intergrain barriers…

The equation proposed in the work seems can fit data plots in reported publications. My concern is that even though the fitting looks good, its universality is still uncertain to me since the number of samples is still small.

We do not claim universality. This is already emphasized at the beginning of the abstract. A necessary condition for the applicability of our model is the formation of nano-sized intergrain barriers. In the beginning, we had considered this as a special case, but we were surprised by the number of experimental data fitting this model. This was the motivation to publish this paper.

Round 2

Reviewer 1 Report

The additional statement in equation (14) can explain the detail of mechanism of this model for readers. It is also useful for building models of similar systems for other researchers.

Reviewer 2 Report

My question has been well answered. I would suggest the publicaiton of this paper in present version.