Physical and Mathematical Models of Quantum Dielectric Relaxation in Electrical and Optoelectric Elements Based on Hydrogen-Bonded Crystals

Round 1

Reviewer 1 Report

The manuscript “Physical and Mathematical Model of Quantum Dielectric relax-2 ation in Electrical and Optoelectric Elements Based on Hydro-3 gen Bonded Crystals” by Valeriy Kalytka, Ali Mekhtiyev, Yelena Neshina et al. devoted to theoretical investigation of the quantum statistical properties of the proton subsystem in hydrogen-bonded crystals. A quantum kinetic equation is constructed for the ensemble of non-interacting protons. Using full quantum mechanical averaging of the polarization operator, the theoretical frequency-temperature spectra of the complex dielectric permittivity was calculated. An analytical study was performed of the dielectric loss tangent in the region of quantum nonlinear relaxation in hydrogen bonded crystals.

However, the following details might help the readers to appreciate the significance of the work:

1.     It would be better if there would be more illustrations (Figures)

2.     It is stated that the frequency-temperature spectra of the complex dielectric permittivity were calculated. Please provide some figures illustrating the example of such spectra.

3.     What physical mechanism are responsible for the four tand peaks in talc and gypsum? Do they correspond to the dielectric relaxation processes in the material?

4.     The authors state that they numerically calculated the tangent of the dielectric loss angle in the HBC. Please provide a Figure where the calculations are compared with experimental data.

5.     Table 1, first column (Layer thickness), last row. Probably, there is a misprint and there should be 3 instead of 30000?

6.     Page 20. “As can be seen from Figures 1-5, the thickness of the crystalline layer has absolutely no effect on the course of the graphs of functions 1,2,3 at different activation energies…”. According to the Figure captions, d=10^-9m everywhere.

7.     Please provide more details in the Figure captions.

Author Response

Dear colleague,

Our author team have made corrections according to your review. Please see the comments below:  

1.In the main text of the article in Section 3 "Results," figures 1,2 were added, containing temperature experimental and temperature experimental spectra measured by us, under certain external conditions, calculated from quasi-classical and quantum-mechanical models, theoretical spectra of the tangent of the dielectric loss angle in gypsum and Onot talc crystals. Theoretical spectra of 2-5 tangent of loss angle are given against the background of experimental spectra.

2. Calculation of frequency-temperature spectra of complex dielectric constant is performed only in analytical form, in the form of equations arising from quantum-mechanical model of dielectric relaxation in HBC. Numerical calculation and, on its basis, plotting are performed for theoretical temperature spectra of dielectric loss angle tangent.

 

3. Analysis of physical nature and calculation of parameters of four "peaks" in temperature experimental spectrum of dielectric loss angle tangent is performed in section "Results" and is reflected in Tables 1,2. The revealed dielectric loss mechanisms correspond to the dielectric relaxation processes in HBC.

 

4. Comparative analysis of calculation results with experimental data was performed based on the data in Tables 1,2.

 

5. The correction in Table 3 has been completed.

 

6. The results of calculations of temperature spectra of functions 1,2,3, at different activation energies are presented in Appendix C (Figures C.1-25). Note that in all the figures, inside each of the C.1-C.5 blocks, the course of the temperature graphs of the 1,2,3 functions remains unchanged, since they do not depend on the crystal thickness at a given activation energy. At the same time, the course of the temperature graph of function 4 at a given activation energy depends on the crystal thickness, which is especially noticeable on the example of small activation energies (0.01 eV (block C.1), 0.03 eV (block C.2)) and, to a lesser extent, for large activation energies (0.07 eV (block C.4), 0.1 eV (block C.5)). Despite the slightly visible visual differences in the graph of function 4 in the figures C.16-20 within block C.4 and in the figures C.21-25 within block C.5, the effect of these differences on the quantum proton relaxation mechanism is very significant in terms of the effects of dimensional effects on the configuration of the degenerate proton energy spectrum in HBC.

 

7. Detailed comments on the temperature experimental and theoretical spectra in Figures 1,2 are given in the form of an analysis of the properties and parameters of these spectra, which is reflected in the text, after the Figures 1,2 themselves.

 

During the corrections and additions in subsection 2.3, it became necessary at the very beginning of subsection 2.3 to introduce formulas for the Hamiltonian elements of the model. Due to the fact that time is extremely limited for technical edits, these formulas are not numbered by us, since this procedure would require changes in formula numbers throughout the text. I believe that the absence of indicated numbering does not distort the quality of material perception

Reviewer 2 Report

The manuscript represents a notable study of Z-scan on “Physical and Mathematical Model of Quantum Dielectric relax-2 ation in Electrical and Optoelectric Elements Based on Hydro-3 gen Bonded Crystals” for the development of NLO materials. Overall, this is a good paper. 

 

Some grammarly mistake is there, do correct it.

Author Response

Dear colleague,

Our author team corrected  all grammar mistakes according to your recommendations.

Reviewer 3 Report

In the Paper, a physical and mathematical model of quantum dielectric relaxation in HBC (Hydrogen Bonded Crystals) is constructed on the basis of non-stationary Liouville operator equation for the crystal proton subsystem using two key assumptions:

– Quantum kinetic equation can be constructed for an ideal proton gas moving in the crystal potential image perturbed by external electric field; and

– Non-balanced density matrix for the perturbed proton subsystem can be calculated from balanced one for the unperturbed subsystem (constructed by the quantum canonical Gibbs distribution) using solutions of the nonlinear quantum kinetic equation in linear approximation by a perturbation theory method.

Full quantum mechanical averaging of the crystalline layer polarization operator makes it possible to study theoretically the influence of its thickness on frequency- and temperature-spectra of material complex dielectric permittivity. Quantum relaxation parameters used in these calculations significantly differ from their semi-classical counterparts. There is presented a special analytical study scheme for the dielectric loss tangent in HBCs in the quantum nonlinear relaxation region.

The obtained results are both of academic and practical interests in developing the theoretical foundations of various prospective technologies and techniques based on materials with proton conduction, in particular, important class of HBC materials with anomalously long residual polarization relaxation time and anomalously high apparent dielectric permittivity.

Their mechanism of spontaneous polarization is believed to be due to quantum tunneling transitions (displacements) of hydrogen ions (protons) inside the hydrogen sublattice near the second-order phase transition point.

* * *

There is detected a number of technical problems.

■ Actually, Keywords section is “Keyphrases” section. Keywords section should be completely rewritten to meet the Journal standard.

■ Lines 339–340: “Note that from the qualitative and numerical analysis of expressions (18), (19) it is established” – This unfinished phrase should be deleted or completed and then moved after expression (19).

■ Lines 384–385: “… from (22) we have … (22)” – Correct the first expression number.

■ Lines 396–398: “… expression (25) transform to a form … (23)” – Correct the first expression number.

■ Lines 478–479: “… 0,5∙10⁻⁴ to 0,39∙10⁻⁸.” – Replace the comma "," with a dot "." in decimal representations here and everywhere below.

■ Lines 868–703: “… (36) … (37) … (38) … (39)” and Lines 849–857: “… (36) … (37) … (38) … (39)” – Renumber al the formulas as well as formula references in the text starting from Line 849.

■ References. [94, 95] are presented in the list, but not in the text. Insert them in text as well.

* * *

Two major recommendations can be given.

■ As it is a fully theoretical work, “2. Materials and Methods” section is better to rename to: “2. Models and Methods”.

Besides, this very lengthy, but not structured, section should be divided into a number subsections devoted to each of special assumptions of the proposed model in regard to:

– Modeling proton subsystem as an ideal gas;

– Which from types of chemical bonds presented in the crystal are taken into account in physical model of the proton quantum motion in the crystal lattice field;

– Choosing the dimensions for the geometric model elements;

– Choosing the numerical values for the mathematical model based on experimental maxima of the dielectric loss tangent;

– Considering the proton oscillations in the model of an isolated one-dimensional parabolic potential well;

– Calculation of the crystal quantum-mechanical model Hamiltonian by the secondary quantization method without taking into account the proton–phonon interaction;

– Construction of solutions of quantum kinetic equation in the perturbation theory linear approximation; etc.

Introducing of all the listed and rest assumptions of the proposed theory should be rigorously argued by providing corresponding experimental data and/or theoretical estimations for any of them.

■ In the Paper, relaxers parameters numerically are estimated using so-called MCF (Minimizing Comparison Function) method – by minimizing the comparison function, which means comparing the theoretical calculated dielectric loss tangent temperature-graphs and its experimentally measured spectra in the experimental maxima vicinity in two test materials. Consequently, obtained agreement between proposed theory and available experimental data does not proves this theory, but only demonstrates its possibility do explain such data.

Meanwhile, it can be imagined an alternative model to explain extra-high apparent dielectric permittivity in layered crystals spontaneously volume-charge polarizable near a phase transition point. On the one hand, planar defects, such as stacking faults and twins, are characteristic of crystals with layered structures – see e.g. [A. P. Rooney, et al. Nat, Commun., 2018, 9, 3597, 1-7, https://doi.org/10.1038/s41467-018-06074-8]: “For nearly six decades, graphite has been used as a textbook example of twinning with illustrations showing atomically sharp interfaces between parent and twin.” On the other hand, beta-rhombohedral boron, the not-layered but prone to the planar defects crystal, reveals [O. A. Tsagareishvili, et al. Semiconductors, 2009, 43, 1, 14-20, https://doi.org/10.1134/S1063782609010047] a “giant” apparent charge capacity related to stacking faults and twins and corresponding local phase transitions.

Therefore, it would be appropriate to add “4. Discussion” with a brief overview of why alternative physical mechanisms seem less likely.

Author Response

Dear colleague,

Our author team have made corrections according to your recommendations. 

Please, see the comments at the attached file 

Author Response File: Author Response.docx

Round 2

Reviewer 1 Report

Major revision have been made. The article can be published in the present form.

Reviewer 3 Report

In his reply, the corresponding Author has rigorously addressed almost all the Reviewer’s comments. So, Paper can be published in revised form.

 

Only a single problem remains unsolved: actually, the provided Keywords section is a “Keyphrases” section consisting not of separate words or, at least, composite terms of 2-3 words, but lengthy phrases. I don’t insist on their reformulation, but (1) doubt that it isn’t allowed by the Journal rules and (2) believe that not adequately formulated keywords will serve as a significant barrier for most potential readers to discover this Paper after it is published.

 

Let’s ask the Academic Editor to resolve the Keywords’ issue.