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The thermoelectric conversion efficiency problem: Insights from the electron gas thermodynamics close to a phase transition
by I. Khomchenko, A. Ryzhov, F. Maculewicz, F. Kurth, R. Hühne, A. Golombek, M. Schleberger, C. Goupil, Ph. Lecoeur, A. Böhmer, G. Benenti, G. Schierning, H. Ouerdane
This Submission thread is now published as
Submission summary
Authors (as registered SciPost users): | Giuliano Benenti · Ilia Khomchenko · Henni Ouerdane · Gabi Schierning |
Submission information | |
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Preprint Link: | https://arxiv.org/abs/2110.11000v3 (pdf) |
Date accepted: | 2024-10-14 |
Date submitted: | 2023-07-25 10:44 |
Submitted by: | Ouerdane, Henni |
Submitted to: | SciPost Physics |
Ontological classification | |
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Academic field: | Physics |
Specialties: |
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Approaches: | Theoretical, Experimental |
Abstract
The bottleneck in modern thermoelectric power generation and cooling is the low energy conversion efficiency of thermoelectric materials. The detrimental effects of lattice phonons on performance can be mitigated, but achieving a high thermoelectric power factor remains a major problem because the Seebeck coefficient and electrical conductivity cannot be jointly increased. The conducting electron gas in thermoelectric materials is the actual working fluid that performs the energy conversion, so its properties determine the maximum efficiency that can theoretically be achieved. By relating the thermoelastic properties of the electronic working fluid to its transport properties (considering noninteracting electron systems), we show why the performance of conventional semiconductor materials is doomed to remain low. Analyzing the temperature dependence of the power factor theoretically in 2D systems and experimentally in a thin film, we find that in the fluctuation regimes of an electronic phase transition, the thermoelectric power factor can significantly increase owing to the increased compressibility of the electron gas. We also calculate the ideal thermoelectric conversion efficiency in noninteracting electron systems across a wide temperature range neglecting phonon effects and dissipative coupling to the heat source and sink. Our results show that driving the electronic system to the vicinity of a phase transition can indeed be an innovative route to strong performance enhancement, but at the cost of an extremely narrow temperature range for the use of such materials, which in turn precludes potential development for the desired wide range of thermoelectric energy conversion applications.
Author comments upon resubmission
On behalf of all coauthors, I thank you for your time reviewing our work and the very useful remarks and criticism.
We believe that our manuscript contains much interesting physics about a very challenging problem, and we are grateful for an opportunity to revise our manuscript and resubmit it.
The work is essentially of theoretical nature but we also complemented it with experimental data to make a case for more efforts and attention on the problem of the electron gas fluctuating regimes in thermoelectricity. The manuscript should thus not be viewed as a "theory vs. experiment" report but rather as a work that provides theoretical grounds and experimental data to further explore fluctuating regimes near phase transition and their influence on thermoelectric energy conversion.
Our reply to the Referees' report is provided separately, just below their reports.
Sincerely,
Dr. Henni Ouerdane
List of changes
The main new change in our work is the discussion on nematic fluctuations to describe the experimental data.
To address the points raised by the Referees we revised our manuscript as follows.
1/ We added text and references on nematic fluctuations in the Introduction;
2/ We clarified the specific aims of our work in the Introduction;
3/ We expanded the Section 2 - Theory, to introduce and explain all the basic ingredients of our thermodynamic model;
4/ We modified our notations for the figures of merit to better distinguish those pertaining to the noninteracting electron gas and those pertaining to the 2D fluctuating Cooper pairs;
5/ We give more detail on the 2D fluctuating Cooper pairs, especially in the new Section 2.2.3;
6/ The new Section 3 is now about our results, numerical and experimental - and it contains parts of the former section 4;
7/ Parts of the former Section 3, which was dedicated to the experimental, notably former sections 3.1 and 3.2, have been moved to the appendix D;
8/ The new Section 4 is dedicated to our discussion, which now includes nematic fluctuations;
9/ The Section Conclusion has been expanded to provide a sharper recap of the work done, and include additional remarks to stress the importance of the fluctuating regimes and briefly indicate the potential for follow-up works as themoelectricity in fluctuating regimes near a phase transition is clearly a path to explore with dedicated experiments and the development of realistic models;
10/ Appendix A has been completed with detail on the parameters used for our numerical simulations.
11/ The bibliography section contains 17 new references.
Published as SciPost Phys. Core 8, 015 (2025)
Reports on this Submission
Report #2 by Anonymous (Referee 1) on 2024-6-11 (Invited Report)
- Cite as: Anonymous, Report on arXiv:2110.11000v3, delivered 2024-06-11, doi: 10.21468/SciPost.Report.9226
Strengths
This work merits publication., because I believe that the experiment is new, even if I recently discovered that nearly all the theory is already published in
"Enhanced thermoelectric coupling near electronic phase transition: The role of fluctuation Cooper pairs", Henni Ouerdane, Andrey A. Varlamov, Alexey V. Kavokin, Christophe Goupil, and Cronin B. Vining, Phys. Rev. B 91, 100501(R) (2015).
Weaknesses
The manuscript is not clear about whether there is good agreement between the theory and the experiment, or not. Or indeed, maybe there are regimes where they agree and regimes where they do not; the manuscript does not say.
It is odd to present the theory (already published elsewhere) in so much detail unless one wants to make careful quantitative fits between theory and experiment, extracting and explaining the relevant fitting parameters.
This weakness is immediately overcome if the authors add a fit of the theory to the experimental data for alpha in Fig 2, and include an explanation of how the fitting was done, what fitting parameters that reveals, and what those parameters imply (see point A of my report below).
Report
Before starting this report, I want to say that I fear that I am not reviewing the latest version of this manuscript ! This is because Henni Ouerdane's response to previous report 2 (at link: https://scipost.org/submissions/2110.11000v2/#comment_id3836) listed a bunch of changes in the manuscript, but those changes are NOT in the version that I have access to via the SciPost website. To be specific, the SciPost website directs me to the version here (https://arxiv.org/abs/2110.11000v3), but it that does not contain info that the response to previous report 2 said is there,
such as:
- Fermi energies such as E_F^{1D}= 0.94meV, E_F^{3D}=3.64meV, etc.
- Density of states for 0D.
- the energy dependence of the velocity.
Thus I suspect that this is not the most recent version of the manuscript. That means that some of my comments below may be out-of-date. However I give them all here, and suggest that the authors ignore any comments that are not relevant to the most recent version of the manuscript.
MAIN COMMENT:
I cannot recommend this work for publication until the authors address the following two comments.
(A) The experimental work is of value, but nearly all of the theoretical part (for normal state and fluctuating cooper pairs) seems to be reproduced from an earlier work by the same authors:
"Enhanced thermoelectric coupling near electronic phase transition: The role of fluctuation Cooper pairs,", Henni Ouerdane, Andrey A. Varlamov, Alexey V. Kavokin, Christophe Goupil, and Cronin B. Vining, Phys. Rev. B 91, 100501(R) (2015).
This reproducing of earlier theory would be worthwhile if the goal is to fit that theory to the experiment. However, the manuscript gives no indication how the theory fits the experiment. This must be rectified.
Fig 2 gives an experimental curve for thermoelectric response, alpha, but it does not appear to look like the theory given above Eq. (23), which predicts alpha_{cp} ~ ln[epsilon] = ln[ln[T/Tc]].
Is the reader supposed to understand that it is a more complicated function of T? If the experiment should be fitted by a sum of alpha_{cp} and alpha_{e}),
then the manuscript need to explain this.
In short, do the theories presented in the earlier sections describe the experiment in Fig 2 or not Provide the fit, and the fitting parameters, please!
Fig 3 shows theory curves on top of the experimental plot of alpha^2 sigma, where sigma is conductivity. The divergence of alpha^2 sigma is fitted by the theory, however I believe that this divergence is entirely due to the divergence of conductivity at T=Tc,
and nothing to do with the thermoelectric response, alpha. Thus this fit is a poor test of a theory trying to explain the thermoelectric response, it would be much more discriminating test to fit alpha in Fig 2.
(B) I find the discussion in section 4.2 is confusing, because it never mentions the contradiction between using Z_{th}T or Z_{e}T to define the thermodynamic efficiency of a thermoelectric machine.
It states that the thermodynamic efficiency is given by Eq. (25) which depends on Z_{th}T (recalling that gamma= 1+Z_{th}T). But the manuscript does not mention that this contradicts the standard approaches, which state that the thermodynamic efficiency is given by the same formula with Z_{e}T or Z_{cp}T INSTEAD of Z_{th}T or Z_{th,cp}T (see e.g. the review of Benenti et al (2017) = Ref [75] of this manuscript).
The earlier parts of the manuscript (specifically Fig 1) shows that Z_{th}T is different from ZT, and this difference is up to five orders of magnitude (for fluctuating Cooper pairs).
Hence, the formula in Eq. (25) will give completely different results from the standard formula.
So which is correct?
I would like the manuscript to discuss the contradiction,
and give arguments why one may be better than the other.
My personal opinion is that Z_{th}T and Z_{th,cp}T are WRONG for predicting the thermodynamic efficiency of a real thermoelectric system, and Z_{e}T and Z_{cp}T are CORRECT (assuming phonon effects are neglected).
However, I am not sure about this, so I would appreciate any arguments that the authors have about which one is correct.
MINOR COMMENTS ON THE PRESENTATION
Addressing the following comments will greatly help the reader. However, I leave it up to the authors to decide how to do so.
(1) There is no reference to appendix A in the main text. I recommend that the authors should add two equations to section 2.2.2, with Z_{th}T and Z_{e}T, stating that Z_{e}T is calculated in appendix A.
Placing these two equations, Z_{th}T and Z_{e}T, one after the other would allow readers to see the similarities and differences, and so allow them to appreciate Fig 1a (a figure which currently has no explanation).
(2) The manuscript does not give all parameters necessary to understand Figs 1 and 5. One essential difference between Z_{th}T and Z_{e}T is that
- Z_{th}T depends on the E dependence of the density of states g(E)
- Z_{e}T depends on the E dependence of tau(E) (v(E))^2 g(E)
Hence the shape of the plots in Figs 1 and 5 depend critically on the choice of energy dependence of tau(E) (v(E))^2. Different choices of tau(E) (v(E))^2 will give very different plots. Appendix A assumes that tau(E) is independent of E, which seems reasonable. However, the energy-dependence of v(E) is intimately related to g(E), but it also depends on the choice of band-structure (e.g. parabolic bands or something else). Hence it would really help the reader to explicitly explain this, state what assumptions are made (parabolic bands, or something else).
(3) If I assume parabolic bands, then
v(E) ~ E^{1/2} (as in H. Ouerdane's response to previous report 2)
d(E) ~ E^{(d/2-1}
Hence assuming tau(E) is E-independent as stated in appendix A, the term in the integrands for Z_{e}T is
tau(E) (v(E))^2 g(E) ~ E^{d/2},
while the equivalent term in integrands for Z_{th}T is
g(E) ~ E^{d/2-1}.
Thus, the integrands in the two quantities always differ by a single power of E. Hence, it seems paradoxical that Fig 5 shows a straight line indicating Z_{e}T=Z_{th}T in 1D, when the integrals are clearly different. The authors should explain the details of how that happens.
(3) The authors must add the explanation of 0D to the manuscript, the fact it is a quantum dot with a Lorentizian transmission, as explained in the response to the previous report 2.
(4)It is also worth adding a comment to the manuscript that mentions the other difference between Z_{th}T and Z_{e}T. The denominator of Z_{e}T contains an extract term proportional the square of the thermoelectric response (the L_{21}L_{12} term in equation for kappa_{e} in Eq. (31)). There is no analogue of this term in Z_{th}T.
(5) I cannot find the place in the manuscript that gives the value of the electro-chemical potential for the examples in Fig. 4 (these values were given in the response to previous report 2). These values are crucial to understand the curves in Fig 4,
because for given d(E) and v(E), the parameter that matters in the ratio of temperature to electro-chemical potential. For example, if one takes a sample with more charge carriers per unit volume so its electro-chemical potential is larger (e.g. larger Fermi energy), then one needs a larger temperature to achieve the same Z_{e}T (and hence achieve the same maximum thermoelectric conversion efficiency).
Requested changes
See my report.
Points A and B are strong recommendations, I cannot recommend publication before they are addressed.
The other points are optional, but they should be easy to do, and would be a great help to readers.
Recommendation
Ask for minor revision
Report #1 by Alexei Vagov (Referee 3) on 2024-5-22 (Contributed Report)
- Cite as: Alexei Vagov, Report on arXiv:2110.11000v3, delivered 2024-05-22, doi: 10.21468/SciPost.Report.9108
Strengths
1. The manuscript contains a fairly large amount of theoretical work and some experimental data also.
2. A basic thermodynamic analysis that gives very good insights into the thermoelectric conversion process as the thermoelastic properties and the transport properties are related. Parametric plots are shown.
3. A theoretical demonstration that in a 2D system near Tc the Wiedeman-Franz law is violated and that the power factor shows a diverging behavior as T goes closer to Tc.
4. Very interesting experimental results showing how the Seebeck coefficient and the electrical resistivity vary with temperature and that their combination into the power factor also diverges near Tc, thus showing a violation of the Wiedeman-Franz law as predicted by the model.
5. The experimental data is shown for a sample with high structural quality and the same sample with degraded structure because of ion bombardment.
6. An interesting discussion about the efficiency in "ideal cases" for simplified electron gas models. Only for the fluctuating regime near Tc, the efficiency can rise to Carnot efficiency.
7. A link between the compressibility of the electron gas and how it is efficient when it increases, which is the case when the gas shows density fluctuations.
Weaknesses
1. The lack of a model for thermoelectric conversion in the nematic fluctuation regime is a disadvantage for the manuscript, but given the complexity of the physical phenomena, such a model could be the object of a full separate work.
2. The manuscript is quite long and has several useful appendices. This shows that the authors want to give the reader as much detail as possible. That can make the reader lose sight of the main results, but given the complexity of the problem, a long text is unavoidable.
Report
The main idea of the manuscript is that the fluctuation regime close to a phase transition is beneficial for thermoelectric conversion efficiency. The thermodynamic analysis aims to explain why the transport coefficients near a phase transition allow for better thermoelectric transport. Some of the Authors already published the idea thermoelectric conversion near a superconducting phase transition in Ref. [15]. In their previous work, they focused on the thermoelastic properties of the conduction electron gas and showed that a quantity they call the thermodynamic figure of merit, which is more or less the isentropic expansion factor, diverges near the transition point. As for classical heat engines, using a working fluid that has a high heat capacity ratio, is beneficial for the heat-to-work conversion. In the mansucript, the authors relate the thermoelastic properties to the transport properties, with a focus on the power factor and the electronic zT (figure of merit without the contribution of the phonons to the thermal conductivity).
The manuscript contains a fairly large amount of theoretical work and some experimental data also. The authors explained in their replies to previous reviewers that the work is not a theory vs experiment, but a theoretical work to which experimental data has been added not to support the calculations but to support the idea that fluctuating regimes can enhance the conversion efficiency. The mathematical model is developped for two-dimensional fluctuating Cooper pairs very close to the superconducting transition temperature Tc and cannot be applied to interpret the experimental data that cover a wide temperature interval, and which shows an interesting behavior largely above Tc. The experimental data results from the measurement of the Seebeck coefficient and the electrical conductivity in a pnictide thin film with 100 nm thickness, which while not bulk is not 2D either. The authors suggest that the increase of the power factor is due to nematic fluctuations for which they provide no model to compute the thermoelastic properties and the transport properties. The suggest nonetheless that very close to Tc superconductive fluctuations may play a role.
The lack of a model for thermoelectric conversion in the nematic fluctuation regime is a disadvantage for the manuscript, but given the complexity of the physical phenomena, such a model could be the object of a full separate work.
What are we left with after reading the manuscript on the positive side:
- a basic thermodynamic analysis that gives very good insights into the thermoelectric conversion process as the thermoelastic properties and the transport properties are related. Parametric plots are shown.
- a theoretical demonstration that in a 2D system near Tc the Wiedeman-Franz law is violated and that the power factor shows a diverging behavior as T goes closer to Tc.
- very interesting experimental results showing how the Seebeck coefficient and the electrical resistivity vary with temperature and that their combination into the power factor also diverges near Tc, thus showing a violation of the Wiedeman-Franz law as predicted by the model.
- the experimental data is shown for a sample with high structural quality and the same sample with degraded structure because of ion bombardment.
- an interesting discussion about the efficiency in "ideal cases" for simplified electron gas models. Only for the fluctuating regime near Tc, the efficiency can rise to Carnot efficiency.
- a link between the compressibility of the electron gas and how it is efficient when it increases, which is the case when the gas shows density fluctuations.
On the negative side: the lack of a model to better support the description and interpretation of the experimental data.
The manuscript is quite long and has several useful appendices. This shows that the authors want to give the reader as much detail as possible. That can make the reader lose sight of the main results, but given the complexity of the problem, a long text is unavoidable.
I believe that thermoelectricity with phase transitions in the conduction electron gas can provide new valuable theoretical problems to consider. The nematic phase transition and the fluctuating regime can be the object of interesting works. From the experimental viewpoint, this work can also inspire new work where for example very thin films or 2D materials are studied.
The authors might comment on this recent paper: Nat Commun 15, 776 (2024) doi: 10.1038/s41467-024-45093-6 by Zhao et al. where the authors work on the modeling of critical thermoelectric transports.
I recommend publication of the manuscript as the physics is interesting and well discussed, and because it provides good ground for future theoretical works and perhaps experimental work, which will fill the gaps of this manuscript.
Requested changes
The authors might comment on this recent paper: Nat Commun 15, 776 (2024) doi: 10.1038/s41467-024-45093-6 by Zhao et al. where the authors work on the modeling of critical thermoelectric transports.
Recommendation
Publish (easily meets expectations and criteria for this Journal; among top 50%)
Author: Henni Ouerdane on 2024-10-15 [id 4868]
(in reply to Report 1 by Alexei Vagov on 2024-05-22)
Dear Professor Vagov,
On behalf of my coauthors, I thank you for your supportive comments and recommendation for publication. Here, I provide a reply to the points your raised.
Concerning weaknesses:
- The lack of a model for thermoelectric conversion in the nematic fluctuation regime is a disadvantage for the manuscript, but given the complexity of the physical phenomena, such a model could be the object of a full separate work.
2. The manuscript is quite long and has several useful appendices. This shows that the authors want to give the reader as much detail as possible. That can make the reader lose sight of the main results, but given the complexity of the problem, a long text is unavoidable.
Our reply:
1.We see that more as a limitation of the scope of our work than a disadvantage though we understand that having such a model would strengthen the manuscript. In fact, while we could perform calculations using the works of Varlamov and Larkin [Theory of Fluctuations in Superconductors, Oxford University Press (2005)] for 2D fluctuating Cooper pairs systems, establishing a transport theory in the nematic fluctuation regime is indeed a task beyond the scope of the present work, which initially was meant to be restricted to a thermodynamic regime very close to $T_{\rm c}$ and away in the normal regime where models of the noninteracting electron gas are applicable. We agree that there is a need to have a dedicated model able to describe the experimental data we have above $T_{\rm c}$, beyond the superconducting fluctuating regime and where noninteracting electron gas models are no longer adequate, like in the nematic fluctuation regime. Instead, we described the temperature-dependent Seebeck coefficient and electrical conductivity data, and discussed their behavior assuming that nematic fluctuations play an important role. While this is insufficient for a fully-fledged theory vs experiment type of paper, our manuscript is not quite of such a type and already contains much physics, simulation and experimental results, which allow to explain that fluctuating regimes close to a phase transition can foster a sizeable increase of the thermoelectric energy conversion efficiency.
2.The first version v1 (available on arXiv) was much shorter and as we wrote the version v2, we saw a need to extend the paper to provide more information in the main text for clarity, referring to our previous work [Phys. Rev. B 91, 100501(R) (2015)]. It turned out that one of the reviewers of the version v2 did not find the thermodynamic approach clear enough and considered this as a weakness [Report 2 on 2022-9-2]:
Theory part is badly explained. I have no reason to think there is anything wrong, but certain crucial information is missing [...],
so we extended the presentation of the theory part. As we rewrote the manuscript, we attempted to produce a text that would not be misleading in the sense that the paper is not an actual theory vs experiment one, and we meant to highlight the limit of validity for the use of the 2D fluctuating Cooper pairs model. To ease the reading, a detailed table of content is available just below the abstract so that any reader at a glance can get a overall grasp of the work we report in the manuscript.
Requested changes
The authors might comment on this recent paper: Nat Commun 15, 776 (2024) doi: 10.1038/s41467-024-45093-6 by Zhao et al. where the authors work on the modeling of critical thermoelectric transports.
Our reply
Thank you for mentioning this very recent paper that also considers that critical phenomena can be beneficial for thermoelectric energy conversion. The authors claim that a quantitative and comprehensive model of thermoelectricity close to a phase transition is still lacking, and rightly so. Their work is a contribution to fill that gap with a strong focus on the provision of formulas to compute transport coefficients using the Landau theory and the Boltzmann transport equation to make their model tractable. They get quantitative results but not a comprehensive model. In their work, the authors consider structural phase transitions and the effects of band broadening and carrier-soft TO phonon interactions on the Seebeck coefficient and on the electrical conductivity respectively. The lambda shape they obtain for the Seebeck and the electrical conductivity, with a peak at the critical point confirms that phase transitions foster the desired transport properties for thermoelectric energy conversion efficiency. Interestingly, the band broadening is caused by structural fluctuations near the critical phase transition temperature.
In our work, we tackle the problem of thermoelectricity close to a phase transition of the conduction electron gas (working fluid) with a thermodynamic analysis. Unlike in the work reported in Nat Commun 15, 776 (2024), we do not consider structural phase transitions but the fluctuation regimes in the electron gas above the superconducting and the nematic phase transition temperature. We seek to find what regime can enhance the electron gas isentropic expansion factor first, and we try and see if that correlates with an improvement of the transport coefficients. While in the paper mentioned above they calculate zT
accounting for the thermal conductivity of the lattice, in our work we consider the power factor. Our theoretical model shows that in the fluctuating regime in the close vicinity of the superconducting phase transition the power factor can be greatly enhanced. Our experimental results, for which there is a need for a dedicated model, show that in the nematic fluctuating regime the power factor is enhanced too. The isentropic expansion factor increase is due to the increase of the compressibility of the electron gas in the fluctuating regime.
To summarize, while the types of critical phenomena considered in our manuscript and in Nat Commun 15, 776 (2024), are different, the rationale is the same: placing a thermoelectric material in a thermodynamic regime close to a phase transition can favor a strong enhancement of its energy conversion performance due to the influence of critical phenomena on the transport coefficients. Note that it would be interesting to compute the temperature dependence of the isentropic expansion factors of the electron gas in Cu2Se and in Cu2Se{1-x}Sx, accounting for band broadening and carrier-soft TO phonon interactions and see theoretically how each of these two phenomena influence it when combined and when separated.
We now cite this paper in the Introduction section.
Author: Henni Ouerdane on 2025-01-31 [id 5172]
(in reply to Henni Ouerdane on 2024-10-15 [id 4868])
Dear Referee,
Please note that for unclear reasons, the reply that I posted on October 15, 2024, were vetted only on January 30, 2025. I have no idea why it took 3,5 months for the vetting to be done, even more so that I sent reminders.
Further, the system did not allow me to submit the revised version that contains all the changes according to your comments and criticism. The revised version is v4 on the arXiv website, which was posted on October 14, 2024.
If you would like to read the revised version, please consider the latest arXiv version (October 14, 2024) rather than the previous one you already reviewed accessible on the SciPost website.
Thank you again for your time and useful reviewing work. I am glad that at long last, you may read my reply.
Sincerely,
Henni
Author: Henni Ouerdane on 2024-10-15 [id 4869]
(in reply to Report 2 on 2024-06-11)Dear Referee,
Thank you very much for your report, comments and criticism. I provide our reply below.
Weaknesses
Reply:
1.The main hypothesis of our work is that fluctuating regimes near a phase transition foster a sizeable increase of the thermoelectric conversion efficiency. We show this considering on the one hand a theoretical model of a 2D fluctuating Copper pairs gas close to $T_{\rm c}$, and on the other hand with experimental data that we aim to fit only close to $T_{\rm c}$ where the model is supposed to be applicable. As explained in the main text, we are unable to account for the observed trends for the Seebeck and power factor starting from 50 K down to $\approx T_{\rm c}$. Our interpretation is that nematic fluctuations can play a role here.
Hence, since its initial version v1, available on arXiv, the model, based on a simple tractable approach, is meant to show, considering the superconducting phase transition, that critical phenomena may be of interest to increase the thermoelectric conversion efficiency; and it is used to partly interpret the observed behavior of the power factor obtained from the experimental data \emph{only} in a very restricted temperature range $[T_{\rm c}; T_{\rm c} + \delta T]$, with $\delta T \ll 1$ K, where the fluctuation regime extremely close to the superconducting phase transition.
The 2D fluctuating Cooper pair model cannot be used to explain the power factor growth with the temperature decrease, from its onset at around 50 K down to temperatures close to $T_{\rm c}$ but outside the superconducting fluctuating regime, and we do not claim it can. That is why in v3, we discuss the possibility of the effects of the nematic fluctuations to describe the observed behavior, though we currently do not have a model of transport coefficients in the nematic fluctuating regime.
2.The version v1 was much shorter than the subsequent ones as the theory part was meant to include a a brief recap of previous works (which we systematically cited) together with comments aiming at clarity, as well as new theory content on the correlation between the thermoelastic properties of the electron gas and its transport properties, and how this relates to thermoelectric conversion performance evaluation. Some details were either in appendixes or in the cited papers, notably in [Phys. Rev. B 91, 100501(R) (2015)]. We decided that for ease of reading of the theoretical part, the main text in the manuscript should be completed with more theoretical elements in the version v2. However, it turned out, as explained in our reply to Prof. Vagov's comments [Report on 2024-5-31], that one of the reviewers of the version v2 did not find the thermodynamic approach clear enough, and considered this as a weakness [Report 2 on 2022-9-2]:
"Theory part is badly explained. I have no reason to think there is anything wrong, but certain crucial information is missing [...]",
so we reshaped and extended the presentation of the theory part in the version v3, hence your comment:
"I recently discovered that nearly all the theory is already published."
That said, it is important to notice that in [Phys. Rev. B 91, 100501(R) (2015)], the focus is only on the thermoelastic properties of noninteracting electron systems and those of the 2D fluctuating Cooper pairs. In the current work, which extends the scope of the previous one, we aim to understand what thermodynamic conditions may favor the desired transport properties and we establish a correlation between the thermoelastic properties and the transport properties: the larger $Z_{\rm th}T$, the larger $zT$, which confirms the hypothesis that a large electronic isentropic expansion factor favors the energy conversion efficiency in thermoelectric systems. We also introduce an analogue of the isentropic expansion factor $\gamma_{\rm tr}$ for an out-of-equilibrium system. The new theoretical material is definitely not marginal.
As shown in Fig. 3 of the manuscript (version v3), the power factor obtained from the measurement data undergoes a very steep increase as the temperature $T$ decreases and approaches $T_{\rm c}$. The 2D fluctuating Cooper pair model, which is only valid for $T$ close to $T_{\rm c}$ yields a power factor $\sigma_{\rm cp}\alpha_{\rm cp}^2$ proportional to $\ln^2(\varepsilon)/\varepsilon$, with $\varepsilon = (T - T_{\rm c})/T_{\rm c}$. Again, in the limit of validity of the model, i.e. as $\varepsilon \rightarrow 0$, the calculated power factor diverges. On Fig. 3 of v3, we indicate with an arrow the quasi vertical line, which has been computed with the formula of the power factor $\sigma_{\rm cp}\alpha_{\rm cp}^2$ scaled with the value of the power factor at 300 K (which is obtained using the 2D electron gas model at room temperature)..
3.The only fit of the Seebeck coefficient that we can do to a good approximation is for $T\approx T_{\rm c}$ using the following formula
And for temperatures $T$ in the non-degenerate regime, we can approximately fit the Seebeck coefficient with the formula
Main Comments
Preamble
Reply: Formulas and data are now provided in full in the Appendix A.
Point (A):
Reply:
1.We explained above how and why the writing of the theoretical section evolved since v1. Note that all notions and results already introduced and discussed in [Phys. Rev. B 91, 100501(R) (2015)] have been systematically referred to the 2015 paper, in the manuscript versions v1, v2, and v3. The theoretical model has also been more thoroughly presented and, importantly, extended in relation to the transport properties and conversion efficiency calculations, which were missing in the 2015 paper.
The main goal of the theory part is two-fold: 1/ to show with a thermodynamics approach that no material and no system for which noninteracting electron gas models provide a good description, can be good candidates for high-efficiency thermoelectric conversion even under ``ideal'' working conditions (no detrimental phonon effects and no other source of dissipation); 2/ to show that harnessing thermoelectric energy conversion near critical phenomena, like, e.g., the superconducting fluctuating regime in 2D systems, could be a promising venue in terms of efficiency. Calculations show that for the latter, efficiency can be high, i.e. approach the Carnot efficiency, albeit in a very restricted range of temperature.
The analysis of the experimental data across the full temperature range, from the critical temperature $T_{\rm c}$ to 300 K, necessitates more sophisticated models to account notably for the band structure, the electronic density of states, interaction with phonons, and the thickness of the thin film (which does not make it a 2D system, but not bulk either). In the manuscript, we suggest an interpretation of the increase of the power factor, which stems from the increase of the thermoelectric coupling: below 50 K nematic fluctuations can play a role, and close to the superconducting phase transition fluctuating Cooper pairs also play a role, albeit in a very small temperature range above $T_{\rm c}$.
As explained in previous replies, our work is based on the hypothesis that fluctuating regimes can be beneficial for thermoelectric conversion efficiency. We showed this, focusing on the electron gas alone with a tractable theoretical model of 2D fluctuating Cooper pairs valid in a very restricted temperature range, and, independently, with experimental results that also indicate that nematic fluctuations could play a role well above $T_{\rm c}$. As mentioned in the Conclusion section, we relate our calculations and interpretations to the fluctuation-compressibility theorem: the larger the fluctuations are, the larger the compressibility of the working fluid is, and, in turn, the larger the isentropic expansion factor. The thermodynamic analysis adapted to the thermoelectric problem shows that to increase the figure of merit $zT$, one has to find the conditions for the electron gas to be a good working fluid.
2.We may provide a fit of the Seebeck coefficient $\alpha$ only in a very narrow range above $T_{\rm c}$, where we assume that fluctuating Cooper pairs play a dominant role. Given the scale of Fig. 2, the simulated curve would appear as a very small vertical segment near $T_{\rm c}$. A zoom on that part would simply reveal a logarithmic behavior. The formula is given in the main text, section 2.4: $\alpha_{\rm cp} = \nabla \mu/q\nabla T = \alpha_{\rm GL} k_{\rm B} \ln\varepsilon/2e$, valid in the $[T_{\rm c}; T_{\rm c} + \delta T]$, with $\delta T \ll 1$ K and the parameters in the Appendix A. A realistic fit beyond $T_{\rm c} + \delta T$ necessitates to compute $\alpha$ using the system's electronic band structure and density of states. Clearly, the ferropnictide thin film experimentally studied in the manuscript cannot be described by a simple noninteracting electron gas model with parabolic bands. That said, one key simulation result is the quasi-vertical line over a tiny temperature range in the vicinity of $T_{\rm c}$ (our temperature regime of interest) in Figure 3 as we want to highlight the influence of critical phenomena on the power factor.
3.We disagree with you here. Figures 2 and 3 show experimental results across a large temperature range for $\alpha(T)$, $\rho(T)\equiv 1/\sigma(T)$, and $\sigma(T)\alpha^2(T)$, under two thin film sample structural conditions: high-structural quality, and low-structural quality after ion bombardment. The degraded sample shows in Fig. 2b that as $T\rightarrow T_{\rm c}$, $\rho(T) \rightarrow 0$, and hence that $\sigma(T)$ diverges while $\alpha(T)$ barely increases in magnitude before saturation. On Fig. 3, the degraded sample shows that the power factor $\sigma(T)\alpha^2(T)$ remains finite too in spite of a resistivity that goes to 0 as $T\rightarrow T_{\rm c}$. Conversely, Fig. 3 shows that the power factor for the high-structural-quality sample increases very largely when as $T\rightarrow T_{\rm c}$, which is, as shown on Fig. 2, thanks to the large increase of the Seebeck (which is modelled theoretically with a formula that shows a diverging as $T\rightarrow T_{\rm c}$). If the Seebeck had no effect on the observed behavior, one could suppose that the diverging resistivity alone would allow for a steep increase of the power factor, which is not the case here. Therefore the effect of the large increase of the Seebeck coefficient on the power factor, is shown by the measurement data by direct comparison of the high-structural quality and the low-structural quality cases. The theoretical fit close to $T_{\rm c}$ is relevant. The observed behavior has indeed something to do with the thermoelectric response in the close vicinity of $T_{\rm c}$.
Point (B)
Reply:
1.The quantities $Z_{\rm th}T$ and $Z_{\rm th,cp}T$ are not contradictory to $Z_{\rm e}T$ and $Z_{\rm cp}T$ respectively, but perfectly complementary. We developed a thermodynamic approach to gain more insights into the thermodynamic conditions that may favor a significant increase of the thermoelectric conversion efficiency. Using $Z_{\rm th}T$ and $Z_{\rm th,cp}T$ for a study of the electron gas, here considered as a working fluid, is definitely appropriate. Saying that this is wrong amounts to stating that the isentropic expansion factor is a meaningless quantity. In fact, the larger $Z_{\rm th}T$ and $Z_{\rm th,cp}T$ are, the less entropy is produced during the energy conversion, which is exactly what one wants. The analysis of $Z_{\rm th}T$ and $Z_{\rm th,cp}T$ points to the most favorable conditions to minimize entropy production. Once these are known, one may study transport and conversion under these conditions. The correlations between $Z_{\rm th}T$ and $z_{\rm e}T$ on the one hand, and $Z_{\rm th,cp}T$ and $z_{\rm cp}T$ on the other hand, depicted in Fig.~1, show that all these quantities are not contradictory.
2.While the maximum theoretical efficiency which can be reached can be calculated with the traditional formula, we show that from the thermodynamic viewpoint, knowledge of the thermoelastic coefficients of the working fluid, here the electron gas, yields the same ranges of efficiency. In fact, the formulas for the maximum efficiency using $Z_{\rm th}T$ and $Z_{\rm th,cp}T$ on the one hand, and $Z_{\rm e}T$ and $Z_{\rm cp}T$, are formally the same -- See Eqs. (25) and (26) of the revised manuscript. This is shown and discussed in the revised Section 4.2. The results are very similar as shown on the revised Fig. 4.
3.As discussed just above the use of $Z_{\rm th}T$ and $Z_{\rm th,cp}T$ is not wrong at all; it gives additional insights into the physics of thermoelectric energy conversion. Both approaches are correct. We hope that the above clarifies the matter.
Minor comments on the presentation
Reply: Actually, there was a reference to Appendix A in the main text, page 9, in the sentence just above the section Results. But following the comment, we found it better to rather have references to Appendix A, below Eq. (2), below Eq. (15) and in the caption of Fig.~1.
$Z_{\rm th}T$ for the systems of noninteracting electrons is already given by Eq. (21) in Section 2.3 devoted to the thermodynamic figure of merit, and the formulas of the thermoelastic coefficients to calculate it given in Eqs. (13), (14) and (15) in Section 2.2.2 where it is announced that they will be used for that purpose.
$z_{\rm e}T$ is also already given by Eq. (2) in the preamble of the section Theory. We note though that we could have referred to Appendix A to indicate that expressions for the transport coefficients are given there. This is now done. We have also modified Eq. (2) including a definition of $z_{\rm e}T$ with the Seebeck coefficient and the Lorenz number.
Just below Eq. (21) we mention the thermoelastic counterpart $\ell$ of the Lorenz number $L$ and discuss their physical interpretation as $\ell$ enters the definition of $Z_{\rm th}T$ and $L$, the definition of $z_{\rm e}T$ as shown in the main text: $Z_{\rm th}T = \alpha_{\rm th}^2/\ell$ and $z_{\rm e}T = \alpha^2/L$.
In the case of the 2D fluctuation Cooper pairs, $Z_{\rm th,cp}T$ and $z_{\rm cp}T$ are given by Eqs. (22) and (23) and one sees that the main difference is in the dependence on $\varepsilon$, the latter having a prefactor $1/\varepsilon$. We also comment on it now in the text below Eq. (23).
Reply: The information can now be found in the revised Appendix A. Yes, it is true that $Z_{\rm th}T$ being defined from the thermoelastic properties, depends only on the density of states as it characterizes the thermostatic properties of the electron gas, while $z_{\rm e}T$ also depends on $v(E)$ and $\tau(E)$ thus reflecting the transport properties. We wrote a small remark below Eq. (15) to highlight this.
Reply: We thank the reviewer for this important observation. Indeed, we initially plotted the results for a quantum chain model based on the 0D model. Then we modified the text without updating the figure, hence the confusion. Following this comment, we replaced the previous Fig.~5 with a new one obtained using the 1D electron gas in the Boltzmann approach. The new plot now displays a nonlinear correlation between $Z_{\rm e}T$ versus $Z_{\rm th}T$, as indeed should be expected.
Reply: The necessary details have been added to the Appendix A.
Reply: The Onsager kinetic coefficients reflect the transport properties of the electron system in an out-of-equilibrium situation because of, e.g., a temperature difference. Hence they are needed to compute $z_{\rm e}T$. As regards $Z_{\rm th}T$, these coefficients are irrelevant as the thermodynamic figure of merit characterizes the quality of the working fluid, since it is akin to the isentropic expansion factor, which can be defined using the thermoelastic coefficients.
Reply: For the chemical potentials we used analytical expressions given in Ref. [71]. Note that for the 1D and 0D cases, we used the numerical values obtained for the calculation of the 2D electrochemical potential. Now, as regards the remark on the impact of the carrier concentration on the value of $z_{\rm e}T$, indeed it decreases since with an increase of carrier concentration, the Lorenz number remains nearly constant (because of the increase of electrical conductivity and of the thermal conductivity too) but the Seebeck coefficient (or entropy per particle in thermodynamics) decreases (low carrier concentrations favoring a higher Seebeck coefficient).
Requested changes
Reply: We thank the referee for a thorough reading and the useful criticism provided in the report. We have addressed points A and B hoping to clarify any confusing point and showing the added value of our work. We also addressed all the minor points raised in the report, which we treated as seriously as the major ones. We hope we have lifted all possible points of confusion and clarified the rationale of our approach.
The manuscript contains much information and though we endeavored to give a coherent and clear account of the work, we understand that it is not straightforward to read. That said, we hope that the work will attract attention and trigger new experiments as well as (and importantly) theoretical works which will address the problems with less simplifying assumptions, which will allow to go deeper in the physics of thermoelectricity close to a phase transition.