The thermoelectric conversion efficiency problem: Insights from the electron gas thermodynamics close to a phase transition

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).

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).

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 That means that some of my comments below may be out-of-date.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).

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.

Ask for minor revision

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.

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.

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.

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.

Publish (easily meets expectations and criteria for this Journal; among top 50%)