Thermodynamics of the thermoelectric working fluid close to the superconducting phase transition

We thank the Referee for an overall positive evaluation. We also take good note of the criticism and the various points raised, which we address in our reply.

EXPERIMENT

We appreciate a very in-depth look into and discussion of our experimental data. Since the manuscript has meanwhile become relatively long, in particular also due to necessary additions by the revision, we have not further deepened these very interesting aspects that can be considered in future works.

THEORY

(A) Confusion between $ZT$ and $Z_{\rm th}T$

Our reply:

The traditional notations used for the thermoelectric figure of merit are usually $ZT$ or $zT$, and they include the thermal conductivity of the lattice in the definition. In our work, as in some others where authors adapt their notations accordingly, we focus solely on the conversion performance of the electron gas. To avoid confusion we adopt the following notations in the revised version of the manuscript:

* $zT$ for the standard thermoelectric figure of merit, given by Eq. (1) of the manuscript;

* $z_{\rm e}T$ for the thermoelectric figure of merit \emph{disregarding} the phonon effects (i.e. without $\kappa_{\rm lat}$). It is a measure of the electron gas performance alone. Hence, the quantity $z_{\rm e}T$ includes only the Seebeck coefficient, the electrical conductivity, and the electronic thermal conductivity; and, of course, $z_{\rm e}T > zT$. In fact, in a given material, $z_{\rm e}T$ is an upper bound for $zT$ that would be reached in a situation where the so-called ``phonon glass'' (absence of heat transport by the lattice) would be managed. It is given by Eq. (2) of the revised manuscript;

* $Z_{\rm th}T$ is what we call the thermodynamic figure of merit defined from the electron gas heat capacity ratio, in Eq. (21). In the present work, we related the thermoelastic properties of the electrons to their transport properties -- see the parametric plots $z_{\rm e}T$ vs $Z_{\rm th}$ in Fig. 1 of the revised manuscript (Fig. 4 of the former version);

* $z_{\rm cp}T$ is the figure of merit of the 2D fluctuating Cooper pairs system, when the system's temperature $T$ is in the close vicinity of $T_{\rm c}$. It is given by Eq. (23);

* $Z_{\rm th,cp}T$ is what we call the thermodynamic figure of merit defined from the 2D FCP system's heat capacity ratio. It is given by Eq. (22).

(B) Missing information for $ZT$

Our reply:

In Fig. 4 of the former version of the manuscript (now Fig. 1), the $ZT$s'' and the$Z_{\rm th}T$s'' are those computed with the transport coefficients and thermoelastic coefficients of the 2D electron gas and the 2D FCP respectively. All the transport and thermoelastic coefficients used for the computation of the electron gas $ZT$'' (now $z_{\rm e}T$ and $Z_{\rm th}T$) are, indeed, completely different from those used to compute the 2D FCP$ZT$'' (now $z_{\rm cp}T$) and $Z_{\rm th}T$ (now $Z_{\rm th,cp}$). The same applies for the power factors in Fig. 5 of the former version of the manuscript (now Fig. 3), though we kept a generic notation as the arrows in the plot clearly indicate to what model the curves correspond.

The transport distribution function $\Sigma(E)$ has three terms: the relaxation time $\tau$, which for simplicity we take as constant, the velocity $v \propto \sqrt{E}$, and the density of states $g(E)$ for noninteracting electron systems. More detail is now given in Appendix A of the revised version.

For our numerical simulations, we consider low-density electron gases with concentrations $n_{\rm 3D}=10^{18} \mathrm{cm}^{-3}$, $n_{\rm 2D}=10^{12} \mathrm{cm}^{-2}$, and $n_{\rm 1D}=10^{6} \mathrm{cm}^{-1}$ for the three-, two-, and one-dimensional systems, respectively.

(i) The temperature ranges from close to $0$ K to $300$ K for the numerical calculations of the electron gases, and from $T_{\rm c}$ to $T_{\rm c}$ + 0.2 K for the 2D FCP.

(ii) The Fermi energy for the two-dimensional electron gas, is $E_{\rm F}^{\rm 2D}=2.39$ meV. The value of the Fermi energy in the three-dimensional case is $E_{\rm F}^{\rm 3D}=3.64$ meV, and $E_{\rm F}^{\rm 1D}=0.94$ meV for the one-dimensional case equals.

(iii) We use a single-level quantum dot to model 0D electron gas and a Lorentzian form for the density of states:

$g_{0D}(E) = \frac{\Gamma}{(E - E_{0})^{2} +(\Gamma/2)^{2}}$

where $\Gamma$ is the energy level width. The central energy of the channel is $E_{0}=0.99E_{F}^{\rm 2D}$, and the channel coupling energy $\Gamma = 0.1k_{B}T$.

(C) Missing information for $Z_{th}T$ for fluctuating Cooper pairs (FCP).

Our reply:

In the revised version, we transferred parts of the appendices of the previous version to the main text. In fact, as this work contains much material that pertains to thermodynamics, materials science, coupled transport, modeling, experiments etc., we wanted, for ease of reading, to avoid an overload of definitions and formulas that can be found in the cited published references.

To answer more specifically the point on the $Z_{\rm th}$ for the 2D FCP, we use and adapt the definition of $\gamma$ in Eq. (21) of the revised version, which explicitly reads:

\[ \gamma_{\rm cp} = 1 + \frac{\alpha_{\rm th,cp}}{\ell_{\rm cp}} = 1 +Z_{\rm th,cp}T \]

\noindent where $\ell_{\rm cp} = C_{N_{\rm cp}}/(q^2\chi_{T_{\rm cp}} T) = -C_{\rm GL} k_{\rm B}^2\ln\epsilon/q^2$, with $C_{\rm GL} = \hbar^2/(2m_{\rm cp}k_{\rm B}T_{\rm c}\xi^2)$ a dimensionless parameter in the Ginzburg-Landau free energy functional. Here $\xi$ is the coherence length. This is how we get to $Z_{\rm th,cp}T = \gamma_{\rm cp} - 1 = -\ln\epsilon$. See also the calculations shown in the appendix of Ref. [15] of the manuscript (Ouerdane et al PRB, 2015).

(D) Physics of fluctuating Cooper pairs

Our reply:

In the new section 2.2.3 of the revised manuscript, we give more information on the nature of the fluctuating Cooper pairs. The physics of fluctuating Cooper pairs is introduced and discussed at length in Ref. [42] of the revised manuscript: A. Larkin and A. Varlamov, Theory of Fluctuations in Superconductors, revised edition, (Oxford Science publications, 2009).

(E) Condition for neglecting lattice phonons $kappa_{lat}$

Our reply:

Interesting point here, which we now discuss in the new Section 2.4. Though lattice phonons are to be considered for a complete evaluation of a thermoelectric material's performance, they are primarily a cause of heat leaks, hence energy loss. It has been stated many times in thermoelectricity papers that the ideal thermoelectric material would behave as an electron crystal–phonon glass''\footnote{This paradigm was first expressed in G. A. Slack, CRC Handbook of Thermoelectrics, edited by D. M. Rowe (CRC Press, Boca Raton, FL, 1995), p. 407.} system, meaning high electrical conductivity and very low, nay zero thermal conductivity. So, more specifically, in the present work where we relate the thermoelastic properties to the transport properties of the electronic working fluid alone in Fig. 4 of the former manuscript (now Fig. 1 in the revised version with the new notations), we evaluate what the maximum conversion efficiency can ideally be in absence of heat leaks. In the case of 2D FCP, yes $\kappa_{\rm cp} \rightarrow 0$ in the limit $T \rightarrow T_{\rm c}$, but this does not pose any problem, quite the contrary in fact, as this shows what happens if the system would tend to become like anelectron crystal'', making $z_{\rm cp}T$ larger. Sure, accounting for $\kappa_{\rm lat}$ would negatively impact on the increase of $z_{\rm cp}T$, but here we see that without heat leaks, the conversion efficiency can tend to Carnot's efficiency in the limit $T\rightarrow T_{\rm c}$. The range over which the numerical calculations are performed in the normal phase is $\left]T_{\rm c} = 25.6 ~;~ 300 \right]$ K, and in the superconducting fluctuation regime $\left]T_{\rm c}~;~ T_{\rm c}+\delta T\right]$, with $\delta T = 0.2$ K.

First part of the comments by the Referee -- overall assessment and basic criticism

Our reply:

We thank the Reviewer for a very critical assessment of our work, which is useful , as we have sought to find a plausible interpretation of the experimental signatures we have found, especially the observed rise of the magnitude of the Seebeck coefficient starting near 50 K. In particular, we involved in these discussions a recognized expert on this material, Prof. Dr. Anna E. Boehmer, who is now also co-author of this paper. After intensive discussions, our interpretation is that the experimental signature in the Seebeck coefficient away from $T_{\rm c}$ could well be due to nematic fluctuations in the electron subsystem.

As a matter of fact, there is no universal model to simulate the thermoelectric properties of pnictide thin films across a wide range of temperatures. The approaches are essentially phenomenological. In our work we developed a thermodynamic model to discuss a particular experimental feature: the divergence of the power factor very close to $T_{\rm c}$, and in principle, the motif of fluctuations in the electron subsystem, at least in its phenomenology, translates very well from fluctuating Cooper pairs to nematic fluctuations. However, one has to say that for fluctuating Cooper pairs there have been established models for decades, whereas the mathematical description of nematic fluctuations is not yet at this established level. We have therefore chosen a compromise for the paper. In the theoretical description, we now make it clear that the crucial aspect comes from the fluctuation regime itself. In the experiment, we explain our data in terms of nematic fluctuations, which we substantiate with several references.

Next part of the comments by the Referee -- main complaint

Our reply:

We agree with the reviewer that consideration of fluctuating Cooper pairs alone here is insufficient. We have changed all text passages accordingly and interpret the data - in view of the literature - by nematic fluctuations.

In principle, we also agree with the reviewer that more experimental data would be necessary. However, this paper has a very strong theoretical ground. The experimental data should show that in principle the phenomenon of electronic fluctuations near phase transitions, shows up clearly in experimental transport data. A more detailed experimental study will follow in the future.

Third part of the comments by the Referee -- conclusion

Our reply:

The main idea behind Ref. [15] was to show quantitatively by means of numerical simulation of different models of electrically charged working fluids that if a system that can undergo a phase transition, its thermoelectric coupling strongly can be enhanced near the critical point. We illustrated this with the case of the 2D fluctuating Cooper pairs. The metric then introduced and used for comparison among different working fluids is the thermodynamic figure of merit $Z_{\rm th}T$, which is a combination of thermoelastic coefficients derived from the definition of the isentropic expansion factor $\gamma$.

In the present work, we significantly go beyond the scope of the previous work first by relating the thermodynamic figure of merit to the electronic thermoelectric figure of merit, i.e. the thermoelastic properties of the working fluid to its transport properties. To do so, we introduce an equivalent of the isentropic expansion factor based on transport coefficients, $\gamma_{\rm tr}$. The latter is a measure of the deviation from the thermostatic $\gamma$ when the system is out-of-equilibrium. In fact, thermoelectric conversion performance improves when $\gamma$ becomes large, which implies that the working fluid transport properties foster a high power factor if the working fluid's compressibility is larger. We see that $\gamma$ becomes large in the fluctuation regimes, and this is a nice illustration of the fluctuation-compressibility theorem in the context of thermoelectricity.

Further, contrary to Ref. [15], we present experimental data, although the development of a complete model that can addresses all aspects of the observed behaviors (Seebeck, electrical resistivity, power factor) in the clean and degraded samples is beyond the scope of the present work. What we had initially set out to do is to interpret and discuss the observed behavior close to $T_{\rm c}$ in light of our thermodynamic analysis, rather than provide a full model. It was clear that given the fact that we made a number of simplifying assumptions for clarity of the electronic working fluid models, we could not explain the trend starting at around 50 K down to close to $T_{\rm c}$. In the revised version, however, we provide a qualitative interpretation of the observed behavior in this range as a manifestation of nematic fluctuations.

We thus believe that in addition to the experimental data and their analysis, our thermodynamic study is not only non-trivial but also sheds light on the basic properties of the electron gas in thermoelectric materials. Further, we believe our work paves the way to the development of theoretical and experimental developments in the field of thermoelectricity as a number of questions remain to be answered.

Additional remarks by the Referee

our reply:

We have reshaped our manuscript to better report our work which has a large theoretical component.

True, the Introduction part is quite long, but we feel that it is necessary to properly lay out the rationale for this work, which is at the crossroads of various fields: thermodynamics, transport, solid-state physics, materials science, energy conversion. Same applies for the abstract that we tried to keep as concise yet as informative as possible.

Interestingly, the format of SciPost papers shows the content of the paper so any reader interested in particular aspects of the work can skip other parts that they find not essential for the understanding of the work. So, a long introduction provides more benefits than drawbacks here.

Close to $T_{\rm c}$ the rise is so sharp and the temperature range so small (to account for the limit of validity of the model) that a logarithmic scale does not provide a better view. In fact, this was our very first intent, but we decided that the linear scale provides a more satisfactory display of the curves.