# Thermoelectric coefficients and the figure of merit for large open quantum dots

### Submission summary

 As Contributors: Robert Whitney Arxiv Link: https://arxiv.org/abs/1805.05166v3 Date accepted: 2018-12-19 Date submitted: 2018-11-30 Submitted by: Whitney, Robert Submitted to: SciPost Physics Domain(s): Theoretical Subject area: Quantum Physics

### Abstract

We consider the thermoelectric response of chaotic or disordered quantum dots in the limit of phase-coherent transport, statistically described by random matrix theory. We calculate the full distribution of the thermoelectric coefficients (Seebeck $S$ and Peltier $\Pi$), and the thermoelectric figure of merit $ZT$, for large open dots at arbitrary temperature and external magnetic field, when the number of modes in the left and right leads ($N_{\rm L}$ and $N_{\rm R}$) are large. Our results show that the thermoelectric coefficients and $ZT$ are maximal when the temperature is half the Thouless energy, and the magnetic field is negligible. They remain small, even at their maximum, but they exhibit a type of universality at all temperatures, in which they do not depend on the asymmetry between the left and right leads $(N_{\rm L}-N_{\rm R})$, even though they depend on $(N_{\rm L}+N_{\rm R})$.

###### Current status:

We thank the referee for a very thorough reading of this manuscript,
and for their insightful recommendation about the results and their presentation.
We have implemented all the changes that the referee suggested;
we explain these changes point by point below.

We added new figures 2 and 4, to help the reader's comprehension.
We have also moved an intuitive explanation of our results from the body of the text into the introduction
(the new section 1.1); this explains why such systems always have small thermoelectric effects,
and allows one to guess their approximate magnitude.

DETAILED RESPONSE TO THE MAIN QUESTIONS IN THE REPORT

1) The referee raises a very interesting point, when he asks why there is so much cancellation between contributions. It is possible that there is a deeper principle at play here, but we do not know what. We see no such principle at the level of the sums over semiclassical trajectories considered here. However, it may be that treating random matrices as formal mathematical objects (without the crutch of the intuition gained from trajectories traversing a real chaotic or disordered system) would lead to such deeper principles.

2) Electron-electron interactions are beyond the scope of the semiclassical single-particle method considered here, and in general much less in known about such semiclassics in the presence of such interactions.
The problem of interactions + disorder is one of the toughest ones in theoretical physics, and we know of few works on the thermoelectric responses of systems with interactions and disorder. We have added some sentences to Section 1.2 to mention two works in this direction in similar (large N) mesoscopic structures.
+ P.M. Chaikin and G. Beni, Thermopower in the correlated hopping regime, Phys. Rev. B 13, 647 (1976).
+ J. W. P. Hsu, A. Kapitulnik, and M. Yu. Reizer, Effect of electron-electron interaction on the thermoelectric power in disordered metallic systems, Phys. Rev. B 40, 7513 (1989).
However, we mention that it is rather easy to break <S>=0, even without electron-electron interactions,
by considering systems with N ~ 1, so the conduction is of order (or less than) a single channel.
Our recent review contains many examples of such systems; both non-interacting (mostly sections 4-6), and systems with electron-electron and electron-phonon interactions (mostly sections 7-9).

### List of changes

RESPONSE TO REQUESTED CHANGES

1) We worry that the words "additive asymmetry" will not mean much to readers,
so we decided to be more explicit in the abstract. Saying
"... they do not depend on the asymmetry between the left and right leads (N_L − N_R),
even though they depend on (N_L + N_R)."
We also modified section 3 "Universality of the Seebeck coefficient and figure of merit" to clarify this point.

2) The dependence <S^2> on N is non-linear in both T and N, because as the referee points out E_{Th} goes like N, and <S^2> is a non-linear function of E_{Th}. We have added a discussion about the 4 regimes
i) very small T where it goes like N^{-4}T^2 as in Ref [10]
ii) the referee's regime where it goes like N^{-3}T
iii) the peak where it goes like N^{-2}
iv) high temperatures where it goes like (NT)^{-1}
and cite the (anonymous) referee for suggesting regime (ii).

3) The referee is correct that the peak makes a small excursion for finite fields. We have added a plot to fig 3 which shows how the peak moves from x=0.48 at b=1 up to nearly x=0.6 at b~2 before dropping back to x=0.48 at b=infinity, see fig 3b.

4) The referee is 100% correct, in this regime of weak energy dependence the ratio G/K is the Lorentz ratio (with corrections of order 1/N discussed in Ref [10]). We have added a new section 2.2 about the Wiedemann-Franz ratio.

5) The referee is correct, there was a typo, we now refer to the "next-to-leading" term, not the leading term.
We have taken the opportunity to calculate the prefactor on this term, now given in a new Eq. (17).

6) We have made the correction that the referee suggested.

7) We have fixed this typo.

8) The referee writes "On p. 12, Fig. 3: the rules mention an encounter inside the dot. I cannot make out from the diagrams why lines I-IV do not run through the dot (kappa=0) and V-VI do run through it (kappa neq 0). Where is the dot to be seen? It is not stated in the caption where the dot is to be seen and that the boxes are the encounters."

We have clarified the connection between physical trajectories in a dot, and the slightly abstract diagrams, in a new Fig.4. We have also modified the diagrams to clarify which encounters are touching the leads, and which are inside the dot, the former are now marked by a triangular box and the latter by a rectangular box.
The value of kappa for the encounters marked by rectangular boxes are then given in Fig. 7, and depend on the energies of the trajectories entering the encounter, and the magnetic field.

9) We have done our best to put the figures as close as possible to the place where they are cited in the text.
10) We have fixed this typo.
11) We have fixed these typos in the conclusions.

### Submission & Refereeing History

Resubmission 1805.05166v3 (30 November 2018)
Submission 1805.05166v2 (2 August 2018)

## Invited Reports on this Submission

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### Report

I'm fully content with the changes made by the authors and their answers to my optional questions.

I maintain my previous list of strengths of the paper and recommend its publication in the present form.

• validity: high
• significance: good
• originality: good
• clarity: top
• formatting: excellent
• grammar: excellent