How to measure the entropy of a mesoscopic system via thermoelectric transport

Quantized Spin Pumping in Topological Ferromagnetic-Superconducting Nanowires

Semiconducting nanowires with strong spin-orbit coupling in the presence of induced superconductivity and ferromagnetism can support Majorana zero modes. We study the pumping due to the precession of the magnetization in single-subband nanowires and show that spin pumping is robustly quantized when the hybrid nanowire is in the topologically nontrivial phase, whereas charge pumping is not quantized. Moreover, there exists one-to-one correspondence between the quantized conductance, entropy change and spin pumping in long topologically nontrivial nanowires but these observables are uncorrelated in the case of accidental zero-energy Andreev bound states in the trivial phase. Thus, we conclude that observation of correlated and quantized peaks in the conductance, entropy change and spin pumping would provide strong evidence of Majorana zero modes, and we elaborate how topological Majorana zero modes can be distinguished from quasi-Majorana modes potentially created by a smooth tunnel barrier at the lead-nanowire interface. Finally, we discuss peculiar interference effects affecting the spin pumping in short nanowires at very low energies.

Entropy Measurement of a Strongly Coupled Quantum Dot

The spin 1/2 entropy of electrons trapped in a quantum dot has previously been measured with great accuracy, but the protocol used for that measurement is valid only within a restrictive set of conditions. Here, we demonstrate a novel entropy measurement protocol that is universal for arbitrary mesoscopic circuits and apply this new approach to measure the entropy of a quantum dot hybridized with a reservoir. The experimental results match closely to numerical renormalization group (NRG) calculations for small and intermediate coupling. For the largest couplings investigated in this Letter, NRG calculations predict a suppression of spin entropy at the charge transition due to the formation of a Kondo singlet, but that suppression is not observed in the experiment.

Fractional Entropy of Multichannel Kondo Systems from Conductance-Charge Relations

Fractional entropy is a signature of nonlocal degrees of freedom, such as Majorana zero modes or more exotic non-Abelian anyons. Although direct experimental measurements remain challenging, Maxwell relations provide an indirect route to the entropy through charge measurements. Here we consider multichannel charge-Kondo systems, which are predicted to host exotic quasiparticles due to a frustration of Kondo screening at low temperatures. In the absence of experimental data for the charge occupation, we derive relations connecting the latter to the conductance, for which experimental results have recently been obtained. Our analysis indicates that Majorana and Fibonacci anyon quasiparticles are well developed in existing two- and three-channel charge-Kondo devices, and that their characteristic k_{B}logsqrt[2] and k_{B}log[(1+sqrt[5])/2] entropies are experimentally measurable.

A Robust Protocol for Entropy Measurement in Mesoscopic Circuits

Previous measurements utilizing Maxwell relations to measure change in entropy, S, demonstrated remarkable accuracy in measuring the spin-1/2 entropy of electrons in a weakly coupled quantum dot. However, these previous measurements relied upon prior knowledge of the charge transition lineshape. This had the benefit of making the quantitative determination of entropy independent of scale factors in the measurement itself but at the cost of limiting the applicability of the approach to simple systems. To measure the entropy of more exotic mesoscopic systems, a more flexible analysis technique may be employed; however, doing so requires a precise calibration of the measurement. Here, we give details on the necessary improvements made to the original experimental approach and highlight some of the common challenges (along with strategies to overcome them) that other groups may face when attempting this type of measurement.

Controlling the Entropy of a Single-Molecule Junction

Single molecules are nanoscale thermodynamic systems with few degrees of freedom. Thus, the knowledge of their entropy can reveal the presence of microscopic electron transfer dynamics that are difficult to observe otherwise. Here, we apply thermocurrent spectroscopy to directly measure the entropy of a single free radical molecule in a magnetic field. Our results allow us to uncover the presence of a singlet to triplet transition in one of the redox states of the molecule, not detected by conventional charge transport measurements. This highlights the power of thermoelectric measurements which can be used to determine the difference in configurational entropy between the redox states of a nanoscale system involved in conductance without any prior assumptions about its structure or microscopic dynamics.

A Thermodynamic Approach to Measuring Entropy in a Few-Electron Nanodevice

The entropy of a system gives a powerful insight into its microscopic degrees of freedom; however, standard experimental ways of measuring entropy through heat capacity are hard to apply to nanoscale systems, as they require the measurement of increasingly small amounts of heat. Two alternative entropy measurement methods have been recently proposed for nanodevices: through charge balance measurements and transport properties. We describe a self-consistent thermodynamic framework for applying thermodynamic relations to few-electron nanodevices-small systems, where fluctuations in particle number are significant, whilst highlighting several ongoing misconceptions. We derive a relation (a consequence of a Maxwell relation for small systems), which describes both existing entropy measurement methods as special cases, while also allowing the experimentalist to probe the intermediate regime between them. Finally, we independently prove the applicability of our framework in systems with complex microscopic dynamics-those with many excited states of various degeneracies-from microscopic considerations.

Spin Seebeck effect of correlated magnetic molecules

In this paper we investigate the spin-resolved thermoelectric properties of strongly correlated molecular junctions in the linear response regime. The magnetic molecule is modeled by a single orbital level to which the molecular core spin is attached by an exchange interaction. Using the numerical renormalization group method we analyze the behavior of the (spin) Seebeck effect, heat conductance and figure of merit for different model parameters of the molecule. We show that the thermopower strongly depends on the strength and type of the exchange interaction as well as the molecule's magnetic anisotropy. When the molecule is coupled to ferromagnetic leads, the thermoelectric properties reveal an interplay between the spin-resolved tunneling processes and intrinsic magnetic properties of the molecule. Moreover, in the case of finite spin accumulation in the leads, the system exhibits the spin Seebeck effect. We demonstrate that a considerable spin Seebeck effect can develop when the molecule exhibits an easy-plane magnetic anisotropy, while the sign of the spin thermopower depends on the type and magnitude of the molecule's exchange interaction.

Complete mapping of the thermoelectric properties of a single molecule

Theoretical studies suggest that mastering the thermocurrent through single molecules can lead to thermoelectric energy harvesters with unprecedentedly high efficiencies.^1-6 This can be achieved by engineering molecule length,⁷ optimizing the tunnel coupling strength of molecules via chemical anchor groups⁸ or by creating localized states in the backbone with resulting quantum interference features.⁴ Empirical verification of these predictions, however, faces considerable experimental challenges and is still awaited. Here we use a novel measurement protocol that simultaneously probes the conductance and thermocurrent flow as a function of bias voltage and gate voltage. We find that the resulting thermocurrent is strongly asymmetric with respect to the gate voltage, with evidence of molecular excited states in the thermocurrent Coulomb diamond maps. These features can be reproduced by a rate-equation model only if it accounts for both the vibrational coupling and the electronic degeneracies, thus giving direct insight into the interplay of electronic and vibrational degrees of freedom, and the role of spin entropy in single molecules. Overall these results show that thermocurrent measurements can be used as a spectroscopic tool to access molecule-specific quantum transport phenomena.

Thermoelectric Transport in a Three-Channel Charge Kondo Circuit

We theoretically investigate the thermoelectric transport through a circuit implementation of the three-channel charge Kondo model quantum simulator [Z. Iftikhar et al., Science 360, 1315 (2018)SCIEAS0036-807510.1126/science.aan5592]. The universal temperature scaling law of the Seebeck coefficient is computed perturbatively approaching the non-Fermi liquid strong coupling fixed point using the Abelian bosonization technique. The predicted T^{1/3}logT scaling behavior of the thermoelectric power sheds light on the properties of Z_{3} emerging parafermions and gives access to exploring prefractionalized zero modes in the quantum transport experiments. We discuss a generalization of approach for investigating a multichannel Kondo problem with emergent Z_{N}→Z_{M} crossovers between "weak" non-Fermi liquid regimes corresponding to different low-temperature fixed points.

Entropy of Conduction Electrons from Transport Experiments

The entropy of conduction electrons was evaluated utilizing the thermodynamic definition of the Seebeck coefficient as a tool. This analysis was applied to two different kinds of scientific questions that can—if at all—be only partially addressed by other methods. These are the field-dependence of meta-magnetic phase transitions and the electronic structure in strongly disordered materials, such as alloys. We showed that the electronic entropy change in meta-magnetic transitions is not constant with the applied magnetic field, as is usually assumed. Furthermore, we traced the evolution of the electronic entropy with respect to the chemical composition of an alloy series. Insights about the strength and kind of interactions appearing in the exemplary materials can be identified in the experiments.