Entropy has a unique role amongst thermodynamic variables in the connection it provides between the macroscopic averaged quantities, such as volume, magnetisation and temperature, and microscopic dynamics. This connection is particularly powerful in small systems with few degrees of freedom, in which the value of entropy allows one to deduce their internal dynamics, such as energy level structures and degeneracies, as well as the presence of quantum correlations. However, as the size of a system decreases, the standard approach to measuring entropy, based on heat flows and its thermodynamic definition dS=dQ/T, becomes increasingly difficult, as it requires measuring microscopic heat flows. To solve this problem, several alternative methods for measuring entropy in nanodevices have recently been developed. These methods do not rely on measuring heat flows, but utilize electrical measurements of charge or current instead.

We have harmonized the charge- and current-based methods within a single thermodynamic framework for treating nanoscale systems exchanging electrons with thermal reservoirs and applied this framework to develop a new way of measuring entropy in thermoelectric molecular devices. From magnetic field-dependent entropy measurements of using our “thermocurrent spectroscopy” method, we are able to infer the presence of a singlet-triplet transition in the reduced state of a free-radical molecule that cannot be detected by conventional charge transport measurements. Direct entropy measurements further enable us to determine the energy of the singlet-triplet splitting in the single-molecule device.

Our approach offers a novel and powerful tool for exploring quantum states of a wide variety of nanoscale systems as well as the ability to use well-developed experimental techniques for studying quantum thermodynamics. Molecular devices are of particular interest because their properties, including entanglement, high-spin states, localisation, and quantum interference, can be controlled via rational chemical design. Molecular thermoelectric devices therefore have the potential to play a key role in the experimental verification of quantum thermodynamics.

Single-molecule topological insulators are promising candidates as conducting wires over nanometer length scales. In past, most conjugated molecular wires exhibit low conductance that decays as the wire length increases. To overcome this limitation, we studied a family of oligophenylene-bridged bis(triarylamines) with tunable and stable (mono-/di-)radicaloid character. The wires can undergo one- and two-electron chemical oxidations to the corresponding monocation and dication, respectively. We found that the oxidized wires exhibit high reversed conductance decay with increasing length, consistent with the expectation for the Su-Schrieffer-Heeger-type one-dimensional (1D) topological insulators. The champion 2.6 nm long dication displays a significantly high conductance greater than 0.1 G0 (2e2/h, the conductance quantum), 5400-fold greater than the neutral form. The observed conductance-length relationship is similar between monocation and dication series. DFT calculations elucidate how the frontier orbitals and delocalization of radicals facilitate the observed nonclassical quasi-metallic behavior. These findings offer new insights into molecular design of highly conducting 1D topological insulators.