A superconductor, when exposed to a spin-exchange field, can exhibit spatial modulation of its order parameter, commonly referred to as the Fulde–Ferrell–Larkin–Ovchinnikov state. Such a state can be induced by controlling the spin-splitting field in Josephson junction devices, allowing access to a wide range of the phase diagram. Here we demonstrate that a Fulde–Ferrell–Larkin–Ovchinnikov state can be induced in Josephson junctions based on the two-dimensional dilute magnetic topological insulator (Hg,Mn)Te. We do this by observing the dependence of the critical current on the magnetic field and temperature. The substitution of Mn dopants induces an enhanced Zeeman effect, which can be controlled with high precision by using a small external magnetic field. We observe multiple re-entrant behaviours of the critical current as a response to an in-plane magnetic field, which we assign to transitions between ground states with a phase shifted by π. This will enable the study of the Fulde–Ferrell–Larkin–Ovchinnikov state in much more accessible experimental conditions.

Entropy is a fundamental thermodynamic quantity indicative of the accessible degrees of freedom in a system. While it has been suggested that the entropy of a mesoscopic system can yield nontrivial information on emergence of exotic states, its measurement in such small electron-number system is a daunting task. Here we propose a method to extract the entropy of a Coulomb-blockaded mesoscopic system from transport measurements. We prove analytically and demonstrate numerically the applicability of the method to such a mesoscopic system of arbitrary spectrum and degeneracies. We then apply our procedure to measurements of thermoelectric response of a single quantum dot, and demonstrate how it can be used to deduce the entropy change across Coulomb-blockade valleys, resolving, along the way, a long-standing puzzle of the experimentally observed finite thermoelectric response at the apparent particle-hole symmetric point.

Fluctuations are strong in mesoscopic systems and have to be taken into account for the description of transport. We show that they can even be used as a resource for the operation of a system as a device. We use the physics of single-electron tunneling to propose a bipartite device working as a thermal transistor. Charge and heat currents in a two-terminal conductor can be gated by thermal fluctuations from a third terminal to which it is capacitively coupled. The gate system can act as a switch that injects neither charge nor energy into the conductor, hence achieving huge amplification factors. Nonthermal properties of the tunneling electrons can be exploited to operate the device with no energy consumption.

We investigate theoretically the dynamics of a Josephson junction in the framework of the resistively shunted junction model. We consider a junction that hosts two supercurrent contributions: a 2π and a 4π periodic in phase, with intensities I2π and I4π, respectively. We study the size of the Shapiro steps as a function of the ratio of the intensity of the mentioned contributions, i.e., I4π/I2π. We provide detailed explanations where to expect clear signatures of the presence of the 4π-periodic contribution as a function of the external parameters: the intensity ac bias Iac and frequency ωac. On the one hand, in the low ac-intensity regime (where Iac is much smaller than the critical current Ic), we find that the nonlinear dynamics of the junction allows the observation of only even Shapiro steps even in the unfavorable situation where I4π/I2π1. On the other hand, in the opposite limit (IacIc), even and odd Shapiro steps are present. Nevertheless, even in this regime, we find signatures of the 4π supercurrent in the beating pattern of the even step sizes as a function of Iac.

Frequency analysis of the rf emission of oscillating Josephson supercurrent is a powerful passive way of probing properties of topological Josephson junctions. In particular, measurements of the Josephson emission enable the detection of topological gapless Andreev bound states that give rise to emission at half the Josephson frequency f_j rather than conventional emission at f_j. Here, we report direct measurement of rf emission spectra on Josephson junctions made of HgTe-based gate-tunable topological weak links. The emission spectra exhibit a clear signal at half the Josephson frequency f_j=2. The linewidths of emission lines indicate a coherence time of 0.3-4 ns for the f_j=2 line, much shorter than for the f_j line (3-4 ns). These observations strongly point towards the presence of topological gapless Andreev bound states and pave the way for a future HgTe-based platform for topological quantum computation.

In recent years, Majorana physics has attracted considerable attention because of exotic new phenomena and its prospects for fault-tolerant topological quantum computation. To this end, one needs to engineer the interplay between superconductivity and electronic properties in a topological insulator, but experimental work remains scarce and ambiguous. Here, we report experimental evidence for topological superconductivity induced in a HgTe quantum well, a 2D topological insulator that exhibits the quantum spin Hall (QSH) effect. The a.c. Josephson effect demonstrates that the supercurrent has a 4π periodicity in the superconducting phase difference, as indicated by a doubling of the voltage step for multiple Shapiro steps. In addition, this response like that of a superconducting quantum interference device to a perpendicular magnetic field shows that the 4π-periodic supercurrent originates from states located on the edges of the junction. Both features appear strongest towards the QSH regime, and thus provide evidence for induced topological superconductivity in the QSH edge states.

We theoretically investigate the propagation of heat currents in a three-terminal quantum dot engine. Electron-electron interactions introduce state-dependent processes which can be resolved by energy-dependent tunneling rates. We identify the relevant transitions which define the operation of the system as a thermal transistor or a thermal diode. In the former case, thermal-induced charge fluctuations in the gate dot modify the thermal currents in the conductor with suppressed heat injection, resulting in huge amplification factors and the possible gating with arbitrarily low energy cost. In the latter case, enhanced correlations of the state-selective tunneling transitions redistribute heat flows giving high rectification coefficients and the unexpected cooling of one conductor terminal by heating the other one. We propose quantum dot arrays as a possible way to achieve the extreme tunneling asymmetries required for the different operations.

The proximity-induced superconducting state in the three-dimensional topological insulator HgTe has been studied using electronic transport of a normal metal-superconducting point contact as a spectroscopic tool (Andreev point-contact spectroscopy). By analyzing the conductance as a function of voltage for various temperatures, magnetic fields, and gate voltages, we find evidence, in equilibrium, for an induced order parameter in HgTe of 70 μeV and a niobium order parameter of 1.1 meV. To understand the full conductance curve as a function of applied voltage we suggest a non-equilibrium-driven transformation of the quantum transport process where the relevant scattering region and equilibrium reservoirs change with voltage. This change implies that the spectroscopy probes the superconducting correlations at different positions in the sample, depending on the bias voltage.

This article reviews recent thermoelectric experiments on quantum dot (QD) systems. The experiments focus on two types of inter-dot coupling: tunnel coupling and Coulomb coupling. Tunnel-coupled QDs allow particles to be exchanged between the attached reservoirs via the QD system. Hence, an applied temperature bias results in a thermovoltage. When being investigated as a function of QD energies, this leads to the thermopower stability diagram. Here, largest thermovoltage is observed in the regions of the triple points. In a QD system which exhibits only capacitive inter-dot coupling, electron transfer is suppressed. Such a device is studied in a three-terminal geometry: while one QD connects to the heat reservoir, the other one can exchange electrons with two reservoirs at a lower temperature. When the symmetry of the tunneling coefficients in the cold system is broken, the device becomes an energy harvester: thermal energy is extracted from the heat reservoir and is converted into a directed charge current between the two cold reservoirs. This review illustrates the large potential of multi-QD devices for thermoelectrics and thermal management at the nanometer-scale. In this article, the authors review the thermoelectric properties of a coupled quantum dot system which can be viewed as an artificial molecule. The first part presents the measurement of the thermopower generated by such a system located between a hot and a cold reservoir. In the second part it is discussed how coupled quantum dots can be used to extract energy from the hot reservoir and convert it into a directed current without particle exchange.

We have observed thermal gating, i.e. electrostatic gating induced by hot electrons. The effect occurs in a device consisting of two capacitively coupled quantum dots. The double dot system is coupled to a hot electron reservoir on one side (QD1), while the conductance of the second dot (QD2) is monitored. When a bias across QD2 is applied we observe a current which is strongly dependent on the temperature of the heat reservoir. This current can be either enhanced or suppressed, depending on the relative energetic alignment of the QD levels. Thus, the system can be used to control a charge current by hot electrons.

Rectification of thermal fluctuations in mesoscopic conductors is the key idea behind recent attempts to build nanoscale thermoelectric energy harvesters to convert heat into useful electric power. So far, most concepts have made use of the Seebeck effect in a two-terminal geometry, where heat and charge are both carried by the same particles. Here, we experimentally demonstrate the working principle of a new kind of energy harvester, proposed recently, using two capacitively coupled quantum dots. We show that, due to the novel three-terminal design of our device, which spatially separates the heat reservoir from the conductor circuit, the directions of charge and heat flow become decoupled. This enables us to manipulate the direction of the generated charge current by means of external gate voltages while leaving the direction of heat flow unaffected. Our results pave the way for a new generation of multi-terminal nanoscale heat engines.