The thermoelectric effects in three-terminal structures with a quantum dot are considered. We propose the experimentally consistent protocol for determination of the transport coefficients in terms of the local and non-local conductances and thermopowers that can be measured in two steps, applying the `four probe technique'. This proposal is compared with other approaches discussed so far in the literature. As an example we study in detail the thermopower induced by the superconducting electrode in a subgap regime which might be useful for analysis of novel hybrid devices.
We study the thermopower of a three-terminal setup composed of a quantum dot attached to three electrodes, one of which is a topological superconductor. In the model, superconductivity is explicitly taken into account. We compare the results for s-wave (trivial) and p-wave (topological) superconductors and observe that for small temperatures the thermopower has different sign in the two cases. This behavior is strongly dependent on temperature and we estimate an energy scale that controls the sign in the p-wave case, which results proportional to the square root of the gap and the coupling to …
We present measurements of current noise and cross-correlations in three-terminal Superconductor-Normal metal-Superconductor (S-N-S) nanostructures that are potential solid-state entanglers thanks to Andreev reflections at the N-S interfaces. The noise correlation measurements spanned from the regime where electron-electron interactions are relevant to the regime of Incoherent Multiple Andreev Reflection (IMAR). In the latter regime, negative cross-correlations are observed in samples with closely-spaced junctions.
We model and evaluate the Peltier and Seebeck effects in magnetic multilayer nanostructures by a finite-element theory of thermoelectric properties. We present analytical expressions for the thermopower and the current-induced temperature changes due to Peltier cooling/heating. The thermopower of a magnetic element is in general spin-polarized, leading to spin-heat coupling effects. Thermoelectric effects in spin valves depend on the relative alignment of the magnetization directions and are sensitive to spin-flip scattering as well as inelastic collisions in the normal metal spacer.
We show that the quasi-continuous gapless spectrum of Andreev bound states in multi-terminal semi-classical superconducting nanostructures exhibits a big number of topological singularities. We concentrate on Weyl points in a 4-terminal nanostructure, compute their density and correlations in 3D parameter space for a universal RMT model as well as for the concrete nanostructures described by the quantum circuit theory. We mention the opportunities for experimental observation of the effect in a quasi-continuous spectrum.
It is shown that for the hopping regime, the thermopowers in both finite two-terminal and three-terminal systems are governed by the edges of the samples. This is due to the fact that the energy transfer between a transport electron and a conducting terminal is determined by the site most strongly coupled to that terminal. One-dimensional systems with both nearest-neighbor and variable-range transport as well as certain types of two-dimensional systems, are considered. For a given sample, the changes in the thermopowers due to modifying the bulk are quite limited, compared with those of the co…
We report a theoretical model for a thermal transistor in dielectric four-terminal nanostructures based on mesoscopic ballistic phonon transport, in which a steady thermal flow condition of system is obtained to set up the temperature field effect of gate. In the environment, thermal flow shows the transisting behaviors at low temperatures: saturation, asymmetry, and rectification. The phenomena can be explained reasonably by the nonlinear variation of the temperature dependence of propagating phonon modes in terminals. The results suggest the possibility of the novel nano-thermal transistor f…
We investigate thermoelectric efficiency of systems with broken time reversal symmetry under a three-terminal transport. Using a model of Aharonov-Bohm interferometer formed with three noninteracting quantum dots, we show that Carnot efficiency can be achieved when the thermopower is a symmetric function of the applied magnetic field. On the other hand, the maximal value of the efficiency at maximum power is obtained for asymmetric thermopower. Indeed, we show that Curzon-Ahlborn limit is exceeded within the linear response regime in our model. Moreover, we investigate thermoelectric efficienc…
We present and review an efficient method to calculate the retarded Green's function in multi-terminal nanostructures; which is needed in order to calculate the conductance through the system and the local particle densities within it. The method uses the recursive Green's function method after the discretized Hamilton matrix has been properly partitioned. We show that this method, the circular slicing scheme, can be modified to accommodate multi-terminal systems as well as the traditional two-terminal systems. Furthermore, we show that the performance and robustness of the circular slicing sc…
The recent proposals of experiments with single Andreev bound states make relevant a detailed analysis of these states in multi-terminal superconducting nanostructures. We evaluate the energy splitting of degenerate Andreev bound states, that overlap in a superconducting lead, and find that the splitting is reduced in comparison with their energy by a small factor $\sqrt{R G_Q}$, $R G_Q$ being the dimensionless resistance of the overlap region in the normal state. This permits quantum manipulation of the quasiparticles in these states. We provide a simple scheme of such manipulation.
Recently, it has been shown that multi-terminal superconducting nanostructures may possess topological properties that involve Berry curvatures in the parametric space of the superconducting phases of the terminals, and associated Chern numbers that are manifested in quantized transconductances of the nanostructure. In this Article, we investigate how the continuous spectrum that is intrinsically present in superconductors, affects these properties. We model the nanostructure within scattering formalism deriving the action and the response function that permits a re-definition of Berry curvatu…
We show theoretically that in the superconducting nanostructures the gapped states of different topology are not always protected by separating gapped states. Depending on the structure design parameters, they can be either protected or not, with a protection-unprotection transition separating these two distinct situations. We build up a general theoretical description of the transition vicinity in the spirit of Landau theory. We speculate that similar protection-unprotection transitions may also occur for other realizations of topological protection in condensed matter systems.
We use the concept of resonant tunneling to calculate the thermopower of molecular nanosystems. It turns out that the sign of the thermovoltage under resonant tunneling conditions depends sensitively on the participating molecular orbital, and one finds a sign change when the transport channel switches from the highest occupied molecular orbital to the lowest unoccupied molecular orbital. Comparing our results to recent experimental data obtained for a BDT molecule contacted with an STM tip, we observe good agreement.
The usually negligibly small thermoelectric effects in superconducting heterostructures can be boosted dramatically due to the simultaneous effect of spin splitting and spin filtering. Building on an idea of our earlier work [Phys. Rev. Lett. $\textbf{110}$, 047002 (2013)], we propose realistic mesoscopic setups to observe thermoelectric effects in superconductor heterostructures with ferromagnetic interfaces or terminals. We focus on the Seebeck effect being a direct measure of the local thermoelectric response and find that a thermopower of the order of $\sim200$ $μV/K$ can be achieved in a …
We have studied the spin-splitting effect in a four-terminal two-dimensional (2D) electron gas system with two potential barriers generated by two surface metal gates and an external perpendicular magnetic field. The calculations show that by tuning the voltage applied on the gates, the injected spin-unpolarized current can be split into different spin currents with a high efficiency. The split currents flow out of the geometry from different output leads separately. The spin freedom of the outputs can be controlled by simply tuning voltage on gates. This phenomenon is a result of the combinat…
Boolean logic is the foundation of modern digital information processing. Recently, there has been a growing interest in phenomena based on pure spin currents, which allow to move from charge to spin based logic gates. We study a proof-of-principle logic device based on the ferrimagnetic insulator Yttrium Iron Garnet (YIG), with Pt strips acting as injectors and detectors for nonequilibrium magnons. We experimentally observe incoherent superposition of magnons generated by different injectors. This allows to implement a fully functional majority gate, enabling multiple logic operations (AND an…
The thermoelectric voltage of a quantum dot connected to leads is calculated using the scattering R-matrix method. Our approach takes into account a temperature gradient between the contacts beyond the linear regime. We obtain sign changes of the thermopower when varying the temperature or the chemical potential around the resonances. The influence of the coupling strength of the contacts and of the thermoelectric field on the thermoelectric voltage is discussed.
The thermoelectric transport through a benzene molecule with three metallic terminals is discussed. Using general local and non-local transport coeffcients, we investigated different conductance and thermopower coefficients within the linear response regime. Based on the Onsager coefficient which depend on the number of terminal efficiencies, efficiency at maximum power is also studied. In the three-terminal set up with tuning temperature differences, a great enhancement of the figure of merit is observed. Results also show that the third terminal model can be useful to improve the efficiency …
All-electrical generation and detection of pure spin currents is a promising way towards controlling the diffusive magnon transport in magnetically ordered insulators. We quantitatively compare two measurement schemes, which allow to measure the magnon spin transport in a three-terminal device based on a yttrium iron garnet thin film. We demonstrate that the dc charge current method based on the current reversal technique and the ac charge current method utilizing first and second harmonic lock-in detection can both efficiently distinguish between electrically and thermally injected magnons. I…
A semi-classical analysis of magneto-thermopower behaviour, namely, the \textit{Seebeck} and \textit{Nernst} effect (NE) in quantum wells of IV-VI lead salts with significant extrinsic Rashba spin-orbit coupling (RSOC) is performed in this report. In addition to the spin-dependent Seebeck effect that has been observed before, we also theoretically predict a similar spin-delineated behaviour for its thermal analog, the spin-dependent NE. The choice of lead salts follows from a two-fold advantage they offer, in part, to their superior thermoelectric properties, especially PbTe, while their low b…
We study the subgap spectrum of the interacting single-level quantum dot coupled between two superconducting reservoirs, forming the Josephson-type circuit, and additionally hybridized with a metallic normal lead. This system allows for the phase-tunable interplay between the correlation effects and the proximity-induced electron pairing resulting in the singlet-doublet (0-$π$) crossover and the phase-dependent Kondo effect. We investigate the spectral function, induced local pairing, Josephson supercurrent, and Andreev conductance in a wide range of system parameters by the numerically exact …
We demonstrate that the flow of a longitudinal spin current with different spin polarization will induce different patterns of charge accumulation in a two-terminal strip, or electric current distribution in a four-terminal Hall-bar structure, of two-dimensional electron gas with Rashba spin-orbit coupling (RSOC). For an in-plane polarized spin current, charges will accumulate either by the two lateral edges or around the center of the strip structure while, for an out-of-plain polarized spin current, charge densities will show opposite signs by the two lateral edges leading to a Hall voltage.…
We report electrical conductance and thermopower measurements on InAs nanowires synthesized by chemical vapor deposition. Gate modulation of the thermopower of individual InAs nanowires with diameter around 20nm is obtained over T=40 to 300K. At low temperatures (T< ~100K), oscillations in the thermopower and power factor concomitant with the stepwise conductance increases are observed as the gate voltage shifts the chemical potential of electrons in InAs nanowire through quasi-one-dimensional (1D) sub-bands. This work experimentally shows the possibility to modulate semiconductor nanowire's t…
Significant progress has been made in recent studies of thermal and thermoelectric transport phenomena in nanostructures and low-dimensional systems. This article reviews several intriguing quantum and classical size effects on thermal and thermoelectric properties that have been predicted by theoretical calculations or observed in experiments. Attention is focused on the Casimir limit in phonon boundary scattering and the effect of phonon confinement on the lattice thermal conductivity of semiconductor nanowires (NWs) and nanomeshes; the effects of thickness, lateral size, and interface inter…
Using the Büttiker-Landauer formulation of transport theory in the linear response regime, the valley currents and non-local resistances of bilayer graphene nanostructures with broken inversion symmetry are calculated. It is shown that broken inversion symmetry in bilayer graphene nanostructures leads to striking enhancement of the non-local 4-terminal resistance and to valley currents several times stronger than the conventional electric current when the Fermi energy is in the spectral gap close to the energy of Dirac point. The scaling relation between local and non-local resistances is inve…
We discuss out-of-equilibrium population imbalances between different orbital states due to applied thermal gradients. This purely thermoelectric orbital effect appears quite generically in nanostructures with a pseudospin degree of freedom. We establish an orbital Seebeck coefficient that characterizes the induced orbital bias in response to a temperature difference between reservoirs coupled to a quantum conductor. We analyze a two-terminal strongly interacting quantum dot with two orbital states and find that the orbital thermopower acts as an excellent tool to describe the transition betwe…
A generic semiclassical superconducting nanostructure connected to multiple superconducting terminals hosts a quasi-continuous spectrum of Andreev states. The spectrum is sensitive to the superconducting phases of the terminals. It can be either gapped or gapless depending on the point in the multi-dimensional parametric space of these phases. Special points in this space correspond to setting some terminals to the phase 0 and the rest to the phase of $π$. For a generic nanostructure, three distinct spectra come together in the vicinity of a special point: two gapped phases of different topolo…
We investigate basic properties of the thermopower (Seebeck coefficient) of phase-coherent conductors under the influence of dephasing and inelastic processes. Transport across the system is caused by a voltage bias or a thermal gradient applied between two terminals. Inelastic scattering is modeled with the aid of an additional probe acting as an ideal potentiometer and thermometer. We find that inelastic scattering reduces the conductor's thermopower and, more unexpectedly, generates a magnetic-field asymmetry in the Seebeck coefficient. The latter effect is shown to be a higher-order effect…
The efficiency and cooling power of a two-terminal thermoelectric refrigerator are analyzed near the limit of vanishing dissipation (ideal system), where the optimal efficiency is the Carnot one, but the cooling power then unfortunately vanishes. This limit, where transport occurs only via a single sharp electronic energy, has been referred to as "strong coupling" or "the best thermoelectric". It follows however, that "parasitic" effects that make the system deviate from the ideal limit, and reduce the efficiency from the Carnot limit, are crucial for the usefulness of the device. Among these …
A two-site nanostructure (e.g, a "molecule") bridging two conducting leads and connected to a phonon bath is considered. The two relevant levels closest to the Fermi energy are connected each to its lead. The leads have slightly different temperatures and chemical potentials and the nanos- tructure is also coupled to a thermal (third) phonon bath. The 3 x 3 linear transport ("Onsager") matrix is evaluated, along with the ensuing new figure of merit, and found to be very favorable for thermoelectric energy conversion.
Coupled semiconductor nanostructures with a high degree of tunability are fabricated using local oxidation with a scanning force microscope. Direct oxidation of the GaAs surface of a Ga[Al]As heterostructure containing a shallow two-dimensional electron gas is combined with the local oxidation of a thin titanium film evaporated on top. A four-terminal quantum dot and a double quantum dot system with integrated charge readout are realized. The structures are tunable via in-plane gates formed by isolated regions in the electron gas and by mutually isolated regions of the Ti film acting as top ga…
A detailed investigation of the non-equilibrium steady-state electric and thermoelectric transport properties of a quantum dot coupled to the normal metallic and s-wave superconducting reservoirs (N-QD-S) are provided within the Coulomb blockade regime. Using non-equilibrium Keldysh Green's function formalism, initially, various model parameter dependence of thermoelectric transport properties are analysed within the linear response regime. It is observed that the single-particle tunnelling close to the superconducting gap edge can generate a relatively large thermopower and figure of merit. M…
We propose a near-field inelastic thermoelectric heat engine where quantum-dots are used to effectively rectify the charge flow of photo-carriers. The device converts near-field heat radiation into useful electrical power. Heat absorption and inelastic transport can be enhanced by introducing two continuous spectra separated by an energy gap. The thermoelectric transport properties of the heat engine are studied in the linear-response regime. Using a small band-gap semiconductor as the absorption material, we show that the device achieves very large thermopower and thermoelectric figure-of-mer…
A confined system of non-interacting electrons, subject to the combined effect of a time-dependent potential and different external chemical-potentials, is considered. The current flowing through such a system is obtained for arbitrary strengths of the modulating potential, using the adiabatic approximation in an iterative manner. A new formula is derived for the charge pumped through an un-biased system (all external chemical potentials are kept at the same value); It reproduces the Brouwer formula for a two-terminal nanostructure. The formalism presented yields the effect of the chemical pot…
We investigate a mesoscopic refrigerator based on chiral quantum Hall edge channels. We discuss a three-terminal cooling device in which charge transport occurs between a pair of voltage-biased terminals only. The third terminal, which is to be cooled, is set as a voltage probe with vanishing particle flux. This largely prevents the generation of direct Joule heating which ensures a high coefficient of performance. Cooling operation is based on energy-dependent quantum transmissions. The latter are implemented with the aid of two tunable scattering resonances (quantum dots). To find the optima…
The performance of thermoelectric energy harvesters can be improved by nanostructures that exploit inelastic transport processes. One prototype is the three-terminal hopping thermoelectric device where electron hopping between quantum-dots are driven by hot phonons. Such three-terminal hopping thermoelectric devices have potential in achieving high efficiency or power via inelastic transport and without relying on heavy-elements or toxic compounds. We show in this work how output power of the device can be optimized via tuning the number and energy configuration of the quantum-dots embedded in…
We propose a quantum heat engine based on an Aharonov-Bohm interferometer in a two-terminal geometry, and investigate its thermoelectric performances in the linear response regime. Sizeable thermopower (up to $\sim 0.3\,\text{mV}$/K) as well as $ZT$ values largely exceeding unity can be achieved by simply adjusting parameters of the setup and temperature bias across the interferometer leading to thermal efficiency at maximum power approaching $30\%$ of the Carnot limit. This is close to the optimal efficiency at maximum power achievable for a two-terminal heat engine. Changing the magnetic flu…
The wave-like nature of electrons leads to the existence of upper bounds on the thermoelectric response of nanostructured devices [R. S. Whitney, Phys. Rev. Lett. 112, 130601 (2014); Phys. Rev. B 91, 115425 (2015)]. This fundamental result, not present in classical thermodynamics, was demonstrated exploiting a two-terminal device modelled by non-linear scattering theory. In the present paper, we consider non-linear quantum transport through the same type of device working both as thermal machine and as refrigerator. For both operations, starting from charge and heat current expressions, we pro…
Valley currents and non-local resistances of graphene nanostructures with broken inversion symmetry are considered theoretically in the linear response regime. Scattering state wave functions of electrons entering the nanostructure from the contacts represented by groups of ideal leads are calculated by solving the Lippmann- Schwinger equation and are projected onto the valley state subspaces to obtain the valley velocity fields and total valley currents in the nanostructures. In the tunneling regime when the Fermi energy is in the spectral gap around the Dirac point energy, inversion symmetry…
Crossed Andreev reflection (CAR) in metallic nanostructures, a possible basis for solid-state electron entangler devices, is usually investigated by detecting non-local voltages in multi-terminal superconductor/normal metal devices. This task is difficult because other subgap processes may mask the effects of CAR. One of these processes is the generation of charge imbalance (CI) and the diffusion of non-equilibrium quasi-particles in the superconductor. Here we demonstrate a characteristic dependence of non-local CI on a magnetic field applied parallel to the superconducting wire, which can be…
We study all-carbon-hydrogen molecular transistors where zigzag graphene nanoribbons play the role of three metallic electrodes connected to a ring-shaped 18-annulene molecule. Using the nonequilibrium Green function formalism combined with density functional theory, recently extended to multiterminal devices, we show that the proposed nanostructures exhibit exponentially small transmission when the source and drain electrodes are attached in a configuration that ensures destructive interference of electron paths around the ring. The third electrode, functioning either as an attached infinite-…
We discuss the temperature-dependent thermoelectric transport properties of semiconductor nanostructures comprising a quantum dot coupled to quantum wires: the thermal dependence of the electrical conductance, thermal conductance, and thermopower. We explore the universality of the thermoelectric properties in the temperature range associated with the Kondo crossover. In this thermal range, general arguments indicate that any equilibrium property's temperature dependence should be a universal function of the ratio $T^{*}=T/T_{K}$, where $T_{K}$ is the Kondo temperature. Considering the particl…
We study the thermoelectric properties and heat-to-work conversion performance of an interacting, multi-level quantum dot (QD) weakly coupled to electronic reservoirs. We focus on the sequential tunneling regime. The dynamics of the charge in the QD is studied by means of master equations for the probabilities of occupation. From here we compute the charge and heat currents in the linear response regime. Assuming a generic multi-terminal setup, and for low temperatures (quantum limit), we obtain analytical expressions for the transport coefficients which account for the interplay between inter…
Bi$_{2}$Te$_{3}$ and Bi$_{2}$Se$_{3}$ are well known 3D-topological insulators. Films made of these materials exhibit metal-like surface states with a Dirac dispersion and possess high mobility. The high mobility metal-like surface states can serve as channel material for TI-based field effect transistors. While such a transistor offers superior terminal characteristics, they suffer from an inherent zero band gap problem. The absence of a band gap for the surface states prevents an easy turn-off mechanism. In this work, techniques that can be employed to easily open a band gap for the TI surfa…
The oxide heterostructure LaAlO3/SrTiO3 supports a two-dimensional electron liquid with a variety of competing phases including magnetism, superconductivity and weak antilocalization due to Rashba spin-orbit coupling. Further confinement of this 2D electron liquid to the quasi-one-dimensional regime can provide insight into the underlying physics of this system and reveal new behavior. Here we describe magnetotransport experiments on narrow LaAlO3/SrTiO3 structures created by a conductive atomic force microscope lithography technique. Four-terminal local transport measurements on ~10-nm-wide H…
We study three-terminal thermoelectric transport in a two-dimensional Quantum Point Contact (QPC) connected to left and right electronic reservoirs, as well as a third one represented by a scanning probe tip. The latter acts as a voltage probe exchanging heat with the system but no charges on average. The thermoelectric coefficients are calculated numerically within the Landauer-Büttiker formalism in the low-temperature and linear response regimes. We find tip-induced oscillations of the local and non-local thermopowers and study their dependence on the QPC opening. If the latter is tuned on a…
Proximity induced quantum coherence of electrons in multi-terminal voltage-driven hybrid normal-superconducting nanostructures may result in a non-trivial interplay between topology-dependent Josephson and Aharonov-Bohm effects. We elucidate a trade-off between stimulation of the voltage-dependent Josephson current due to non-equilibrium effects and quantum dephasing of quasiparticles causing reduction of both Josephson and Aharonov-Bohm currents. We also predict phase-shifted quantum coherent oscillations of the induced electrostatic potential as a function of the externally applied magnetic …
We propose a multiterminal nanostructure for electrical probing of the quantum spin Hall effect (QSHE) in two-dimensional (2D) topological insulators. The device consists of a ferromagnetic (FM) island with precessing magnetization that pumps (in the absence of any bias voltage) pure spin current symmetrically into the left and right adjacent 2D TIs modeled as graphene nanoribbons with the intrinsic spin-orbit (SO) coupling. The QSH regime of the six-terminal TI|FM|TI nanodevice, attached to two longitudinal and four transverse normal metal electrodes, is characterized by the SO-coupling-induc…
We demonstrate that thermoelectric signal as well as dc Josephson current may be severely enhanced in multi-terminal superconducting hybrid nanostructures exposed to a temperature gradient. At temperatures $T$ strongly exceeding the Thouless energy of our device both the supercurrent and the thermo-induced voltage are dominated by the contribution from non-equilibrium low energy quasiparticles and are predicted to decay slowly (algebraically rather than exponentially) with increasing $T$. We also predict a non-trivial current-phase relation and a transition to a $π$-junction state controlled b…
We compare non-local magnetoresistance measurements in multi-terminal Ni nanostructures with corresponding local experiments. In both configurations, the measured voltages show the characteristic features of anisotropic magnetoresistance (AMR). However, the magnitude of the non-local AMR signal is up to one order of magnitude larger than its local counterpart. Moreover, the non-local AMR increases with increasing degree of non-locality, i.e., with the separation between the region of the main current flow and the voltage measurement region. All experimental observations can be consistently mod…
Herein we discuss the fabrication of ballistic suspended graphene nanostructures supplemented with local gating. Using in-situ current annealing, we show that exceptional high mobilities can be obtained in these devices. A detailed description is given of the fabrication of bottom and different top-gate structures, which enable the realization of complex graphene structures. We have studied the basic building block, the p-n junction in detail, where a striking oscillating pattern was observed, which can be traced back to Fabry-Perot oscillations that are localized in the electronic cavities fo…
Local spin and charge densities on a two-dimensional honeycomb lattice are calculated by the Landauer-Keldysh formalism (LKF). Through the empirical tight-binding method, we show how the realistic band structure can be brought into the LKF. Taking the Bi(111) surface, on which strong surface states and Rashba spin-orbit coupling are present [Phys. Rev. Lett. 93, 046403 (2004)], as a numeric example, we show typical intrinsic spin-Hall accumulation (ISHA) patterns thereon. The Fermi-energy-dependence of the spin and charge transport in two-terminal nanostructure samples is subsequently analyzed…
The influence of nanostructuring and quantum confinement on the thermoelectric properties of materials has been extensively studied. While this has made possible multiple breakthroughs in the achievable figure of merit, classical confinement, and its effect on the local Seebeck coefficient has mostly been neglected, as has the Peltier effect in general due to the complexity of measuring small temperature gradients locally. Here we report that reducing the width of a graphene channel to 100 nm changes the Seebeck coefficient by orders of magnitude. Using a scanning thermal microscope allows us …
