Theoretical approaches for nanoscale thermoelectric phenomena
Carmine Antonio Perroni, Giuliano Benenti
备注：
20 pages, 7 figures, to be published as chapter in the Proceedings of Varenna Summer Course 207 - Advances in thermoelectricity: foundational issues, materials and nanotechnology
Focus of the chapter is on the theoretical approaches aimed to analyze thermoelectric properties at the nanoscale. We discuss several relevant theoretical approaches for different set-ups of nano-devices providing estimations of the thermoelectric parameters in the linear and non-linear regime, in particular the thermoelectric figure of merit and the power-efficiency trade-off. Moreover, we analyze the role of not only electronic, but also of vibrational degrees of freedom. First, nanoscale thermoelectric phenomena are considered in the quantum coherent regime using the Landauer-Büttiker method and focusing on effects of energy filtering. Then, we analyze the effects of many-body couplings between nano-structure degrees of freedom, such as electron-electron and electron-vibration interactions, which can strongly affect the thermoelectric conversion. In particular, we discuss the enhancement of the thermoelectric figure of merit in the Coulomb blockade regime for a quantum dot model starting from the master equation for charge state probabilities and the tunneling rates through the electrodes. Finally, within the non-equilibrium Green function formalism, we quantify the reduction of the thermoelectric performance in simple models of molecular junctions due to the effects of the electron-vibration coupling and phonon transport at room temperature.
We investigate the transport through a nanoscale device consisting of a degenerate double-orbital Anderson dot coupled to two uncorrelated leads. We determine the thermoelectric transport properties close to the one-electron regime and compare them to a corresponding single-orbital dot. The linear and nonlinear regimes are addressed, the latter via a non-equilibrium generalization of the non-crossing approximation based on the Keldysh formalism. Power output and efficiency in the Kondo regime are shown to be strongly enhanced through the presence of a second orbital. We predict an experimental…
We show that the thermoelectric properties of nanoscale junctions featuring states near the Fermi level strongly depend on the type of resonance generated by such states, which can be either Fano or Breit-Wigner-like. We give general expressions for the thermoelectric coefficients generated by the two types of resonances and calculate the thermoelectric properties of these systems, which encompass most nanoelectronics junctions. We include simulations of real junctions where metalloporphyrin molecules bridge gold electrodes and prove that for some metallic elements the thermoelectric propertie…
We present thermoelectric measurements of the heat dissipated due to ferromagnetic resonance of a Permalloy strip. A microwave magnetic field, produced by an on-chip coplanar strip waveguide, is used to drive the magnetization precession. The generated heat is detected via Seebeck measurements on a thermocouple connected to the ferromagnet. The observed resonance peak shape is in agreement with the Landau-Lifshitz-Gilbert equation and is compared with thermoelectric finite element modeling. Unlike other methods, this technique is not restricted to electrically conductive media and is therefore…
The quest to identify the best heat engine has been at the center of science and technology. Thermoelectric nanoscale heat engines convert heat flows into useful work in the form of electrical power and promise the realization of on-chip power production. Considerable studies have so far revealed the potentials to yield an enhanced efficiency originating from quantum confinement effects and energy-dependent transport properties. However, the full benefit of many-body interactions in thermoelectric is yet to be investigated; identifying the optimal interaction is a hard problem due to combinato…
We propose a thermoelectric cooling device based on an atomic-sized junction. Using first-principles approaches, we investigate the working conditions and the coefficient of performance (COP) of an atomic-scale electronic refrigerator where the effects of phonon's thermal current and local heating are included. It is observed that the functioning of the thermoelectric nano-refrigerator is restricted to a narrow range of driving voltages. Compared with the bulk thermoelectric system with the overwhelmingly irreversible Joule heating, the 4-Al atomic refrigerator has a higher efficiency than a b…
We theoretically study how one can control and enhance nonlinear thermoelectricity by regulating quantum coherence in nanostructures such as a quantum dot system or a single-molecule junction. In nanostructures, the typical temperature scale is much smaller than the resonance width, which largely suppresses thermoelectric effects. Yet we demonstrate one can achieve a reasonably good thermoelectric performance by regulating quantum coherence. Engaging a quantum-dot interferometer (a quantum dot embedded in the ring geometry) as a heat engine, we explore the idea of thermoelectric enhancement in…
We study quantum transport through two-terminal nanoscale devices in contact with two particle reservoirs at different temperatures and chemical potentials. We discuss the general expressions controlling the electric charge current, heat currents and the efficiency of energy transmutation in steady conditions in the linear regime. With focus in the parameter domain where the electron system acts as a power-generator, we elaborate workable expressions for optimal efficiency and thermoelectric pameters of nanoscale devices. The general concepts are set at work in the paradigmatic cases of Lorent…
Despite its intrinsic non-equilibrium origin, thermoelectricity in nanoscale systems is usually described within a static scattering approach which disregards the dynamical interaction with the thermal baths that maintain energy flow. Using the theory of open quantum systems we show instead that unexpected properties, such as a resonant structure and large sign sensitivity, emerge if the non-equilibrium nature of this problem is considered. Our approach also allows us to define and study a local temperature, which shows hot spots and oscillations along the system according to the coupling of t…
We analyze an electronic nanoscale thermal machine driven by time-dependent environment: besides bias and gate voltage variations, we consider also the less prevailing time modulation of the couplings between leads and dot. We provide energy and heat current expressions in such situations, as well as expressions for the power exchanged between the dot+leads system and its outside. Calculations are made in the Keldysh nonequilibrium Green's function framework. We apply these results to design a cyclic refrigerator, circumventing the ambiguity of defining energy flows between subsystems in the c…
This mini-review is intended as a short introduction to electron flow in nanostructures. Its aim is to provide a brief overview of this topic for people who are interested in the thermodynamics of quantum systems but know little about nanostructures. We particularly emphasize devices that work in the steady-state, such as simple thermoelectrics, but also mention cyclically driven heat engines. We do not aim to be either complete or rigorous, but use a few pages to outline some of the main ideas in the topic.
We propose a possible route to achieve high thermoelectric efficiency in molecular junctions by combining a local chemical tuning of the molecular electronic states with the use of semiconducting electrodes. The former allows to control the position of the HOMO transmission resonance with respect to the Fermi energy while the latter fulfills a twofold purpose: the suppression of electron-like contributions to the thermopower and the cut-off of the HOMO transmission tails into the semiconductor band gap. As a result a large thermopower can be obtained. Our results strongly suggest that large fi…
Based on the Landauer-Büttiker theory, we explore the thermal regimes of two-terminal nanoscale systems with an energy-peaked transmission function. The device is in contact with two reservoirs held at different temperatures and chemical potentials. We identify the operation regions where the system acts as energy pump (thermal machine) or heat pump (refrigerator machine), or where it is working in dissipative modes. The corresponding thermoelectric parameters are obtained without numerical calculations. The recent literature, by focusing on systems with box-like or step-like shapes of the tra…
In this work, we aim to design a heterostructure based nanoscale thermoelectric generator that can maximize the waste-heat conversion efficiency at a given output power. The primary objective to be achieved for this is to realize a boxcar-shaped (bandpass) electronic transmission function (R. S. Whitney, Phys. Rev. Lett. 112, 130601 (2014)). In order to achieve that, we propose the use of an electronic analog of optical Fabry-Pérot cavity over a central resonant tunneling structure. We further explore the optimum design possibilities by varying the geometry of the cavity wall to ensure a nearl…
The addition of porosity to thermoelectric materials can significantly increase the figure of merit, ZT, by reducing the thermal conductivity. Unfortunately, porosity is also detrimental to the thermoelectric power factor in the numerator of the figure of merit ZT. In this manuscript we derive strategies to recoup electrical performance in nanoporous Si by fine tuning the carrier concentration and through judicious design of the pore size and shape so as to provide energy selective electron filtering. In this study, we considered phosphorus doped silicon containing discrete pores that are eith…
Thermoelectric transport in nanoscale conductors is analyzed in terms of the response of the system to a thermo-mechanical field, first introduced by Luttinger, which couples to the electronic energy density. While in this approach the temperature remains spatially uniform, we show that a spatially varying thermo-mechanical field effectively simulates a temperature gradient across the system and allows us to calculate the electric and thermal currents that flow due to the thermo-mechanical field. In particular, we show that, in the long-time limit, the currents thus calculated reduce to those …
Research on thermoelectrical energy conversion, the reuse of waste heat produced by some mechanical or chemical processes to generate electricity, has recently gained some momentum. The calculation of the electronic parameters entering the figure of merit of this energy conversion, and therefore the discovery of efficient materials, is usually performed starting from the Landauer's approach to quantum transport coupled with the Onsager's linear response theory. As it is well known, that approach suffers of certain serious drawbacks. Here, we discuss alternative dynamical methods that can go be…
We propose here a feasible and effective design of new thermoelectric systems based on physical gaps in nanoscale junctions. We show that, depending on the type of features, i.e. the character of surface/edge states, on both sides of the gap, it is possible to achieve rather high figures of merit. In particular, we show that for configurations that have localized states at the surfaces/edges, which translate into sharp resonances in the transmission, it is possible to achieve large Seebeck coefficients and figures of merit by carefully tuning their energy and their coupling to other states. We…
We have experimentally studied the role of thermoelectric effects in nanoscale nonlocal spin valve devices. A finite element thermoelectric model is developed to calculate the generated Seebeck voltages due to Peltier and Joule heating in the devices. By measuring the first, second and third harmonic voltage response non locally, the model is experimentally examined. The results indicate that the combination of Peltier and Seebeck effects contributes significantly to the nonlocal baseline resistance. Moreover, we found that the second and third harmonic response signals can be attributed to Jo…
We study thermoelectric phenomena in a system consisting of strongly correlated quantum dot coupled to external leads in the Kondo regime. We calculate linear and nonlinear electrical and thermal conductance and thermopower of the quantum dot and discuss the role of asymmetry in the couplings to external electrodes. In the linear regime electrical and thermal conductances are modified, while thermopower remains unchanged. In the nonlinear regime the Kondo resonance in differential conductance develops at non-zero source-drain voltage, which has important consequences on thermoelectric properti…
Thermoelectric material provides a high hope for converting harmful and useless heat into useful energy, electricity. It can also be used as solid state Peltier coolers in integrated circuits, an outstanding challenge for electronic engineers. The desire for a high efficient thermoelectric material has never been so keen. Although many works have been done, we are still far from having a recipe for thermoelectric materials. Nano-structuring provides an effective way to increase figure of merit (ZT) by reducing the thermal conductivity without affecting electronic property.1 Here, we propose a …
Thermodynamic currents can fluctuate significantly at the nanoscale. But some currents fluctuate less than others. Hyperaccurate currents are those which fluctuate the least, in the sense that they maximize the signal-to-noise ratio (precision). In this letter we analytically determine the hyperaccurate current in the case of a quantum thermoelectric, modeled by the Landauer-Büttiker formalism.
Thermoelectric effects in a double quantum dot system coupled to external magnetic/nonmagnetic leads are investigated theoretically. The basic thermoelectric transport characteristics, like thermopower, electronic contribution to heat conductance, and the corresponding figure of merit, have been calculated in terms of the linear response theory and Green function formalism in the Hartree-Fock approximation for Coulomb interactions. An enhancement of the thermal efficiency (figure of merit ZT) due to Coulomb blockade has been found. The magnitude of ZT is further considerably enhanced by quantu…
Nanoscale structuring holds promise to improve thermoelectric properties of materials for energy conversion and photodetection. We report a study of the spatial distribution of the photothermoelectric voltage in thin-film nanowire devices fabricated from single metal. A focused laser beam is used to locally heat the metal nanostructure via a combination of direct absorption and excitation of a plasmon resonance in Au devices. As seen previously, in nanowires shorter than the spot size of the laser, we observe a thermoelectric voltage distribution that is consistent with the local Seebeck coeff…
We theoretically consider the spin Seebeck effect, the charge Seebeck coefficient, and the thermoelectric figure of merit in superconducting hybrid structures including either magnetic textures or intrinsic spin-orbit coupling. We demonstrate that large magnitudes for all these quantities are obtainable in Josephson-based systems with either zero or a small externally applied magnetic field. This provides an alternative to the thermoelectric effects generated in high-field ($\sim 1$ T) superconducting hybrid systems, which were recently experimentally demonstrated. The systems studied contain …
We investigate transport and thermoelectric properties of hybrid systems based on a single-level quantum dot and one superconducting lead. The other lead is generally normal-metallic ferromagnet. In the latter case single-particle transport is spin-polarized. Our main interest is in the interplay of Andreev tunneling of Cooper pairs and single-particle tunneling. The latter is responsible for relatively large thermopower and figure of merit due to a diverging density of single-particle states at the superconducting gap edges. System with ferromagnetic and superconducting leads can also reveal …
The thermoelectric transport coefficients are calculated in a generic lattice model of multi-Weyl semimetals with a broken time-reversal symmetry by using the Kubo's linear response theory. The contributions connected with the Berry curvature-induced electromagnetic orbital and heat magnetizations are systematically taken into account. It is shown that the thermoelectric transport is profoundly affected by the nontrivial topology of multi-Weyl semimetals. In particular, the calculation reveals a number of thermal coefficients of the topological origin which describe the anomalous Nernst and th…
We experimentally verify hitherto untested theoretical predictions about the thermoelectric properties of Kondo correlated quantum dots (QDs). The specific conditions required for this study are obtained by using QDs epitaxially grown in nanowires, combined with a recently developed method for controlling and measuring temperature differences at the nanoscale. This makes it possible to obtain data of very high quality both below and above the Kondo temperature, and allows a quantitative comparison with theoretical predictions. Specifically, we verify that Kondo correlations can induce a polari…
We predict an enormous order-dependent quantum enhancement of thermoelectric effects in the vicinity of a higher-order `supernode' in the transmission spectrum of a nanoscale junction. Single-molecule junctions based on 3,3'-biphenyl and polyphenyl ether (PPE) are investigated in detail. The nonequilibrium thermodynamic efficiency and power output of a thermoelectric heat engine based on a 1,3-benzene junction are calculated using many-body theory, and compared to the predictions of the figure-of-merit ZT.
The reduction of the thermal conductivity in nanostructures opens up the possibility of exploiting for thermoelectric purposes also materials such as silicon, which are cheap, available and sustainable but with a high thermal conductivity in their bulk form. The development of thermoelectric devices based on these innovative materials requires reliable techniques for the measurement of thermal conductivity on a nanometric scale. The approximations introduced by conventional techniques for thermal conductivity measurements can lead to unreliable results when applied to nanostructures, because h…
We review recent developments in nonlinear quantum transport through nanostructures and mesoscopic systems driven by thermal gradients or in combination with voltage biases. Low-dimensional conductors are excellent platforms to analyze both the thermoelectric and heat dynamics beyond linear response because due to their small size a small temperature difference applied across regions gives rise to large thermal biases. We offer a theoretical discussion based on the scattering approach to highlight the differences between the linear and the nonlinear regimes of transport. We discuss recent expe…
Recent advances in creating complex oxide heterostructures, interfaces formed between two different transition metal oxides, have heralded a new era of materials and physics research, enabling a uniquely diverse set of coexisting physical properties to be combined with an ever- increasing degree of experimental control. Already, these systems have exhibited such varied phenomena as superconductivity, magnetism, and ferroelasticity, all of which are gate-tunable, demonstrating their promise for fundamental discovery and technological innovation alike. To fully exploit this richness, it is neces…
Bismuth chalcogenides and lead telluride/selenide alloys exhibit exceptional thermoelectric properties which could be harnessed for power generation and device applications. Since phonons play a significant role in achieving these desired properties, quantifying the interaction between phonons and electrons, which is encoded in the Eliashberg function of a material, is of immense importance. However, its precise extraction has in part been limited due to the lack of local experimental probes. Here we construct a method to directly extract the Eliashberg function using Landau level spectroscopy…
Topological insulators (TI) realize a novel state of quantum matter that are distinguished by topological invariants of bulk band structure rather than spontaneously broken symmetries. A number of exotic quantum phenomena have been predicted to exist in multiply-connected geometries which require an enormous amount of materials flexibility. We have extended our previous search for TI materials from binary (Bi2X3 series) to the thermoelectric ternary compounds. We discover that the distorted LuPtSb is the first ternary compound harboring a 3D topological insulator state. We also show that the h…
We review the progress that has been recently made in the application of time-dependent density functional theory to thermoelectric phenomena. As the field is very young, we emphasize open problems and fundamental issues. We begin by introducing the formal structure of \emph{thermal density functional theory}, a density functional theory with two basic variables -- the density and the energy density -- and two conjugate fields -- the ordinary scalar potential and Luttinger's thermomechanical potential. The static version of this theory is contrasted with the familiar finite-temperature density…
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…
Thermoelectric effects allow the generation of electrical power from waste heat and the electrical control of cooling and heating. Remarkably, these effects are also highly sensitive to the asymmetry in the density of states around the Fermi energy and can therefore be exploited as probes of distortions in the electronic structure at the nanoscale. Here we consider two-dimensional graphene as an excellent nanoscale carbon material for exploring the interaction between electronic and thermal transport phenomena, by presenting a direct and quantitative measurement of the Peltier component to ele…
Recent developments in nanoscale experimental techniques made it possible to utilize single molecule junctions as devices for electronics and energy transfer with quantum coherence playing an important role in their thermoelectric characteristics. Theoretical studies on the efficiency of nanoscale devices usually employ rate (Pauli) equations, which do not account for quantum coherence. Therefore, the question whether quantum coherence could improve the efficiency of a molecular device cannot be fully addressed within such considerations. Here, we employ a nonequilibrium Green function approac…
An expression describing the controlling parameters involved in short range nanoscale dissipation is proposed and supported by simulations and experimental findings. The expression is deconstructed into the geometrical, dynamic, chemical and mechanical properties of the system. In atomic force microscopy these are translated into 1) tip radius and tip-sample deformation, 2) resonant frequency and oscillation amplitude and 3) hysteretic and viscous dissipation. The latter are characteristic parameters defining the chemical and mechanical properties of the tip-sample system. Long range processes…
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 …
Thermoelectric transport phenomena in a single-level quantum dot coupled to ferromagnetic leads are considered theoretically in the Kondo regime. The dot is described by the Anderson model with Rashba type spin-orbit interaction. The finite-U mean field slave boson technique is used to describe the transport characteristics, like heat conductance, thermopower, thermoelectric efficiency (figure of merit). The role of quantum interference effects in thermoelectric parameters is also analyzed.
The thermoelectric properties of a nanoscale germanium segment connected by aluminium nanowires are studied using scanning thermal microscopy. The germanium segment of 168\,nm length features atomically sharp interfaces to the aluminium wires and is surrounded by an Al$_2$O$_3$ shell. The temperature distribution along the self-heated nanowire is measured as a function of the applied electrical current, for both Joule and Peltier effects. An analysis is developed that is able to extract the thermal and thermoelectric properties including thermal conductivity, the thermal boundary resistance to…
The magnetic skyrmion is a nanoscale topological object characterized by the winding of the magnetic moments, appearing in magnetic materials with broken inversion symmetry. Because of its low current threshold for driving, the skyrmions have been intensely studied toward novel storage applications by using electron-beam, X-ray, and visible light microscopies. Here, we show that the skyrmions can be imaged via thermoelectric signals of spin-caloritronic phenomena in combination with focused heating. We applied a local temperature gradient to a CoFeB/Ta/W multilayer film with scanning a heating…
We study nonequilibrium thermoelectric transport properties of a correlated impurity connected to two leads for temperatures below the Kondo scale. At finite bias, for which a current flows across the leads, we investigate the differential response of the current to a temperature gradient. In particular, we compare the influence of a bias voltage and of a finite temperature on this thermoelectric response. This is of interest from a fundamental point of view to better understand the two different decoherence mechanisms produced by a bias voltage and by temperature. Our results show that in thi…
Enhanced thermoelectric effects in a driven one-dimensional system
NEST, Istituto Nanoscienze-CNR and Scuola Normale Superiore（NEST，意大利国家研究委员会纳米科学研究所和比萨高等师范学院）；Institute for Quantum Studies, Chapman University（查普曼大学量子研究学院）；Department of Physics, Lancaster University（兰开斯特大学物理系）
We study transport phenomena cross-correlated among the heat and electric currents of magnons and Dirac electrons on the surface of ferromagnetic topological insulators. For a perpendicular magnetization, we calculate magnon- (electron-) drag anomalous Nernst/Seebeck (anomalous Ettingshausen/Peltier) effects and magnon-/electron-drag thermal Hall effects. The magnon-drag thermoelectric effects are interpreted to be caused by magnon-induced electromotive force. When the magnetization has in-plane components, there arise thermal/thermoelectric analogs of anisotropic magnetoresistance (AMR). In t…
We present a method, based on characterizing efficiency fluctuations, to asses the performance of nanoscale thermoelectric junctions. This method accounts for effects typically arising in small junctions, namely, stochasticity in the junction's performance, quantum effects, and nonequilibrium features preventing a linear response analysis. It is based on a nonequilibrium Green's function (NEGF) approach, which we use to derive the full counting statistics (FCS) for heat and work, and which in turn allows us to calculate the statistical properties of efficiency fluctuations. We simulate the lat…
Understanding the electronic and phononic transport properties of junctions consisting of a scattering region such as a nanoscale matters or molecules connected to two or more electrodes is the central basis for future nano and molecular scale applications. The theoretical and mathematical techniques to treat electron and phonon transport are leading to model the physical properties of nano and molecular scale junctions. My aim in this paper is to review the theoretical and mathematical techniques to treat electron and phonon transport in nano and molecular scale junctions leading to models of…
Atomically thin layers of two-dimensional (2D) materials such as graphene, MoS2 and h-BN have immense potential as sensors and electronic devices thanks to their highly desirable electronic, mechanical, optical and heat transport properties. In particular their extreme stiffness, tensile strength and low density allows for high frequency electronic devices, resonators and ultra-sensitive detectors providing realistic avenues for down-scaling electronic devices and nanoelectromechanical systems (NEMS). Whilst nanoscale morphology and electronic properties of 2D materials can be studied using ex…
Quantum dots (QDs) are good model systems for fundamental studies of mesoscopic transport phenomena using thermoelectric effects because of their small size, electrostatically tunable properties and thermoelectric response characteristics that are very sensitive to small thermal biases. Here we provide a review of experimental studies on thermoelectric properties of single QDs realized in two-dimensional electron gases, single-walled carbon nanotubes and semiconductor nanowires. A key requirement for such experiments is methods for nanoscale thermal biasing. We briefly review the main techniqu…
Understanding transport processes in complex nanoscale systems, like ionic conductivities in nanofluidic devices or heat conduction in low dimensional solids, poses the problem of examining fluctuations of currents within nonequilibrium steady states and relating those fluctuations to nonlinear or anomalous responses. We have developed a systematic framework for computing distributions of time integrated currents in molecular models and relating cumulants of those distributions to nonlinear transport coefficients. The approach elaborated upon in this perspective follows from the theory of dyna…
In the past decade, a new research frontier emerges at the interface between physics and renewable energy, termed as the inelastic thermoelectric effects where inelastic transport processes play a key role. The study of inelastic thermoelectric effects broadens our understanding of thermoelectric phenomena and provides new routes towards high-performance thermoelectric energy conversion. Here, we review the main progress in this field, with a particular focus on inelastic thermoelectric effects induced by the electron-phonon and electron-photon interactions. We introduce the motivations, the b…
