Thermoelectric effects in correlated quantum dots and molecules
V. Schoeps, V. Zlatic, T. A. Costi
备注：
4 pages, 3 figures; to appear in the Proceedings of the International Conference on Strongly Correlated Electron Systems, Santa Fe 2010; revised version: typos corrected and references updated
We investigate thermoelectric properties of correlated quantum dots and molecules, described by a single level Anderson model coupled to conduction electron leads, by using Wilson's numerical renormalization group method. In the Kondo regime, the thermopower, $S(T)$, exhibits two sign changes, at temperatures $T=T_{1}$ and $T=T_{2}>T_{1}$. We find that $T_{2}$ is of order the level width $Γ$ and $T_{1}> T_{p}\approx T_{K}$, where $T_{p}$ is the position of the Kondo induced peak in the thermopower and $T_{K}$ is the Kondo scale. No sign change is found outside the Kondo regime, or, for weak correlations, making a sign change in $S(T)$ a particularly sensitive signature of strong correlations and Kondo physics. For molecules, we investigate the effect of screening by conduction electrons on the thermoelectric transport. We find that a large screening interaction enhances the figure of merit in the Kondo and mixed valence regimes.
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 study thermoelectric transport through double quantum dots system with spin-dependent interdot coupling and ferromagnetic electrodes by means of the non-equilibrium Green function in the linear response regime. It is found that the thermoelectric coefficients are strongly dependent on the splitting of interdot coupling, the relative magnetic configurations and the spin polarization of leads. In particular, the thermoelectric efficiency can achieve considerable value in parallel configuration when the effective interdot coupling and tunnel coupling between QDs and the leads for spin-down ele…
In this Letter we study thermoelectric effects in ultra small quantum dots. We study the behaviour of the thermopower, Peltier coefficient and thermal conductance both in the sequencial tunneling regime and in the regime where Kondo correlations develope. Both cases of linear response and non-equilibrium induced by strong temperature gradients are considered. The thermopower is a very sensitive tool to detect Kondo correlations. It changes sign both as a function of temperature and temperature gradient. We also discuss violations of the Wiedemann-Franz law.
We review recent theoretical work on thermoelectric energy harvesting in multi-terminal quantum-dot setups. We first discuss several examples of nanoscale heat engines based on Coulomb-coupled conductors. In particular, we focus on quantum dots in the Coulomb-blockade regime, chaotic cavities and resonant tunneling through quantum dots and wells. We then turn towards quantum-dot heat engines that are driven by bosonic degrees of freedom such as phonons, magnons and microwave photons. These systems provide interesting connections to spin caloritronics and circuit quantum electrodynamics.
We propose a scheme of multilayer thermoelectric engine where {\em one} electric current is coupled to {\em two} temperature gradients in three-terminal geometry. This is realized by resonant tunneling through quantum dots embedded in two thermal and electrical resisting polymer matrix layers between highly conducting semiconductor layers. There are two thermoelectric effects, one of which is pertaining to inelastic transport processes (if energies of quantum dots in the two layers are different) while the other exists also for elastic transport processes. {These two correspond to the transver…
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 investigate the finite-frequency thermal transport through a quantum dot subject to strong interactions, by providing an exact, nonperturbative formalism that allows us to carry out a systematic analysis of the thermopower at any frequency. Special emphasis is put on the dc and high-frequency limits. We demonstrate that, in the Kondo regime, the ac thermopower is characterized by a universal function that we determine numerically.
We study the effects caused by Rashba and Dresselhaus spin-orbit coupling over the thermoelectric transport properties of a single-electron transistor, viz., a quantum dot connected to one-dimensional leads. Using linear response theory and employing the numerical renormalization group method, we calculate the thermopower, electrical and thermal conductances, dimensionless thermoelectric figure of merit, and study the Wiedemann-Franz law, showing their temperature maps. Our results for all those properties indicate that spin-orbit coupling drives the system into the Kondo regime. We show that …
We consider the steady-state thermoelectric transport through a vibrating molecular quantum dot that is contacted to macroscopic leads. For moderate electron-phonon interaction strength and comparable electronic and phononic timescales, we investigate the impact of the formation of a local polaron on the thermoelectric properties of the junction. We apply a variational Lang-Firsov transformation and solve the equations of motion in the Kadanoff-Baym formalism up to second order in the dot-lead coupling parameter. We calculate the thermoelectric current and voltage for finite temperature differ…
The power factor of a thermoelectric device is a measure of the heat-to-energy conversion efficiency in nanoscopic devices. Yet, even as interest in low-dimensional thermoelectric materials has increased, experimental research on what influences the power factor in these systems is scarce. Here, we present a detailed thermoelectric study of graphene quantum dot devices. We show that spin-degeneracy of the quantum dot states has a significant impact on the zero-bias conductance of the device and leads to an increase of the power factor. Conversely, we demonstrate that non-ideal heat exchange wi…
Quantum dots are model systems for quantum thermoelectric behavior because of the ability to control and measure the effects of electron-energy filtering and quantum confinement on thermoelectric properties. Interestingly, nonlinear thermoelectric properties of such small systems can modify the efficiency of thermoelectric power conversion. Using quantum dots embedded in semiconductor nanowires, we measure thermovoltage and thermocurrent that are strongly nonlinear in the applied thermal bias. We show that most of the observed nonlinear effects can be understood in terms of a renormalization o…
We have theoretically studied the thermoelectric properties of serially coupled quantum dots (SCQD) embedded in an insulator matrix connected to metallic electrodes. In the framework of Keldysh Green's function technique, the Landauer formula of transmission factor is obtained by using the equation of motion method. Based on such analytical expressions of charge and heat currents, we calculate the electrical conductance, Seebeck coefficient, electron thermal conductance and figure of merit (ZT) of SCQD in the linear response regime. The effects of electron Coulomb interactions on the reduction…
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…
We demonstrate high-temperature thermoelectric conversion in InAs/InP nanowire quantum dots by taking advantage of their strong electronic confinement. The electrical conductance G and the thermopower S are obtained from charge transport measurements and accurately reproduced with a theoretical model accounting for the multi-level structure of the quantum dot. Notably, our analysis does not rely on the estimate of co-tunnelling contributions since electronic thermal transport is dominated by multi-level heat transport. By taking into account two spin-degenerate energy levels we are able to eva…
Coupled double quantum dots (c-2QD) connected to leads have been widely adopted as prototype model systems to verify interference effects on quantum transport at the nanoscale. We provide here an analytic study of the thermoelectric properties of c-2QD systems pierced by a uniform magnetic field. Fully analytic and easy-to-use expressions are derived for all the kinetic functionals of interest. Within the Green 0 s function formalism, our results allow a simple inexpensive procedure for the theoretical description of the thermoelectric phenomena for different chemical potentials and temperatur…
Molecular junctions comprise small molecules on the order of one or a few nanometers in length, chemically attached to two electrodes of metal or semiconductors. They have recently been identified as promising candidates for materials/devices for thermoelectric energy conversion. Molecules have the potential to act as sharp energy filters for electrical currents and could outperform other materials considered for thermoelectric energy conversion so far, thereby providing a wide-spread technological relevance thermoelectrics could not achieve to date. However, there is a clear gap between predi…
It was recently suggested that molecular junctions would be excellent elements for efficient and high-power thermoelectric energy conversion devices. However, experimental measurements of thermoelectric conversion in molecular junctions have indicated rather poor efficiency, raising the question of whether it is indeed possible to design a setup for molecular junctions that will exhibit enhanced thermoelectric performance. Here we suggest that hybrid single-molecule nanoparticle junctions can serve as efficient thermoelectric converters. The introduction of a semiconducting nanoparticle introd…
We study the thermoelectrical transports for an interacting dot attached to two graphene electrodes. Graphene band structure shows a pseudogap density of states that affects strongly the transport properties. In this work, we focus on the Coulomb blockade regime and derive the expression for Onsager matrix O_{ij} that relates the electrical and heat currents with electrical and thermal biases in the linear response regime. Our findings show double peak structures for the electrical and thermal conductances versus the dot level in accordance with the Coulom blockade phenomenon. Remarkably, howe…
Thermoelectric effects in a quantum dot coupled to the source and drain charge reservoirs are explored using a nonequilibrium Green's functions formalism beyond the Hartree-Fock approximation. Thermal transport is analyzed within a linear response regime. A transition from Coulomb blockade regime to Kondo regime in thermoelectric transport through a single-level quantum dot is traced using unified approximations for the relevant Green's functions.
We report a numerical renormalization-group study of the thermoelectric effect in the single-electron transistor (SET) and side-coupled geometries. As expected, the computed thermal conductance and thermopower curves show signatures of the Kondo effect and of Fano interference. The thermopower curves are also affected by particle-hole asymmetry.
The energy dependent thermoelectric response of a single molecule contains valuable information about its transmission function and its excited states. However, measuring it requires devices that can efficiently heat up one side of the molecule while being able to tune its electrochemical potential over a wide energy range. Furthermore, to increase junction stability devices need to operate at cryogenic temperatures. In this work we report on a new device architecture to study the thermoelectric properties and the conductance of single molecules simultaneously over a wide energy range. We empl…
In this article we review the thermoelectric properties of three terminal devices with Coulomb coupled quantum dots (QDs) as observed in recent experiments [1,2]. The system we consider consists of two Coulomb-blockade QDs one of which can exchange electrons with only a single reservoir (heat reservoir) while the other dot is tunnel coupled to two reservoirs at a lower temperature (conductor). The heat reservoir and the conductor interact only via the Coulomb-coupling of the quantum dots. It has been found that two regimes have to be considered. In the first one heat flow between the two syste…
We investigate thermal transport in a serial asymmetric double quantum dot (DQD) coupled to two electron reservoirs with different temperatures. The inter- and intra-Coulomb interactions are taken into account in a Coulomb blockade DQD where the electron sequential tunneling via four different master equation approaches is considered. In the absence of Coulomb interactions, a neglectable thermoelectric and heat currents is found identifying as the Coulomb blockade DQD regime. In the presence of Coulomb interactions, intra- and inter-Coulomb interactions, crossings energies between the intra- a…
We report on the single-molecule electronic and thermoelectric properties of strategically chosen anthracene-based molecules with anchor groups capable of binding to noble metal substrates, such as gold and platinum. Specifically, we study the effect of different anchor groups, as well as quantum interference, on the electric conductance and the thermopower of gold/single-molecule/gold junctions and generally find good agreement between theory and experiment. All molecular junctions display transport characteristics consistent with coherent transport and a Fermi alignment approximately in the …
In the present work, we theoretically analyze the steady-state thermoelectric transport through a single-molecule junction with a vibrating bridge. Thermally induced charge current in the system is explored using a nonequilibrium Green's functions formalism. We study combined effects of Coulomb interactions between charge carriers on the bridge and electron-phonon interactions on the thermocurrent beyond the linear response regime. It is shown that electron-vibron interactions may significantly affect both magnitude and direction of the thermocurrent, and vibrational signatures may appear.
We investigate with the aid of numerical renormalization group techniques the thermoelectric properties of a molecular quantum dot described by the negative-U Anderson model. We show that the charge Kondo effect provides a mechanism for enhanced thermoelectric power via a correlation induced asymmetry in the spectral function close to the Fermi level. We show that this effect results in a dramatic enhancement of the Kondo induced peak in the thermopower of negative-U systems with Seebeck coefficients exceeding 50$μV/K$ over a wide range of gate voltages.
We show that coherent electron transport through zero-dimensional systems can be used to tailor the shape of the system's transmission function. This quantum-engineering approach can be used to enhance the performance of quantum dots or molecules in thermal-to-electric power conversion. Specifically, we show that electron interference in a two-level system can substantially improve the maximum thermoelectric power and the efficiency at maximum power by suppressing parasitic charge flow near the Fermi energy, and by reducing electronic heat conduction. We discuss possible realizations of this a…
We study thermoelectric effects in Coulomb-coupled quantum-dot (CCQD) systems beyond lowest-order tunneling processes and the often applied wide-band approximation. To this end, we present a master-equation (ME) approach based on a perturbative $T$-matrix calculation of the charge and heat tunneling rates and transport currents. Applying the method to transport through a non-interacting single-level QD, we demonstrate excellent agreement with the Landauer-B{ü}ttiker theory when higher-order (cotunneling) processes are included in the ME. Next, we study the effect of cotunneling and energy-depe…
We report results of theoretical studies of thermoelectric efficiency of single-molecule junctions with long molecular linkers. The linker is simulated by a chain of identical sites described using a tight-binding model. It is shown that thermoelectric figure of merit ZT strongly depends on the bridge length, being controlled by the lineshape of electron transmission function within the tunnel energy range corresponding to HOMO/LUMO transport channel. Using the adopted model we demonstrate that ZT may significantly increase as the linker lengthens, and that gateway states on the bridge (if any…
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 …
Thermoelectric properties of a two-level molecule attached to the metallic electrodes are analyzed using the equation of motion technique within the Green function formalism. Results show that the electrical conductance is strongly dependent on the electron and phonon temperatures and the electron-phonon coupling strength. In addition, it is observed that the thermal conductance peaks in the electron-hole symmetry points are vanished in the presence of the strong electron-phonon interaction. It is also found that the figure of merit is strongly suppressed in the strong electron-phonon interact…
We consider the thermoelectric response of chaotic or disordered quantum dots in the limit of phase-coherent transport, statistically described by random matrix theory. We calculate the full distribution of the thermoelectric coefficients (Seebeck $S$ and Peltier $Π$), and the thermoelectric figure of merit $ZT$, for large open dots at arbitrary temperature and external magnetic field, when the number of modes in the left and right leads ($N_{\rm L}$ and $N_{\rm R}$) are large. Our results show that the thermoelectric coefficients and $ZT$ are maximal when the temperature is half the Thouless …
We study thermoelectric devices where a single 18-annulene molecule is connected to metallic zigzag graphene nanoribbons (ZGNR) via highly transparent contacts that allow for injection of evanescent wave functions from ZGNRs into the molecular ring. Their overlap generates a peak in the electronic transmission, while ZGNRs additionally suppress hole-like contributions to the thermopower. Thus optimized thermopower, together with suppression of phonon transport through ZGNR-molecule-ZGNR structure, yield the thermoelectric figure of merit ZT ~ 0.5 at room temperature and 0.5 < ZT < 2.5 below li…
We discuss the nonlinear current of an interacting quantum dot coupled to normal and superconducting reservoirs with applied voltage and temperature differences. Due to the particle-hole symmetry introduced by the superconducting lead, the pure (subgap) thermoelectric response vanishes. However, we show that the Andreev bound states shift as the thermal gradient increases. As a consequence, the $I$--$V$ characteristic can be tuned with a temperature bias if the system is simultaneously voltage biased. This is a cross effect that occurs beyond linear response only. Furthermore, we emphasize the…
The electrical conductance, thermal conductance, thermal power and figure of merit (ZT) of semiconductor quantum dots (QDs) embedded into an insulator matrix connected with metallic electrodes are theoretically investigated in the Coulomb blockade regime. The multilevel Anderson model is used to simulate the multiple QDs junction system. The charge and heat currents in the sequential tunneling process are calculated by the Keldysh Green function technique. In the linear response regime the ZT values are still very impressive in the small tunneling rates case, although the effect of electron Co…
Thermoelectric energy harvesters can have a much higher conversion efficiency by implementing quantum dots/wells between the high temperature region and the low temperature region. However they still suffer a limitation of the maximum output power, represented by the maximum $ΔE$ (maximum energy gap of two quantum dots/wells layers). In this work, we use the piezoelectric material in the high temperature region, which has conceptually addressed the problem of the maximum power limitation. Full analysis of device physics including comparison with the existing technology and quantum simulation h…
A theoretical study of the thermoelectric current and energy harvesting in an interacting double quantum dot system, connected to reservoirs held at different chemical potentials and temperatures is presented. Using a rate-equation approach, the current is evaluated for different energetic configurations of the double quantum dot. We discuss in detail the current-temperature gradient relations (the thermoelectric analog to current-voltage relations), and demonstrate that, due to interactions, the current is non-monotonically dependent on thermal bias. This interaction-induced non-monotonicity …
Using the conformal field theory approach to the thermoelectric characteristics of fractional quantum Hall states, previously developed in Nucl. Phys. B 894 (2015) 284, we show that the thermoelectric power factor of Coulomb-blockaded islands, realized by point contacts in Fabry--Pérot interferometers in the $\mathbb{Z}_k$ parafermion Hall states, could give reliable signatures for distinguishing the topological orders of different quantum Hall states having identical electric properties. For example, while the conductance peak patterns in the Coulomb blockade regime for such states are practi…
Noble metals break preferably exposing (111)-oriented surfaces, that host Shockley type surface states (SSs). Nevertheless, the relevance of SSs on the electrical properties of gold-based molecular junctions has not been explored in detail yet. Here, we present ab initio simulations that show how the gold (111) SS, that lies approximately 0.5 eV below the Fermi energy, is key to determining correctly the electrical and thermoelectric response of the above junctions. We show how the ability to shift in a controlled way the energy position of gold SS enables us to tune the electrical and thermoe…
Quantum interference (QI) effects in molecular junctions may be used to obtain large thermoelectric responses. We study the electrical conductance G and the thermoelec- tric response of a series of molecules featuring a quinoid core using density functional theory (DFT), as well as a semi-empirical interacting model Hamiltonian describing the π-system of the molecule which we treat in the GW approximation. Molecules with a quinoid type structure are shown to have two distinct destructive QI features close to the frontier orbital energies. These manifest themselves as two dips in the transmissi…
Here we present a theoretical study of the thermoelectric transport through {[}2,2{]}para\-cyclo\-phane-based single-molecule junctions. Combining electronic and vibrational structures, obtained from density functional theory (DFT), with nonequilibrium Green's function techniques, allows us to treat both electronic and phononic transport properties at a first-principles level. For the electronic part, we include an approximate self-energy correction, based on the DFT+$Σ$ approach. This enables us to make a reliable prediction of all linear response transport coefficients entering the thermoele…
The thermoelectric transport through a molecular bridge is discussed, with an emphasis on the effects of inelastic processes of the transport electrons caused by the coupling to the vibrational modes of the molecule. In particular it is found that when the molecule is strongly coupled to a thermal bath of its own, which may be at a temperature different from those of the electronic reservoirs, a heat current between the molecule and the electrons can be converted into electric current. Expressions for the transport coefficients governing this conversion and similar ones are derived, and a poss…
Quantum dots are useful model systems for studying quantum thermoelectric behavior because of their highly energy-dependent electron transport properties, which are tunable by electrostatic gating. As a result of this strong energy dependence, the thermoelectric response of quantum dots is expected to be nonlinear with respect to an applied thermal bias. However, until now this effect has been challenging to observe because, first, it is experimentally difficult to apply a sufficiently large thermal bias at the nanoscale and, second, it is difficult to distinguish thermal bias effects from pur…
Motivated by recent experiments, we present here an ab initio study of the impact of the phonon transport on the thermal conductance and thermoelectric figure of merit of C$_{60}$-based single-molecule junctions. To be precise, we combine density functional theory with nonequilibrium Green's function techniques to compute these two quantities in junctions with either a C$_{60}$ monomer or a C$_{60}$ dimer connected to gold electrodes, taking into account the contributions of both electrons and phonons. Our results show that for C$_{60}$ monomer junctions phonon transport plays a minor role in …
Since the concepts for the implementation of data storage and logic gates used in conventional electronics cannot be simply downscaled to the level of single molecule devices, new architectural paradigms are needed, where quantum interference (QI) effects are likely to provide an useful starting point. In order to be able to use QI for design purposes in single molecule electronics, the relation between their occurrence and molecular structure has to be understood at such a level that simple guidelines for electrical engineering can be established. We made a big step towards this aim by develo…
The present work discusses a non-synthetic strategy to achieve a favorable thermoelectric response in pentacene via strain. It is found that a uni-axial strain is capable of inducing spatial anisotropy in the molecule. As a result, the transmission spectrum becomes highly asymmetric under a particular strained scenario, which is the primary requirement to get a favorable thermoelectric response. Different thermoelectric quantities are computed for the strain-induced pentacene using Green's function formalism following the Landauer-Buttiker prescription. Various scenarios are considered to make…
We theoretically investigate the thermoelectric and spin thermoelectric properties of a T-shaped double quantum dot strongly coupled to two ferromagnetic leads, focusing on transport regime where the system exhibits the two-stage Kondo effect. We study the dependence of the (spin) Seebeck coefficient, the corresponding power factor and the figure of merit on temperature, leads' spin polarization and dot level position. We show that the thermal conductance fulfills a modified Wiedemann-Franz law. We also demonstrate that the spin thermopower is enhanced at temperatures corresponding to the seco…
The quantum interference and orbital filling effects on the thermoelectric (TE) properties of quantum dot molecules with high figure of merit are illustrated via the full solution to the Hubbard- Anderson model in the Coulomb blockade regime. It is found that under certain condition in the triangular QD molecule (TQDM), destructive quantum interference (QI) can occur, which leads to vanishing small electrical conductance, while the Seebeck coefficient is modified dramatically. When TQDM is in the charge localization state due to QI, the Seebeck coefficient is seriously suppressed at low temper…
We overview nonequilibrium Green function combined with density functional theory (NEGF-DFT) modeling of independent electron and phonon transport in nanojunctions with applications focused on a new class of thermoelectric devices where a single molecule is attached to two metallic zigzag graphene nanoribbons (ZGNRs) via highly transparent contacts. Such contacts make possible injection of evanescent wavefunctions from ZGNRs, so that their overlap within the molecular region generates a peak in the electronic transmission. Additionally, the spatial symmetry properties of the transverse propaga…
Although it was demonstrated that discrete molecular levels determine the sign and magnitude of the thermoelectric effect in single-molecule junctions, full electrostatic control of these levels has not been achieved to date. Here, we show that graphene nanogaps combined with gold microheaters serve as a testbed for studying single-molecule thermoelectricity. Reduced screening of the gate electric field compared to conventional metal electrodes allows control of the position of the dominant transport orbital by hundreds of meV. We find that the power factor of graphene-fullerene junctions can …
The nonlinear electron and heat currents of quantum dot molecules (QDMs) under a temperature bias are theoretically investigated, including all correlation functions arising from electron Coulomb interactions in QDMs. Unlike the case of double QDs, the maximum efficiency of serially coupled triple QDs (SCTQD) occurs in the orbital depletion regime owing to the interdot Coulomb blockade. The electron current in SCTQD shows a bipolar oscillatory behavior with respect to the variation of QD energy levels, whereas the heat current does not show such a behavior. This is mainly attributed to thermal…
The thermoelectric transport through a ring threaded by an Aharonov-Bohm flux, with a molecular bridge on one of its arms, is analyzed. The transport electrons also interact with the vibrational excitations of that molecule. This nano-system is connected to three terminals: two are electronic reservoirs, which supply the transport electrons, and the third is the phonon bath which thermalizes the molecular vibrations. Expressions for the transport coefficients, relating all charge and heat currents to the temperature and chemical potential differences between the terminals, are derived to secon…
