Thermoelectric Effects in Nanoscale Junctions

Abstract

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 the latter to the electrodes. This demonstrates that Fourier's law -a paradigm of statistical mechanics -is generally violated in nanoscale junctions. PACS numbers: 72.15.Jf,73.63.Rt,65.80.+n Non-equilibrium (NE) processes at the nanoscale are receiving a great deal of attention due in large part to the advancements in fabrication and manipulation of these systems.[1] An especially interesting class of NE phenomena pertain to energy transport and the conversion of thermal to electrical energy. When a thermal gradient ∆T is applied to a finite system, electrons respond by departing from their ground state to partially accumulate at one end of the system, thus creating a measurable voltage difference ∆V . The ratio S = − ∆V ∆T is called thermopower [2], and has been measured in a variety of nano-scale systems such as quantum point contacts [3], atomic-size metallic wires [4], quantum dots [5], Si nanowires [6] and recently in molecular junctions [7]. In a bulk material, when S < 0 the transient current is carried by electrons; when S > 0 it is carried by holes.In nanoscale systems this NE problem has recently received a lot of attention [3,[8][9][10][11][12][13]. In these theories the single-particle scattering formalism [14] is used to relate the thermopower to single-particle transmission probabilities. This approach, however, does not take into account the dynamical formation of the thermopower and neglects the fact that even at steady state, when the charge current is zero an energy current is still present, like, e.g., in insulators [15]. Another effect neglected by such theories, which is now within reach of experimental verification [16], is the formation of local temperature variations along the structure. In order to study all these effects one needs to describe a nanoscale system interacting with an environment that maintains the thermal gradient, namely one needs to resort to a theory of NE open quantum systems.In this letter we introduce such a theory, based on a generalization of quantum master equations, and use it to study the dynamical formation of thermo-electric effects in nanojunctions. We show that the thermopower is a highly non-linear function of the thermal gradient and it is very sensitive to the junction geometry, even in the simplest case of non-interacting electrons. This precludes an easy interpretation of its sign in terms of electrons or holes as it has been argued in some literature [3,[8][9][...

Thermodynamic meaning of local temperature of nonequilibrium open quantum systems

“…Such local heating affects crucially the device properties [13][14][15][16], and have significant influence on some physical processes, such as thermoelectric conversion [17][18][19], heat dissipation [8,19], and electron-phonon interactions [20,21]. All these studies, however, leave open the question of what precisely is a "local temperature" in a nonequilibrium system, a concept that has a well-established meaning only in global equilibrium.Over the past decade, numerous experimental [15,[22][23][24][25][26][27] and theoretical [28][29][30][31][32][33][34][35][36][37][38] efforts have been made to provide practical and meaningful definitions of local temperature for nonequilibrium systems that bear a close conceptual resemblance to the thermodynamic one. However, it has remained largely unclear how to physically interpret the defined local temperature, and how to associate the measured value with the magnitudes of local excitations and local heating at a quantitative level.…”

“…In particular, we shall focus on the definition of local temperature based on the zero-current condition (ZCC) proposed by Engquist and Anderson [39], and that based on the minimal-perturbation condition (MPC) as proposed in Refs. [30,37]. Here, we will focus only on the electronic contribution to the local temperature and leave to future studies the analysis of the effects of phonons.…”

“…By locally coupling a probe (p) to the QD, the expectation value of a local observable O = ⟨Ô⟩ is subjected to a perturbation that depends explicitly on T p and µ p . Within the MPC, the local temperature T * of the QD is determined by varying T p so that the perturbation δO p (T p , µ p ) is minimized [30,37,45]:During the variation of T p , µ p is kept aligned with a preset local chemical potentialwhere coefficients ζ L and ζ R are determined by measuring the electric currents [45].At zero bias T * determined by Eq.(1) recovers exactly the physical equilibrium temperature. While in many cases the MPC-defined T * is numerically close to that obtained by the ZCC [37], the former does not require the measurement of heat currents, and hence its experimental realization is feasible.…”

Scattering theory of thermocurrent in quantum dots and molecules

“…However, application of these advanced formalisms to realistic models simulating molecular junctions is extremely difficult. Several simplified approaches based on scattering theory [6] and on quantum rate equations [5,14,20,31,32] were developed and used to study thermoelectric properties of molecular junctions taking into account contributions of vibrational phonons and electron-vibron interactions. Very recently, a scattering theory based approach was suggested to analyze weakly nonlinear thermoelectric transport in mesoscopic systems [25][26][27].…”

Spin Thermoelectric Effects in Transport through a Two-Level Quantum Dot Coupled to Ferromagnetic Leads

“…When the spin accumulation exist in the external leads, one can introduce a spin analog of the gure of merit, given by ZT spin = G s S 2 s T /κ, where G s and S s denote the spin conductance (normalized to /2e) and the spin Seebeck coecient, respectively. The system is a good heat to spin-voltage converter when ZT spin > 1 [8].…”