Low-dimensional electronic systems in thermoelectrics have the potential to achieve high thermalto-electric energy conversion efficiency. A key measure of performance is the efficiency when the device is operated under maximum power conditions. Here we study the efficiency at maximum power of three low-dimensional, thermoelectric systems: a zero-dimensional quantum dot (QD) with a Lorentzian transmission resonance of finite width, a one-dimensional (1D) ballistic conductor, and a thermionic (TI) power generator formed by a two-dimensional energy barrier. In all three systems, the efficiency at maximum power is independent of temperature, and in each case a careful tuning of relevant energies is required to achieve maximal performance. We find that quantum dots perform relatively poorly under maximum power conditions, with relatively low efficiency and small power throughput. Ideal one-dimensional conductors offer the highest efficiency at maximum power (36% of the Carnot efficiency). Whether 1D or TI systems achieve the larger maximum power output depends on temperature and area filling factor. These results are also discussed in the context of the traditional figure of merit ZT .

Quantum dots are model systems for quantum thermoelectric behavior because of their 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 7 These authors contributed equally to this work. 8

Quantum dots are an important model system for thermoelectric phenomena, and may be used to enhance the thermal-to-electric energy conversion efficiency in functional materials, by tuning the Fermi energy relative to the dots' transmission resonances. It is therefore important to obtain a detailed understanding of a quantum dot's thermopower as a function of the Fermi energy. However, so far it has proven difficult to take the effects of interactions into account in the interpretation of experimental data. In this paper, we present detailed measurements of the thermopower of quantum dots defined in heterostructure nanowires. We show that the thermopower lineshape is described well by a Landauer-type transport model that uses as its input experimental values of the dot conductance, which contains information about interaction effects. 6

When a quantum dot is subjected to a thermal gradient, the temperature of electrons entering the dot can be determined from the dot's thermocurrent if the conductance spectrum and background temperature are known. We demonstrate this technique by measuring the temperature difference across a 15 nm quantum dot embedded in a nanowire. This technique can be used when the dot's energy states are separated by many kT and will enable future quantitative investigations of electron-phonon interaction, nonlinear thermoelectric effects, and the efficiency of thermoelectric energy conversion in quantum dots.

The Seebeck coefficient, a key parameter describing a material's thermoelectric performance, is generally difficult to measure, and no intrinsic calibration standard exists. Quantum dots and single electron tunneling devices with sharp transmission resonances spaced by many kT have a material-independent Seebeck coefficient that depends only on the electronic charge and the average device temperature T. Here we propose the use of a quantum dot to create an intrinsic, nanoscale standard for the Seebeck coefficient and discuss its implementation.

We present a method for the measurement of a temperature differential across a single quantum dot that has transmission resonances that are separated in energy by much more than the thermal energy. We determine numerically that the method is accurate to within a few percent across a wide range of parameters. The proposed method measures the temperature of the electrons that enter the quantum dot and will be useful in experiments that aim to test theory which predicts quantum dots are highly-efficient thermoelectrics.Comment: 3 pages, 4 Figure

Nanoscale thermoelectric materials are at the center of current thermoelectric research because they offer higher efficiency than bulk thermoelectrics, which has been attributed to energy selectivity via a strongly modulated electron density of states and lattice thermal conductivity reduction [1]. Our research aims to understand better the efficiency of thermallyinduced electron transport using a quantum dot as a model system and to progress toward engineering high-efficiency nanoscale thermoelectric devices for power generation. We discuss here a device used to measure the electronic efficiency of nanoscale thermoelectric processes.Result 1: In order to quantify device efficiency, we measure thermovoltage as a function of back-gate voltage. The back-gate shifts the resonant energy levels of the dot relative to the source and drain electrochemical potentials. When the source electrochemical potential is near a resonant energy level, the device converts heat to electricity more efficiently than when far from resonance. This efficiency, in principle, approaches the Carnot limit. We infer device efficiency by studying the rate of change in thermovoltage as a function of energy. The device has an efficiency which agrees with the theoretical prediction within a factor of 2.Result 2: In bulk thermoelectric materials, the thermocurrent is linear in the applied temperature differential, AT. Here we measure thermocurrents that become nonlinear in AT when ATIT is roughly 10%. At higher AT, the thermocurrent is strongly nonlinear and even makes complete sign reversals. Understanding these novel nonlinear processes helps engineer superior bulk thermoelectrics, such as composite materials with embedded quantum dots.Result 3: In the past, measurements of sub-Kelvin temperature differences across submicron distances have been made using quantum point contacts as a kind of pre-calibrated thermocouple [2]. Recently, a novel thermometry technique has been proposed theoretically [3], which measures the electron temperature rises relative to the background cryostat temperature, T, on both the source and drain sides of the quantum dot, separately, and needs no pre-calibration. We present here, for the first time, experimental validation of this theory.Method: We use an InAs nanowire -50 nm in diameter and up to 2 pm long grown using chemical beam epitaxy. Two InP barriers, 5 nm thick, have been embedded in the nanowire, defining an InAs quantum dot between the two barriers. The resulting InAs/InP heterostructure nanowire is deposited onto a SiO, substrate below which lies a conducting back-gate. Ni/Au source and drain contacts are connected to the nanowire using electron beam lithography. With source, drain, and gate, the quantum dot operates as a single electron transistor. To investigate the electronic aspect of thermoelectric efficiency, the device temperature is varied between 0.5 -4 K, where electron-phonon interactions should be negligible. Joule heating from an ac heating current establishes a temperature differential ranging...

The Seebeck coefficient S is an important performance characteristic of thermoelectric materials. In this paper we establish the fact that quantum dots and single-electron tunneling devices with narrow, well-spaced energy levels and sharp transmission resonances have a Seebeck coefficient independent of material parameters. By employing a delta function for the transmission resonances we arrive at an intrinsic expression for S in terms of the fundamental electronic charge e. We further confirm the validity of our result in the case of a transmission resonance with finite width.