Efficient heat-energy conversion from a non-thermal Tomonaga-Luttinger liquid

Abstract

Energy harvesting is a technique that generates useful work from waste heat. Conventional energy harvesters acting on local thermal equilibrium states are constrained by thermodynamic limits, such as the Carnot efficiency. Quantum heat engines with non-thermal reservoirs are expected to exceed such limits. Here, we demonstrate energy harvesting from a nonthermal Tomonaga-Luttinger (TL) liquid in quantum Hall edge channels, where the non-thermal state is naturally formed due to the absence of thermalization. The scheme is tested with a quantum-dot energy harvester working on a non-thermal TL liquid supplied with waste heat from a quantum-point-contact transistor. Compared to the quasi-thermalized TL liquid, the non-thermal state prepared under the same heat is capable of a larger electromotive force and higher conversion efficiency. These characteristics can be understood by considering a binary Fermi distribution function of the non-thermal state induced by entropy-conserving equilibration. TL liquids are attractive non-thermal carriers for excellent energy harvesting.

Figures (11)

• Figure 1: Setup for energy-harvesting experiment. a, Schematic heat flow from an active device through a Tomonaga-Luttinger (TL) liquid to a heat engine. The heat power $J_{\mathrm{T}}$ generated from the device is converted to electric power $P$ through a non-thermal TL liquid. b, Schematic device structure with emitter (E), source (S), and drain (D) regions. Electrons and heat travel unidirectionally in the edge channels C$_{r,\uparrow }$ and C$_{r,\downarrow }$ of $r\in \left\{ \mathrm{E,S,D}\right\}$. The quantum point contact (QPC) with bias voltage $V_{\mathrm{S}}$ and transmission coefficient $g$ generates total heat current $J_{\mathrm{T}}$ in the channels (magenta lines). The quantum dot (QD) with an effective bias voltage, $V_{\mathrm{eff}}$, works as a heat engine. Thermoelectricity for non-thermal (NT) and quasi-thermal (QT) states is evaluated by measuring the drain current $I_{\mathrm{D}}$. c, Schematic energy diagram of the QD heat engine for NT and QT distribution functions $f_{\mathrm{NT}}$ and $f_{\mathrm{QT}}$, respectively, in the source. As compared to the QT state, the NT state under the same $J_{\mathrm{T}}$ provides higher electromotive force and higher thermoelectric efficiency. d, Scanning electron micrograph of a control device with false color for setup I. The distance between the QPC and the QD is $L=$ 2 $\mu$m. e, Conductance $G$ of the QPC. The NT and QT states are prepared at $G\sim$ 0.03 $e^{2}/h$ and $\sim$ 0.5 $e^{2}/h$, respectively. f, Coulomb diamond characteristics of the QD. The QD current $I_{\mathrm{D}}$ is plotted as a function of the gate voltage $V_{\mathrm{GD}}$ for various effective bias voltage $V_{\mathrm{eff}}$. Each trace is offset for clarity. Current steps are associated with transport through the ground state (GS) and excited states (ES, ES', and ES” ) of the QD.
• Figure 2: Heat-engine characteristics for non-thermal (NT) and quasi-thermal (QT) states.a, b, Drain current $I_{\mathrm{D}}$ as a function of the gate voltage $V_{\mathrm{GD}}$ at various effective energy bias $eV_{\mathrm{eff}}$. Positive power generation is highlighted by red. The upper-right and lower-left insets in a show the energy diagrams for positive power generation at $eV_{\mathrm{eff}}>0$ ($\varepsilon >\mu _{\mathrm{D},\uparrow }>\mu _{\mathrm{S},\uparrow }$) and $eV_{\mathrm{eff}}<0$ ($\varepsilon <\mu _{\mathrm{D},\uparrow }<\mu _{\mathrm{S},\uparrow }$), respectively, where $\varepsilon$ is the electrochemical potential of the quantum dot (QD), and $\mu _{\mathrm{D},\uparrow }$ and $\mu _{\mathrm{S},\uparrow }$ are the chemical potentials of the drain and source spin-up channels, respectively. c, d, Color plot of electric power $P=-I_{\mathrm{D}}V_{\mathrm{eff}}$ as a function of $eV_{\mathrm{eff}}$ and $\varepsilon -\bar{\mu}$, where $\bar{\mu}=\left( \mu _{\mathrm{D},\uparrow }+\mu _{\mathrm{S},\uparrow }\right) /2$ is the average chemical potential. Comparable heat power $J_{\mathrm{T}}\simeq$ 400 fW was used in a and c for the NT state, and b and d for the QT state. The NT state provides high electromotive force $V_{\mathrm{emf}}\simeq$ 130 $\mu$V and high efficiencies $\bar{\eta}_{\mathrm{Z}}\simeq$ 0.65 in the zero power limit and $\bar{\eta}_{\mathrm{M}}\simeq$ 0.45 at maximum power in c, as compared to the QT state ($V_{\mathrm{emf}}\simeq$ 50 $\mu$V, $\bar{\eta}_{\mathrm{Z}}\simeq$ 0.58 and $\bar{\eta}_{\mathrm{M}}\simeq$ 0.35) in d. e, f, Color plot of $P$ calculated for an idealized QD by using the binary Fermi distribution function $f_{\mathrm{bin}}$ with the fraction $p=$ 0.12, the high thermal energy $k_{\mathrm{B}}T_{\mathrm{S}}=$ 116 $\mu$eV, the low thermal energy $k_{\mathrm{B}}T_{\mathrm{L}}=$ 15.2 $\mu$eV, and the heat transfer factor $\kappa =$ 0.2 for the NT state in e and the thermalized Fermi distribution function $f_{\mathrm{th}}$ with the thermal energies $k_{\mathrm{B}}T_{\mathrm{th}}=$ 42.3 $\mu$eV in the source and $k_{\mathrm{B}}T_{\mathrm{th}}^{\prime }=$ 23.2 $\mu$eV in the drain for the thermalized state in f. The conditions for $\varepsilon =\mu _{\mathrm{D},\uparrow }$ and $\varepsilon =\mu _{\mathrm{S},\uparrow }$, and the idealized efficiency $\bar{\eta}=$ 0.2, 0.4, and 0.6 are shown by the dashed and solid lines, respectively, in c - f. The estimated $\bar{\eta}_{\mathrm{Z}}$ and $\bar{\eta}_{\mathrm{M}}$ are shown by red and blue dashed lines in c-f, while those in f coincide with the Carnot efficiency $\eta _{\mathrm{C}}$ and the Curzon-Ahlborn efficiency $\eta _{\mathrm{CA}}$, respectively.
• Figure 3: Analysis of heat-engine characteristics.a, Schematic diagram of the analysis. The total heat power $J_{\mathrm{T}}$ generated at the quantum point contact (QPC) is divided into non-thermal heat currents $J_{\mathrm{S,\uparrow }}$, $J_{\mathrm{S,\downarrow }}$, $J_{\mathrm{E,\uparrow }}$, and $J_{\mathrm{E,\downarrow }}$. The long-range interaction with heat transfer factor $\kappa$ induces heat current $J_{\mathrm{D,\uparrow }}$ in the drain channel. The quantum dot (QD) heat engine is attached to the heat source with a distribution function $f_{\mathrm{S,\uparrow }}$ and a chemical potential $\mu _{\mathrm{S,\uparrow }}$, and the heat drain with $f_{\mathrm{D,\uparrow }}$ and $\mu _{\mathrm{D,\uparrow }}$. The net power $P$ generated by the QD is estimated from the effective bias $V_{\mathrm{eff}}$ and the measured current $I_{\mathrm{D}}$. b, c, Distribution functions $f_{\mathrm{S,\uparrow }}$ and $f_{\mathrm{D,\uparrow }}$. The current $I_{\mathrm{D}}=-e\Gamma \left[ f_{\mathrm{S,\uparrow }}\left( \varepsilon \right) -f_{\mathrm{D,\uparrow }}\left( \varepsilon \right) \right]$ measures the difference at QD level $\varepsilon$, where $\Gamma$ is the overall tunnel rate. The difference in $I_{\mathrm{D}}$ between b and c with the same $\varepsilon -\mu _{\mathrm{D,\uparrow }}$ but different $V_{\mathrm{eff}}$ [($=\left( \mu _{\mathrm{D,\uparrow }}-\mu _{\mathrm{S,\uparrow }}\right) /e$)] is used to estimate the derivative $\frac{d}{dE}f_{\mathrm{S,\uparrow }}$. The integrated current $K_{I}=\int I_{\mathrm{D}}d\varepsilon$ measures the area enclosed by $f_{\mathrm{S,\uparrow }}$ and $f_{\mathrm{D,\uparrow }}$. d, $K_{I}$ as a function of $\mu _{\mathrm{D,\uparrow }}$ ($=-eV_{\mathrm{D}}$), from which $\mu _{\mathrm{S,\uparrow }}$ is obtained. e, The integrated heat currents $K_{J,\varepsilon -\bar{\mu}}=\int \left( \varepsilon -\bar{\mu}\right) I_{\mathrm{D}}d\varepsilon$ (the red dashed line) and $K_{J,\varepsilon _{\mathrm{G}}}=\int \varepsilon _{\mathrm{G}}I_{\mathrm{D}}d\varepsilon$ (the red solid line) as a function of $\mu _{\mathrm{D,\uparrow }}$. $K_{J,\varepsilon -\bar{\mu}}$ provides the difference of the heat currents $\Delta J$ ($=J_{\mathrm{S,\uparrow }}-J_{\mathrm{D,\uparrow }}$) independent of $\mu _{\mathrm{D,\uparrow }}$, and $K_{J,\varepsilon _{\mathrm{G}}}$ provides $\Delta J$ only at $\mu _{\mathrm{D,\uparrow }}=\mu _{\mathrm{S,\uparrow }}$, from which $\varepsilon -\bar{\mu}$ can be determined.
• Figure 4: Conversion from voltage-dependent current $I_{\mathrm{D}}\left( V_{\mathrm{GD}},V_{\mathrm{D}}\right)$ to energy-dependent current $I_{\mathrm{D}}\left( \varepsilon -\bar{\mu},eV_{\mathrm{eff}}\right)$.a, The original data $I_{\mathrm{D}}\left( V_{\mathrm{GD}},\mu _{\mathrm{D,\uparrow }}^{\prime }\right)$ taken as a function of the gate voltage $V_{\mathrm{GD}}$ and the nominal drain potential $\mu _{\mathrm{D,\uparrow }}^{\prime }$ providing the actual drain potential $eV_{\mathrm{D}}=\mu _{\mathrm{D,\uparrow }}^{\prime }+\mu _{\mathrm{D,0}}$ with predicted offset $\mu _{\mathrm{D,0}}\simeq$ 30 $\mu$eV in setup I. b, The integrated current $K_{I}$ as a function of $\mu _{\mathrm{D,\uparrow }}^{\prime }$. The overall tunnel rate $\Gamma$ and the nominal source potential $\mu _{\mathrm{S,\uparrow }}^{\prime }$ is obtained from the fit (the red line) with the relation $K_{I}=e\Gamma \left( \mu _{\mathrm{D,\uparrow }}^{\prime }-\mu _{\mathrm{S,\uparrow }}^{\prime }\right)$. The inset shows the magnified plot around $K_{I}=0$ (the arrow). c, The energy current $K_{J,\varepsilon _{\mathrm{G}}}$ as a function of $\mu _{\mathrm{D,\uparrow }}^{\prime }$. The heat-current difference $\Delta J$ is obtained from $K_{J,\varepsilon _{\mathrm{G}}}$ value at $\mu _{\mathrm{D,\uparrow }}^{\prime }=\mu _{\mathrm{S,\uparrow }}^{\prime }$ (the vertical dashed line). d, The fluctuating quantum-dot level $\tilde{\varepsilon}$ measured from $\bar{\mu}=\left( \mu _{\mathrm{S,\uparrow }}+\mu _{\mathrm{D,\uparrow }}\right) /2$, $\tilde{\varepsilon}-\bar{\mu}$, as a function of $\mu _{\mathrm{D,\uparrow }}^{\prime }$. The black trace is obtained by using the relation $\tilde{\varepsilon}-\bar{\mu}=eh\Gamma \left( K_{J,\varepsilon _{\mathrm{G}}}-\Delta J\right) /K_{I}$ with the data $K_{I}$ and $\Gamma$ in b and $K_{J,\varepsilon _{\mathrm{G}}}$ and $\Delta J$ in c. The conversion from $\varepsilon _{\mathrm{G}}$ ($=-\alpha V_{\mathrm{GD}}$ with the lever-arm factor $\alpha$) to $\varepsilon -\bar{\mu}$ ($=\varepsilon _{\mathrm{G}}+\tilde{\varepsilon}-\bar{\mu}$) is performed by the red trace of $\tilde{\varepsilon}-\bar{\mu}$, which was obtained from the correction of switching events at the vertical bars and linear fitting between them. The red trace is shifted vertically for clarity. e, The converted plot of $I_{\mathrm{D}}\left( \varepsilon -\bar{\mu},eV_{\mathrm{eff}}\right)$. The initial drift X and jump Y in a are removed in e.
• Figure 5: Energy distribution functions of non-thermal (NT) and quasi-thermal (QT) states.a, b, Distribution functions $f_{\mathrm{S,\uparrow }}(E)$ of the source spin-up channel C$_{\mathrm{S},\uparrow }$ and $f_{\mathrm{D,\uparrow }}(E)$ of the drain spin-up channel C$_{\mathrm{D},\uparrow }$ extracted from the data obtained with a NT state (the transmission coefficient $g=$ 0.058, the total heat power $J_{\mathrm{T}}=71$ fW, and the source voltage $V_{\mathrm{S}}=$ 183 $\mu$V) in a and a QT state ($g=$ 0.5, $J_{\mathrm{T}}=61$ fW, and $V_{\mathrm{S}}=$ 79 $\mu$V) in b. The empirical binary Fermi distribution function $f_{\mathrm{bin}}(E)$ with the fraction $p=$ 0.21, the high thermal energy $k_{\mathrm{B}}T_{\mathrm{S}}=$ 36 $\mu$eV, and the low thermal energy $k_{\mathrm{B}}T_{\mathrm{L}}=$ 15.5 $\mu$eV in a, the thermalized distribution function $f_{\mathrm{th}}(E)$ with thermal energy $k_{\mathrm{B}}T_{\mathrm{th}}=$ 22 $\mu$eV in a and 22 $\mu$eV in b, and the Fermi distribution function $f_{\mathrm{B}}$ at thermal energy $k_{\mathrm{B}}T_{\mathrm{B}}=$ 15 $\mu$eV for the base temperature $T_{\mathrm{B}}=$ 175 mK in a are also shown. c, Comparison of initial double-step distribution functions $f_{\mathrm{stp}}(E)$, the binary Fermi function $f_{\mathrm{bin}}(E)$, and the thermalized function $f_{\mathrm{th}}(E)$ for $g=$ 0.03, $J_{\mathrm{T}}=$ 410 fW, and $V_{\mathrm{S}}=$ 600 $\mu$V. d, Comparison of initial entropy $S_{\mathrm{stp}}$ for $f_{\mathrm{stp}}$, the non-thermal entropy $S_{\mathrm{bin}}$ for $f_{\mathrm{bin}}$, and the thermalized entropy $S_{\mathrm{th}}$ for $f_{\mathrm{th}}$ as a function of normalized conductance $g$ of the QPC. $S_{\mathrm{bin}}\simeq S_{\mathrm{stp}}$ suggests entropy-conserving equilibration. The dashed line indicates the condition $g=0.03$ for c.