Nanowire-based thermoelectric ratchet in the hopping regime

Riccardo Bosisio, Geneviève Fleury, Jean-Louis Pichard, and Cosimo Gorini

Phys. Rev. B 93, 165404 – Published 4 April 2016

Abstract

We study a thermoelectric ratchet consisting of an array of disordered nanowires arranged in parallel on top of an insulating substrate and contacted asymmetrically to two electrodes. Transport is investigated in the Mott hopping regime, when localized electrons can propagate through the nanowires via thermally assisted hops. When the electronic temperature in the nanowires is different from the phononic one in the substrate, we show that a finite electrical current is generated even in the absence of driving forces between the electrodes. We discuss the device performance both as an energy harvester, when an excess heat from the substrate is converted into useful power, and as a refrigerator, when an external power is supplied to cool down the substrate.

Received 22 October 2015

DOI:https://doi.org/10.1103/PhysRevB.93.165404

Authors & Affiliations

Riccardo Bosisio^1,2, Geneviève Fleury^3,*, Jean-Louis Pichard³, and Cosimo Gorini⁴

¹SPIN-CNR, Via Dodecaneso 33, 16146 Genova, Italy
²NEST, Instituto Nanoscienze-CNR and Scuola Normale Superiore, I-56127 Pisa, Italy
³SPEC, CEA, CNRS, Université Paris-Saclay, CEA Saclay 91191 Gif-sur-Yvette Cedex, France
⁴Institut für Theoretische Physik, Universität Regensburg, D-93040 Regensburg, Germany

^*genevieve.fleury@cea.fr

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Vol. 93, Iss. 16 — 15 April 2016

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Figure 1
(a) Sketch of the NW-based ratchet studied in this paper. A large array of parallel disordered NWs (in blue) is deposited onto an insulating substrate (in red) acting as a phonon bath. The NWs are attached to metallic electrodes (yellow) via asymmetric contacts. (b) Schematic of the three-terminal transport. Electrons inside the NWs are localized within their impurity band (blue) and can exchange phonons with the substrate (held at temperature $T_{P}$ ). Under condition of asymmetric couplings (gray strips) to the electrodes (yellow), this can generate net currents flowing through the NWs. The particle and heat currents between the NWs and the three reservoirs are represented by the arrows, assuming they are positive when they enter the system.
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Figure 2
Illustration of the ratchet effect powered by $δ T_{P} > 0$ , for various types of contacts. The total particle current $I_{L}^{N}$ [see Eq. (7)], obtained with $δ T_{p} = 10^{- 3} ε$ and $δ μ = 0$ , is plotted (in units of $10^{5} ε / ℏ$ and divided by $M$ ) in top panels (a-c) for different configurations shown below (a1-c3) and listed at the end of Sec. III A. (Left) Energy-independent tunnel barriers. If the contacts are symmetric (a1) or if $μ = 0$ (a2), $I_{L}^{N} = 0$ on average. If both symmetries are broken (a3) a net current is generated. (a) $I_{L}^{N} / M$ as a function of $γ_{e L}$ (in units of $ε / ℏ$ ) for fixed $γ_{e R} = ε / ℏ$ and various positions of $μ$ [ and $- 2.5$ ( $▵$ )]. (Middle) Energy filter. When $E_{d} = 0$ electron-hole symmetry makes the total current vanish (b1), whereas $E_{d} > 0$ (b2) and $E_{d} < 0$ (b3) correspond to negative and positive current, respectively. (b) $I_{L}^{N} / M$ at $μ = 0$ , for constant $γ_{e R} = ε / ℏ$ on the right and an energy filter on the left with $γ_{e} = ε / ℏ$ , opening $Γ = ε$ , and tunable $E_{d}$ (in units of $ε$ ). (Right) (c1) Single barrier configuration. The Schottky barrier prevents electron- and hole-like excitations with energy below $E_{s}^{R}$ to escape to the right contact. (c2) Double barrier configuration. A low energy filter is added at the left contact, forbidding electron- and hole-like excitations with energy above $E_{s}^{L}$ to tunnel leftward. (c3) Hybrid configuration. A Schottky barrier on the right is combined with an energy filter on the left. (c) $I_{L}^{N} / M$ at $μ = 0$ , as a function of the barrier height $E_{s}$ (in units of $ε$ ), with $γ_{e} = ε / ℏ$ , for the single barrier ( $▪, E_{s} \equiv E_{s}^{R}$ ), double barrier (, $E_{s} \equiv E_{s}^{R} = - E_{s}^{L}$ ), and hybrid cases ( $E_{s} \equiv E_{s}^{R}$ ) with $E_{d} = 0$ ( $□$ ) and .
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Figure 3
(Top) Nonlocal thermopower $S$ (in units of $k_{B} / e$ ), electronic figure of merit $Z T$ , and power factor $Q$ (in units of $k_{B}^{2} / ℏ$ ) in the single barrier (black line), double barrier (red line) and hybrid [with and ] configurations. Data are plotted as functions of the (right) barrier height $E_{s}^{R} \equiv E_{s}$ (in units of $ε$ ). For the double barrier case, $E_{s}^{R} = - E_{s}^{L}$ . For the hybrid one, $Γ = ε$ . (Bottom) $S, Z T$ , and $Q / M$ in the double barrier configuration as functions of the left and right barrier heights $E_{s}^{L}$ and $E_{s}^{R}$ (in units of $ε$ ). In all panels, $γ_{e} = ε / ℏ$ and $μ = 0$ .
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Figure 4
(a) Phase diagram showing for which values of $δ μ$ (in units of $ε$ ) and $δ T_{P}$ (in units of $ε / k_{B}$ ) the NW-based ratchet operates as an energy harvester or as a phonon bath refrigerator, given that $δ T = 0$ . The plot is drawn for the hybrid configuration (with $E_{d} = - ε, Γ = ε$ , and $E_{s}^{R} = ε$ ), the single and double barrier ones being qualitatively similar. (b) Heat-to-work conversion efficiency $η$ (normalized to $η_{C}$ ) as function of the output power $P$ (in units of $ε^{2} / ℏ$ ) when $δ μ$ is varied, at fixed $δ T_{P} = 10^{- 3} ε / k_{B}$ . The three loops correspond to the single barrier ( $E_{s}^{R} = ε$ , black line), double barrier ( $E_{s}^{R} = - E_{s}^{L} = ε$ , red line), and hybrid [same as in (a), red dashed line] cases. The red dots highlight the values of the efficiency at maximum power $η (P_{\max})$ . In both panels, $μ = 0$ and $γ_{e} = ε / ℏ$ .
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Figure 5
Nonlocal thermopower $S$ (in units of $k_{B} / e$ ), electronic figure of merit $Z T$ and power factor $Q$ (in units of $k_{B}^{2} / ℏ$ ) for various temperatures $[k_{B} T / ε = 0.05$ ( $\circ$ ), $0.1$ ( $r e d □$ ), $0.5$ ( $g r e e n ⋄$ ), and 1 ( $b l u e ▵$ )]. Data are plotted for the hybrid configuration as functions of the (right) barrier height $E_{s}^{R} \equiv E_{s}$ (in units of $ε$ ). Other parameters are $E_{d} = - ε, Γ = ε, γ_{e} = ε / ℏ$ , and $μ = 0$ .
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Figure 6
Nonlocal (full lines) versus local (dashed lines) transport coefficients, shown for the single barrier (black), double barrier (red), and hybrid (blue) configurations. Data are plotted as functions of the (right) barrier height $E_{s}^{R} \equiv E_{s}$ (in units of $ε$ ). For the double barrier case, $E_{s}^{R} = - E_{s}^{L}$ . For the hybrid one, $E_{d} = - ε$ and $Γ = ε$ . The thermopowers and power factors are given in units of $k_{B} / e$ and $k_{B}^{2} / ℏ$ , respectively. In all panels, $γ_{e} = ε / ℏ$ and $μ = 0$ .
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