Circuit approach to photonic heat transport

L. M. A. Pascal, H. Courtois, and F. W. J. Hekking

Phys. Rev. B 83, 125113 – Published 23 March 2011

Abstract

We discuss the heat transfer by photons between two metals coupled by a circuit containing linear reactive impedances. Using a simple circuit approach we calculate the spectral power transmitted from one metal to the other and find that it is determined by a photon transmission coefficient which depends on the impedances of the metals and of the coupling circuit. We study the total photonic power flow for different coupling impedances both in the linear regime where the temperature difference between the metals is small and in the nonlinear regime of large temperature differences.

Received 20 March 2010

DOI:https://doi.org/10.1103/PhysRevB.83.125113

Authors & Affiliations

L. M. A. Pascal¹, H. Courtois¹, and F. W. J. Hekking²

¹Institut Néel, CNRS and Université Joseph Fourier, 25 Avenue des Martyrs, BP 166, F-38042 Grenoble, France
²LPMMC, Université Joseph Fourier and CNRS, 25 Avenue des Martyrs, BP 166, F-38042 Grenoble, France

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Vol. 83, Iss. 12 — 15 March 2011

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Images

Figure 1
Top left panel circuit A: two impedances $Z_{1}$ and $Z_{2}$ are coupled by a purely reactive coupling impedance $Z_{c}$ . Top right panel circuit B: two impedances $Z_{1}$ and $Z_{2}$ are coupled via mutual inductances to a purely reactive central circuit of total impedance $Z_{c}$ (here an $LC$ resonator). Bottom panel circuit C: two impedances $Z_{1}$ and $Z_{2}$ are coupled via a transmission line.Reuse & Permissions
Figure 2
Case of an inductive coupling. Spectrum of the thermal noise power density $x^{2} / \sinh^{2} x$ (black dots), the photon transmission coefficient $T_{L} (x) = α_{L}^{2} / (α_{L}^{2} + x^{2})$ (dotted blue line), and of the product of the two namely the photonic heat (thin red full line) as a function of the frequency for values of the parameter $α_{L}$ $=$ 0.03, 1, and 30 from top to bottom. The frequency is plotted in units of the thermal frequency $2 k_{B} T / ℏ$ . We consider the case of perfect resistance matching $R_{1} = R_{2} = R$ .Reuse & Permissions
Figure 3
Case of a capacitive coupling. Spectrum of the thermal noise power density $x^{2} / \sinh^{2} x$ (black dots), the photon transmission coefficient $T_{C} (x) = x^{2} / (x^{2} + α_{C}^{2})$ (dotted blue line), and of photonic heat (thin red full line) as a function of the frequency for values of the parameter $α_{C} = 0.03$ , 1, and 30 from top to bottom. The frequency is plotted in units of the thermal frequency $2 k_{B} T / ℏ$ . We consider the case of perfect resistance matching $R_{1} = R_{2} = R$ .Reuse & Permissions
Figure 4
Case of a mutual coupling via an $LC$ -resonator circuit. Spectrum of the thermal noise power density $x^{2} / \sinh^{2} x$ (black dots), the spectral transmission factor $T_{M} (x)$ (dotted blue line), and of the photonic heat (thin red full line) as a function of the frequency for values of the parameter $γ$ $=$ 0.01, 0.1, and 1. The frequency is plotted in units of the thermal frequency $2 k_{B} T / ℏ$ . We consider the case of perfect resistance matching $R_{1} = R_{2}$ and a parameter $α_{M} = 30$ .Reuse & Permissions
Figure 5
Case of a coupling through a transmission line. Spectrum of the thermal noise power density $x^{2} / \sinh^{2} x$ (black dots), the photon transmission coefficient $T_{TL} (x)$ (dotted and full blue lines), and of the photonic heat (dotted and full red lines) as a function of the frequency. The full lines stand for a number of cells $N =$ 6 and the dotted lines for $N$ infinite. The frequency is plotted in units of the cut-off frequency $ω_{c}$ . We consider a temperature $T$ so that $α_{TL} =$ 0.2 and the case of perfect resistance matching $R_{1} = R_{2}$ with $R = 0.1 \sqrt{L / C}$ .Reuse & Permissions
Figure 6
Dependence of the photonic power through a capacitive or inductive coupling impedance on the parameters $α_{C}$ or $α_{L}$ for different values of the relative temperature difference $Δ T / T$ in units of $K_{Q} Δ T$ , the maximum photonic power in the case of a linear response.Reuse & Permissions

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