Heat rectification, heat fluxes, and spectral matching

Heat rectifiers would facilitate energy management operations such as cooling or energy harvesting, but devices of practical interest are still missing. Understanding heat rectification at a fundamental level is key to helping us find or design such devices. The match or mismatch of the phonon band spectrum of device segments for forward or reverse temperature bias of the thermal baths at device boundaries was proposed as the mechanism behind rectification. However, no explicit theoretical relation derived from first principles had been found so far between heat fluxes and spectral matching. We study heat rectification in a minimalistic chain of two coupled ions. The fluxes and rectification can be calculated analytically. We propose a definition of the matching that sets an upper bound for the heat flux. In a regime where the device rectifies optimally, matching and flux ratios for forward and reverse configurations are found to be proportional. The results can be extended to a system of $N$ particles in arbitrary traps with nearest-neighbor linear interactions.

Figure 1

Scheme of the model described in Sec. 2. Each mass $m_i$ is at temperature ${\tau }_i$ ($i=L$ and $R$ is the generic index for “left” or “right”), trapped by a harmonic potential, and connected to a thermal bath at temperature $T_i$ and friction coefficient ${\gamma }_i$. The masses interact by a harmonic potential with each other.

Figure 2

Rectification, $R$, given by Eq. (26) as a function of the ratios $a$ and $g$. The arrows give the gradient direction.

Figure 3

Rectification for different values of $g={\gamma }_R/{\gamma }_L$. The blue solid line gives the maximal rectification, which is found when $a=g$ [see Eq. (25)]. For the red dashed line the mass ratio is kept constant, $a=1.648$ (corresponding to ${\mathrm{Ca}}^{+}$ and ${\mathrm{Mg}}^{+}$ ions). The blue squares correspond to the values of $C=1$ (the two ions and the friction coefficients are equal, so $R\sim 0$) and $C=10, R\sim 0.8.$ The spectra for $C=10$ are depicted in Fig. 4.

Figure 4

Spectral densities for both ions multiplied by their masses, $m_L{S}_L$ (red dashed line) and $m_R{S}_R$ (blue solid line) vs $\omega$ for $C=10$ corresponding to forward and reverse configurations. $T_L={\tilde{T}}_R=2\mathrm{mK}$ and $T_R={\tilde{T}}_L=1\mathrm{mK}$. (${\tilde{T}}_i$ are the temperatures of the reverse configuration.) The vertical lines are the real part of the frequencies of the dissipative normal modes of the system [8]. The areas are proportional to the particle kinetic energies (or temperatures). The rectification coefficient is $R\approx 0.8$: in the forward configuration the spectra match well, while in the reverse configuration there is a clear mismatching.

Figure 5

Flux and matching versus $C=a=g$. $T_L=1\mathrm{mK}$ and $T_R=0.1\mathrm{mK}$. Other parameters are as explained in Sec. 2c.

Figure 6

Flux and matching versus $g$ with a constant mass ratio $a$. $T_L=1\mathrm{mK}$ and $T_R=0.1\mathrm{mK}$. Other parameters are as explained in Sec. 2c.

Figure 7

Forward to reverse ratios $|J/\tilde{J}|, \mathcal{S}/\tilde{\mathcal{S}}$, and $M/\tilde{M}$ versus $C$. $T_L=2\mathrm{mK}$ and $T_R=1\mathrm{mK}$. The blue squares show the results of $|J/\tilde{J}|$ using the analytical expression $|J/\tilde{J}|=(1+C^2)/2C$.

Figure 8

Flux ratio $J/\tilde{J}$ vs the matching ratio $M/\tilde{M}$ for different temperature intervals. The line points depend parametrically on $C$. The slope depends on the ratio of temperatures $T_L/T_R$. Dots and lines of the same color indicate same temperatures' ratio but different values of the temperatures (in mK) as indicated in the legend. For instance, blue dots correspond to $T_L=1\mathrm{mK}$ and $T_R=0.5\mathrm{mK}$ and the blue line corresponds to $T_L=2\mathrm{mK}$ and $T_R=1$ mK; both cases verify $T_L/T_R=2$.

Figure 9

$N$-particle linear chain with interaction between nearest neighbors.