Efficient thermal diode with ballistic spacer

Thermal rectification is of importance not only for fundamental physics, but also for potential applications in thermal manipulations and thermal management. However, thermal rectification effect usually decays rapidly with system size. Here, we show that a mass-graded system, with two diffusive leads separated by a ballistic spacer, can exhibit large thermal rectification effect, with the rectification factor independent of system size. The underlying mechanism is explained in terms of the effective size-independent thermal gradient and the match or mismatch of the phonon bands. We also show the robustness of the thermal diode upon variation of the model's parameters. Our finding suggests a promising way for designing realistic efficient thermal diodes.

Figure 1

Schematic drawing of our model. It consists of three parts: one central ballistic channel with $N_C$ particles of mass $m_C$, and two leads $({\varphi }^4$ lattices), with $N_L\left(N_R\right)$ particles of mass $m_L\left(m_R\right)$ and on-site potential strength ${\gamma }_L\left({\gamma }_R\right)$ in the left (right) lead.

Figure 2

Thermal rectification factor $f_r$ versus system size $N_{\mathrm{tot}}$ for chains with or without ballistic channel (squares are for the ${\varphi }^4$-harmonic-${\varphi }^4$ model, i.e., with ballistic channel, and triangles are for the ${\varphi }^4-{\varphi }^4-{\varphi }^4$ model, i.e., without ballistic channel). Here, $T_{+}=9.5,T_{-}=0.5,N_L=N_R=10,m_L=1,m_C=4.5,m_R=10,{\gamma }_L={\gamma }_R=1$, and ${\gamma }_C=0\left({\gamma }_C=1\right)$ for the model with (without) ballistic channel.

Figure 3

(a) Temperature profiles, both for forward (left-right, full symbols) and backward (right-left, empty symbols) temperature bias, for the model with a ballistic channel $({\varphi }^4$-harmonic-${\varphi }^4$ model) and (b) for the model without a ballistic channel $({\varphi }^4-{\varphi }^4-{\varphi }^4$ model). In both cases $N_{\mathrm{tot}}=84$ (triangles), 148 (squares), and 276 (diamonds), respectively. Here $T_{+}=9.5,T_{-}=0.5,m_L=1,m_R=10,m_C=4.5,N_L=N_R=10,{\gamma }_L={\gamma }_R=1$, and ${\gamma }_C=0$ [panel (a)] or ${\gamma }_C=1$ [panel (b)].

Figure 4

Vibrational power spectrum for the ${\varphi }^4$ lattice, calculated at thermal equilibrium at temperature $T$. (a) $m=1,T=0.5$, system size in equilibrium simulations $N=500$ (solid black line), 1000 (dash-dotted olive line), and 2000 (dotted red line), respectively; the perfect overlap of the data for different sizes indicates that the results are not affected by finite-size effects; the vertical dashed black lines correspond to the boundaries of the analytically estimated phonon band. (b) Analytically estimated phonon band as a function of temperature, for $m=1$ (solid red lines) and $m=10$ (dashed blue lines). (c) Power spectrum for $m=10,T=9.5$ (left dashed blue line) and for $m=1,T=0.5$ (right solid red line); $N=1000$. (d) Power spectrum for $m=10,T=0.5$ (left dashed blue line) and for $m=1,T=9.5$ (right solid red line); $N=1000$. In all cases the nonlinearity strength $\gamma =1$.

Figure 5

Dependence of the rectification factor on (a) temperature difference $(m_L=1,m_R=10,m_C=4.5$, average temperature $T=5)$, (b) mass gradient $\left(m_R=10,m_C=(m_L+m_R)/2,T_{+}=9.5,T_{-}=0.5\right)$, and (c) mass of the particles in the ballistic channel $\left(m_L=1,m_R=10,T_{+}=9.5,T_{-}=0.5\right)$. In all panels, $N_L=N_R=10,N_C=64$, total system size $N_{\mathrm{tot}}=84,{\gamma }_L={\gamma }_R=1$.

Figure 6

(a) Dependence of the rectification factor on the strength ${\gamma }_L$ of the on-site potential in the left lead for (a) the equal-mass system $\left(m_L=m_R=m_C=1\right)$ and (b) the mass-graded system $\left(m_L=1,m_R=10,m_C=5.5\right)$. In both panels, $N_L=N_R=10,N_C=64$, total system size $N_{\mathrm{tot}}=84,T_{+}=9.5,T_{-}=0.5,{\gamma }_R=1$.

Figure 7

Effect of the anharmonicity (FPU-$\beta$ model) of the central channel on the rectification. (a) Rectification factor versus system size for different values of $\beta (\beta =0$ corresponds to the ballistic, harmonic channel); the case of a diffusive, ${\varphi }^4$ central channel is also shown for a comparison. (b) Rectification factor versus anharmonicity strength for size of the central channel $N_C=64$. In both panels, $N_L=N_R=10,m_L=1,m_R=10,m_C=4.5,T_{+}=9.5,T_{-}=0.5,{\gamma }_L={\gamma }_R=1$.