Microscopic model of a phononic refrigerator

We analyze a simple microscopic model to pump heat from a cold to a hot reservoir in a nanomechanical system. The model consists of a one-dimensional chain of masses and springs coupled to a back gate through which a time-dependent perturbation is applied. The action of the gate creates a moving phononic barrier by locally pinning a mass. We solve the problem numerically using a nonequilibrium Green's function technique. For low driving frequencies and for sharp traveling barriers, we show that this microscopic model realizes a phonon refrigerator.

Figure 1

The one-dimensional microscopic model for acoustic phonons consists of a chain of particles (atoms of molecules) with equal mass $m_c$ coupled by springs with elastic constants $k_0$. The semi-infinite left portion of the chain is kept at a temperature $T_L$ which is lower than the temperature $T_R$ of the semi-infinite right portion. A traveling harmonic potential is applied locally to the central portion of the chain. This potential restricts the longitudinal motion of the particles—see the last term on the right-hand side of Eq. (2). To perform a cooling cycle, the local potential is turned on and moved from near the cold to the hot reservoir at a constant speed $v$. When it reaches the hot reservoir, the potential is turned off and moved back to near the cold reservoir.

Figure 2

The transmission function $\mathcal{T}\left(\omega \right)$ defined in Eq. (42) for a central system of $N=40$ atoms without any barrier (dashed lines) and with a Gaussian barrier with height $k_1=1.5k_0$ and $\sigma =a$ (solid lines) centered at $l=20$. The frequency is shown in units of the chain's characteristic frequency ${\omega }_c=\sqrt{k_0/m_c}$.

Figure 3

The average energy flowing through the contacts in a cycle, ${\overline{J}}_{\alpha }^Q\tau$, as functions of the temperature $T_L$ of the left (cold) reservoir (given in units of ${\omega }_c$). The temperature of the right reservoir is kept fixed $T_R=0.04{\omega }_c$ and ${\Omega }_0=1\times {10}^{-5}{\omega }_c$. The central system contains $N=80$ atoms. The barrier has a height $k_1=1.5k_0$, $\sigma =a$, and moves from left to right with a speed $v=aN{\Omega }_0/\left(2\pi \right)$. The arrows on the horizontal axis indicate two temperature scales, $T^{*}$ and $T_{\min }$, defined in the main text.

Figure 4

The relative temperature interval $(T_R-T_{\min })/T_R$ as a function of the right reservoir temperature $T_R$, where $T_{\min }$ is the minimum temperature of the left reservoir for which cooling is possible. Systems of sizes $N=40$, 80, 120, and 160 corresponding to black circles, red squares, green downward triangles, and blue upward triangles, respectively, are considered (the solid lines are guides to the eye). For each system size, a fitting to Eq. (17) is performed (dashed lines). Other parameters are the same as in Fig. 3.