Thermal conduction and interface effects in nanoscale Fermi-Pasta-Ulam conductors

We perform classical nonequilibrium molecular dynamics simulations to calculate heat flow through a microscopic junction connecting two larger reservoirs. In contrast to earlier papers, we also include the reservoirs in the simulated region to study the effect of the bulk-nanostructure interfaces and the bulk conductance. The scalar Fermi-Pasta-Ulam (FPU) model is used to describe the effects of anharmonic interactions in a simple manner. The temperature profile close to the junction in the low-temperature limit is shown to exhibit strong directional features that fade out when temperature increases. Simulating both the FPU chain and the two bulk regions is also shown to eliminate the nonmonotonous temperature variations found for simpler geometries and models. We show that, with sufficiently large reservoirs, the temperature profile in the chain does not depend on the details of thermalization used at the boundaries.

Figure 1

(a) The square lattice model and the device setup. The positions of the black atoms are fixed, red (dark gray in the black and white version) atoms are in a Langevin bath at temperature $T_{+}$, cyan (light gray) atoms are in a bath at temperature $T_{-}$, and the white atoms are free (apart from interactions with neighbors). Atoms located at neighboring sites are connected by anharmonic springs. The size of the reservoirs is defined by the parameters $W^R$ and $L^R$, and the size of the constriction is defined by $W^C$ and $L^C$. (b) A bath-terminated chain of $N$ atoms.

Figure 2

Kinetic temperature profile at (a) low bath temperatures ($T_{+}=0.20,T_{-}=0.05$) and (b) high bath temperatures ($T_{+}=2.0,T_{-}=0.5$). The bulk size is $W^R=161,L^R=80$, and the constriction size is $W^C=5,L^C=9$. The labels on the horizontal and vertical axes mark the atom indices. The separations of isolines are (a) 0.005 and (b) 0.05.

Figure 3

Vertical local heat current $-J_i^y$ at (a) low bath temperatures ($T_{+}=0.20,T_{-}=0.05$) and (b) high bath temperatures ($T_{+}=2.0,T_{-}=0.5$). The labels on the horizontal and vertical axes mark the atom indices.

Figure 4

Heat current flowing through a $W^C=5,L^C=9$ constriction as a function of the size of the reservoirs at (a) low bath temperatures ($T_{+}=0.20,T_{-}=0.05$) and (b) high bath temperatures ($T_{+}=2.0,T_{-}=0.5$). The transversal size of the bulk regions is $W^R=2L^R+1$. The heat currents flowing through the bridge have been averaged over the $L^C=9$ cross sections of the bridge during the simulation, and the error bars $5\delta$ were calculated in the end using Eq. (14).

Figure 5

Schematic of the simple model used for explaining the dependence of heat current on the bulk size for large systems. The central region, which consists of two disks of radius $r_0$ and the bridge, contributes a constant factor $R_0$ to the total thermal resistance. Heat transfer in the two bulk regions away from the central region is assumed to obey Fourier's law and, therefore, to contribute $R_1=\ln (r_1/r_0)/\pi \kappa$ to the thermal resistance. The current flowing through the structure is then given by Eq. (16).

Figure 6

Local kinetic temperatures along the chain for a chain coupled to the bulk (dashed lines, $L^C=8$, $W^C=1$, $L^R=30$, $W^R=61$) and for the bath-terminated $N=8$ chain (solid lines) at (a) low bath temperatures ($T_{+}=0.20,T_{-}=0.05$) and (b) high bath temperatures ($T_{+}=2,T_{-}=0.5$) for two Langevin friction parameters $\gamma$. Note that the vertical scale of the temperature profile for the extended structure has an offset for clarity and the curves for $\gamma =0.5$ and $\gamma =2.0$ lie on top of each other. The error bars $5\delta$ are smaller than the sizes of the markers. The insets in (a) are for clarification.

Figure 7

Thermal conductance $K$ versus temperature $T$ in chains of length (a) $N=8$ and (b) $N=50$ for two extended structures and the bath termination. The Langevin friction parameter is $\gamma =2$. At high temperatures, the thermal conductances of the extended structures scale as black, dashed line: $T^{0.22}$ and $T^{0.21}$ for the two chain lengths. The heat current has been averaged over the length of the chain. The error bars are of length $5\delta$ [Eq. (14)].