Anomalous thermal response of silicene to uniaxial stretching

Silicene—the silicon counterpart of graphene—has a two-dimensional structure that leads to a host of interesting physical and chemical properties of significant utility. We report here an investigation with nonequilibrium molecular dynamics simulations of thermal transport in a single-layer silicene sheet under uniaxial stretching. We discovered that, contrary to its counterpart of graphene and despite the similarity of their honeycomb lattice structure, silicene exhibits an anomalous thermal response to tensile strain: The thermal conductivity of silicene and silicene nanoribbons first increases significantly with applied tensile strain rather than decreasing and then fluctuates at an elevated plateau. By quantifying the relative contribution from different phonon polarizations, we show first that the phonon transport in silicene is dominated by the out-of-plane flexural modes, similar to graphene. We attribute subsequently the unexpected and markedly different behavior of silicene to the interplay between two competing mechanisms governing heat conduction in a stretched silicene sheet, namely, (1) uniaxial stretching modulation in the longitudinal direction significantly depressing the phonon group velocities of longitudinal and transverse modes (phonon softening) and hindering heat conduction, and (2) phonon stiffening in the flexural modes counteracting the phonon softening effect and facilitating thermal transport. The abnormal behavior of the silicene sheet is further correlated to the unique deformation characteristics of its hexagonal lattice. Our study offers perspectives of modulating the thermal properties of low-dimensional structures for applications such as thermoelectric, photovoltaic, and optoelectronic devices.

Figure 1

Snapshot of a typical zigzag silicene sheet used as model system in the MD simulations. The $x$ axis (transverse direction) is along the armchair edge; the $y$ axis (longitudinal direction) is along the zigzag edge, and the uniaxial tension along the $y$ direction is indicated by blue arrows; the $z$ axis is the out-of-plane flexural direction. The total length of the silicene is 1000 unit cells. The two fixed ends served as “rigid walls” and are colored in red.

Figure 2

(a) Comparison of the relative thermal conductivities ($\kappa /{\kappa }_0$) of silicene monolayer (width $=$ 10 unit cells), silicene nanoribbon (width $=$ 5 and 10 unit cells), graphene monolayer (width $=$ 10 unit cells), and graphene nanoribbon (width $=$ 5 and 10 unit cells) as a function of uniaxial strain along the longitudinal direction. All silicene and graphene layers are zigzag and are 1000 and 1500 unit cells long, respectively. The reference values (${\kappa }_0$) are 40.1, 31.4, and 35.0 W/mK for silicene monolayer (w $=$ 10 u.c.) and silicene nanoribbons (w $=$ 5 and 10 u.c.), respectively. ${\kappa }_0=2002.2$, 1323.3, and 1531.9 W/mK for graphene monolayer (w $=$ 10 u.c.) and graphene nanoribbons (w $=$ 5 and 10 u.c.), respectively. For comparison, the results of the same silicene monolayer with EDIP interatomic potential (filled magenta squares) and silicon nanowire extended in [110] direction with Tersoff potential (filled black squares) are also shown, with ${\kappa }_0$ of 26.0 and 14.8 W/mK, respectively. (b) The mean total (kinetic plus potential) energy of atoms in the system as a function of strain. The symbols are the same as those in panel (a).

Figure 3

Comparison of vibrational density of states between single-layer (a) silicene and (b) graphene at typical strains of 0$\%$, 3$\%$, 8$\%$, 12$\%$, and 16$\%$. For each column the top, middle, and bottom panel correspond to the density of states contributed by the lattice vibrations in $x$ (transverse), $y$ (longitudinal), and $z$ (flexural) directions, respectively.

Figure 4

(a) Phonon dispersion curves of silicene monolayer at some typical tensile strains. The six branches of longitudinal acoustic (LA)/optical (LO), transverse acoustic (TA)/optical (TO), and flexural acoustic (ZA)/optical (ZO) are labeled. The LO and TO branches after stretching are also indicated for clarification. (b) Corresponding phonon group velocity vs frequency. The same group lines and color code are used. Note that different lattice constant “$a$” should be used when calculating group velocity at different strains, which results in significantly different group velocity even if the phonon dispersion curves are visually the same.

Figure 5

(a) Phonon dispersion curves and (b) corresponding phonon group velocities of graphene monolayer. The same labels, group lines, and color codes are used as in Fig. 4.

Figure 6

Relative contribution (percentage) to total heat flux from vibrations in $x$, $y$, and $z$ directions as a function of uniaxial strain for single-layer silicene and graphene sheet. The $x$, $y$, and $z$ correspond to transverse, longitudinal, and flexural directions, respectively.

Figure 7

Comparison of Poisson's ratio (left axis) and transverse strain (right axis) as a function of uniaxial stretching strain between single-layer silicene and graphene sheet. The two horizontal lines denote the zero point of Poisson ratio and transverse strain.

Figure 8

Comparison of (a), (c) bond length and (b), (d) bond angle at some typical stretching strains between single-layer silicene (top) and graphene (bottom). All occurrence probabilities are normalized by the total number of occurrences so that the integral across the entire range of the bond length or angle is equal to unity. For comparison, the bond length in (a) and (c) is normalized by the unstrained values, corresponding to 0$\%$ strain, of 2.309 and 1.438 Å for silicene and graphene, respectively. The same color codes are used for all figures. Inset in (a) shows the deformation of a hexagonal cell undergoing a uniaxial stretching along the zigzag direction. Filled circles with solid lines and open circles with dashed lines are structures before and after stretching, respectively.