panel (c), the direction of the vibrational displacement is the same for all atoms, while the direction changes for the second order mode shown in panel (d). The twisting mode reflects the rod-like structure of the SiNW and is determined by the shear stiffness of the SiNW. It can serve to sense the tiny torsion movement of a nano device.[32] This mode can be used to extract the internal information of the SiNW, as it carries the information of the rigid rotational invariance symmetry.[33–35] Panels (e) and (f) are the first and second order longitudinal vibrational phonon modes. In very long SiNW, this mode is actually the well-known longitudinal acoustic mode. Fig. 3 shows frequencies of the three lowest-energy phonon modes of SiNW in different directions with different sizes. Frequencies for all phonon modes decrease with increasing structure ratio γ = length/diameter . This result indicates that longer SiNW are more flexible and elastic. As a result, SiNW with large γ may be thermally more unstable, and possibly show a coiling-up behavior like a single-walled carbon nanotube.[36] Typically, we note that the frequency of bending mode in (a) is the lowest one compared with the other two modes shown in panels (b) and (c). This is because the bending mode turns to be the elastic flexure mode in the limit of γ → ∞ . An important feature of the flexure mode is its quadratic phonon spectrum: ω ∝ k $^{2}$. While the other acoustic phonon modes are linearly dependent on the wave vector. So the flexure mode (also bending mode) always has the lowest frequency. This figure also shows that the bending mode has lowest frequency in [100] direction and highest frequency in [110] direction with the same γ . So the SiNW in [100] direction are easiest to be bend, which may be due to its regular surface structure. We calculate the CTE along the axial direction of the SiNW using the nonequilibrium Green’s function method, which includes all phonon modes automatically and can be applied to very low temperatures as it is a quantum approach.[24] In this method, firstly, the average vibrational displacement of atom j , 〈 u$_{j}$ 〉 , is calculated through: 〈 u$_{j}$ 〉 = i ℏ G$_{j}$ , where G$_{j}$ is the one-point Green’s function and can be calculated from its Feynman diagram expansion in terms of the nonlinear interaction. We consider the nonlinear interaction as, where the nonlinear force constant is extracted from the Tersoff potential in GULP by finite H$_{n}$ = ∑ lmn k$_{lmn}$ 3 u$_{l}$u$_{m}$u$_{n}$, (1) 4