some small deviation. To depress this small deviation, we do not use these atoms on the surface and boundaries. It seems that Si atoms in the figure do not distribute uniformly in the figure. That is because the structure is viewed from the upper front direction, which is better for illustrating a three-dimensional structure. Other configuration figures in this paper may also have this visual effect. The frequencies and eigen vectors of phonon modes are obtained from the Tersoff in- teratomic potential for silicon implemented in the GULP. The Tersoff potential can well describe bonding in covalently bonded solids such as silicon and carbon system, and also includes the nonlinear interaction. Its accuracy has been confirmed by comparison with re- sults from ab initio density functional theory.[26] In the investigation of CTE in the SiNW, we will use the clamped boundary condition, where one end (left, for example) is fixed and another end (right) is free. For consistency, we also apply this boundary condition in the study of phonon, which is different from Refs. 27, 28. Fig. 2 shows the vibrational morphology of three important phonon modes for CTE in SiNW. In the figure the SiNW is in [111] growth direction with ( L , D )=(3, 1) nm. The left-most 0.5 nm region (blue online) is fixed (clamped). Arrows (red online) attached onto each atom display the vibrational displacement for the atom.[29] The length of the arrow is not the exact value of vibrational displacement to show this displacement more clearly. It has been enlarged by a constant factor to all atoms. Panels (a) and (b) are the first and second order bending modes, which is a lateral mode with all atoms vibrate in the direction perpendicular to the axial direction. It is doubly degenerate due to two independent lateral vibrational directions. As the SiNW is clamped on the left, the vibrational displacement increases from left to right in both modes. In panel (a), the direction of the vibrational displacement does not change, while this direction changes in the second order mode in panel (b). The bending mode is actually a characteristic property for rod-like systems (eg. SiNW) or two-dimensional plane sheet (eg. graphene). Using its relation to the Young’s modulus of the material, the bending mode has been used to detected the value of the Young’s modulus of the low-dimensional systems such as single-walled carbon nanotubes[30] or graphene.[31] For longer SiNW, this mode will gradually turn to the standard flexure mode in a rod- like systems. Panels (c) and (d) are the first and second order twisting modes. Because of its special rotating morphology, the vibrational displacement increases from inner to the surface. This is consistent with the vanishing surface stress condition in a rod.[27] In 3