difference method. Then the CTE can be calculated through its definition, where x$_{j}$ is the lattice position of atom j along the axial direction. We can get the final value of CTE for SiNW by averaging α$_{j}$ over all atoms. In Fig. 4, we compare the CTE in three different directions. All three SiNW have the same structure parameters ( L , D ) = (1, 4) nm. It shows that the CTE in all directions are negative in very low temperature region. Then it increases with further increasing temperature and reach a saturate value in high temperature limit. The saturate value is positive for these SiNW shown in the figure. We mention that before reaching the saturate value, all three curves still show obvious increasing behavior at about T = 500 K, which implies that the Debye temperature in SiNW is roughly about 500 K. From this figure, we also learn that the CTE in [110] direction is distinctly larger than the other two directions. While the CTE in [100] direction is close yet a little smaller than that in [111]. The above behaviors for the CTE at different temperatures and in different directions can be understood from the various phonon modes in SiNW discussed in the previous part. Because of different vibrational morphology, these phonon modes have very different con- tribution to the CTE. The bending mode has negative effect on the CTE of the rod-like SiNW.[37, 38] The twisting mode only takes up some degrees of freedom, without making any contribution to CTE. The longitudinal vibration mode has positive effect on the CTE. So the CTE is mainly determined by the competition between the bending mode and lon- gitudinal mode. Since the bending mode has the lowest energy among all phonon modes, it is the only mode excited at low temperature, leading to negative CTE. With the increase of temperature T , the longitudinal mode will be excited gradually and competes with bend- ing mode. This competition slows down the decreasing of CTE, and results in a minimum point. As the longitudinal mode becomes more and more important, CTE will increase with further increasing temperature. In high temperature limit, CTE reaches a saturate value after sufficient competition between all phonon modes. The positive saturate value in these SiNW implies the leading contribution of longitudinal mode for these SiNW. Also the bending mode is anisotropic with the frequencies in [110], [111] and [100] directions as 53.1, 41.2, 39.3 cm − $^{1}$, respectively. This anisotropic property leads to the anisotropic effect of CTE, i.e. α$_{[110]}$ > α$_{[111]}$ > α$_{[100]}$ . 5 α$_{j}$ = d 〈 u$_{j}$ 〉 dT × 1 x$_{j}$ , (2)