We now turn to the discussion of the size dependence for CTE in SiNW [111]. There are similar effects for the other two directions. Fig. 5 shows the dependence of CTE on the structure ratio γ in whole temperature range. The value of CTE decreases quickly with increasing γ . This is because the frequency of bending mode decreases quickly with increas- ing γ (see Fig. 3). As a result, the negative effect from bending mode on CTE increases. For SiNW with γ = 0 . 25, the transition of CTE from negative to positive happens around 120 K, and CTE reaches a saturate value of 10 × 10 − 6 K − 1 in high temperature limit. A large positive saturate value of CTE indicates that the longitudinal mode makes dominant contribution rather than the bending mode in this SiNW. The transition temperature in- creases from 120 K to 270 K with the change of γ from 0.25 to 1.0. The saturate value of CTE for SiNW with γ = 1 . 0 decreases for more than 50% compared with that of γ = 0 . 25. This result is reasonable and has also been observed in single-walled carbon nanotubes,[24] which has similar one-dimensional rod-like structure. It seems to be a common phenomenon in one-dimensional system that the CTE is much smaller in longer and thinner structures. When γ = 1 . 3, CTE is negative in whole temperature range, which indicates that the SiNW with these structure ratio undergoes thermal contraction at all temperatures. For these type of SiNW, the bending mode has played an dominant role over the longitudinal mode. To summarize, we have studied the thermal expansion of SiNW in different growth di- rections by the nonequilibrium Green’s function method. We find that at low temperatures, all SiNW studied exhibit thermal contraction effect due to the lowest energy of the bending mode which has negative effect on CTE. The CTE in SiNW [110] is larger than the other two directions, because of higher frequency of bending mode in SiNW [110]. The CTE decreases quickly with increasing of the structure ratio γ , and is negative at all temperatures with γ = 1 . 3. The work is supported in part by a Faculty Research Grant of R-144-000-257-112 of NUS, and Grant R-144-000-203-112 from Ministry of Education of Republic of Singapore, 6 IV. ACKNOWLEDGEMENTS III. CONCLUSION