Dr. Aniruddha Chakraborty

The availability of top-down and bottom-up nano-fabrication capabilities has initiated a new area of nano-mechanics [1,2], in which ultra-small mechanical systems are used to explore both fundamental and applied phenomena. Here we consider a suspended elastic rod under longitudinal compression [3]. The compression can be used to adjust potential energy for transverse displacements from harmonic to double well regime. As compressional strain is increased to the buckling instability [4], the frequency of fundamental vibrational mode drops continuously to zero (first buckling instability).
As one tunes the separation between ends of a rod, the system remains stable beyond the instability and develops a double well potential for transverse motion. The two minima in potential energy curve describe two possible buckled states at a particular strain. From one buckled state it can go over to the other by thermal fluctuations or quantum tunnelling. Using a continuum approach [4] and transition state theory (TST) we have calculated the rate of conversion from one state to other. Saddle point for the change from one state to other is the straight rod configuration. Using TST we find exact expressions for rate under the harmonic approximation [5].
The expression, however, diverges at the second buckling instability. At this point, the straight rod configuration, which was a saddle till then, becomes hill top and two new saddles are generated. The new saddles have bent configurations and as rod goes through further instabilities, they remain stable and the rate calculated according to harmonic approximation around transition state remains finite. However, rate diverges near the second buckling instability. Therefore, we have included anharmonic corrections and derived expressions which are well behaved through the second buckling instability. Using these expressions, we calculate rate of passage from one buckled state to the other, for Si rods of two sizes. First size that we choose is 500nm x 20nm x 10nm. This size has been fabricated by Carr et al. [4]. We also did calculation for a hypothetical situation where dimensions were taken to be 50nm x 2nm x 1nm. It is not yet possible to fabricate Si rod of these dimensions. However it should be possible to synthesize molecular rods (not of Si) of similar size and elasticities.
We find that quantum effects are rather small and difficult to observe for a system of dimensions 500nm x 20nm x 10nm. But for a rod of dimensions 50nm x 2nm x 1nm, quantum effects are significant at very low temperature (1K) [6]. A calculation including friction has been carried out, by assuming that each segment of the rod is coupled to its own collection of harmonic oscillators. We find that friction lowers the rate of conversion [7].

References

1. H. Park, J. Park, A. K. L. Lim, E. H. Anderson, A. P. Alivisatos and P. L. McEuen, Nature, 407, 57 (2000).
2.  M. Roukes, Scientific American (September), 48 (2001).
3.  S. M. Carr, W. E. Lawrence, and M. N. Wybourne, Phys. Rev. B 64, 220101(R) (2001).
4.  L. Euler, in Elastic Curves, translated and annotated by W. A. Oldfather, C. A. Ellis, and D. M. Brown, reprinted from ISIS No. 58    XX(1), 1774 (Saint Catherine Press,   Bruges, Belgium).
5.  A. Chakraborty, S. Bagchi & K. L. Sebastian, J. Comput. Theor. Nanosci. 4, 504 (2007).
6.  A. Chakraborty, Mol. Phys. 107, 1777 (2009).
7.  A. Chakraborty, Mol. Phys., 109, 517 (2011).