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16 Results cross section. For a wire with a quadratic cross section we have 2π2 En = 2md2(n21+n2), (2.2) 0 where n1 and n2 is the quantum numbers = 1,2,3,... . The side in the square is denoted d0, and m is the electron mass. The degeneration is twofold (not counting spin) unless n1 = n2. We bend a nanowire, which initially has a square cross section. This leads to the deformation of the nanowire cantilever cross section as shown in Figure B.1. To calculate Ω we must sum over all energy levels, from the ground level up to just below the Fermi level EF . The dependency of the two quantum numbers are somewhat complicated because their sum n21 + n2 is restricted so that the energy of the wave does not exceed EF . Also, after its deformation, due to the wire bending, the shape of the cross section is no longer square. To be able to proceed analytically we use the formula for the asymptotic eigenenergies for high eigenvalues of confined waves on a two-dimensional flat surface of area S, which may have any shape. We have [28] En(x) = 2 4π n, (2.3) 2m S (x) where n is now the only ”quantum number” in this approximation. The higher n becomes, the better the approximation. S (x) is given in Paper I by , (2.4) S(x)=d201+ 0 8L6 9ν2d2 (L − x)2 Z2 where ν is the Poisson’s ratio of the metal in the wire. S thus varies along the wire: if the cross section is allowed to expand and contract freely, the most deformed part of the cross section is closer to the fixed end at x = 0. At the free tip at x = L it assumes its undeformed value S = d20. Calculating, this leads to a bending dependent spring constant given by N 2 m 6 d 20 ν 2 E0n π22nE0 4L3 EF− 9ν2d20Z2 9ν2d2Z22, (2.5) (2.6) To illustrate the changes in equation (2.5) with different values of N = EF /E0 we made plots. The function to be plotted is N(1+CZ2) n n k′(Z) = N−1+CZ2 ×(1+CZ2)2, (2.7) n=1 k= using E0 = 2π2/(md20) and N = E 1+ 8L4 . n=1 1+4 1+0 8L 8L4 EF 9ν2d20Z2 0PDF Image | Nanomechanics Quantum Size Effects, Contacts, and Triboelectricity
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