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2.1 Quantum effects in nanomechanics 17 k (a.u.) 24.5 24.0 23.5 23.0 22.5 22.0 21.5 0 Figure 2.1: Spring constant k as a function of normalized bending Z′ obtained from our model for T = 0. For the curve with EF /E0 = 5.9 no electron state will be added for small bending. Bending will instead decrease all energy levels making the spring softer to bend further. For the curve with EF /E0 = 6.1 bending the wire will add an electron state making the spring harder to bend further at small bending. For small Z we have approximately k = k0 + K × Z2 for the upper and lower curve with different signs of the konstant K. where C = 9ν2d20/(8L4), which has the same type of dependency on n and Z as equation (2.5) has. We then obtain curves as shown in Figure 2.1. Taylor expansion for small bending Z yields ′ N √ nN−n+CZ n=1 Because of the weak non-linearity the resonance frequency of the oscillation is shifted from ω0 to ω02 + 34βA2 , see ref. [29]. A shift in the amplitude maximum is ob- tained, towards a higher frequency for positive β and towards a lower frequency for negative β. For small Z we have that k = k0 + K × Z2 except for the case when N = integer. The constant K may be positiv as in the upper curve in Figure 2.1, or negative as in the lower curve. We may then rewrite the harmonic equation with a bending dependent spring constant. The Duffing equation reads d2Z+ω02Z+βZ3+2γdZ = Fsin(ωt), (2.9) dt2 dt where ω02 = k0/M and the effective mass M is about 0.24 of the cantilever mass for an evenly thick cantilever, see [30, 31]. We show that the relative frequency shift ∆f /f0 would be given by ∆f 27 2d02A2 f ≈ 64ξν L L . (2.10) 0 k(Z)= n=1 2√N−n +.... (2.8) 0.2 0.4 0.6 0.8 E F = 6.1 E0 EF = 6.0 E0 EF = 5.9 E0 1.0 Z’ 2 N 5 n 2 − 4 N n PDF Image | Nanomechanics Quantum Size Effects, Contacts, and Triboelectricity
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