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46 Metallic nanowires: differencies between stretching and bending Side view y Top view z z y F Z F x x r z Δ θ/2 A l = A - A’ R A’ θ d0 Δ θ/2 d0 S Figure B.1: A cross section S of the cantilever before and after a perpendicular force is applied on the wire’s end. Inset: Side view and top view of the cantilever nanowire. The expansions and contractions of the cross sections are larger nearer the fixed end if the cross section is allowed to contract and expand freely. using Legendre transforms [52] of the energy to get Ω were Stot is the entropy of the electron gas. We have for the electron gas the heat capacity Cv [53] π2 T Cv = 2NtotkBT , (B.6) F where EF = kB TF . By using dStot = dQ/T = Cv dT /T and integrating we obtain π2 T Stot = 2 NtotkB T , F sousingμ≈EF atroomtemperaturewefind π2 T 2 TStot=μNtot2T . F (B.7) (B.8) BecauseT/TF ≈1/100atroomtemperature,thecontributionfromtheentropyterm TStot to Ω may be neglected and we obtain using Eq. (B.5) the grand canonical potential given by [10, 8] for the T = 0 case Ω = − L 4 2m (EF −En (x))3/2 dx, (B.9) n 0 3 π22 where L is the length of the wire. This equation is the starting point of Paper I and also of Blom et al. [8]. xyPDF Image | Nanomechanics Quantum Size Effects, Contacts, and Triboelectricity
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