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1.2 Quantum effects at the nanoscale 3 Figure 1.1: Examples of small animals that have their feet covered by nano-sized hair with terminal elements (circles) to ensure high adhesion. Feet with a larger area would only have a true contact area from a small number of nanocontacts, whereas if there is a large number of nanocontacts, all of them could potentially adhere to the target surface. The heavier the animal the smaller is the diameter of the top of the hair. From ref [6]. ⃝c 2003 by National Academy of Sciences are usually in the quantum regime while the much heavier atoms are in the classical regime. If we are in the classical regime the law of equipartition is valid. It tells us that for each excited degree of freedom, like the kinetic energy in the three translation directions, rotation or vibration kinetic energy components or potential energy com- ponents, there will be an equal contribution to the internal energy. If the dependence is quadratic like in 21 mv2 or 21 kx2, then the contribution for each such term will be 12 kB T . New degrees of freedom become excited as the temperature rises and we are leaving the quantum regime for this degree of freedom. The law of equipartition is also valid for a nanowire cantilever. If its potential energy is given by 12 kA2 where A is the amplitude of oscillation and k is the spring constant, the cantilever has the energy 12 kB T . This means that we can calculate the thermally excited oscillation amplitude from kA2 = kB T . At room temperature this amplitude has been observed, see Figure 1.3 (2A). The de Broglie relation λ = h/p between wavelength λ and linear momentum p as well as the relation between energy E and frequency f , E = hf are universally valid for all particles and quasi particles: for photons (light), electrons, phonons (vibration quanta) and even nanowire cantilevers [7]. We can thus write its energy, assuming that the cantilever is an harmonic oscillator, as 21 kA2 = (n + 12 )hf where the integer n is the quantum number. For a cantilever, however n is very large, so the difference in amplitude A for different n can not be observed as it could be in the thermalPDF Image | Nanomechanics Quantum Size Effects, Contacts, and Triboelectricity
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