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4 Introduction d = 3λ 2 d = λ λ E3 λ= h Ε=mv2 mv 2 λ = de Broglie wavelength E2 h = Plank’s constant m = electron mass E1 v = velocity d = 2 Figure 1.2: Standing waves corresponding to an integer number of half wavelength will fit. d E = kinetic energy d = box diameter excitation case above. Through the existing quantum-size effect in the electron gas even a metallic nanosized cantilever will indirectly be in the quantum regim. The electrons in a metal can either be bound to individual atoms or be delocalized and free to move in the entire volume of the metal. These free electrons form the gas that makes electrical conduction possible. An electron in motion has a de Broglie wavelength associated with it. In a given direction, this wavelength depends on the electron velocity component in this particular direction. The higher the velocity of the electron, the shorter the de Broglie wavelength becomes (electron size). Because an electron in a volume is described by standing waves between the walls of the volume, only certain wavelengths will fit, see Figure 1.2. For a wire of length L with a rectangular cross section we have the energies 2π2 n1 2 n2 2 n3 2 En = 2m d + d + L (1.3) 12 where d1 and d2 are the sides in the rectangle, m is the electron mass and n1, n2 and n3 are the quantum numbers = 1, 2, 3, . . . There is thus a minimum energy the electron can have when it is confined to move between the walls. The smaller the volume the larger this minimum energy. If one dimension is much smaller than the other two this distance will determine the electron minimum energy, see Eq. (1.3). Electrons also have a maximum energy depending on the metal used. This max- imum energy is called the Fermi energy. If we then make a wire of the metal and reduce its diameter the minimum energy of the electrons will increase. When this minimum energy exceeds the Fermi energy of the metal, the wire will no longer be able to contain any free electrons and would no longer be a conductor. What will then happens if we stretch a nanowire? This has been calculated by Stafford et al. and Blom et al. [10, 8]. When a nanowire is stretched, its cross section decreases. All electron energy levels are then shifted to higher energies, see FigurePDF Image | Nanomechanics Quantum Size Effects, Contacts, and Triboelectricity
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