(Solved): (a) The classical Drude theory of conductivity of metals states that electrons behave as a gas, dif ...

(a) The classical Drude theory of conductivity of metals states that electrons behave as a gas, diffusing through a crystal and scattering from atoms in the lattice. How does quantum band-theory explain why electron waves don't constantly scatter off the lattice atoms in the crystal? (b) Explain, with sketches of the \\( E(k) \\) relationship in the reduced zone scheme, the concept of effective mass and how it relates quantitatively to the bandstructure. (c) A piece of pure metal has a conductivity of \\( 3.65 \\times 10^{5} \\mathrm{ohm}^{-1} \\mathrm{~cm}^{-1} \\), a carrier concentration of \\( 5.85 \\times 10^{22} \\mathrm{~cm} \\) , and an effective mass of \\( 0.89 \\mathrm{~m}_{\\mathrm{e}} \\). At time \\( t=0 \\) an electric field \\( E=1 \\mathrm{~V} / \\mathrm{m} \\) is applied for \\( 1 \\mathrm{ps}\\left(10^{-12} \\mathrm{~s}\\right) \\), then turned off. i) Paying attention to units, calculate the maximum electron velocity ii) Sketch a numerically labelled graph of the electron velocity as a function of time. iii) Sketch an equivalent graph for an electron in free space, and explain the difference