Question 1. What is the minimum possible curvature of the glass layer of the thickness L before it cracks (fig. 3.33), if the critical tensile stress is σ* and the Young modulus is E? On which surface does the crack initiate?
Question 2.
(a) Explain why the energy band E(k→)=?2k→2/(2m) for a bulk 3D free electron metal is split into a number of sub-bands En(k→?)=En(0)+?2k→?2/(2m), where En(0)=π2?2n22m⋅L2 when the metal is a thin slab of thickness L.
(b) Calculate the density of states within each sub-band as a function of energy.
(c) Show that the sum of the densities of states at a given energy E over sub-bands for which En(0) < E is equal to the total density of states for the slab.
(d) Show that the density of states per unit length in a quantum wire with square cross-section LxL is ρ(E)=mh∑En,i(0)
(e) Show that the energy of an electronic state in a box of size LxLxL (a cubic quantum dot) is En,i,j(0)=π2?22mL2(n2+i2+j2),????n,i,j=1,???2,???3,???...,???La. where a is lattice constant of the box material.