Tonomura experiment to prove wave-particle duality of matter

Question

Question 1)a) Compute packing fraction of FCC lattice. State two crystals having FCC crystal structure.

b) Cobalt is usually found with hexagonal crystal structure. Recently it was found that it crystallizes with simple cubic structure now called ε-cobalt having a= 6.097 Å. The density is 8.635 g/cm³. The unit cell contains 20 atoms. Compute the number of atoms in 2nM diameter nanocrystal.

c) What is the significance of surface to volume ratio of nanostructures?

Question 2)a) How do you distinguish a nanoparticle in terms of its physical size in the case of optical properties of semiconductor materials?

b) Derive the expression for exciton Bohr radius. How is it related to Bohr radius.

c) Compute exciton Bohr radius of the following: InAs with m_e = 0.02 m₀ , m_h = 0.4 m_o and ε = 14.5.

Question 3)a) What are the types of excitons? Derive the expression for the total energy of an exciton in terms of Rydberg’s constant.

b) How do you define the confinement regimes for semiconductor nanoparticle.

Question 4)a) Explain possible solutions of Schrodinger’s equation for 1D well of arbitrary form depending on the boundary conditions in terms of energy of the particle and strength of potential well.

b) Define energy density of 1D continuous medium. Show that energy density for travelling plane wave is independent of coordinate and is proportional to ω²

c) Define phase velocity of travelling wave.

Question 5)a) Explain Tonomura experiment to prove wave-particle duality of matter.

b) Explain the electron motion in a 1D potential well of penetrable walls and having finite potential walls.

c) Consider an electron which is placed into a quantum well of width L = 10^-6 barrier height V cm and barrier height V_b = 300 mev. Compute the lowest energy level.

Question 6) Explain different parts of STM with a neat diagram. How the STM tips are fabricated. How can it be used to understand LDOS of a given nanostructured material?

b) Write short notes on:

i) Atom lithography.

ii) Scanning probe nanofabrication.