Order 1238142: Condensed matter

Senior Condensed Matter Physics

Assignment A, 2018

This assignment is due Thursday 18 October 2018 (11:59pm). It should be submitted online through the Canvas or Turnitin link provided on the Canvas site for this unit.

Question 1. The Sommerfeld theory of metals improves upon the Drude theory of metals in which Drude applied the highly successful kinetic theory of gases to a metal, which was considered as a gas of electrons. Explain how the improvement was achieved and describe shortcomings of Sommerfeld’s theory.

Question 2. The density of copper is 8.92x103 kg/m3. The resistivity is 1.73x10-8 Ω m and the weight of a copper atom is 63.5 amu (atomic mass unit). (i) Assuming each Cu atom contributes one conduction electron, calculate the electron mobility and the average time between collisions (relaxation time). (ii) Calculate the Fermi energy, Fermi wavevector, Fermi velocity and Fermi temperature.

Question 3. Sodium atoms occupy a volume of 4 x 10-29 m3. Each atom contributes one free conduction electron. Calculate the Fermi velocity of sodium and compare it to that of copper above.

Question 4.

(i) A face-centred-cubic lattice with conventional cubic cell has a side length a. Determine its reciprocal lattice unit cell (first Brillouin zone) and reciprocal lattice vectors.

(ii) A hexagonal lattice has lattice constants a and c. Determine its reciprocal lattice unit cell and reciprocal lattice vectors. In both cases show full working.

Question 5. Consider a primitive cubic crystal of side a where the crystal potential becomes vanishingly small (the empty lattice approximation).

(a) Draw a diagram in real space showing the (001), (110), (110), and (111) planes. (b) Draw an equivalent diagram in reciprocal space showing the [001], [110], [110] and [111]

reciprocal lattice vectors. (c) Write down all possible reciprocal lattice vectors of the type ⟨001⟩. (d) An electron is propagating along the [111] direction in real space. It has then wavevector

components kx=ky=kz=k/√3. Prove, that in this direction, the first Brilloun zone boundary occurs at (2π/a)[1/2,1/2,1/2].

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Question 6. (Normal only) Sketch the reciprocal lattice for the three lattices below. Include units in your answers. The filled and empty circles represent different atomic species.

Question 6. (Advanced only) Consider a two-dimensional solid measuring LxL in which the atoms are arranged in a square lattice with lattice parameter a. There is only a single atomic species in the solid, and the valence is equal to one, that is, each atom contributes one electron to the electron “sea”.

(i) Sketch the reciprocal lattice for this crystal and include units. (ii) Compute the Fermi wave-vector kF. (iii) Make a two-dimensional sketch and indicate, the Fermi circle (i.e. the occupied states at

T=0K) and the reciprocal lattice. (iv) Sketch a plot of Energy versus k for nearly-free electrons travelling in the i and i+j

directions. Indicate on each plot the occupied states at T=0 and comment on the electrical conductivity of the system. Explain your reasoning.