Condensed-Matter-2 Homeworks

Keivan Esfarjani, Sharif University of Technology

Saturdays and Mondays 14:00-16:00 

Series 1: The Stoner-Wohlfarth model

Consider the following energy functional for a ferromagnetic grain in an external  field and with anisotropy:

E= -M B cos(q-j) - K cos² q

q represents the angle of M with the easy axis B and (q-j) its angle with the magnetic field. Compute the ground state configurations by minimizing the energy for each magnetic field and plot the hysteresis curves for the 3 cases j=0, p/4 and p/2.

Series 2: Magnons

Calculate the magnon dispersion relation for spins located on atomic sites of graphene: a bipartite lattice with antiferromagnetic exchange coupling.

Series 3: Magnon-phonon interaction

Consider the Heisenberg Hamiltonian where the exchange coupling is a function of  interatomic distances. Expand J in powers of the atomic displacements and deduce the magnon-phonon interaction Hamiltonian. Compute to second order in perturbation theory the shift in magnon frequencies due to this interaction.

Series 4: Phase transitions

Calculate the critical temperature from ferromagnetic to paramagnetic phase for the Hubbard model in one dimension within the mean field approximation.

Series 5: Fluctuation-dissipation theorem

Relate the imaginary part of the retarded response function c_AB (w) to the correlation functions S_AB (w) and S_BA (w) defined as

S_AB (w)= ∫ <A(t)B(0)> e^iwt dt.

Series 6: Semiclassical Transport

Derive the four matrix elements L_ij  relating the electrical and thermal currents to the applied electrochemical and temperature gradient fields. Do the explicit calculation for metals and doped semiconductors with one type of carriers.