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 cAB (w) to the correlation functions SAB (w) and SBA (w) defined as

SAB (w)= <A(t)B(0)> eiwt dt.

Series 6: Semiclassical Transport

Derive the four matrix elements Lij  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.