In the tight-binding model for a 1D chain with one orbital per site, derive the band energy (E(k)).
An n-type semiconductor has donor concentration (N_d). Find the Fermi level at low (T).
Explain the origin of ferromagnetism in the mean-field Heisenberg model. condensed matter physics problems and solutions pdf
London eq: (\nabla^2 \mathbfB = \frac1\lambda_L^2 \mathbfB), with (\lambda_L = \sqrt\fracm\mu_0 n_s e^2). Solution: (\mathbfB(x) = \mathbfB_0 e^-x/\lambda_L).
Calculate the electronic specific heat (C_V) in the free electron model. In the tight-binding model for a 1D chain
Elastic scattering: (\mathbfk' = \mathbfk + \mathbfG). (|\mathbfk'| = |\mathbfk| \Rightarrow |\mathbfk + \mathbfG|^2 = |\mathbfk|^2 \Rightarrow 2\mathbfk\cdot\mathbfG + G^2 = 0). For a cubic lattice, (|\mathbfG| = 2\pi n/d), leading to (2d\sin\theta = n\lambda). 2. Lattice Vibrations (Phonons) Problem 2.1: For a monatomic linear chain with nearest-neighbor spring constant (C) and mass (M), find the dispersion relation.
At low (T), (n \approx \sqrtN_d N_c e^-E_d/(2k_B T)), then (E_F = \fracE_c + E_d2 + \frack_B T2 \ln\left(\fracN_d2N_c\right)). 6. Magnetism Problem 6.1: Derive the Curie law for a paramagnet of spin-1/2 moments in a magnetic field. Explain the origin of ferromagnetism in the mean-field
At low (T), only electrons within (k_B T) of (E_F) contribute: (C_V = \frac\pi^22 N k_B \fracTT_F), where (T_F = E_F/k_B). 4. Band Theory & Nearly Free Electrons Problem 4.1: A weak periodic potential (V(x) = 2V_0 \cos(2\pi x / a)) opens a gap at (k = \pi/a). Find the gap magnitude.