Condensed Matter Physics Problems — And Solutions Pdf

Degenerate perturbation theory at Brillouin zone boundary: Matrix element (\langle k|V|k'\rangle = V_0). Gap (E_g = 2|V_0|).

(n_i = \sqrtN_c N_v e^-E_g/(2k_B T)), with (N_c = 2\left(\frac2\pi m_e^* k_B Th^2\right)^3/2), similarly for (N_v). condensed matter physics problems and solutions pdf

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. At low (T), only electrons within (k_B T)

Compute the density of states in 1D, 2D, and 3D Debye models. Find the gap magnitude

In the tight-binding model for a 1D chain with one orbital per site, derive the band energy (E(k)).

Calculate the electronic specific heat (C_V) in the free electron model.