3. Consider a one-dimensional monatomic chain with lattice parameter a. Each atom possesses two localised orbitals: an s orbital, and a $p_z$ orbital oriented along the chain.
In the tight-binding approximation a Bloch state of this atomic chain can be written
$$|\psi_k\rangle = \sum_{n = -\infty}^{+\infty} e^{inka} (c_s |s_n\rangle + c_p |p_{z,n}\rangle),$$
where $|s_n\rangle$ and $|p_{z,n}\rangle$ respectively denote the s and $p_z$ orbitals localised on atom n.
Define the orbital energies $\langle s_n |H| s_n \rangle = \epsilon_s$ and $\langle p_{z,n} |H| p_{z,n} \rangle = \epsilon_p$, and define the nearest-neighbour hopping parameters between s orbitals as $\langle s_n |H| s_{n \pm 1} \rangle = -t_{ss}$, and between $p_z$ orbitals as $\langle p_{z,n} |H| p_{z,n \pm 1} \rangle = -t_{pp}$. Defining the nearest-neighbour hopping parameters between an s orbital and a $p_z$ orbital requires that we account for the $p_z$ orbital localised on atom n, at position $x = na$, possessing a positive (negative) lobe for $x < na$ ($x > na$). The hopping parameter involving this $p_z$ orbital and a nearest-neighbour s orbital thus changes sign if the s orbital is localised on atom n-1 vs. atom n + 1. We therefore define $\langle s_n |H| p_{z, n \pm 1} \rangle = \pm t_{sp}$.
Assume that all hopping parameters are real-valued, that orbitals localised on the same atom are non-interacting, $\langle s_n |H| p_{z,n} \rangle = 0$, and that all orbitals are orthonormal, $\langle \alpha_m | \beta_n \rangle = \delta_{\alpha \beta} \delta_{mn}$, where $\alpha, \beta$ and m, n respectively index orbitals and atoms.
(a) Show that Schrödinger's equation $H |\psi_k\rangle = E |\psi_k\rangle$ for this system can be written as the nearest-neighbour tight-binding model
$$ \begin{pmatrix} \epsilon_s - 2t_{ss}\cos(ka) & 2it_{sp}\sin(ka) \\ -2it_{sp}\sin(ka) & \epsilon_p - 2t_{pp}\cos(ka) \end{pmatrix} \begin{pmatrix} c_s \\ c_p \end{pmatrix} = E(k) \begin{pmatrix} c_s \\ c_p \end{pmatrix}, $$
where the 2 x 2 matrix H(k) on the left-hand side is the tight-binding Hamiltonian.
(b) Write down the Hamiltonian matrix H(k) at $k = \pi/2a$ and compute the band gap $E_g(\pi/2a)$ at $k = \pi/2a$.
(c) How is $H(-\pi/2a)$ related to $H(\pi/2a)$? What does that imply about $E_g(-\pi/2a)$?
