Past exam of the condensed matter physics major option course of the University of Oxford / 2019 / 1 / d

Apply the same change of reciprocal lattice coordinates to the phonon wave vectors:Although $(1,0{)}_r$ is an integer coordinate of the rectangular reciprocal grid, it is not a reciprocal lattice vector of the actual primitive crystal. Thus $P_1$ is not equivalent to the Brillouin zone centre.

For ordinary one-phonon infrared optical absorption, both an energy and a momentum selection rule must hold. The optical photon has negligible crystal momentum on the scale of the Brillouin zone, so crystal momentum conservation selects $\mathbf{q}\simeq 0$ modulo a genuine reciprocal lattice vector. In addition, the mode must change the macroscopic electric polarization, with a component along the incident polarization of an electromagnetic wave.

In terms of atomic displacements ${\mathbf{u}}_j$ and Born effective charge tensors $Z_j^{\ast }$, the relevant derivative is proportional to ${\sum }_j{Z}_j^{\ast }{\mathbf{u}}_j$ per primitive unit cell. Thus an optical phonon at $\mathbf{q}=0$ is not automatically infrared active: its dipole contributions may cancel. Disorder-assisted and multiphonon processes relax these simple first-order selection rules.

$P_1$ has primitive coordinates $(1/2,1/2)$ and therefore fails the first-order optical momentum condition; its local $V_1$ pattern also has zero dipole derivative. $P_2$ is a zone-centre mode with the dipole-active $V_2$ pattern, so its molecular dipole changes add coherently.The allowed absorption for $P_2$ uses an polarization of an electromagnetic wave with an out-of-plane component. At normal incidence on an ideal free-standing layer the electric field is in the plane, so that specific measurement geometry would suppress it despite the mode's intrinsic vibrational infrared activity.