Collective Interference of Phonon Spin and Dipole Moment Rotation Induced Circular Dichroism

Abstract

The classical field description of phonon spin relies on the invariance of a continuous elastic field under infinitesimal rotation. However, a local medium element in the continuous field may contain large numbers of vibrational particles at microscopic level, like for complex lattices with many atoms in a unit cell. We find this causes the phonon spin in real materials no longer a simple sum of each atom rotation, but a collective interference of many atoms, since phonons are phase-coherent vibrational modes across unit cells. We demonstrate the collective interference phonon spin manifested as the dipole moment rotating (DMR) of charge-polarized unit cell, by deriving the infrared circular dichroism (ICD) with phonon-photon interaction in complex lattices. We compare the DMR with the local atom rotation without interference, and exemplify their distinct ICD spectrum in a chiral lattice model and two realistic chiral materials. Detectable ICD measurements are proposed in quartz with Weyl phonon near Gamma point. Our study underlies the important role of collective interference and uncovers a deeper insight of phonon spin in real materials with complex lattices.

Figures (4)

• Figure 1: Schematic diagram of elastic spin, phonon spin and dipole moment rotating (DMR). (a) Elastic spin in continuum. Composition of two out-of-phase linearly-polarized elastic wave will contribute to circularly-polarized elastic wave whose spin AM is nonzero. (b) Phonon spin in simple and complex lattices. The unit cell is marked by dashed parallelogram and the rotation of mass centers are shown by orange trajectories. The rotation of atoms are shown by red and blue trajectories in complex lattice at right panel. (c) Schematic illustration of DMR. Dipole moments shown by yellow arrows are caused by the positive- ($+$) or negative- ($-$) charged atoms' displacements from equilibrium position shown as red and blue arrows. The left panel shows the rotation of single dipole moment like the case of simple lattice and right panel shows the rotation of unit cell's DMR $\bm P_{\textrm{uc}}$ composited by every single dipole moment ($\bm P_\pm$) like the case of complex lattice. (d) Three cases of DMR in the diatomic unit cell. Case 1 shows nonzero DMR caused by two dipole moments rotating with the same phase. Case 2 shows DMR vanishes when two dipole moments rotating with phase difference $\pi$. Case 3 shows two linear motions of dipole moments with phase difference $\pi/2$ can contribute to nonvanish DMR.
• Figure 2: DMR mapped on phonon dispersion relation of 3-fold helical chain. (a) Schematic diagram of three-fold screw helical chain. The unit cell is marked by black dotted box and the top view of this structure is shown above. (b) $S_z^{\textrm{local}}$ component of AM of atoms' local rotation mapped on the dispersion relation. The motion of atoms at $\Gamma$ point for one of the two-degenerate optical states is shown on the top where three atoms rotate clockwise. (c) $\mathcal{R}_z$ component of DMR mapped on the dispersion relation with charge distribution $[+1,+1,+1]$. The $\mathcal{R}_z$ mainly exists at acoustic branches while keeps zero at optical branches since atoms' motion at optical branches are out-of-phase resulting zero dipole moment of unit cell which is shown above by orange solid arrows. (d) $\mathcal{R}_z$ component of DMR mapped on the dispersion relation with charge distribution $[+1,-1,0]$. Nonzero DMRs arise at optical branches due to the rotating of unit cell's dipole moment.
• Figure 3: Comparison between the AM of local atom rotation $S_z^{\textrm{local}}$ and the collective interference DMR $\mathcal{R}_z$ of real chiral materials possessing different charge distribution. (a),(d) Atomic structure of right-handed tellurium and $\alpha$-quartz, respectively. (b)-(c) Phonon dispersion relation of tellurium together with band resolved $S^{\textrm{local}}_z$ and $\mathcal{R}_z$, respectively. (e)-(f) Same as (b)-(c) but for $\alpha$-quartz. The results clearly demonstrate that the DMR, as a manifestation of collective interference phonon spin in ICD, is for a coherent vibrational mode involving collective motion of many particles, so that is distinct from the local rotation of individual atom. See details of the calculation methods in section S2 of SM SM.
• Figure 4: Experimental measurable signature for the DMR, as a manifestation of collective interference phonon spin in ICD. (a) Phonon dispersion for the Weyl phonon [$\Psi_{1,2}$ states in Figs. \ref{['fig3']}(e)-(f)] around $\Gamma$. $\Delta \nu$ denotes the light frequency difference referenced to the frequency of Weyl phonon. (b) $\mathcal{R}^{\textrm{local}}$ (left panel) and total DMR $\mathcal{R} = \mathcal{R}^{\textrm{local}} + \mathcal{R}^{\textrm{nonlocal}}$ (right panel) corresponding to state $\Psi_1$. The degenerate partner state $\Psi_2$ has opposite $\mathcal{R}^{\textrm{local}}$ and $\mathcal{R}$ as $\Psi_1$. (c) Calculated spectrum of $\boldsymbol{\mathcal{R}}$-induced ICD for right-handed $\alpha$-quartz as a function of incident angle $\theta$ and frequency $\Delta \nu$. The incident angle is defined as the angle between the incident direction ($\boldsymbol{e_q}$) and the skew rotation symmetry axis as shown in the inset. (d) $\boldsymbol{\mathcal{R}}$-induced ICD averaged over incident angle $\theta$. $\Delta \nu_+$ ($\Delta \nu_-$) refers to the frequency of positive (negative) peak. (e) $\theta$-averaged ICD spectrum as function of effective refractive index $n_{\textrm{eff}}$.