Chiral phonons in tellurium

A phonon mode in which atoms trace circular (rather than linear) orbits carries angular momentum and a definite handedness — a chiral phonon. In chiral crystals the handedness is pinned by the structure: the screw symmetry of the lattice quantizes a phonon pseudo-angular momentum, fixes selection rules at high-symmetry points, and routes the crystal’s chirality axis through its lattice dynamics. Trigonal tellurium — helical chains in the chiral space groups $P{3}_{1}21$ / $P{3}_{2}21$ — is the canonical platform. This is a frontier note: thinner than the L2 lattice-dynamics notes, but it carries the phonon angular-momentum operator and the $\mathbf{q}/-\mathbf{q}$, time-reversal structure that the selection rules rest on.

Prerequisites

dynamical matrix & ASR · crystal symmetry

1. Phonon angular momentum

Solve the dynamical matrix eigenproblem $D(\mathbf{q}){\mathbf{ϵ}}_{\mathbf{q}\nu }={\omega }_{\mathbf{q}\nu }^2{\mathbf{ϵ}}_{\mathbf{q}\nu }$ for the (complex) polarization vectors $\mathbf{ϵ}$. A mode is circularly polarized when, within a sublattice block, the two transverse Cartesian components are $\pm i$ out of phase, e.g. $\mathbf{ϵ}\propto (\hat{\mathbf{x}}\pm i\hat{\mathbf{y}})/\sqrt{2}$ — each atom executes a circular orbit. The phonon angular momentum along $\hat{z}$ (per mode, in the occupied phonon gas) is the expectation of the angular-momentum operator built from the displacement and its conjugate momentum,

$$\boxed{{\ell }_{\mathbf{q}\nu }^z=\hslash {\mathbf{ϵ}}_{\mathbf{q}\nu }^{†}\hat{\mathcal{M}}{\mathbf{ϵ}}_{\mathbf{q}\nu },\hat{\mathcal{M}}=1_N\otimes \left(\begin{array}{ccc}0& -i& 0\\ i& 0& 0\\ 0& 0& 0\end{array}\right)}$$

the $\hat{z}$ generator of $SO(3)$ rotations acting on each atom’s Cartesian triple ($N$ atoms in the cell). ${\ell }^z$ vanishes for a linearly polarized (real $\mathbf{ϵ}$) mode and saturates at $\pm \hslash$ for a fully circular single-atom mode; for a multi-atom cell it is the sublattice-summed circular weight, generally a non-quantized real number. Summed over the thermally populated spectrum, $J_{\text{ph}}^z={\sum }_{\mathbf{q}\nu }{\ell }_{\mathbf{q}\nu }^z[n({\omega }_{\mathbf{q}\nu })+\frac{1}{2}]$, this is the lattice’s intrinsic phonon angular momentum — the carrier of the phonon contribution to the Einstein–de Haas effect and to phonon-driven (ultrafast) magnetization.

2. Time-reversal and the $\mathbf{q}\to -\mathbf{q}$ structure

Phonon angular momentum is time-reversal odd. Under $\mathcal{T}$ the dynamical matrix obeys $D(\mathbf{q}{)}^{\ast }=D(-\mathbf{q})$ (real force constants), so the eigenvectors satisfy ${\mathbf{ϵ}}_{\mathbf{q}\nu }^{\ast }={\mathbf{ϵ}}_{-\mathbf{q}\nu }$ and consequently

$${\ell }_{-\mathbf{q}\nu }^z=-{\ell }_{\mathbf{q}\nu }^z.$$

A mode at $+\mathbf{q}$ rotating clockwise has a partner at $-\mathbf{q}$ rotating counterclockwise — a relation enforced by $\mathcal{T}$ alone, hence present in any non-magnetic crystal. Since ${\omega }_{-\mathbf{q}\nu }={\omega }_{\mathbf{q}\nu }$ as well, the equilibrium Brillouin-zone sum $J_{\text{ph}}^z={\sum }_{\mathbf{q}\nu }{\ell }_{\mathbf{q}\nu }^z[n+\frac{1}{2}]$ vanishes by this $\mathcal{T}$ relation regardless of chirality; a net macroscopic phonon angular momentum requires breaking $\mathcal{T}$ (a magnetic field or magnetic order) or driving the lattice out of equilibrium (a thermal gradient — the phonon Hall effect — or optical pumping). What chirality controls is the per-mode angular momentum: an improper operation (inversion, a mirror, or $S_n$) combined with $\mathcal{T}$ forces ${\ell }_{\mathbf{q}\nu }^z=0$ for every mode, so a centrosymmetric crystal hosts no chiral phonons at all. Chirality removes every improper operation, so a single mode at $\mathbf{q}$ carries a definite, nonzero ${\ell }^z$ — well-defined chiral phonons, with a handedness pinned to the lattice’s. This is the structural reason chiral crystals, not arbitrary lattices, are the natural hosts of robust chiral phonons.

3. Pseudo-angular momentum and selection rules at high-symmetry points

At a high-symmetry point invariant under an $n$-fold (screw) rotation $C_n$, a phonon eigenvector is also an eigenvector of $C_n$ with eigenvalue $e^{-i2\pi l_{\text{ph}}/n}$, defining the phonon pseudo-angular momentum (PAM) $l_{\text{ph}}\in \{0,\pm 1,\dots \}\mathrm{mod}n$. The PAM splits into an intrinsic part (the circular character of the polarization vector within a sublattice) and an orbital part (the phase the screw operation imprints by permuting sublattices), reflecting that a screw rotation combines a rotation with a fractional lattice translation. Conservation of total PAM — lattice (the umklapp $\mathrm{mod}n$), electronic, and photon — gives the selection rules: in a circularly polarized optical (infrared / Raman) or phonon-assisted process the change in electronic PAM plus the photon’s $\pm 1$ must match the phonon PAM modulo $n$. This is what makes chiral phonons valley/helicity selective — a given circular polarization excites only one PAM channel — and is the experimental handle (helicity-resolved Raman, transient infrared) on the handedness. The detailed PAM bookkeeping is set by the little group of the high-symmetry $\mathbf{k}$ and its irreducible representations.

4. Tellurium — the chiral lattice

Trigonal Te is the textbook chiral crystal: helical chains of covalently bonded atoms wind around the $c$ axis with a three-atom repeat, stacked on a hexagonal lattice and held by weaker interchain bonds. The handedness of the helix selects between the two enantiomorphic, non-centrosymmetric space groups

$$P{3}_{1}21(\#152)\text{and}P{3}_{2}21(\#154),$$

related by mirror reflection and distinguished only by the sense ($3_1$ vs. $3_2$ screw) of the chains — there is no inversion, no mirror, no $S_n$, so the crystal is structurally chiral, and its $3_1/3_2$ screw axis is exactly the $C_n$ of §3 with $n=3$. The helical chains make Te a natural host for chiral phonons: zone-center and zone-boundary optical modes inherit a definite PAM from the screw, the chirality of the lattice imprints a chirality on the phonon eigenvectors, and the sign of the phonon angular momentum tracks the enantiomer. Te thereby ties the lattice-dynamics chirality to its better-known electronic cousins — the Weyl-like radial spin texture and the chiral charge transport of the same crystal — making the phonon sector a probe of, and potentially a knob on, the structural handedness. (Quantitative mode-by-mode PAM assignments and the helicity-resolved spectroscopy of Te are an active, partly unsettled, literature — TODO(cite) for the specific Te mode table beyond the canonical chiral-phonon references below.)

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Builds toward: phonon-driven magnetization / Einstein–de Haas · phonon Hall & thermal Hall effects · chiral charge/spin transport in Te · valley-selective phonon spectroscopy

Key references

• Phonon angular momentum — L. Zhang & Q. Niu, Phys. Rev. Lett. 112, 085503 (2014).
• Chiral phonons (concept & observation) — L. Zhang & Q. Niu, Phys. Rev. Lett. 115, 115502 (2015); H. Zhu, J. Yi, M.-Y. Li, J. Xiao, L. Zhang, C.-W. Yang, R. A. Kaindl, L.-J. Li, Y. Wang & X. Zhang, Science 359, 579 (2018).
• Chiral crystals & tellurium structure — see space-group entries $P{3}_{1}21$ (#152) / $P{3}_{2}21$ (#154); A. Koma & S. Tanaka, Phys. Status Solidi B 40, 239 (1970) for Te lattice dynamics TODO(verify cite); chirality/transport review: TODO(cite) for a current chiral-Te phonon review.