Abnormal phonon angular momentum due to off-diagonal elements in the density matrix induced by a temperature gradient

A temperature gradient in chiral materials can generate a nonzero mean value of phonon angular momentum (PAM). We used the Kubo formula to investigate the contribution of both intraband and interband terms of PAM to the mean PAM. Interestingly, the interband term was found to be as important as the intraband term, indicating that the quantum transition between different phonon branches induced by a temperature gradient strongly affects locally atomic rotation. This discovery opens up an alternative mechanism for generating PAM and phonon magnetic moments.

Figure 1

Crystal structures: (a) wurtzite AlN and (b) MoSSe. (c) and (d) The first Brilliouin zones of AlN and MoSSe.

Figure 2

(a) Phonon dispersion of the wurtzite AlN. (b) Distribution of PAM of the sixth branch on the plane $k_z=0$. (c) Trajectories of the four atoms in the unit cell of the sixth phonon branch at $\mathbf{k}=(0,0.5828,0)\left({\mathring{\text{A}}}^{-1}\right)$. The axes are expressed in dimensionless units.

Figure 3

PAM response tensor ${\mathrm{\Lambda }}_{xy}$ of the wurtzite AlN. (a) The intraband part ${\mathrm{\Lambda }}_{xy}^{\text{D}}$ (red solid line), the interband part ${\mathrm{\Lambda }}_{xy}^{\text{OD}}$ (blue dashed line), and the total contribution ${\mathrm{\Lambda }}_{xy}$ (green dotted line) as functions of the relaxation time $\tau$ when $T=300$ K. (b) ${\mathrm{\Lambda }}_{xy}^{\text{D}}$ (red solid line), ${\mathrm{\Lambda }}_{xy}^{\text{OD}}$ (blue dashed line), and ${\mathrm{\Lambda }}_{xy}$ (green dotted line) versus the temperature $T$ when $\tau =1$ ps. (c) The contribution of acoustic branches (black dotted line) and optical branches (red dotted line) to the intraband part ${\mathrm{\Lambda }}_{xy}^{\text{D}}$. (d) The contribution to the interband part ${\mathrm{\Lambda }}_{xy}^{\text{OD}}$ from the transitions between different phonon branches.

Figure 4

(a) Phonon dispersion of the monolayer MoSSe. (b) Distribution of the PAM of the sixth branch. (c) Trajectories of tree atoms in the unit cell of the sixth phonon branch at high-symmetry point K, which represents the normalized polarization vector, and the axes are expressed in dimensionless units.

Figure 5

Calculated PAM response tensor ${\mathrm{\Lambda }}_{xy}$ of the monolayer MoSSe. (a) The intraband part ${\mathrm{\Lambda }}_{xy}^{\text{D}}$ (red solid line), the interband part ${\mathrm{\Lambda }}_{xy}^{\text{OD}}$ (blue dashed line), and ${\mathrm{\Lambda }}_{xy}$ (green dotted line) as functions of the relaxation time $\tau$ when $T=300\mathrm{K}$. (b) ${\mathrm{\Lambda }}_{xy}^{\text{D}}$ (red solid line), ${\mathrm{\Lambda }}_{xy}^{\text{OD}}$ (blue dashed line), and ${\mathrm{\Lambda }}_{xy}$ (green dotted line) versus $T$ when $\tau =1$ ps. (c) The contribution from acoustic branches (black dotted line) and optical branches (red dotted line) to ${\mathrm{\Lambda }}_{xy}^{\text{D}}$. (d) The contribution to the interband part ${\mathrm{\Lambda }}_{xy}^{\text{OD}}$ from the transitions between different phonon branches.