Dynamical Orbital Angular Momentum Induced by Chiral Phonons

Abstract

We show that the orbital angular momentum (OAM) of electrons is dynamically induced by chiral phonons. The induced OAM originates from the adiabatic evolution in which electrons dynamically acquire Berry phase. By introducing a tight-binding model with p orbitals on a honeycomb lattice, we show a microscopic picture that chiral phonons modulate orbital overlaps of electrons, and calculate the generated OAM, whose sign depends on phonon chirality. We then construct an effective model for valley phonons with different phonon pseudoangular momenta (PAM) and identity their distinct intervalley-scattering channels. Our model obeys the selection rule between phonons and electrons with the orbital degree of freedom. Extending this framework to d-orbital electrons, our model is applied to describe the induced OAM in monolayer transition metal dichalcogenides. Our results reveal a direct lattice-OAM transfer mechanism that emerges even in materials with weak spin-orbital coupling, opening a new promising way for orbitronics applications.

Figures (4)

• Figure 1: Electronic states with phonon dynamics on 2D honeycomb lattice. (a) Schematic view of the 2D honeycomb lattice with $p_x$ and $p_y$ orbitals. (b) Electronic band structure without phonons blue lines) and that with the chiral phonon at $\Gamma$ point (red lines) after taking time average. (c) and (d) Dynamical OAM induced by the optical-$\Gamma$ chiral phonons. We show its dependence on the dimensionless time $\tau$ with the CCW phonon mode in the inset of (c) and CW phonon mode in the inset of (d). Red and blue solid lines represent the contribution from the first and second band shown in (b) below $E=0$, and the black dot-dashed line is their sum. Here we employ the parameters: $\Delta=0.2t_{\sigma}$ and $u_r=0.05a_0$.
• Figure 2: Valley chiral phonons and selection rule. (a) Phonon dispersion between the high-symmetry points $\Gamma$ and $K$ in the phonon BZ. (b) Phonon eigenmodes at $K$ point with PAM. Here we set the location of the atom A marked in (b2) as the center of $C_3$ rotation. We assume that the springs only exist between the NN atoms. We take the longitudinal and transverse spring constants as $K_T=K_L/4$, and the masses of atoms A and B satisfies $m_B=0.8m_A$ (see Supplemental Material SM for details). (c) Chiral phonons with different PAM lead to intervalley scattering of electrons, which satisfy the selection rules between phonons and electrons with orbital degree of freedom.
• Figure 3: Band structure and OAM calculated from the effective Hamiltonian with $p$ orbitals. Band structure after taking time average calculated from the effective Hamiltonian plotted by (a) green lines with $l_{\text{ph},K}=1$ and $l_{\text{ph},K'}=-1$, (b) red lines with $l_{\text{ph},K}=0$ and $l_{\text{ph},K'}=0$, and (c) blue lines with $l_{\text{ph},K}=-1$ and $l_{\text{ph},K'}=1$. (d) Time-averaged OAM induced by chiral phonons with different PAM. Here the horizontal axis is the wavenumber near the $\Gamma$ point. The parameters are employed as $\Delta=0.2t_{\sigma}$, $t'_{\sigma}=0.1t_{\sigma}$, $u_A=3u_B=0.05a_0$ with $a_0=1$.
• Figure 4: Electronic OAM induced by the phonon at $K$ point with $l_{\text{ph}}=1$ in monolayer TMDs MoS$_2$, MoSe$_2$, WS$_2$, and WSe$_2$. (a) Schematic picture of a TMD crystal structure with $t_{\text{M-X}}$ and $a_0$ being the hopping parameter and the distance between the atoms M and X. The left side is the lattice from the top view. (b) OAM as a function of wavenumber $q$ near the $\Gamma$ point. (c) Total OAM as a function of $u_M/a_0$, which represent the size that the atom M displaced from the equilibrium.