Effect of the inversion symmetry breaking on the orbital Hall effect: A model study

The orbital Hall effect (OHE) is the transverse flow of orbital moment in a solid in response to an applied electric field, analogous to the flow of spin moment in the spin Hall effect (SHE). Although the effect has not been directly observed, there is ample indirect evidence for its existence in a number of experiments. Here, we show that the OHE is enhanced in solids with broken inversion symmetry, which may be more suitable to observe the effect. The mechanism of the OHE is fundamentally different in solids with inversion symmetry, where the orbital moment is quenched in the Brillouin zone (BZ), from that in a solid with broken inversion symmetry, where an intrinsic orbital moment is already present, the motion of which under the applied electric field could lead to a robust OHE. Using a tight-binding model Hamiltonian of a simple cubic lattice with two atoms in the unit cell, we study the effect of the inversion symmetry breaking on the OHE. We show that with the increase in the strength of the broken symmetry, the magnitude of the intrinsic orbital moment in the Brillouin zone increases. This, in turn, enhances the orbital Hall conductivity, in particular, the part that is directly proportional to the orbital moment in the BZ, which we call the “noncentrosymmetric contribution.” If the spin-orbit coupling is present, which couples the orbital and spin moments, the OHE leads to the SHE, which also becomes enhanced by the broken inversion symmetry. Our work has important implications for experimenters, suggesting that noncentrosymmetric solids may be more suitable for direct observation of the OHE.

Figure 1

Left: Schematic illustration of the orbital Hall effect. The $+$ and $-$ signs indicate the accumulation of opposite orbital moments on the two edges of the sample. Right: The orbital moment $M_z\left(\vec{k}\right)$ (in units of ${\mu }_B$) on the $k_x\text{−}{k}_y$ plane in the Brillouin zone in a noncentrosymmetric crystal. The anomalous velocity ${\vec{V}}_{\vec{k}}$ indicated by the arrows is opposite for opposite moments leading to an orbital Hall current transverse to the electric field.

Figure 2

Band structure, orbital moment, and orbital Hall conductivity for the Hamiltonian equation (1). (a) Band structure along the $k$ direction $-M(-\pi /a,\pi /a,0)\to \mathrm{\Gamma }(0,0,0)\to M(\pi /a,-\pi /a,0)$ for the atom displacement along the cube diagonal, $\tau =0.2$ Å (see inset). The orbital characters are indicated in (a) as well as the Fermi energy ${\varepsilon }_F$, corresponding to the electron occupancy $n_e=6$. (b) The orbital moment $M_z\left(\vec{k}\right)$, computed from Eq. (6) and summed over the occupied bands. The maximum value occurs close to the $M$ point. (c) The variation of this maximum value $M_z^{\max }$ as a function of the broken $\mathcal{I}$ symmetry, parametrized by the displacement $\tau$. (d) The orbital Hall conductivity ${\sigma }_{yx}^{z,\mathrm{orb}}$ [in units of ${10}^3(\hslash /2e){\mathrm{\Omega }}^{-1}{\mathrm{cm}}^{-1}$] as a function of $\tau$, indicating larger OHC for noncentrosymmetric systems. For (a)–(d), spin-orbit coupling $\lambda =0$.

Figure 3

Berry curvature ${\mathrm{\Omega }}^z\left(\vec{k}\right)$ for a noncentrosymmetric crystal on the $k_x\text{−}{k}_y$ plane in the BZ, obtained from Eq. (8) by summing over the occupied bands. Units are in ${\AA }^2$, and $\tau =0.2$. The resulting anomalous velocity ${\vec{V}}_{\vec{k}}=(e/\hslash )\vec{E}\times \vec{\mathrm{\Omega }}\left(\vec{k}\right)$ was shown in Fig. 1 along with the computed orbital moments using Eq. (6) for the same value of $\tau$.

Figure 4

Centrosymmetric (C) and noncentrosymmetric (NC) contributions to the OHC. (a) Total and partial contributions to the OHC as a function of the symmetry-breaking parameter $\tau$. (b)–(d) The momentum space distribution of the orbital Berry curvature in the $k_z=0$ plane for $\tau =0.4$ ($k_x$ and $k_y$ are in ${\AA }^{-1}$). (b) shows the total ${\mathrm{\Omega }}_{yx}^z\left(\vec{k}\right)$, while (c) is the centrosymmetric part ${\mathrm{\Omega }}_{yx}^{z,\mathrm{C}}\left(\vec{k}\right)$ (Centro) and (d) is the noncentrosymmetric part ${\mathrm{\Omega }}_{yx}^{z,\mathrm{NC}}\left(\vec{k}\right)$ (Non-Centro).

Figure 5

Variation of the OHC [in units of ${10}^3(\hslash /2e){\mathrm{\Omega }}^{-1}{\mathrm{cm}}^{-1}$] for three different values of $V_{sp\sigma }$. In all cases, the OHC increases with $V_{sp\sigma }$. For the centrosymmetric system ($\tau =0$), there is no OHC if $V_{sp\sigma }$ is zero as discussed in the text.

Figure 6

Orbital and spin Hall conductivities as a function of the spin-orbit coupling strength $\lambda$. (a) The spin angular momentum $S_z\left(\vec{k}\right)$ (units of $\hslash$), summed over all occupied bands at $\vec{k}$, in the $k_z=0$ plane with $\lambda =0.05\mathrm{eV}$. (b) The same for the orbital angular momentum $L_z\left(\vec{k}\right)$. (c) The largest spin and orbital values along the $\mathrm{\Gamma }\text{−}M$ line, $S_z^{\max }\equiv \max [S_z\left(\vec{k}\right):\vec{k}\in \mathrm{\Gamma }\text{−}M]$ and $L_z^{\max }\equiv \max [L_z\left(\vec{k}\right):\vec{k}\in \mathrm{\Gamma }\text{−}M]$, as a function of the SOC strength $\lambda$. (d) The OHC and SHC, in units of ${10}^3(\hslash /2e){\mathrm{\Omega }}^{-1}{\mathrm{cm}}^{-1}$, as a function of $\lambda$. Note, from (a) and (b), the negative correlation between $L_z\left(\vec{k}\right)$ and $S_z\left(\vec{k}\right)$, which leads to the opposite signs of the OHC and SHC in (d). This correlation could be opposite in other situations as discussed in the text. In (a)–(d), $\tau =0.4$, and $n_e=6$.

Figure 7

Effect of the distance-dependent Slater-Koster TB parameters on OHC and SHC: distance-independent parameters (blue lines) and distance-dependent parameters (red lines). (a) Band structure, (b) orbital moment, (c) OHC, and (d) SHC. The Harrison scaling equation (16) was used for the distance dependence of the Slater-Koster TB parameters, $V_{ss\sigma }$, etc. Other parameters were $\lambda =0.05\mathrm{eV}, n_e=6$, and, in (a) and (b), $\tau =0.2$.

Figure 8

Spin Hall conductivity: Effect of the broken inversion symmetry, parametrized by $\tau$, and the spin-orbit coupling strength $\lambda$. (a) The momentum space distribution of the spin Berry curvature ${\mathrm{\Omega }}_{yx}^{z,\mathrm{spin}}$ (units of ${\AA }^2$) in the $k_z=0$ plane for $\tau =0.4$. (b) Centrosymmetric and noncentrosymmetric contributions to the SHC, ${\sigma }_{xy}^{z,\mathrm{spin}}={\sigma }_{\mathrm{spin}}^{\mathrm{C}}+{\sigma }_{\mathrm{spin}}^{\mathrm{NC}}$, as a function of the symmetry-breaking parameter $\tau$. (c) Centrosymmetric and noncentrosymmetric contributions as a function of $\lambda$. (d) Variation of SHC with $V_{sp\sigma }$. In (a)–(d), $\lambda =0.05\mathrm{eV}$ and $n_e=6$, except in (c), where $n_e=6$ but $\lambda$ is varied. The units are ${\AA }^{-1}$ for $k_x$ and $k_y$ and ${10}^3(\hslash /2e){\mathrm{\Omega }}^{-1}{\mathrm{cm}}^{-1}$ for the conductivities.

Figure 9

Variation of the OHC and SHC with electron occupancy $n_e$ ($n_e=16$ when all bands are full). Parameters are $\lambda =0.05\mathrm{eV}$ and $\tau =0.4$, and the conductivities are in units of ${10}^3(\hslash /2e){\mathrm{\Omega }}^{-1}{\mathrm{cm}}^{-1}$.

Figure 10

The correlation $\langle \vec{L}·\vec{S}\rangle$. Plotted are the components of the orbital and spin angular momenta $\vec{L}\left(\vec{k}\right)$ and $\vec{S}\left(\vec{k}\right)$ along the $\mathrm{\Gamma }\text{−}M$ line in the BZ, indicating a positive correlation $\langle \vec{L}·\vec{S}\rangle >0$ for the case $n_e=10$. Parameters are $\tau =0.4$ and $\lambda =0.05\mathrm{eV}$.