Polarization and third-order Hall effect in III-V semiconductor heterojunctions

We study Berry connection polarizability (BCP) induced electric polarization and third-order Hall (TOH) effect in a two-dimensional electron/hole gas (2DEG/2DHG) with Rashba-Dresselhaus (RD) spin-orbit couplings in III-V semiconductor heterostructures. The electric polarization decreases with the increase of the Fermi energy and is responsive to the electric field orientation in the presence of RD spin-orbit couplings for both systems. We determine the BCP-induced TOH conductivity ($\chi_{\perp}^{\text{I}}$) along with the TOH conductivity associated with the band velocity ($\chi_{\perp}^{\text{II}}$). We find that the presence of an infinitesimal amount of Dresselhaus coupling in addition to the dominant Rashba coupling results in finite TOH responses. These conductivities vanish when the field is aligned with and/or orthogonal to the symmetry lines $k_x\pm k_y=0$ in both systems. For typical system parameters in a 2DEG with $k$-linear RD interactions, the magnitude of $\chi_{\perp}^{\text{I}}$ is smaller than that of $\chi_{\perp}^{\text{II}}$. On the other hand, when both the SO couplings are comparable, $\chi_{\perp}^{\text{I}}$ shows a notable increase in magnitude, owing to the distinctive characteristics of BCP. The TOH conductivity of 2DEG remains unchanged when Rashba and Dresselhaus spin-orbit couplings are exchanged. For 2DHG with $k$-cubic RD interactions, $\chi_{\perp}^{\text{I},h}$ exhibits a larger magnitude compared to $\chi_{\perp}^{\text{II},h}$. Unlike the electron case, the BCP induced $\chi_{\perp}^{\text{I},h}$ alters under the exchange of spin-orbit coupling parameters, whereas $\chi_{\perp}^{\text{II},h}$ remains the same.