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We study Berry connection polarizability (BCP) induced electric polarizationand third-order Hall (TOH) effect in a two-dimensional electron/hole gas(2DEG/2DHG) with Rashba-Dresselhaus (RD) spin-orbit couplings in III-Vsemiconductor heterostructures. The electric polarization decreases with theincrease of the Fermi energy and is responsive to the electric fieldorientation in the presence of RD spin-orbit couplings for both systems. Wedetermine the BCP-induced TOH conductivity ($\chi_{\perp}^{\text{I}}$) alongwith the TOH conductivity associated with the band velocity($\chi_{\perp}^{\text{II}}$). We find that the presence of an infinitesimalamount of Dresselhaus coupling in addition to the dominant Rashba couplingresults in finite TOH responses. These conductivities vanish when the field isaligned with and/or orthogonal to the symmetry lines $k_x\pm k_y=0$ in bothsystems. For typical system parameters in a 2DEG with $k$-linear RDinteractions, the magnitude of $\chi_{\perp}^{\text{I}}$ is smaller than thatof $\chi_{\perp}^{\text{II}}$. On the other hand, when both the SO couplingsare comparable, $\chi_{\perp}^{\text{I}}$ shows a notable increase inmagnitude, owing to the distinctive characteristics of BCP. The TOHconductivity of 2DEG remains unchanged when Rashba and Dresselhaus spin-orbitcouplings 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 couplingparameters, whereas $\chi_{\perp}^{\text{II},h}$ remains the same.

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