Abstract

We investigate the planar Hall effect (PHE) in topological insulators (TIs) in proximity to an altermagnet (AM). We find that, due to the interplay between altermagnetism and spin-momentum locking, in-plane magnetic fields can induce PHE in the TI surface states by modulating the anisotropy of the Fermi velocity, the characteristic vectors associated with spin-momentum locking, and the tilt of the Dirac cone. Both linear and nonlinear PHE can be significantly enhanced by the adjacent AM, particularly when the Néel vector lies within the TI surface. Notably, the planar Hall conductivities are tunable by the spin-momentum locking properties and change sign as the Néel vector rotates by 180° in the TI surface. Therefore, the PHE can serve as a powerful tool for detecting both the Néel vector and the spin-momentum locking characteristics.

Variables

FIG1B

The magnetic-field renormalized band structure. This dataset contains 8 variables, kx and ky represent kₓ and kᵧ, Ek1_B0 and Ek2_B0 represent the dispersion relations under B = 0, while Ek1_B1 and Ek2_B1 correspond to the dispersion relations for B₁ ≠ 0. Similarly, Ek1_delta0 and Ek2_delta0 denote the dispersion relations when δ₀ ≠ 0.

FIG1C

Spin textures for Rashba-type spin-momentum interaction. This dataset contains 5 variables, kx and ky represent kₓ and kᵧ, c_Sx and c_Sy correspond to the spin polarization components sₓ and sᵧ, c_Ek denotes the energy dispersion εₖ.

FIG1D

Spin textures for Dresselhaus-type spin-momentum interaction. This dataset contains 5 variables, kx and ky represent kₓ and kᵧ, d_Sx and d_Sy correspond to the spin polarization components sₓ and sᵧ, d_Ek denotes the energy dispersion εₖ.

FIG2A

Spin textures on a constant energy surface without the AM for right-handed spin-momentum locking with ϕₛ₋ₚ⁺ = π/4. This dataset contains 5 variables, kx and ky represent kₓ and kᵧ, a_Sx and a_Sy correspond to the spin polarization components sₓ and sᵧ, a_Ek denotes the energy dispersion εₖ.

FIG2B

Spin textures on a constant energy surface without the AM for left-handed spin-momentum locking with ϕₛ₋ₚ⁻ = π/4. This dataset contains 5 variables, kx and ky represent kₓ and kᵧ, b_Sx and b_Sy correspond to the spin polarization components sₓ and sᵧ, b_Ek denotes the energy dispersion εₖ.

FIG2C

Spin textures on a constant energy surface without the AM for nonorthogonal e₁ and e₂ with ϕₛ₋ₚ⁻ = π/4. This dataset contains 5 variables, kx and ky represent kₓ and kᵧ, c_Sx and c_Sy correspond to the spin polarization components sₓ and sᵧ, c_Ek denotes the energy dispersion εₖ

FIG2D

Spin texture with AM of out-of-plane Néel vector. This dataset contains 6 variables, kx and ky represent kₓ and kᵧ, d_Sx, d_Sy and d_Sz correspond to the spin polarization components sₓ, sᵧ and s_z, d_Ek denotes the energy dispersion εₖ.

FIG3A

The data exhibits the evolution of σ_H^{intra with different right-handed spin-momentum locking. Comprising 4 variables, phiB denotes φ_B, with the first/second/third rows of phiB_sigmaH_RH indicating σ_H}intra at φ_{s-p}^+ = 0 / 0.25π / 0.5π respectively. The associated parameters are λ = 0, α = 0.2, φ_E = 0.

FIG3B

The data exhibits the evolution of σ_H^{intra with different left-handed spin-momentum locking. Comprising 4 variables, phiB denotes φ_B, with the first/second/third rows of phiB_sigmaH_LH indicating σ_H}intra at φ_{s-p}^- = 0 / 0.25π / 0.5π respectively. The associated parameters are λ = 0, α = 0.2, φ_E = 0.

FIG3C

The data exhibits the evolution of σ_H^{intra with different φ_E. Comprising 4 variables, phiB denotes φ_B, with the first/second/third rows of phiB_sigmaH_phiE_C indicating σ_H}intra at φ_E = 0 / 0.25π / 0.5π respectively. The associated parameters are λ = 0, α = 0.2, φ_{s-p}^+ = 0.5π.

FIG3D

The data exhibits the evolution of σ_H^{intra with different φ_E. Comprising 4 variables, phiB denotes φ_B, with the first/second/third rows of phiB_sigmaH_phiE_D indicating σ_H}intra at φ_E = 0 / 0.25π / 0.5π respectively. The associated parameters are λ = 0.2, α = 0, φ_{s-p}^+ = 0.5π, ϑ_N = 0.

FIG4A

AM-modulated linear intraband planar Hall conductivity σ_H^{intra. This dataset contains 5 variables, phiB denotes φ_B, with the first/second/third/fourth rows of phiB_sigmaH_thetaN indicating σ_H}intra at ϑ_N = 0 / 0.1π / 0.2π / 0.5π. The associated parameters are α = 0, λ = 0.2, φ_N = 0.

FIG4B

AM-modulated linear intraband planar Hall conductivity σ_H^{intra. This dataset contains 4 variables, phiB denotes φ_B, with the first/second/third rows of phiB_sigmaH_phiN indicating σ_H}intra at φ_N = 0 / 0.5π / π. The associated parameters are α = 0, λ = 0.2, ϑ_N = π / 2.

FIG4C

AM-modulated linear intraband planar Hall conductivity. The amplitude Δσ_H^{intra of the planar Hall conductivity as function of B. This dataset contains 4 variables, B denotes B, with the first/second/third rows of B_sigmaH_thetaN indicating σ_H}intra at ϑ_N = 0 / 0.1π / 0.5π. The associated parameters are α = 0, λ = 0.2, φ_N = 0.

FIG5A

The linear interband planar Hall conductivity as functions of φ_B. This dataset contains 5 variables, phiB denotes φ_B, with the first/second/third/fourth rows of phiB_sigmaxyBC_thetaN indicating σ_xy^{BC at ϑ_N = 0 / 0.3π / 0.4π / 0.5π. The associated parameters are E_F = 0, φ_s-p}+ = π/2, φ_N = 0.

FIG5B

The linear interband planar Hall conductivity as functions of φ_B. This dataset contains 4 variables, phiB denotes φ_B, with the first/second/third rows of phiB_sigmaxyBC_RH indicating σ_xy^{BC at φ_s-p}+ = 0 / 0.25π / 0.5π. The associated parameters are E_F = 0, ϑ_N = 0, φ_N = 0.

FIG5C

The linear interband planar Hall conductivity as functions of φ_B. This dataset contains 5 variables, phiB denotes φ_B, with the first/second/third/fourth rows of phiB_sigmaxyQM_thetaN indicating σ_xy^{QM at ϑ_N = 0 / 0.1π / 0.3π / 0.5π. The associated parameters are E_F = 0.21, φ_s-p}+ = π/2, φ_N = 0.

FIG5D

The linear interband planar Hall conductivity as functions of φ_B. This dataset contains 5 variables, phiB denotes φ_B, with the first/second/third rows of phiB_sigmaxyQM_phiN indicating σ_xy^{QM at φ_N = 0 / 0.5π / π. The associated parameters are E_F = 0.21, φ_s-p}+ = π/2, ϑ_N = π/2.

FIG6A

Distribution of the integral kernel for the nonlinear conductivity σ_xyy^intra. This dataset contains 4 variables, kx and ky represent k_x and k_y. Ek1 denotes ε_k1. ky_kx_Exyy corresponds to the distribution associated with the second partial derivative of ε_kη with respect to k_x and k_y multiplied by the partial derivative of ε_kη with respect to k_y. The constant E_F is the Fermi energy. The associated parameters are η = +1, φ_B = 0, φ_N = 0, ϑ_N = π/2.

FIG6B

Distribution of the integral kernel for the nonlinear conductivity σ_yxx^intra. This dataset contains 4 variables, kx and ky represent k_x and k_y. Ek1 denotes ε_k1. ky_kx_Eyxx corresponds to the distribution associated with the second partial derivative of ε_kn with respect to k_x and k_y multiplied by the partial derivative of ε_kn with respect to k_x. The constant E_F is the Fermi energy. The associated parameters are η = +1, φ_B = 0, φ_N = 0, ϑ_N = π/2.

FIG6C

Distribution of the integral kernel for the nonlinear conductivity σ_xyy^{BC. This dataset contains 4 variables, kx and ky represent k_x and k_y. Ek1 denotes ε_k1. ky_kx_Dxyy corresponds to the distribution associated with 𝒟_k,xyy}η. The constant E_F is the Fermi energy. The associated parameters are η = +1, φ_B = 0, φ_N = 0, ϑ_N = 0.

FIG6D

Distribution of the integral kernel for the nonlinear conductivity σ_yxx^{BC. This dataset contains 4 variables, kx and ky represent k_x and k_y. Ek1 denotes ε_k1. ky_kx_Dyxx corresponds to the distribution associated with 𝒟_k,yxx}η. The constant E_F is the Fermi energy E_F. The associated parameters are η = +1, φ_B = 0, φ_N = 0, ϑ_N = 0.

FIG6E

Distribution of the integral kernel for the nonlinear conductivity σ_xyy^{QM. This dataset contains 4 variables, kx and ky represent k_x and k_y. Ek1 denotes ε_k1. ky_kx_2Myyx_Mxyy corresponds to the distribution associated with 2𝓜_k,yyx}η − 𝓜_k,xyy^η. The constant E_F is the Fermi energy E_F. The associated parameters are η = +1, φ_B = 0, φ_N = 0, ϑ_N = π/2.

FIG6F

Distribution of the integral kernel for the nonlinear conductivity σ_yxx^{QM. This dataset contains 4 variables, kx and ky represent k_x and k_y. Ek1 denotes ε_k1. ky_kx_2Mxxy_Myxx corresponds to the distribution associated with 2𝓜_k,xxy}η − 𝓜_k,yxx^η. The constant E_F is the Fermi energy E_F. The associated parameters are η = +1, φ_B = 0, φ_N = 0, ϑ_N = π/2.

FIG7A

Distribution of σ_xyy^{intra for φ_N = 0 in the ϑ_N–φ_B parameter space. This dataset contains 3 variables, thetaN denotes ϑ_N and phiB denotes φ_B, thetaN_phiB_sigmaxyy represents σ_xyy}intra values under different combinations of ϑ_N and φ_B conditions.

FIG7B

σ_xyy^{intra as functions of φ_B for ϑ_N = π/2 and different φ_N. This dataset contains 4 variables, phiB denotes φ_B, with the first/second/third rows of phiB_sigmaxyy_phiN indicating σ_xyy}intra at φ_N = 0 / 0.5π / π.

FIG7C

Evolution of σ_xyy^{intra with respect to ϑ_N for φ_B = 0. This dataset contains 4 variables, phiN denotes φ_N, with the first/second/third rows of phiN_sigmaxyy_thetaN indicating σ_xyy}intra at ϑ_N = 0.1π / 0.3π / 0.5π.

FIG7D

Evolution of σ_yxx^{intra with respect to ϑ_N for φ_B = 0. This dataset contains 4 variables, phiN denotes φ_N, with the first/second/third rows of phiN_sigmayxx_thetaN indicating σ_yxx}intra at ϑ_N = 0.1π / 0.3π / 0.5π.

FIG8A

Distribution of σ_xyy^{BC for φ_N = 0 in the ϑ_N - φ_B parameter space. This dataset contains 3 variables, thetaN denotes ϑ_N and phiB denotes φ_B, thetaN_phiB_sigmaxyyBC represents σ_xyy}BC values under different combinations of ϑ_N and φ_B conditions.

FIG8B

Distribution of σxyy^{QM for φN = 0 in the θN-φB parameter space. This dataset contains 3 variables, thetaN denotes θN and phiB denotes φB, thetaN_phiB_sigmaxyyQM represents σxyy}QM values under different combinations of θN and φB conditions.

FIG8C

σxyy^{QM as functions of φN for φB = 0 and different φs-p}+. This dataset contains 4 variables, phiN denotes φN, with the first/second/third rows of phiN_sigmaxyy_thetaN indicating σxyy^{QM at φs-p}+ = 0 / 0.25π / 0.5π.

FIG8D

σyxx^{QM as functions of φN for φB = 0 and different φs-p}+. This dataset contains 4 variables, phiN denotes φN, with the first/second/third rows of phiN_sigmayxx_thetaN indicating σyxx^{QM at φs-p}+ = 0 / 0.25π / 0.5π.

Software

The dataset plotted in FIG1-FIG8 were obtained by running MATLAB software.