DOI: 10.5061/dryad.fbg79cp6q
title: Data from: Enhancement of the linear and nonlinear planar Hall effect by altermagnets on the surface of topological insulators

description: 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.
description: # Data from: Enhancement of the linear and nonlinear planar Hall effect by
 altermagnets on the surface of topological insulators **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.