Kinetic magnetoelectric effect in topological insulators

The study investigates the kinetic magnetoelectric effect (KME), also termed the orbital Edelstein effect, which converts an electric current into orbital magnetization in noncentrosymmetric crystals. KME is analogous to the spin Edelstein effect but does not require spin–orbit coupling and can appear in systems with broken inversion symmetry, including chiral and polar crystals. It differs from the conventional magnetoelectric (ME) effect, which arises from equilibrium band-structure changes and typically requires both broken inversion and time-reversal symmetry. Because orbital magnetoelectric responses can depend on boundaries, the work focuses on topological insulators (TIs) where current flows via surface states. The purpose is to formulate and compute KME in 3D Chern insulators and Z₂ TIs, clarify its surface dependence, derive a surface-theory description, and identify material platforms exhibiting a giant KME compared with metals.

Recent proposals have highlighted current–magnetization interconversion mechanisms in low-symmetry crystals, including KME/orbital Edelstein and gyrotropic magnetic effects. Prior work established KME in chiral and polar systems, including observations in elemental tellurium and theoretical treatments without spin–orbit coupling. The Edelstein effect requires broken inversion but not necessarily broken time-reversal symmetry, while the conventional ME effect requires both. Although equilibrium orbital magnetization is a bulk quantity independent of boundaries, general orbital magnetoelectric responses can be boundary-sensitive. These insights motivate examining KME in systems with conducting surfaces, such as topological insulators, where boundary conditions and surface terminations are crucial.

The authors compute current-induced orbital magnetization in a long cylindrical geometry (axis along z) using the orbital magnetization operator M ∝ sum over occupied states of (r × v)z/S, with v = [r, H]. Within the Boltzmann relaxation-time approximation at zero temperature, an applied Ez modifies the distribution f(E) = f0(E) − (eτEz/ħ) ∂f0/∂E, yielding a linear-response expression for the KME magnetization MKME. They contrast Z₂ TIs (TRS preserved, zero equilibrium orbital magnetization; electric-field–induced state modification does not contribute at linear order) and Chern insulators (TRS broken; both nonequilibrium distribution and field-induced state modification may contribute). A 3D tight-binding model of a layered Chern insulator is constructed: 2D Wilson–Dirac layers (square lattice, lattice constants a in-plane, c between layers) are stacked with right-handed chiral interlayer hoppings forming solenoid-like structures. Numerical KME is computed in a one-dimensional quadrangular prism (periodic along z) with xz and yz surfaces, varying system sizes Lx and Ly to probe boundary effects. A surface-theory formulation is then derived. For a slab with surfaces at y = y± (periodic in x, z), the KME along x induced by Ez is expressed as an integral over the surface Brillouin zone of Fermi-surface velocities, MKME x,slab ∝ ∫∫ dkx dkz (∂E/∂kx) sgn(∂E/∂kz), explicitly depending on surface-state dispersion and termination. Extending to a cylinder/prism with four side surfaces (I–IV), the authors match surface eigenstates using energy equality, current conservation at edges, periodic boundary conditions, and normalization, obtaining a general expression for MKME in terms of surface velocities vI = (1/ħ)∂E/∂kx and vII = (1/ħ)∂E/∂ky. When vx, vy are weakly k-dependent, MKME in the prism is well approximated by the average of slab responses for the xz and yz faces: MKME ≈ 1/2 (MKME z,slab + MKME y,slab). A complementary circuit picture relates MKME to the circulating surface current density and anisotropic surface conductivity tensor σ, showing dependence on the aspect ratio Lx/Ly. Finite-size corrections from finite penetration depth are modeled via a fitting function including O(1/L) and O(1/L²) terms. For material assessment, first-principles DFT (VASP, PBE, Γ-centered 10×10×10 k-mesh, 460.8 eV cutoff) and Wannier90 tight-binding are used to compute bulk bands, surface states, Fermi surfaces, and KME coefficients. Z₂ topology in non-centrosymmetric materials is diagnosed using S4 symmetry-based indicators. Surface terminations on specific orientations are analyzed to extract anisotropic conductivities and KME coefficients.

• KME in TIs is a surface phenomenon: calculated MKME depends strongly on surface terminations and system size/aspect ratio, demonstrating it cannot be defined as a bulk quantity. Even for sizes much larger than the surface-state penetration depth, residual size dependence persists due to surface geometry and matching at edges.
• Surface-theory formula: For a slab, MKME can be written as a Fermi-surface integral over surface-state velocities, capturing termination-dependent KME. For a prism, the KME is well reproduced by averaging slab responses of the xz and yz faces when surface velocities are nearly k-independent.
• Circuit/transport picture: MKME relates to a circulating surface current jcirc and anisotropic surface conductivity, MKME = jcirc (σxx−1 + σyy−1)−1 + σxy−1 E (schematically), linking KME to measurable surface transport and aspect ratio Lx/Ly.
• Finite-size effects: Leading corrections scale as O(1/L) from finite penetration depth and O(1/L²) from corner effects; a fitting function accurately extrapolates to the large-size limit.
• Material candidates: Using symmetry indicators and database screening, Cu₂ZnSnSe₄ (space group 82, S4 symmetry) and CdGeAs₂ (space group 122) are identified as Z₂ TI candidates. Detailed calculations for Cu₂ZnSnSe₄ show insulating bulk bands and Dirac-like surface states with termination-dependent Fermi surfaces on [001] and [010] faces.
• Quantitative KME for Cu₂ZnSnSe₄: On [001], the magnetoelectric susceptibility is α ≈ −1.804 × 10⁻¹² Ω⁻¹·τ (Cu–Sn termination, surface A) and α ≈ −2.565 × 10⁻¹² Ω⁻¹·τ (Se termination, surface B). On [010] (surface C), α ≈ −2.324 × 10⁻¹¹ Ω⁻¹·τ. Larger Fermi surfaces (e.g., on surface B) yield larger α. Example anisotropic sheet conductances on [001]: σzz ≈ −3.6 × 10⁵ Ω⁻¹·τ, σxz ≈ 7.3 × 10⁸ Ω⁻¹·τ, σxx ≈ −5.1 × 10⁻¹² Ω⁻¹·τ, σyy ≈ 2.0 × 10¹⁰ Ω⁻¹·τ, giving tan θ ≈ −0.49 and −0.26 for different terminations.
• Scaling and magnitude: In bulk metals, both α and σ are intensive (∝ L⁰), so M/j is size independent and small. In TIs, α remains intensive while σ scales as L⁻¹ (current carried by surfaces), so M/j ∝ L, becoming macroscopic for millimeter-scale samples. For a prism, M/j = (L/4) tan θ. Compared with p-doped Te, where M/j ≈ 1.85 × 10⁻⁹ m at j = 1000 A/cm², Cu₂ZnSnSe₄ exhibits responses larger by roughly 5–8 orders of magnitude.
• Orbital vs spin response: In TIs, current-induced orbital magnetization (interatomic contribution) grows with system size (α/σ ∝ L), while spin magnetization scales inversely with size (∝ 1/L) and is much smaller (by 5–8 orders) for millimeter-sized samples. Table-of-scaling shows only the interatomic orbital component in TIs exhibits L¹ scaling for response to current.

The work addresses whether and how KME can be enhanced in systems with surface-dominated transport. By formulating KME in terms of surface Hamiltonians and validating against tight-binding numerics, the authors show that TIs provide a natural route to giant KME because surface currents produce macroscopic circulating loops, unlike the microscopic loops in bulk metals. The explicit boundary sensitivity explains why KME is not a bulk-defined quantity, aligning with known boundary dependence of orbital magnetoelectric responses. The scaling analysis demonstrates that, for a given current density, induced orbital magnetization increases with system size in TIs, offering a practical route to large magnetization control via current. The material case study of Cu₂ZnSnSe₄ confirms that realistic TI surfaces and terminations can yield sizable α and associated anisotropic conductivities, with termination engineering providing an additional tuning knob. The marked contrast between orbital and spin responses in TIs underscores distinct mechanisms and highlights the interatomic orbital contribution as the dominant channel for current-induced magnetization in topological materials.

The study proposes and formulates a giant kinetic magnetoelectric effect in topological insulators with chiral crystal structures, showing it is a surface-driven, boundary-sensitive response not definable as a bulk quantity. A surface-state Fermi-surface formula captures KME and agrees with direct tight-binding computations. In prism geometries, KME relates to circulating surface currents and anisotropic surface conductivities and scales linearly with system size when viewed as a response to current, enabling macroscopic orbital magnetization. First-principles calculations identify Cu₂ZnSnSe₄ (and CdGeAs₂) as promising Z₂ TIs, with termination-dependent KME magnitudes that surpass those in metals like Te by 5–8 orders of magnitude. The analysis distinguishes interatomic orbital magnetization as uniquely exhibiting size-enhanced response in TIs compared with spin and intraatomic components. Future work could extend to other TI classes (topological crystalline insulators) and topological semimetals where surface states coexist with bulk carriers, explore experimental verification via surface engineering, and quantify contributions beyond the relaxation-time approximation including field-induced band modifications in TRS-broken phases.

• Boundary and termination sensitivity: The KME depends on specific surface terminations and facet-dependent dispersions; results are not bulk universal and require careful surface control.
• Approximations: The Boltzmann relaxation-time approximation with constant τ is used; scattering anisotropy, energy dependence of τ, and vertex corrections are neglected.
• Finite-size treatment: Surface-state penetration depth is neglected in the core surface theory and included phenomenologically via O(1/L) fits; edge/corner effects are treated via simplified matching conditions.
• Symmetry assumptions: Some derivations assume C2 or related symmetries to relate surfaces and simplify formulas; deviations may modify quantitative results.
• Field-induced state modifications: In Chern insulators (TRS broken), additional linear contributions analogous to ME effects may exist; these are not fully quantified within the presented KME formalism.
• DFT specifics: First-principles results depend on exchange-correlation functional choice, slab terminations, and Wannierization; many-body effects and temperature/disorder are not explicitly included.