Flexocurrent-induced magnetization: Strain gradient-induced magnetization in time-reversal symmetric systems

Technical Summary: Flexocurrent-Induced Magnetization in Time-Reversal Symmetric Systems

Problem Statement
In condensed matter physics, cross-correlation responses—where an observable's symmetry differs from that of the applied external field—are typically restricted to systems with broken time-reversal symmetry (TRS). Conventional magnetization responses to strain fields, such as the piezomagnetic and flexomagnetic effects, are equilibrium phenomena that strictly require broken TRS because strain (and its gradient) is even under time reversal, while magnetization is odd. While current-induced magnetization (CIM), or the Edelstein effect, demonstrates that nonequilibrium magnetization can exist in TRS-preserving systems via spin-momentum locking, a similar mechanism driven by mechanical strain gradients has not been formally established for nonmagnetic materials. The central problem addressed is whether a nonuniform strain field can induce a finite magnetization in nonmagnetic metals and semiconductors that preserve TRS, and if so, what the theoretical framework and symmetry constraints for such a response are.

Methodology
The authors develop a general theoretical framework for "flexocurrent-induced magnetization" (FCIM), defined as the nonequilibrium magnetization response to a strain gradient.

1. Hamiltonian Formulation: Starting with a free-fermion Hamiltonian, the authors introduce an external strain field $\epsilon(r)$ that couples to local electric quadrupole operators $Q_\lambda(r)$. This coupling is treated as a perturbation $V = \sum g_\lambda \int dr \epsilon_\lambda(r) Q_\lambda(r)$.
2. Semiclassical Analysis: Using Luttinger's method and semiclassical wave-packet theory, the authors demonstrate that a spatially varying strain field (strain gradient $\nabla \epsilon_\lambda$) acts as a driving force on electrons. This force accelerates the wavepackets, creating an asymmetric distribution in momentum space. In the presence of dissipation (e.g., impurity scattering), this leads to a nonequilibrium steady state.
3. Linear Response Theory: The authors derive a Kubo formula for the magnetization response tensor $f^\lambda_{\alpha\beta}$, relating macroscopic magnetization $\bar{M}_\alpha$ to the strain gradient $\nabla_\beta \epsilon_\lambda$. They explicitly separate the response into dissipative and nondissipative components, focusing on the dissipative part which corresponds to the nonequilibrium FCIM.
4. Symmetry Analysis: A group-theoretical analysis is performed to determine the conditions under which FCIM is allowed. The authors analyze the transformation properties of the strain gradient, electric quadrupole current, torque, and magnetization under spatial inversion ($P$) and time-reversal ($T$) operations.
5. Numerical Application: The formalism is applied to three representative nonmagnetic, non-centrosymmetric systems using tight-binding models:
   • A decorated square lattice (buckled Lieb lattice) with $C_{4v}$ symmetry.
   • Monolayer $MoS_2$ with $D_{3h}$ symmetry.
   • Monolayer Janus $MoSSe$ with $C_{3v}$ symmetry.
     For each system, the authors calculate the chemical potential dependence of the response coefficients for both orbital and spin magnetization components.

Key Contributions and Results

• Theoretical Mechanism: The paper establishes that a strain gradient acts as a driving force analogous to an electric field in CIM. It generates an electric-quadrupole current ("flexocurrent") which, in systems with spin-momentum locking, induces a finite magnetization. This mechanism is distinct from the equilibrium flexomagnetic effect and is inherently a nonequilibrium, dissipative response.
• Symmetry Constraints: The authors prove that FCIM is symmetry-allowed in systems that preserve TRS, provided they lack spatial inversion symmetry.
  • In systems with spatial inversion symmetry, the response vanishes because the integrand in the Kubo formula is odd under inversion.
  • In systems with broken inversion symmetry but preserved TRS, the response is finite. The analysis shows that FCIM is allowed in all 21 non-centrosymmetric crystallographic point groups.
• Numerical Findings:
  • Decorated Square Lattice ($C_{4v}$): Finite magnetization responses ($\bar{L}_x, \bar{L}_y, \bar{S}_x, \bar{S}_y$) are induced by specific strain gradients ($\nabla_x \epsilon_{xy}, \nabla_x \epsilon_{x^2-y^2}$, etc.), consistent with symmetry arguments. The response arises from Rashba-type couplings enabled by the lattice structure.
  • Monolayer $MoS_2$ ($D_{3h}$): In-plane magnetization components ($\bar{L}_x, \bar{L}_y, \bar{S}_x, \bar{S}_y$) vanish due to symmetry constraints, while out-of-plane components ($\bar{L}_z, \bar{S}_z$) are finite and induced by $\nabla_x \epsilon_{x^2-y^2}$. The response is significantly enhanced near the valence band edge.
  • Monolayer Janus $MoSSe$($C_{3v}$): The reduction of symmetry from $D_{3h}$ to $C_{3v}$ allows for additional non-zero components. Both in-plane and out-of-plane magnetization responses are finite, with specific dependencies on various strain gradient components, fully consistent with the derived symmetry rules.

Significance
The paper claims that FCIM offers a novel pathway for controlling magnetization in nonmagnetic materials using strain fields, without requiring the breaking of time-reversal symmetry. This distinguishes it from conventional piezomagnetic and flexomagnetic effects. The authors suggest that this effect could be experimentally detectable in nonmagnetic metals and semiconductors (such as $MoS_2$ and $MoSSe$) via techniques like nuclear magnetic resonance or magneto-optical Kerr effect, as it represents the only magnetization response allowed in these systems under slowly varying strain fields. The work provides a unified theoretical description linking mechanical strain gradients to magnetic responses in TRS-preserving systems, expanding the scope of multiferroic and spintronic phenomena to nonmagnetic materials.