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

Symmetry constraints determine which physical responses are allowed in a given system. Magnetization induced by strain fields, such as in piezomagnetic and flexomagnetic effects, has typically been considered in materials that break time-reversal symmetry. Here, we propose that nonuniform strain can induce magnetization even in nonmagnetic metals and semiconductors that preserve time-reversal symmetry. This mechanism differs from the conventional flexomagnetic effect: the strain gradient acts as a driving force on the electrons, generating magnetization in a manner closely analogous to current-induced magnetization. Treating the strain field as an external field, we derive a general expression for the magnetization induced by a strain gradient and demonstrate that this response is symmetry-allowed even in time-reversal symmetric systems. We apply our formulation to nonmagnetic systems that lack spatial inversion symmetry while preserving time-reversal symmetry, using a decorated square lattice, monolayer MoS$_2$, and monolayer Janus MoSSe as representative examples. We find a finite magnetization response to strain gradients, which is consistent with symmetry arguments, supporting the validity of our theoretical framework. These results offer a pathway for controlling magnetization in nonmagnetic materials using strain fields.

💡 Research Summary

The paper introduces a novel mechanism by which non‑uniform strain (a strain gradient) can generate a finite magnetization in non‑magnetic metals and semiconductors that preserve time‑reversal symmetry (TRS). Traditionally, strain‑induced magnetization effects such as piezomagnetism and flexomagnetism have been considered only in systems that already break TRS, because both strain and its spatial gradient are even under time reversal while magnetization is odd. The authors overturn this conventional view by focusing on nonequilibrium responses: a strain gradient acts as an effective driving force on the electronic degrees of freedom, analogous to an electric field in the Edelstein (current‑induced magnetization, CIM) effect.

Starting from a generic free‑fermion Hamiltonian, they couple the strain tensor ϵ(r) to electric‑quadrupole operators Qλ(r) via a local interaction V = ∑λ gλ∫d r ϵλ(r) Qλ(r). By expanding the strain field around the centre of a wave packet, they show that the gradient ∇ϵλ enters the semiclassical equation of motion for the crystal momentum as ℏ ·k̇ = −∑λ gλ (Qλ,k) ∇ϵλ. Thus, a spatially varying strain accelerates electrons, creating an asymmetric distribution in momentum space—a “flexocurrent”. In systems with spin‑momentum locking (e.g., Rashba‑type bands), this asymmetric distribution produces a net spin polarization, which translates into a macroscopic magnetization.

To quantify this effect, the authors develop a linear‑response formalism based on Luttinger’s method for thermal‑gradient responses. They define an adiabatic magnetization‑gradient tensor fλ αβ through M̄α = fλ αβ ∇β ϵλ and derive a Kubo formula for fλ αβ in the transport limit (q → 0 followed by ω → 0). This limit isolates the dissipative, nonequilibrium component of the response, distinguishing it from the equilibrium flexomagnetic effect obtained in the opposite limit (ω → 0 then q → 0). The resulting tensor is allowed by symmetry even when TRS is intact, because the nonequilibrium steady state itself breaks TRS.

A systematic symmetry analysis shows that the strain gradient (a rank‑3 tensor) can couple to magnetization in any crystal class lacking inversion symmetry, provided spin‑orbit coupling yields spin‑momentum locking. The authors illustrate the theory with three concrete two‑dimensional models:

1. A decorated square lattice with C₄ᵥ symmetry, where orbital degrees of freedom generate electric quadrupoles that couple to strain. Numerical evaluation of fλ αβ reveals a finite magnetization response to ∇x ϵxx.

2. Monolayer MoS₂ (D₃h symmetry), a well‑studied transition‑metal dichalcogenide with intrinsic Rashba‑type spin splitting. Strain gradients such as ∇x ϵxy produce a measurable flexocurrent‑induced magnetization (FCIM), with both spin and orbital contributions quantified.

3. Janus monolayer MoSSe (C₃ᵥ symmetry), which lacks vertical mirror symmetry and thus exhibits enhanced electric‑quadrupole activity. The calculations show larger FCIM amplitudes than in MoS₂, confirming the role of structural asymmetry.

In each case, the authors decompose the magnetization into spin and orbital parts, analyze the band‑structure origins, and confirm that the response vanishes when either spin‑momentum locking or the strain‑quadrupole coupling is switched off, thereby validating the physical picture.

The paper also discusses experimental prospects. Strain gradients of the required magnitude can be generated in nanoscale films, nanowires, or via piezoelectric actuators, reaching values that produce observable magnetizations. Detection schemes include magneto‑optical Kerr effect, spin‑polarized scanning tunneling microscopy, or transport measurements of anomalous Hall signals induced by the flexocurrent. Because FCIM is driven mechanically rather than electrically, it opens a new route for spin‑tronic control that could be combined with conventional electric‑field techniques for multifunctional device architectures.

In summary, the authors provide a comprehensive theoretical framework—semiclassical dynamics, Kubo linear response, and symmetry analysis—for a strain‑gradient‑induced, nonequilibrium magnetization that is allowed in time‑reversal‑symmetric, non‑magnetic materials. Their numerical examples demonstrate that the effect is robust and potentially sizable, suggesting that flexocurrent‑induced magnetization could become a valuable tool for manipulating spin and orbital degrees of freedom in future spintronic and multiferroic technologies.