[Paper Review] Symmetry, spin-texture, and tunable quantum geometry in a WTe$_2$ monolayer

Li-kun Shi, Justin C. W. Song|arXiv (Cornell University)|May 2, 2018

2D Materials and Applications4 citations

TL;DR

This paper proposes that monolayer 1T’-WTe₂ exhibits electrically tunable quantum geometry due to its low-symmetry crystal structure, where an out-of-plane electric field induces a canted spin texture, lifting spin degeneracy and generating tunable Berry curvature and magnetic moment. The key result is the realization of a current-induced out-of-plane magnetization and a quantum nonlinear anomalous Hall effect, enabled by the material’s dipolar Berry curvature and magnetic moment distribution in momentum space.

ABSTRACT

The spin orientation of electronic wavefunctions in crystals is an internal degree of freedom, typically insensitive to electrical knobs. We argue from a general symmetry analysis and a $\vec k \cdot \vec p$ perspective, that monolayer 1T'-WTe$_2$ possesses a gate-activated canted spin texture that produces an electrically tunable bulk band quantum geometry. In particular, we find that due to its out-of-plane asymmetry, an applied out-of-plane electric field breaks inversion symmetry to induce both in-plane and out-of-plane electric dipoles. These in-turn generate spin-orbit coupling to lift the spin degeneracy and enable a bulk band Berry curvature and magnetic moment distribution to develop. Further, due to its low symmetry, Berry curvature and magnetic moment in 1T'-WTe$_2$ possess a dipolar distribution in momentum space, and can lead to unconventional effects such as a current induced magnetization and quantum non-linear anomalous Hall effect. These render 1T'-WTe$_2$ a rich two-dimensional platform for all-electrical control over quantum geometric effects.

Motivation & Objective

• To understand how the low-symmetry crystal structure of monolayer 1T’-WTe₂ enables electrically tunable quantum geometric effects.
• To investigate the role of out-of-plane electric fields in breaking inversion symmetry and inducing spin-orbit coupling.
• To demonstrate the emergence of a dipolar distribution of Berry curvature and magnetic moment in momentum space.
• To explore the emergence of unconventional effects such as current-induced magnetization and nonlinear anomalous Hall effect.
• To establish 1T’-WTe₂ as a platform for all-electrical control of spin and magnetic degrees of freedom in 2D materials.

Proposed method

• Developed a six-band k·p model based on symmetry analysis, incorporating time-reversal symmetry, mirror symmetry (xz plane), and broken inversion symmetry.
• Used symmetry-allowed terms in the k·p Hamiltonian to describe spin-splitting and spin-texture evolution under an out-of-plane electric field.
• Calculated the intrinsic magnetic moment as a sum of orbital and spin contributions: $ m_{n}^{ m tot}({f k}) = m_{n}^{ m int}({f k}) + (ge/2m_0)raket{u_{f k}|s_z|u_{f k}} $, with $ g \sim 5 $.
• Derived the kinetic magneto-electric effect via $ M_z = \sum_i \tilde{\alpha}_{zi} j_i $, where $ \tilde{\alpha}_{zi} = \left[ \frac{e}{\hbar} \sum_{n,{\bf k}} f_{n{\bf k}}^{(0)} \frac{\partial m_{n}^{{\rm tot}}({\bf k})}{\partial k_i} \right] (D_{ii})^{-1} $, linking current flow to out-of-plane magnetization.
• Computed the dipolar distribution of Berry curvature and magnetic moment in momentum space using the full six-band model.
• Simulated the emergence of net $ M_z $ under in-plane current flow, showing non-zero $ \tilde{\alpha}_{zy} $ when current is driven along the y-direction.

Experimental results

Research questions

• RQ1 How does the low-symmetry crystal structure of 1T’-WTe₂ enable electrically tunable quantum geometry?
• RQ2 What is the origin of out-of-plane spin polarization in 1T’-WTe₂ under an out-of-plane electric field?
• RQ3 How does the dipolar distribution of Berry curvature and magnetic moment in momentum space lead to novel transport effects?
• RQ4 Can a current-induced out-of-plane magnetization be generated in 1T’-WTe₂, and what is its symmetry-protected nature?
• RQ5 What is the role of the Drude weight and distribution function shift in enabling the nonlinear anomalous Hall effect?

Key findings

• An out-of-plane electric field induces both in-plane and out-of-plane spin components due to broken inversion symmetry and non-aligned Te atoms, lifting spin degeneracy.
• The system exhibits a dipolar distribution of Berry curvature and magnetic moment in momentum space, a direct consequence of low crystal symmetry.
• A current-induced out-of-plane magnetization $ M_z $ emerges when an in-plane current is driven along the y-direction, with $ \tilde{\alpha}_{zy} \neq 0 $, while $ \tilde{\alpha}_{zx} = 0 $, consistent with symmetry.
• The kinetic magneto-electric effect $ M_z = \sum_i \tilde{\alpha}_{zi} j_i $ is finite only when the chemical potential is in the conduction band, indicating a dissipative, non-equilibrium origin.
• The nonlinear anomalous Hall effect arises from the asymmetric, dipolar distribution of Berry curvature, leading to a quantum non-linear response without external magnetic fields.
• The estimated $ M_z $ is sizeable and detectable via Kerr effect microscopy, making 1T’-WTe₂ a viable platform for all-electrical control of magnetic degrees of freedom.