Field-Tunable Quantum Metric in Few-Layer Phosphorene

TL;DR

This work addresses the challenge of directly detecting and controlling the quantum metric in quantum materials by showing that few-layer phosphorene provides a tunable platform where an out-of-plane electric field modulates the band structure and thereby the quantum metric tensor $g_{ij}^n(\mathbf{k})$ and quantum weight $K_{ij}$. Using tight-binding models derived from ab initio inputs and a gauge-invariant Kubo formalism, the authors demonstrate layer-dependent band-gap tuning and a field-induced giant enhancement of the QMT components, with critical fields around $E_c \sim 0.3$–$1.0$ V Å$^{-1}$ depending on the layer count. The QMT distributions develop ring-like features and concentrate near the Γ point as the gap closes, indicating strong coupling to electric-field-induced band inversion. The results propose experimental routes via circular-dichroism ARPES (CD-ARPES) and scattering-based probes to directly access the quantum metric, highlighting few-layer phosphorene as a practical platform for quantum geometry in real materials.

Abstract

The quantum metric -- which quantifies the distance between quantum states -- is a fundamental component of the quantum geometric tensor, playing a crucial role in a wide range of physical phenomena. Its direct detection and control remains a challenge, requiring suitable material candidates. In this work, we present the emergence of a tunable quantum metric in a versatile two-dimensional material platform, namely, few-layer phosphorene. Using ab-initio-derived models, we show how electric fields can be used to substantially enhance the quantum metric as well as the associated quantum weight. Furthermore, we present a layer-dependent evolution of the quantum metric and its interplay with the electric field in this material. Our results establish few-layer phosphorene as a promising platform for exploring control over the quantum metric and the resulting metric responses in real materials.

Figures (5)

• Figure 1: Crystal structure and electronic band structure of few-layer phosphorene. (a) Side view of the few-layer phosphorene showing four layers. The electric field, $E$, shown by the black arrow, is applied perpendicular to the layers along the stacking direction. (b) The Brillouin zone with the high symmetry points shown. Band structure of four-layer phosphorene (c) without and (d) with an electric field $E = 0.30~\mathrm{V\,\AA^{-1}}$. We note that with increasing electric field, the band gap decreases and closes at a critical value.
• Figure 2: Tunable quantum metric in quadruple-layer phosphorene with varying electric field. (a) Trace of the quantum metric for quadruple-layer phosphorene with increasing electric field, $E$. (b) Variation of the quantum weight components $K_{xx}$ and $K_{yy}$ with the electric field. Note the pronounced increase near the critical electric field, $E\approx 0.3~\mathrm{V\,\AA^{-1}}$. The quantum weight and the quantum metric components can be directly tuned by an applied electric field.
• Figure 3: Distribution of the quantum metric components of quadruple-layer phosphorene under varying electric field. The quantum metric components $g_{xx}$, $g_{xy}$, and $g_{yy}$ are plotted column-wise, while the electric field increases vertically downward. We note that with increasing electric field, each component attains large values with order of magnitude enhancements seen near $E = 0.30~\mathrm{V\,\AA^{-1}}$.
• Figure 4: Quantum metric for bi- and tri-layer phosphorene under varying electric field. (a) Variation of the trace of the quantum metric with the electric field, $E$, for bi- and tri-layer phosphorene. (b) The quantum weight for bi- and tri-layer phosphorene with the electric field. We note that the magnitude of the trace of the QMT and the quantum weight increases with $E$ and reaches a maximum at the critical fields $E = 0.99~\mathrm{V\,\AA^{-1}}$ and $E = 0.47~\mathrm{V\,\AA^{-1}}$, where the conduction and valence bands touch for the bi-layer and tri-layer cases, respectively. The corresponding critical band structures are shown in panels (c) and (d) for bi- and tri-layer phosphorene, respectively. Thus, the quantum weight and the quantum metric components in can be tuned directly by the applied electric field.
• Figure Appendix 1: Hopping model and electrostatic potential distribution under a transverse electric field. Schematic illustration of bilayer phosphorene showing atomic sites connected by various intralayer and interlayer hopping parameters given in Table-I. The electrostatic potential $U$ varies between atoms located at different heights along the direction of the applied electric field $\mathbf{E}$, which is perpendicular to the phosphorene layers. The intralayer hopping amplitudes in phosphorene are $t_{10}^{\parallel}, t_{20}^{\parallel}, t_{30}^{\parallel}, t_{40}^{\parallel},$ and $t_{50}^{\parallel}$, corresponding to the site pairs $A_1-A_2$, $A_1-A_3$, $A_1-A_4$, $A_1-A_5$, and $A_2-A_5$, respectively. For multilayer phosphorene, the interlayer hopping parameters are also present and denoted by $t_{10}^{\perp}, t_{20}^{\perp}, t_{30}^{\perp},$ and $t_{40}^{\perp}$, associated with $A_4-B_1$, $A_6-B_2$, $A_6-B_4$, and $A_4-B_4$, respectively.