Anomalous transition magnetic moments in two-dimensional Dirac materials

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Anomalous transition magnetic moments in two-dimensional Dirac materials

Sanghita Sengupta, Madalina I. Furis, Oleg P. Sushkov, and Valeri N. Kotov

Phys. Rev. B 102, 024432 – Published 20 July 2020

Abstract

We show that the magnetic response of atomically thin materials with a Dirac spectrum and spin-orbit interactions can exhibit strong dependence on electron-electron interactions. While graphene itself has a very small spin-orbit coupling, various two-dimensional (2D) compounds “beyond graphene” are good candidates to exhibit the strong interplay between spin-orbit and Coulomb interactions. Materials in this class include dichalcogenides (such as ${MoS}_{2}$ and ${WSe}_{2}$ ), silicene, germanene, and 2D topological insulators described by the Kane-Mele model. We present a unified theory for their in-plane magnetic field response leading to “anomalous,” i.e., electron interaction dependent, transition moments. Our predictions can be potentially used to construct unique magnetic probes with high sensitivity to electron correlations.

Received 2 January 2020
Accepted 2 July 2020

DOI:https://doi.org/10.1103/PhysRevB.102.024432

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Kane-Mele model Magnetic interactions Spin-orbit coupling Surface states Topological insulators Topological materials

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Sanghita Sengupta¹, Madalina I. Furis^2,3, Oleg P. Sushkov⁴, and Valeri N. Kotov^3,2

¹Institut Quantique and Département de Physique, Université de Sherbrooke, Sherbrooke, Québec, Canada J1K 2R1
²Materials Science Program, University of Vermont, Burlington, Vermont 05405, USA
³Department of Physics, University of Vermont, Burlington, Vermont 05405, USA
⁴School of Physics, University of New South Wales, Sydney 2052, New South Wales, Australia

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Issue

Vol. 102, Iss. 2 — 1 July 2020

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Images

Figure 1
Left: Feynman diagram for the bare transition moment with uniform (zero momentum, $q \to 0$ ) in-plane magnetic field $B_{x}$ , corresponding to field coupling of the form $B_{x} (q \to 0) S_{x}$ . We set the field coupling prefactor $g μ_{B} = 1$ in this field direction for simplicity (it is known that the $g$ factor can be strongly material dependent and should be restored when comparison with experiment is made). Right: Vertex diagram associated with the one-loop Coulomb interaction correction shown by the wiggly line $V (p) = 2 π e^{2} / p$ .
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Figure 2
Top: Variation of the correction function $W (k / λ)$ with rescaled momentum $k / λ$ . The magnitude of the correction is large and maximum at $k = 0$ . Inset: Low-energy band structure for the Kane-Mele model. Bands are spin degenerate with a gap that is generated by the spin-orbit interaction $λ = 1 μ eV$ . Bottom: Variation of the total spin-flip transition moment, $μ + δ μ = \frac{k}{\sqrt{k^{2} + λ^{2}}} [1 + α W (k / λ)]$ , with rescaled momentum ( $k / λ$ ). With the increase in $α$ we see an enhancement in the total transition moment. We relate this increase to the enlarged correction effects from Coulomb interactions.
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Figure 3
Top: A comparison of the correction function $W (k / λ)$ with rescaled momentum $k / λ$ for the one-loop case and the RPA. The magnitude of the correction is seen to decrease with the inclusion of self-consistent screening. Bottom: Comparative plots of the variation of the spin-flip transition moment, $μ + δ μ$ , with rescaled momentum ( $k / λ$ ) for the suspended case within the formalism of one-loop, RPA, and no-Coulomb correction effects.
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Figure 4
Top: Correction function $H (2 k / λ)$ for the dichalcogenides. Here, we have used the rescaled momentum: $2 k / Δ$ . As can be seen, at $2 k / Δ = 0$ , the value of the correction function $H (2 k / Δ)$ is very small ( $\sim 0.08$ .) The inset in the top panel shows the corresponding dispersion relation for the dichalcogenides. The conduction bands are degenerate at $k = 0$ and are also seen to undergo a band inversion. Bottom: Bare transition moment for the dichalcogenides for various values of $λ / Δ$ . For TMDCs with the relevant value of $λ / Δ ≲ 0.15$ , we see that the value of $μ$ is almost a constant $\approx 1$ and shows negligible variation with the momentum.
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Figure 5
For the purposes of illustration we plot the evolution of energy dispersion curves for silicene-germanene-type materials. With the proper tuning of the ratio of the spin-independent gap to spin-orbit coupling ( $Δ / λ$ ), the band structure shows a transition from a topological insulator ( $Δ ≪ λ$ ) to a band insulator ( $Δ ≫ λ$ ) via the quantum critical VSPM state ( $Δ = λ$ ). Subsequent removal of the spin degeneracy is also observed for the bulk insulator regime ( $Δ ≫ λ$ ).
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Figure 6
Top: Variation of the bare transition moment $μ = 〈 ↑ | 2 S_{x} | ↓ 〉$ with the rescaled momentum $2 k / λ$ for several values of $Δ / λ$ . As the coupling parameter $Δ / λ$ increases, the system makes a transition from TI to BI via the VSPM state ( $Δ / λ = 1$ ). Bottom: Variation of the correction function $F (2 k / λ)$ with $2 k / λ$ for various values of coupling $0 < Δ / λ < 2$ . The correction term is seen to be large for the topological insulators ( $Δ / λ ≪ 1$ ) compared to the VSPM ( $Δ / λ = 1$ ) or bulk insulator states ( $Δ / λ ≫ 1$ ).
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