Emergent Correlated Phases in Rhombohedral Trilayer Graphene Induced by Proximity Spin-Orbit and Exchange Coupling

Yaroslav Zhumagulov, Denis Kochan, and Jaroslav Fabian

Phys. Rev. Lett. 132, 186401 – Published 1 May 2024

Abstract

The impact of proximity-induced spin-orbit and exchange coupling on the correlated phase diagram of rhombohedral trilayer graphene (RTG) is investigated theoretically. By employing ab initio-fitted effective models of RTG encapsulated by transition metal dichalcogenides (spin-orbit proximity effect) and ferromagnetic ${Cr}_{2} {Ge}_{2} {Te}_{6}$ (exchange proximity effect), we incorporate the Coulomb interactions within the random-phase approximation to explore potential correlated phases at different displacement fields and doping. We find a rich spectrum of spin-valley resolved Stoner and intervalley coherence instabilities induced by the spin-orbit proximity effects, such as the emergence of a spin-valley-coherent phase due to the presence of valley-Zeeman coupling. Similarly, proximity exchange removes the phase degeneracies by biasing the spin direction, enabling a magnetocorrelation effect—strong sensitivity of the correlated phases to the relative magnetization orientations (parallel or antiparallel) of the encapsulating ferromagnetic layers.

Received 25 May 2023
Revised 28 November 2023
Accepted 22 March 2024

DOI:https://doi.org/10.1103/PhysRevLett.132.186401

Physics Subject Headings (PhySH)

Electronic structure Exchange interaction Magnetic phase transitions Phase diagrams Phase transitions Spin-orbit coupling Spintronics Valley degrees of freedom van Hove singularity

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Yaroslav Zhumagulov ^1,*, Denis Kochan ^2,1,3,†, and Jaroslav Fabian ^1,‡

¹Institute for Theoretical Physics, University of Regensburg, 93040 Regensburg, Germany
²Institute of Physics, Slovak Academy of Sciences, 84511 Bratislava, Slovakia
³Center for Quantum Frontiers of Research and Technology (QFort), National Cheng Kung University, Tainan 70101, Taiwan

^*iaroslav.zhumagulov@ur.de
^†denis.kochan@savba.sk
^‡jaroslav.fabian@ur.de

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Vol. 132, Iss. 18 — 3 May 2024

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Images

Figure 1
(a) Scheme of an RTG-based heterostructure encapsulated by transition-metal dichalcogenide (TMDC) or CGT monolayers, which induce, respectively, SO or EX interactions in RTG. (b) RTG unit cell with the relevant interlayer orbital hoppings $γ$ . The colors distinguish the three layers and $A_{l}$ and $B_{l}$ denote sublattice sites of the $l$ th layer. (c) The top-down view of the RTG lattice. (d) Calculated single-particle low-energy electronic dispersions at $K$ for pristine RTG and indicated heterostructures (red marks spin-up and blue spin-down states).
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Figure 2
Calculated phase diagram of pristine RTG ( ${\hat{h}}_{prox} = 0$ ) for varying doping $n_{e}$ and displacement field $u_{d}$ . There are two dominant phases: the intervalley coherent phase (IVC) and Stoner instability. The white background corresponds to a stable Fermi liquid (FL).
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Figure 3
(a) Calculated phase diagram of ${MoSe}_{2} / RTG / {WSe}_{2}$ heterostructure ( ${\hat{h}}_{R} \neq 0$ , ${\hat{h}}_{I} \neq 0$ , and ${\hat{h}}_{ex} = 0$ ) in the space spanned by displacement field $u_{d}$ and doping density $n_{e}$ . Six symmetry-breaking phases: ${VC}_{\pm}$ (valley coherent state), ${SVC}_{\pm}$ (spin-valley coherent state), and ${SVP}_{\pm}$ (spin-valley polarized state), marked by different colors, are predicted apart from the Fermi liquid state (white). Hartree-Fock excitation dispersions for (b) $u_{d} = 40 meV$ and $n_{e} = - 0.7 \times 10^{12} {cm}^{- 2}$ , and (c) $u_{d} = - 40 meV$ and $n_{e} = 0.3 \times 10^{12} {cm}^{- 2}$ . Red, blue, and gray lines mark spin-up, spin-down, and spin-unpolarized states.
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Figure 4
(a) Calculated phase diagram of the $CGT / RTG / CGT$ system with ferromagnetic CGT configuration ( ${\hat{h}}_{R} = 0$ , ${\hat{h}}_{I} = 0$ , and ${\hat{h}}_{ex} \neq 0$ ) in a parameter space spanned by displacement field $u_{d}$ and electron doping $n_{e}$ . Six symmetry broken phases, ${VC}_{\pm}$ —valley coherent state, ${VP}_{\pm}$ —valley-polarized state, and ${SP}_{\pm}$ —in-plane spin-polarized state, are predicted and displayed by different colors, along with the Fermi liquid (FL) state. (b),(c) Correlated band structures of the $CGT / RTG / CGT$ system for ferromagnetic CGT alignment at displacement field $u_{d} = 40 meV$ with two different electron dopings: (b) $n_{e} = - 0.7 \times 10^{12} {cm}^{- 2}$ and (c) $n_{e} = 0.3 \times 10^{12} {cm}^{- 2}$ . The color code represents spin expectation values: red for spin-up, blue for spin-down, and grey for spin-unpolarized.
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Figure 5
(a) Phase diagram of the $CGT / RTG / CGT$ electron system with antiferromagnetic CGT alignments ( ${\hat{h}}_{R} = 0$ , ${\hat{h}}_{I} = 0$ , and ${\hat{h}}_{ex} \neq 0$ ) in a parameter space of displacement field $u_{d}$ and electron doping $n_{e}$ . Six phases with symmetry breaking, ${VC}_{\pm}$ —valley coherent state, ${VP}_{\pm}$ —valley-polarized state, and ${SP}_{\pm}$ —in-plane spin-polarized state, are predicted along with the Fermi Liquid (FL) phase. (b), (c) Correlated band structure of the $CGT / RTG / CGT$ system with antiferromagnetic CGT layers at displacement field $u_{d} = 40 meV$ and electron doping level (b) $n_{e} = - 0.7 \times 10^{12} {cm}^{- 2}$ and (c) $n_{e} = 0.3 \times 10^{12} {cm}^{- 2}$ . The color code represents spin expectation values: red for spin-up, blue for spin-down, and grey for spin-unpolarized.
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