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Proximity spin-orbit coupling in graphene and its multilayers on transition-metal dichalcogenides

LI Yang 東北大学

2020.09.25

概要

Recently, proximity-induced phenomena in van der Waals (vdW) heterostruc- tures [1] consisting of several 2D materials have attracted considerable exper- imental and theoretical interests. Since the physical properties of the 2D materials are generally sensitive to the substrate, we can select a specific kind of substrates to manipulate the properties of the 2D materials. A typ- ical example is graphene on a substrate of transition-metal dichalcogenides (TMDC) [2, 3, 4, 5], where negligibly small spin-orbit coupling (SOC) of in- trinsic graphene is significantly enhanced by the interlayer interaction with the TMDC substrate. Similarly a magnetic exchange field is induced on graphene when it is placed on a ferromagnet substrate [6, 7, 8].

Although great efforts have been devoted to the research of the proximity effects in the 2D materials, the previous studies have mainly considered com- mensurate structures, where the effect of lattice mis-orientation is neglected. In particular, the previous works on the proximity SOC in the graphene/T- MDC systems only focused on non-rotated configurations [3, 4, 5, 9, 10], where the lattice constant is adjusted to obtain a commensurate structures with a finite unit cell. Meanwhile, the effects of the lattice mis-orientation have been extensively studied in other 2D heterostructures [11, 12]. In graphene/hBN (hexagonal boron nitride) [13, 14, 15, 16, 17, 18, 19, 20, 21], the moir´e in- terference pattern gives rise to the formation of a miniband structure with the secondary Dirac cones [17]. The twisted bilayer graphene also exhibits dramatic angle-dependent phenomena, such as the flat band formation [22, 23, 24, 25, 26, 27, 28, 29] and the emergent superconductivity [30, 31].

For graphene/TMDC heterostructures, the band structure was theoret- ically calculated for several commensurate angles using the density func- tional theory (DFT). [32, 33, 34]. It was also experimentally probed by angle-resolved photoemission spectroscopy [35, 36], photoluminescence, Ra- man spectroscopy [37], and scanning tunneling spectroscopy[38]. However, the effect of lattice mis-orientation and the twist-angle dependence of prox- imity induced SOC on graphene still remain unclear. It is generally hard to consider an arbitrary twist angle in the DFT calculation, because the system is essentially incommensurate and does not have a finite unit cell.

In this thesis, we study the proximity SOC effect in incommensurate graphene-TMDC heterostructures by using a perturbational approach based on the tight-binding model, which do not need the commensurate lattice matching. We obtain an effective proximity-SOC potential for graphene as a continuous function of θ, and reveal its angle dependence for several types of TMDCs. We find that the relative rotation of graphene to TMDC greatly enhances the spin splitting of graphene, typically by a few to ten times com- pared to the non-rotated geometry, and the maximum splitting is achieved around 20◦. Furthermore, we demonstrate that the spin-splitting is sensitive to the relative band energy between graphene and TMDC, and it sharply rises when the graphene’s Dirac point is shifted toward TMDC band by applying the gate voltage. In the latter part, we study the proximity SOC effect in multilayer graphenes on TMDC. We apply the above theoretical analysis to AB-stacked bilayer graphene (BLG), ABA-stacked trilayer graphene (TLG) and ABC-stacked TLG. The multilayer graphene is distinct from the mono- layer in that the band structure is tunable by the perpendicular gate electric field [39, 40, 41, 42, 43, 44, 45]. We will demonstrate that the proximity induced SOC on multilayer graphenes can be controlled by the electric field. The theoretical method proposed here is applicable to any incommensurate bilayer systems where the DFT calculation cannot be used, and therefore it considerably extends the applicability of the theoretical framework to a wide variety of van der Waals heterostructures.

The thesis is organized as follows. In the rest of this chapter, we re- view the previous works on graphene, multilayer graphenes, TMDC and two dimensional moir´e superlattices . Chapter 2 presents the essential theoreti- cal tools to study graphene on TMDC heterostructures, where we introduce the electronic band models of monolayer graphene, multilayer graphenes and TMDC, and then present the theoretical treatment to describe the interlayer interaction between two layers with different periodicities. Using these theo- retical methods, we study the angular dependence of proximity induces SOC for graphene on TMDC in Chapter 3 and that for multilayer graphenes in Chapter 4. Chapter 5 concludes the thesis.

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