Proximity spin-orbit and exchange coupling in ABA and ABC trilayer graphene van der Waals heterostructures

We investigate the proximity spin-orbit and exchange couplings in ABA and ABC trilayer graphene encapsulated within monolayers of semiconducting transition-metal dichalcogenides and the ferromagnetic semiconductor ${\mathrm{Cr}}_2{\mathrm{Ge}}_2{\mathrm{Te}}_6$. Employing first-principles calculations, we obtain the electronic structures of the multilayer stacks and extract the relevant proximity-induced orbital and spin interaction parameters by fitting the low-energy bands to model Hamiltonians. We also demonstrate the tunability of the proximity effects by a transverse electric field. Using the model Hamiltonians, we also study mixed spin-orbit/exchange coupling encapsulation, which allows us to tailor the spin interactions very efficiently by the applied field. We also summarize the spin-orbit physics of bare ABA, ABC, and ABB trilayers, and we provide—along with the first-principles results of the electronic band structures, density of states, spin splittings, and electric-field tunabilities of the bands—a qualitative understanding of the observed behavior and realistic model parameters as a resource for model simulations of transport and correlation physics in trilayer graphene.

Figure 1

(a) Lattice structure of the ABA TLG. Due to $z$-mirror symmetry, the sublattice atoms with the same color (red, blue, green, and gray) belong together. Gray rhombuses define the unit cells of the layers. (b) The DFT-calculated band structure in the vicinity of the $K$ point. The bands are color-coded by their atomic projections as defined in (a). (c) The energy level diagram of states at the $K$ point without SOC, with SOC, and with SOC in the presence of a perpendicular electric field of 0.2 V/nm.

Figure 2

(a) Lattice structure of the ABC TLG. Due to space inversion symmetry, the sublattice atoms of the same color (red, blue, and green) contribute equally to the energy bands. Gray rhombuses define the unit cells of the layers. (b) The DFT-calculated band structure in the vicinity of the $K$ point. The bands are color-coded by their atomic projections as defined in (a). The inset in (b) shows a zoom to the low-energy bands near the Fermi level, indicating the conelike touching point, where a gap of about $25\mu \mathrm{eV}$ arises due to SOC. (c) The energy level diagram of the low- and high-energy states at the $K$ point without SOC, with SOC, and with SOC in the presence of a perpendicular electric field of 0.2 V/nm.

Figure 3

(a) Lattice structure of the ABB TLG. Due to the absence of spatial inversion and $z$-mirror symmetry, all sublattice atoms are different (red, green, blue, purple, black, orange). Gray rhombuses define the unit cells of the layers. (b) The energy level diagram of states at the $K$ point without SOC, with SOC, and with SOC in the presence of a transverse electric field of 0.2 V/nm. (c) The DFT-calculated band structure in the vicinity of the $K$ point without SOC. From left to right, we project onto the different basis atoms from the three graphene layers, with the color code as defined in (a).

Figure 4

Top: Zooms to the ABC TLG bands in the vicinity of the $K$ point. We compare DFT data (symbols) with model Hamiltonian fits (solid lines) employing parameters from Table 3 in Appendix pp1. In the dispersion, we specify the band gap at the Fermi level. Bottom: The corresponding calculated density of states (DOS). From left to right, we increase the transverse electric field from 0 to 1 V/nm.

Figure 5

Geometries of the encapsulated TLG heterostructures. Left: TMDC (${\mathrm{MoSe}}_2$ and ${\mathrm{WSe}}_2$) encapsulated ABA and ABC TLG. Right: CGT encapsulated ABA and ABC TLG. The relaxed interlayer distances are indicated. We also specify the direction of a positive electric field.

Figure 6

Schematic illustration of the ABA (left) and ABC (right) TLG lattices, showing the relevant intra- and interlayer hoppings ${\gamma }_j, j=\{0,1,2,3,4,5,6\}$ (dashed lines). In addition, the bottom (top) graphene layer is placed in the potential $V_1$ ($-V_1$), while the middle layer is placed in potential $V_2$. The asymmetries $\mathrm{\Delta }$ and $\eta$ arise due to vertical hoppings.

Figure 7

(a) DFT-calculated band structure of the ${\mathrm{MoSe}}_2$/ABA-TLG/${\mathrm{WSe}}_2$ heterostructure along the $M\text{−}K\text{−}\mathrm{\Gamma }$ path. The color of the bands corresponds to the $s_z$ spin expectation value. We specify the 12 relevant energy bands, ${\varepsilon }_{1-12}$, corresponding to TLG, which seem to be pairwise spin-degenerate. (b) Zoom to the calculated low-energy bands (symbols) around the $K$ point, corresponding to the band structure in (a), with a fit to the model Hamiltonian (solid lines). The bands are spin-split due to proximity-induced SOC. (c) The spin expectation values of the eight low-energy bands as labeled in (b). The $s_x$ and $s_y$ values are multiplied by a factor of 5 for better visualization.

Figure 8

(a) DFT-calculated band structure of the ${\mathrm{MoSe}}_2$/ABC-TLG/${\mathrm{WSe}}_2$ heterostructure along the $M\text{−}K\text{−}\mathrm{\Gamma }$ path. The color of the bands corresponds to the $s_z$ spin expectation value. We specify the 12 relevant energy bands, ${\varepsilon }_{1-12}$, corresponding to TLG, which seem to be pairwise spin-degenerate. (b) Zoom to the calculated low-energy bands (symbols) around the $K$ point, corresponding to the band structure in (a), with a fit to the model Hamiltonian (solid lines). The bands are spin-split due to proximity-induced SOC. (c) The spin expectation values of the four low-energy bands as labeled in (b). The $s_x$ and $s_y$ values are multiplied by a factor of 5 for better visualization.

Figure 9

First-principles calculated spin-orbit fields around the $K$ point of the ${\mathrm{MoSe}}_2$/ABC-TLG/${\mathrm{WSe}}_2$ heterostructure, corresponding to the four low-energy bands ${\varepsilon }_5$–${\varepsilon }_8$ in Fig. 12 at zero electric field. The dashed white lines represent the edges of the hexagonal Brillouin zone with the $K$ point at the center ($k_x=k_y=0$). The $s_z$ spin expectation value determines the color, while in-plane spins are depicted as arrows. Here, we enhanced in-plane spin expectation values by a factor of 10 for better visualization.

Figure 10

(a) Band structure of the CGT/ABA-TLG/CGT heterostructure along the $M\text{−}K\text{−}\mathrm{\Gamma }$ path. Bands in solid red (dashed blue) correspond to spin up (spin down). The two CGT layers have parallel magnetizations along the $z$ direction. (b) Zoom to the calculated low-energy bands (symbols) around the $K$ point, corresponding to the band structure in (a), with a fit to the model Hamiltonian (solid lines). (c),(d) Same as (a),(b) but for antiparallel magnetization of the two CGT layers (bottom layer along $z$, top layer along $-z$).

Figure 11

(a) Band structure of the CGT/ABC-TLG/CGT heterostructure along the $M\text{−}K\text{−}\mathrm{\Gamma }$ path. Bands in solid red (dashed blue) correspond to spin up (spin down). The two CGT layers have parallel magnetizations along the $z$ direction. (b) Zoom to the calculated low-energy bands (symbols) around the $K$ point, corresponding to the band structure in (a), with a fit to the model Hamiltonian (solid lines). (c),(d) Same as (a),(b) but for antiparallel magnetization of the two CGT layers (bottom layer along $z$, top layer along $-z$).

Figure 12

Top: Electric field evolution of the low-energy bands of the ${\mathrm{MoSe}}_2$/ABC-TLG/${\mathrm{WSe}}_2$ heterostructure near the $K$ point. The color of the bands corresponds to the $s_z$ spin expectation value. From left to right, we tune the electric field from $-0.25$ to 0.25 V/nm. The calculated low-energy bands (symbols) match well with the model Hamiltonian fits (solid lines). Bands are plotted with respect to the Dirac point energy $E_D$. For the $\pm 0.25$ V/nm cases, we fixed the parameters from Table 1 and refitted the potential $V_1$ and asymmetry $\eta$, since mainly these parameters are affected by a small electric field (see Table 3). We obtained $V_1=6.259(-3.839)$ meV and $\eta =-6.782(-0.771)$ meV for the negative (positive) field. Bottom: The corresponding electric field evolution of the DOS, as calculated from the model Hamiltonian.

Figure 13

Top: Electric field evolution of the low-energy bands of the ${\mathrm{MoSe}}_2$/ABA-TLG/${\mathrm{WSe}}_2$ heterostructure near the $K$ point. The color of the bands corresponds to the $s_z$ spin expectation value. From left to right, we tune the electric field from $-0.25$ to 0.25 V/nm. The calculated low-energy bands (symbols) match well with the model Hamiltonian fits (solid lines). For the $\pm 0.25$ V/nm cases, we fixed the parameters from Table 1, and refitted the potentials $V_1$ and $V_2$, since mainly these parameters are affected by a small electric field (see Table 2). We obtained $V_1=13.748(-12.021)$ meV and $V_2=-4.984(-6.195)$ meV for the negative (positive) field. Bottom: The corresponding electric field evolution of the DOS, as calculated from the model Hamiltonian.

Figure 14

Electric field evolution of the low-energy bands of a CGT/ABC-TLG/${\mathrm{WSe}}_2$ heterostructure near the $K$ (top row) and $K^{\prime }$ (bottom row) points. The color of the bands corresponds to the $s_z$ spin expectation value. From left to right, we tune the electric field from $-0.25$ to 0.25 V/nm. Next to the bands, we list the spin splitting at the $K/K^{\prime }$ point. The dispersions are calculated from the model, assuming reasonable parameters as explained in the text.

Figure 15

Electric field evolution of the low-energy bands of a CGT/ABA-TLG/${\mathrm{WSe}}_2$ heterostructure near the $K$ (top row) and $K^{\prime }$ (bottom row) points. The color of the bands corresponds to the $s_z$ spin expectation value. From left to right, we tune the electric field from $-0.25$ to 0.25 V/nm. The bands are calculated from the model, assuming reasonable parameters as explained in the text.

Figure 16

Top: Zooms to the ABA TLG bands in the vicinity of the $K$ point. We compare DFT data (symbols) with the model fits (solid lines) employing parameters from Table 2. In the dispersion, we list the band gap at the Fermi level. Bottom: The corresponding calculated density of states (DOS). From left to right, we increase the transverse electric field from 0 to 1 V/nm.

Figure 17

Schematic illustration of the ABB TLG lattice, showing the relevant intra- and interlayer hoppings ${\gamma }_j, j=\{0,1,3,4,5,7\}$ (dashed lines). In addition, the bottom (top) graphene layer is placed in the potential $V_1$ ($-V_1$), while the middle layer is placed in the potential $V_2$. The asymmetry $\mathrm{\Delta }$ arises due to vertical hoppings. In general, the direct interlayer couplings ${\gamma }_1$ and ${\gamma }_1^{\prime }$ can be different. However, when both interlayer distances are the same (in our case fixed to $d=3.3$ Å), ${\gamma }_1={\gamma }_1^{\prime }$ holds.

Figure 18

(a) DFT-calculated low-energy bands of ABB TLG with SOC (black solid line) and additionally with a transverse electric field of 0.2 V/nm (red dashed line). (b) The spin splitting of valence (red) and conduction (blue) bands. Solid lines are with SOC only, and dashed lines are with SOC in the presence of the electric field.

Figure 19

Top: Zooms to the ABB TLG bands in the vicinity of the $K$ point. We compare DFT data (symbols) with the model fits (solid lines) employing parameters from Table 4. Bottom: The corresponding calculated density of states (DOS). From left to right, we increase the transverse electric field from 0 to 1 V/nm.

Figure 20

Top: Zooms to the TLG dispersions in the vicinity of the $K$ point. We compare unrelaxed (black) and fully relaxed (red) structures employing DFT-D2 vdW corrections. Bottom: The corresponding calculated density of states (DOS). The red lines can be reproduced (not shown) by model Hamiltonian parameters summarized in Table 5.

Figure 21

Zooms to the TLG bands in the vicinity of the $K$ point of the ${\mathrm{MoSe}}_2$/ABA-TLG/${\mathrm{WSe}}_2$ heterostructure. From left to right, we project onto the sublattice atoms ($A$ = red-filled circles, $B$ = blue-open circles) of the bottom, middle, and top graphene layers. The sublattice atoms are indicated in the sketch of the ABA TLG geometry in the lower panels.

Figure 22

Zooms to the TLG bands in the vicinity of the $K$ point of the ${\mathrm{MoSe}}_2$/ABC-TLG/${\mathrm{WSe}}_2$ heterostructure. From left to right, we project onto the sublattice atoms ($A$ = red-filled circles, $B$ = blue-open circles) of the bottom, middle, and top graphene layers. The sublattice atoms are indicated in the sketch of the ABC TLG geometry in the lower panels.