Tuning correlated states of twisted mono-bilayer graphene with proximity-induced spin-orbit coupling

Abstract

We study the correlated ground states of twisted mono-bilayer graphene with and without proximity-induced spin-orbit coupling (SOC) from a transition-metal dichalcogenide layer placed on top. We perform self-consistent Hartree-Fock calculations that allow the variational space to include multi-$Q$ translational symmetry broken states for all integer and half-integer fillings of the conduction bands, where signatures of correlated, topological states have been reported experimentally. We find interaction-induced insulators that retain moiré translational symmetry at integer fillings, but that break this symmetry at half-integer fillings. We argue that translational symmetry breaking arises from half-filled polarized bands, even when SOC is present. Yet, we find that small SOC can already crucially affect the spin nature of correlated states. Generally, Ising SOC favors out-of-plane spin polarization and spin-valley locking, while Rashba SOC favors in-plane spin order. If only one of these two terms is present, we find that, depending on the type of SOC, it drives a transition from a tetrahedal antiferromagnet to either a coplanar, non-coplanar, or collinear spin-density wave state for half-integer fillings. The frustration associated with the simultaneous presence of both types of SOC can induce chiral, non-coplanar order in parameter ranges where the ground state in the absence of SOC is collinear.

Figures (10)

• Figure 1: Schematic cross-section of SOC-TMBG: a van der Waals heterostructure composed of a graphene monolayer ($l = 1$) and Bernal-stacked graphene bilayer ($l=2$ and $l=3$) with a relative twist angle of $\theta$, as well as a TMD layer. The TMD layer, such as WSe${}_2$, is located on top of the graphene monolayer. We study both the case with and without the TMD layer. A displacement field is applied along the $\boldsymbol{z}$ direction and the associated electrostatic potential is designated as $U,0,-U$ for layer $l=1,2,3$, respectively.
• Figure 2: The single-particle miniband dispersions, DOS and Fermi surfaces for SOC-TMBG with different SOC parameters. (a-c) Single-particle band dispersion for the lowest conduction and valence minibands as well as the corresponding DOS. The dispersions and the DOS are for $K$ valley only. These data were obtained by diagonalizing Eq. (\ref{['eq:Ham']}) for $K$ valley. The different panels are for different SOC parameters: (a) $(\lambda_I, \lambda_R) = (5,0)$, (b) $(\lambda_I, \lambda_R) = (0,5)$, (c) $(\lambda_I, \lambda_R) = (5,5)$. The SOC parameters are in units of meV. The dispersions are plotted along the momentum path depicted in the inset of panel ($\textbf{c}$) covering the high symmetry points ($K_L\rightarrow K_U\rightarrow\Gamma\rightarrow M \rightarrow\Gamma\rightarrow K_L$). At each momentum, the miniband dispersion is color coded according to the spin expectation value $\langle s_z \rangle$. The color legend is shown on the right of panel ($\textbf{f}$). (d-f) Fermi contours for the same SOC parameter sets as in panels (a-c) at integer fillings and near fillings where the Fermi surface topology changes due to the emergence of VHS. The color coding is again based on the spin expectation value $\langle s_z\rangle$ in the mBZ.
• Figure 3: HF data for translation-invariant (TI) and $1Q$ states with different SOC parameters. (a) HF energy difference between $1Q$ and TI states per moiré unit cell as a function of filling-factor $\nu$ with different SOC parameters. (b) Filling-factor dependence of moiré translation symmetry breaking order parameter $O_{\textrm{MTSB}}$ with different SOC parameters. (c) Filling-factor dependence of $\langle\boldsymbol{s}\rangle^2=\langle s_x \rangle^2+\langle s_y \rangle^2+\langle s_z \rangle^2$, $\langle \tau_z \rangle$ and $\langle s_z \tau_z \rangle$ with different SOC parameters. See Eq. (\ref{['eq:SDWorder']}) and Eq. (\ref{['eq:expectation']}) for the definitions of order parameters. (d) Table showing which flavor polarized states emerge for each HF ground state at different fillings $\nu$ and SOC parameters. The unit of SOC parameters is meV. Here, VP, SP, and SVP stand for valley polarized, spin polarized, and spin-valley polarized order, respectively. The gray (non)-shaded sectors of the table denote $1Q$ (TI) states. The SOC parameters are in units of meV. The system size is $12\times12$.
• Figure 4: Correlator matrix and spin order parameter of $1Q$ states (a) Correlator matrix structure with an absolute value $|\tilde{\textbf{P}}_{1Q}|$ at filling factor $\nu = 1/2$ with different SOC parameters. The matrix structure of $\tilde{\textbf{P}}_{1Q}$ is built such that the smallest red-outlined block denotes valley flavor, the next green-outlined block denotes spin (or band depending on presence of Rashba SOC) flavor, and the largest blue-outlined block denotes the additional degrees of freedom from broken moiré translation symmetry. (b) Filling-factor dependence of in-plane $\textbf{S}^2_{xy}$ and out-of-plane $\textbf{S}^2_z$ spin order parameter for the HF ground state with different SOC parameters. The SOC parameters are in units of meV. The system size is $N=12\times12$.
• Figure 5: GL phase diagram and the ground state configuration (a) GL phase diagram for zero SOC for different parameter sets $(c_2, c_3)$. (b) GL phase diagram for nonzero Ising SOC for different parameter sets $(\lambda_I, a)$. (c) GL phase diagram for nonzero Rashba SOC for different parameter sets $(\lambda_R, a)$. (d) Configuration of net spin expectation value $\boldsymbol{S}(\boldsymbol{R})$ for a collinear SDW ($\textrm{clSDW}_{3Q}$) state, coplanar SDW (cpSDW) state, non-coplanar SDW (ncpSDW) state, and tetrahedral antiferromagnetic (TAF) state in moiré real space, which are obtained from a GL analysis. For the figures, we choose (i) a $\textrm{clSDW}_{3Q}$ state with $\boldsymbol{\phi}_1 = \boldsymbol{\phi}_2=\boldsymbol{\phi}_3\neq0$, (ii) a cpSDW state with $m_{12}=m_{23}=m_{31}=-\frac{1}{2}$ and (iii) a ncpSDW state with $m_{12}=-m_{23}=-m_{31}=-\frac{1}{2}, \ \chi_{\textrm{GL}}\simeq0.38$. Black arrows denote $(\boldsymbol{S}(\boldsymbol{R})_x, \boldsymbol{S}(\boldsymbol{R})_y)$ at position $\boldsymbol{R}$, red dots denote AA sites in moiré real space and green arrows denote the direction of $(\boldsymbol{S}(\boldsymbol{R}=\boldsymbol{R}_{AA})_x, \boldsymbol{S}(\boldsymbol{R}=\boldsymbol{R}_{AA})_y)$ at AA sites. For ncpSDW and TAF, we also show the $z$ component of net spin expectation value $\boldsymbol{S}(\boldsymbol{R})_z$ for each position $\boldsymbol{R}$, which illustrates the non-coplanar nature of the spin configurations.