Uncovering the spin ordering in magic-angle graphene via edge state equilibration

Abstract

The flat bands in magic-angle twisted bilayer graphene (MATBG) provide an especially rich arena to investigate interaction-driven ground states. While progress has been made in identifying the correlated insulators and their excitations at commensurate moiré filling factors, the spin-valley polarizations of the topological states that emerge at high magnetic field remain unknown. Here we introduce a technique based on twist-decoupled van der Waals layers that enables measurement of their electronic band structure and-by studying the backscattering between counter-propagating edge states-the determination of the relative spin polarization of their edge modes. We find that the symmetry-broken quantum Hall states that extend from the charge neutrality point in MATBG are spin unpolarized at even integer filling factors. The measurements also indicate that the correlated Chern insulator emerging from half filling of the flat valence band is spin unpolarized and suggest that its conduction band counterpart may be spin polarized.

Fig. 1. Twist-decoupled monolayer graphene (MLG) and magic-angle twisted bilayer graphene (MATBG).

a Optical image of the device. The scale bar is 2 μm. b Schematic of the device structure and interlayer angles. The twisted trilayer graphene is encapsulated in hexagonal boron nitride (hBN) and has graphite top and bottom gates. c Band diagram of the combined MLG-MATBG system. The displacement field D modifies the energies of states in each subsystem and, therefore, tunes the relative chemical potential μ_i of each subsystem i at fixed total carrier density n_tot. d, e Longitudinal resistance R_xx as a function of n_tot and D, at zero magnetic field B and at B = 2 T, respectively. Black solid (white dashed) lines denote where the MLG (MATBG) is at its charge neutrality point (CNP). Parentheses indicate which carrier types are present in the MLG and MATBG, respectively: e indicates electrons and h indicates holes. f R_xx as a function of moiré filling factor s at B = 0 and at various temperatures T where the MLG is at its CNP (solid black curve in d). g μ_MATBG as a function of s at B = 0, as extracted from (d) and analogous data at other temperatures.

Fig. 2. Spin polarization of MATBG quantum Hall states near the CNP.

a Schematic illustration of two possible scenarios for a single pair of counter-propagating edge modes. If the spins of each edge mode are aligned (top), backscattering is allowed (orange circle). Backscattering is suppressed when the spins are anti-aligned (bottom), leading to quantum spin Hall-like behavior with R_xx = h/2e². b R_xx as a function of the total filling factor ν_tot = ν_MLG + ν_MATBG and D at B = 8 T. c, d R_xx and R_NL, respectively measured in the configurations shown in the top left insets, as a function of D when ν_tot = 0. The filling factors of each subsystem for each regime of D are indicated in the bottom inset of c. Insets in (d) schematically represent the inferred relative spin orientations (black arrows) of edge modes in MLG (blue arrows) and MATBG (purple arrows), with orange circles indicating backscattering between a given pair. e, f R_xx and R_NL for ν_MATBG = ±2/∓2 (red and blue, respectively) averaged over 0.1 < ∣D∣ < 0.25 V nm⁻¹. Error bars correspond to one standard deviation. The straight lines connecting data points are guides for the eye. g Schematic diagram of CNP MATBG Landau levels (LLs) and their spin characters. Gaps between LLs are depicted schematically and do not represent experimentally measured field dependence.

Fig. 3. Landau fans demonstrating correlated Chern Insulators (ChIs).

a, b R_xx and R_xy as a function of s and B at fixed top gate voltage V_t = 3 V. c Wannier diagram indicating the strongest quantum Hall and ChI states determined from (a, b). The Chern numbers t of the MATBG states are labeled. At high fields, the total Chern numbers of each state are offset by 2 because ν_MLG = 2. Black, red, orange, and blue lines correspond to states with zero-field intercepts s = 0, s = ∣1∣, s = ∣2∣, and s = ∣3∣, respectively. For states with s = 0, t ≡ ν_MATBG. Black dashed lines label the MATBG symmetry-broken quantum Hall states −4 < ν_MATBG <4. d–f Same as a–c, but for V_t = −3, where ν_MLG = −2 at high fields. Data were collected at T ≈ 300 mK.

Fig. 4. Spin polarization of the ChIs in MATBG.

a R_xx as a function of n_tot and D at B = 8 T (see Supplementary Fig. 8 for the equivalent map in a non-local contact configuration). Black dashed circle: ν_MLG = 1, (t, s) = (−1, −3). Red dashed box: ν_MLG = 2, (t, s) = (−2, −2). Blue dashed box: ν_MLG = −2, (t, s) = (2, 2). b R_xx for the ν_MLG = 1, (t, s) = (−1, −3) state as a function of B. c, d R_xx and R_NL, respectively, measured in the configurations shown in the top left insets, for ν_MLG = ±2, (t, s) = (∓2, ∓2) states (red and blue, respectively) as a function of B. Data are averaged over 0.325 < ∣D∣ < 0.525 V nm⁻¹. Error bars correspond to one standard deviation. Insets in (d) schematically represent the inferred relative spin orientations (black arrows) of edge modes in MLG (blue arrows) and MATBG (purple arrows), with orange circles indicating backscattering between a given pair. The straight lines connecting data points are guides for the eye.