Spin-orbit-driven ferromagnetism at half moiré filling in magic-angle twisted bilayer graphene
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REPORTS
Cite as: J.-X. Lin et al., Science
10.1126/science.abh2889 (2022).
Spin-orbit–driven ferromagnetism at half moiré filling in
magic-angle twisted bilayer graphene
Jiang-Xiazi Lin1, Ya-Hui Zhang2, Erin Morissette1, Zhi Wang1, Song Liu3, Daniel Rhodes3,
K. Watanabe4, T. Taniguchi4, James Hone3, J. I. A. Li1*
1
Department of Physics, Brown University, Providence, RI 02912, USA. 2Department of Physics, Harvard University, Cambridge, MA 02138, USA. 3Department of
Mechanical Engineering, Columbia University, New York, NY 10027, USA. 4National Institute for Materials Science, 1-1 Namiki, Tsukuba 305-0044, Japan.
*Corresponding author. Email: jia_li@brown.edu
Strong electron correlation and spin-orbit coupling (SOC) can have a profound influence on the electronic
properties of materials. We examine their combined influence on a 2-dimensional electronic system at the
atomic interface between magic-angle twisted bilayer graphene and a tungsten diselenide crystal. Strong
electron correlation within the moiré flatband stabilizes correlated insulating states at both quarter and
half filling, and SOC transforms these Mott-like insulators into ferromagnets, evidenced by robust
anomalous Hall effect with hysteretic switching behavior. The coupling between spin and valley degrees of
freedom is demonstrated through the control of the magnetic order with an in-plane magnetic field, or a
perpendicular electric field. Our findings establish an experimental knob to engineer topological properties
of moiré bands in twisted bilayer graphene and related systems.
The van der Waals (vdW) moiré structures are an intriguing (hBN) substrate (16, 20). Unlike bulk materials, where tun-
platform for exploring the interplay of correlation, topology ing the chemical composition is required to produce spin-
and broken symmetry in 2-dimensional (2D) electronic sys- orbit locking, an alternative route through the proximity
tems. The rotational alignment between two sheets of vdW effect exists in vdW structures. Close proximity between
crystal gives rise to a flat moiré energy band where strong graphene and transition metal dichalcogenide crystals, such
Coulomb correlation plays a dominating role in a rich land- as tungsten diselenide (WSe2), allows electron wavefunc-
scape of emergent quantum phenomena (1–7). In a graphene tions from both crystals to overlap and hybridize, endowing
moiré structure, breaking the C2T symmetry was shown to graphene with strong SOC (21–28). In this work, we use
stabilize spontaneous orbital ferromagnetism at quarter and transport measurement to examine the effect of proximity-
three-quarter filling, which is manifested in robust anoma- induced SOC on properties of the moiré band and its associ-
lous Hall effect (AHE) with hysteretic switching transitions ated quantum phases.
(8–11). Unlike one and three-quarter filling, a potential or- The geometry of the twisted bilayer graphene
bital ferromagnetic state at half-filled moiré band would (tBLG)/WSe2 heterostructure is shown in Fig. 1A. An atomic
feature a spin-unpolarized edge mode that is able to proxi- interface is created by stacking a few-layer WSe2 crystal on
mate superconducting pairing along a ferromag- top of magic angle tBLG, which is further encapsulated with
net/superconducting interface (12). Such construction has dual hBN and graphite crystals on top and bottom to
been proposed to be key in realizing the Majorana mode. achieve optimal sample quality (29). Transport measure-
However, an orbital ferromagnet is predicted to be energeti- ment indicates excellent sample quality with low charge
cally unfavorable in twisted graphene structures, owing to fluctuation δn ~ 0.08 (1012 cm–2) (see fig. S10). Longitudinal
the inter-valley Hund’s coupling (13–16). resistance Rxx measured from tBLG exhibits a series of well-
As an essential ingredient in forming certain topological defined resistance peaks emerging at partial filling of the
phases, spin-orbit coupling (SOC) provides an additional moiré band, ν = –2, +1, +2, and +3, which are associated
experimental knob to engineer the topological properties of with the correlated insulator (CIs) states. The positions of
moiré structures (17–19). It was recently proposed that SOC these peaks are consistent with a twist angle of θ ≈ 0.98°.
endows the moiré energy band with nonzero Berry curva- The tBLG and hBN substrate are maximally misaligned ac-
ture, making ferromagnetic order at half moiré filling a pos- cording to the optical image of the heterostructure (see fig.
sibility without alignment with the hexagonal boron nitride S2) (16), which is consistent with the fact that the sample
First release: 6 January 2022 science.org (Page numbers not final at time of first release) 1
appears gapless at the charge neutrality point (CNP) [see the complex conjugation. Both the Ising and Rashba SOC
(16) for more discussions regarding the coupling between break the C2 symmetry, while preserving the time reversal.
tBLG and hBN] (8, 9). Transverse resistance measurements Therefore, C2T is broken in the presence of proximity-
reveal large Hall resistance Rxy at ν = +1 and +2; the Hall induced SOC.
resistance exhibits hysteretic switching behavior as the The combination of Rashba SOC and time reversal
field-effect induced doping in tBLG, ntBLG, is swept back and symmetry gives rise to a valley-contrasting spin texture
forth (Fig. 1C). Hysteresis in magnetization reversal is also within the mini Brillouin Zone (MBZ) for each band: for any
observed while sweeping an external magnetic field aligned momentum k, spin for the two valleys points in opposite
perpendicular to the 2D interface, B⊥ (Fig. 1, D and E). We directions, sK(k) = –sK´(–k). The average value of in-plane
note that the resistance peak at ν = +2 vanishes at large in- spin moment of each band is obtained by integrating over
plane B-field (see fig. S9) (16), indicating a spin-unpolarized the MBZ for valley K (K´),
isospin configuration. As such, the ground state is likely
valley-polarized and the ferromagnetic order is orbital. This 1
is further illustrated by a schematic representation of the S K , K ′ =
N
∑
k∈MBZ
sK , K ′ ( k ) (2)
band structure (right panels of Fig. 1, D and E), where the
two lowest conduction bands feature the same valley index
where N is the system size. We note that a broken C3 rota-
with nonzero Chern number of C = –3 and +1.
A valley polarized state at ν = 2 is unfavorable in the ab- tion symmetry could lead to nonzero SK and SK ′ . In
sence of SOC because of the influence of inter-valley Hund’s this scenario, time reversal symmetry is preserved by the
coupling [see (16) for a more detailed discussion] (13–16). As valley-contrasting spin texture SK = − SK ′ . On the other
a result, observing orbital ferromagnetism at the half-filled
moiré band has remained elusive (8, 9, 30). To this end, the hand, an orbital ferromagnetic state emerges because valley-
AHE at ν = +2 in our sample provides strong evidence that polarized Chern bands are occupied, spontaneously break-
the moiré band structure is transformed by proximity- ing time reversal symmetry. Most remarkably, the combina-
induced SOC, which is more dominant compared to inter- tion of SOC, valley polarization and a broken C3 rotation
valley Hund’s coupling (20). Although the presence of SOC symmetry allows an in-plane magnetic field to couple to the
endows the moiré flatband with a nonzero Chern number, orbital magnetic order through the in-plane Zeeman energy.
as shown in Fig. 1, D and E, the observed Hall resistance is This control of the out-of-plane magnetic order using an
much smaller than the expected value of the quantum in-plane magnetic field B|| is demonstrated in Fig. 2, B and
anomalous Hall effect. We ascribe this behavior to the pres- C: as B|| is swept back and forth, Rxy exhibits hysteretic
ence of bulk conduction channels parallel to the chiral edge switching behavior at both ν = +1 and +2. Using Hall re-
conduction, which could result from the presence of mag- sistance measurements, we show that the out-of-plane com-
netic domain walls (8) or sample disorder. A fully developed ponent of the B-field is negligible compared to the out-of-
Chern insulator state could be realized in a sample with plane coercive field, confirming that the magnetic order is
lower disorder or stronger SOC, hence a larger energy gap. indeed controlled by the in-plane component of the B-field,
To better understand the influence of proximity- B|| (see fig. S4). At the same time, direct coupling between
induced SOC, we note that the introduction of SOC adds an the valley order and B|| is shown to be absent in tBLG sam-
extra term to the Hamiltonian: ples without SOC (31), which rules out the orbital effect as a
possible origin for the observed B||-dependence. Taken to-
gether, we conclude that B⊥ and B|| couple to the magnetic
λ I τ z sz + λ R ( τ z σ x s y − σ y sx )
1 1
hSOC ( k ) = (1) order through different mechanisms: B⊥ directly controls
2 2 the magnetic ground state through valley Zeeman coupling,
EZv = − γ v B⊥ τ z , whereas the influence of B|| arises from the
Here τ, σ and s denote the Pauli matrix for valley, sublattice
and spin at each momentum k, whereas λI and λR represent combination of SOC and spin Zeeman coupling,
the Ising and Rashba SOC coefficients, respectively (See eqs. E = − γ s τ z B ⋅ ⟨S ⟩ . Here τz corresponds to the valley polari-
s
Z
S3 to S7 for more detailed discussions) (16). The Ising SOC zation, γv and γs are valley and spin gyromagnetic ratio, re-
locks the valley moment τz with the spin moment sz, where- spectively.
as the Rashba term λR locks the in-plane spin sx, sy with the We note a few intriguing properties of the B-induced
sublattice σx, σy (the locking depends on the valley τz). For hysteresis behavior in Figs. 1 and 2: (i) a large B|| stabilizes
tBLG without SOC, there is a C2 symmetry defined as C2:τxσx opposite magnetic orders at ν = +1 and +2, which is evi-
and time reversal symmetry defined as T:iτxsyK, where K is denced by Rxy with opposite signs (Fig. 2, B and C). This in-
First release: 6 January 2022 science.org (Page numbers not final at time of first release) 2
dicates that ⟨S||⟩ points in opposite directions for ν = +1 and also manifested in the sign change of coercive field B↑ and
+2. On the other hand, an out-of-plane B-field stabilizes the
same magnetic order at different fillings (Fig. 1, D and E), B↓ at D ~ –100 mV/nm (Fig. 3E). A possible explanation for
providing further confirmation that ν = +1 and +2 feature this unique D-dependence is obtained by examining the av-
the same valley index and the ground state at half-filling is erage in-plane spin moment for valley K, SK . Calculations
valley polarized; (ii) although the out-of-plane coercive
using the 4-band model show that SK changes sign with
fields are similar, the in-plane coercive field at ν = 2 is much
bigger compared to ν = 1. This suggests that the average in- varying D at ν = 2 (Fig. 3F), indicating that an in-plane B-
plane spin moment ⟨S||⟩ is much smaller at ν = 2, which is field favors opposite valley polarizations at different D. By
consistent with a predominantly spin unpolarized ground comparison, Fig. 3F shows that no sign change in SK is
state (fig. S9) (16). It is worth pointing out that the observed
expected at ν = 1, which is consistent with our experimental
behavior at ν = +2 does not rule out alternative isospin con- observation (Fig. 3E). Because carrier density remains the
figurations. Our results provide important constraints for same when changing D, the D-controlled magnetization re-
future theoretical work to examine such possibilities. The versal represents electric field control of the magnetic order,
effective control of in-plane B-field on the magnetic order which is made possible by proximity induced SOC.
not only provides further validation that the orbital ferro- Next, we turn our attention to the effect of SOC on the
magnetic order is stabilized by proximity-induced SOC, it isospin polarization and superconductivity. In the absence
also reveals that a broken C3 rotational symmetry gives rise
of SOC, the ground state at ν = –1 was shown to be isospin
to a nonzero ⟨S||⟩. A spontaneously broken C3 symmetry nat-
unpolarized at B = 0 (38, 39). The application of a large in-
urally derives from the combination of strong SOC and ne-
plane magnetic field lifts the isospin degeneracy, stabilizing
matic charge order (32–34). Alternatively, a preferred in-
an isospin ferromagnetic state (IF3) near ν = –1, which is
plane direction for spin could result from small amount of
separated from the unpolarized state (IU) by a resistance
uniaxial strain in the moiré lattice (Fig. 2D), which is shown
peak in Rxx and a step in the Hall density (38, 39). Figure 4,
to be common for tBLG samples (35, 36).
A and B, shows that this phase boundary between IF3 and
The proximity-induced SOC arises from wavefunction
IU extends to B|| = 0 in the presence of proximity-induced
overlap across the interface (22). The role of wavefunction
SOC. Taken together, the influence of SOC on the iso-spin
overlap was recently demonstrated experimentally, as SOC
degeneracy is comparable to a large in-plane magnetic field
strength was shown to depend on interlayer separation,
(see fig. S7) (16). Notably, we show that the superconducting
which is tunable with hydrostatic pressure (37). In the same
phase is unstable against proximity-induced SOC in our
vein, we show that SOC strength can be controlled with a
sample. As shown in Fig. 4C, no zero-resistance state is ob-
perpendicular electric field D: under a positive (negative) D,
served over the full density range of the moiré band. In ad-
charge carriers are polarized toward (away from) the WSe2
dition, Rxx – T traces exhibit no clear downturn with
crystal, resulting in increased (decreased) wavefunction
overlap and stronger (weaker) SOC (Fig. 3A) (22, 37). Figure decreasing T at moiré filling ν = –2 – δ, where a robust su-
3 demonstrates the effect of D by plotting the evolution of B- perconducting phase usually emerges in magic-angle tBLG
induced hysteresis loops: with increasing D, hysteresis loops (Fig. 4D). Because strong Ising and Rashba SOC break C2T
symmetry, the absence of superconductivity, combined with
exhibit larger coercive fields ( B↑⊥ and B↓⊥ ) for both ν = +1 the emergence of AHE are potentially consistent with a re-
and +2 (Fig. 3, B and C). Transport measurements near the cent theoretical proposal that C2T symmetry is essential for
CNP show that the width of the disorder regime remains the stabilizing the superconducting phase in tBLG (40). Our
same over a wide range of D-field (see fig. S10B), suggesting results are distinct from another experimental report show-
that changes in the coercive field are not caused by the in- ing robust superconductivity stabilized by SOC in tBLG
fluence of disorder across two graphene layers. Because the away from the magic angle (28). Apart from the difference
value of the coercive field reflects the robustness of magnet- in twist angle range, these distinct observations could result
ic order, the D-dependence shown in Fig. 3C provides an- from a few other factors that will be discussed in the follow-
other indication that the orbital ferromagnetism is ing. The D-dependence shown in Fig. 3 suggests that a dual-
stabilized by proximity-induced SOC. gated geometry, which allows independent control on D and
Most interestingly, the effect of varying D at ν = +2 is ntBLG, is key to investigating the influence of proximity-
drastically different in the presence of an in-plane magnetic induced SOC in WSe2/tBLG samples. When carrier density is
field B||: changing D induces a magnetization reversal, controlled with only the bottom gate electrode (28), doping
which is evidenced by hysteresis loops with opposite signs tBLG with electron also gives rise to a D-field that pulls elec-
in Rxy (Fig. 3D and fig. S12A) (16). The D-induced reversal is trons away from the WSe2 crystal (see fig. S1) (16). This re-
First release: 6 January 2022 science.org (Page numbers not final at time of first release) 3
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First release: 6 January 2022 science.org (Page numbers not final at time of first release) 5Fig. 1. Emerging ferromagnetic order from the tBLG/WSe2 interface. (A) Schematic of the heterostructure consisting of a tBLG/WSe2 interface, which is doubly encapsulated with hBN and graphite. (B) Calculated dispersion of moiré bands for a single valley. λI and λR indicate the strength of Ising and Rashba SOC, respectively in units of meV. Red and blue color denote the out-of-plane component of spin moment of each band. Chern number is expected to be zero for all energy bands in the absence of λR (left and middle panel). The combination of strong Ising and Rashba SOC gives rise to nonzero valley Chern number. The value of λI and λR are taken from previous measurements of proximity induced SOC (27, 44). (C) Longitudinal and transverse resistance, Rxx and Rxy, as carrier density ntBLG is swept back and forth. Carrier density ntBLG and moiré filling ν are denoted as the bottom and top axis, respectively. The measurement is performed at B = 10 mT, T = 20 mK and D = 252 mV/nm. (D and E) Rxy measured at (D) ntBLG = 0.55 × 1012 cm–2 near ν = 1 and (E) ntBLG = 1.22 × 1012 cm–2 near ν = 2, as B⊥ is swept back and forth. These measurements are performed at D = 0. The hysteresis loop disappears at high temperature. Right panels: schematic band structure at (D) ν = +1 and (E) ν = +2. For (E), the two lowest bands feature the same valley index and have nonzero Chern numbers of C = –3 and +1. As a result, the ground state at ν = 2 is valley polarized with net Chern number Cnet = –2. First release: 6 January 2022 science.org (Page numbers not final at time of first release) 6
Fig. 2. Controlling magnetic order using an in-plane magnetic field. (A) Schematic showing the effect of SOC, which couples the in-plane component of spin, ±S||, with the out-of-plane component of valley, τz = K and K´. (B and C) Rxy as a function of in-plane B field, which is aligned within 0.5° of the tBLG/WSe2 interface. Traces and retraces are shown as blue and red solid lines, respectively. The measurement is performed at (B) ν = +1 and D = –167 mV/nm, (C) ν = +2 and D = 0. (D) The orientation of in-plane spin momentum over the MBZ for valley K, which is obtained by diagonalizing the single particle Hamiltonian [Fig. 1B, see (16) for a more detailed discussion]. Here we show the first conduction band with valley index K above the neutrality. In the presence of C3 symmetry, SK averages to zero (left panel), whereas a uniaxial strain breaks C3, resulting in nonzero SK (right panel). First release: 6 January 2022 science.org (Page numbers not final at time of first release) 7
Fig. 3. Displacement-field dependence. (A) Schematic demonstrating the effect of electric displacement D on layer
polarization. (B) B-induced hysteresis loops of Rxy measured at different D with B-field aligned perpendicular to the 2D
⊥
layers. (C) Out-of-plane coercive fields B↑ and B↓⊥ for both ν = +1 and +2 as a function of D. (D) Same as (B), but for B-
field parallel to the 2D layers. (E) Same as (C), but for in-plane coercive fields B↑ and B↓ . B↑ (B↓) is defined as the value
of B where the sign of Rxy switches from negative to positive (positive to negative), whereas the superscript denotes the
orientation of the B-field (See fig. S11) (16). Both in-plane and out-of-plane magnetic hysteresis behaviors for ν = +1 are
measured at T = 20 mK. At ν = +2, out-of-plane magnetic hysteresis is measured at T = 20 mK, whereas in-plane
hysteresis is measured at T ≤ 3 K (see fig. S11 for more details) (16). (F) ⟨S||⟩ calculated from the 4-band model as a
function of D for the two lowest energy bands, band 1 and 2 for the valley K. Both bands feature large ⟨S||⟩ that are mostly
independent of D, shown as red and pink traces, respectively. The combination of band 1 and 2 yields a nonzero, albeit
small, ⟨S||⟩, which changes sign around D = 0.
First release: 6 January 2022 science.org (Page numbers not final at time of first release) 8Fig. 4. Isospin order and the absence of superconductivity. (A) Rxx as a function of moiré filling ν and in-plane magnetic field B|| measured at T = 20 mK. Carrier density is controlled by sweeping bottom gate voltage while top gate voltage is kept at zero. (B) Renormalized Hall density, νH – ν0, expressed in electrons per superlattice unit cell, measured at different B|| and B⊥ for D = 252 mV/nm. Circles in (A) denote the phase boundary between symmetry breaking isospin ferromagnets (IF2 and IF3) and an isospin unpolarized state (IU), which are defined as the peak position in d(νH – ν0)/dν. The expected Hall density steps of tBLG without SOC are shown in light blue for each panel in (B) (38). (C) Longitudinal resistance Rxx as a function of temperature and moiré filling measured at B = 0 and D = 252 mV/nm. (D) R – T line trace extracted from (C) along the vertical dashed lines. Inset: Rxx increases slightly with increasing B⊥, but decreases with B||, displaying no clear indication of Zeeman induced Cooper pair breaking. First release: 6 January 2022 science.org (Page numbers not final at time of first release) 9
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