this behavior to the presence of bulk con-
duction channels parallel to the chiral edge
conduction, which could result from the pres-
ence of magnetic domain walls ( 8 ) or sample
disorder. A fully developed Chern insulator
state could be realized in a sample with lower
disorder or stronger SOC, hence a larger en-
ergy gap.
To better understand the influence of
proximity-induced SOC, we note that the in-
troduction of SOC adds an extra term to the
Hamiltonian:
hSOCð Þ¼k1
2lItzszþ1
2lRðtzsxsy sysxÞð 1 Þwheret,s, andsdenote the Pauli matrix for
valley, sublattice, and spin at each momentum
k, andlIandlRrepresent the Ising and Rashba
SOC coefficients, respectively (eqs. S3 to S7)
( 16 ).TheIsingSOClocksthevalleymomenttz
with the spin momentsz, whereas the Rashba
termlRlocks the in-plane spinsx,sywith the
sublatticesx,sy(the locking depends on the
valleytz). For tBLG without SOC, there is aC 2
symmetry defined asC 2 :txsxand time-reversal
symmetry defined asT:itxsyK, whereKis the
complex conjugation. Both the Ising and Rashba
SOC break theC 2 symmetry while preserving
the time reversal. Therefore,C 2 Tis broken in
the presence of proximity-induced SOC.
The combination of Rashba SOC and time-
reversal symmetry gives rise to a valley-
contrasting spin texture within the mini–
Brillouin zone (MBZ) for each band: For any
momentumk, spin for the two valleys points
in opposite directions,sK(k)=–sK ́(–k). The
average value of the in-plane spin moment
of each band is obtained by integrating over
the MBZ for valleyK(K ́),S∥K;K′DE
¼
1
NXk∈MBZs∥K;K′ðÞkDE
ð 2 ÞwhereNis the system size. We note that a
brokenC 3 rotation symmetry could lead to
nonzerohSK∥iandhSK∥′i. In this scenario, time-
reversal symmetry is preserved by the valley-
contrasting spin texturehS
∥
Ki ¼ hS∥
K′i. On
the other hand, an orbital ferromagnetic state
emerges because valley-polarized Chern bands
are occupied, spontaneously breaking time-
reversal symmetry. Most remarkably, the combi-
nation of SOC, valley polarization, and a broken
C 3 rotation symmetry allows an in-plane mag-
netic field to couple to the orbital magnetic
order through the in-plane Zeeman energy.This control of the out-of-plane magnetic
order using an in-plane magnetic field,B||, is
demonstrated in Fig. 2, B and C: AsB||is
swept back and forth,Rxyexhibits hysteretic
switching behavior at bothn= +1 and +2.
Using Hall resistance measurements, we show
that the out-of-plane component of theBfield
is negligible relative to the out-of-plane coercive
field, confirming that the magnetic order is
indeed controlled by the in-plane component
of theBfield,B||(fig. S4). At the same time,
direct coupling between the valley order and
B||is shown to be absent in tBLG samples
without SOC ( 31 ), which rules out the orbital
effect as a possible origin for the observed
B||dependence. These results suggest thatB⊥
andB||couple to the magnetic order through
different mechanisms:B⊥directly controls the
magnetic ground state through valley Zeeman
coupling,EvZ¼ gvB⊥tz, whereas the influence
ofB||arises from the combination of SOC and
spin Zeeman coupling,EZs¼ gstzB∥hS∥i.
Here,tzcorresponds to the valley polarization,
andgvandgsare the valley and spin gyromag-
netic ratio, respectively.
We note a few intriguing properties of the
B-induced hysteresis behavior in Figs. 1 and
2: (i) A large value ofB||stabilizes opposite
magnetic orders atn=+1and+2,asisevi-
denced byRxywith opposite signs (Fig. 2, B
and C). This indicates thathS||ipoints in
opposite directions forn= +1 and +2. On the
other hand, an out-of-planeBfield stabilizes
the same magnetic order at different fillings
(Fig. 1, D and E), providing further confirma-
tion thatn= +1 and +2 feature the same valley
index and the ground state at half-filling is
valley-polarized. (ii) Although the out-of-plane
coercive fields are similar, the in-plane coercive
field atn= 2 is much bigger than atn= 1. This
suggests that the average in-plane spin moment
hS||iis much smaller atn= 2, which is con-
sistent with a predominantly spin-unpolarized
ground state (fig. S9) ( 16 ). It is noteworthy that
the observed behavior atn= +2 does not rule
out alternative isospin configurations. Our re-
sults provide important constraints for future
theoretical work to examine such possibilities.
The effective control of in-planeBfield on
the magnetic order not only provides further
validation that the orbital ferromagnetic order
is stabilized by proximity-induced SOC, it also
reveals that a brokenC 3 rotational symmetry
gives rise to a nonzerohS||i. A spontaneously
brokenC 3 symmetry naturally derives from
the combination of strong SOC and nematic
charge order ( 32 – 34 ). Alternatively, a preferred
in-plane direction for spin could result from
a small amount of uniaxial strain in the moiré
lattice (Fig. 2D), which is known to be com-
mon for tBLG samples ( 35 , 36 ).
The proximity-induced SOC arises from
wave function overlap across the interface
( 22 ). The role of wave function overlap wasSCIENCEscience.org 28 JANUARY 2022•VOL 375 ISSUE 6579 439
K
K’
-S// S//
SOC
SOC
A
ν = 1-0.8 -0.4 0 0.4 0. 8-0.6-0.30.30.00.6R(kΩ)xyB total(T)B-1.0-0.50.00.51.0-6 -4 -2 0246R(kΩ)xyB total(T)Bν = 2BC
D‹S //‹= 0 ‹S //‹= 0/
Fig. 2. Control of 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,tz=K
andK′.(BandC)Rxyas a function of in-planeBfield, which is aligned within 0.5° of the tBLG/WSe 2 interface.
Traces and retraces are shown as blue and red solid lines, respectively. The measurement is performed
atn=+1andD=–167 mV/nm (B) and atn=+2andD= 0 (C). (D) The orientation of in-plane spin
momentum over the MBZ for valleyK, which is obtained by diagonalizing the single-particle Hamiltonian
(Fig. 1B) ( 16 ). Here we show the first conduction band with valley indexKabove the neutrality.
In the presence ofC 3 symmetry,hSK∥iaverages to zero (left), whereas a uniaxial strain breaksC 3 ,
resulting in nonzerohSK∥i(right).
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