Science - USA (2022-01-28)

recently demonstrated experimentally, as SOC
strength has been shown to depend on inter-
layer separation, which is tunable with hydro-
static pressure ( 37 ).Inthesamevein,weshow
that SOC strength can be controlled with a

perpendicular electric fieldD: Under a posi-

tiveD, charge carriers are polarized toward

the WSe 2 crystal, resulting in increased wave

function overlap and stronger SOC; contrari-

wise, under a negativeD, charge carriers are

polarized away from the WSe 2 crystal, resulting

in decreased wave function overlap and weaker

SOC (Fig. 3A) ( 22 , 37 ). Figure 3 demonstrates

the effect ofDby plotting the evolution of

B-induced hysteresis loops: With increasing

440 28 JANUARY 2022¥VOL 375 ISSUE 6579 science.orgSCIENCE

Fig. 3. Displacement-field dependence.(A) Sche-

matic demonstrating the effect of electric displacement

Don layer polarization. (B)B-induced hysteresis

loops ofRxymeasured at different values ofD,

withBfield aligned perpendicular to the 2D layers.

(C) Out-of-plane coercive fieldsB⊥↑andB⊥↓for both

n= +1 and +2 as a function ofD.(D) Same as (B), but

forBfield parallel to the 2D layers. (E) Same as

(C), but for in-plane coercive fieldsB∥↑andB∥↓.B↑and

B↓are defined as the value ofBwhere the sign

ofRxyswitches from negative to positive and from

positive to negative, respectively; the superscript

denotes the orientation of theBfield (fig. S11) ( 16 ).

Both in-plane and out-of-plane magnetic hysteresis

behaviors forn= +1 are measured atT= 20 mK.

Atn= +2, out-of-plane magnetic hysteresis is

measured atT= 20 mK, whereas in-plane hysteresis

is measured atT≤3 K (fig. S11) ( 16 ). (F)hS||i

calculated from the four-band model as a function of

Dfor the two lowest-energy bands 1 and 2 for the

valleyK. Both bands feature largehS||ithat are mostly

independent ofD, shown as red and pink traces,

respectively. The combination of bands 1 and 2 yields

a nonzero, albeit small,hS||i, which changes sign

aroundD= 0.

D< 0

D> 0

band 1+2 (ν=2)

band 2

band 1 (ν=1)

ST S

S//

-40 -20 0 20 40

-0.2

-0.1

0.0

0.1

0.2

X 20

D (mV/nm)

S

//

‹

‹

D E

AB C

F

-400 0 400

-200

-300

-100

0

200

300

100

D (mV/nm)

B

T (mT)

B

ν=1

ν=2

B

T

B

T

B

T

B

T

-400 0 400

-2

-4

0

2

4

D (mV/nm)

B

(T)//

B

B//

B//

B//

B//

ν=1

ν=2

ν = 2

-0.5

0.0

0.5

D = 84 mV/nm

-6 -4 -2 0 2 4 6

-0.5

0.0

0.5

B (T)//

D = -336 mV/nm

B//

B//

B//

B//

R

(k

Ω

)

xy

B

-200 0 200

-1

0

1

-1

0

1

D = 366 mV/nm

-336 mV/nm

B (mT)T

R

(k

Ω

)

xy

ν = 2

B

T

B

T

B

Fig. 4. Isospin order and the absence of

superconductivity.(A)Rxxas a function of

moiré fillingnand in-plane magnetic fieldB||

measured atT= 20 mK. Carrier density is

controlled by sweeping bottom-gate voltage while

top-gate voltage is kept at zero. Circles denote

the phase boundary between symmetry-breaking

isospin ferromagnets (IF 2 and IF 3 ) and an isospin-

unpolarized state (IU), which is defined as the peak

position ind(nH–n 0 )/dn.(B) Renormalized Hall

density,nH–n 0 , expressed in electrons per

superlattice unit cell, measured at different values

ofB||andB⊥forD= 252 mV/nm. The expected

Hall density steps of tBLG without SOC are shown in

light blue for each condition ( 38 ). (C) Longitudinal

resistanceRxxas a function of temperature and

moiré filling measured atB= 0 andD= 252 mV/nm.

(D)R-Tline trace extracted from (C) along the

vertical dashed lines. Inset:Rxxincreases slightly

with increasingB⊥but decreases withB||, displaying

no clear indication of Zeeman-induced Cooper

pair breaking.

C

D

A B

-3 -2 -1 0 1 2 3

2

4

6

ν

T

(K)

R (kxx Ω)^012

-3 -2 -1 0 1 2 3

0

2

4

6

ν

B

(T)//

R xx(kΩ)^025

IF 2 IF 3 IU IU IF 3

B = 0.2TT

B = 4T//

B = 0

ν = -2.42

R

(k

Ω

)

xx

T (K)

0246

0

2

4

6

0 10 20

0

4

8

ν

-2.68

-2.42

R

(k

Ω

)

xx

T (K)

-3 -2 -1 0 1 2 3

-2

0

2

-2

0

2

-2

0

2

ν

ν − νH

0

B = 0.4 TT B = 7T//

B = 0.4 TT B = 0//

B = 0.23 TT B = 4T//

RESEARCH | REPORTS