of 13
Tracing out Correlated Chern Insulators in Magic Angle
Twisted Bilayer Graphene
Youngjoon Choi
1
,
2
,
3
, Hyunjin Kim
1
,
2
,
3
, Yang Peng
4
,
3
, Alex Thomson
2
,
3
,
5
, Cyprian Lewandowski
2
,
3
,
5
,
Robert Polski
1
,
2
, Yiran Zhang
1
,
2
,
3
, Harpreet Singh Arora
1
,
2
, Kenji Watanabe
6
, Takashi Taniguchi
6
,
Jason Alicea
2
,
3
,
5
, Stevan Nadj-Perge
1
,
2
1
T. J. Watson Laboratory of Applied Physics, California Institute of Technology, 1200 East Cali-
fornia Boulevard, Pasadena, California 91125, USA
2
Institute for Quantum Information and Matter, California Institute of Technology, Pasadena, Cal-
ifornia 91125, USA
3
Department of Physics, California Institute of Technology, Pasadena, California 91125, USA
4
Department of Physics and Astronomy, California State University, Northridge, California 91330,
USA
5
Walter Burke Institute for Theoretical Physics, California Institute of Technology, Pasadena, Cal-
ifornia 91125, USA
6
National Institute for Materials Science, Namiki 1-1, Tsukuba, Ibaraki 305 0044, Japan
*These authors contributed equally to this work
Correspondence: s.nadj-perge@caltech.edu
Magic-angle twisted bilayer graphene (MATBG) exhibits a range of correlated phenomena
that originate from strong electron-electron interactions. These interactions make the Fermi
surface highly susceptible to reconstruction when
±
1
,
±
2
,
±
3
electrons occupy each moir
́
e
unit cell and lead to the formation of correlated insulating, superconducting and ferromag-
netic phases
1–4
. While some phases have been shown to carry a non-zero Chern number
5, 6
,
the local microscopic properties and topological character of many other phases remain elu-
sive. Here we introduce a set of novel techniques hinging on scanning tunneling microscopy
(STM) to map out topological phases in MATBG that emerge in finite magnetic field. By
following the evolution of the local density of states (LDOS) at the Fermi level with electro-
static doping and magnetic field, we visualize a local Landau fan diagram that enables us
to directly assign Chern numbers to all observed phases. We uncover the existence of six
topological phases emanating from integer fillings in finite fields and whose origin relates to
a cascade of symmetry-breaking transitions driven by correlations
7, 8
. The spatially resolved
and electron-density-tuned LDOS maps further reveal that these topological phases can form
only in a small range of twist angles around the magic-angle value. Both the microscopic
origin and extreme sensitivity to twist angle differentiate these topological phases from the
Landau levels observed near charge neutrality. Moreover, we observe that even the charge-
neutrality Landau spectrum taken at low fields is considerably modified by interactions and
exhibits prominent electron-hole asymmetry and an unexpected splitting between zero Lan-
dau levels that can be as large as
3
5
meV, providing new insights into the structure of
flat bands. Our results show how strong electronic interactions affect the band structure of
MATBG and lead to the formation of correlation-enabled topological phases.
1
arXiv:2008.11746v1 [cond-mat.str-el] 26 Aug 2020
When two graphene sheets are rotationally misaligned (twisted), the interlayer coupling leads
to the formation of an effective triangular moir
́
e lattice with spatial periodicity
L
m
=
a/
(2 sin(
θ/
2))
set by the twist angle
θ
and graphene lattice constant
a
= 0
.
246
nm
9, 10
. At small twist angles, the
moir
́
e period is hundreds of times larger than the inter-atomic distance, and the electronic bands
of the bilayer, by virtue of the moir
́
e interlayer coupling, are substantially modified. Near the
magic angle (
θ
M
1
.
1
°), the electronic structure consists two maximally flat bands that give rise
to strongly correlated physics and are separated by
20
30
meV gaps from the more disper-
sive remote bands. In addition to modifying the band structure, the periodic moir
́
e potential also
affects the band topology. As shown in experiments where graphene and hexagonal boron nitride
(hBN) are aligned
11–14
, the added spatial periodicity combined with the orbital motion of electrons
in high (
15
T) magnetic fields generates Chern insulating phases characteristic of Hofstadter’s
spectrum
15
. While similar effects are expected in MATBG
16
, the impact of strong correlations on
Hofstadter physics and the new phases that may emerge in finite magnetic fields is to a large degree
unexplored.
Figure 1a shows a schematic of our experiment. MATBG is placed on a structure consisting
of a tungsten diselenide (WSe
2
) monolayer, thick hBN dielectric layer and graphite gate (see Ex-
tended Data Fig. 1 and Methods, section 1, for fabrication details). We use monolayer WSe
2
as an
immediate substrate for MATBG since previous transport studies
17
indicate that WSe
2
improves
the sample quality and does not change the magic-angle condition. As in previous STM studies
18–20
twist angle can be directly determined by measuring the distance between neighbouring AA sites,
where the density of states is highly localized, in topographic data (Fig. 1b). Figure 1c shows the
tunneling conductance (
dI
/
dV
) corresponding to the local density of states (LDOS) taken at an
AB site at zero magnetic field, as a function of sample bias (
V
Bias
) and gate voltage (
V
Gate
) that
tunes electrostatic doping. At
V
Gate
>
+5
V, the two LDOS peaks originating from the Van Hove
singularities (VHS) of the flat bands are below the Fermi energy (
E
F
, corresponding to
V
Bias
= 0
mV), indicating that flat bands are completely filled with electrons. As
V
Gate
is reduced, the first
VHS corresponding to the conduction flat band crosses
E
F
several times, resetting its position
around gate voltages corresponding to occupations of
ν
= 1
,
2
,
and 3
electrons per moir
́
e unit cell
(see Methods, section 4, for assigning of filling factor
ν
to
V
Gate
). A similar cascade of transitions
was previously observed in STM measurements of MATBG placed directly on hBN
8
; our observed
cascade demonstrates that the addition of WSe
2
significantly changes neither the spectrum nor the
cascade mechanics. Around the charge neutrality point (CNP;
ν
= 0
, corresponding to
V
Gate
0
.
1
V) the splitting between two VHS is maximized by interactions as discussed previously
19–22
.
When a perpendicular magnetic field is applied, the overall spectrum changes as Landau
levels (LLs) develop around charge neutrality and
E
F
. In addition, the onsets of the cascade transi-
tions shift away from the CNP (Fig. 1d,e, marked by black arrows) and are accompanied by a very
low LDOS at the corresponding Fermi energies as well as by nearly horizontal resonance peaks,
indicating the presence of gapped states
23
. We first focus on the LLs. A linecut close to the CNP
at
B = 8
T shows four well-resolved peaks (Fig. 1f); the inner two correspond to LLs. The large
intensity of VHS peaks on the AA sites obscures some LL-related features (see Extended Data
2
Fig. 2 for spectrum on an AA site), so we instead study the AB sites to maximize visibility.
The phenomenological ten-band model
24
with parameters chosen to semi-quantitatively match
the data (Fig. 1g) suggests that the inner peaks are zero LLs (zLLs) originating from Dirac points
while the outer peaks (VHS) form from LLs descending from other less-dispersive parts of the
band structure and thus can not be individually resolved (see Methods, section 7, for discussion
of the modeling). Importantly, both zLLs and VHS are expected to carry non-zero Chern number
(
+1
and
1
respectively), and to be four-fold spin-valley degenerate. This assignment of the Chern
numbers takes into account the energetic splitting of the two Dirac cones that is experimentally ob-
served (see discussion on Fig. 4). Note that this splitting naturally accounts for the reduction of
an eight- to four-fold degeneracy observed in previous transport MATBG experiments. With this
interpretation in mind, and as discussed in the remainder of the paper, we attribute the shifting of
the zero-field cascades (black arrows in Fig. 1d and e) and the accompanying gaps to the formation
of Chern insulating phases at high fields, enabled by correlations.
The evolution of LLs with magnetic field in MATBG has so far been studied using trans-
port measurements that can only provide information about electronic structure close to the Fermi
energy
1, 3, 4, 25
. As a starting point, we relate these experiments to our STM findings by utilizing
a novel approach that enables us to measure a full Landau fan diagram via LDOS. This LDOS
Landau fan is taken by measuring the tunneling conductance without feedback, with tip-sample
bias voltage (
V
Bias
) fixed at
0
mV, such that the STM probes the system at the Fermi energy as it is
tuned by changing the electron density (though
V
Gate
)
20
and magnetic field. The resulting signal
is directly proportional to the LDOS and thus it is suppressed in certain regions of carrier densities
where gaps develop in the energy spectrum (Fig. 2a and b).
Our LDOS Landau fan measurements (Fig. 2c), taken at one particular AB point on the
sample, reproduce many features previously established by magneto-transport in MATBG
1, 3, 4, 17
despite the fact that here we record a fundamentally different quantity, LDOS, instead of the typ-
ically measured longitudinal resistance. This approach further enables comparing the value of
the twist angle extracted from the Landau fan with the corresponding local twist angle seen in
topography (agreement is within 0.01°). However, in addition to previously identified Landau
levels, we also observe a strong suppression of the LDOS in certain regions, indicating the for-
mation of insulating phases emanating from
ν
=
±
1
,
±
2
, and
+3
that abruptly appear in finite
fields (
B
>
3
T for
ν >
0
and
B
>
6
T for
ν <
0
; red dashed lines in Fig. 2c)
26–28
. The Chern
numbers corresponding to these phases,
C
=
±
3
,
±
2
, and
+1
, respectively, are assigned directly
from the observed slopes using the gap positions as a function of
ν
and the Diophantine equation
29
,
ν
(B) =
C
×
A
m
×
B
0
+
ν
(B = 0)
(see Extended Data Fig. 6 for data showing also
C
=
1
state). Here
A
m
is the moir
́
e unit cell area,
φ
0
is the flux quantum, and
ν
(B = 0)
denotes the filling
(
±
1
,
±
2
,
+3
) from which the phases emanate.
To better understand the formation of the observed Chern insulating phases, we now turn to
3
spectroscopic measurements at fixed magnetic field, focusing on the conduction flat band (Fig. 2d).
As the gate voltage is increased, starting from
V
Gate
2
V, the four-fold degenerate VHS ap-
proaches the Fermi level. Just before crossing (
V
Gate
3
V), a series of small gaps open up
accompanied by set of nearly horizontal resonance peaks. These resonances are attributed to quan-
tum dot formation in the sample, indicating formation of a fully gapped insulating phase around
the tip (see Methods, section 5, for more details). As we increase gate voltage further, part of the
VHS is abruptly pushed up in energy (seen at higher
V
Bias
in Fig. 2d) reducing its spectral weight
as the number of unfilled bands (each featuring one VHS) decreases. Similar transitions are ob-
served at around
V
Gate
4
V and
V
Gate
5
.
1
V with the spectral weight reducing after each
transition. This sequence of transitions is an analogue of the
B
= 0
cascade. Most importantly
however, the onsets of these finite-field cascade transitions are now shifted to new
V
Gate
positions,
and hence fillings, that trail the location of nearby Chern insulating phases. This is demonstrated
in Extended Data Fig. 3, where as magnetic field changes, positions of the Chern insulating phases
shift, and the onsets of the cascade shift accordingly for the conduction band VHS. For the valence
band VHS, the onsets of the cascade are hardly affected until
B = 6
T where the Chern insulating
phases start to form (see Methods, section 6, for additional discussion).
These observations can be explained within the Hofstadter picture described in Fig. 1f and
g, where each zero LL and VHS respectively carry total Chern number
C
= +4
and
C
=
4
(the factor 4 reflects spin-valley degeneracy). Figure 2e schematically illustrates the evolution of
VHS upon changing
ν
. When the conduction band VHS is empty and all the other bands are
filled, the total Chern number of the occupied bands is
C
= +4
(
4
from the valence band VHS
combined with
+4
×
2
from CNP LLs). Consequently, the gap between the LL and VHS will
follow the slope corresponding to
C
= +4
in the LDOS Landau fan. As
E
F
increases, all four-
fold degenerate conduction bands start to become populated equally, until
E
F
reaches the VHS
for the first time. At this point, interactions underlying the cascade
7
shift all carriers to one band
only (seen as only one of the four bands crossing the Fermi energy) and then the other three bands
become unfilled and hence are pushed to higher energies. Since the added band carries
C
=
1
,
the total Chern number is now
C
= +3
, and the next corresponding gap in the LDOS Landau fan
follows an accordingly reduced slope. The sequence is repeated again, creating a cascade.
To verify the role of correlations on the observed Chern insulating phases, we extended the
technique introduced in Fig. 2a-c to directly visualize the evolution of the Chern insulating phases
with the twist angle. In areas where twist angle is slowly evolving over hundreds of nanometers
(many moir
́
e periods), the local angle is well-defined and the strain level is low (
<
0.3
%
) (Fig. 3a).
By measuring the LDOS at the Fermi energy against
V
Gate
and spatial position, we image the
evolution of the Chern insulating phases as well as LLs from the CNP with twist angle at finite
magnetic field (Fig. 3b). Gaps between LLs originating from the CNP and corresponding to LL
filling factor
ν
LL
=
±
4
,
±
2
,
0
appear at fixed
V
Gate
and do not change with the twist angle, as they
depend only on electron density. The Chern insulating phases, however, move outward from the
CNP with increasing angle as expected from the change of the moir
́
e unit cell size. Also, while
the gaps between LLs from charge neutrality persist at all angles (0.98-1.3° in our experiment),
4
the Chern insulating phases are only observed in a certain narrow range around the magic angle
(Fig. 3b). For example, the
C
=
3
state emanating from
ν
=
1
is present only for
1
.
02
°
<
θ <
1
.
14
° at
B = 7
T while the
C
=
2
state is stable for a larger angle range. Moreover, as
the magnetic field is lowered, the angle range where the gaps are observed reduces (Fig. 3c-e).
Figure 3f shows the onset in field where we observe the
C
=
3
and
C
=
2
insulating phases,
summarizing their angle sensitivity.
The observed evolution of Chern insulators with twist angle reflects a competition between
Coulomb interactions and kinetic energy, similarly to the cascade at
B
= 0
T. Here the charac-
teristic scale of electron-electron interactions
U
is approximately set by
U
e
2
/
4
π
L
m
(with
e
,

being the electron charge and the dielectric constant, respectively) and increases with increasing
twist angle. On the other hand, the typical kinetic energy scale, taken to be the bandwidth
W
of
the
C
=
1
band that forms the VHS peak seen in measurements, shows non-monotonic behavior
with twist angle. The bandwidth
W
is minimal at the magic angle and further narrows with in-
creasing magnetic field. We thus expect that magnetic-field-induced Chern insulating phases will
occur most prominently close to the magic angle, with larger fields required for their onset away
from the magic angle as seen in the phase diagram of Fig. 3f. Indeed theoretical estimates of the
ratio
U/W
as a function of magnetic field based on a continuum model uphold this reasoning and
reproduce the trends observed in the experimental phase diagram (see Methods, section 7). Since
the correlated Chern insulators occur only when
U/W
is large, their existence serves as an alter-
native measure of correlation strength. We note that there is a general asymmetry of angle range
between the electron and hole side (Fig. 3b). The
C
=
2
phase emanating from
ν
=
2
on the
hole side starts at 1.19° and ends at
1
.
01
°, while the
C
= 2
state emanating from
ν
= 2
on the
electron side starts at
1
.
15
° and continues to
0
.
98
°. This observation indicates that the ‘magic an-
gle’ condition where correlations are strongest differs between the conduction and valence bands,
since the system lacks particle-hole symmetry, and highlights the sensitivity of MATBG physics
to tiny twist-angle changes.
Aside from their implications for Chern insulating phases, LLs near charge neutrality ob-
served at low magnetic fields (
B
<
2
T) in Fig. 2c also shed light on the MATBG band struc-
ture. Some of the early observations in MATBG remain poorly understood—e.g., the appearance
of four-fold degenerate LLs around charge neutrality
1
instead of eight-fold as expected from the
presence of eight degenerate Dirac cones of the two stacked monolayers, and anomalously large
bandwidth (
40
meV) of the flat band
19–22
deviating from the
5
10
meV widths expected from
continuum models
10
. This is largely due to difficulties in band structure calculations that incorpo-
rate all relevant effects such as electronic correlations
30–32
, strain
33
, and atomic reconstruction
34
.
In particular, several mechanisms were proposed to explain the origin of four-fold LLs formed at
charge neutrality
33, 35, 36
but to date there is no general consensus.
The evolution of LLs from the CNP at
θ
= 1.04° at low fields appears in Fig. 4a-f. We name
the two LLs closest to the CNP as 0+ and 0- (green and red lines in Fig. 4b,d,f; they develop into
the two zero LLs in Fig. 1f at high fields), and then we label the remaining levels sequentially. As
5
already revealed in Fig. 2c, the four-fold degeneracy of each level can also be seen here by the
equal separation between LLs in
V
Gate
at the Fermi level (corresponding
ν
LL
marked in Fig. 4f),
indicating no hidden LLs from poor resolution or layer sensitivity. Having identified LLs, we now
obtain the LL energy spectrum from linecuts fixing
V
Gate
at various magnetic fields (Fig. 4g-i).
The relative energy separation between LLs changes with B, and different LLs are more visible
for different electron densities. When
E
F
resides in between LLs, the energy separation between
those LLs becomes larger due to an exchange interaction (Fig. 4h, between
0+
and
0
). By
avoiding such interaction-magnified regions and cumulating their relative separations, we compile
the single-particle energy spectrum shown in Fig. 4j (error bars come from small changes in the
separation measured at different electron densities). Note that for a slightly different strain and
angle, fine details of the spectrum become different from point to point in the sample, but the
energy separations are of similar values.
The observed LL spectrum is consistent with a scenario wherein the two moir
́
e Brillouin zone
Dirac cones are shifted in energy, either by strain (0.3
%
in this area)
33, 36
or layer polarization due to
a displacement field
37
in our measurements (see Methods, section 9 for more detailed discussion).
In this scenario,
0+
and
0
(Fig. 4j) come form the two Dirac points. The LL spectrum can be
compared to expectations from a Dirac-like dispersion
E
n
=
sgn
(
n
)
v
D
2
e
~
|
n
|
B
. The observed
energy separations (e.g.,
8
meV for
0
to
1
and
5
meV for
1
to
2
at B
= 2
T) far exceed
those predicted by a non-interacting continuum model. In particular, our measurements yield a
Dirac velocity
v
D
1
.
5
2
×
10
5
m/s that is an order of magnitude larger than the continuum-
model prediction (
10
4
m/s), suggesting strong interactions near charge neutrality. Electron-hole
asymmetry is also clearly present, as the energy differences between the first few LLs on the hole
side are comparably larger than their electron-side counterparts. Moreover, upon doping away
from the CNP, LLs move together toward the VHS while the separation between them is hardly
affected (Fig. 4a,c,e). This signals that the shape of dispersive pockets within the flat bands do
not change significantly as flatter parts of the bands are deformed due to interactions
30, 31
. Taken
together, these observations put strict restrictions on the overall band structure of the MATBG and
provide guidence for further theoretical modeling. Looking ahead, we anticipate that the novel
STM spectroscopic techniques developed here will enable the exploration of other exotic phases
in MATBG and related moir
́
e systems.
Note: In the course of preparation of this manuscript we became aware of the related work
38
.
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Strongly Correlated Chern Insulators in Magic-Angle Twisted Bilayer
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Acknowledgments:
We acknowledge discussions with Andrea Young, Gil Refael and Soudabeh
Mashhadi. The device nanofabrication was performed at the Kavli Nanoscience Institute (KNI) at
Caltech.
Funding:
This work was supported by NSF through grants DMR-2005129 and DMR-
1723367 and by the Army Research Office under Grant Award W911NF-17-1-0323. Part of the
initial STM characterization has been supported by CAREER DMR-1753306. Nanofabrication
performed by Y.Z. has been supported by DOE-QIS program (DE-SC0019166). J.A. and S.N.-
P. also acknowledge the support of IQIM (an NSF Physics Frontiers Center with support of the
Gordon and Betty Moore Foundation through Grant GBMF1250). A.T., C.L., and J.A. are grateful
for support from the Walter Burke Institute for Theoretical Physics at Caltech and the Gordon
and Betty Moore Foundation’s EPiQS Initiative, Grant GBMF8682. Y.C. and H.K. acknowledge
support from the Kwanjeong fellowship.
Author Contribution:
Y.C. and H.K. fabricated samples with the help of with the help of R.P.,
Y.Z., and H.A., and performed STM measurements. Y.C., H.K., and S.N.-P. analyzed the data. Y.P.
and A.T. implemented models. Y.P., A.T., C.L., provided theoretical analysis supervised by J.A. .
K.W. and T.T. provided materials (hBN). S.N-P supervised the project. Y.C, H.K, Y.P., A.T., C.L.,
J.A. and S.N-P wrote the manuscript.
Data availability:
The data that support the findings of this study are available from the corre-
sponding authors on reasonable request.
9
Figure 1
|
Spectroscopy of MATBG with magnetic field at
2
K. a
, Schematic experimental
setup. MATBG is placed on a monolayer WSe
2
and supported by hBN. A graphite gate resides
underneath. Inset shows details of WSe
2
and graphene crystal structure.
b
, Typical topography
showing a moir
́
e pattern at the magic angle (
V
Bias
=
400
mV,
I = 20
pA).
c-e
, Point spectra on an
AB site at
θ
= 1
.
03
° as a function of
V
Gate
for three different magnetic fields applied perpendicular
to the sample. (
c
),
B = 0
T, the evolution of two peaks in density of states originating from flat-
band VHSs. As each of the peaks crosses the Fermi energy, it creates a cascade of transitions,
appearing here as splitting of the VHSs into multiple branches close to integer filling factors
ν
.
(
d, e
),
B = 4
T and
B = 8
T, respectively. Landau levels form around charge neutrality (
ν
= 0).
Black arrows indicate the newly formed gaps that appear in magnetic field and are visible as a
suppression of
dI/dV
conductance.
f
, Conductance linecuts at
V
Gate
= 0
.
6
V and
B = 8
T on AA
and AB sites. Landau levels are more visible on the AB site due to reduced VHS weight.
g
, Energy
spectrum calculated from the continuum model with parameters chosen such that the relative peak
positions match the experimental data in (
f
). Most LLs merge into the electron and hole VHSs that
each carry Chern number
C
=
1
, while two isolated LLs at charge neutrality remain around zero
energy and carry
C
= +1
.
10
Figure 2
|
Local Density of States (LDOS) Landau fan and gaps induced by Chern insulating
phases. a
, Principle of acquiring LDOS Landau fan. Conductance
d
I
/d
V
LDOS is measured
while sweeping
V
Gate
to change carrier density at fixed
V
Bias
= 0
mV.
b
, Example linecut taken
at
B = 8
T and -6 V
<
V
Gate
<
6
V. Insulating phases appear as LDOS dips. Chern insulating
phases and LLs are indicated by grey regions and blue vertical lines, respectively. The position of
these lines is obtained from the slopes in (
c
).
c
, LDOS Landau Fan diagram at an AB site for
θ
=
1.02°. LDOS data, taken by sweeping
V
Gate
, is normalized by an average LDOS value for each
magnetic field (separately for flat and remote bands). Black solid (blue dashed) lines indicate gaps
between LLs originating from the CNP (half-filling). Purple lines on the yellow background show
the LL gaps in remote (dispersive) bands. Magnetic-field-activated correlated Chern insulator gaps
are marked by red dashed lines. The signal in the flat band region is multiplied by five to enhance
visibility in relation to the remote bands.
d
, Point spectra on the same AB point as in (
c
) as a
function of
V
Gate
taken at
B = 7
T, highlighting the crossing of the electron-side VHS in magnetic
field. Chern insulator gaps corresponding to
C
= 3
,
C
= 2
and
C
= 1
are indicated by white
arrows. The gaps are accompanied by resonances (horizontal features) originating from quantum
dots formed within the insulating bulk (see Methods section 5). (
e
), Schematic of the cascade
in magnetic field. Each time the VHS crosses the Fermi energy, a Chern gap appears and the
corresponding band Chern number changes.
11
Figure 3
|
Angle and magnetic-field dependence of the LDOS and identification of Chern
insulators. a
, Topography of a 40nm x 520nm area where twist angle gradually changes from
1
.
3
°
(top yellow line) to
0
.
99
° (bottom yellow line); tunneling conditions are
V
Bias
= 100
mV,
I = 20
pA.
b
, Conductance at
V
Bias
= 0
mV and
B = 7
T taken at different spatial points—characterized
by different local twist angles
θ
—and for different
V
Gate
. Measurements were spatially averaged
over the horizontal direction within the green dashed box shown in (
a
). Chern insulating gaps
develop only in a certain range of angles near the magic angle. The
V
Gate
positions of integer filling
factors shift as a function of a local twist angle
θ
due to the change of the Chern insulating positions
as they depend on the moir
́
e unit cell area
A
m
=
3
L
2
m
/
2
. Landau level gaps originating from the
CNP persist for the whole range of angles and do not shift.
c-e
, Magnetic-field dependence of the
LDOS for hole doping (-5 V
<
V
Gate
<
0 V) at
B = 6
T (
c
),
B = 5
T(
d
), and
B = 4
T (
e
). The
Chern insulating gaps disappear as the magnetic field is lowered.
f
, Reconstructed phase diagram
showing the range of fields and angles where
C
=
2
and
C
=
3
Chern insulators are observed.
In the shaded regions the suppression of LDOS due to Chern insulators is prominent. See also
Extended Data Fig. 6 for high-resolution data resolving the
C
=
1
state.
12
Figure 4
|
Evolution of Landau levels from charge neutrality
.
a-f
, Point spectra as a function
of
V
Gate
(
a,c,e
) and schematics tracking the evolution of Landau levels (
b,d,f
) for
B = 0
.
5T
(
a,b
),
B = 1T
(
c,d
), and
B = 2
.
5T
(
e,f
). At the Fermi energy, each Landau level is equally separated in
electron density (
V
Gate
) as marked by the Landau-level filling
ν
LL
, indicating four-fold degeneracy
of each level.
g-i
, Linecuts at
V
Gate
=
1
V (
g
),
V
Gate
= 0
.
2
V (
h
), and
V
Gate
= 1
.
1
V (
i
)
illustrating the LL spectrum change with magnetic field. A smooth background was subtracted
to enhance visibility. The indicated LLs were identified in (
b,d,f
).
j
, Combined energy spectrum
for LLs around charge neutrality. Zero energy is set as the midpoint between
0+
and
0
levels at
B = 0
.
5
T.
13