of the moiré lattice that favors the lower-
energy ( 16 ) AtA configuration. Close exam-
ination of Fig. 1D reveals that this moiré
lattice reconstruction (MLR) produces a peri-
odic warping of the AAA site positions to en-
force AtA stacking over the entire sample area.
Theobservedwarpingofthemoirélatticecan
be understood at the atomic scale as arising
from variations in the local twist angle (qx)
and strain (ex) of the individual graphene
layers.qx∼a=
ffiffiffiffiffiffi
Axp
is plotted in Fig. 1, E and F,
for two nearby sample regions, whereAxis the
area of the moiré unit cell centered on position
x. For small-angle mismatchdq, the system
segregates into highly uniform triangular do-
mains (Fig. 1, E and F, blue areas) bounded by
sharp point-like irregularities in the local twist
angle (Fig. 1, E and F, red areas).
ForL≳30nm, the average twist angle inter-
nal to each triangular domain (qI) saturates to
a common value of ~1.5° that is independent
of the moiré of moiré wavelength (Fig. 1G).
This implies that the MLR not only enforces
AtA stacking but also tends to lock the lattice
to a constant local twist angle, even asqTMor
qBMis varied. To shed light on this behavior,
we performed structural relaxation calcula-
tions for TTG at a range of interlayer twists.
The results (fig. S10) indicate that fordq≲ 0 :5°,
qIlocks to the smaller ofqTMandqBMbecause
a stronger interlayer coupling exists at a lower
twist angle interface. The additional twist
angle degree of freedom therefore enables
TTG to locally conform to the mirror symmet-
ric magic angle structure while“absorbing”
twist angle inhomogeneity at the larger moiré
of moiré length scale. The effect of the MLR on
the local electronic structure is profound, as
evidenced by the Fermi-level local density of
states (LDOS) map (Fig. 1D, inset), which shows
large modulations in the tunneling conductiv-
ity across regions of the MLR. It was therefore
necessary, in considering the potentialities of
TTG as a platform for correlated phases, toanalyze the electronic structure on both the
sub- and supra-Llength scales.
In Fig. 2A, we present STM topography of a
250-nm^2 area, which is part of an even larger
region with only a single-moiré wavelength
corresponding to a twist angle ofq= 1.55°.
The extreme degree of homogeneity in this
area is conveyed by the local twist angle his-
togram (Fig. 2A, inset), showing a standard
deviation of 0.03° over the entire field of view.
This indicates a twist angle mismatch ofdq<
0.05°, which provided us with the opportu-
nity to study a single domain of the MLR as
well as to investigate the spectroscopic proper-
ties of a large patch of magic-angle TTG that
approaches the size of a transport device.
The high energy resolution of STS permitted
us to directly probe the structure of the flat
bands. A series of STS measurements acquired
at 7.2 K on a single AAA site is shown in Fig. 2B
for a range of voltages (Vg) applied to the
graphite back gate (additional twist angles are194 8 APRIL 2022•VOL 376 ISSUE 6589 science.orgSCIENCE
AtB AtAAMLR
TM AA site
BM AA siteBT
M
BNorm. HeightHiLoABBAAB
BAC AAAABA
BABD1.5 ̊1.7 ̊1.9 ̊R230 40 50 60 70 80 90
Λ (nm)1.451.501.551.60θ( ̊I)R1
R2θBMθTMMC MR1
EFGΛ
Lo LDOS (EF) Hiλ
0 pm250 pmFig. 1. STM on three twisted graphene layers.(A) Illustration of the moiré of
moiré pattern in TTG forqTM≠qBMin the absence of lattice relaxation. Local
AtA and AtB domains are formed, creating two characteristic length scales. (Inset)
Illustration of the two independent twist angles expected in a general three-layer
stack. (B) Normalized out-of-plane corrugation calculated ( 18 ) for AtB and AtA
stacking configurations, showing the local domain structure of each configuration.
(C) Schematic of the atomic stacking structure of AtA and AtB TTG. The two
configurations are related by translation of the top layer. In real devices, a MLR makes
it energetically favorable for AtB domains to warp into AtA. (D) STM topography of
TTG at an average twist of 1.56°. (Inset) Charge-neutral local density of states map
acquired at the Fermi level, showing electronic inhomogeneity caused by the MLR.
Set voltage (Vset) = 300 mV, set current (Iset) = 120 pA, and modulation voltage
(Vmod)=2mV.(EandF) Local twist-angle maps over the region shown in (D) (R1)
and a nearby sample area (R2). The local twist angle is extracted from the cell
areas of the Voronoi tessellation generated by the AAA site positions. (G)Plotof
the internal twist angle (qI)withinaMLRdomainasafunctionofdomainsize(L) for
regions R1 (E) and R2 (F). Error bars represent 1 SD of the local twist angle within
a given domain. Scale bars, 50 nm.RESEARCH | REPORTS