of 32
Articles
https://doi.org/10.1038/s41567-019-0606-5
Electronic correlations in twisted bilayer graphene
near the magic angle
Youngjoon Choi
1,2,3
, Jeannette Kemmer
1,2
, Yang Peng
2,3,4
, Alex Thomson
2,3,4
, Harpreet Arora
1,2
,
Robert Polski
1,2
, Yiran Zhang
1,2,3
, Hechen Ren
1,2
, Jason Alicea
2,3,4
, Gil Refael
2,3,4
, Felix von Oppen
2,5
,
Kenji Watanabe
6
, Takashi Taniguchi
6
and Stevan Nadj-Perge
1,2
*
1
T. J. Watson Laboratory of Applied Physics, California Institute of Technology, Pasadena, CA, USA.
2
Institute for Quantum Information and Matter,
California Institute of Technology, Pasadena, CA, USA.
3
Department of Physics, California Institute of Technology, Pasadena, CA, USA.
4
Walter Burke
Institute for Theoretical Physics, California Institute of Technology, Pasadena, CA, USA.
5
Dahlem Center for Complex Quantum Systems and Fachbereich
Physik, Freie Universität Berlin, Berlin, Germany.
6
National Institute for Materials Science, Tsukuba, Ibaraki, Japan. *e-mail: s.nadj-perge@caltech.edu
SUPPLEMENTARY INFORMATION
In the format provided by the authors and unedited.
NATuRE PHYSiCS
|
www.nature.com/naturephysics
Supplementary information for “
Electronic
Correlations
in Twisted Bilayer Graphene
Near
the Magic Angle”
Authors
:
Youngjoon Choi, Jeannette Kemmer, Yang Peng, Alex Thomson, Harpreet Arora,
Robert Polski, Yiran Zhang, Hechen Ren, Jason Alicea,
Gil Refael, Felix von Oppen, Kenji
Watanabe, Takashi Taniguchi and Stevan Nadj
-
Perge
Table of c
ontents
Section 1: Sample fabrication and measurement details
................................
................................
.............
1
Section 2: Data on 4.8nm moir
é
pattern
................................
................................
................................
......
1
Section 3: Tip
-
related effects
................................
................................
................................
........................
2
Section
4: Tip
-
gating correction at the CNP and the enhancement of the width
................................
........
7
Section 5: Areas with different twist angle
θ
................................
................................
................................
9
Section 6: Additional data
................................
................................
................................
...........................
10
Section 7: Theoretical modeling
................................
................................
................................
.................
12
Refere
nces:
................................
................................
................................
................................
.................
29
Supplementary Text
Section 1
:
Sample fabrication and measurement details
Figure S1 outlines the fabrication steps for twisted bilayer
graphene (TBG). We
us
ed
the
tear
-
and
-
twist
technique
following similar procedures outlined in Refs
1
-
3
. After picking up a 30
-
50nm
-
thick
boron nitride
flake using
Polydimethylsiloxane
(PDMS) coated with a P
oly(bisphenol
A carbonate)
(PC) polymer, the graph
ene is torn into two parts which were subsequently picked
up while controlling the twist between the parts. The entire stack is then transferred onto a
separate PDMS film in order to flip the order of the layers (Fig. S1 left panel). In this step, PC is
di
ssolved in
N
-
Methyl
-
2
-
pyrrolidone
(NMP). Afterward the PDMS with the inverted stack structure
is transferred onto a prepared chip
(silicon chip covered with 300nm of silicon
-
oxide)
with pre
-
patterned electrodes and a 10nm thick metallic graphene multilayer
that is used as the back gate
(Fig. S1 right panel). The twisted bilayer graphene (TBG) is then contacted to the gold electrodes
using additional few
-
layer graphene contacts. During device fabrication special care is taken so
that the temperature of the s
ample never exceeds 150
̊
C to avoid untwisting of the TBG
4
.
Fig
ure
S1
. Critical steps of
the
device fabrication
sequence. After picking up the BN and graphene
flakes, the stack is transferred and flipped onto a
second PDMS stamp that is used for a second
transfer onto the graphite back gate. In a separate
step, few
-
layer graphene is transferred to contact
TBG with a bias electrode.
Section
2
:
Data on
4.8nm
moir
é
pattern
Fig
ure
S2 shows a dI/dV map for the L
M
= 4.8nm (device D2) moir
é
pattern. The charge neutrality
point and both vHs peaks move in parallel
following the quadratic relation between charge
density
n
and chemical potential
μ
, |
n
|
=
α
μ
2
(up to an offset in V
Bias
and V
Bg
). This suggests that
in this regime the density of states between the vHs peaks follows approximately a linear
dispersion as expected for large twist angles. The vertical offset between the two branches is a
1
consequence of tip
-
induced gating, which is ex
plained in the following section. We note that this
effect may explain the apparent small increase in the distance between the van Hove singularities
at the CNP observed in Ref.
5
. Importantly, the relative enhancement of the peak splitting for
angles clos
e to the magic angle value is too large to be explained by this effect
as
dis
cu
ssed
in
S
ection 4
.
Fig
ure
S2. Spectroscopy
data taken on
the AA site for
θ
3
°
, device D2. Dashed lines correspond
to a fit
|
n
|
=
α
μ
2
where
μ =
-
eV
Bias
and
n
= C
Bg
V
Bg
.
The small vertical offset of the two branches
that give rise to a small enhancement in splitting in the V
Bias
direction
that
is due to tip gating as
discussed in
the
following section.
Section
3
:
Tip
-
related effects
Dynamic strain effects
O
ccasionally
we
observe
topographies with pronounced hysteresis, especially for parameters
when the STM tip
-
sample distance is reduced (for example V
Bias
=
±
200mV, set point current
100pA or more)
.
Similar STM
-
tip
-
induced strain effects have been previously
reported
for
moir
é
patterns in single
-
layer graphene/boron nitride heterostructures
6
as well as in graphene on
silicon oxide
7
. The intuitive understanding of this effect is that when the interaction between the
tip and the sample is strong, the tip can slightly displace the carbon atoms as it scans over the
surface and consequently shift the moir
é
pattern boundari
es. We have performed a similar
analysis as in Ref.
6
and obtained strain maps illustrating this effect (see Fig. S3).
In addition to
2
th
ese
strain
-
related effects,
we have also observed other signatures of tip
-
graphene interaction,
such as hysteresis in th
e
current vs height
relation
I(z).
Figure S3.
(a)
T
he t
opography of the strained region close to the magic angle (L
M
= 13nm) in device
D1 and the corresponding strain
map (b).
The map represents the local lattice constant obtained
by coarsening
the topography grid into discrete pixels and performing a Fourier analysis of the
3nmx3nm area around each pixel and selecting one particular direction. Other directions show a
similar dependence. Scale bar is 10nm for both plots.
The dynamic strain effects are not observed consistently throughout both samples. The strain is
typically observed in areas close to the magic angle in device D1 and in certain areas away from
the magic
angle in device D2.
Also
,
in regions close to the mag
ic angle in device D2, the moir
é
superlattice is more uniform and there is overall less strain. From these observations we conclude
that the presence of intrinsic external strain is needed in order to observe the dynamic ti
p
-
induced strain effect.
D
ifferent STM microtips may have different sensitivities to these effects.
I
mportantly
,
we d
o
not see
a
qualitative difference in
the
spectroscopy data between the two
devices indicating that th
ese
effects
do not play a significant role
in determining the l
ocal density
of states
. This is consistent with
the
previous
ly
reported observation
s in Ref.
6
.
While this effect
produces artifacts in the topography, it does not change it permanently so the distance between
AA sites and hence the twist angle
θ
can still
be obtained accurately.
3
E
ffects of tip screening
Due to the close proximity between the STM tip and the sample, one might expect that the
presence of metallic objects will facilitate screening of electronic correlations. In this section, we
estimate
the s
trength of the electron
-
electron interaction in this system. Indeed, in the presence
of the metal tip which provides additional screening the interaction energy scale can be greatly
suppressed compared to the pristine unscreened case. A naïve estimate for
the electrostatic
energy for two electrons placed L
M
=
13nm apart is given by e
2
/(4
πεε
BN
L
M
) = 36meV. This estimate
is enhanced further to approximately 50meV if we consider that we have the boron nitride
dielectric only on one side. However, in the presenc
e of a metallic tip this interaction is locally
screened considerably. For a tip
-
sample distance of 1nm (taken as an upper limit), the Coulomb
-
energy scale would scale as (
1
/
1
/
(
+
4
))
resulting in an interaction energy of
0.5meV for L
M
=
13nm. More precise estimates for the decay of the Coulomb interaction are
presented in Figure S4 as obtained by electrostatic simulations that take into account the
presence of the metallic tip and the surrounding dielectric. The values of
~
10mV extracted
from
comparing our experimental results to the model calculations presented in the main text are
consistent with these estimates.
Figure S4
. (a) 3D
Electrostatic simulation of the potential that takes into account the tip
-
sample
geometry. Blue circle re
presents a metallic tip with fixed potential V
=
0 and the rectangle
corresponds to the dielectric slab. The small circle was charged by 1.6e
-
19C and the slice
represents the decay of the potential along the y
-
z
direction. (b) Decay
of the potential as a
f
unction of distance. Different lines correspond to numerical calculations with and without the
tip as well as theoretical estimates corresponding to 1/d decay and (1/d
-
1/(d
2
+4z
tip
2
)
1/2
) decay.
4
Tip
-
induced
gating and
work function difference
between TBG and the tip
It is established that
the
STM tip can change
the
local potential in semiconducting samples due
to
a
finite screening length as observed in InAs
8
,
monolayer and bilayer
g
raphene in magnetic
field
9
,
as well as other semiconducting systems
. We have systematically observed
the
formation
of quantum dots in the regions close to
the
magi
c angle
indicative of the formation of insulating
states
.
The induced quantum dots introduce a series of sharp resonances observed as almost
horizontal lines crossing the features in V
Bias
vs V
B
g
conductance maps.
The observed resonance can be utilized t
o characterize the electrostatic properties of the
quantum dot and determine the capacitance of the tip C
Tip
and the work
-
function difference
between the tip and the twisted bilayer graphene.
To this end
, we take measurements at
different tip heights, i.e.
,
different set points. Figure S5 shows a typical spectrum on an AA site
for two different set currents of 100pA and 1nA. Lines (indicated by arrows) mark some of the
resonances originating from the quantum dots. The slope of the lines directly measures th
e ratio
between the tip and the back
-
gate capacitances C
Tip
/C
Bg
, and it is approximately 20 for device D2.
The tip capacitance changes when the tip moves closer to the sample. This is reflected
in
a
change
of
the
V
Bias
/V
Bg
slope. Another effect observed is the overall shift of the p
ositions in
point
spectroscopy
V
Bias
vs.
V
Bg
plots
for which
the upper
flat band touch
es
the Fermi level and
also the
position of
charge neutrality. The shift of these point
s
indicates that tip
-
i
nduced
gating change
s
the electron density underneath the ti
p
in a manner that depends on the
tip
-
sample distance.
This effect is a consequence of
a
difference between the work functions of the metallic tip and
the twisted bilayer
graphene.
In a simple mo
del, the charge density of the TBG underneath the tip can be written as:
(
,
)
=
(
,
)
(
)
,
where
is the work
-
function difference between the tip and the sample. Specifically, when the
tip moves closer, the charge
neutrality point moves towards more negative voltages (V
B
g
=
-
6.8 V
at CNP for 100pA setpoint and V
B
g
=
-
8.6 V at CNP for 1nA). Also, the slope of the lines changes
,
reflecting a change in the capacitance (C
Tip
/C
B
g
= 20 for 100pA setpoint and 24 for 1nA setpoi
nt).
5
By solving the equation for the charge density at charge neutrality for two setpoints, one gets an
estimate
for
= 150
-
200mV. We note that the STM tips in our measurements are prepared on
a silver crystal that has a smaller work function compared
to graphene which results in the
observed n
-
type doping.
Figure S
5
: Spectroscopy of the AA site of device D2
for 100pA (a) and 1nA (b).
Black
arrows
indicate resonances originating due to
a
tip
-
induced quantum dot
.
Note that
the
overall position
of
the
point
where the flat bands
start
cross
ing
the Fermi level (V
Bias
=
0mV line)
as well as the
charge neutrality point
shift towards more negative back
-
gate voltages.
The data are taken in the
area corresponding to
θ
= 1.03
̊
(L
MA
=
13.9nm
;L
MB
=
13.7nm
; L
MC
=
13.6nm
)
6
Section 4
:
Tip
-
gating
correction
at the
CNP
and the enhancement of the width
The slope of the lines identified
to correspond to constant density can be used
to correct
the
effects o
f tip gating by explicitly offsetting
the
back
-
gate voltage.
Fig S6 (a,b) shows the example
of
this procedure
applied for
device D1
data.
While in the uncorrected image (Fig. S6a) th
e
constant
-
density
lines are sloped
(see the lines starting from V
BG
0V;
V
Bias
=
-
100mV
i
n the Fig.
Figure S6: Device D1 data
before (a) and after the linear offset correction (b). The corrected
electron density is not affected by the gating of the tip. (c) Line traces from (b): Blue trace
taken at the CNP; Red trace at full filling. Both the peak
-
to
-
peak distance as well as over
all
width of the bands increases near the CNP. (d) Energy difference between threshold widths
at the CNP and full filling as a function of the angle
θ.
7
S6
a), in
the
corrected data the
constant density lines
are horizontal
indicating
decoupling
between
the bias voltage
and
the carrier density. The correction cancels the influence of the tip
gating on the splitting between the TD
OS peaks as well as the peak bandwidth. Importantly, even
after this correction
correlations
effects
i.e. enhanced splitting of the TDOS peaks and increased
bandwidth (defined as width at certain threshold dI/dV value
, see Fig. S6(c,d)
) near
the
CNP
,
are
still present
. We note that precise position of the TDOS peaks
(data plotted in the main text Fig.
2e)
are highly spatially dependent while the width at the threshold value set to be 20%
-
30% of
the peak
(at full filling)
depends weakly on the spatial
position around AA site
. Regardless
of the
details such as
threshold
value
,
both
the difference between threshold widths
Δ
Th
=w
ThCNP
-
w
ThFF
(
shown in Fig. S6d)
and the difference between TDOS peak
positions
Δ
PP
=w
PP
CNP
-
w
PP
FF
(shown in
Fig. 2e of the main text
)
are maximized around the magic angle value
θ
=1.1
̊±
0.1.
8
Section
5
:
Areas with different twist angle
θ
Figure S
7
. Examples of different areas of device D2 with twist angle
θ
= 2.92
°
(a);
θ
= 1.31
°
(b);
θ
= 0.97
°
(c);
θ
= 1.07
°
(d).
Set point
conditions: V
s
=
-
200mV and I
s
= 30pA (a,b)
;
I
s
= 100pA (c,d)
. In all
samples we typically find small clean areas with the lateral size of 50nm for which the angle
θ is
constant
. Larger
-
scale areas show significant amount of fabrication residues and strain
.
9