# Tuning the bandstructure of electrons in a two-dimensional artificial
electrostatic crystal in GaAs quantum wells

Daisy Q. Wang  School of Physics, University of New South Wales, Kensington,
NSW 2052, Australia  Australian Research Council Centre of Excellence in
Future Low-Energy Electronics Technologies,The University of New South Wales,
Sydney 2052, Australia  Zeb Krix  School of Physics, University of New South
Wales, Kensington, NSW 2052, Australia  Australian Research Council Centre of
Excellence in Future Low-Energy Electronics Technologies,The University of New
South Wales, Sydney 2052, Australia  Olga A. Tkachenko  Rzhanov Institute of
Semiconductor Physics, Novosibirsk, 630090, Russia  Vitaly A. Tkachenko
Rzhanov Institute of Semiconductor Physics, Novosibirsk, 630090, Russia
Novosibirsk State University, Novosibirsk, 630090, Russia  Chong Chen
Cavendish Laboratory, J. J. Thomson Avenue, Cambridge, CB3 0HE, United Kingdom
Ian Farrer***Present Address: Department of Electronic and Electrical
Engineering, The University of Sheffield, Mappin Street, Sheffield, S1 3JD,
United Kingdom. Cavendish Laboratory, J. J. Thomson Avenue, Cambridge, CB3
0HE, United Kingdom  David A. Ritchie  Cavendish Laboratory, J. J. Thomson
Avenue, Cambridge, CB3 0HE, United Kingdom  Oleg P. Sushkov  School of
Physics, University of New South Wales, Kensington, NSW 2052, Australia
Australian Research Council Centre of Excellence in Future Low-Energy
Electronics Technologies,The University of New South Wales, Sydney 2052,
Australia  Alexander R. Hamilton  School of Physics, University of New South
Wales, Kensington, NSW 2052, Australia  Australian Research Council Centre of
Excellence in Future Low-Energy Electronics Technologies,The University of New
South Wales, Sydney 2052, Australia  Oleh Klochan  School of Physics,
University of New South Wales, Kensington, NSW 2052, Australia  Australian
Research Council Centre of Excellence in Future Low-Energy Electronics
Technologies,The University of New South Wales, Sydney 2052, Australia  School
of Science, The University of New South Wales, Canberra, ACT 2612, Australia
School of Physics, University of New South Wales, Kensington, NSW 2052,
Australia  Australian Research Council Centre of Excellence in Future Low-
Energy Electronics Technologies,The University of New South Wales, Sydney
2052, Australia  School of Physics, University of New South Wales, Kensington,
NSW 2052, Australia  Australian Research Council Centre of Excellence in
Future Low-Energy Electronics Technologies,The University of New South Wales,
Sydney 2052, Australia  Rzhanov Institute of Semiconductor Physics,
Novosibirsk, 630090, Russia  Rzhanov Institute of Semiconductor Physics,
Novosibirsk, 630090, Russia  Novosibirsk State University, Novosibirsk,
630090, Russia  Cavendish Laboratory, J. J. Thomson Avenue, Cambridge, CB3
0HE, United Kingdom  Cavendish Laboratory, J. J. Thomson Avenue, Cambridge,
CB3 0HE, United Kingdom  Cavendish Laboratory, J. J. Thomson Avenue,
Cambridge, CB3 0HE, United Kingdom  School of Physics, University of New South
Wales, Kensington, NSW 2052, Australia  Australian Research Council Centre of
Excellence in Future Low-Energy Electronics Technologies,The University of New
South Wales, Sydney 2052, Australia  School of Physics, University of New
South Wales, Kensington, NSW 2052, Australia  Australian Research Council
Centre of Excellence in Future Low-Energy Electronics Technologies,The
University of New South Wales, Sydney 2052, Australia  School of Physics,
University of New South Wales, Kensington, NSW 2052, Australia  Australian
Research Council Centre of Excellence in Future Low-Energy Electronics
Technologies,The University of New South Wales, Sydney 2052, Australia  School
of Science, University of New South Wales, Canberra, ACT 2612, Australia

The electronic properties of solids are determined by the crystal structure
and interactions between electrons, giving rise to a variety of collective
phenomena including superconductivity, strange metals and correlated
insulators. The mechanisms underpinning many of these collective phenomena
remain unknown, driving interest in creating artificial crystals which
replicate the system of interest while allowing precise control of key
parameters. Cold atoms trapped in optical lattices provide great flexibility
and tunability Tarruell  _et al._ (2012); Goldman  _et al._ (2016), but cannot
replicate the long range Coulomb interactions and long range hopping that
drive collective phenomena in real crystals. Solid state approaches support
long range hopping and interactions, but previous attempts with laterally
patterned semiconductor systems were not able to create tunable low disorder
artificial crystals, while approaches based on Moiré superlattices in twisted
two-dimensional (2D) materials Kennes  _et al._ (2021); Mak and Shan (2022)
have limited tunability and control of lattice geometry. Here we demonstrate
the formation of highly tunable artificial crystals by superimposing a
periodic electrostatic potential on the 2D electron gas in an ultra-shallow
(25 nm deep) GaAs quantum well. The 100 nm period artificial crystal is
identified by the formation of a new bandstructure, different from the
original cubic crystal and unique to the artificial triangular lattice:
transport measurements show the Hall coefficient changing sign as the chemical
potential sweeps through the artificial bands. Uniquely, the artificial
bandstructure can be continuously tuned from parabolic free-electron bands
into linear graphene-like and flat kagome-like bands in a single device. This
approach allows the formation arbitrary geometry 2D artificial crystals,
opening a new route to studying collective quantum states.

A key challenge in condensed-matter physics is understanding strongly
interacting quantum systems, where many-body correlated states such as
superconductivity emerge. Artificial crystals where the key parameters can be
controlled in-situ can provide a powerful tool to simulate and study these
complex systems. Compared to optical systems, the pioneer in the field of
artificial crystals, solid-state based artificial crystals exhibit a pivotal
advantage: easy integration of the long-range Coulomb interactions that are
critical to emulating collective behaviour in real materials. However creating
solid state artificial crystals is non-trivial. The major challenges lie in
fabricating a highly uniform periodic potential $U(r)$ with an amplitude much
larger than the Fermi energy $E_{F}$, while maintaining very low levels of
disorder $\Gamma\ll U$.

Artificial solid state crystals have been explored in doped semiconductor
heterostructures where nanolithography techniques allows lattices of arbitrary
geometry to be created with excellent control. Early studies of one-
dimensional lateral superlattices revealed periodic magnetoresistance
oscillations when the electron’s cyclotron orbit matches the 1D lattice
spacing Weiss  _et al._ (1989); Winkler  _et al._ (1989), and in two
dimensional lattices signatures of Hofstadter physics were observed Albrecht
_et al._ (1999); Deutschmann  _et al._ (2001); Geisler  _et al._ (2004).
However the weak artificial lattice potential ($U(r)<<E_{F}$) and disorder
prevented the formation of an artificial solid state crystal with different
bandstructure and electronic properties than the host semiconductor. Optical
studies of honeycomb lattices etched into GaAs quantum wells Gibertini  _et
al._ (2009); Singha  _et al._ (2011) have revealed characteristics of the
honeycomb bands Wang  _et al._ (2018), although etched systems do not allow
tuning of the lattice potential, making it difficult to unambiguously identify
the bandstructure. More recently periodic lattice potentials have been created
by stacking and twisting two layers of atomically thin materials relative to
one another Geim and Grigorieva (2013); Andrei and MacDonald (2020), revealing
a rich variety of correlated electronic phases Cao  _et al._ (2018a, b); Lu
_et al._ (2019); Balents  _et al._ (2020). Despite these advances, these Moiré
systems face their own set of challenges: the symmetry and strength of the
artificial lattice potential are constrained by the host crystal, while twist
angle disorder remains a difficult obstacle to overcome Uri  _et al._ (2020).

In this work we demonstrate a new route to creating low-disorder two-
dimensional artificial crystals in semiconductor systems through electrostatic
gating. Electron beam lithography is used to define a 100 nm period triangular
lattice in a metal gate electrode only 25 nm above the electrons in a GaAs
quantum well. This setup substantially amplifies the superlattice potential,
allowing us to reach the regime where $U(r)>>E_{F}$, while we eliminate random
disorder from dopant atoms by using entirely undoped crystals so
$\Gamma<<E_{F},U$. We detect formation of the artificial crystal by probing
its unique bandstructure, with low-field transport measurements of the Hall
effect directly revealing electron-like and hole-like charge carriers in the
artificial bands. Independent confirmation of the formation of the artificial
crystal comes from the magnetoresistance at higher magnetic field, which shows
periodic oscillations as individual flux quanta are threaded through the
artificial unit cell. Remarkably we can continuously tune the energy bands of
the artificial crystal from the original parabolic free-electron dispersion to
graphene-like and kagome-like bands in a single device, with signs of a
correlated insulating state in the kagome flat band regime. We highlight that
this electrostatic gating technique is not material specific: here it is
demonstrated with GaAs/Al0.6Ga0.4As hetertostructures, but it can be used with
a multitude of semiconductors and atomically thin 2D materials. In the future
it could be used to engineer artificial solid state crystals to study
collective quantum phenomena such as correlated insulators, superconductivity,
and magnetism in lattices of arbitrary geometry.

##  I Calculating the bandstructure of the artificial crystal

Figure 1: Bandstructure of the artificial crystal. The system we consider is a
two-dimensional electron system in a GaAs/Al0.6Ga0.4As quantum well with a
periodic potential imposed on it. A strong modulation potential $U(r)$ is
essential for creating an artificial crystal with a well defined
bandstructure. (a) Schematic of the device, showing the double-layer gate
design. A surface metal patterned gate (PG, closest to the 2DEG) is patterned
with a triangular array of holes and positively biased to induce electrons at
the heterointerface 25 nm below the gate. The lattice constant is
$a=100~{}\textrm{nm}$ and the hole diameter is 45 nm. The overall top gate
(TG), separated by a thin $\mathrm{AlO_{x}}$ dielectric, controls the strength
of the superlattice potential through the holes etched in the patterned gate.
(b) The modulation potential in the weak $U_{p-p}<E_{F}$ and strong
($U_{p-p}>E_{F}$) limits. (c, d, e) Evolution of the calculated bandstructure
for a 100 nm lattice spacing with different peak-to-peak modulation amplitudes
$U_{p-p}$. (c) For a weak modulation potential ($U<E_{F}$, typically
$E_{F}\sim 2.5$ meV), the mini-bands have mostly parabolic energy dispersion,
with small splittings near the artificial Brillouin zone boundary. (d) As the
strength of the modulation potential increases the splittings at the zone
boundaries become larger, although only the lowest bands are well defined;
higher bands overlap and form a ’spaghetti’ of intersecting levels. (e) Only
at very strong potential modulation do the mini-bands become distinct and non-
overlapping, with a set of two graphene-like bands at lower energies (red),
and three kagome-like bands at higher energies (blue). Special points in the
bandstructure can be identified in experiments: the van Hove singularities at
the band edges (VH1-VH4), the Dirac points at band crossings (DP1 and DP2),
and also the flat band (F). (f) A surface plot of the bandstructure in (e),
and cross-sectional plots of the Fermi surfaces (i), (ii), (iii) and (iv),
show how the carriers in the artificial bands change from electron-like to
hole-like as the Fermi energy is increased. The shaded regions in panels
(i)-(iv) indicate occupied momentum states. (i) Fermi energy is between DP1
and VH2 (electron-like carriers). (ii) The Fermi energy is just above VH2
(hole-like carriers). (iii) Fermi energy is between F and VH3 (electron-like
carriers). (iv) Fermi energy is between VH3 and DP2 (hole-like carriers).

Determining the unique band structure of the artificial crystal is crucial for
understanding its electronic properties. We thus begin with a comprehensive
analysis of the 2D band structure and corresponding Fermi surface topology,
which highlights the importance of a strong superlattice potential. Our
experiments use a high-quality GaAs/Al0.6Ga0.4As heterostructure with a two-
dimensional electron gas (2DEG) subjected to a periodic triangular anti-dot
potential. A double-layer gate architecture, shown schematically in Fig. 1(a)
and detailed in the Methods, allows independent control over the amplitude of
superlattice potential and the electron density (the doping). We choose a
triangular symmetry of the lattice potential since this creates a unique
bandstructure that can be detected in measurements.

We model the electrostatic potential as a periodic potential shown in Fig.
1(b)) and described by Eqn. 1 in Methods (self consistent numerical modelling
has also been performed in Refs. Tkachenko  _et al._ (2015); Sushkov and
Castro Neto (2013)). We perform an exact numerical solution of the single-
particle Schr$\mathrm{\ddot{o}}$dinger equation to obtain the artificial
bandstructure. The key parameter that determines the effective strength of the
periodic potential is the peak-to-peak magnitude $U_{p-p}$ with respect to the
Fermi energy $E_{F}$, as sketched in Fig. 1(b). When the superlattice
potential is weak the mini-bands are essentially parabolic (Fig. 1(c)),
corresponding to nearly-free electrons, a regime that has been studied
extensively in experiments with GaAs superlattices Albrecht  _et al._ (1999);
Deutschmann  _et al._ (2001); Geisler  _et al._ (2004); Wang  _et al._ (2023).
To create an artificial crystal it is necessary for the superlattice potential
to be strong ($U_{p-p}\gg E_{F}$). In this regime (Fig. 1(e)) the energy bands
become non-overlapping with two graphene-like bands at low energies (red) and
three kagome-like bands (blue) at higher energies (by ‘graphene-like’ and
‘kagome-like’ we mean that the energy dispersion matches that of the
corresponding lattice with nearest neighbour hopping). The major difference
between the artificial bands in Fig. 1(e) and those in natural materials is
the smaller bandwidth (meV instead of eV) due to the larger lattice constant,
$a=100~{}\mathrm{nm}$. This necessitates very low levels of disorder, but
means that the Fermi level can easily be swept through the different bands
simply by tuning the bias on the patterned gate.

To understand the dynamics of electrons in this artificial crystal we look at
the change in the Fermi surface topology as the bands are filled: if the Fermi
surface expands as energy increases then charge carriers are electron-like,
while if the Fermi surface shrinks then charge carriers are hole-like. The
full bandstructure for a strong superlattice potential is shown in Fig. 1(f).
Charge carriers can change from electron-like to hole-like (or vice versa)
only at certain points in the band structure: at van Hove singularities or at
Dirac points, labelled in Fig. 1(f) as VH and DP, respectively. The insets
(i-iv) of Fig. 1(f) show the Fermi surfaces between these special points, with
shaded areas corresponding to occupied states. Panels (i) and (iii) show an
electron-like Fermi surface, while panels (ii) and (iv) show a hole-like Fermi
surface. A key experimental signature of the artificial crystal will be to
detect transitions between electron-like and hole-like behaviour.

##  II Measuring the artificial bandstructure

Figure 2: Measuring artificial bandstructure in graphene-like bands. The 100
nm period triangular superlattice imposed by the negative bias
$V_{TG}=-0.5~{}\mathrm{V}$ on the top-gate causes the electrons in the host
semiconductor to alternately behave as electrons and holes as the Fermi energy
is swept through the artificial bands. This provides unique features in the
low field Hall effect and the longitudinal resistance measured at $T=1.5~{}$K.
(a) In contrast to the host semiconductor the low field Hall resistance
$R_{xy}$ is no longer featureless, but shows striking features as a function
of $V_{PG}$ (proportional to the 2DEG density) and magnetic field, $B$. (b)
Line cuts of the Hall resistance at four different carrier densities indicated
by the dashed lines in (a). The Hall slope changes sign near $B=0$: the black
and green lines have a negative slope indicative of electron-like carriers,
while the red and blue lines have a positive slope indicative of hole-like
carriers. The transition points between electron-like and hole-like behaviour
are highlighted in panel (c), which shows a colour map of the Hall coefficient
$R_{H}=dR_{xy}/dB$. Electron-like and hole-like behaviour is indicated by the
colour (blue and red, respectively). Three transition points separate four
regions of different carrier type, alternating between electron-like, hole-
like, electron-like and hole-like as $V_{PG}$ is increased. This sequence of
transitions is consistent with the calculated band structure (panel (d)),
where the two graphene-like bands at low energy have three transition points:
VH1, DP1, and VH2. The electron-like (blue) and hole-like (red) behaviour on
either side of a transition is colour-coded, with sub panels (i-iv) showing
the corresponding Fermi surfaces. (e) The calculated charge distribution
corresponding to $E_{F}=1.5~{}\mathrm{meV}$ shows a honeycomb lattice. (f)
Additional evidence for the formation of a Dirac point comes from the
longitudinal resistance $R_{xx}$, plotted as a function of $V_{PG}$ and
magnetic field, $B$. A line cut of $R_{xx}$ at $B=0$ in panel (g) shows a
clear resistance peak at the position of the Dirac point DP1 (indicated by the
black arrow).

Experimentally we probe the formation of the artificial bandstructure using
measurements of the low-field magnetoresistance at $T=1.5~{}\textrm{K}$ (Fig.
2). The sample has a Van der Pauw geometry with approximately $2,600$ lattice
sites which allows four terminal measurements of the longitudinal $R_{xx}$ and
transverse (Hall) $R_{xy}$ resistances as a function of magnetic field and
gate biases. The superlattice potential is controlled with a negative bias
$V_{TG}$ on the top-gate, while the carrier density, or band filling, is
proportional to the positive bias $V_{PG}$ on the patterned gate. The sign of
the Hall slope probes whether charge carriers are electron-like or hole-like,
and can hence detect the formation of artificial bands. We highlight that the
results are highly reproducible, not only between devices made on the same
heterostructure, but even between devices made with different lattice
constants and from different wafers (Supplementary section III).

When the superlattice potential is weak the system behaves as a conventional
2DEG. In this well-explored regime Weiss  _et al._ (1989); Wang  _et al._
(2023) the Hall resistance is featureless over the entire range of 2DEG
density, with a negative Hall slope $R_{H}=dR_{xy}/dB$ indicating that charge
carriers are electron-like (Fig. S1 of Supplementary section I). To create the
artificial crystal we increase the strength of the superlattice potential by
making $V_{TG}$ more negative. The formation of the artificial crystal
modifies the Hall effect: the carriers change behaviour from electron-like to
hole-like as the Fermi energy is swept through the artificial bands, causing a
change in sign of the Hall coefficient as $V_{PG}$ is increased (Fig.
2(a,b,c)). The observation of a positive Hall slope is an unambiguous
signature of hole-like carriers, which can only arise due to the formation of
mini-bands in the artificial crystal.

The measured Hall coefficient in Fig. 2(c) is related to the calculated
bandstructure by counting the number of transitions between electron-like
($R_{H}<0$, blue) and hole-like ($R_{H}>0$, red) behaviour with increasing
carrier density. These transition points must correspond to either a van Hove
singularity (VH) or a Dirac point (DP). We observe four regions of different
carrier type in Fig. 2(c), with three transitions, which match well with the
VH1, DP1, and VH2 points in the calculated band diagram (Fig. 2(d)). For high
densities (above the last hole-like region, $V_{PG}>0.875$ V) $R_{H}$ is
strongly suppressed as the mini-bands merge together into a continuum at high
energies $E_{F}\sim U_{p-p}$. We use this point at which $R_{H}$ is suppressed
to estimate the value of $U_{p-p}$ (Supplementary section V).

Additional evidence for the formation of graphene-like bands comes from the
evolution of the longitudinal resistance $R_{xx}$ as a function of carrier
density (controlled by $V_{PG}$, Fig. 2(f)). There is a clear resistance peak
at $V_{PG}=0.843$ V centered at the position of the Dirac Point DP1 (DP1 was
previously identified from the sequence of the Hall coefficient sign changes).
This resistance peak is expected for a graphene-like system. We estimate the
mobility of the charge carriers near the artificial Dirac cone using the
resistance and effective density at VH1 or VH2 ($0.6\times
10^{10}~{}\textrm{cm}^{-2}$) to be $\mu_{D}\approx
100,000~{}\mathrm{cm^{2}/Vs}$, approximately $10$ times higher than that of
the host 2DEG at the same carrier density. This high mobility is consistent
with the ‘massless’ nature of Dirac carriers in the graphene-like bands. We
use the width of the Dirac peak to estimate the disorder, $\Gamma\approx
0.1~{}\textrm{meV}$, which is 40 times smaller than the superlattice potential
$U_{p-p}$.

Previous studies of gate defined artificial lattices were unable to reach the
strong modulation regime $U_{p-p}\sim E_{F}$ without the system breaking up
into ’puddles’ due to long range disorder Liang  _et al._ (1994); Melinte  _et
al._ (2004). Optical studies of honeycomb lattices etched into GaAs quantum
wells have revealed signatures of linearly dispersing bands Wang  _et al._
(2018), but etched systems do not allow continuous tuning of the supelattice
potential. In contrast, here we have tuned continuously from an unperturbed
system with massive electrons in parabolic bands to electrons and holes in an
artificial graphene bandstructure.

##  III Tuning from graphene-like to kagome-like bands

Figure 3: Formation of kagome-like bands. The top-gate bias is increased to
$V_{TG}=-3~{}\mathrm{V}$ to create a strong superlattice potential,
$U_{p-p}\gg E_{F}$. (a) Measurements of the low field Hall effect show
additional features in the colourmap of the Hall resistance $R_{xy}$ at
$V_{PG}>0.94~{}$V due to the formation of kagome bands. (b) Transitions
between electron-like (blue) and hole-like (red) Hall coefficient $dR_{xy}/dB$
calculated from the data in (a) as the carrier density is tuned with $V_{PG}$.
Black dashed lines indicate the positions of the special points VH2, VH3, and
DP2 where the Hall coefficient changes sign. The red dashed lines demarcate
the flat band region F where the Hall coefficient is not well defined
theoretically. (c) The calculated bandstructure for
$U_{p-p}=13.5~{}\mathrm{meV}$ shows both graphene-like and kagome-like bands
(see Supplementary Section V for details of the estimation of the superlattice
potential and the effects of Coulomb screening due to filling of multiple
bands). Insets show the Fermi surfaces, indicating either electron-like (blue)
or hole-like (red) charge carriers as the bands are filled. Comparison with
the measured Hall data in (b) give the width of a full band as $\Delta
V_{PG}=24~{}\textrm{mV}$. (d) The measured longitudinal resistance $R_{xx}$ as
a function of $V_{PG}$ and magnetic field $B$ shows an increased resistance in
the flat band region F. (e) Longitudinal resistance as a function of $V_{PG}$
at $B=0$ (taken along the dashed line in panel (d)). A strong resistance peak
occurs in the band gap and flat band region (red dashed lines demarcate the
flat band F). This resistance peak is not due to sample disorder, since it
disappears as the density is _decreased_. The width of the peak corresponds to
filling a complete artificial band, $\Delta V_{PG}=24~{}\textrm{mV}$, as
indicated by the black arrows. (f) The calculated charge density distribution
of the electron states integrated within the energy range
$0.5~{}\textrm{meV}<E<2.7~{}\textrm{meV}$, see panel (c), shows the kagome
lattice structure.

Having shown that we can use the electrostatic gate to transform the 2D
electrons in the GaAs quantum well into holes and graphene-like particles we
now exploit the ability to tune the strength of the superlattice potential to
create a kagome lattice with flat bands. We continuously increase the
modulation strength with $V_{TG}$ and track the resulting evolution of the
bandstructure through the experimentally determined transition points that
separate the electron-like and hole-like behaviour (Supplementary Section II).
As $V_{TG}$ is made more negative, these transition points in the Hall
resistance move to more positive $V_{PG}$. In Fig. 3(a) we show the measured
Hall resistance at the strongest superlattice potential $V_{TG}=-3$ V (the
most negative bias that could be applied while keeping the gate leakage below
0.1 nA). The increased modulation causes the sign changes of $R_{H}$
associated with the VH1, VH2 and DP1 points in the lower graphene-like bands
at low $V_{PG}$ to be suppressed (although their evolution can be followed in
Supplementary Section II), with new sign changes emerging at higher $V_{PG}$
associated with the formation of well-defined kagome-like bands. We relate the
experimental data to theoretical calculations by mapping electron-like and
hole-like regions of the Hall coefficient (Fig. 3(b)) to the calculated
bandstructure in Fig. 3(c). This allows the identification of the second and
third van Hove singularities VH2 and VH3, as well as the second Dirac point
DP2. The change in $V_{PG}$ from the top of the graphene-like bands to DP2
corresponds to filling two bands, which is used to extract the width of a
single band $\Delta V_{PG}\approx 0.024~{}\mathrm{V}$. This provides a
calibration that will later be used to count how many artificial bands are
filled for a given change in $V_{PG}$, and to locate the flat band region.

A key feature of the kagome-like bands is the flat band. Such bands often host
correlated states as the kinetic energy is quenched. The calculated band
diagram in Fig. 3(c) shows a well-developed flat band F with a width of $\sim
0.25$ meV separated from the lower graphene-like bands by a $\sim 1$ meV band
gap. Using the method in Ref. Krix  _et al._ (2022) we estimate the ratio
between Coulomb energy and kinetic energy in the upper kagome-like bands to be
approximately 5. This implies that the upper kagome-like bands are still in
the Fermi liquid regime. Within the flat band, however, the ratio between
Coulomb and kinetic energy is enhanced by about 10 times due to its narrow
bandwidth, meaning that the flat band is within the strongly correlated
regime. Therefore, the Hall coefficient within the flat band is not defined by
the single particle bandstructure and cannot be used to identify this band. We
can see a signature of the flat band, however, in the longitudinal resistance
$R_{xx}$ (Fig. 3(d,e)). The resistance $R_{xx}$ at $B=0$ shows a strong peak,
which starts from where $R_{xy}$ identifies the position of the band-gap, and
peaks at $V_{PG}=0.941~{}\mathrm{V}$ in the flat band region. We emphasise
that this resistance peak cannot be caused by a Mott-Anderson disorder
transition unrelated to the artificial lattice, because the resistance drops
from the peak value of $15~{}\mathrm{k}\Omega$ to $3~{}\mathrm{k}\Omega$ at
both higher _and_ lower carrier densities. In contrast trivial disorder would
cause the resistance to always increase with decreasing carrier density.
Furthermore, the width of the peak $\approx 0.024$ V is consistent with the
electron density required to fill one band noted previously. This suggests the
peak is due to localisation of electrons in a band rather than a single
particle insulator due to a band gap. The existence of this re-entrant
resistance peak in the flat band region, together with the large ratio between
Coulomb and kinetic energy, strongly hints toward correlation effects that
could be of interest for future studies.

##  IV Artificial crystal in ultra-high effective magnetic field

Figure 4: Magnetoresistance in the high magnetic field regime. Data taken for
a strong superlattice potential, $V_{TG}=-2~{}\mathrm{V}$ at
$T=1.5~{}\mathrm{K}$ show periodic features caused by the artificial lattice.
Similar oscillations are also observed at higher $V_{TG}$. (a-c) The Hall
resistance shows periodic oscillations both as a function of magnetic field
and as a function of band filling controlled by $V_{PG}$. (a) Colour map of
the Hall resistance $R_{xy}$ as a function of $V_{PG}$ and magnetic field in
flux quanta per artificial unit cell $\Phi/\Phi_{0}$ (the equivalent magnetic
field in Tesla is shown on the top axis). (b) Horizontal linecut of Hall
resistance along the black dashed line in (a) showing oscillations with a
period of one flux quantum per artificial unit cell. (c) Vertical linecut of
Hall resistance along the red dashed line in (a) showing oscillations with
period corresponding to the filling of one artificial band. This period
$\Delta V_{PG}\approx 0.024~{}\mathrm{V}$ is consistent with the band
assignments from Fig. 3. (d) The Hofstadter butterfly spectrum, and a phase
shift in the Hall signal at $V_{PG}=0.915~{}\mathrm{V}$ corresponding to the
kagome flat band, are revealed by subtracting a slowly varying background from
the Hall resistance in panel (a) to give $\Delta R_{xy}$. (e) The colourmap of
the longitudinal resistance $R_{xx}$ as a function of $V_{PG}$ and flux quanta
per unit cell shows a resistance peak (red) corresponding to the kagome flat
band, $V_{PG}\approx 0.915~{}\mathrm{V}$.

We provide additional confirmation of the formation of an artificial crystal
by studying the magnetoresistance at higher magnetic fields. The large lattice
constant in our system ($a=100$ nm) allows access to ultra-high effective
magnetic fields with multiple flux quanta per unit cell ($\Phi/\Phi_{0}=1$,
with $\Phi_{0}=h/e$ corresponds to $B_{0}=0.48$ T), which is not possible in
natural materials such as graphene (where $B_{0}\approx 80,000$ T) or Moiré
superlattices such as magic-angle twisted bilayer graphene ($B_{0}\approx 30$
T). First, $R_{xy}$ measured up to $B=2$ T (Fig. S7(a,b)) shows oscillations
which are periodic in the number of magnetic flux quanta per superlattice unit
cell. The observation of oscillations with period $\Delta\Phi=\Phi_{0}$ proves
that electrons are hopping between localised sites on the artificial lattice:
gauge invariance requires that physical properties must be invariant under
addition one flux quantum per unit cell for electrons on a lattice. Second,
$R_{xy}$ shows regular oscillations in $V_{PG}$, with a period $\Delta
V_{PG}=0.024$ V (Fig. 3(c)). This periodicity perfectly matches the value
obtained previously for the filling of one artificial mini-band, confirming
that the features in the magnetoresistance with increasing $V_{PG}$ correspond
to electrons filling bands in the artificial lattice. This also validates the
assignment of features in the experimental data to the theoretically
calculated bandstructure in Figs. 2, 3. The measured data here is, with some
exceptions noted below, in remarkably good agreement with numerical
calculations of the energy spectrum presented in Supplementary section VII.

Finally we use the oscillatory magnetoresistance to probe the flat band
region. To highlight the periodic pattern in $R_{xy}$ (Fig. S7a) we subtract a
slowly varying background to obtain $\Delta R_{xy}$ (Fig. S7(d)). This $\Delta
R_{xy}$ reveals a distinct shift in the phase of the oscillations when
carriers start to populate the flat band at $V_{PG}\approx 0.915$ V. Although
the single-particle calculation described in Supplementary section VII
perfectly reproduces the periodicities along $\Phi$ and $V_{PG}$ it cannot
explain the observed phase shift. Since the phase shift in $\Delta R_{xy}$
also coincides with the $R_{xx}$ resistance peak in Fig. S7(e), it is
plausible that both signatures are due to correlations, with charge carriers
becoming immobilised as the kagome flat band is populated. These high magnetic
field data confirm that it is possible to create artificial crystals where
electron-electron interactions are much stronger than the disorder potential,
which will enable the study of correlated states in arbitrary lattice
geometries.

##  V Discussion

In this work we have established a new approach for creating solid state
artificial crystals to allow studies of physical phenomena driven by long-
range hopping and strong Coulomb interactions. Our approach exploits both the
flexibility of electron beam lithography to create lattices of arbitrary
geometry and a double-layer gate architecture that allows the amplitude of the
superlattice potential and the chemical potential of electrons to be
continuously tuned. Experimentally, we detected the artificial crystal via
formation of a new bandstructure, with the Hall coefficient changing sign as
the carrier behaviour alternated from electron-like to hole-like when the
chemical potential was swept through the artificial bands. The artificial
bandstructure was continuously tuned from parabolic free-electron bands into
graphene-like and kagome-like bands simply by tuning a gate voltage. The
experimental data was in excellent agreement with theoretical calculations of
the predicted bandstructure. Further studies will be required to determine the
exact nature of the re-entrant insulating phase in the kagome flat band
regime, and the role of many-body interactions.

We emphasise that our approach not only allows lattices of any geometry to be
created, but is material agnostic and can be applied to a variety of 2D
systems, including atomically thin materials Brey and Fertig (2009); Zhang
_et al._ (2021); Yang  _et al._ (2022); Krix and Sushkov (2023). Furthermore,
the technique can be extended to create topological systems by introducing a
spin-orbit interaction by using valence band holes instead of conduction band
electrons Sushkov and Castro Neto (2013), or to study exotic phases in
honeycomb and kagome systems Li  _et al._ (2020); Ghorashi  _et al._ (2023)
including ferrielectric and topological ferromagnetic states in the high
magnetic field regime Mishra  _et al._ (2016); Mai  _et al._ (2023). Overall,
the ability to create arbitrary crystal geometries, with unprecedented control
over topology, doping, spin-orbit interaction, and superlattice potential will
allow an enormous variety of systems to be fabricated and studied.

Acknowledgements This work was funded by the Australian Research Council
Centre of Excellence for Future Low Energy Electronics Technologies
(CE170100039) and EP/R029075/1 Non-Ergodic Quantum Manipulation, UK. Device
fabrication was partially carried out at the Australian National Fabrication
Facility (ANFF) at the UNSW node.

Author contributions D.Q.W. fabricated samples and performed transport
measurements. D.Q.W., Z.K., O.P.S., A.R.H. and O.K. performed data analysis
and discussed the results. A.R.H., O.P.S. and O.K. supervised the project.
Z.K., O.P.S., O.A.T. and V.A.T. performed numerical calculations. C.C., I.F.
and D.A.R. provided the GaAs/Al0.6Ga0.4As heterostructures. D.W., Z.K.,
O.P.S., A.R.H. and O.K. co-wrote the manuscript with input from all co-
authors.

##  VI Methods

Theoretical methods. To model the artificial crystal we need to know the shape
of the potential experienced by electrons in the 2DEG. A full three-
dimensional numerical model of the device in Fig. 1(a), including Hartree
screening, has been used in Ref. Tkachenko  _et al._ (2015) to calculate the
artificial bandstructure. To simplify the analysis we construct a model
Hamiltonian which depends on only a single parameter $W$ which represents the
amplitude of the applied potential, yet captures the essential physics and
produces a bandstructure in very good agreement with the full 3D numerical
solution:

$\displaystyle\begin{split}H&=\frac{p^{2}}{2m^{*}}+U(\bf{r})\\\
U(\bf{r})&=2W\left[\cos({\bf G}_{1}\cdot{\bf r})+\cos({\bf G}_{2}\cdot{\bf
r})+\cos({\bf G}_{3}\cdot{\bf r})\right]\end{split}$ (1)

Where $\bf{G}_{1,2}$ are the basic reciprocal vectors of the triangular
lattice and $\bf{G}_{3}=\bf{G}_{2}-\bf{G}_{1}$ with
$|{\bf{G}_{i}}|=4\pi/\sqrt{3}a$. Note that the minimum of the potential is
$U=-3W$ and the maximum is $U=6W$, such that the peak-to-peak potential
amplitude $U_{p-p}$ is

$\displaystyle U_{p-p}=9W$ (2)

The lattice constant $a$ is 100 nm in all calculations. Here, higher harmonics
of the potential have been neglected. The validity of this Hamiltonian has
been checked against fully numerical 3D finite element Poisson calculations in
Ref. Tkachenko  _et al._ (2015). It is possible, however, to understand the
form of Eqn. 1 in terms of the following considerations: (i) at the level of
the patterned gate the potential has the shape of a triangular array of
(circular) hat functions. (ii) According to the Poisson equation, electrons in
the plane of the 2DEG experience this potential with Fourier components
modified from $U_{k}\to e^{-kz}U_{k}$, where $z$ is the distance to the gate.
(iii) The fundamental harmonics have the form $\cos(\bf{G_{i}}\cdot\bf{r})$,
with $|\bf{G}|=4\pi/\sqrt{3}a$, and higher harmonics are suppressed relative
to these by a factor $e^{-4\pi/\sqrt{3}a}(\approx 0.1$). (iv) For $W=0$
electrons in the 2DEG are described by a quadratic dispersion, $p^{2}/2m^{*}$,
where $m^{*}=0.0667m_{e}$ is the effective mass of electrons in GaAs.

The band structures were computed by exact numerical diagonalisation of the
Hamiltonian in Eqn. 1. To do this we write Eqn. 1 in the basis of plane wave
states, $|{\bf k}\rangle=e^{i{\bf k}\cdot{\bf r}}$. Since $U({\bf r})$ only
mixes states which differ in momentum by $\pm\bf{G}_{i}$ or $\pm{\bf G}_{2}$
this basis can be restricted to states, $|{\bf k}+{\bf g}_{i}\rangle$, where
${\bf k}$ is a momentum within the first Brillouin zone and ${\bf g}_{i}$ is
an arbitrary reciprocal lattice vector. We thus diagonalise the following
matrix

$\displaystyle\langle{\bf k}+{\bf g}_{i}|H|{\bf k}+{\bf
g}_{j}\rangle=\frac{({\bf k}+{\bf
g}_{i})^{2}}{2m}\delta_{ij}+W\sum_{n=1}^{3}\delta({\bf g}_{i}-{\bf g}_{j}-{\bf
G}_{n})$

Where $\delta_{ij}$ is the Kronecker symbol. This matrix must be truncated
such that the $|{\bf g}_{i}|$ are smaller than some upper limit; the limit is
chosen such that eigenvalues and eigenvectors are independent of the limit
(for example, the limit ${\bf g}=n{\bf G}_{1}+m{\bf G}_{2}$ with $|n|,\
|m|\leq 10$ is more than large enough for the strongest potentials that we
consider). Numerical diagonalisation gives a set of energy levels
$\varepsilon_{n}({\bf k})$ (these are plotted in Fig. 1) and a set of
corresponding Bloch functions $\psi_{n,{\bf k}}({\bf r})$. The charge
densities shown in Figs. 2e and 3f are equal to
$\sum_{\mu_{0}<\varepsilon_{n}({\bf k})<\mu}|\psi_{n,{\bf k}}({\bf r})|^{2}$.
For Fig. 2e we sum over states from the bottom of band 1 to the mid-point of
band 2 ($\mu=1.5$ meV). For Fig. 3f we sum over states from the bottom of band
3 to the top of band 4, -0.5 meV $<\epsilon<$ 2.7 meV . Of course, in the
latter case, bands 1 and 2 also contribute to the total charge density,
however, these states are not physically relevant when $\mu$ is within the
kagome-like bands and we exclude them for the sake of clarity.

Experimental methods. The devices in this study are fabricated on an ultra-
shallow high quality undoped GaAs/Al0.6Ga0.4As heterostructure comprising of a
3 nm GaAs cap, a 22 nm AlGaAs layer, and a thick GaAs buffer layer, grown by
molecular beam epitaxy. N-type ohmic contacts (AuGe) to the heterostructure
were thermally evaporated into etched pits and then annealed. The triangular
lattice is patterned into the surface metal gate by electron-beam lithography
and reactive ion etching with $\mathrm{SF_{6}}$. A 15 nm thick
$\mathrm{AlO_{x}}$ is deposited via atomic layer deposition to isolate the
metal gates from each other and from the ohmic contacts Wang  _et al._ (2020).
The mean free path of electrons in the unpatterned wafer is
$3~{}\mu\mathrm{m}$ at $\mathrm{n=2\times 10^{11}~{}cm^{-2}}$, much larger
than the lattice constant. Transport measurements were performed with standard
lock-in techniques at 33 Hz. Both Hall and longitudinal resistance are
measured in a Van der Pauw geometry over a patterned area of $5\times
5~{}\mu\mathrm{m}$.

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Supplementary Information for Tuning the bandstructure of electrons in a two-
dimensional artificial electrostatic crystal in GaAs quantum wells

Daisy Q. Wang

Zeb Krix

Olga A. Tkachenko

Vitaly A. Tkachenko

Chong Chen

Ian Farrer†††Present Address: Department of Electronic and Electrical
Engineering, The University of Sheffield, Mappin Street, Sheffield, S1 3JD,
United Kingdom.

David A. Ritchie

Oleg P. Sushkov

Alexander R. Hamilton

Oleh Klochan

##  VII Measurements in the weak modulation regime

To access the very weak modulation regime, it is necessary to reduce the
superlattice potential strength by using a deeper heterostructure. Details
about the effect of the wafer depth on the strength of the superlattice
potential are discussed in Supplementary section III. Here we show
measurements performed on a device fabricated using a 37 nm deep
heterostructure (W916) and show how the device can be continuously tuned from
a plain 2DEG to an artificial crystal.

Figure S1: Hall measurements of a device on the 37 nm deep heterostructure
W916 with a lattice constant of 120 nm at $T=1.5$ K. First column shows the
evolution of Hall resistance as a function of the patterned gate voltage
$V_{PG}$ and perpendicular magnetic field B with increasing modulation
strength (a) $V_{TG}=0.5~{}\mathrm{V}$, (b) $V_{TG}=-0.5~{}\mathrm{V}$ and (c)
$V_{TG}=-1~{}\mathrm{V}$. Dashed lines indicate the position of the linecuts
for the second column. (d)-(f) Hall resistance as a function of B field at (d)
$V_{PG}=0.83~{}\mathrm{V}$, (e) $V_{PG}=0.858~{}\mathrm{V}$ and (f)
$V_{PG}=0.87~{}\mathrm{V}$ for three modulation strengths respectively,
showing the change in Hall resistance near zero-field as the modulation
strength is increased. (g)-(i) Hall coefficient as a function of as a function
of $V_{PG}$ and B at three different modulation strengths corresponding to
(a)-(c). Red regions indicate where the Hall slope changes sign and the
carrier type becomes hole-like.

In the regime of very weak modulation strength $V_{TG}=+0.5~{}\mathrm{V}$, the
system bears resemblance to a conventional 2D electron gas, as demonstrated by
the featureless Hall resistance in Fig. S1(a),(d) and a negative Hall
coefficient in Fig. S1(g). This slope is inversely proportional to the density
of the 2D electron gas (2DEG) $n=1/(edR_{xy}/dB)$, which in turn is directly
proportional to the applied voltage on the patterned gate $V_{PG}$.

When the modulation potential is increased, by setting a top-gate voltage to
$V_{TG}=-0.5~{}\mathrm{V}$, we observe suppression of the Hall resistance
around $B=0$ (Fig. S1(b)). Increasing the potential modulation further to
$V_{TG}=-1~{}\mathrm{V}$ causes the Hall resistance around $B=0$ to change
sign, as shown in Fig. S1(c,f). As a result, the Hall coefficient switches
sign as highlighed by the red-colored region in Fig. S1(i), a direct signature
of the existence of hole-like carriers. The periodic oscillations of the Hall
resistance as a function of the magnetic field $B$ seen in Fig. S1(f) are due
to gauge invariance when integer numbers of flux quanta are threaded through
each artificial unit cell, as discussed in the main text.

##  VIII Bandstructure evolution as a function of modulation strength

As the modulation strength gradually increases by applying a more negative
gate voltage on the top-gate, the mini-bands in the system gradually move
apart and become separate. As the bands become discrete, more sign changes due
to transitions between electron-like and hole-like carriers start to appear.
On the other hand, as the modulation strength increases, we gradually lose
transport in the first band and the sign changes in the first two graphene-
like bands become suppressed. This is probably due to disorder in modulation
potential and localization of electrons in the strong modulation regime. As
shown in Fig. S2, at $V_{TG}=-0.5~{}\mathrm{V}$, the first two graphene-like
bands are fully developed with clear hole-like and electron-like regions
observed both below and above the Dirac point. It is also worth noting that
when the hole-like part of the second band overlaps with the third band, the
Hall resistance can still have a positive slope due to the existence of both
electron-like and hole-like carriers. This means the region of positive Hall
slope above the second van Hove singularity can be larger than that
corresponding to one band density since it contains overlapping bands. Only
when the second band is fully separate from the third band, i.e. a bandgap has
properly developed, the size of the positive Hall coefficient region above the
first Dirac point can be used to calibrate the gate voltage for one band
density in our system. As shown in Fig. S2(e)-(f), the band density $n_{0}$ of
the system in $V_{PG}$ is $\sim 24~{}\mathrm{mV}$, which for a $a=100$ nm
triangular lattice, is equivalent to a 2DEG density of $2.3\times
10^{10}~{}\mathrm{cm^{-2}}$ with 2 electrons per unit cell.

Figure S2: Hall resistance $R_{xy}$ as a function of patterned gate voltage
$V_{PG}$ and perpendicular magnetic field $B$ for the device on 25 nm deep
heterostructure W1740 with a lattice constant of 100 nm at $T=1.5$ K. The
modulation strength is gradually increased by stepping up the top-gate voltage
$V_{TG}$ (a) $V_{TG}=0~{}\mathrm{V}$, (b) $V_{TG}=-0.5~{}\mathrm{V}$, (c)
$V_{TG}=-1~{}\mathrm{V}$, (d) $V_{TG}=-1.5~{}\mathrm{V}$, (e)
$V_{TG}=-2~{}\mathrm{V}$, (f) $V_{TG}=-2.5~{}\mathrm{V}$,
(g)$V_{TG}=-3~{}\mathrm{V}$. Black dashed lines indicate the positions of the
Dirac points and van Hove singularities where Hall slope changes sign and/or
the positions where each band terminates. Red dashed lines illustrate the
evenly spaced bands in patterned gate voltage $V_{PG}$.

##  IX Reproducibility across multiple devices and wafers

Figure S3: Hall resistance $R_{xy}$ as a function of patterned gate voltage
$V_{PG}$ and perpendicular magnetic field $B$ for the device on 33 nm deep
heterostructure W1735 with a lattice constant of 100 nm at $T=1.5$ K. The
modulation strength is gradually increased by stepping up the top-gate voltage
$V_{TG}$: (a) $V_{TG}=~{}\mathrm{0V}$, (b) $V_{TG}=-0.5~{}\mathrm{V}$, (c)
$V_{TG}=-1~{}\mathrm{V}$, (d) $V_{TG}=-1.5~{}\mathrm{V}$, (e)
$V_{TG}=-2~{}\mathrm{V}$, (f) $V_{TG}=-2.5~{}\mathrm{V}$, (g)
$V_{TG}=-3~{}\mathrm{V}$. Black dashed lines indicate the positions of the
Dirac points and van Hove singularities where Hall slope changes sign and/or
the positions where each band terminates. Red dashed lines illustrate the
evenly spaced bands in the patterned gate voltage $V_{PG}$. Figure S4: Hall
resistance $R_{xy}$ as a function of patterned gate voltage $V_{PG}$ and
perpendicular magnetic field $B$ for the device on 37 nm deep heterostructure
W916 with a lattice constant of 120 nm. The top row (a,b,c,d) are measured at
$T=1.5$ K and the bottom row (e,f,g,h) are measured at $T=20$ mK. Features
related to the first Dirac point becomes more visible as the temperature is
reduced. Modulation strength is gradually increased by stepping ug the top-
gate voltage $V_{TG}$: (a,e) $V_{TG}=-1~{}\mathrm{V}$, (b,f)
$V_{TG}=-1.5~{}\mathrm{V}$, (c,g) $V_{TG}=-2~{}\mathrm{V}$ and (d,h)
$V_{TG}=-2.5~{}\mathrm{V}$. Dashed lines indicate the positions of the first
Dirac point and the first van Hove singularity where Hall slope changes sign.

The strength of the modulation potential is predominantly determined by two
experimental factors that are related to the device architecture and design:
1) the distance between the patterned gate and the 2DEG $d$ and 2) the lattice
constant $a$ (see Ref. Tkachenko  _et al._ , 2015). When $d$ is reduced, the
modulation strength increases exponentially. Increasing $a$ also increases
potential modulation. However, a larger lattice constant results in a smaller
band density which hinders the observation of the mini-bands. Therefore, the
optimal range for the lattice constant is $\sim 100$ nm with a band density of
$\sim 2\times 10^{10}~{}\mathrm{cm^{-2}}$ which is easily accessible in
GaAs/AlGaAs heterostructures.

We have measured multiple devices on three different heterostructures: W1740
which is 25 nm deep (Figs in the main text and Fig.S2), W1735 which is 33 nm
deep (Fig.S3) and W916 which is 37 nm deep (Fig.S1 and Fig.S4). The devices on
25 nm and 33 nm deep wafers were made with a 100 nm lattice while the device
on the 37 nm deep heterostructure has a 120 nm lattice. Even though the
accessible range of modulation strength is different for different wafers, we
still observe very similar behaviour across all devices. As shown in Fig.S2
and Fig.S3 we can extract a band width of $\Delta V_{PG}\approx 0.024$ V from
the 25 nm deep device and a band width of $\Delta V_{PG}\approx 0.033$ V from
the 33 nm deep device. This ratio of the two band widths in $V_{PG}$ exactly
reflects the change in depth by assuming a simple parallel capacitor model
$\Delta V_{PG}/d\propto n_{0}=2.3\times 10^{10}~{}cm^{-2}$.

##  X Coulomb screening, effect on modulation strength

The actual potential ($\widetilde{U}$) experienced by electrons differs from
the applied potential ($U_{0}$) because of Coulomb screening. We can account
for this difference using the Hartree equation

$\displaystyle\widetilde{U}_{q}=U_{0q}+\frac{2\pi e^{2}}{\varepsilon q}n_{q}$
(3)

Where $n_{q}$ is the Fourier transform of the spatially varying particle
density of a system described by Eqn.(1) in the main text with
$U=\widetilde{U}$. Here $\varepsilon=13$ is the dielectric constant of GaAs.
The subscripts $q$ denote Fourier coefficients. The Hartree equation must be
solved at a particular Fermi energy, $E_{F}$. In real space, the particle
density is defined by

$\displaystyle n(r)=\sum_{\begin{subarray}{c}k,n\\\
E_{k,n}<E_{F}\end{subarray}}|\psi_{k,n}(r)|^{2}$ (4)

Where $\psi_{k,n}(r)$ are the Bloch functions obtained via numerical
diagonalisation (see Methods). To solve the Hartree equation we assume that
$\widetilde{U}$, like $U_{0}$, is sinusoidal, meaning that its only non-zero
Fourier component is $\widetilde{U}_{g_{i}}$. Evaluating the Hartree equation
at $q=g_{i}$ we obtain

$\displaystyle\widetilde{W}=W_{0}+\frac{2\pi e^{2}}{\varepsilon
g_{i}}\frac{n_{g_{i}}}{A_{cell}}$ (5)

Where $A_{cell}$ is the unit cell area in real space and

$\displaystyle n_{g_{i}}=\int_{cell}d^{2}r\ n(r)e^{ig_{i}\cdot r}$ (6)

Using Eqn. 5 we can find the strength of the applied potential $W_{0}$
necessary to obtain a self-consistent potential strength $\widetilde{W}$. The
results of this calculation are presented in Fig. S5. We remind that the
parameter $W$ is related to the peak-to-peak width $U_{p-p}$ of the potential
by $U_{p-p}=9W$.

Figure S5: Effect of Coulomb screening on the potential amplitude. Each curve
is the applied potential $W_{0}$ necessary to obtain a given self-consistent
potential $\widetilde{W}$ at different electron densities. Electron density is
given on the horizontal axis in units of the number of filled electron bands.
Each curve corresponds to a different value of $\widetilde{W}$.

##  XI Estimation of modulation strength

Before we can estimate the potential amplitude a few points must first be made
clear.

  1. 1.

Potential amplitude decreases as electron density increases. For example, an
imposed potential of $W_{0}=3$ meV is reduced to $\widetilde{W}=1$ meV when
four energy bands are fully filled (at $n=4n_{0}=9.2\times 10^{10}\
\text{cm}^{-2}$). This is the result of Fig. S5.

  2. 2.

At large potential amplitudes each energy band is distinct, there is no
overlap between bands. As the potential amplitude is reduced bands begin to
overlap starting with the higher-energy bands first (see Figs. 1c-e in the
main text).

  3. 3.

If the bands of interest (the graphene-like and kagome-like bands) are
distinct for a given imposed potential, $W_{0}$, then there will be a
‘critical’ density $n_{c}$ such that for all $n>n_{c}$ the energy bands are
overlapping. This occurs because the potential strength decreases with
density.

  4. 4.

When the energy bands are distinct there is only ever one kind of charge
carrier present for a given electron density (either electron-like or hole-
like, see the inset to Fig. 1f in the main text). When the energy bands
overlap both kinds of charge carrier can exist simultaneously. In the latter
case no clear signature of the bands is expected in the Hall slope.

We use these points to estimate the size of the potential amplitude at
$V_{TG}=-0.5$ V (Fig. 2) and $V_{TG}=-3$ V (Fig. 3). In particular, we
identify the value of $V_{PG}$ which corresponds to $n_{c}$. At voltages
larger than this $V_{PG}$ no Hall slope transitions are observed. We assign a
value of $\widetilde{W}$ at this $V_{PG}$ using our band structure
calculation: the band structure must give the correct number of Hall slope
transitions below $n_{c}$.

Starting with Fig. 2 we note that three transitions in the Hall slope are
observed below $V_{PG}=0.87$ V. Above $V_{PG}=0.87$ V there are no further
transitions. We identify the point $V_{PG}=0.87$ V with the density $n_{c}$
defined above. Using the notation defined in this section the band structure
of Fig. 2d was computed for $\widetilde{W}(n_{c})=0.45$ meV. This band
structure is consistent with the observed Hall slope transitions and with our
estimation that the electron density is less than
$n_{0}/2=1/A_{cell}=1.15\times 10^{10}\ \text{cm}^{-2}$ at the lowest value of
$V_{PG}$ in Fig. 2a ($V_{PG}=0.825$ V). Using Fig. S5 we find that the imposed
potential amplitude must be $W_{0}=1$ meV.

We can repeat this analysis for the data in Fig. 3. In this figure the lowest
value of $V_{PG}$ corresponds to an electron density just above the first
Dirac point. The first transition above this point corresponds to the second
van Hove singularity ($V_{PG}=0.927$ V). The next transition after this
($V_{PG}=0.935$ V) is the top of the graphene-like bands. From the second
transition point, the subsequent $0.024$ V is identified with the third energy
band (the flat band). Above the third band there are two further transitions
in Hall slope. Again, the difference, $\Delta V_{PG}$, between the top of the
flat band and the second subsequent transition is $0.024$ V. Each of these
transitions, and the spacings between them are consistent with a potential
amplitude $\widetilde{W}=1$ meV at complete filling of four energy bands. At
lower fillings the value of $\widetilde{W}$ is larger (band structure shown in
Fig. 3e). Using Fig. S5 we find that $\widetilde{W}$ is $1.25$ meV when three
bands are fully filled and $1.75$ meV when 2 bands are fully filled. Finally,
the value of the imposed potential amplitude is $W_{0}=3$ meV. Note that in
Fig. 3(c) we plot the bandstructure for $\widetilde{W}=1.5$ meV which
corresponds to a filling in the flat band region.

##  XII Estimation of disorder and mobility

There are 3 kinds of disorder in our device. (i) A short-range disorder
related to the presence of impurities in the host semiconductor. (ii) A short-
range disorder related to imperfections in nano-lithography process. (iii) A
long-range disorder (puddles) related to imperfections in nano-lithography
process. The disorder of kind (i) can be neglected because the host
semiconductor is very clean, having a mobility of
$500,000~{}\mathrm{cm^{2}/Vs}$. Disorder of kinds (ii) and (iii) have been
simulated numerically in detail in Ref. Tkachenko  _et al._ , 2015. It was
shown that suppression of short wave-length harmonics due to the Poisson
equation make type (ii) less important; puddle formation (type iii) is the
most dangerous one, with the size of the puddles ranging from a few to several
lattice periods. We thus concentrate on type (iii). First, we assume that
puddles result in a Lorentzian broadening of the chemical potential

$\displaystyle\frac{\Gamma/2}{\pi(\delta\mu^{2}+\Gamma^{2}/4)}\ ,$ (7)

Using the width of the resistance peak at the first Dirac point DP1 we
estimate the full-width-half-maximum $\Gamma$. Fig. S6 shows this peak at two
temperatures temperatures, $T=550$ mK and $T=1.5$ K. The $T=1.5$ K peak is
slightly broader due to additional temperature broadening. To estimate the
disorder broadening $\Gamma$ we use the $T=550$ mK data. If we take the base
of the peak to be the resistance values at VH1 and VH2 then the full-width-
half-maximum at this temperature is $\delta V_{PG}=7.9$ mV. The difference in
$V_{PG}$ between VH1 and VH2 is $16$ mV. Thus, the full-width-half-maximum
$\delta V_{PG}$ accounts for half of the change in electron density between
VH1 and VH2. To translate this to an energy scale we take half of the energy
difference between VH1 and VH2 in the computed bandstructure of Fig. 2d. Doing
this we find

$\displaystyle\Gamma=0.2~{}meV$ (8)

Next, mobility is estimated using the standard equation

$\displaystyle\mu=\frac{1}{n_{eff}e\rho_{xx}}$ (9)

Where $n_{eff}$ is the effective density of charge carriers and $\rho_{xx}$ is
resistivity. Note that both the chemical potential and the mobility are
denoted by the same letter $\mu$, we hope that the meaning of $\mu$ will be
clear from its context. To estimate mobility near van Hove singularities VH1
and VH2 we have to remember that the effective density at the Dirac point is
zero and the effective density at the van Hove singularities is about
$n_{e}\approx n_{0}/4=5.7\times 10^{9}~{}\mathrm{cm^{-2}}$. The measured
longitudinal resistance presented in Fig. 2(g) and in Fig. S6 is about
$2400~{}\Omega$; to obtain resistivity we must multiply by the Van der Pauw
contant $\frac{\pi}{\ln{2}}=4.53$, appropriate for a device with Van der Pauw
geometry. Hence, $\rho_{xx}=4.53R_{xx}\approx 11,000~{}\Omega$. The mobility
close to VH1 and VH2 is about

$\displaystyle\mu=100,000~{}\mathrm{cm^{2}/Vs}$ (10)

This has to be compared with the mobility of a free electron gas in GaAs at
density $n_{e}=n_{eff}=5.7\times 10^{9}~{}\mathrm{cm^{-2}}$ which is about
$10,000~{}\mathrm{cm^{2}/Vs}$.

Figure S6: $R_{xx}$ as a function of $V_{PG}$ at $V_{TG}=-0.5$ V. Two traces
at different temperatures 1.5 K and 550 mK are plotted for comparison. The
peak in $R_{xx}$ corresponds to the same Dirac peak in Fig. 2 in the main
text. Black arrows indicate the positions the Dirac point and van Hove
singularities from the Hall measurement.

##  XIII Hofstadter’s butterfly and density of states

To understand the results of Fig. 4 of the main text we computed the density
of states of the simple Hamiltonian Eqn. (1) (Methods section) in the presence
of a magnetic field. Using the gauge $\bm{A}=(0,Bx,0)$ the Hamiltonian is

$\displaystyle H=p_{x}^{2}/2m^{*}+(p_{y}-eBx)^{2}/2m^{*}+U(x,y)$ (11)

The matrix elements of this Hamiltonian can be computed in the basis of Landau
level eigenstates, given by

$\displaystyle|k_{y},n\rangle=e^{ik_{y}y}\chi_{n}(x-k_{y}/eB)$ (12)

Where $\chi_{n}(x)$ is the $n$th wave-function of the harmonic oscillator with
frequency $\omega=eB/m^{*}$. Using a method analogous to that used in the main
text to address the zero-field limit, we take the matrix elements, $\langle
k_{y}^{\prime},n|H|k_{y},m\rangle$, and then numerically diagonalise. These
matrix elements have a complicated form and have already been computed in
previous works (see, for example, Ref. Krix and Sushkov, 2020). Importantly,
$k_{y}$ can only be changed by $\pm 2\pi/a$ (see Ref. Krix and Sushkov, 2020),
so that we can set $k_{y}=k+j2\pi/a$ and $k_{y}^{\prime}=k+i2\pi/a$ where $k$
now varies from $-\pi/a$ to $+\pi/a$. To perform the numerical diagonalisation
we must place restrictions on $i,j$ and $n,m$ such that $|i|,|j|<i_{max}$ and
$n,m<n_{max}$. Again, the values of $i_{max}$ and $n_{max}$ are chosen such
that all relevant energy levels are independent of them.

Using the explicit form for the matrix elements of Eqn. 11 given in Ref. Krix
and Sushkov, 2020, we can numerically diagonalise the Hamiltonian and obtain a
set of energy levels, $\varepsilon_{n}(k)$. Here, $n$ is a more general
quantum number which encompasses both the traditional band index and the
degrees of freedom normally enumerated by the quasi-momentum $k_{x}$. The
strength of the magnetic field is a free parameter, so we can obtain a set of
energy levels at each value of $B$ that we wish to study. Strictly speaking,
the magnetic field cannot become too small; at small fields the Landau level
spacing becomes small and we must include more Landau level states in our
basis. In other words, when $B$ is too small $n_{max}$ becomes too large for
practical computations.

Figure S7: Theoretical density of states at the Fermi energy (colour) as a
function of particle density (expressed in units of $n_{0}=2/A_{cell}$, the
particle density of a single Bloch band) and magnetic field (expressed in
terms of the number of flux quanta per unit cell). To perform the calculation
we suppose that the total particle density is independent of the magnetic
field. The density of states is given in arbitrary units.

To compute the density of states we fix the total particle density $n$ of the
system; this will be independent of the magnetic field. Then, for each
magnetic field $B$, we compute the total set of energy levels and find the
energy $\mu$ such that number of states below $\mu$ gives the correct particle
density $n$. After finding $\mu$ we compute the density of states in a small
window around $\mu$.

Our results for the density of states are plotted in Fig. S7. Here, we express
magnetic field in terms of flux quanta per unit cell (for $a=100$ nm one flux
quantum corresponds to $0.475$ Tesla). Particle density is expressed in terms
of the number of filled Bloch bands (for $a=100$ nm one filled band contains
$n_{0}=2/A_{cell}=2.3\times 10^{10}$ $\text{cm}^{-2}$ states). In this figure
the dark lines correspond to minima in the density of states, they appear when
the chemical potential, $\mu$, is within a band gap. We draw a connection
between the parrallel lines observed in this figure (showing density of
states) and in Fig. 4 of the main text (showing $R_{xx}$). Since these lines
are parallel there are two associated periods: (i) the horizontal period
$\Delta\phi/\phi_{0}=1$, and (ii) the vertical period $\Delta n/n_{0}=1$.