These authors contributed equally to this work. These authors contributed
equally to this work. [4]Ben J. Powell [2,5]Nikhil V. Medhekar [1,2]Agustin
Schiffrin 1]School of Physics and Astronomy, Monash University, Clayton,
Victoria 3800, Australia 2]ARC Centre of Excellence in Future Low-Energy
Electronics Technologies, Monash University, Clayton, Victoria 3800, Australia
3]Quantum Information Science and Technology Unit, Okinawa Institute of
Science and Technology Graduate University, Onna-son, Okinawa 904-0495, Japan
4]School of Mathematics and Physics, The University of Queensland, Brisbane,
Queensland 4072, Australia 5]Department of Materials Science and Engineering,
Monash University, Clayton, Victoria 3800, Australia

# Local gate control of Mott metal-insulator transition in a 2D metal-organic
framework

Benjamin Lowe  Bernard Field  Jack Hellerstedt  Julian Ceddia  Henry L. Nourse
[  [  [  [  [<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Electron-electron interactions in materials lead to exotic many-body quantum
phenomena including Mott metal-insulator transitions (MITs), magnetism,
quantum spin liquids, and superconductivity. These phases depend on electronic
band occupation and can be controlled via the chemical potential. Flat bands
in two-dimensional (2D) and layered materials with a kagome lattice enhance
electronic correlations. Although theoretically predicted, correlated-electron
Mott insulating phases in monolayer 2D metal-organic frameworks (MOFs) with a
kagome structure have not yet been realised experimentally. Here, we
synthesise a 2D kagome MOF on a 2D insulator. Scanning tunnelling microscopy
(STM) and spectroscopy reveal a MOF electronic energy gap of
$\mathord{\sim}$200 meV, consistent with dynamical mean field theory
predictions of a Mott insulator. Combining template-induced (via work function
variations of the substrate) and STM probe-induced gating, we locally tune the
electron population of the MOF kagome bands and induce Mott MITs. These
findings enable technologies based on electrostatic control of many-body
quantum phases in 2D MOFs.

###### keywords:

Strongly correlated electrons, metal-organic frameworks, kagome, dynamical
mean-field theory, scanning probe microscopy, Mott insulator, metal-insulator
transition

Strong electronic correlations arise in a material at specific electron
fillings of its bands, provided that the on-site Coulomb repulsion
(characterised by the Hubbard energy, $U$) is of the order of, or larger than,
the bandwidth, $W$. These electronic correlations can result in a wide range
of exotic many-body quantum phases. Examples include correlated insulating
phases, quantum spin liquids, correlated magnetism, and superconductivity –
phenomena which have been realised in monolayer transition metal-
dichalcogenides nakata_monolayer_2016 ; liu_monolayer_2021 ; lin_scanning_2020
; chen_strong_2020 ; ruan_evidence_2021 ; chen_evidence_2022 , twisted few-
layer graphene cao_unconventional_2018 ; park_tunable_2021 , inorganic kagome
crystals zheng_emergent_2022 ; ye_massive_2018 ; yin_quantum-limit_2020 , and
organic charge transfer salts powell_quantum_2011 ; kanoda_mott_2011 ;
miksch_gapped_2021 .

Tuning of the chemical potential via electrostatic gating can allow for
control over such band electron filling, enabling reversible switching between
correlated phases cao_unconventional_2018 . This makes these systems amenable
to integration as active materials in voltage-controlled devices, offering
enticing prospects for applications in electronics, spintronics, and
information processing and storage powell_quantum_2011 ; shao_recent_2018 .

Two-dimensional (2D) materials have emerged as particularly promising
candidates for realising strongly correlated phenomena as the absence of
interlayer hopping and screening can contribute to decreasing $W$ and
increasing $U$ chen_strong_2020 . Additionally, some 2D crystal geometries –
such as the kagome structure – give rise to intrinsic flat electronic bands
balents_superconductivity_2020 ; neupert_charge_2022 . When these extremely
narrow bands are half-filled, even weak Coulomb repulsion can open an energy
gap and give rise to a Mott insulating phase powell_quantum_2011 . Away from
half-filling, the gap closes and the system becomes metallic.

Metal-organic frameworks (MOFs) are a broad class of materials whose
properties are highly tunable through careful selection of constituent organic
molecules and metal atoms cui_metalorganic_2016 . There has been growing
interest in 2D MOFs for their electronic properties zhang_intrinsic_2016 ;
jin_large-gap_2018 ; jiang_exotic_2021 . In particular, layered 2D MOF
structures have been recently shown to host strongly correlated
superconductivity takenaka_strongly_2021 . Monolayer 2D MOFs have attracted
attention for their magnetism yamada_designing_2017 ; zhang_two-
dimensional_2019 ; kumar_manifestation_2021 , including ferromagnetism
resulting from exchange interaction between unpaired metal centre electrons
lobo-checa_ferromagnetism_2024 . Despite theoretical predictions
fuchs_kagome_2020 ; nourse_multiple_2021 , correlated Mott phases have not yet
been realised experimentally in monolayer 2D MOFs, however.

Here, we demonstrate local electrostatic control over a Mott metal-insulator-
transition (MIT) in a single-layer 2D kagome MOF, in excellent agreement with
theoretical predictions.

##  1 Results

###  1.1 A monolayer 2D metal-organic framework on an atomically thin
insulator

We synthesised the monolayer MOF – consisting of 9,10-dicyanoanthracene (DCA)
molecules coordinated to copper (Cu) atoms – on monolayer hexagonal boron
nitride (hBN) on Cu(111) (see Methods for sample preparation). A scanning
tunnelling microscopy (STM) image of a crystalline single-layer MOF domain
grown seamlessly across the hBN/Cu(111) substrate is shown in Fig. 1a. We
observe some defects within the MOF domain, as well as some DCA-only regions
(discussed in Supplementary Note 11). The long-range modulation of the MOF STM
apparent height follows the hBN/Cu(111) moiré pattern, which arises due to
mismatch between the hBN and Cu(111) lattices (giving rise to pore, P, and
wire, W, regions – see upper inset) joshi_boron_2012 ; zhang_tuning_2018 ;
auwarter_hexagonal_2019 . This moiré pattern has been shown to affect the
electronic properties of adsorbates auwarter_hexagonal_2019 , including one
previous example of a MOF urgel_controlling_2015 .

The MOF is characterised by a hexagonal lattice (lattice constant: 2.01 $\pm$
0.06 nm), with a unit cell including two Cu atoms (honeycomb arrangement;
bright protrusions in Fig. 1b) and three DCA molecules (kagome arrangement,
with protrusions at both ends of anthracene backbone in STM image in Fig. 1b),
similar to previous reports yan_synthesis_2021 ; yan_two-dimensional_2021 ;
kumar_manifestation_2021 ; hernandez-lopez_searching_2021 .

We calculated the band structure of this monolayer DCA3Cu2 MOF on hBN/Cu(111)
by density functional theory (DFT; with $U=0$); Fig. 1d. Projection of the
Kohn-Sham wavefunctions onto MOF states shows the prototypical kagome energy
dispersion with two Dirac bands and a flat band, consistent with prior
theoretical calculations for the freestanding MOF zhang_intrinsic_2016 ;
fuchs_kagome_2020 ; field_correlation-induced_2022 . This near-Fermi band
structure has predominantly molecular DCA character, and is well described by
a nearest-neighbour tight-binding (TB) model (see corresponding density of
electronic states, DOS, as a function of energy $E$ in Fig. 1e
field_correlation-induced_2022 ). The hBN monolayer, a 2D insulator with a
bandgap $>5$ eV auwarter_hexagonal_2019 , prevents electronic hybridization
between the underlying Cu(111) surface and the 2D MOF auwarter_hexagonal_2019
. This allows the MOF to preserve its intrinsic electronic properties, in
contrast to previous findings on metal surfaces pawin_surface_2008 ;
hernandez-lopez_searching_2021 ; kumar_manifestation_2021 ; field_correlation-
induced_2022 . These $U=0$ calculations predict that the MOF on hBN/Cu(111) is
metallic, with some electron transfer from substrate to MOF leading to the
chemical potential lying above the Dirac point, close to half-filling of the
three kagome bands (Fig. 1d).

Strong electronic interactions have been theoretically predicted in DCA3Cu2
fuchs_kagome_2020 , and a signature was recently detected experimentally
kumar_manifestation_2021 . We therefore calculated the many-body spectral
function $A(E)$ – analogous to the DOS in the non-interacting regime – of the
free-standing MOF via dynamical mean-field theory (DMFT). In contrast to the
TB model or DFT, DMFT explicitly captures local electronic correlations caused
by the Hubbard energy $U$ (see Methods) georges_dynamical_1996 ;
kotliar_strongly_2004 ; vollhardt_dynamical_2012 ; adler_correlated_2018 . In
Fig 1e, $A(E)$ is shown for $U$ = 0.65 eV (consistent with previous
experimental estimates yan_synthesis_2021 ) and for a chemical potential which
matches the DFT-predicted occupation of the kagome bands for the MOF on
hBN/Cu(111) (see Supplementary Note 2 and Supplementary Note 23 for further
DMFT calculations, including temperature dependence). We observe two broad
peaks (lower and upper Hubbard bands) separated by an energy gap of
$\mathord{\sim}$200 meV, dramatically different from the non-interacting
kagome DOS, and indicative of a Mott insulating phase ohashi_mott_2006 ;
fuchs_kagome_2020 .

###  1.2 Observation of $\sim$200 meV Mott energy gap

To experimentally probe the electronic properties of DCA3Cu2/hBN/Cu(111), we
conducted differential conductance ($\textrm{d}I/\textrm{d}V$) scanning
tunnelling spectroscopy (STS); $\textrm{d}I/\textrm{d}V$ is an approximation
of the local DOS [$A(E)$] in the non-interacting (interacting, respectively)
picture. We performed STS at the ends of the DCA anthracene moiety and at the
Cu sites of the MOF – locations where we expect the strongest signature of the
kagome bands based on the spatial distribution of the orbitals that give rise
to these bands yan_synthesis_2021 ; yan_two-dimensional_2021 ;
kumar_manifestation_2021 . These spectra (Fig. 2a), taken at a pore region of
the hBN/Cu(111) moiré pattern, both show broad peaks at bias voltages
$V_{\textrm{b}}\approx-0.2$ and 0.2 V. In a bias voltage window of $\sim$0.2 V
around the chemical potential $E_{\textrm{F}}$ ($V_{\textrm{b}}=0$), the
$\textrm{d}I/\textrm{d}V$ signal is low, significantly smaller than that for
bare hBN/Cu(111).

STM images acquired within this low-$\textrm{d}I/\textrm{d}V$ bias voltage
window (Fig. 2b, c) show mainly the topography of the MOF, with the molecules
appearing as ellipses of uniform intensity and the Cu atoms as weak
protrusions. Outside the low-$\textrm{d}I/\textrm{d}V$ bias voltage window
($|V_{\textrm{b}}|>200$ mV), Cu sites and the ends of the DCA anthracene
moieties appear as bright protrusions (Fig. 2d, e), similar to the spatial
distribution of the electronic orbitals of the DCA3Cu2 MOF associated with the
near-Fermi kagome bands (right inset of Fig. 2d; see Supplementary Figs. 11,
12 for more $V_{\textrm{b}}$-dependent STM images and
$\mathrm{d}I/\mathrm{d}V$ maps) yan_synthesis_2021 ; yan_two-dimensional_2021
; kumar_manifestation_2021 .

This suggests that the $\textrm{d}I/\textrm{d}V$ peaks at
$|V_{\textrm{b}}|\approx 0.2$ V in Fig. 2a are related to intrinsic MOF
electronic states near $E_{\textrm{F}}$, with the
low-$\textrm{d}I/\textrm{d}V$ bias voltage window of $\sim$0.2 V around
$E_{\textrm{F}}$ representing an energy gap, $E_{\mathrm{g}}$, between these
states. This is consistent with in-gap topographic STM imaging in Fig. 2b, c
repp_molecules_2005 . These $\textrm{d}I/\textrm{d}V$ peaks cannot be
attributed to inelastic tunnelling (e.g., MOF vibrational modes) as they are
not always symmetric about $E_{\textrm{F}}$ (see Fig. 3). The gap is much
larger than that predicted from spin-orbit coupling in such a system
zhang_intrinsic_2016 , and the monocrystalline growth of the MOF domains (Fig.
1a) makes a large disorder-related gap unlikely. Furthermore, the gap is
inconsistent with DFT and TB calculations (Fig. 1d, e).

The MOF spectra in Fig. 2a strongly resemble the DMFT-calculated spectral
function $A(E)$ in Fig. 1e, including an energy gap of the same magnitude
between two similar peaks (Supplementary Fig. 9). This suggests that these
$\mathrm{d}I/\mathrm{d}V$ spectra are hallmarks of a Mott insulator.

###  1.3 Template-assisted Mott metal-insulator transition

We further measured $\mathrm{d}I/\mathrm{d}V$ spectra at Cu sites of the
DCA3Cu2 MOF across the hBN/Cu(111) moiré pattern (Fig. 3a, b). In Fig. 3b,
$E_{\mathrm{g}}$ is centred symmetrically about $E_{\mathrm{F}}$ for spectra
taken in the middle of a pore region, while those taken closer to the wire
region show the Hubbard bands shifting upwards in energy (lowering the barrier
to creation of a hole). At the centre of the wire region, the gap at the Fermi
level vanishes with a clear increase in Fermi-level $\mathrm{d}I/\mathrm{d}V$
signal (Fig. 3e). The same behaviour was observed for DCA lobe sites of the
MOF (see Supplementary Note 14).

The hBN/Cu(111) moiré pattern consists of a modulation of the local work
function $\Phi$ (with little structural corrugation), where the quantity
$\Delta\Phi=\Phi_{\textrm{wire}}-\Phi_{\textrm{pore}}$ depends on the period
of the moiré superstructure, $\lambda$ joshi_boron_2012 ; zhang_tuning_2018 ;
auwarter_hexagonal_2019 . For the hBN/Cu(111) domain in Fig. 3a with
$\lambda\approx 12.5$ nm, $\Delta\Phi\approx 0.2$ eV (Fig. 3c; see
Supplementary Note 15 for moiré domains with different periods)
zhang_tuning_2018 . Due to energy level alignment joshi_control_2014 ;
zimmermann_self-assembly_2020 ; portner_charge_2020 , this corrugation of
$\Phi$ affects substrate-to-MOF electron transfer and hence the effective
electron filling of the MOF bands, with this filling smaller at wire than pore
regions joshi_control_2014 ; kumar_mesoscopic_2023 . This is consistent with
the effective reduction of the hole creation barrier at the wire relative to
the pore in Fig. 3b.

To capture the effect of this moiré-induced modulation of $\Phi$ on the MOF
electronic properties, we conducted further DMFT calculations. Using $U=$ 0.65
eV (the same as Fig. 1e), we calculated $A(E)$ for a range of $E_{\mathrm{F}}$
assuming a uniform system. We considered a sinusoidal variation of
$E_{\mathrm{F}}$ from a minimum value corresponding to half-filling of the
kagome MOF bands, with an amplitude of 0.2 eV, to match the experimental
$\Delta\Phi$ for this specific hBN/Cu(111) moiré domain kumar_mesoscopic_2023
; zhang_tuning_2018  (Fig. 3c; see Methods). The obtained $A(E)$ (Fig. 3d)
reproduce the experimental spectral features in Fig. 3b, including the
shifting of the lower Hubbard band maximum (LHBM) and upper Hubbard band
minimum (UHBM) (Fig. 3f), and the and the vanishing of the gap and increase of
the spectral function at the Fermi level for the wire region (Fig. 3e, g).

The DMFT-calculated spectral functions $A(E)$ with no gap at the Fermi level
at the smallest electron filling ($\Delta\Phi_{\mathrm{DMFT}}=0.2$ eV) show
peaks near $E_{\mathrm{F}}$ (Fig. 3d), however, which were not observed in the
experimental spectra at the wire region (Fig. 3b). These peaks are indicative
of coherent quasiparticles georges_dynamical_1996 , with their width
associated with the quasiparticle lifetime and quasiparticle mean free path
$\ell$. Via our DMFT and TB calculations (Supplementary Note 3), we estimate
$\ell\approx 10$ nm, much larger than the wire region width of
$\mathord{\sim}$4 nm. We hypothesize that the coherence peaks are suppressed
in the experiment as quasiparticles are strongly scattered by the pore regions
(where the MOF remains insulating). This is a key difference between Fig. 3b
and d: the experimental measurements represent changes in electron population
at finite size MOF regions due to a locally varying work function, whereas
each theoretical spectrum corresponds to an infinite uniform system with a
constant chemical potential.

Furthermore, some of the experimental spectra in the vicinity of the wire
region in Fig. 3b feature narrow peaks at energies of $\sim$0.4 eV, not
present in the theoretical $A(E)$ in Fig. 3d. We claim that these peaks do not
represent intrinsic electronic states, but are instead related to charging
phenomena not captured by the DMFT calculations (see Fig. 4 and Supplementary
Note 17).

Despite these discrepancies with experiment, our DMFT calculations capture the
fundamental electronic properties of the 2D DCA3Cu2 MOF, hosting a Mott
insulating phase with $E_{\textrm{g}}\approx 200$ meV, and a metal-like phase
(with no gap at the Fermi level) at the wire region (for the specific tip-
sample distance considered in Fig. 3). DMFT is a well-established method for
understanding the Mott insulator and Mott MITs georges_dynamical_1996 ;
kotliar_strongly_2004 ; vollhardt_dynamical_2012 ; adler_correlated_2018 .
This claim of strong electronic correlations and of a Mott insulating phase
for the DCA3Cu2 kagome MOF is consistent with previous literature
kumar_manifestation_2021 ; fuchs_kagome_2020 ; field_correlation-induced_2022
; ohashi_mott_2006 .

###  1.4 Tip-assisted Mott metal-insulator transition

To explore the nature of the metal-like phase observed at the wire region, we
conducted further $\mathrm{d}I/\mathrm{d}V$ STS of the MOF as a function of
tip-sample distance $\Delta z+z_{0}$ (where $z_{0}$ is set by tunnelling
parameters), at a DCA lobe site within the wire region (Fig. 4d). For large
$\Delta z+z_{0}$, these spectra feature an energy gap $E_{\textrm{g}}$, with a
small $\mathrm{d}I/\mathrm{d}V$ signal at the Fermi level (Fig. 4f), similar
to spectra in the pore regions (Figs. 2a, 3b; Supplementary Note 20), with a
sharp peak at positive $V_{\textrm{b}}$ (purple circles in Fig. 4d) and a
subtler band edge (red squares; similar to band features in Fig. 3b) at
negative $V_{\textrm{b}}$. As $\Delta z$ decreases, the energy position of the
sharp peak decreases linearly (Fig. 4e). Conversely, the energy position of
the subtler band edge increases with decreasing $\Delta z$, non-linearly and
at a lower rate (Fig. 4e). These features cross the Fermi level at
intermediate $\Delta z$, and the spectrum becomes gapless
($E_{\textrm{g}}=0$), with a significant increase in $\mathrm{d}I/\mathrm{d}V$
signal at the Fermi level (Fig. 4f). Note that an intermediate $\Delta z$ was
also used for all spectra in Fig. 3b where similar metal-like signatures were
observed at the wire region. We found a similar $\Delta z$-dependent trend for
wire region Cu sites (Supplementary Note 21). Notably, moiré pore regions
remain gapped for all $\Delta z$ values (for both Cu and DCA lobe sites;
Supplementary Note 20 and Supplementary Note 21).

As $V_{\textrm{b}}$ is applied between the tip and Cu substrate, the STM
double-barrier tunnel junction (DBTJ) – where the vacuum between tip and MOF
is a first tunnel barrier and the insulating hBN is a second one – causes a
voltage drop at the MOF location. This can lead to energy shifts of MOF states
and/or charging of such states when they become resonant with the Cu(111)
Fermi level (Fig. 4a) kumar_electric_2019 ; portner_charge_2020 ;
yan_synthesis_2021 . These phenomena have previously been observed for the
DCA3Cu2 MOF on a decoupling graphene surface yan_synthesis_2021 . In this
scenario, the bias voltages corresponding to an intrinsic electronic state,
$V_{\mathrm{state}}$, and to charging of such a state, $V_{\mathrm{charge}}$,
vary as a function of $\Delta z$ as:

$V_{\mathrm{state}}(\Delta
z)=\frac{d_{\mathrm{eff}}\left(V_{\infty}+\Delta\Phi_{\mathrm{ts}}\right)}{(\Delta
z+z_{0})}+V_{\infty},$ (1)

$V_{\mathrm{charge}}(\Delta z)=-\frac{V_{\infty}(\Delta
z+z_{0})}{d_{\mathrm{eff}}}-(V_{\infty}+\Delta\Phi_{\mathrm{ts}}),$ (2)

where $d_{\mathrm{eff}}$ is the effective width of the hBN tunnel barrier,
$V_{\infty}$ is the bias voltage corresponding to the electronic state as
$\Delta z\rightarrow\infty$, and $\Delta\Phi_{\mathrm{ts}}$ is the difference
between tip and sample work functions kumar_electric_2019 .

We fit the $\Delta z$-dependent bias voltage associated with the subtle band
edge (red squares) in Fig. 4d with Eq. (1), and the bias voltage associated
with the sharp peak (purple circles) with Eq. (2) (Fig. 4e). The agreement
between experimental data and fits indicate that the subtle spectral band edge
(red) represents an intrinsic MOF electronic state, with its energy shifting
as $\Delta z$ varies, and with the sharp peak (purple) corresponding to
charging of such a state.

##  2 Discussion

We interpret these results as follows. For large $\Delta z$ (Fig. 4c), the MOF
electronic states are strongly pinned to the substrate, and the MOF near-Fermi
kagome bands are approximately half-filled. Here, the
$\mathrm{d}I/\mathrm{d}V$ spectra feature an energy gap at the Fermi level,
both at moiré pore (Figs. 2a, 3b; Supplementary Note 20 and Supplementary Note
21) and wire regions (bottom spectra of Fig. 4d; Supplementary Note 21): the
entire monolayer 2D MOF is intrinsically a Mott insulator featuring localised
electrons. This is consistent with DMFT calculations, which show that the
system can remain Mott insulating even with a 0.2 eV modulation of
$E_{\mathrm{F}}$ (corresponding to the experimental moiré work function
corrugation; Supplementary Fig. 4).

Due to the hBN/Cu(111) moiré work function corrugation, the LHBM at a wire
region is very close to the Fermi level (in comparison to the pore region). As
$\Delta z$ is reduced, the MOF states become less pinned to the Cu(111)
substrate and more pinned to the tip via the DBTJ effect (Fig. 4;
Supplementary Note 19, Supplementary Note 20, Supplementary Note 21). Given
that $\Phi_{\textrm{tip}}>\Phi_{\textrm{wire}}$ (see Supplementary Note 18),
this leads to an energy upshift of the MOF states with respect to the Cu(111)
Fermi level (with the LHBM susceptible to charging as $V_{\textrm{b}}$ becomes
more positive). At intermediate $\Delta z$, this energy upshift depopulates
the LHB and leads to the transition from the Mott insulating phase (only
existing at half-filling of the three near-Fermi kagome bands; Fig. 1) to the
metallic phase. This is concomitant with a dramatic change in the
$\mathrm{d}I/\mathrm{d}V$ spectra, including the vanishing of the energy gap
and the increase in $\mathrm{d}I/\mathrm{d}V$ signal at the Fermi level. As
$\Delta z$ is further reduced (Fig. 4a), the near-Fermi kagome bands become
fully depopulated (with the bottom of these bands susceptible to charging as
$V_{\textrm{b}}$ becomes more negative; top spectra of Fig. 4d) and the MOF
becomes a trivial insulator (with a gap between these near-Fermi bands and
lower energy bands; Supplementary Fig. 1). The DBTJ effect also manifests
itself at the pore region, with energy shifts of the LHBM and UHBM as $\Delta
z$ varies (Supplementary Note 20, Supplementary Note 21). Yet, due to the
smaller $\Phi_{\mathrm{pore}}$ (Fig. 3c), the Fermi level lies close to the
centre of the energy gap (Fig. 3b), making the Mott insulating phase robust
for the considered $\Delta z$ range, consistent with DMFT (Fig. 3d,
Supplementary Figs. 2b, 4). In the wire region, the STM tip, via the DBTJ
effect, in combination with the large $\Phi_{\textrm{wire}}$, acts as a local
electrostatic gate, switching the 2D MOF from Mott insulator to metal
(Supplementary Fig. 26).

This tip-induced gating at the wire region is inherently local, occurring
within the cross section of the DBTJ (typically with a diameter of $\sim$10 nm
given by the tip radius of curvature). This locality could lead to
$\mathrm{d}I/\mathrm{d}V$ features (e.g., due to electronic confinement) not
captured by DMFT. Whether these local changes produce truly delocalised
metallic states across an extended area of the sample remains an open
question, beyond the scope of this investigation. Future work could
investigate the DCA3Cu2 MOF on single-crystal exfoliated hBN, within a gated
heterostructure, allowing for uniform electrostatic control and bulk (e.g.,
transport) measurements.

Our assertion that the MOF is a Mott insulator at the moiré pore regions for
all tip-sample distances, and at the wire regions for large tip-sample
distances, is well supported by DMFT calculations, and by previous literature
kumar_manifestation_2021 ; fuchs_kagome_2020 ; field_correlation-induced_2022
; ohashi_mott_2006 . The DMFT spectral functions demonstrate excellent
agreement with $\mathrm{d}I/\mathrm{d}V$ spectra at the pore regions,
including the $\sim$200 meV gap at the Fermi level, and the energy modulation
of the LHB and UHB due to variations in electrostatic potential. Our assertion
of a metallic phase for the MOF at the wire region for intermediate tip-sample
distances is well supported by the DBTJ model, and by the qualitative
agreement with DMFT, including the absence of an energy gap at the Fermi level
and an increase in Fermi-level $\mathrm{d}I/\mathrm{d}V$ signal and spectral
function.

Mott insulating – with intrinsically localised electronic states – and
metallic phases have been observed in TaS2 and TaSe2 monolayers, which feature
frustrated lattice geometries and lattice constants (due to charge density
wave distortions) similar to our work lin_scanning_2020 ; chen_strong_2020 ;
vano_artificial_2021 ; bu_possible_2019 ; fei_understanding_2022 , for finite
domain sizes as small as $\sim$10x10 nm2 vano_artificial_2021 . Also, local
topological phase transitions induced by an STM tip have been demonstrated
collins_electric-field-tuned_2018 . Our interpretation of a local population-
induced Mott metal-insulator transition is consistent with these findings.

Note that the vanishing of the gap at the Fermi level for the metallic phase
in the moiré wire region me (Fig. 3b) is reminiscent of spectral features for
the pseudogap phase in cuprates and other transition metal oxides
Cai_NatPhys_2016 ; Battisti_NatPhys_2017 ; Zhong_PRL_2020 . Future work on
controllable Mott MITs in MOFs might contribute to the general understanding
of doped Mott insulators.

Monolayer DCA-based MOFs have been studied on other substrates
pawin_surface_2008 ; yan_synthesis_2021 ; yan_two-dimensional_2021 ;
hernandez-lopez_searching_2021 ; kumar_manifestation_2021 ; kumar_two-
dimensional_2018 ; yan_-surface_2019 ; lobo-checa_ferromagnetism_2024 ;
vano_emergence_2023 , without observing a Mott phase. In our case, the
combination of the wide bandgap hBN as a template (allowing the MOF to retain
its intrinsic electronic properties), and of the adequate energy level
alignment given by the hBN/Cu(111) substrate (resulting in half-filling of
kagome bands; Fig. 1d), plays a key role in the realisation of the correlated-
electron Mott phase.

We have demonstrated that single-layer DCA3Cu2 not only hosts a robust Mott
insulating phase (with $E_{\textrm{g}}\gg k_{\textrm{B}}T$ at $T=300$ K), but
also that Mott MITs can be achieved via the combination of template- (Fig. 3)
and tip- (Fig. 4) induced gating, consistent with DMFT and the DBTJ model.
This shows that such phase transitions can be controlled in monolayer MOFs via
electrostatic tuning of the chemical potential.

Our findings represent a promising step towards incorporation of 2D MOFs as
active materials in device-like architectures (e.g., van der Waals
heterostructures based on 2D materials), benefiting from efficient synthesis
approaches and versatility offered by MOFs, and allowing for access and
control of correlated-electron phases therein via electrostatic gating
riss_imaging_2014 . Our work establishes single-layer 2D MOFs – with crystal
geometries allowing for flat bands – as promising platforms for controllable
switching between diverse many-body quantum phenomena, potentially including
correlated magnetism, superconductivity, and quantum spin liquids.

##  3 Methods

###  3.1 Sample preparation

The monolayer DCA3Cu2 kagome MOF was synthesised on hBN/Cu(111) in UHV (base
pressure $\sim$2 $\times 10^{-10}$ mbar). The Cu(111) surface was first
cleaned via 2-3 cycles of sputtering with Ar+ ions and subsequent annealing at
$\sim$770 K. A hBN monolayer was synthesised on Cu(111) via the thermal
decomposition of borazine auwarter_hexagonal_2019 . We dosed a partial
pressure of borazine of ${\sim}9\times 10^{-7}$ mbar for 45 minutes with the
Cu(111) sample maintained at 1140 K. We kept the Cu(111) sample at this
temperature for a further 20 minutes to ensure a complete reaction. We then
cooled the sample to room temperature and deposited the DCA molecules via
sublimation at 390 K, corresponding to a deposition rate of 0.007 ML/sec. In
our experiments we considered DCA coverages of $\sim$0.4-0.6 ML. We then
further cooled the sample to $\sim$77 K before depositing Cu via sublimation
at 1250 K (Cu deposition rate: $\sim$0.002 ML/sec; typical Cu coverages in our
experiments: $\sim$0.05 ML). Finally, the sample was annealed to $\sim$200 K
for 15 minutes. Further details in Supplementary Note 6.

The DCA3Cu2 MOF crystalline structure was found to be commensurate with the
hBN lattice but incommensurate with the long-range hBN/Cu(111) moiré patterns
of different sizes, across which the MOF grows without disruption (see
Supplementary Fig. 8). The hBN/Cu(111) moiré pattern is clearly visible in
large-scale STM images of the MOF (Fig. 1a). This is consistent with the
modulation of the MOF’s electronic properties illustrated in Fig. 3 (also see
$\mathrm{d}I/\mathrm{d}V$ maps in Supplementary Fig. 12).

###  3.2 STM and STS measurements

All STM and d$I/$d$V$ STS measurements were performed at 4.5 K (except
measurements in Supplementary Note 23 performed at 77 K), at a base pressure
${<}1\times 10^{-10}$ mbar, with a hand-cut Pt/Ir tip. All STM images were
acquired in constant-current mode with tunnelling parameters as reported in
the text (bias voltage applied to sample). All d$I/$d$V$ spectra were obtained
by acquiring $I(V)$ at a constant tip-sample distance (stabilised by a
specified setpoint tunnelling current and bias voltage), and by then
numerically differentiating $I(V)$ to obtain d$I/$d$V$ as a function of bias
voltage. Tips were characterised on regions of bare hBN/Cu(111) prior to
spectroscopy measurements, where the Shockley surface state of Cu(111) could
be observed (grey curve in Fig. 2a, onset shifted upwards in energy due to
confinement by hBN monolayer) joshi_boron_2012 .

###  3.3 DFT calculations

We calculated the non-spin-polarised band structure of DCA3Cu2 on hBN on
Cu(111) via DFT (Fig. 1d), using the Vienna Ab-Initio Simulation Package
(VASP) kresse_efficiency_1996  with the Perdew-Burke-Ernzerhof (PBE)
functional under the generalised gradient approximation
perdew_generalized_1996 . We used projector augmented wave (PAW)
pseudopotentials blochl_projector_1994 ; kresse_ultrasoft_1999  to describe
core electrons, and the semi-empirical potential DFT-D3 grimme_consistent_2010
to describe van der Waals forces.

The substrate was modelled as a slab three Cu atoms thick, with the bottom
layer fixed at the bulk lattice constant rumble_crc_2019 . A layer of
passivating hydrogen atoms was applied to the bottom face to terminate
dangling bonds.

A 400 eV cut-off was used for the plane wave basis set. The threshold for
energy convergence was $10^{-4}$ eV. The atomic positions of the
DCA3Cu2/hBN/Cu(111) were relaxed until Hellmann-Feynman forces were less than
0.01 eV/Å, using a $3\times 3\times 1$ k-point grid for sampling the Brillouin
zone and 1st order Methfessel-Paxton smearing of $0.2$ eV. The charge density
for the relaxed structure was calculated using an $11\times 11\times 1$
k-point grid, Blochl tetrahedron interpolation, and dipole corrections. The
band structure was determined non-self-consistently from the charge density.

Note that small (${<}1$ Å) perturbations in the height of DCA3Cu2 above the
hBN/Cu(111) do not appreciably affect the calculated band structure. As such,
small perturbations in height related to the hBN/Cu(111) moiré pattern (of at
most 0.7 Å schwarz_corrugation_2017 ) were not captured by these calculations
field_correlation-induced_2022 .

###  3.4 DMFT calculations

We performed dynamical mean-field theory (DMFT) calculations on the free-
standing DCA3Cu2 kagome MOF. We used the Hubbard model for a kagome lattice
with nearest-neighbour hopping,

$\displaystyle H$ $\displaystyle=-t\sum_{\langle
i,j\rangle,\sigma}\hat{c}^{\dagger}_{i,\sigma}\hat{c}_{j,\sigma}+U\sum_{i}\hat{n}_{i,\uparrow}\hat{n}_{i,\downarrow},$
(3)

where the first term is the tight-binding (TB) Hamiltonian with nearest-
neighbour hopping energy $t$, $\sum_{\langle i,j\rangle}$ is a sum over
nearest-neighbour sites, and the second term is the interaction Hamiltonian
with on-site Coulomb repulsion $U$. The operator
$\hat{c}^{\dagger}_{i,\sigma}$ ($\hat{c}_{i,\sigma}$) creates (annihilates) an
electron at site $i$ with spin $\sigma$;
$\hat{n}_{i,\sigma}=\hat{c}^{\dagger}_{i,\sigma}\hat{c}_{i,\sigma}$ is the
density operator. We take $t=0.05$ eV to match prior DFT calculations of
DCA3Cu2 field_correlation-induced_2022 ; zhang_intrinsic_2016 ;
fuchs_kagome_2020 .

We first calculated the non-interacting ($U=0$) density of states (DOS; blue
curve in Fig. 1e) by numerically integrating over all momenta in the first
Brillouin zone. The chemical potential $E_{\mathrm{F}}$ in Fig. 1e was chosen
to be consistent with the electron filling predicted by DFT (Fig. 1d). We
applied a thermal broadening ($k_{\textrm{B}}T=2.5$ meV) to this non-
interacting TB DOS to make it consistent with the thermal broadening of the
DMFT-generated (see below) spectral function $A(E)$ in Fig. 1e.

To account for electronic correlations, we then implemented the DMFT formalism
georges_hubbard_1992 ; georges_dynamical_1996 ; georges_strongly_2004  using
the Toolbox for Researching Interacting Quantum Systems (TRIQS)
parcollet_triqs_2015 , with the continuous-time hybridization expansion solver
(CTHYB) seth_triqscthyb_2016 ; gull_continuous-time_2011  at a temperature of
$\sim$29 K ($k_{\textrm{B}}T\approx 0.05t$; unless specified otherwise, see
Supplementary Note 23 for temperature-dependent calculations), with $U=0.65$
eV. To use a single-site DMFT formalism georges_dynamical_1996  with the
kagome band structure, the non-interacting DOS of the three kagome bands were
combined into a single function for use as the input into the DMFT procedure.

We calculated the many-body spectral functions $A(E)$ (analogous to the DOS,
but in the interacting regime; Fig. 3d) via analytic continuation using the
maximum entropy method (MaxEnt) as implemented in TRIQS kraberger_maximum_2017
. The meta-parameter, $\alpha$, was determined from the maximum curvature of
the distance between the MaxEnt fit and data, $\chi^{2}$, as a function of
$\alpha$ bergeron_algorithms_2016 .

Each DMFT calculation assumed a spatially uniform work function $\Phi$; long-
range modulation of $\Phi$ is beyond the capabilities of DMFT. As such, the
spatially varying sample work function $\Phi$ resulting from the experimental
hBN/Cu(111) moiré pattern was not explicitly captured in the individual $A(E)$
spectra in Fig. 3d. This spatial variation of $\Phi$ was approximated by
varying the uniform $E_{\mathrm{F}}$ of the system for each individual $A(E)$
spectrum. Each of these $A(E)$ spectra was then associated to a specific
location of the hBN/Cu(111) moiré pattern (and hence to a specific
experimental $\textrm{d}I/\textrm{d}V$ curve) based on how this variation of
$E_{\mathrm{F}}$ would translate to a local $\Phi$. These calculations assume
that the theoretical $A(E)$ spectra, calculated with a uniform $E_{F}$ (and
hence uniform $\Phi$), are reasonable representations of the locally acquired
experimental d$I$/d$V$ curves, which are affected by a spatially varying
$\Phi$. This assumption is reasonable for the insulating phase. Indeed, in the
Mott insulating phase, electronic states are localised at the kagome sites,
confined within areas that are small in length
fazekas_lecture_electron_correlation_1999  compared to the distance between
nearest-neighbour kagome sites ($\sim$1 nm) and to the periodicity $\lambda$
of the hBN/Cu(111) moiré domains considered in our experiments ($\lambda>5$
nm). DMFT indicates that this Mott insulating phase is robust to variations in
chemical potential $E_{\textrm{F}}$ larger than 0.2 eV, with the spectral
function $A(E)$ shifting in energy as $E_{\textrm{F}}$ is varied within this
range, without other significant qualitative changes (see Supplementary Fig.
2b for $E_{\textrm{F}}=$ 0.25 to 0.5 eV). In our experiments, the hBN/Cu(111)
moiré pattern imposes a periodic modulation of the local work function $\Phi$,
with a peak-to-peak modulation amplitude of $\sim$0.2 eV and a modulation
periodicity $\lambda\approx 12.5$ nm (Fig. 3c; this amplitude becomes smaller
with decreasing $\lambda$, see Supplementary Fig. 22). That is, $\Phi$ varies
_slowly_ across the MOF kagome lattice. The effect of such long-range
modulation of $\Phi$ on the MOF localised electronic states is to shift the
energy of these localised states accordingly. As long as the $\Phi$ modulation
period is larger than the distance between nearest-neighbour kagome sites and
the $\Phi$ modulation amplitude is smaller than a critical value inducing the
transition to the metallic phase, there is no other dramatic qualitative
effect on these localised electronic states. This explains the excellent
agreement between experimental $\mathrm{d}I/\mathrm{d}V$ spectra for the MOF
at the hBN/Cu(111) moiré pore region and DMFT-calculated spectral functions
$A(E)$ for the system in the Mott insulating phase (Fig. 3b, d), with LHB and
UHB modulated in energy following the variation in electrostatic potential.
For the MOF in the metallic phase at the moiré wire region (Figs. 3, 4),
discrepancies between theory and experiment can be explained (as discussed in
the main text) by long electronic coherence lengths (Supplementary Note 3),
and by effects of long-range moiré $\Phi$ modulation and of finite DBTJ cross
section on the potentially delocalised metallic MOF states.

###  3.5 DBTJ model

In Figs. 4d, e, the bias voltage, $V_{\textrm{state}}$, corresponds to the
energy level of an intrinsic MOF frontier electronic state (which is
susceptible to charging), at a band edge in either the Mott or trivial
insulator regime. Red square markers indicating $V_{\textrm{state}}$ were
assigned based on method outlined in Supplementary Note 13. The bias voltage,
$V_{\mathrm{charge}}$, corresponds to the peak associated with charging of
such a MOF state. Purple circle markers indicating $V_{\mathrm{charge}}$ were
assigned by finding a local maximum in $\mathrm{d}I/\mathrm{d}V$.

In Fig. 4e, we considered four experimental datasets for fitting with Eqs. (1)
and (2): $V_{\textrm{state}}(\Delta z)$ (red squares) and
$V_{\textrm{charge}}(\Delta z)$ (purple circles) for small values of $\Delta
z$ (trivial insulator phase), and $V_{\textrm{state}}(\Delta z)$ and
$V_{\textrm{charge}}(\Delta z)$ for large values of $\Delta z$ (Mott insulator
phase). Given the Mott MIT, we considered two different intrinsic MOF band
edges susceptible of charging, embodied in two different values of
$V_{\infty}$: one for the trivial insulator phase (small values of $\Delta z$)
and one for the Mott insulator phase (large $\Delta z$). This phase transition
is evident from the offset in $V_{\textrm{state}}$ (red squares in Fig. 4e)
observed when $\Delta z$ varies from small to large (through the metallic
phase at intermediate $\Delta z$). Accordingly, we used a global fitting
approach to obtain the same fitting parameters $d_{\mathrm{eff}}$, $z_{0}$,
and $\Delta\Phi_{\mathrm{ts}}$ (characteristic of the DBTJ and the acquisition
location) for these four experimental datasets, and a separate $V_{\infty}$
value for each regime.

Schematics in Fig. 4a-c represent cartoon illustrations of our proposed
physical mechanism for tip-induced gating.

It is important to note that the DBTJ inherently affects all
$\mathrm{d}I/\mathrm{d}V$ measurements in this work, including those in Figs.
2 and 3, for both pore and wire regions of the hBN/Cu(111) moiré pattern. The
DBTJ effect does not cause phase transitions at the pore regions, however (see
Supplementary Note 20, Supplementary Note 21). The measurements in Fig. 3 were
performed with an intermediate tip-sample distance – which is why metallic
properties were observed at the wire region.

##  4 Data availability

The data supporting the findings of this study is available from the authors
upon request.

##  5 Code availability

All codes relating to DMFT and MaxEnt are available at
https://doi.org/10.5281/zenodo.7439858. Code for data analysis and theoretical
calculations can be made available from the authors upon request.

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##  6 Acknowledgements

A.S. acknowledges funding support from the ARC Future Fellowship scheme
(FT150100426). B.J.P. acknowledges funding support from the ARC Discovery
Project scheme (DP180101483). H.L.N. acknowledges funding support from the
MEXT Quantum Leap Flagship Program (JPMXS0118069605). B.L., J.H., J.C., and
N.V.M. acknowledge funding support from the Australian Research Council (ARC)
Centre of Excellence in Future Low-Energy Electronics Technologies
(CE170100039). B.L., B.F., and J.C. are supported through Australian
Government Research Training Program (RTP) Scholarships. B.F. and N.V.M.
gratefully acknowledge the computational support from National Computing
Infrastructure and Pawsey Supercomputing Facility. The authors also thank
Prof. Michael S. Fuhrer, Prof. Jaime Merino Troncoso, and Dr. Daniel Moreno
Cerrada for valuable discussions.

##  7 Author Contributions

B.L., J.H., and A.S. conceived and designed the experiments. B.L., J.H., and
J.C. performed the experiments. B.F. and H.L.N. performed the theoretical
calculations with guidance from B.J.P. and N.V.M. All authors contributed to
writing the manuscript.

##  8 Competing Interests

The authors declare no competing interests.

##  9 Additional Information

###  9.1 Supplementary Information

Further DFT calculations; further DMFT calculations; discussion of
quasiparticle peaks in Fig. 3; MOF growth and structure details; comparison
between DMFT and STS; bias-dependent STM topography images and
$\mathrm{d}I/\mathrm{d}V$ maps; further STS measurements; STS measurements
across moiré patterns with different periods; method for determining band
edges; further charging signatures; estimate of tip work function; further
DBTJ model details including tip-induced gating at Cu sites of the MOF;
$\mathrm{d}I/\mathrm{d}V(\Delta z)$ measurements at the pore region; further
evidence of metallic character; temperature-dependent STS measurements and
DMFT calculations; comparison of STS measurements at different moiré pore
regions.

###  9.2 Correspondence & Requests for Materials

To be addressed to Ben J. Powell, Nikhil V. Medhekar, or Agustin Schiffrin.

Figure 1: A 2D kagome MOF on an atomically thin insulator: DCA3Cu2 on single-
layer hBN on Cu(111). a, STM image of MOF (with organic DCA-only regions;
$V_{\textrm{b}}$ = -1 V, $I_{\textrm{t}}$ = 10 pA). ‘P’ and ‘W’ indicate pore
(dashed white circle) and wire regions of hBN/Cu(111) moiré pattern. Lower
inset: Fourier transform of STM image; sharp spots correspond to MOF hexagonal
periodicity (scale bar: 0.25 nm-1). Upper inset: STM image of bare hBN/Cu(111)
moiré pattern ($V_{\textrm{b}}$ = 4 V, $I_{\textrm{t}}$ = 100 pA, scale bar: 4
nm). Red arrows: examples of crack, vacancy, and Cu cluster defects within MOF
domain. b, Top-view of MOF model overlaid upon small-scale STM image of region
within red box in (a) ($V_{\textrm{b}}$ = -1 V, $I_{\textrm{t}}$ = 10 pA). MOF
unit cell indicated in light blue. Blue dashed lines and solid circles: kagome
pattern formed by DCA molecules, with inter-site electron hopping, $t$, and
on-site Coulomb repulsion, $U$. c, Model of MOF/hBN/Cu(111) (side view).
Hydrogen: white; carbon: grey; boron: pink; nitrogen: blue; copper: orange. d,
Electronic band structure calculated by DFT (with $U=0$) field_correlation-
induced_2022 . Blue circles: projections onto MOF states. Grey curves: Cu(111)
states (hBN states do not contribute within the shown energy range).
Hybridisation between MOF and Cu(111) is hindered by the hBN monolayer; the
MOF band structure retains its kagome character. Chemical potential
$E_{\textrm{F}}$ (blue dashed line) is close to half-filling of kagome bands.
e, Density of states, DOS($E$) (tight-binding model with thermal broadening,
$U$ = 0, blue), and spectral function, $A(E)$ (DMFT, $U=$ 0.65 eV, orange), of
free-standing MOF. Local Coulomb interaction opens a significant Mott energy
gap $E_{\textrm{g}}$ between lower (LHB) and upper Hubbard bands (UHB).
Figure 2: Bandgap E${}_{\textbf{g}}$ $\mathbf{\approx 200}$ meV in DCA3Cu2 MOF
on hBN/Cu(111). a, $\textrm{d}I/\textrm{d}V$ spectra at MOF positions
indicated by orange markers in (b-e) and on bare hBN/Cu(111) (grey dots)
(setpoint: $V_{\textrm{b}}$ = -500 mV, $I_{\textrm{t}}$ = 500 pA). Gray dotted
vertical lines indicate bias voltages at which STM images were acquired in
(b-e). Spectra offset for clarity. Dashed horizontal lines indicate
$\mathrm{d}I/\mathrm{d}V=0$ reference for each curve. Spectra reveal a bandgap
$E_{\textrm{g}}\approx 200$ meV. The non-zero $\textrm{d}I/\textrm{d}V$ within
gap is due to states of underlying Cu(111) leaking through hBN. Red squares
(circles): lower Hubbard band maxima, LHBM (upper Hubbard band minima, UHBM).
b-e, STM images of MOF on hBN/Cu(111) at specified bias voltages
($I_{\textrm{t}}$ = 10 pA). White dashed circle: hBN/Cu(111) moiré pore. MOF
unit cell indicated in light blue. Scale bars: 1 nm. Insets: zoom-in of region
within red box, with overlaid Cu-DCA-Cu chemical structure, showing
significant contributions from the ends of the DCA anthracene moiety (green
arrow) and Cu (orange arrow) for $V_{\textrm{b}}<\mathrm{LHBM}$ and
$V_{\textrm{b}}>\mathrm{UHBM}$. Right inset in (d): charge density isosurface
(0.0025 e-Å-3) of DCA3Cu2 obtained by integration of near-Fermi ($\pm 0.5$ eV)
DFT wavefunctions. Figure 3: Variation of Mott energy gap in DCA3Cu2 MOF
induced by hBN/Cu(111) moiré modulation of local work function. a, STM image
of MOF ($V_{\textrm{b}}$ = -1 V, $I_{\textrm{t}}$ = 10 pA). White dashed
circles (P): hBN/Cu(111) moiré pores, separated by wire (W). Grey arrow
indicates moiré period $\lambda\approx$ 12.5 nm. b, d$I/$d$V$ spectra acquired
at MOF Cu sites, at positions indicated by coloured markers in (a) (tip 190 pm
further from STM setpoint $V_{\textrm{b}}$ = 10 mV, $I_{\textrm{t}}$ = 10 pA).
Energy gap $E_{\textrm{g}}\approx 200$ meV at P regions, vanishing at W region
(LHBM: lower Hubbard band maximum; UHBM: upper Hubbard band minimum). c,
Sinusoidal variation of work function, $\Delta\Phi=\Delta
E_{\textrm{vac}}-E_{\textrm{F}}$ ($E_{\textrm{vac}}$: vacuum energy level),
across hBN/Cu(111) moiré domain with periodicity $\lambda\approx$ 12.5 nm
zhang_tuning_2018 , affecting the MOF electron filling. d, Spectral functions
$A(E)$ calculated via DMFT ($U=0.65$ eV, $t=0.05$ eV) for isolated uniform
DCA3Cu2, for different values of $E_{\textrm{F}}$. We account for experimental
corrugation $\Delta\Phi$ by varying $E_{\textrm{F}}$ sinusoidally with an
amplitude of 0.2 eV as per (c) (see Methods). e, Experimental
$\mathrm{d}I/\mathrm{d}V$ signal at Fermi level ($V_{\mathrm{b}}=0$) as a
function of $x$ position, from (b). Increased
$\mathrm{d}I/\mathrm{d}V(V_{\mathrm{b}}=0$) indicates metallic phase. f, g,
Experimental (red; from b) and DMFT (blue; from d) LHBM (squares), UHBM
(circles) and energy gap $E_{\textrm{g}}$ (triangles), as a function of x in
(a) (experiment) or corresponding $E_{\textrm{F}}$ (DMFT).  Figure 4: Mott
insulator-to-metal transition controlled via tip-induced gating. a-c,
Schematics and energy diagrams of tunnelling and charging processes at a
double-barrier tunnel junction (DBTJ), consisting of STM tip, vacuum barrier
(vac.), MOF, hBN barrier and Cu(111), for small, intermediate (int.) and large
tip-sample distances ($\Delta z+z_{\mathrm{0}}$). When $V_{\textrm{b}}$ is
applied, voltage drop at MOF location enables tip-controlled charging, energy
level shifts and gating of MOF transitions, from (correlated) Mott insulator
to metal to trivial insulator. These schematics qualitatively illustrate our
proposed physical mechanism of tip-induced gating. d, d$I$/d$V$ spectra at MOF
DCA lobe site, at hBN/Cu(111) moiré wire, for different $\Delta
z+z_{\mathrm{0}}$ ($z_{\mathrm{0}}$ given by STM setpoint $V_{\textrm{b}}=$ 10
mV, $I_{\textrm{t}}=$ 10 pA). Purple circles (red squares): MOF charging peak
(intrinsic electronic state at MOF band edge). Spectra normalised and offset
for clarity. e, $V_{\textrm{charge}}$ (purple circles in d) and
$V_{\textrm{state}}$ (red squares in d) as a function of $\Delta z$. Black
solid lines: global fits to Eqs. (1) and (2). f, $\mathrm{d}I/\mathrm{d}V$
signal at Fermi level ($V_{\mathrm{b}}=0$) as a function of $\Delta z$, from
(d). Increased $\mathrm{d}I/\mathrm{d}V(V_{\mathrm{b}}=0$) indicates metallic
phase at intermediate $\Delta z+z_{0}$. Inset: STM image of MOF on hBN/Cu(111)
showing site (blue triangle marker) where $\mathrm{d}I/\mathrm{d}V(\Delta z)$
measurements were performed ($V_{\mathrm{b}}=-1$ V, $I_{\mathrm{t}}=10$ pA,
scale bar: 2 nm).

See pages - of Lowe_et_al_DCA-Cu_hBN_Mott_SI.pdf


For every clause gadget, we have, once again, an execution
$\sigma^{\prime}\in\mathord{\operatorname{\text{L}}_{!}}$. Hence, we see that
the input-language of every run can be restricted to the bounded language
$(\mathsf{1}^{*}\bar{\mathsf{1}}^{*}\ldots\mathsf{n}^{*}\bar{\mathsf{n}}^{*}\\#^{*})^{m+1}$.
Furthermore, while there are loops in the FIFO machine, it can be seen that no
loop can be executed infinitely often. We can also see that along every run,
the channel is bounded and the size of the channel does not exceed $n+1$.

The given 3-CNF formula is satisfiable iff the control state of the cleanup
gadget can be reached with the channel being empty. Hence, this constitutes a
reduction to the reachability problem. Furthermore, if we add a self loop to
the state of the cleanup gadget, such that it sends the letter $\\#$ to the
channel, then this loop can be iterated infinitely often to add unboundedly
many occurrences of the letter $\\#$ to the channel. Now, the given 3-CNF
formula is satisfiable iff the constructed FIFO machine is unbounded iff
channel is unbounded iff there is a non-terminating run. Hence reachability,
unboundedness and non-termination are all NP-hard.

######  Remark 34.

We can remove the $\varepsilon$-transitions in the above construction by
instead non-deterministically choosing one of the three variables per clause.
This would eliminate all $\varepsilon$-transitions, and corresponds to the
model we define, which does not have them.

We can adapt the proof above to the more restrictive case of FIFO machines
whose input language is restricted to a distinct-letter-bounded language, by
modifying the transitions in the gadget for clause $C_{i}$ as follows: For
every transition sequence of the form $?\mathsf{a}!\mathsf{a}$, we replace it
by $?\mathsf{a}_{i-1}!\mathsf{a}_{i}$, thereby ensuring that we write
different letters to the channel in every gadget. The transitions for the
gadgets for each variable $x_{k}$ (when it is set initially) would be modified
by $!\mathsf{k}_{0}$ and $?\bar{\mathsf{k}}_{0}$.

Furthermore, we can modify gadgets for the clauses (see Figure 7) in order to
have the following corollary, which improves a similar result for flat FIFO
machines with _multiple_ channels in [FP19].

######  Corollary 35.

For flat FIFO machines with a single channel, reachability, unboundedness, and
non-termination are NP-hard, hence, they are also NP-complete.

$\varepsilon$$\varepsilon$$\varepsilon$$?\mathsf{1}!\mathsf{1}$$?\mathsf{2}!\mathsf{2}$$?\bar{\mathsf{2}}!\bar{\mathsf{2}}$$\ldots$$?\mathsf{n}!\mathsf{n}$$?\bar{\mathsf{n}}!\bar{\mathsf{n}}$$?\\#!\\#$$?\mathsf{1}!\mathsf{1}$$?\bar{\mathsf{1}}!\bar{\mathsf{1}}$$?\bar{\mathsf{2}}!\bar{\mathsf{2}}$$\ldots$$?\mathsf{n}!\mathsf{n}$$?\bar{\mathsf{n}}!\bar{\mathsf{n}}$$?\\#!\\#$$?\mathsf{1}!\mathsf{1}$$?\bar{\mathsf{1}}!\bar{\mathsf{1}}$$?\mathsf{2}!\mathsf{2}$$?\bar{\mathsf{2}}!\bar{\mathsf{2}}$$?\bar{\mathsf{3}}!\bar{\mathsf{3}}$$\ldots$$?\mathsf{n}!\mathsf{n}$$?\bar{\mathsf{n}}!\bar{\mathsf{n}}$$?\\#!\\#$$?\mathsf{1}$$\varepsilon$$?\bar{\mathsf{1}}$$\varepsilon$$\ldots$$?\\#$
(a) Gadget for clause $C_{1}=x_{1}\lor\neg x_{2}\lor\neg x_{3}$ (b) Gadget for
cleanup Figure 7. Showing NP-hardness for flat systems with a single channel

######  Remark 36.

When considering FIFO systems of communicating processes with two FIFO
channels (and not FIFO machines with a single channel), the deadlock and the
unboundedness are PSPACE for systems of two processes with two one-directional
FIFO channels if (at least) one channel is letter-input-bounded [GGLR87]. We
conjecture that this result can be upgraded to input-bounded FIFO systems.

##  8\. Conclusion and Perspectives

We extend recent results of the _bounded verification_ of communicating
finite-state machines (equivalently FIFO machines) [EGM12] and of _flat_ FIFO
machines [FP19] by using bounded languages for controlling the input-languages
of FIFO channels (and not for controlling the runs of the machine). We extend
old and recent results about input-bounded FIFO machines (see Table 1). In
particular, we introduce the rational-reachability problem, which subsumes
most of the well-known variants of reachability problems like: the (classical)
reachability problem, the control-state reachability problem, and the deadlock
problem. We also unify the terminology to facilitate the comparison between
results. Moreover, note that, for most problems (except general/rational
reachability), we can reduce _output-bounded_ reachability to an equivalent
input-bounded problem. There are still many open problems and challenges:

  * •

What is the precise complexity of the five problems for input-bounded FIFO
machines with a fixed number of channels?

  * •

What is the precise complexity of control-state reachability, deadlock,
unboundedness, and termination for input-bounded FIFO machines?

  * •

The size of the counter machine associated with a FIFO machine and a tuple of
bounded languages is exponential, but only polynomial when we start from a
normal form. It will be interesting to see whether the use of existing tools
for counter machines is feasible for the verification of FIFO machines from
case studies. Case studies shall also reveal how many FIFO machines/systems
are actually boundable and/or flattable.

Table 1. Summary of key results; results for all other extensions are subsumed by these results (D stands for decidable). | Flat | Letter-bounded | Bounded
---|---|---|---
UNBOUND | NP-C ([FP19]) | D ([GGLR87]) | D ([JJ93])
TERM | NP-C ([FP19]) | D | D
REACH | NP-C ([FP19]) | D | D, not ELEM
CS-REACH | NP-C ([EGM12, FP19]) | D | D
DEADLOCK | D | D ([GGLR87]) | D

Towards a theory of boundable FIFO machines. In Example 3, we have seen that
all configurations that are reachable in the CDP protocol are already
reachable in presence of a suitable collection $\operatorname{\text{L}}$ of
bounded input-languages. By analogy with the well-established theory of
_flattable_ machines [BFLP08, DFGvD10, CFS11], we propose the following
definition. Let ${M}$ be a FIFO machine and let $\operatorname{\text{L}}$ be a
tuple of regular bounded languages. We say that ${M}$ is
_$\operatorname{\text{L}}$ -boundable_ if
$\textit{Reach}_{{M}}=\mathit{Reach}_{{M}}(\operatorname{\text{L}}_{!})$. We
say that ${M}$ is _boundable_ if there exists a tuple
$\operatorname{\text{L}}$ of regular bounded languages such that ${M}$ is
_$\operatorname{\text{L}}$ -boundable_. Hence, we deduce that reachability is
decidable for $\operatorname{\text{L}}$-boundable FIFO machines, which is a
_strictly larger_ class than input-bounded machines. CDP is not input-bounded
but it is $\operatorname{\text{L}}_{\mathit{CDP}}$-boundable with
$\operatorname{\text{L}}_{\mathit{CDP}}=((ab)^{*}(a+\varepsilon)(ab)^{*},e^{\ast})$.
Let us also remark that CDP is flattable by using the bounded set of runs
$(!a!b)^{*}!a!e?e(!a!b)^{*}+(!a!b)^{*}$ (where we omit channel information for
readability), because it covers the reachability set which is equal to
$(ab)^{*}(a+\varepsilon)(ab)^{*}$ on control-state $(0,0)$. It is not clear
whether reachability is decidable for boundable machines. A strategy that
would fairly enumerate _all_ regular bounded families
$\operatorname{\text{L}}_{1},\operatorname{\text{L}}_{2},\ldots,\operatorname{\text{L}}_{n},\ldots$
will necessarily find the good one, if ${M}$ is boundable, but this is not
sufficient because we must be able to _recognize_ $Reach_{M}$. Observe that
boundable machines are more robust than flat machines. Consider a system
$\mathcal{S}=(\mathcal{A}_{1},\mathcal{A}_{2},\ldots,\mathcal{A}_{n})$ of $n$
flat finite automata $\mathcal{A}_{i}$ communicating peer to peer (P2P)
through one-directional FIFO channels. Let $M_{\mathcal{S}}$ denote FIFO the
machine obtained as the Cartesian product of all automata $\mathcal{A}_{i}$ of
$\mathcal{S}$; there is no reason to assume that $M_{\mathcal{S}}$ is
flattable but it is input-bounded and thus $M_{\mathcal{S}}$ is
$\operatorname{\text{L}}$-boundable where $\operatorname{\text{L}}$ is easily
computable from $\mathcal{S}$.

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