# Correlation-induced magnetism in substrate-supported 2D metal-organic
frameworks

Bernard Field  School of Physics and Astronomy, Monash University, Clayton,
Victoria 3800, Australia  ARC Centre of Excellence in Future Low-Energy
Electronics Technologies, Monash University, Clayton, Victoria 3800, Australia
Agustin Schiffrin  School of Physics and Astronomy, Monash University,
Clayton, Victoria 3800, Australia  ARC Centre of Excellence in Future Low-
Energy Electronics Technologies, Monash University, Clayton, Victoria 3800,
Australia  Nikhil V. Medhekar  ARC Centre of Excellence in Future Low-Energy
Electronics Technologies, Monash University, Clayton, Victoria 3800, Australia
Department of Materials Science and Engineering, Monash University, Clayton,
Victoria 3800, Australia<EMAIL_ADDRESS>
###### Abstract

Two-dimensional (2D) metal-organic frameworks (MOFs) in a kagome lattice can
exhibit strong electron-electron interactions, which can lead to tunable
quantum phases including many exotic magnetic phases. While technological
developments of 2D MOFs typically take advantage of substrates for growth,
support, and electrical contacts, investigations often ignore substrates and
their dramatic influence on electronic properties of MOFs. Here, we show how
substrates alter the correlated magnetic phases in kagome MOFs using
systematic density functional theory and mean-field Hubbard calculations. We
demonstrate that MOF-substrate coupling, MOF-substrate charge transfer,
strain, and external electric fields are key variables, activating and
deactivating magnetic phases in these materials. While we consider the example
of kagome-arranged 9,10-dicyanoanthracene molecules coordinated with copper
atoms, our findings should generalise to any kagome lattice. This work offers
useful predictions for tunable interaction-induced magnetism in surface-
supported 2D organic materials, opening the door to solid-state electronic and
spintronic technologies based on such systems.

## Introduction

Two-dimensional (2D) materials with flat electronic bands, such as the kagome
lattice, can host localised electronic wavefunctions, which can lead to strong
electron-electron interactions. These strong interactions lead to diverse
electronic phases such as fractional quantum Hall states1, 2,
superconductivity3, 4, or different magnetic phases, including ferromagnetism,
antiferromagnetism, and spin glasses5, 6. Such magnetic phases are critical
for emerging technologies such as spintronics and magnetic information
storage7. Kagome lattices which can host such phases have been realised in
many different systems. While inorganic systems such as herbertsmithite and
FeSn are well-known examples, a class of materials of growing interest are 2D
metal-organic frameworks (MOFs), which combine transition metal ions with
planar organic molecules into a self-assembled crystal. 2D MOFs have potential
applications in functional electronics, leveraging the intrinsic electrical,
sensing, or catalytic properties of these systems8, 9, with promise for
increased versatility, tunability, and affordability. For practical solid-
state applications, these 2D MOFs are required to interface with substrates
and solid interconnects, either for structural support, to enable bottom-up
synthesis10, 11, 12, or as metallic contacts. However, the interplay between
the various influences of the substrate and the intrinsic properties of the
MOF is not understood.

The kagome lattice is an archetypal 2D flat-band crystal structure, with a
three-site basis arranged in a six-pointed star motif (Fig. 1b). Its
electronic band structure has a pair of Dirac bands, similar to graphene. The
frustrated geometry from the triangular three-atom basis allows for
destructive interference which localises the electronic wavefunctions and
creates an electronic flat band (in addition to the 2 Dirac bands). This
localisation and narrow bandwidth enhances electronic interactions13, which
can lead to phenomena driven by many-body electron correlations, including
magnetism (due to energy cost, from Coulomb interactions, of two electrons
occupying the same site exceeding the kinetic energy). This geometry also
allows kagome lattices to host exotic spin behaviour, such as spin liquids,
frustrated antiferromagnetism, intrinsic Kondo resonances, and spin density
waves6, 14, 15, 16, which can be explored by tuning the interactions16, 17.
Furthermore, as in other Dirac materials, spin-orbit coupling can lead to
topological quantum spin Hall phases18. The combination of geometry, topology,
and interactions makes the kagome lattice a rich playground of electronic
quantum phenomena.

Of the material systems which kagome lattices have been realised in, an
emerging class of materials are 2D MOFs. In MOFs, organic ligands connect to
transition metal atoms via coordination chemistry and form an extended crystal
lattice. By using organic chemistry to alter the ligands and choosing between
a large number of transition metal ions, MOFs demonstrate incredible
versatility and tunability. 2D MOFs can be electrically conducting, giving
them applications in functional electronics8, 9. MOFs also offer room-
temperature synthesis8 which makes them cheaper to produce than inorganic
materials. Kagome MOFs have attracted much attention in the search for
topological insulators19, 20, 21, 22. However, such searches rarely
investigate interactions and the resultant magnetism which can compete with
topology23.

In particular, the competing influences of the substrate to interactions in
MOFs are often overlooked. Substrates support, template, and enable the
bottom-up synthesis of 2D MOFs10, 11, 12. These substrates can facilitate new
functionalities via effects such as charge doping24, 25, hybridisation26, 27,
28, strain29, symmetry breaking30, rearrangement of bonds facilitating
magnetism31, and the Kondo effect32, 33, 34, 35, 36, 37. Substrates are
sometimes deleterious, such as through excessive hybridisation or unfavourable
symmetry breaking30, and sometimes beneficial, such as through mediating
magnetic interactions34, 38 or favourable charge transfer39. Clearly, the
influence of substrates on strongly-correlated MOFs or on correlated magnetism
is important, but it has not been thoroughly investigated.

An ideal example of a correlated kagome MOF is 9,10-dicyanoanthracene-copper
(DCA-Cu) — a 2D MOF consisting of kagome-arranged DCA molecules coordinated
with Cu atoms, where each Cu atom binds to 3 DCA molecules in a trigonal
planar geometry (Fig. 1b). DCA-Cu has been experimentally synthesised on
Ag(111)37, Cu(111)40, 41, 42, graphene24, and NbSe243. DCA-Cu has clean kagome
bands, meaning it is well-described by a simple nearest-neighbour tight-
binding model. DCA-Cu has been predicted to be a topological insulator, with
substitution of metal atoms enhancing this effect44. DCA-Cu has also been
predicted to have strong electron-electron interactions 45, 24, making it a
good system to investigate correlated physics. A sign of the electronic
interactions in DCA-Cu are the magnetic moments it hosts when on Ag(111)37.
Unlike MOFs with intrinsically magnetic components, where the magnetic moments
are fixed, the variable magnetism in DCA-Cu arises due to electronic
interactions which cause otherwise mobile electrons to become spin-polarised,
as described by the Hubbard model37. However, magnetic moments in DCA-Cu have
not been observed on other substrates and it is unclear why some substrates
support magnetism in this MOF while others suppress it.

Here we show how the coupling between a kagome lattice and a substrate
influences electron-electron interactions, leading to the emergence or
suppression of magnetic moments and magnetic order. We investigate the
emergent magnetism in DCA-Cu on several substrates (Ag(111), Cu(111), Au(111),
Al(111), graphite, and hexagonal boron nitride (hBN)) using a combination of
density functional theory (DFT) and a mean-field Hubbard model. The systematic
DFT calculations provide a quantitative description of electronic and magnetic
properties of DCA-Cu, including the emergence of magnetically ordered phases.
The Hubbard model distills this complicated system into a small number of key
ingredients, including MOF-substrate coupling, while being generic to any
kagome lattice on a substrate. Both DFT and Hubbard approaches consistently
show that certain substrate parameters, namely the density of states, MOF-
substrate charge transfer, MOF-substrate electronic hybridisation, and
substrate-induced MOF tensile strain, can dramatically affect the MOF magnetic
properties (Fig. 1c). Our work quantitatively isolates the effect of each of
these phenomena on the system’s magnetic properties. Sensitivity to charge
transfer allows switching the magnetic phases by an external electric field,
which we explicitly demonstrate in DCA-Cu, suggesting possible applications in
energy-efficient electric-field–controlled spintronics46, 47. Understanding
how different parameters affect the magnetism in this MOF allows us to tune
the electron-electron interactions and magnetic phases, a feature not offered
by intrinsically magnetic systems. By developing a framework for analysing the
effects of a substrate on an adsorbed 2D system, we present useful tools for
the rational design of novel electronic devices whose functionality relies on
correlations-induced quantum phenomena.

Figure 1: Schematic depiction of interaction-induced magnetism in a 2D MOF and
how substrates influence it. a Ball-and-stick models of DCA and Cu, the
components of the MOF, and their isolated electronic structures. The copper(I)
ions and DCA molecules in the MOF are not intrinsically magnetic. b Ball-and-
stick model of DCA-Cu kagome MOF. After self-assembly, intra-MOF electron-
electron Coulomb interactions induce magnetic moments in the MOF. The MOF has
a kagome electronic band structure derived from DCA LUMO (non-spin-polarised
schematic shown). c Ball-and-stick model of DCA-Cu on a substrate and
schematic band structures. The magnetism in the MOF is influenced by coupling
to the substrate, charge transfer either out of or into the MOF (such as by an
applied electric field), and strain, which alters the bandwidth. The strength
of the magnetic moments can be enhanced by using a weakly interacting
substrate, favourable electron filling of the MOF by choice of substrate work
function or application of electric fields, and by applying tensile strain to
the MOF such as by lattice mismatch. (HOMO: highest occupied molecular
orbital. LUMO: lowest unoccupied molecular orbital)  Figure 2: DFT+$U$ band
structures and magnetization densities of DCA-Cu (a,d) without a substrate,
(b,e) on Ag(111), and (c,f) on hBN on Cu(111). Strong electron-electron
Coulomb interactions separate spin up and down bands (red and blue
respectively) and lead to local magnetic moments on DCA molecules. On Ag(111)
(b), substrate coupling distorts the bands, while on hBN (c) coupling is
negligible so bands retain free-standing character (a). Circle radius in band
structures is proportional to projection of bands onto the atoms of DCA-Cu.
Inset to a is the Brillouin zone. Overlaid in d is the atomic structure of
free-standing DCA-Cu. Figure 3: Charge transfer and local magnetic moments of
DCA-Cu MOF on different substrates and under different conditions, calculated
using DFT+$U$. These local magnetic moments lead to magnetically ordered
phases (Figs. 2 and 5). Two different charge partitioning methods set bounds
for the MOF-substrate charge transfer: Bader48 and DDEC49, 50. A third method,
integrating the projected density of states, gives values between the Bader
and DDEC results (Supplementary Fig. A.1). E refers to the electric field
applied normal to the substrate. The grey shaded region marks the results
calculated within the mean-field Hubbard model with a single unit cell,
without a substrate, and with interaction parameter $U_{H}$ between $6t$ and
$7t$, presented for comparison.

## Results

### Electronic and magnetic structure of DCA-Cu

To understand the effect of real substrates on real materials, we use DCA-Cu
as an illustrative example and perform systematic DFT+$U$ calculations for
free-standing DCA-Cu as well as for different substrates (see Methods). The
use of Hubbard $U$ corrections treats the correlations in $d$ electrons and
are essential for an accurate description of magnetic moments in DCA-Cu on
Ag(111)37. We report results for $U=$3\text{\,}\mathrm{eV}$$; besides the
Ag(111) results, varying $U$ within a reasonable range does not cause any
qualitative changes (see Supplementary Fig. A.1 and Ref. 37). From these
calculations, we determine which substrates enhance or suppress local magnetic
moments in DCA-Cu and identify the contribution of charge transfer. We also
address how an externally applied electric field affects these local magnetic
moments.

We first investigate the electronic structure of free-standing DCA-Cu without
a substrate. For calculations without spin polarisation, the band structure of
free-standing DCA-Cu matches an ideal nearest-neighbour tight-binding kagome
band structure (Fig. 1b and Supplementary Fig. A.2a). Electronically, each Cu
atom donates one electron to the DCA lowest unoccupied molecular orbitals
(Fig. 1a), which form the kagome bands at the Fermi energy. This integer
charge transfer pins the Fermi level at the Dirac point and leaves the Cu+
ions in the nominally non-magnetic [Ar]$3d^{10}$ configuration. When spin is
included, electron-electron interactions lead to spin-splitting of bands and
magnetization of DCA orbitals (Fig. 2a,d). The system prefers a frustrated
antiferromagnetic configuration. This magnetization cannot be explained by
simple counting of unpaired spins as the magnetic moments are a fraction of a
Bohr magneton, instead requiring electronic interactions as in the Hubbard
model to partially hinder electron pairing37. As we show next, these
interactions are altered drastically when the 2D MOF is in contact with a
substrate.

### Emergent magnetism in DCA-Cu on substrates

To elucidate the substrate’s influence on the electronic and magnetic
structure of DCA-Cu, we sample a broad selection of different substrates. We
choose substrates with a range of work functions and lattice constants and a
mixture of metals and gapped materials (Supplementary Table A.1). For metals,
we consider Ag(111), Cu(111), Au(111), and Al(111), which are structurally
similar but have differing work functions. We consider graphite, which is a
semimetal so has a much lower density of states and can be regarded as multi-
layer graphene in this context. And we consider hexagonal boron nitride (hBN),
a wide band gap insulator commonly used in 2D heterostructures with potential
for electronic devices, on Cu(111), a common surface for growing hBN51. While
not a comprehensive survey of all possible substrates, this selection
represents a sufficient sample size for identifying the essential trends.

We find that of the six substrates considered, only two retain local magnetic
moments in DCA-Cu, as shown in Fig. 2. One of them is Ag(111), consistent with
previous experiments37. The band structure of DCA-Cu on Ag(111) shows
substantial distortion of the kagome bands, indicating non-negligible MOF-
substrate coupling (Fig. 2b). In spite of this, magnetization of DCA orbitals
persists on Ag(111) (Fig. 2e). The other substrate to retain magnetic moments
is hBN on Cu(111). The kagome band structure does not exhibit any avoided
crossings, behaving similarly to free-standing DCA-Cu (Fig. 2c, Supplementary
Fig. A.2b). Compared to free-standing DCA-Cu, the shifted Fermi level
indicates that DCA-Cu receives electrons from the substrate, and also has
local magnetic moments with greater magnitude (Fig. 2f).

The other substrates which were considered — graphite, Cu(111), Au(111), and
Al(111) — all suppress the local magnetic moments in DCA-Cu (Supplementary
Fig. A.3). This observation is consistent with experiments of DCA-Cu on
Cu(111), which also did not find magnetic moments37, 40. The avoided crossings
in the band structures indicate that Cu(111), Au(111), and Al(111) are
strongly interacting (just like Ag(111)) while graphite is weakly interacting.
The bands also clearly indicate that Au(111) fully depopulates the DCA-Cu
kagome bands, Al(111) fully populates the DCA-Cu kagome bands, and graphite
slightly reduces the electron occupation of DCA-Cu.

This raises the important question: why do some substrates allow magnetism in
the MOF while others do not? Figure 3 shows the influence of charge transfer
between the substrate and MOF on the local magnetic moments on DCA. Free-
standing DCA-Cu with varying charge is also included as a control. Determining
an unambiguous charge transfer can be challenging due to electron density
partitioning effects. Therefore we use multiple methods to estimate plausible
bounds for the charge transfer (see Methods and Supplementary Notes). Free-
standing DCA-Cu shows that local magnetic moments are maximised when the
orbitals are half-filled (one electron per orbital) and weaken rapidly away
from half-filling. This trend is corroborated on substrates. Substrates with
electron fillings closer to half-filling (Ag(111), hBN/Cu(111)) show magnetic
moments, while substrates which deplete electrons (Cu(111), Au(111)) or
completely fill the bands (Al(111), see also Supplementary Fig. A.3c) do not
show any magnetic moments. Graphite, while having much weaker MOF-substrate
coupling than bare metals, also has very weak magnetic moments due to a low
electron occupation. Overall, it is clear that electron occupation of the MOF
is a significant factor in determining whether or not magnetic moments occur
in the MOF. However, this does not address the effect which MOF-substrate
hybridisation plays on the magnetic moments. For that we turn to a simplified
model.

Figure 4: Schematic representation of our physical model of emergent magnetism
in substrate-supported 2D MOFs. The 2D lattice is treated within the Hubbard
model with nearest-neighbour hopping $t$ (tunable by strain) and Coulomb
repulsion $U_{H}$. The substrate is modelled as having plane wave states with
dispersion $\epsilon(\mathbf{k})$ (derived from ab initio calculations).
Coupling between the substrate and the lattice is assumed to be point-like
hopping in real-space, with strength $V$. The lattice sites have an on-site
energy, which alters the electron occupation of the lattice based on its
alignment with the substrate chemical potential. Figure 5: Emergent magnetism
in the kagome lattice from our mean-field Hubbard model. a Select real-space
magnetization densities in a free-standing kagome lattice. b Magnetic phase
diagram of the free-standing kagome lattice in a supercell. The dotted region
indicates the transition from antiferromagnetic to ferromagnetic phase, with a
partial net polarisation. ‘Pin’ refers to a pinned metal droplet phase16.
Triangular markers indicate phase space locations corresponding to a. c Effect
of effective hybridisation $V^{2}\rho$ and MOF hopping parameter $t$
(proportional to lattice bandwidth) on the strength of magnetic moments for
different substrates. The substrate density of states $\rho$ is taken as the
mean DOS of the substrate dispersions over $\pm 0.2\text{\,}\mathrm{eV}$
around the Fermi level. The kink in the $30\text{\,}\mathrm{meV}$ graphite
curve is due to an antiferromagnetic to ferromagnetic phase transition (see
Supplementary Note A.1). Data obtained using
$U_{H}=$300\text{\,}\mathrm{meV}$$ and an electron occupation of approximately
2/3. d Effect of electron occupation and coupling to a substrate on the
strength of magnetic moments. Magnetic phases can withstand higher coupling
with the substrate at higher electron occupations than lower electron
occupations. Data obtained using a single kagome unit cell,
$U_{H}=$300\text{\,}\mathrm{meV}$$, $t=$40\text{\,}\mathrm{meV}$$, and the
Ag(111) dispersion.

### Mean-field Hubbard model for emergent magnetism in a kagome lattice on a
metal

To explore and rationalise the essential ingredients which lead to the physics
observed via the DFT calculations, we develop a simplified mean-field model
(Fig. 4). We model the kagome MOF as a tight-binding lattice with nearest-
neighbour hopping $t$ and on-site Coulomb interactions $U_{H}$, as in the
Hubbard model13. In this model, electrons hop between adjacent sites due to
$t$ and are repelled if they share a site with another electron due to
$U_{H}$, which will be of the opposite spin due to the Pauli exclusion
principle. Note that the $U_{H}$ used here is different from $U$ in the
DFT+$U$ calculations; the former describes all electronic interactions
including the DCA molecular orbitals, while the latter is a correction to
electronic correlations in Cu $d$ orbitals. In our model, electrons in the
substrate propagate as plane waves with periodic energy dispersion
$\epsilon(\mathbf{k})$ which we derive from DFT calculations for metal slabs.
Electrons can hop between the point-like MOF sites and the substrate with
coupling amplitude $V$, as in the Anderson impurity model52. This simplified
model allows for capturing and exploring different magnetically ordered
phases, interpreting our ab initio DFT results, and identifying general trends
in parameter space.

First we demonstrate the emergent magnetism in a free-standing kagome lattice
using our Hubbard model. As shown in Figure 5a-b, when $U_{H}$ is small
compared to the bandwidth ($6t$) the system is paramagnetic, while the local
magnetic moments have large magnitude when $U_{H}$ is large or $t$ is small.
The local magnetic moments are maximal near half-filling ($n=1$) with whole
spins sitting on each site in an Ising-like configuration. The frustrated
antiferromagnetic phase consists of intricate spin textures, as indicated in
Fig. 5a, which arise due to geometric frustration from the kagome lattice and
energetically favourable antiferromagnetic interactions. Some of these spin
textures are periodic spin-density-wave phases53, 23, while others are
disordered, potentially resembling spin liquids6 or spin ices. Precise
classification of disordered magnetic phases in the kagome lattice is beyond
the scope of this study; we focus instead on the strength of individual
magnetic moments. Away from half-filling the system transitions towards a
ferromagnetic phase, passing an intermediate, partially polarised phase with
pinned minority spins16. Fundamentally, larger interactions $U_{H}$, smaller
hopping $t$, or electron occupations $n$ closer to one per site enhance the
magnetic moments. Stronger electron-electron interactions within the kagome
lattice lead to less pairing of electron spins, and electron occupations
closer to one per site allow more unpaired electrons.

With the substrate-free results as a baseline, we next investigate magnetism
in a kagome lattice coupled to metallic substrates, specifically Ag(111),
Cu(111), and graphite (Fig. 5c). Figure 5c shows the local magnetic moments of
kagome lattices with different hopping $t$ coupled to different substrates
with respect to the effective hybridisation between the substrate and the
lattice. This effective hybridisation can be expressed, to first order, as
$V^{2}\rho$, where $\rho$ is the substrate density of states (DOS) near the
Fermi level, similar to the Anderson impurity model52 (also see Supplementary
Fig. A.4). When this effective hybridisation is increased, either with larger
MOF-substrate coupling $V$ or higher DOS, the local magnetic moments decrease
until the system becomes paramagnetic above some critical hybridisation. Small
deviations between substrates come from finer details of the band structure,
such as non-uniform DOS over the kagome bandwidth. Nevertheless, this scaling
brings substrates with very different DOS to similar values, indicating that
it captures the main contributions — greater coupling with the substrate or
higher substrate DOS reduces the local magnetic moments in the kagome lattice.
It is also evident from Fig. 5c that increasing intra-kagome hopping $t$
lowers the critical hybridisation at which the system becomes paramagnetic. As
the hopping between sites is proportional to the overlap between the localised
orbitals and is inversely proportional to the distance between molecules, this
observation points to a tantalising prospect for tuning the magnetic phases
with externally induced strain, as we show in the next section.

We next investigate the influence of electron occupation on the magnetic
moments, as shown in Fig. 5d. Reducing the electron occupation significantly
below half-filling ($n=1$) substantially lowers the critical coupling (the
transition from magnetic to paramagnetic). The critical coupling does not
decrease when $n$ is increased above half-filling until the bands are almost
entirely filled, a difference which may come from particle-hole asymmetry.
Figure 5d does not contain a ‘pin’ phase like Fig. 5b because it was obtained
with a single kagome unit cell rather than a supercell for computational ease.
See Supplementary Fig. A.5 for the phase diagram for a free-standing single
cell for a more direct comparison. Overall, electron occupation is an
important variable for controlling local magnetic moments, which can be tuned
by choice of substrate work function or application of electric fields (i.e.
electrostatic gating), as we illustrate in the next section.

We estimate values for parameters in our mean-field Hubbard model from the DFT
calculations. (See Supplementary Fig. A.6 for consistency between our mean-
field Hubbard model and DFT.) The hopping parameter $t$ can be obtained from
the bandwidth of non-spin-polarised free-standing DCA-Cu, giving
$t=$49.2\text{\,}\mathrm{meV}$$, although this varies on substrates with
different lattice constants as described in the next section. Interactions can
be estimated from fitting the magnetic moments between DFT and mean-field
Hubbard results. We obtained $U_{H}\simeq 0.29-$0.35\text{\,}\mathrm{eV}$$ by
fitting the magnetic moments between DFT and mean-field Hubbard results for
free-standing DCA-Cu. Electron occupation is obtained directly from DFT (Fig.
3). MOF-substrate coupling can be estimated by fitting tight-binding band
structures ($U_{H}=0$) with non-spin-polarised DFT band structures
(Supplementary Fig. A.7). While this fitting is very coarse due to simplified
modelling of the substrate in our toy model, on Ag(111) and Cu(111) we
obtained $V\simeq$0.2\pm 0.05\text{\,}\mathrm{eV}$$. These fitted parameters
provide a connection between our simple model and concrete ab initio results.

Figure 6: Tailoring magnetism in DCA-Cu on substrates by strain and electric
fields. a Variation of hopping constants and local magnetic moments of free-
standing DCA-Cu due to strain. Vertical lines mark lattice constants of
different substrate supercells for reference. b Band structure and
magnetization density of DCA-Cu on graphite before and after applying strain.
c Band structure and magnetization density of DCA-Cu on Cu(111) before and
after applying an electric field normal to the plane pointed towards the
substrate, increasing electron filling. d Band structure and magnetization
density of DCA-Cu on hBN on Cu(111) before and after applying an electric
field in the opposite direction, decreasing electron filling. Legends in b-d
have the same meaning as in Fig. 2. Purple circles are spin degenerate.

### Tunable control over emergent magnetism: strain and electric fields

We have shown that the strength of the local magnetic moments in 2D MOFs
depends strongly on the molecule-molecule hopping $t$ and the electron
occupation in MOFs. Here we demonstrate how we can exploit this knowledge for
tunable external control over the magnetism in a substrate-supported 2D MOF.
First, the mean-field Hubbard model showed that decreasing molecule-molecule
hopping $t$ increases the local magnetic moments. Hopping is proportional to
orbital overlap, which is reduced by increasing the intermolecular distance
and lattice constant, that is, by applying tensile strain. Strain in a 2D MOF
can be introduced by either lattice mismatch with the substrate or by
mechanically straining the MOF-substrate system. We consider how strain
affects $t$ and local magnetic moments for free-standing DCA-Cu in Fig. 6a. As
expected, tensile strain reduces the bandwidth and hopping parameter $t$,
increasing the local magnetic moments. For example, 2% tensile strain reduces
$t$ in free-standing DCA-Cu from 49 to $43\text{\,}\mathrm{meV}$ and increases
the average magnetic moment from 0.26 to 0.36$\mu_{B}$. Conversely,
compressive strain increases $t$ and causes the system to become paramagnetic.

We illustrate how externally applied strain can be used to activate magnetism
in a substrate-supported MOF where magnetism would not otherwise be present.
Consider a graphite substrate, which in its equilibrium state exerts
compressive strain on DCA-Cu due to lattice mismatch. Due to this substrate-
induced compressive strain, DCA-Cu on graphite has considerably weaker local
magnetic moments than in its free-standing state (Fig. 3). If we apply tensile
strain, e.g. 2.7%, to the DCA-Cu/graphite system the local magnetic moments
double (Fig. 6b). This demonstrates that strain from either lattice mismatch
or external means is an important factor in tailoring the local magnetic
moments in 2D MOFs.

The results presented earlier showed that the strength of the local magnetic
moments depended strongly on the electron occupation of DCA. This electron
occupation can be altered by charge transfer to or from the substrate, either
by choice of a substrate with a suitable work function or by applying an
external electric field. By considering two different substrates, namely
Cu(111) and hBN on Cu(111), we illustrate how electric fields can alter the
magnetic moments.

Earlier, we showed that DCA-Cu on Cu(111) does not have any magnetism due to
its low electron occupation. However, when we apply an electric field of
$0.5\text{\,}\mathrm{eV}\text{\,}{\mathrm{\text{\r{A}}}}^{-1}$ towards the
substrate, the direction of charge transfer is reversed and DCA-Cu undergoes a
magnetic phase transition as shown in Fig. 6c, with local magnetic moments
similar to DCA-Cu on Ag(111) (Fig. 3). While DCA-Cu favours a ferromagnetic
configuration over a frustrated antiferromagnetic configuration in this case
(by $0.4\text{\,}\mathrm{meV}$; and by $1.4\text{\,}\mathrm{meV}$ over
paramagnetism), this may be due to substrate-mediated coupling of local
magnetic moments (Supplementary Note A.1). Conversely, we can also use
electric fields to drive a transition towards paramagnetism. By applying a
sufficiently strong electric field to DCA-Cu/hBN/Cu(111) in the opposite
direction, the MOF is depleted of charge (Fig. 3) and the local magnetic
moments vanish. These results are remarkable because electric fields can be
easily applied in a transistor device and do not rely on altering intrinsic
material parameters. The electric field allows for an external control knob
that tunes the local magnetization and magnetic moment ordering, opening the
doors for control of magnetic phases in 2D kagome materials by electrostatic
gating.

## Discussion

Our results, using DFT and a mean-field Hubbard model, show that substrate-
induced electron occupation, strain, and hybridisation control the emergent
magnetism in strongly correlated 2D MOFs. While we have used DCA-Cu as an
illustrative example in our DFT calculations, the consistency between DFT and
our more general Hubbard model indicates that our results can be generalised
to other kagome MOFs. By choice of the MOF metals and ligands, of the
substrate, and of external controls, it should be possible to explore more of
the phase space than that considered in this work. For example, changing the
components of the MOF will change hopping parameter $t$ (e.g. with longer
ligands, or tuning metal-ligand coordination strength via choice of functional
groups and metal atoms) and on-site interaction $U_{H}$ (e.g. with more
localised orbitals). These parameters can be calculated with a free-standing
DFT calculation, which is far cheaper than a calculation involving the
substrate. The effect of a substrate can then be included in a simple
effective model. This could be our mean-field Hubbard model, an extension to
the Hubbard model (e.g. spin-orbit coupling, longer range interactions), or
the Hubbard model with a more advanced treatment of correlations (e.g.
dynamical mean-field theory). Such modelling allows for easily exploring a
wide range of chemical motifs.

As we have shown, the substrate influences the emergent magnetism in 2D MOFs
by strain, charge transfer, and hybridisation. Consequently, a substrate can
be chosen to provide the desired strain (by lattice mismatch) and charge
transfer (estimated by work function differences). Some layered substrates,
such as hBN on Cu(111), exhibit long-ranged modulation of the work function
due to a Moire pattern54, which could result in spatial modulation of the
electron occupation and magnetic phase. For substrates with strong
hybridisation with MOFs, it may be possible to reduce the coupling strength by
adding decoupling functional groups to the MOF ligands (e.g. 55). By
understanding such trends, we can rapidly narrow the search space of candidate
substrates, aiding rational design.

We have also shown that we can adjust the emergent magnetism by external
controls such as electric fields and mechanical strain. Electrostatic gating
could be used to change the charge transfer, potentially allowing for
switchable magnetic phases. Our results indicate that an electron occupation
change of approximately 1 electron per unit cell, or
$2.5\text{\times}{10}^{13}\text{\,}{\mathrm{cm}}^{-2}$, is needed to change
the magnetic phases. If we consider a $10\text{\,}\mathrm{nm}$ thick parallel-
plate capacitor with hBN as a dielectric with a dielectric constant of 3.7656,
we get a switching voltage of $12\text{\,}\mathrm{V}$ and, for a 10 by
$10\text{\,}\mathrm{nm}$ device, a switching energy of
$24\text{\,}\mathrm{aJ}$. These voltages are experimentally feasible and the
energies are very low46, 47. If higher electric fields or capacitance is
required, electrolytic gates could be employed57. Applied strain, such as from
flexible, piezoelectric, or thermally expanding substrates29, can also
activate or suppress magnetic phases. These controls could be used to induce
magnetism in otherwise non-magnetic MOFs or to dynamically switch magnetism on
and off.

It is worth pointing out that our calculations capture electron correlations
at the mean-field level. The mean-field approximation allows for exploring
magnetic phases in the kagome lattice17, 16. However, effects such as the
Kondo effect37 or Mott transition are not captured in our calculations. Higher
levels of theory, such as dynamical mean-field theory, could capture these
effects and reveal further interesting physics58. Furthermore, next-nearest-
neighbour hopping was ignored in our Hubbard model because for DCA-Cu this is
negligible, as is common in MOFs (Supplementary Fig. A.2). For simplicity we
regarded only on-site Coulomb interactions in the Hubbard model, and although
inter-site Coulomb interactions can slightly alter the phase space45, it is
generally possible to replace inter-site interactions with an effective on-
site interaction59. Our DFT calculations also showed some minor changes to the
substrate due to the presence of the MOF beyond what we include in the Hubbard
model (e.g. graphite in Supplementary Fig. A.7), which may lead to differences
in the chosen magnetic ordering due to substrate-mediated exchange coupling.
These limitations notwithstanding, our mean-field Hubbard model is adequate
for interpreting our DFT results and identifying general trends in phase
space.

In conclusion, we have shown how choice of substrate controls interaction-
induced magnetism in metal-organic frameworks. MOF-substrate coupling,
electron occupation of the MOF, and MOF bandwidth are all critical factors for
interaction-induced magnetism (Fig. 1). These can be tuned by choice of
substrate (with density of states, lattice mismatch, and work function) or by
external controls such as strain or electric field. The use of electric fields
to control magnetism opens the door for electric-field-controlled 2D solid-
state devices for spintronics and quantum information technologies. While our
DFT results model DCA-Cu specifically, our Hubbard results are generalisable
to any kagome lattice on a substrate. MOFs (as well as purely organic
materials like covalent organic frameworks) benefit from synthesis protocols
via supramolecular chemistry, leveraging the versatile toolbox of organic
chemistry for tailoring atomic-scale 2D material structure. We envision
experiments where correlated-electron phases of 2D flat-band organic and
metal-organic materials – including the magnetic phases predicted in this work
as well as metal-to-Mott-insulator quantum phase transitions – can be observed
via electron transport measurements, optical and photoelectron spectroscopies
(including angle-resolved), and scanning probe microscopy techniques
(including spin-resolved). Insights gained here on the physics of substrate-
induced magnetism in MOFs can be expected to aid in the rational design of
MOFs with exotic properties in realistic device setups.

## Methods

### Density Functional Theory Calculations

Our density functional theory (DFT) calculations for the 2D MOF DCA-Cu on
various substrates were performed using Vienna Ab-initio Simulation Package
(VASP)60. The Perdew-Burke-Ernzerhof (PBE) functional under the generalised
gradient approximation (GGA) was used to describe exchange-correlation
effects61. Projector augmented wave (PAW) pseudopotentials were used to
describe core electrons62, 61. A semi-empirical potential developed by Grimme
(DFT-D3) was used to describe van der Waals forces63. For an accurate
description of electron correlations in $d$ electrons, we used Dudarev’s
implementation of DFT+U64, using $U=$3\text{\,}\mathrm{eV}$$ on DCA-Cu Cu
atoms (which is a modest value65, 66) unless otherwise mentioned. See
Supplementary Notes for further details.

For all substrates considered, we used experimental values of the substrate
lattice constants67, as given in Supplementary Table A.1. The charge transfer
between the substrate and the MOF was determined using Bader analysis48 and
DDEC charge analysis68, 49, 50, 69. Two further methods for calculating charge
transfer, namely integrating the projected density of states and fitting the
kagome band structure, were also used (see Supplementary Notes); they gave
values which fell between the Bader and DDEC results. The local magnetic
moments on the MOF were determined by partitioning the spin density then
taking the net spin on each DCA molecule. Both Bader and DDEC analysis methods
gave similar results for the magnetic moments.

Visualisation of atomic positions was performed using VESTA70.

### Mean-Field Hubbard Model

Our model Hamiltonian for the MOF-substrate system is

$\hat{H}=\hat{H}_{0}+\hat{H}_{U}+\hat{H}_{sub}+\hat{H}_{couple}.$ (1)

The first two terms are for the kagome lattice, describing nearest-neighbour
hopping and on-site Coulomb interactions respectively37, defined as

$\displaystyle\hat{H}_{0}$ $\displaystyle=-t\sum_{\langle
ij\rangle,\sigma}\hat{c}^{\dagger}_{i,\sigma}\hat{c}_{j,\sigma}+\varepsilon,$
(2) $\displaystyle\hat{H}_{U}$
$\displaystyle=U_{H}\sum_{i}\hat{n}_{i,\uparrow}\hat{n}_{i,\downarrow}$
$\displaystyle\approx
U_{H}\sum_{i}(\hat{n}_{i,\uparrow}\langle\hat{n}_{i,\downarrow}\rangle+\hat{n}_{i,\downarrow}\langle\hat{n}_{i,\uparrow}\rangle-\langle\hat{n}_{i,\uparrow}\rangle\langle\hat{n}_{i,\downarrow}\rangle),$
(3)

where $t$ is the hopping constant, $\varepsilon$ is the on-site energy,
$U_{H}$ is the Coulomb interaction, $\hat{c}^{\dagger}_{i,\sigma}$
($\hat{c}_{i,\sigma}$) creates (annihilates) an electron with spin $\sigma$ at
site $i$, and
$\hat{n}_{i,\sigma}=\hat{c}^{\dagger}_{i,\sigma}\hat{c}_{i,\sigma}$ is the
density operator. The Coulomb interactions are solved using the Hartree mean-
field approximation. The substrate is described by the term

$\hat{H}_{sub}=\sum_{\mathbf{k},\alpha,\sigma}\epsilon_{\alpha}(\mathbf{k})\hat{\psi}^{\dagger}_{\mathbf{k},\alpha,\sigma}\hat{\psi}_{\mathbf{k},\alpha,\sigma},$
(4)

where $\hat{\psi}^{\dagger}_{\mathbf{k},\alpha,\sigma}$ creates an electron
with spin $\sigma$ and momentum $\mathbf{k}$ as a plane wave in substrate band
$\alpha$ with dispersion $\epsilon_{\alpha}(\mathbf{k})$. We derive the
substrate dispersions from DFT calculations on a substrate slab. The coupling
between the substrate and kagome lattice is given by

$\hat{H}_{couple}=V\sum_{i,\alpha}(\hat{c}^{\dagger}_{i}\hat{e}_{\mathbf{r}_{i},\alpha}+\hat{e}^{\dagger}_{\mathbf{r}_{i},\alpha}\hat{c}_{i}),$
(5)

where $V$ is the coupling constant and $\hat{e}^{\dagger}_{\mathbf{r},\alpha}$
creates a localised electron in substrate band $\alpha$ at position
$\mathbf{r}$, and $\mathbf{r}_{i}$ is the position of kagome lattice site $i$.
We solve this model with a substrate (1) self-consistently in the grand
canonical ensemble. To get a fixed number of electrons in the kagome lattice,
we adjust the relative energy of the kagome lattice and the substrate until
the approximately correct electron number is obtained. Further details of the
Hamiltonian are in Supplementary Note A.5.

## Acknowledgements

We would like to thank Yuefeng Yin for helpful discussions. B.F. and N.V.M.
gratefully acknowledge the computational support from National Computing
Infrastructure and Pawsey Supercomputing Facility. B.F. is supported through
an Australian Government Research Training Program (RTP) Scholarship. N.V.M.
acknowledges funding support from the Australian Research Council (ARC) Centre
of Excellence in Future Low-Energy Electronics Technologies (CE170100039).
A.S. acknowledges funding support from the ARC Future Fellowship scheme
(FT150100426).

## Author Contributions

B.F. performed the calculations and analyses. B.F. wrote the custom codes.
A.S. and N.V.M. conceived of and supervised the project. All authors
participated in the writing of this manuscript.

## Competing Interests

The authors declare no competing interests.

## Data Availability

The data that support the findings of this study are openly available in
figshare at
https://doi.org/10.26180/19210632.v171

## Code Availability

The code which implements the mean-field Hubbard model in this work is openly
available on GitHub and archived in Zenodo at
https://doi.org/10.5281/zenodo.613128572

##  Appendix A Supplementary Information

###  A.1 Coupling-induced antiferromagnetic–ferromagnetic phase transition

In our mean-field Hubbard calculations, we observed that for some substrates
there was an antiferromagnetic–ferromagnetic transition with increasing $V$,
prior to the magnetic–paramagnetic transition. The kink in the
$t=$30\text{\,}\mathrm{meV}$$ graphite curve in Fig. 5c is an example of this
transition. We also observe a preference for ferromagnetic order when an
electric field is applied to DCA-Cu/Cu(111), despite being in a region of
phase space where the free-standing system would prefer a frustrated
antiferromagnetic configuration. This ferromagnetism could be due to a
substrate-mediated interaction, such at the Ruderman-Kittel-Kasuya-Yosida
interaction73. Identifying such interactions is beyond the scope of this study
and would require further investigations.

###  A.2 Parameters used for density functional theory calculations

In all our DFT calculations, a $400\text{\,}\mathrm{eV}$ cut-off was used for
the plane wave basis set. Ionic positions were relaxed until Hellman-Feynman
forces were less than
$0.01\text{\,}\mathrm{eV}\text{\,}{\mathrm{\text{\r{A}}}}^{-1}$, using
$3\times 3\times 1$ k-point grid for sampling the Brillouin zone, and
neglecting spin and $U$. On substrates we did this relaxation using 1st order
Methfessel-Paxton smearing with $\sigma=$0.2\text{\,}\mathrm{eV}$$. For free-
standing DCA-Cu we instead used Gaussian smearing with
$\sigma=$0.05\text{\,}\mathrm{eV}$$ during the relaxation. Charge and spin
density calculations used an $11\times 11\times 1$ k-point grid, energy
convergence criterion of ${10}^{-6}\text{\,}\mathrm{eV}$, Blochl tetrahedron
interpolation, and dipole corrections.

Bulk substrates were modelled as being three atoms thick. The bottom layer of
atoms was frozen at their bulk positions during structural relaxation. A layer
of passivating hydrogen atoms was applied to the bottom face of metallic
substrates (not graphite) to terminate dangling bonds. The 2D systems were
modelled by applying $15\text{\,}\mathrm{\text{\r{A}}}$ of vacuum spacing
between periodic images.

We used $U=$3\text{\,}\mathrm{eV}$$ for all main-text DFT results. Lack of a
quantitative experimental metric sensitive to $U$ such as band gap made
fitting to a precise value of $U$ impractical, such that we rely on comparing
multiple $U$ values. We find $U$ provides a small reduction to $t$ and
enhancement to the magnetic moments (see Supplementary Fig. A.1) but does not
otherwise cause any qualitative changes. This is true even for larger $U$37.
Based on past literature, $U=$3\text{\,}\mathrm{eV}$$ is a reasonable lower
estimate for Cu atoms in MOFs65, 66, especially considering that DCA-Cu is
known to host strong electronic correlations45, 37.

###  A.3 Determining charge from the projected density of states

We used Bader and DDEC analysis to obtain estimates for the charge transfer
between the MOF and substrate. To provide a third independent method for
determining the charge transfer, we also used the projected density of states
(PDOS). Taking the PDOS projected onto DCA-Cu atomic $spd$ orbitals, we
integrated the occupied PDOS and the total PDOS in an energy window around the
Fermi energy, then took their ratio. Multiplying the ratio by two to account
for spin degeneracy yields an estimate for the electron occupation of DCA-Cu.
A non-zero background PDOS due to hybridisation with the substrate renders
this method sensitive to the window, providing some uncertainty in the precise
value of the charge transfer. We integrated the PDOS from $-1$ to
$1\text{\,}\mathrm{eV}$ to get a central estimate for the charge transfer. We
obtained upper and lower estimates by integrating from $-1$ to
$0.5\text{\,}\mathrm{eV}$ and $-0.5$ to $1\text{\,}\mathrm{eV}$. The energy
window was further constrained when shoulders of adjacent bands encroached on
these windows. The charge transfer from the PDOS was found to lie between the
values obtained by Bader and DDEC and is shown in Supplementary Fig. A.1.

###  A.4 Determining charge from band structure

For substrates with negligible hybridisation with the MOF, namely DCA-
Cu/hBN/Cu(111), we also extracted the charge transfer by fitting a non-spin-
polarised band structure. Here we derive how the position of the Fermi level
relative to the band edges gives the electron occupation in an ideal kagome
band structure.

For a nearest-neighbour tight-binding model with hopping constant $t$ and
neglecting any offset, the kagome band structure is given by

$\epsilon_{1/2}(\mathbf{k})=t\left(-1\pm\sqrt{4\sum_{i}\cos^{2}(\mathbf{k}\cdot\mathbf{b}_{i}/2)-3}\right),\quad\epsilon_{3}(\mathbf{k})=2t,$
(A.1)

where the normalised lattice vectors can be given by

$\mathbf{b}_{1}=(\frac{1}{2},\frac{\sqrt{3}}{2}),\quad\mathbf{b}_{2}=(1,0),\quad\mathbf{b}_{3}=\mathbf{b}_{2}-\mathbf{b}_{1}.$
(A.2)

Contours of constant energy in k-space for the dispersive bands are given by

$k_{y}(k_{x},\epsilon)=\frac{2}{\sqrt{3}}\arccos\left(\frac{\epsilon^{2}/4t^{2}+\epsilon/2t-\cos^{2}(k_{x}/2)}{\cos(k_{x}/2)}\right)$
(A.3)

The electron occupation is simply the normalised area bounded by a contour of
constant energy in the Brillouin zone, times two to account for the two spin
channels. Assuming $t>0$,

$N(\epsilon)=\begin{cases}0&\text{ for }\epsilon\leq-4t\\\
2\times\frac{\sqrt{3}}{2}\frac{1}{(2\pi)^{2}}\times
4\int_{0}^{x_{1}(\epsilon)}k_{y}(k_{x},\epsilon)dk_{x}&\text{ for
}-4t\leq\epsilon\leq-2t\\\
2-2\times\frac{\sqrt{3}}{2}\frac{1}{(2\pi)^{2}}\times
4\int_{x_{1}(\epsilon)}^{x_{2}(\epsilon)}k_{y}(k_{x},\epsilon)dk_{x}&\text{
for }-2t\leq\epsilon\leq-t\\\
2+2\times\frac{\sqrt{3}}{2}\frac{1}{(2\pi)^{2}}\times
4\int_{x_{1}(\epsilon)}^{x_{2}(\epsilon)}k_{y}(k_{x},\epsilon)dk_{x}&\text{
for }-t\leq\epsilon\leq 0\\\
4-2\times\frac{\sqrt{3}}{2}\frac{1}{(2\pi)^{2}}\times
4\int_{0}^{x_{1}(\epsilon)}k_{y}(k_{x},\epsilon)dk_{x}&\text{ for
}0\leq\epsilon<2t\\\ 6&\text{ for }\epsilon\geq 2t\end{cases},$ (A.4)

where the integration bounds are

$\displaystyle
x_{1}(\epsilon)=2\arccos\left(-\frac{1}{2}+\sqrt{\frac{\epsilon^{2}}{4t^{2}}+\frac{\epsilon}{2t}+\frac{1}{4}}\right),\quad
x_{2}(\epsilon)=2\arccos\left(-\frac{1}{2}-\sqrt{\frac{\epsilon^{2}}{4t^{2}}+\frac{\epsilon}{2t}+\frac{1}{4}}\right).$
(A.5)

The number of electrons is then simply given by (A.4) at the Fermi level
relative to any overall shifting of the bands (A.1). For a given band minimum
$\epsilon_{min}$, band maximum $\epsilon_{max}$, and Fermi energy
$\epsilon_{F}$, the units are normalised and given to (A.4) as

$N\left(6\frac{\epsilon_{F}-\epsilon_{max}}{\epsilon_{max}-\epsilon_{min}}+2\right)\bigg{|}_{t=1}.$
(A.6)

Number of electrons $N$ can be converted to the electron occupation $n$ by
dividing by 3, the number of sites per unit cell.

###  A.5 Mean-field Hubbard Model

A non-interacting tight-binding model of a lattice with nearest-neighbour
hopping has a Hamiltonian of the form

$\hat{H}_{0}=-t\sum_{\langle
ij\rangle,\sigma}\hat{c}^{\dagger}_{i,\sigma}\hat{c}_{j,\sigma}+\varepsilon,$
(A.7)

where $t$ is the nearest-neighbour hopping constant, $\varepsilon$ is an
energy shift, and $\hat{c}^{\dagger}_{i,\sigma}$ ($\hat{c}_{i,\sigma}$)
creates (annihilates) an electron in the lattice at site $i$ with spin
$\sigma$. Local electron-electron interactions can be included by adding the
Hubbard term

$\hat{H}_{U}=U_{H}\sum_{i}\hat{n}_{i,\uparrow}\hat{n}_{i,\downarrow}$ (A.8)

to the Hamiltonian, where
$\hat{n}_{i,\sigma}=\hat{c}^{\dagger}_{i,\sigma}\hat{c}_{i,\sigma}$ is the
density operator and $U_{H}$ is the on-site Coulomb repulsion. Solving the
Hubbard model is difficult. We apply the Hartee approximation, assuming
collinear spins, giving us the mean-field Hamiltonian

$\hat{H}_{U}\approx
U_{H}\sum_{i}(\hat{n}_{i,\uparrow}\langle\hat{n}_{i,\downarrow}\rangle+\hat{n}_{i,\downarrow}\langle\hat{n}_{i,\uparrow}\rangle-\langle\hat{n}_{i,\uparrow}\rangle\langle\hat{n}_{i,\downarrow}\rangle).$
(A.9)

See Refs. 37, 72 for our implementation of this mean-field Hubbard model.

To model the substrate, we assume we have a substrate with some bands with a
known dispersion,

$\hat{H}_{sub}=\sum_{\mathbf{k},\alpha,\sigma}\epsilon_{\alpha}(\mathbf{k})\hat{\psi}^{\dagger}_{\mathbf{k},\alpha,\sigma}\hat{\psi}_{\mathbf{k},\alpha,\sigma},$
(A.10)

where $\hat{\psi}^{\dagger}_{\mathbf{k},\alpha,\sigma}$ creates an eigenstate
of the substrate with wavevector $\mathbf{k}$, band index $\alpha$, spin
$\sigma$, and eigenenergy $\epsilon_{\alpha}(\mathbf{k})$. The eigenenergies
may be derived from a simple tight-binding model, or they may be directly
calculated by DFT for a given substrate. In our calculations, we used
substrate dispersions obtained from DFT calculations using a single substrate
unit cell and $48\times 48\times 1$ k-points, without spin or $U$, but
otherwise with the same parameters as the other DFT calculations, including
the slab thickness. We assume the substrate eigenstates are simple plane waves
(that is, the substrate is homogeneous in real-space). This gives the relation
between momentum and position eigenstates

$\langle
0|\hat{e}_{\mathbf{r},\alpha}\hat{\psi}^{\dagger}_{\mathbf{k},\beta}|0\rangle=\frac{1}{\sqrt{NN_{R}}}e^{i\mathbf{k}\cdot\mathbf{r}}\delta_{\alpha,\beta},$
(A.11)

where $N$ is the number of substrate cells which fit within the lattice cell,
$N_{R}$ is the number of lattice cells in the entire system, and
$\hat{e}^{\dagger}_{\mathbf{r},\alpha}$ creates an electron in the substrate
in band $\alpha$ at the point $\mathbf{r}$. As a consequence of the substrate
eigenstates being homogeneous and to simplify calculations we also assume that
there is no Coulomb interaction within the substrate. We also assume that the
substrate is commensurate with the lattice supercell and that $N$ primitive
unit cells of the substrate fit within the supercell of the lattice. This
means for each band $\alpha$ in a single substrate cell we have $N$ copies of
that band in the supercell Brillouin zone, each shifted by a reciprocal
lattice vector.

Finally, we assume coupling between the substrate and lattice is point-like,
which gives us the coupling term

$\hat{H}_{couple}=\sum_{i,\alpha}V_{\alpha}(\hat{c}^{\dagger}_{i}\hat{e}_{\mathbf{r}_{i},\alpha}+\hat{e}^{\dagger}_{\mathbf{r}_{i},\alpha}\hat{c}_{i}),$
(A.12)

where $V_{\alpha}$ is the coupling constant between the lattice and the
substrate band $\alpha$, and $\mathbf{r}_{i}$ is the position of lattice site
$i$. To perform calculations in a periodic system, we convert the lattice
basis into a Bloch basis,

$\hat{\phi}^{\dagger}_{\mathbf{k},i,\sigma}=\frac{1}{\sqrt{N_{R}}}\sum_{\mathbf{R}}e^{i\mathbf{k}\cdot\mathbf{R}}\hat{c}^{\dagger}_{\mathbf{R},i,\sigma},$
(A.13)

where $\mathbf{R}$ is a lattice vector. The conversion of $\hat{H}_{0}$ and
$\hat{H}_{U}$ occur by the normal means. The coupling term has matrix elements
given by

$\langle
0|\hat{\phi}_{\mathbf{k},i,\sigma}\hat{H}_{couple}\hat{\psi}^{\dagger}_{\mathbf{k}+\mathbf{G},\alpha,\sigma}|0\rangle=\frac{V_{\alpha}}{\sqrt{N}}e^{i(\mathbf{k}+\mathbf{G})\cdot\mathbf{r}_{i}},$
(A.14)

where $\mathbf{G}$ is a reciprocal lattice vector and $\mathbf{r}_{i}$ here
refers to the position within the home unit cell.

The overall Hamiltonian for the Hubbard model with the substrate is

$\hat{H}=\hat{H}_{0}+\hat{H}_{sub}+\hat{H}_{couple}+\hat{H}_{U}.$ (A.15)

For concreteness, consider a single kagome unit cell and a single substrate
band. In matrix form the Hamiltonian for a single spin channel is

${H}_{\sigma}(\mathbf{k})=\begin{pmatrix}\varepsilon+U_{H}n_{1,-\sigma}&-t(1+e^{-i\mathbf{k}\cdot\mathbf{b}_{1}})&-t(1+e^{-i\mathbf{k}\cdot\mathbf{b}_{2}})&\tfrac{V}{\sqrt{N}}&\tfrac{V}{\sqrt{N}}&\dots\\\
-t(1+e^{i\mathbf{k}\cdot\mathbf{b}_{1}})&\varepsilon+U_{H}n_{2,-\sigma}&-t(1+e^{-i\mathbf{k}\cdot\mathbf{b}_{3}})&\tfrac{V}{\sqrt{N}}e^{i(\mathbf{k}+\mathbf{G}_{1})\cdot\mathbf{b}_{1}/2}&\tfrac{V}{\sqrt{N}}e^{i(\mathbf{k}+\mathbf{G}_{2})\cdot\mathbf{b}_{1}/2}&\dots\\\
-t(1+e^{i\mathbf{k}\cdot\mathbf{b}_{2}})&-t(1+e^{i\mathbf{k}\cdot\mathbf{b}_{3}})&\varepsilon+U_{H}n_{3,-\sigma}&\tfrac{V}{\sqrt{N}}e^{i(\mathbf{k}+\mathbf{G}_{1})\cdot\mathbf{b}_{2}/2}&\tfrac{V}{\sqrt{N}}e^{i(\mathbf{k}+\mathbf{G}_{2})\cdot\mathbf{b}_{2}/2}&\dots\\\
\tfrac{V}{\sqrt{N}}&\tfrac{V}{\sqrt{N}}e^{-i(\mathbf{k}+\mathbf{G}_{1})\cdot\mathbf{b}_{1}/2}&\tfrac{V}{\sqrt{N}}e^{-i(\mathbf{k}+\mathbf{G}_{1})\cdot\mathbf{b}_{2}/2}&\epsilon(\mathbf{k}+\mathbf{G}_{1})&0&\\\
\tfrac{V}{\sqrt{N}}&\tfrac{V}{\sqrt{N}}e^{-i(\mathbf{k}+\mathbf{G}_{2})\cdot\mathbf{b}_{1}/2}&\tfrac{V}{\sqrt{N}}e^{-i(\mathbf{k}+\mathbf{G}_{2})\cdot\mathbf{b}_{2}/2}&0&\epsilon(\mathbf{k}+\mathbf{G}_{2})&\\\
\vdots&\vdots&\vdots&&&\ddots\\\ \end{pmatrix},$ (A.16)

where $n_{i,\sigma}\equiv\langle\hat{n}_{i,\sigma}\rangle$, which are found
self-consistently from the occupied eigenstates of ${H}_{\sigma}$. The total
energy is the sum of occupied eigenenergies from each spin channel plus
$-U\sum_{i}n_{i,\uparrow}n_{i,\downarrow}$. Extra substrate bands can be added
by concatenating the blocks for the coupling term and substrate dispersion to
$H_{\sigma}$.

We solve this substrate model in the grand canonical ensemble, meaning we fix
the chemical potential $\mu$. This affects the occupation of single-particle
eigenstates and thus the electron density in the self-consistency loop. We
also subtract $\mu N_{e}$ from the total energy (where $N_{e}$ is the total
electron number). Using the grand canonical ensemble simulates the large
reservoir of electrons which a real substrate would provide without having to
simulate a large number of bulk bands. Where a fixed number of electrons in
the lattice was desired, we incrementally changed the relative alignment of
the lattice and substrate bands until the electron occupation was close to the
desired value. The chemical potential of the substrate was kept constant.
Conversely, free-standing calculations were performed in the canonical
ensemble where the number of electrons is kept constant. To get the spin
configurations, we initially compare multiple different initial spin
configurations and keep the lowest energy configuration. For calculations with
increasing substrate coupling we use the spin configuration from the previous
value of coupling as the trial spin configuration to speed up convergence. At
each point we also compared to a fully paramagnetic configuration. All
calculations with a substrate considered a single kagome unit cell and a
number of substrate unit cells comparable to the DFT calculations. In single
kagome unit cell calculations, we considered a $25\times 25$ k-point grid.
Kagome supercell calculations (used for free-standing systems) had the k-point
grid scaled to keep a similar k-point density.

Substrate | Lattice ($\mathrm{\text{\r{A}}}$) | Work function ($\mathrm{eV}$) | DOS (atom-1${\mathrm{eV}}^{-1}$)
---|---|---|---
No substrate | 20.022 | 3.95 (This work) | –
Ag(111) ($7\times 7$) | 20.223 | 4.7467 | 0.27
Cu(111) ($8\times 8$) | 20.432 | 4.9467 | 0.30
hBN/Cu(111) ($8\times 8$) | 20.432 | 3.6254 | 0
Al(111) ($7\times 7$) | 20.045 | 4.2667 | 0.41
Au(111) ($7\times 7$) | 20.186 | 5.3167 | 0.28
Graphite ($8\times 8$) | 19.690 | 4.774 | 0.01*
Table A.1: Physical properties of the substrates. Lattice constants are those
used in calculations. Work functions and bulk densities of states (DOS) are
provided to illustrate the range of different substrate parameters considered
and as indicative of possible charge transfer and hybridisation, respectively.
DOS is taken at the Fermi level and calculated using DFT with a $48\times
48\times 48$ k-point grid and no spin or $U$, except graphite which used a
$48\times 48\times 18$ k-point grid due to different unit cell vectors and had
the DOS averaged over $\pm 0.2\text{\,}\mathrm{eV}$ due to the Dirac point.
Note that this bulk DOS is slightly different to the DOS of a thin slab used
in Fig. 5c. Figure A.1: Summary of charge transfer and local magnetic moments
of DCA-Cu on different substrates and under different conditions, calculated
using DFT. Data is as in Fig. 3, but $U=0$ data is included here, as well as
integrating the projected density of states (PDOS) (Supplementary Note A.3)
and fitting to non-spin-polarised or ferromagnetic kagome band structures
(Supplementary Note A.4) to obtain other measures of the charge transfer. The
inclusion of $U$ in DFT+$U$ causes a slight enhancement in the magnetic
moments but otherwise does not cause any qualitative changes. The exception is
Ag(111), where $U$ brings the system from nearly non-magnetic to noticeably
magnetic. This is because DCA-Cu/Ag(111) sits near the phase boundary between
non-magnetic and magnetic so is sensitive to small changes. Figure A.2:
Comparison between non-spin-polarised DFT ($U=0$) band structures of (a) free-
standing DCA-Cu and (b) DCA-Cu/hBN/Cu(111) and a kagome tight-binding model.
Red circles mark DFT projection onto DCA-Cu. Black lines are a nearest-
neighbour tight-binding model fitted to the bandwidth at the Gamma point. Grey
lines are substrate bands from DFT. Free-standing DCA-Cu is well described by
a nearest-neighbour tight-binding model (with
$t=$53.6\text{\,}\mathrm{meV}$$). DCA-Cu/hBN/Cu(111) is well-approximated as
free-standing (with $t=$44.1\text{\,}\mathrm{meV}$$). Note that results with
$U$ have slightly lower values of $t$. Figure A.3: Band structures of (a) DCA-
Cu/Cu(111), (b) DCA-Cu/Au(111), (c) DCA-Cu/Al(111), and (d) DCA-Cu/Graphite,
calculated by DFT. Circle radii are proportional to the projection onto DCA-
Cu. All the bands here are spin degenerate, indicating an absence of magnetic
moments when DCA-Cu is on these substrates. Figure A.4: Comparison between
Hubbard models with substrates which vary only by their bandwidth and DOS. The
substrate is a $5\times 5$ supercell of a triangular nearest-neighbour single-
orbital tight-binding model. The kagome lattice uses
$t=$40\text{\,}\mathrm{meV}$$, $U_{H}=$300\text{\,}\mathrm{meV}$$ and
$n\approx 2/3$. The x-axis plots coupling squared divided by substrate
bandwidth. Because the bandwidth is the only feature which differs between
these substrates, substrate DOS is inversely proportional to bandwidth so the
x-axis is $V^{2}\rho$ up to some scale factor. Changing the substrate DOS
causes only a predictable rescaling of the effective coupling. Figure A.5:
Phase diagram of the Hubbard model without a substrate, with respect to
interaction parameter $U_{H}$ and electron occupation $n$, in a single unit
cell. Note the similar trends to the supercell phase diagram (Fig. 5), but
with no intermediate ‘pin’ phase because the unit cell is too small to
describe that phase. Figure A.6: Local magnetic moments in the Hubbard model
on a substrate at parameters derived from DFT results. Grey bars are the
charge and magnetic moments obtained from DFT calculations on the same
substrate, with the annotated strain or electric field, otherwise without
applied strain or electric field. As illustrated, the Hubbard model reproduces
the DFT-derived local magnetic moments in a portion of the Hubbard model
parameter space. Although not always a perfect match, this affirms that our
simple model is appropriate for modelling the fundamental physics of this
system. Figure A.7: Fitting of tight-binding parameters to DFT results with a
substrate. DFT results are non-spin-polarised, $U=0$. Tight-binding results
are with the Hubbard model with $U_{H}=0$. Initial fit parameters were
obtained manually, then refined by convolving the projected band structures
with a Lorentzian (width $5\text{\,}\mathrm{meV}$) then maximising their
cross-correlation. While quantitative features differ, the model captures
qualitative features, especially for the metallic substrates. The graphite
results (c), however, do not properly reflect the coupling in the DFT results,
possibly due to spatially extended coupling (i.e. not point-like) or MOF-
induced substrate reconstruction.

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