††thanks: Equal contributions††thanks: Equal contributions††thanks: Equal
contributions

# Metamagnetic Transitions in Few-Layer CrOCl Controlled by Magnetic
Anisotropy Flipping

Minjie Zhang  Zhejiang Province Key Laboratory of Quantum Technology and
Device, Department of Physics, and State Key Laboratory of Silicon Materials,
Zhejiang University, Hangzhou 310027, P. R. China  Qifeng Hu  Zhejiang
Province Key Laboratory of Quantum Technology and Device, Department of
Physics, and State Key Laboratory of Silicon Materials, Zhejiang University,
Hangzhou 310027, P. R. China  Chenqiang Hua  Zhejiang Province Key Laboratory
of Quantum Technology and Device, Department of Physics, and State Key
Laboratory of Silicon Materials, Zhejiang University, Hangzhou 310027, P. R.
China  Man Cheng  Zhejiang Province Key Laboratory of Quantum Technology and
Device, Department of Physics, and State Key Laboratory of Silicon Materials,
Zhejiang University, Hangzhou 310027, P. R. China  Zhou Liu  State Key
Laboratory for Mesoscopic Physics, School of Physics, Peking University,
Beijing, 100871, P.R. China  Beijing Key Laboratory for Magnetoelectric
Materials and Devices, Beijing, 100871, P.R. China  Shijie Song  Zhejiang
Province Key Laboratory of Quantum Technology and Device, Department of
Physics, and State Key Laboratory of Silicon Materials, Zhejiang University,
Hangzhou 310027, P. R. China  Fanggui Wang  State Key Laboratory for
Mesoscopic Physics, School of Physics, Peking University, Beijing, 100871,
P.R. China  Beijing Key Laboratory for Magnetoelectric Materials and Devices,
Beijing, 100871, P.R. China  Pimo He  Zhejiang Province Key Laboratory of
Quantum Technology and Device, Department of Physics, and State Key Laboratory
of Silicon Materials, Zhejiang University, Hangzhou 310027, P. R. China
Guang-Han Cao  Zhejiang Province Key Laboratory of Quantum Technology and
Device, Department of Physics, and State Key Laboratory of Silicon Materials,
Zhejiang University, Hangzhou 310027, P. R. China  Collaborative Innovation
Centre of Advanced Microstructures, Nanjing University, Nanjing 210093, P. R.
China  Zhu-An Xu  Zhejiang Province Key Laboratory of Quantum Technology and
Device, Department of Physics, and State Key Laboratory of Silicon Materials,
Zhejiang University, Hangzhou 310027, P. R. China  Collaborative Innovation
Centre of Advanced Microstructures, Nanjing University, Nanjing 210093, P. R.
China  Yunhao Lu<EMAIL_ADDRESS>Zhejiang Province Key Laboratory of Quantum
Technology and Device, Department of Physics, and State Key Laboratory of
Silicon Materials, Zhejiang University, Hangzhou 310027, P. R. China  Jinbo
Yang<EMAIL_ADDRESS>State Key Laboratory for Mesoscopic Physics, School of
Physics, Peking University, Beijing, 100871, P.R. China  Beijing Key
Laboratory for Magnetoelectric Materials and Devices, Beijing, 100871, P.R.
China  Yi Zheng<EMAIL_ADDRESS>Zhejiang Province Key Laboratory of
Quantum Technology and Device, Department of Physics, and State Key Laboratory
of Silicon Materials, Zhejiang University, Hangzhou 310027, P. R. China
Collaborative Innovation Centre of Advanced Microstructures, Nanjing
University, Nanjing 210093, P. R. China

###### Abstract

The pivotal role of magnetic anisotropy in stabilising two-dimensional (2D)
magnetism has been widely accepted [1, 2, 3, 4, 5], however, direct
correlation between magnetic anisotropy and long-range magnetic ordering in
the 2D limit is yet to be explored. Here, using angle- and temperature-
dependent tunnelling magnetoresistance [6, 7, 8, 9], we report unprecedented
metamagnetic phase transitions in atomically-thin CrOCl, triggered by magnetic
easy-axis flipping instead of the conventional spin flop mechanism. Few-layer
CrOCl tunnelling devices of various thicknesses consistently show an in-plane
antiferromagnetic (AFM) ground state with the easy axis aligned along the Cr-
O-Cr direction ($\mathbf{b}$-axis). Strikingly, with the presence of a
magnetic field perpendicular to the easy-axis ($H\|\mathbf{c}$), magnetization
of CrOCl does not follow the prevalent spin rotation and saturation pattern,
but rather exhibits an easy-axis flipping from the in-plane to out-of-plane
directions. Such magnetic anisotropy controlled metamagnetic phase transitions
are manifested by a drastic upturn in tunnelling current, which shows
anomalous shifts towards higher $H$ when temperature increases. By 2D mapping
of tunnelling currents as a function of both temperature and $H$, we determine
a unique ferrimagnetic state with a superstructure periodicity of five unit
cells after the field-induced metamagnetic transitions. The feasibility to
control 2D magnetism by manipulating magnetic anisotropy may open enormous
opportunities in spin-based device applications.

When magnetic anisotropy opens a gap in the low-energy modes of magnon
excitations, local magnetic moments can form long range ordering in the 2D
limit by various interaction mechanisms, such as direct and indirect
exchanges, asymmetric Dzyaloshinskii-Moriya coupling, and Ruderman-Kittel-
Kasuya-Yosida interaction via conduction electrons, etc [10, 11, 12, 13].
Prototypical examples are CrI3 with a uniaxial out-of-plane magnetic
anisotropy energy (MAE) and the Ising type of ferromagnetism (FM) [1], in
contrast to Cr2Ge2Te6 with an isotropic in-plane MAE and vanishing magnetic
ordering when approaching the monolayer (ML) limit [2]. However, the recent
gold rush in discovering divergent magnetic 2D materials [14, 15, 16, 5, 17,
18, 19, 20, 21] reveals that many van der Waals (vdW) magnets may have
competing MAE axes, which induces distinctive metamagnetic transitions
manifested as drastic magnetisation increases when external magnetic field
($H$) is applied along the easy axis of magnetization [17, 18, 19, 21]. These
abundant vdW metamagnets [22], normally antiferromagnetic (AFM) in the bulk
form, are only scantily explored but may open enormous opportunities in
understanding and manipulating magnetism in the 2D limit, as exemplified by
the discovery of magnetic anisotropy flipping controlled metamagnetic states
in atomically thin CrOCl.

Figure 1: Few-layer tunnelling junctions of CrOCl. a, Side- and top-view of
the staggered square lattice of CrOCl consisting of CrO4Cl2 octahedrons,
imposing cubic crystal field on 3d orbitals in Cr3+. b, Schematic cross-
section of FLG/CrOCl/FLG tunnelling devices for TMR measurements. c, NCAFM
image of a representative CrOCl flake, showing ML steps of $\sim 0.8$ nm. Note
that ML on SiO2 exhibits an effective thickness of 1.40 nm. d, Fingerprinting
Raman red shift in $A_{g}^{1}$ for determining CrOCl layer numbers. e, $I_{t}$
vs $H$ characteristics of representative few-layer CrOCl tunnelling devices
with various thicknesses from ML to quintuple layer. Note that high-frequency
tunnelling peaks above 4 T are induced by the Shubnikov-de Haas oscillations
of FLG electrodes.

The bulk magnetism of CrOCl crystals have been systematically investigated in
literatures [23, 24], which confirm an AFM ground state below Néel temperature
$T_{N}$ of $\sim 13.5$ K. Such an AFM ordering was interpreted as a c-easy
axis type in which the magnetisation experiments show an unusual saturation
behaviour with low magnetic moment per Cr atom of 0.6 $\mu_{B}$[25]. Recent
$T$-dependent Raman scattering reveals the existence of an additional phase
transition at $\sim 27$ K, possibly corresponding to the formation of an
incommensurate magnetic superstructure from the paramagnetic high-$T$
phase[25, 26]. These studies infer the critical importance of the unique
quasi-1D staggered lattice of ML CrOCl, which is AA stacked out of plane to
form the bulk. As shown in Figure 1a, ML CrOCl can be viewed as a staggered
CrO double layer sandwiched by two halogen atomic layers. By occupying the
same B-site, every two nearest neighbouring Cl and four O atoms form a
distorted octahedron surrounding an A-site Cr, creating inequivalent upper and
lower quasi-1D Cr chains along the Cl-Cr-Cl direction, i.e. the a axis in Fig.
1a. In perpendicular to the 1D chains, Cr atoms are interconnected solely by O
atoms, making the b axis significantly longer than a (lower panel of Fig. 1a).

Due to the wide semiconducting band gap of $\sim 2.3$ eV, we fabricate
vertical tunnelling junctions (TJs) to study the layer-dependent magnetism in
CrOCl. Fig. 1b illustrates the tunnelling device structure, in which
atomically-thin CrOCl channels are sandwiched by two few-layer graphene (FLG)
electrodes. In brief, different-layer CrOCl flakes were first exfoliated on
SiO2/Si substrates, followed by two dry-transfer steps [27] to assemble a TJ
device (see Methods for details). To minimize detrimental leakage current, we
use narrow FLG flakes of several $\mu$m in width, and thus, confine the
working channel areas to below $1.5$ $\mu$m2. Typical tunnelling current
($I_{t}$) vs bias voltage ($V_{\mathrm{bias}}$) characteristics are shown in
Supplementary Information (SI) Figure S1. After tunnelling experiments, the
layer numbers of each device are determined by non-contact atomic force
microscopy (NCAFM) using a ML step height of 0.8 nm (Fig. 1c). Equally
important, Raman spectroscopy reveals that below six MLs, the $A_{g}^{1}$
phonon vibration mode shows monotonic red shift from the bulk value of $207.5$
cm-1 to the ML wave number of $200.5$ cm-1, allowing us to unambiguously
confirm the layers of our devices in complementary with the NCAFM data (Fig.
1d and extended discussions in SI Note 1).

Figure 2: Angle-dependent tunnelling and magnetic field induced phase
transitions in few-layer CrOCl devices. a, Few-layer CrOCl devices
consistently show steep $I_{t}$ upturns for both $H\|\mathbf{c}$ and
$H\|\mathbf{a}$, which is contradicting with a uniaxial c-easy axis AFM. b,
For in-plane $H$ rotation from the $\mathbf{a}$\- to $\mathbf{b}$-axis, CrOCl
TJ devices exhibit an inverse cosine scaling in the $I_{t}$ transition
thresholds, suggesting Cr magnetic moments are rotated only by the effective
magnetic field along the $\mathbf{a}$-axis. c, Polar plotting of $I_{t}$ as a
function of in-plane $H$ rotation. The evolution of $I_{t}$ from isotropic
circles in low $H$ to distinctive butterfly shapes in high $H$ manifests a
field-induced magnetic phase transition.

Figure 1e summarizes layer-dependent $I_{t}$ as a function of out-of-plane $H$
at 1.6 K in few-layer CrOCl TJs with thicknesses from ML to quintuple layers,
all consistently showing a drastic upturn in tunnelling current at around 3 T.
At first glance, the results seem to suggest field induced spin-flop (SF)
transitions for $H\|\mathbf{c}$, which are expected for a c-easy axis AFM
ground state with negligible interlayer magnetic coupling. However, by
rotating $H\|\mathbf{c}$ towards the $\mathbf{a}$ and $\mathbf{b}$ axes
respectively, angle ($\theta$)-dependent $I_{t}$-$H$ curves unfold a distinct
AFM magnetic ordering for few-layer CrOCl crystals. As shown in Figure 2a,
abrupt $I_{t}$ upturns are observed for both $H\|\mathbf{c}$ and
$H\|\mathbf{a}$, contradicting with the uniaxial c-easy axis AFM in the bulk.
Because the $I_{t}$ upturn thresholds monotonically increase when $H$ rotates
from the $\mathbf{c}$\- to $\mathbf{a}$-axis (SI Figure S2), we can also
exclude an easy-plane AFM configuration for few-layer CrOCl, which would show
a maximum SF threshold when $H$ is aligned with the canting direction of Cr
magnetic moments ($\theta\sim 30^{\circ}$; see extended discussions in SI Note
2). To get insight into the magnetic ground state of few-layer CrOCl TJs, we
further investigated the tunnelling behaviour with in-plane $H$ rotation. As
shown in Figure 2b, when $H$ rotates from the $\mathbf{a}$\- to
$\mathbf{b}$-axis, there is an inverse cosine scaling in the $I_{t}$
transition thresholds, indicating that the effective magnetic field along the
$\mathbf{a}$-axis is solely responsible for the magnetic phase transitions.
Using polar plotting, it is clear to see that $I_{t}$, which is nearly
isotropic in low $H$, is characterized by a distinctive butterfly-shaped two
fold symmetry after the phase transitions (Figure 2c).

These self-contradicting experimental observations with the presumable bulk-
like c-easy axis AFM ordering for few-layer CrOCl, however, can be
consistently explained by an in-plane AFM ground state with its easy axis
aligned along the b crystallographic direction. In this easy-axis
configuration, $H\|\mathbf{c}$ causes out-of-plane spin rotation in low
fields, and eventually induces a magnetic phase transition which flips the
easy magnetization direction from the b\- to c-axis. Using $T$-dependent
tunnelling, we can clearly see that the $I_{t}$ thresholds substantially
increase when TJ devices are warmed up (Figure 3a). Such an anomalous $T$
dependence is opposite to the trend of SF transition-controlled tunnelling,
which also can not explain the low-field hysteresis in all CrOCl TJs with
different thicknesses (Figure 3a and Figure 1e).

Figure 3: Temperature-dependent tunnelling in few-layer CrOCl devices. a,
$T$-dependent tunnelling of few-layer CrOCl devices reveal an opposite $T$
dependence to the trend of SF transition-controlled tunnelling, which should
shift $I_{t}$ transition thresholds to lower $H$ when TJ devices are warmed
up. b, Schematic of partial density of states of three occupied $t_{2g}$-$d$
orbitals in CrOCl. Due to the Cr3+ valence state and an octahedron crystal
field, $d_{zx}$, $d_{xy}$ and $d_{yz}$ orbitals of the $t_{2g}$ manifold are
half-filled. Note that the conventional octahedron coordinates, in which $x$
and $y$ axes are defined by two Cr-Cl axes within the CrO2Cl2 basal plane, are
used here for clarity.

To appreciate such a field-induced MAE flipping mechanism, it is crucial to
comprehend intralayer competition between in-plane exchange anisotropy (EA)
and out-of-plane magnetocrystalline anisotropy (MCA), rooted in the unique
staggered square lattice of CrOCl. In particularly, an in-plane AFM ordering
in CrOCl requires a monoclinic lattice distortion by the rotation of the a
axis within the $a-c$ plane [24], rather different from the in-plane
monoclinic transition in VOCl [25]. Despite sharing the same octahedron
crystal field symmetry as CrOCl, V3+ in VOCl has only two 3$d$ electrons
occupying $d_{xy}$ and $d_{xz}$ orbitals respectively (Figure 3b and SI Note
3). By in-plane monoclinic distortion, AFM ordering between neighbouring upper
and lower V3+ atoms along the compressed diagonal direction
($\frac{1}{2}\mathbf{a}+\frac{1}{2}\mathbf{b}$) is introduced by enhanced
direct exchange interactions to avoid geometrical frustration associated with
a rhombic unit cell (see SI Figure S3). Along the quasi-1D V-Cl-V atomic
chains, direct exchange of filled $d_{xy}$ orbitals also enforce AFM coupling,
leaving V3+ atoms along the elongated
($\frac{1}{2}\mathbf{a}-\frac{1}{2}\mathbf{b}$) direction inevitably FM
aligned.

However, once $d_{yz}$ is occupied in the case of CrOCl, in-plane lattice
distortion becomes unfavourable because lattice compression along the
($\frac{1}{2}\mathbf{a}+\frac{1}{2}\mathbf{b}$) direction directly increases
head-on overlapping between 3$d_{yz}$ orbitals (Figure 3b). Instead, CrOCl
undergoes an out-of-plane lattice distortion by rotating the a axis towards c,
which produces a c-axis monoclinic lattice of $90.06^{\circ}$ as well as
zigzagging in the Cr-Cl-Cr atomic chains [24]. This unique type of monoclinic
lattice distortion, which weakens direct exchange along the a-axis while
leaving Cr-Cl-Cr superexchange interaction unchanged, makes the in-plane AFM
ordering feasible in few-layer CrOCl when device encapsulations slightly
reduce the monoclinic angle.

Figure 4: MAE flipping driven metamagnetism in few-layer CrOCl. a-d, 2D
mapping of $I_{t}$ as a function of both $T$ and $H$ for two representative TJ
devices: a trilayer device TL-S17 (a and b) and a quadruple layer device
QL-S27 (c and d). Few-layer CrOCl exhibits four distinctive magnetic phases,
which are PM, PM′, AFM and FiM, respectively. Note that the AFM dome and the
much larger MAE-flipped dome are separated by the PM′ phaseonly above 7 K. e,
Top panel: Schematic of the in-plane AFM ground phase of few-layer CrOCl,
which has a periodicity of four unit cells and quasi-1D FM chains along the
b-axis. Bottom panel: The field-induced FiM state, which is an out-of-plane
ferrimagnetic state with a five-fold periodicity. Note that there is a
reentrant behaviour from the FiM to the PM′ states above 8 T, when CrOCl
devices are not at the base temperature.

Using 2D mapping of $I_{t}$ as a function of both $T$ and $H$, it is
illuminating to see that few-layer CrOCl exhibits four distinctive magnetic
phases. As shown in Figure 4a-4d for two representative devices (TL-S17 and
QL-S27 respectively), few-layer CrOCl TJs in zero field make phase transitions
at $\sim 12.5$ K from an intemediate paramagnetic state (PM′) to the AFM
ground state, in good agreement with the bulk $T_{N}\sim 13.5$ K. By applying
$H\|\mathbf{c}$, the PM′-to-AFM transitions are gradually suppressed,
producing a magnetic dome structure which is typical for an AFM magnet. Above
4 T, there emerges a second magnetic dome with a distinct nature from a SF
phase, i.e. canted AFM [22, 17, 18, 19, 21]. As shown in Figure 4b and 4d, by
fixing $T$ and sweeping $H$, well-defined phase boundaries between the AFM
ground state and the MAE-flipped domain only exist below 7 K, above which the
two magnetic domes are separated by the PM′ phase. It is also conspicuous
that, for fixed field of $H=8$ T, the MAE-flipped dome has a maximum phase
transition $T$ of 17.5 K, which is significantly higher even when compared
with the bulk $T_{N}$. Utilizing first-principle density functional theory
(DFT) calculations (See detailed discussions in SI Note 4), we determine the
magnetic ordering for the AFM ground state and the MAE-flipped phase,
respectively. For the AFM ground state, we compare different in-plane magnetic
configurations with zigzagging Cr-Cl-Cl atomic chains along the a-axis and
varying c-axis monoclinic angles from $90^{\circ}$ to $90.06^{\circ}$. To meet
the two criteria of minimized magnetic coupling energy and a b-easy axis AFM,
we determine the a unique in-plane AFM state with a periodicity of four unit
cells and quasi-1D FM chains along the b-axis. Based on this AFM
configuration, the SF field is calculated to be exceeding 13 T, which can
self-consistently explain the absence of SF phase transitions for $H\|b$ (Fig.
1a and 1b). For the MAE-flipped phase, DFT suggests an out-of-plane
ferrimagnetic (FiM) state with a five-fold periodicity (Bottom panel of
Figure. 4e).

Noticeably, few-layer CrOCl TJs also show abnormal downturn in $I_{t}$ for
$H>8$ T, which can not be explained by field induced magnetic moment flipping
within the ferrimagnetic state since increasing in magnetic moment alignments
with $H$ generally enhance the tunnelling processes between two FLG
electrodes. Such unusual tunnelling current decrease is due to the reentry of
the FiM phase into the PM′ phase. Indeed, as shown Figure 4a and Figure 4c,
for both TL and QL CrOCl, the high-field $I_{t}$ for $H\|c$ approaches the
tunnelling results of the PM′ state. The trend becomes more conspicuous by
warming up the samples to above 10 K, which clearly demonstrate that the high-
field data at different $T$ set-points are converging to the PM′ tunnelling
curve of 20 K. It is also interesting to notice that 2D mapping of tunnelling
magnetoresistance is able to resolve a definite phase boundary between the PM′
and PM phases, by slow scanning of $T$ with fixed $H$ setpoints (SI Figure
S4). This allows us to complete the magnetic phase diagram of few-layer CrOCl,
as shown in Figure 4b and 4d.

Although atomically thin 2D materials in general exhibit distinctive physical
properties from the bulk counterparts, e.g. interlayer coupling driven
dimensional crossover in mechanical properties from ML to BL graphene and MoS2
[28, 29], the fascinating MAE flipping controlled magnetic ordering and
divergent metamagnetism in few-layer CrOCl unfold the great promises of
exploring 2D magnetism in a wide range of vdW layered AFM materials. Being
isostructural compounds, it will be appealing to compare 2D magnetism in few-
layer TiOCl, VOCl and FeOCl with CrOCl, considering that different numbers of
$d$ electrons can drastically change the competition between in-plane EA and
out-of-plane MCA. There may be other exotic magnetic mechanism in the 2D
limit, such as spin-phonon coupling and magnetorestriction in these unique
systems with staggered lattices.

## Methods

Single Crystal Synthesis. High-quality CrOCl single crystals were synthesised
by chemical vapour transport of stoichiometrically mixed high-purity Cr2O3 and
CrCl3 powder (Alpha Esar, 99.9$\%$). The containing quartz tube was evacuated
to a base pressure of 10-2 Pa when flame-sealed before loading into a tube
furnace. Single crystals were grown for 7 days in a temperature gradient of
1073 K to 1213 K. Then, the furnace was naturally cooled down to room
temperature.

Vibrating-Sample Magnetometer. The bulk magnetisation of CrOCl single crystals
were characterised using a vibrating sample magnetometer (VSM) module in a
Quantum Design Physical Properties Measurement System (PPMS-9T). During the
experiments, a CrOCl single crystal was vibrated sinusoidally, which causes an
electrical signal proportional to the sample magnetisation in a stationary
signal pick-up coil. The VSM results confirm the high quality of our single
crystals, which are in excellent agreement with the literature reports [24].

Magnetic Torque. Magnetic torque measurements are based on a metal-cantilever
capacitor geometry. During the torque experiment, a CrOCl single crystal was
mounted on a cantilever made of Au foil, which has a thickness of 25 $\mu$m.
The width of cantilever is 0.3 mm and the vacuum gap between cantilever and
the bottom Au electrode is 0.5 mm. By applying an external magnetic field to
the sample, torque exerted on the cantilever (${\tau}={\mu}_{0}m{\times}H$)
are detected by measuring the capacitance signals, which are used to deduce
the magnetizations of the sample.

Tunnelling Device Fabrication. Few-layer CrOCl flakes were exfoliated from
single crystals in an argon-filled glove box to protect the samples from air
degradation. Different layer numbers of CrOCl samples were first selected by
optical contrast, which were cross-checked by complementary non-contact atomic
force microscopy and Raman spectroscopy. During tunnelling junction
preparation, top graphene electrodes, few-layer CrOCl and bottom graphene
electrodes were picked up sequentially using PC on PDMS, followed by a final
transferring step onto pre-patterned Cr/Au electrodes on Si/SiO2 substrate.
The resulting junction are free of interfacial contaminations, which is
crucial for probing the intrinsic magnetism of few-layer CrOCl.

Tunnelling Magnetoresistance Measurements. Tunnelling measurements were
performed in an Oxford-14 T cryostat. The system is equipped with a sample
rotator for angle-dependent measurements with in-plane or out-of-plane
magnetic field. The field- and $T$-dependent TMR and $I-V$ curves were
measured using a Keithley 2400.

DFT Calculations. To get the electronic structure of CrOCl, the Vienna ab
initio simulation package (VASP) [30, 31] is used, by the method of the
projector augmented wave [32] and the generalized gradient approximation (GGA-
PBE) [33], in which the exchange-correlation potential was taken into account.
The plane-wave cutoff energy is set at about 500 eV and the k-point sampling
is performed by the Monkhorst-Pack scheme [34]. The total energy is ensured to
be converged within 1$\times$10-6 eV per unit cell and the force threshold is
0.005 eV/Å. For multilayer, the method of SCAN+rVV10 [35] is adopted to
describe the Van der Waals (vdW) interaction. The magnetic ground states of
CrOCl are calculated using the method of PBE plus effective $U$[36]. For
magnetic anisotropy energy (MAE) calculation, spin-orbit coupling was added as
a perturbation term[37]in VASP.

Atomic Force Microscopy. We use a multi-mode Park NX10 AFM to characterise the
surface morphology and measure thin-film thicknesses of freshly exfoliated
CrOCl crystals adn CrOCl TJ devices after tunnelling experiments. Few-layer
microflakes were transferred onto a SiO2/Si substrate and baked in-situ for a
few minutes in order to enhance the contact between samples and the substrate.

Raman Spectroscopy. A Witec Alpha-300R Raman system with 532 nm laser
excitation was used to perform Raman spectroscopy. The laser power was set to
be below 5 mW to avoid excessive heating damage to the samples.

## Data Availability

The authors declare that the main data supporting the findings of this study
are available within the paper and its Supporting Information files. Extra
data are available from the corresponding authors upon request.

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###### Acknowledgements.

This work is supported by the National Key R&D Program of the MOST of China
(Grant Nos. 2016YFA0300204 and 2017YFA0303002), and the National Science
Foundation of China (Grant Nos. 11790313 and 11574264), and Zhejiang
Provincial Natural Science Foundation (D19A040001). Y.Z. acknowledges the
funding support from the Fundamental Research Funds for the Central
Universities.

## Author Contributions

Y.Z. initiated and supervised the project. M.J.Z. and Q.F.H. synthesised and
characterised COC crystals, fabricated COC devices and carried out all the
measurements, assisted by M.C., Z. L. and F. W.. C.Q.H and Y.H.L. carried out
the DFT calculations. M.J.Z., Q.F.H., C.Q.H., J.B.Y., Y.H.L. and Y.Z. analysed
the data and wrote the paper with inputs from all authors.

## Competing financial interests:

The authors declare no competing financial interests.

## Data availability.

The authors declare that the main data supporting the findings of this study
are available within the paper and the Supplementary Information file. Extra
data are available from the corresponding authors upon reasonable request.


# Shallow Art: Art Extension Through Simple Machine Learning

Kyle Robinson, Dan Brown
Cheriton School of Computer Science
Unversity of Waterloo
Waterloo, ON N2L 3G1 Canada
{kyle.robinson<EMAIL_ADDRESS>

###### Abstract

> Shallow Art presents, implements, and tests the use of simple single-output
> classification and regression models for the purpose of art generation.
> Various machine learning algorithms are trained on collections of computer
> generated images, artworks from Vincent van Gogh, and artworks from
> Rembrandt van Rijn. These models are then provided half of an image and
> asked to complete the missing side. The resulting images are displayed, and
> we explore implications for computational creativity.

## Introduction

The use of Machine Learning algorithms as assistive tools for the creation of
visual and auditory artworks is an ever growing area of research (?). In
addition, machine learning methodologies–especially those based on neural
networks–have been used in the creation of wholly new artworks (?). The
creation and use of such tools encourages discussions at the boundary between
computational creativity and creativity support.

Many machine learning computational creativity applications have thus far used
neural networks. While these methods have lead to extremely interesting and
thought-provoking artwork (?; ?), their analysis from a creative perspective
is complicated by an inability to interpret why the model acted as it did.
This black-box effect can in some ways parallel our understanding of human
creativity but differs in many ways, especially as neural networks are
meticulously designed to fulfill their specific purpose.

In contrast to neural network based approaches of algorithmic art generation,
Shallow Art applies more classical Machine Learning algorithms and
methodologies to the problem of art generation. More specifically, Shallow Art
applies computationally efficient classification and regression algorithms to
the problem of art extension. We have developed a methodology that allows any
single-output classification or regression algorithm to be trained on a
dataset of images and then complete a partially provided image. We conclude by
presenting outputs from various models trained using a number of different
types of training images.

## Related Work

Work on machine learning based creativity support is rich in breadth and
depth. One significant source of recent research is the Magenta research
division of Google Brain (?). In the area of music generation, their work
includes a model which creates piano performances featuring expressive changes
in tempo and dynamics, as well as a musical counterpoint generator using a
specifically-designed convolutional neural network (?; ?). Additionally,
Magenta tools transcribe polyphonic music, and synthesize sounds for music
production (?; ?). In visual art, Magenta has developed a recurrent neural
network able to create very simple stick drawings (?). The network is trained
on human drawings with a provided single classification and can generate
drawings with or without a provided topic. These drawings are extremely simple
in nature; consisting of just a few simple lines and shapes.

One interesting computational art generation research paper similar to Shallow
Art attempts to learn the artistic style of one picture and transfer it to
another as if the second image was created in the style of the first (?).
Where all of these research papers differ from Shallow Art is in their use of
black-box or grey-box neural networks for training and generation. Conversely,
Shallow Art prioritizes and enables the use of easily understandable and
traceable machine learning algorithms such as decision trees and seperating
hyperplanes, with each pixel of the resultant project treated independently.

## Methodology

### Image Source

The data for this project is sourced through a combination of black-and-white
computer-generated images of varying complexities, colored computer generated
images, and a collection of artworks created by Vincent van Gogh and Rembrandt
van Rijn. In the black-and-white data set each pixel of each image is either
white or black. In the coloured image data sets each pixel is represented by a
3-tuple $(Red,Green,Blue)$ of integers ranging in value from 0-255.

Figure 1: Image Data. Two examples each of a selection of image types used to
train and test classification and regression models. (a)-(e) were randomly
computer generated, whereas (f) and (g) are from collections of van Gogh and
Rembrandt works respectively.

The computer generated image data sets were created using Python 3 and the
Python Imaging Library through the Pillow distribution (?). Four arbitrary
patterns were selected for initial black and white image testing (figs. 1, 1,
1 and 1): horizontal lines, vertical lines, triangles, and circles. Each image
type eliminates all structure except one abstract concept which is exceedingly
easy for a human to perceive. These generated images can also be sorted into
two groups: those with unique solutions to the right half (that is, where
knowing the left half implies the right half), and those without unique
solutions to the right half. Note that as the resolution of the generated
images decreases, the approximations of lines, and curves drastically decrease
due to rasterization. Black and white images were generated at a resolution of
250x250 pixels and stored in the PNG format. Color images (including van Gogh
and Rembrandt works) were generated or cropped to a resolution of 200x200
pixels due to limitations in the van Gogh and Rembrandt data repositories.

#### Lines

Two types of line-based images were generated: horizontal, and vertical. Each
image begins as a square white image, then horizontal or vertical lines one
pixel thick are drawn at 50 random locations. Given any horizontal line
image’s left side the right side has only one correct solution, whereas the
vertical line images have no left-to-right continuity. Examples of these
images can be seen in Figures 1 and 1.

#### Circles

Generated circle images consist of a white background with 50 randomly placed
black circles. All 50 circles have a diameter of 15 pixels and may be placed
at locations which clip with the edge of the image by up to their diameter.
Circles are drawn as rasterized approximations of circles. Given any circle
image’s left side the right side contains some unique solutions (where circles
are split in half) and many unknown solutions (the rest). Examples of circle
images can be seen in Figure 1

#### Triangles

We generated triangle images with two randomly chosen vertices on the left
side and one on the right. These vertices are then connected and filled in
order to create a black triangle on a white background. Given any left-side
image, predicting the right side is a relatively trivial task of extending the
triangles’ edges until they converge at the third vertex. Examples of these
images can be seen in Figure 1. We also generated a set of triangle images
that were each assigned a random fill colour and placed on a white background.
Examples can be seen in Figure 1

#### Van Gogh & Rembrandt

We obtained Vincent van Gogh and Rembrandt van Rijn works (public domain) by
scraping thumbnails from online galleries (?; ?). In total we collected 2,298
van Gogh images and 1,097 Rembrandt images. These images consist of finished
and unfinished works of differing aspect ratios and sizes. We standardized
these data sets by removing duplicates and non-rectangular images and scaling
all remaining images to 200x200 pixels as seen in Figures 1 and 1. The van
Gogh and Rembrandt images pixel data is stored and accessed in the 3-tuple RGB
format described earlier.

Figure 2: Each pixel on the right is predicted using the values of all pixels
on the left (attributes). This process is repeated so that each pixel on the
right half of the image is the output of one independent model.

### Machine Learning

The task of binary prediction on the black and white data sets is one of
binary classification, whereas the prediction of coloured images is one of
regression, since the output RGB values must be between 0 and 255. In order to
prepare images for machine learning classification and regression, each image
is converted into a flattened one-dimensional array of values. For black and
white images, each value is a binary representation of the pixel. For color
images, each pixel corresponds to three values in the array, one for each
colour shading. Image pixel arrays are split in half so that one half contains
all pixels of the left side of the image and the other contains all pixels
from the right side. The data array corresponding the left of the image is
used as a series of attributes, and the array corresponding to the right side
of the image is used as series of labels. In this way the black and white
images contain $\frac{250\times 250}{2}=31,250$ training attributes each and
the color images contain $\frac{(200\times 200)\times 3}{2}=60,000$ attributes
each.

As traditional regression and classification tasks involve many attributes and
one prediction per data point a method of converting single-output
classification and regression models to multi-output is required. One simple
and computationally efficient method of doing so is to train one model for
each output required as described in Figure 2. Our implementation is
altogether independent: it consumes any classification or regression model and
trains one independent model for each of any number of required outputs. Due
to the halving of images in this work the total number of trained models
required for each image type is equal to the number of attributes. There are
thus 31,250 models for black and white images and 60,000 models for color
images. We refer to a collection of models which has been trained for per
pixel image extension as a Wrapper-Model (WM). Importantly, WM’s do not take
into account any relationships (positional or otherwise) of the pixel data;
each pixel model operates completely independently of all others within a WM.
We isolated training and testing image data from each other, and implemented
models using scikit Learn without hyper-parameter optimization (?).

In general, our WM implementation allows for the use of any
classification/regression model, though computational feasibility is tightly
interwoven with the computational complexity of any underlying model selected.
We focused on easily interpretable models. Black and white image data sets
were used to train four different types of WM’s: decision tree, random forest,
perceptron, and linear SVM. Decision tree and random forest are both tree-
based algorithms capable of n-dimensional decision boundaries, whereas
perceptron and linear SVM learn a linear seperating hyperplane with a singular
linear decision boundary. Only decision tree WM results are presented for
colour images due to model training time constraints.

(a) Dec. Tree

(b) Ran. Forest

(c) Perceptron

(d) Lin. SVM

Figure 3: Horizontal Lines. Images created by classification Wrapper-Models
trained on 50 randomly generated training images as described in Figure 1.

(a) Dec. Tree

(b) Ran. Forest

(c) Perceptron

(d) Lin. SVM

Figure 4: Vertical Lines. Images created by classification Wrapper-Models
trained on 50 randomly generated training images as described in Figure 1.

(a) Dec. Tree

(b) Ran. Forest

(c) Perceptron

(d) Lin. SVM

Figure 5: Circles. Images created by classification Wrapper-Models trained on
50 randomly generated training images as described in Figure 1.

(a) Dec. Tree

(b) Ran. Forest

(c) Perceptron

(d) Lin. SVM

Figure 6: Triangle. Images created by classification Wrapper-Models trained on
50 randomly generated training images as described in Figure 1.

Figure 7: Triangle WM. Outputs from a decision tree WM trained on 1900 images
of triangles as described in Figure 1

Figure 8: Vincent van Gogh WM. Outputs from a decision tree WM trained on 1900
van Gogh works as described in Figure 1

Figure 9: Rembrandt van Rijn WM. Outputs from a decision tree WM trained on
900 Rembrandt works as described in Figure 1

## Results & Discussion

### Black and White

Each black and white image type was used to train four WM’s, one for each
learning approach. Each WM was trained on 50 images111Black and white images
were used as a proof of concept and were therefore trained on small data sets.
and then provided a single left side of the same image type and asked to
predict the missing right side. Figures 3, 4, 5 and 6 present the outputs from
each black and white WM for horizontal lines, vertical lines, circles, and
triangles respectfully. The left half of each image is the raw input image and
the right half is the output prediction from the WM.

Both tree-based WM’s were able to correctly predict the horizontal line images
(figs. 3(a) and 3(b)), whereas the separating hyperplane WM’s were not (figs.
3(c) and 3(d)). The opposite is true for the vertical line images where
perceptron and linear SVM algorithm outputs look much more natural (fig. 4).
The accuracy on the vertical line images is far less important than the
continuity of the lines which are drawn. Outputs from the circle WM show that
none of the models was able to learn the general pattern of circles with the
limited number of training images, but each algorithms attempt looks different
(fig. 5). The triangle image WM’s all show a general inclination towards
convergence, though they differ drastically in their interpretation of how to
complete the image. More testing with larger training data sets is required to
fully interpret how and if the models converge.

### Colour

Colour image types were each used to train a single decision tree WM and then
used to predict two images of every other colour image type. For example, a
decision tree WM was trained on van Gogh images and then provided two image
halves from the coloured triangle, van Gogh, and Rembrandt data sets each to
extend. In this sense each model is trained on one style and then asked to
complete images that come from other styles. The colored triangle and van Gogh
models were trained using 1900 images, and the Rembrandt model was trained
using 900 images. Figures 7, 8 and 9 present the outputs from each of the
three trained colour decision tree WM’s. As with the black and white images,
the left half is the raw input image and the right half is the WM’s output.

In all cases each model shows that it is able to predict the general colours
and shapes of images in the same style despite never having seen them before.
The green triangle WM image shows convergence towards a single point, whereas
the second from the left image shows a large spectrum of dark colours with no
clear convergence (figs. 7 and 7). When tasked with extending van Gogh and
Rembrandt works the triangle WM outputs are expectedly abstract (figs. 7, 7, 7
and 7). Van Gogh and Rembrandt WM’s are both able to complete images in the
style of each other while maintaining a sense of colour and shape; this is
especially seen in Figures 8, 8, 9 and 9. When posed with the abstract
triangle images the van Gogh and Rembrandt models correctly paint the white
background on the fringes and in Figure 8 even extend the dark colours from
the triangle.

## Conclusion

Through the implementation of a wrapper method which can utilize any single-
output classification or regression model to learn from and complete images,
Shallow Art presents a novel new method of computational art generation which
straddles the boundary between computational creativity and creativity
support.

Shallow Art differs from previous approaches in computational creativity
support and art generation in its focus on interpretability. This
interpretability paves the way for a new analytical perspective which neural
network based approaches do not provide. In addition, when framed as an agent
in a co-creative process Shallow Art presents an interesting new perspective
to discussion on co-creativity, and might enable multi-round experimentation
between a human and the Shallow Art system.

Using abstract computer-generated visuals in tandem with artworks from famous
classical artists as training data for simple machine learning methods has
opened the door to future experimentation and analysis on the topic of non-
neural network based approaches to computational art creation.

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