From Stoner to local moment magnetism in atomically thin Cr2Te3

The field of two-dimensional (2D) ferromagnetism has been proliferating over the past few years, with ongoing interests in basic science and potential applications in spintronic technology. However, a high-resolution spectroscopic study of the 2D ferromagnet is still lacking due to the small size and air sensitivity of the exfoliated nanoflakes. Here, we report a thickness-dependent ferromagnetism in epitaxially grown Cr2Te3 thin films and investigate the evolution of the underlying electronic structure by synergistic angle-resolved photoemission spectroscopy, scanning tunneling microscopy, x-ray absorption spectroscopy, and first-principle calculations. A conspicuous ferromagnetic transition from Stoner to Heisenberg-type is directly observed in the atomically thin limit, indicating that dimensionality is a powerful tuning knob to manipulate the novel properties of 2D magnetism. Monolayer Cr2Te3 retains robust ferromagnetism, but with a suppressed Curie temperature, due to the drastic drop in the density of states near the Fermi level. Our results establish atomically thin Cr2Te3 as an excellent platform to explore the dual nature of localized and itinerant ferromagnetism in 2D magnets.


Introduction
Although the quantum theory of magnetism was established in the early days of quantum mechanics, the localized vs. delocalized conundrum remains controversial in many magnetic materials [1].Based on the itinerant picture, the Stoner model successfully predicts the exchange splitting of electron bands and noninteger magnetic moments in ferromagnetic metals [2].On the contrary, the Heisenberg model is usually used to interpret the antiferromagnetic and ferromagnetic order of local moments in magnetic insulators [3].
In the 1960s, the Hubbard model was proposed to combine the "localized" and "delocalized" pictures under a unified framework [4], which principally resolves the dual nature of electrons in magnets.However, an exact solution to the Hubbard model is impossible [5].Instead, as a simplification of the Hubbard model, the spin-fluctuation method is widely used to qualitatively describe the magnetic properties in half-metals [6].Among the rich branches of magnetism, exploring 2D magnetic order is extremely important to the development of phase transition theory and statistical physics.Although Mermin-Wagner theorem prohibits the long-range 2D magnetic order at any finite temperature in an isotropic spin orientation case [7], Onsager and Kosterlitz-Thouless prove the existence of 2D magnetism in a lower symmetry (spin anisotropy) system [8,9].
The research on 2D ferromagnetism has advanced rapidly thanks to the recent development of fabrication methods and characterization tools [10][11][12].Among the material library, the most studied are Cr2Ge2Te6, CrI3, and Fe3GeTe2 [13][14][15][16][17][18].Although the 2D ferromagnetism in insulating Cr2Ge2Te6 and CrI3 is well represented by Heisenberg or Ising-like models [13,14,17,18], no consensus has been achieved for metallic Fe3GeTe2.While transport measurements claim the itinerant ferromagnetism [15,16], an angleresolved photoemission spectroscopy (ARPES) study shows the absence of exchange splitting energy across the Curie temperature (TC) [19].Additionally, scanning tunneling spectroscopy measurement finds a Kondo lattice-like behavior in the ferromagnetic state [20], implying a dual nature of localized and itinerant electrons.A similar situation also occurs in the prototypical ferromagnet SrRuO3 [21,22].These examples demonstrate that ferromagnetic metals contain more physics beyond the simple Stoner model.Furthermore, despite extensive efforts to explore the corresponding macroscopic properties, there is still no ARPES study to elucidate the underlying band structures of 2D magnets, which hinders the comprehensive understanding of the microscopic mechanism.Therefore, large-area atomically thin ferromagnets are highly desired, not only to unveil the nature of 2D ferromagnetism, but also for future practical applications.
Cr2Te3 is a ferromagnetic metal with TC = 170 K [23].The easy axis is along the out-of-plane direction.
It consists of Cr-deficient (Cr1 atom) and Cr-filled layers (Cr2 and Cr3 atoms), as shown in Fig. 1(a).Each Cr atom is surrounded by six Te atoms, forming a corner-shared octahedron.According to the ionic picture, the covalent state of Cr is expected to be +3 with an electronic configuration 3d 3 .In an octahedral structure, the crystal field splits the five d orbitals into a higher energy eg doublet and a lower energy t2g triplet.Thus, the Cr 3+ ion has a total spin of S = 3/2, with three electrons in the t2g manifold to satisfy Hund's rule.
Experimentally, the saturated magnetic moment is 2.65   /Cr [24].This non-integer and large magnetization supports a picture of itinerant ferromagnetism in Cr2Te3.In addition, neutron scattering measurements show -0.14   , 2.78   and 2.53   for Cr1, Cr2, and Cr3, indicating the essential role of the Cr-filled layer in driving the long-range ferromagnetic order in Cr2Te3.Cr2Te3 thin films have been fabricated by molecular beam epitaxy (MBE) method on silicon and sapphire substrates [25,26], displaying a comparable TC to the single crystals as the film thickness exceeds 4 nm.However, to our best knowledge, there is no exploration of the ferromagnetism in the atomically thin 2D limit.
Here, we report a dimensionality-induced ferromagnetic ground state transition in the atomically thin Cr2Te3.We used MBE to prepare Cr2Te3 thin films.2D ferromagnetism was observed in the samples with thickness from monolayer (ML) to 6 ML, although the TC drops to nearly half of the bulk value in the monolayer case.To understand this drastic change, ARPES, x-ray absorption spectroscopy (XAS), and density functional theory (DFT) calculations were utilized to simultaneously track the thickness-dependent electronic structures in the Cr2Te3 system.A suppression of density of states (DOS) near the Fermi level (EF) is consistent with the lower TC in 1 ML Cr2Te3.Most importantly, in contrast to the Stoner-type ferromagnetism exhibited in thicker films, there is a fundamental ground state transition to the localized magnetism in the 2D limit.Hence, our study provides another important example of novel physics at the atomically thin limit, as the indirect to direct gap transition in MoSe2 [27], and the Weyl semimetal to quantum spin Hall insulator transition in WTe2 [28].

Results
From 3D to 2D ferromagnetism Fig. 1b displays the reflection high-energy electron diffraction (RHEED) patterns during the process of sample preparation, in which the sharp streaks indicate the epitaxial nature of the thin film.The singlecrystalline Cr2Te3 phase was confirmed by x-ray diffraction measurement on 6 ML sample.Except for the substrate peaks, a set of Cr2Te3 peaks are clearly observed in the 2- scan (Fig. 1c).In-situ scanning tunneling microscopy measurement further illustrates the atomic-scale structure of 1 ML and 6 ML samples.
Both line-profile cuts and fast Fourier transformation analysis are used to determine the in-plane lattice parameters: the lattice constant of the 1 ML sample is 6.32 Å (Fig. 1d), which is nearly the same as that of the Si (111) substrate.The lattice parameter however grows to 6.81 Å in the 6 ML sample (Fig. 1e), closer to the value of bulk Cr2Te3.In addition, the out-of-plane lattice parameter in the 1 ML sample is 13.3 Å, which is 7% larger than that of the 6 ML sample (Supplementary Fig. 1).All the above observations indicate a significant strain effect in the 1 ML sample, which transforms the pristine CrTe6 octahedral structure to the trigonal symmetry, further splitting the degenerate t2g bands into a1g and eg bands (Fig. 1f).Fourier transformation analysis shows clear 1 × 2 surface reconstructions on the 1 ML and 6 ML samples, consistent to the previous study [25].Combining the in-plane and out-of-plane lattice information, pure Cr2Te3 structure is unambiguously obtained in our thin films, without any impurities or phase separation.
Temperature-dependent magnetization of 6 ML sample was measured by superconducting quantum interference device (SQUID) in Supplementary Fig. 2, showing the same TC as that in literatures [25,26].
However, the elusive ferromagnetic signal of 1 ML sample requires a technique of better sensitivity than that provided by conventional SQUID.X-ray magnetic circular dichroism (XMCD) is a unique element-specific technique to explore the local electronic and magnetic structures of 3d transition-metal compounds, which is the difference spectrum of two polarized absorption spectra taken under an external magnetic field.This method has become a standard tool for investigating ferromagnetism in 2D magnets [15,16,18], even with small net magnetic moments.
Figs. 2a,b display the left-/right-circularly polarized absorption spectra (ρ + and ρ -) and the corresponding difference spectra (XMCD = ρ + -ρ -) on the 1 ML and 3 ML Cr2Te3 samples.For the 3 ML sample, an obvious XMCD signal around Cr L2,3 edges (located at 575 eV and 583 eV) is observed corresponding to long-range ferromagnetic order, consistent with the reported moment of 2.65   .In the monolayer sample case, the ferromagnetic feature is still clear although the dichroic signal is weak.It implies the existence of 2D ferromagnetism in the Cr2Te3 system.
In addition, temperature-dependent XMCD measurements were performed on the 1 ML, 3 ML, and 6 ML samples, as shown in Figs.2c-e.We defined TC as the temperature at which the XMCD signal disappears.While the same TC (= 170 K) was demonstrated in the 3 ML and 6 ML samples, we observed that a drastic suppression of TC (= 90 K) occurs in the monolayer case.Similar phenomena were observed in other 2D magnets, where the 2D ferromagnetic transition temperature drops to half or one-third of the bulk value [13][14][15][16].Furthermore, we can use the ratio of the XMCD and XAS signals to depict the stiffness of ferromagnetism, which is proportional to the amplitude of magnetization [18].Fig. 2f displays the temperature-dependent XMCD percentage on the 1 ML, 3 ML, and 6 ML samples.The signal in the monolayer sample is only 5% of that in the 6 ML sample (nominal value is 1/6), indicating a magnetic ground state change from 3D to 2D limit, which will be discussed later.

From Stoner to Heisenberg ferromagnetism
Temperature-dependent ARPES measurements were performed to investigate the ferromagnetic structure in the 2D limit.For simplicity, we choose 6 ML and 1 ML samples as representatives to illustrate the crossover from 3D to 2D (Fermi surface and band structure of the 3 ML sample are shown in Supplementary On the contrary, the band structure of the 1 ML sample has little change across   (Figs.3d-f), implying the local magnetic moments persisting in the paramagnetic state.Furthermore, the local vs.itinerant features in the 1 ML and 6 ML samples are also illustrated by the spectral weight analysis for the energy distribution curve (EDC) at the  point (Supplementary Fig. 5 and Fig. 6).6 ML sample shows clear spectral weight transfer from the deeper Cr t2g band to the higher Te 5p band, presenting strong evidence for a Stoner-like ferromagnetic exchange interaction in thicker Cr2Te3 films [20].However, there is negligible spectral weight transfer in the monolayer case, implying that the itinerant scenario is insufficient to account for the 2D ferromagnetism in Cr2Te3.

Electronic structure evolution from 3D to 2D
To understand the dramatic change of TC in atomically thin Cr2Te3, thickness-dependent ARPES measurements were performed to illustrate the underlying electronic structures.bands.Similar band structures are observed in 3 ML Cr2Te3 (see Supplementary Fig. 3), explaining the same physical properties in these thicker films.First principle calculations (DFT + U) were carried out to understand the band structure better.By carefully tuning the correlation parameter U, a good match to the experimental result is achieved with U in the range of 1.7-2.2eV (Fig. 4c and Supplementary Fig. 7), which is consistent with the previous study [29].
However, the band structure changes fundamentally in the monolayer case.Only two bands pass through the EF, as shown in Figs.4d,e.The inner Te band moves upward with the band maximum touching the EF.Meanwhile, the Cr t2g bands move downwards in the range of −1.1 eV to −1.9 eV.A simple charge transfer scenario from the substrate can be excluded due to the opposite directions of the Te and Cr bands' shifts.By considering the dimensionality and strain effect in the monolayer case, all the essential features of the band structure can be interpreted by the DFT + U calculation with U in the range of 1.7-3.1 eV (Fig. 4f and Supplementary Fig. 7).The similar Hubbard U indicates that there is no drastic change of correlation strength in the 2D limit of Cr2Te3, which is in sharp contrast to the case of SrRuO3 [30].
One can integrate the intensity of E-k dispersion from  to M to obtain the angle-integrated photoemission spectrum (PES), which is a rough estimate of the density of states (DOS) in a material (Supplementary Fig. 8).The PES spectra of the 3 ML and 6 ML Cr2Te3 are nearly the same, echoing the same TC for the two samples (Fig. 4g).On the contrary, there is a significant DOS suppression near EF in the monolayer sample, implying a direct link to the suppressed TC.DFT + U calculations support the experimental observations: the bulk calculation is similar to the PES spectra of the 3 ML and 6 ML samples, and the monolayer curve repeats the essential features of the monolayer sample.The understanding of the electronic structure evolution from 3D to 2D is further augmented by the Cr L 2,3 -edge XAS measurements (Supplementary Fig. 9).The pre-peak around 572 eV comes from the Te M5 edge (Fig. 2b), reflecting the strength of covalent bonding between the Cr 3d and Te 2p orbitals.For example, previous studies have shown that this Te M5 feature in ferromagnetic insulator Cr2Ge2Te6 is much sharper than that of ferromagnetic metal Cr 2 Te 3 [17,31], which is reminiscent of the thickness-dependent results from the localized regime (1 ML) to the itinerant regime (6 ML).

Discussion
Although our study shows an obvious transition of the ferromagnetic state in the 2D limit, a Stoner model is still a good starting point to qualitatively understand the thickness-dependent ferromagnetism in atomically thin Cr2Te3.First, the Stoner criterion (  ) > 1 claims (  ) as a critical parameter to determine the stiffness of magnetic order in the ground state, where (  ) is the DOS at EF and  is the exchange parameter.The parameters (  ) and  can be obtained from the DFT + U calculations of the spin-split Te 2p bands (Supplementary Fig. 10).The product (  ) is 1.59 in bulk case, confirming the validity of Stoner model in thicker Cr2Te3 film (Supplementary Table 1 The monolayer sample accompanies an octahedral to trigonal structural transformation, which splits the t2g triplets into a1g and eg bands, with the possibility of a low spin state.But, the much smaller trigonal splitting field Dt (0.2 eV) than Hubbard U (1.7 eV) prohibits this scenario [32].According to Mott's seminal paper [33], carrier density (hence bandwidth) has an anti-correlation to the lattice constant.In the monolayer case, the smaller in-plane lattice parameter should increase the carrier density, which is entirely contrary to the experimental observation.Therefore, the consistency of the results with DFT + U calculations leaves the dimensionality as the only candidate for the suppressed   and the localized ferromagnetism in the monolayer Cr2Te3.The bad metal property of monolayer Cr2Te3 may result from the interplay between the itinerant Te 5p bands and the localized Cr 3d bands.Although the energy bands near EF are mainly of delocalized Te orbitals, they can mediate the local spin moments of the Cr bands and realize long-range ferromagnetism.Intrinsically, this coherent picture belongs to the RKKY interaction, which couples the localized and itinerant electrons in a more localized ferromagnetic metal [34].Although previous studies report a similar   reduction in monolayer Fe3GeTe2 and monolayer CrI3, it has been proven to be difficult to illustrate the microscopic origin of such reduction from the electronic structures of the magnetic ground state in the true 2D limit.Our multi-probe study demonstrates an unambiguous magnetic transition from Stoner-type to Heisenberg-type in Cr2Te3.
Our finding provides a hint on how to increase   in 2D ferromagnets.For Cr-Te compounds, we find that the ferromagnetic metals have a higher   than the ferromagnetic insulators:   = 170 K in Cr2Te3 [23], = 60 K in Cr2Ge2Te6 [12], and   = 30 K in Cr2Si2Te6 [35] .Moreover, itinerant ferromagnets, such as Fe, Co, and Ni, possess the highest   above 1000 K.This indicates that delocalized electrons are important to realize a higher   , which mediate the spin moments between local atoms to realize long-range order.
Then, according to the Stoner model, larger (  ) is beneficial to a higher   .3d electrons of transition metal have large effective mass and narrow bandwidth, more likely to induce high   ferromagnets if the chemical potential is tuned to approach the 3d flat band.Such properties have been used to control the emergent ferromagnetism in twisted bilayer graphene [36].
In conclusion, we have demonstrated the dual nature of electrons in an atomically thin ferromagnet Cr2Te3.Even in the strict 2D limit, the monolayer Cr2Te3 still retains the ferromagnetism despite the reduction in   due to the dimensionality effect.Our multi-probe spectroscopic analysis reveals a ferromagnetic ground state transition from the itinerant to the localized picture.These results provide crucial information to optimize the performance of 2D ferromagnetism for future applications.

Methods
Thin film growth and ARPES.Atomically thin Cr2Te3 samples were grown in a custom-built MBE chamber with a base pressure of 2 × 10 -10 Torr.The samples were transferred directly to the analysis chamber for in-situ ARPES

STM measurements.
A low-temperature STM system was used to characterize the as-grown films (Unisoku-1300).
All the STM images were taken in the ultrahigh vacuum chamber ( 2x10 -10 Torr) under liquid nitrogen temperature ( 77 K).Electrochemically etched tungsten tips were used after flashing and silver decoration in situ.
Electronic structure calculations.The density-functional theory DFT + U [37] calculations of the structures, band structures, and density of states are performed by the VASP code [38,39].We employ PAW pseudopotentials with GGA [40,41].The total valence electrons treated explicitly in the DFT calculations are Cr 3 6  5 4 1 and Te 5 2  4 electrons, described by a plane-wave basis set with cut-off energy of 550 eV.The bulk crystal structure was relaxed with an unrestricted variable cell relaxation as implemented in VASP.However, the supercell lattice constants for 1ML structure were fixed upon the consideration of Si substrate, only the internal atomic positions are relaxed.The unit cell of the 1ML structure has a 6% ab plane shrinking and a 13% stretch along the c axis, which are chosen according to the STM measured results.The relaxations were done with the Hubbard U turned off and no spin polarization.To obtain the magnetic properties, we perform the collinear magnetic calculation and specify the initial magnetic moment of each atom as the value referred to in the article [42].We find a ferromagnetic solution for the bulk crystal leads to the lowest energy.The initial magnetic moment was set up as ferromagnetic for 1ML selfconsistent run.All band structures were calculated using the corresponding relaxed structures.To acquire an optimal U, we vary U from 0 to 3.1 eV for Cr in our tests.The Fermi surface and the renormalization ratio of the band structures have been adjusted to best agree with the ARPES data.The total DOS was determined by integrating the projected density of states (PDOS) corresponding to the bands below the Fermi surface.

Fig. 3 )
Fig. 3).Figs.3a,b display the Fermi surface maps of the 6 ML and 1 ML Cr2Te3 samples.Circular hole Fig. 4a displays highresolution s-polarization and p-polarization E-k dispersions for the 6 ML sample.The corresponding second-derivative images are shown in Fig. 4b.There are three bands crossing EF in the 6 ML sample: two dispersive hole-like bands near  point come from Te-5p orbitals, and one shallow band near M point stems from Cr-3d orbital.The quasi-flat dispersions in the binding energy range of −1 eV to −1.6 eV are Cr t2g measurements at the HERS endstation of Beamline 10.0.1, Advanced Light Source, Lawrence Berkeley National Laboratory.Both n-and p-doped Si (111) substrates were used for the epitaxial growth.The 7 × 7 reconstructed surface of Si (111) was obtained by ten cycles of flash annealing at 1200 o C. High-purity Cr and Te sources were simultaneously evaporated onto the Si (111) substrate that was kept at 300 o C during the growth.The flux ratio Cr/Te = 1 : 4 was determined by the quartz crystal microbalance (QCM) measurement before the growth.The growth process was monitored by in situ reflective high energy electron diffraction (RHEED), and the corresponding rate is about 20 min per monolayer.After growth, the sample was annealed at 320 o C for 10 min for improved sample quality.The ARPES data were collected with a Scienta R4000 analyzer.The photon energy was 53 eV during measurements, with energy and angular resolution of 20 meV and 0.1 o , respectively.The p-polarization direction was set to be 72 o out of the plane of incident photons.

Fig. 1 .
photon beam on the fixed external magnetic (H ~ 0.5 T) field along the surface normal direction.The degree of circular polarization was ~ 95%.All spectroscopic experiments (XAS and XMCD) were carried out at beamline 13-3 at the Stanford Synchrotron Radiation Lightsource (SSRL).

Fig. 2 .
Fig. 2. Thickness-dependent ferromagnetism in Cr2Te3 thin films.(a, b) Cr L2,3-edge absorption spectra (ρ + and ρ -) on 1 ML and 3 ML samples.The dashed orange and solid green curves denote the left-hand and right-hand polarized XAS signals, respectively.The blue curve represents the XMCD signal, which is enlarged by 10 times in 1 ML case due to the small magnitude.(c-e) Temperature-dependent XMCD spectra in 1 ML, 3 ML, and 6 ML samples.The disappearance of XMCD intensity near the L3 edge is used to determine the Curie temperature TC.(f) The ratio of XMCD to XAS as a function of temperature on 1 ML, 3 ML, and 6 ML samples.The error bar is the uncertainty of background estimation during the XMCD analysis.

Fig. 3 .
Fig. 3. Signature of the transition from Stoner-type to Heisenberg-type ferromagnetism in atomically thin Cr2Te3.(a, b) Fermi surface maps of 6 ML and 1 ML samples.The integrated energy window is ± 20 mV around EF.

Fig. 4 .
Fig. 4. Experimental and theoretical electronic structures in Cr2Te3 thin films.(a) s-polarized and p-polarized Ek dispersions along -M direction for 6 ML samples.The photon energies in s-and p-polarization geometries are 53 eV and 63 eV, respectively.(b) The corresponding second derivative E-k dispersions.The dashed orange line is the guide for the shadow band from Cr-3d orbital.(c) Calculated band structures of bulk Cr2Te3 by the DFT + U method.(d) s-polarized and p-polarized E-k dispersions along -M direction for 1 ML sample.The photon energies in s-and ).Second,   is dependent on (  ) via a relation   ∝ √1 − 1 (  ) ⁄ in the Stoner model, which supports the suppressed   in the monolayer Cr2Te3.Third, as the Stoner model says that the magnetization M is proportional to ( ), the much smaller XMCD percentage signal in the monolayer sample can be naturally explained.However, the absence of exchange splitting energy in the monolayer Cr2Te3 is beyond the framework of the Stoner model.Actually, the product (  ) is 0.72 in monolayer case (Supplementary Table1), conflicting the Stoner criterion.The Strain and dimensionality are two possibilities to enact this change.