Introduction

Two-dimensional (2D) van der Waals (vdW) magnetic materials have gained great attention for both device applications and fundamental science1,2,3,4,5,6,7,8,9. Among the broader family of chromium tellurides (Cr1+δTe2)10,11,12,13,14,15,16,17, 1T-CrTe2 is particularly compelling, featuring a high Curie temperature (TC) above 300 K in its bulk form18 that remains robust down to the thin-film form19,20,21,22,23,24. Furthermore, a rich variety of spintronic phenomena have been revealed in this system, spanning both experimental demonstrations and theoretical predictions23,24,25,26,27,28,29,30,31,32,33,34,35,36. These combined features establish 1T-CrTe2 as a prime candidate for next-generation spintronic platforms. Nevertheless, to explore its full potential, it is desirable to understand the underlying principles driving these remarkable properties.

In this study, we suggest 1T-CrTe2 as a self-doped double-exchange (DE) ferromagnet, a picture grounded in a dual electronic nature of Cr-d electrons. The origin of its high-TC ferromagnetism has been a source of active debate, which has remained elusive despite previous proposals including Stoner model18,37, superexchange34,37,38,39,40,41, DE42,43 and Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction44,45. By employing density functional theory plus dynamical mean-field theory (DFT+DMFT), we present the many-body electronic structure that has not been reported before. The calculated local spin susceptibility and the analysis based on ‘first Matsubara rule’ clearly indicate the dual nature; namely, itinerant eg electrons coexist with localized t2g moments. Based on it, the self-doped DE picture is well established not only by the relatively weak electronegativity of Te but also by several of our synthetic simulations of varying Hund’s coupling (JH), coherent orbital behavior, and total energy profile.

Further, we demonstrate that 1T-CrTe2 exhibits characteristic Hund metallic behaviors. In a class of materials recently dubbed “Hund metals’’46,47,48,49,50, a variety of intriguing phenomena are known to be hosted, including unconventional superconductivity51,52, spin-freezing/non-Fermi-liquid behavior51,53, spin-orbital separation (SOS)54,55,56,57, a broad valence histogram favoring high-spin multiplets48,58, and orbital differentiation59,60,61,62,63. While this concept is well-established in prominent materials like iron-based superconductors46,48,62 and ruthenates64,65,66, its implications for 2D vdW materials remain largely unexplored67. Our study confirms these hallmark features in 1T-CrTe2 and uncovers intriguing physics reminiscent of the orbital-selective Mott phase (OSMP), distinguishing it from canonical Hund metals.

Finally, extending our study to the monolayer limit, we find that structural deformation, rather than reduced dimensionality, is the dominant factor responsible for the notable reduction in TC. Understanding thickness-dependent magnetic behaviors in 2D materials is crucial, as dimensionality reduction can dramatically alter magnetic properties6,8,9. In the case of 1T-CrTe2, experimental studies have revealed complex thickness-dependent phenomena: while the TC generally decreases in thinner films, an enhancement of spin polarization has also been reported21,23,24,25. Our analysis clarifies the interplay between structural relaxation and electronic correlation by showing how specific geometric changes in the monolayer suppress the ferromagnetic (FM) coupling, while simultaneously uncovering an enhancement of the local magnetic moment. This latter finding, originating from the persistent Hund metal nature of the system, not only explains the observed increase in spin polarization but also highlights the potential of strain engineering for controlling magnetism in correlated 2D vdW materials.

Results

Many-body electronic structure by DFT+DMFT

Bulk 1T-CrTe2 forms a layered hexagonal lattice with space group \(P3\overline{m}1\) (No. 164). Each layer has a triangular network of edge-sharing CrTe6 octahedra. These octahedra generate crystal fields that split Cr-3d levels into eg and t2g, and the latter splits further into a1g and \({e}_{g}^{{\prime} }\) under trigonal distortion (Fig. 1a). Within a simple ionic picture, 1T-CrTe2 consists of Cr4+ (3d2) and Te2− (5p6) ions, which yields doubly-occupied \({e}_{g}^{{\prime} }\) and empty a1g and eg orbitals. In reality, however, significant d-p hybridization makes all Cr-d levels have sizable electron occupations as summarized in Table 1.

Fig. 1: Crystal and electronic structure of the bulk phase.
Fig. 1: Crystal and electronic structure of the bulk phase.
Full size image

a Crystal structure of bulk 1T-CrTe2 (left) and a schematic diagram of the crystal field levels (right). b Upper panel: LDA+DMFT spectral functions A(ω) in the PM phase (solid lines; T = 450 K) compared with the LDA results (shaded areas). Lower panel: Corresponding orbital-resolved A(ω) from the LDA+DMFT calculation. c Upper panel: LDA+DMFT Cr-d spectral functions in the FM phase (purple solid; T = 150 K) compared with the LSDA (red dotted) and LSDA+U (blue dash-dot) results. Positive and negative values correspond to the majority (up) and minority (down) spins, respectively. Lower panel: Same as in (b), but for the FM phase. d LDA+DMFT momentum-dependent spectral function A(k, ω) at T = 150 K for the up-spin (left panel) and down-spin (right panel) states compared with the LSDA (red dotted) and LSDA+U (blue dash-dot) bands.

Table 1 Calculated orbital occupations and magnetic moments
Full size table

As a metallic magnet with partially-filled 3d bands, its correlated electronic structure is of particular interest. While a DFT+DMFT study was recently reported, it focused on the magnetic exchange interactions, leaving the many-body electronic structure unexplored68. Figure 1b and c show our LDA+DMFT (local density approximation plus DMFT) results for paramagnetic (PM) and FM phases, respectively. For computational details, see Supporting Information (SI). In the PM solution, a notable many-body feature is the quasiparticle state forming a narrow peak near the Fermi level (EF; ω = 0 eV) as shown in Fig. 1b. Also, DMFT self-energy transfers a sizable portion of Cr-d spectral weight to the higher energy region. These characteristic features of correlated metallic systems, being clearly distinctive from those of the weakly-correlated electron approximations18,19,68, highlight the importance of accurate description of dynamic correlations69,70.

The result for the FM phase is shown in Fig. 1c where we compare LDA+DMFT with local spin density approximation (LSDA) and LSDA+U71,72. The calculated total magnetic moments are 2.35, 2.66, and 2.26 μB/f.u. for LSDA, LSDA+U, and LDA+DMFT (at T = 150 K), respectively, as shown in Table 1. In the LSDA+U solution, the static mean-field treatment of Hubbard-type correction shifts the occupied up-spin states downward and the unoccupied down-spin states upward, enhancing the spin splitting compared to LSDA. In LDA+DMFT, this enhancement is moderate. Instead, as in the PM solution, the dynamical correlation effects transfer a sizable spectral weight toward the higher energies, which, for instance, manifests as a broad tail in the down-spin states above the EF.

It is interesting to note that the coherent state at ω = 0 eV in the PM phase is absent here. While there is a small peak in the down-spin states just above EF only in LDA+DMFT (see a black arrow marked in Fig. 1c), it is nearly temperature-independent. We also found that it appears only when the spin-flip Coulomb interaction term is included (see Fig. S2), featuring the many-body effect reminiscent of the non-quasiparticle peak in half-metals73.

For comparison with experiments, we present the momentum-dependent spectral function A(k, ω) in Fig. 1d. While the overall dispersion is not much different from LSDA (red dotted lines), the LDA+DMFT result exhibits substantial redistributions of spectral weights. The damping reflects the finite quasiparticle lifetime arising from electron-electron scattering. Noticeable features include the incoherent up-spin spectral weight at around −1 eV as well as the hole pockets centered at Γ point and the electron-like bands near M point below EF, which is in good agreement with recent angle-resolved photoemission spectroscopy (ARPES) studies23,24.

Dual nature of Cr-d electrons and self-doped double-exchange

To better understand ferromagnetism in 1T-CrTe2, here we suggest a new picture of self-doped DE. We first investigate the localization versus delocalization of Cr-3d electrons. Figure 2a shows the calculated DMFT spin susceptibility \({\chi }_{\rm spin}^{\rm loc}\), indicating the Curie-Weiss (CW) behavior rather than Pauli-like (see the grey line). In this regard, it is in contrast to Stoner ferromagnetism18,37,74. Orbital dependent \({\chi }_{\rm spin}^{\rm loc}\) elucidates further details: the blue and orange line in Fig. 2a is \({\chi }_{\rm spin}^{\rm loc}\) for eg and t2g orbitals, respectively. It is clearly noted that eg exhibits the notably different feature (being much closer to itinerancy) from the localized t2g moment46,56,67,75,76. This intriguing orbital dependency is also seen from the analysis based on ‘first-Matsubara-frequency rule’77. This rule states that in Fermi-liquid (FL) regime, the imaginary part of local self-energy at the lowest Matsubara frequency is linearly proportional to temperature: \({\text{Im}}\Sigma (i\pi {k}_{B}T) \sim \lambda T\) where λ is a real constant and Σ is self-energy. Figure 2b reveals that eg orbitals nearly restore FL behavior with itinerant nature at low temperatures. On the other hand, t2g orbitals show a clear deviation: Down to the lowest temperature (T = 150 K) we investigated, the T = 0 intercept remains well below 0. Thus it also supports our suggested picture of ‘dual’ nature of Cr-3d electrons in this material. In the SI, we present another analysis based on the hybridization function (see Fig. S4). This ‘dual’ nature is the basis of self-doped DE.

Fig. 2: Dual electronic nature driving double-exchange ferromagnetism.
Fig. 2: Dual electronic nature driving double-exchange ferromagnetism.
Full size image

a The temperature dependence of orbital-resolved static local spin susceptibility \({\chi }_{\rm spin}^{\rm loc}\) obtained in the PM phase. b The imaginary part of self-energy at the lowest fermionic Matsubara frequency (iω0 = iπkBT, where kB is Boltzmann constant) in the PM phase. The inset shows an enlarged view of the low-temperature region. c Orbital-resolved M = μB(〈n↑〉 − 〈n↓〉) as a function of JH at T = 150 K. The FM phase sets in for JH ≳ 0.5 eV.

In the conventional DE picture, the externally doped carriers mediate the localized spin moments to align ferromagnetically. For the typical case of colossal magneto-resistive manganites La1−xDxMnO3 (D: divalent ion)78,79,80, the doped electrons occupy and hop around through eg band, and the localized t2g spins are ordered ferromagnetically with the aid of strong JH against the crystal field splitting. The picture of ‘self-doped’ DE was suggested to explain the ferromagnetism in CrO281. This intriguingly modified version can be valid when two different natures of Cr-d states coexist. In the original paper by Korotin et al., it was argued based on the flat dxy band (locating below EF) and the higher-lying dispersive dyz±zx with strong d-p hybridization (going across EF). Considering the increased electron affinity for the high oxidation state of Cr4+ ion, the effective charge transfer is expected and the DE mechanism validated even without actual doping.

The aforementioned dual nature is therefore the solid ground for self-doped DE ferromagnetism in 1T-CrTe2. We note, while the original idea81 was suggested by largely relying on the band width and occupancy argument, DFT+DMFT directly accesses the local/delocal nature from \({\chi }_{\rm spin}^{\rm loc}\). Given that Te has weaker electronegativity than O, 1T-CrTe2 is expected to have greater d-p hybridization and is better suited for the self-doped DE picture. In fact, previous ARPES studies identify Te-p-derived hole pockets near the Γ point23,24,28, which is also corroborated by X-ray absorption spectroscopy (XAS) and X-ray magnetic circular dichroism (XMCD) data reporting, respectively, the predominant Cr3+ state and the spin moment exceeding 2 μB23,24.

Figure 2c shows the calculated magnetization M with varying JH at a fixed T = 150 K < TC. We note that non-zero M is obtained only with the sufficiently large JH (\(\gtrsim {J}_{H}^{* } \sim 0.5\) eV). Interestingly, not only t2g but also eg electrons carry sizable moments despite their small weight at EF and the itinerant nature discussed above. Both orbital moments share the common FM on-set value of \({J}_{H}^{* }\) and increase with JH while maintaining their relative ratio (Fig. S5a). The pivotal role of JH is further corroborated by the increasing TC with JH (Fig. S5b)82 and the DFT+U total energy calculation revealing the antiferromagnetic (AFM) to FM transition (Fig. S6). All these observations are well consistent with the suggested picture of self-doped DE.

Hund metallic behaviors

Let us further investigate how JH influences the electronic behavior of 1T-CrTe2. In particular, we explore the possible ‘Hund metal’ characters. Figure 3a shows the imaginary part of self-energy \({\text{Im}}\Sigma (i{\omega }_{n})\) (ωn: fermionic Matsubara frequency). All orbitals exhibit an enlarged \(| \rm Im\Sigma |\) with increasing JH, indicative of the enhanced incoherence by JH. Notably, the t2g self-energy is more significantly affected than the eg’s. As JH approaches 0.85 eV (presumably close to the realistic value for this system), t2g states enter the (so-called) ‘spin-frozen’ regime characterized by a finite scattering rate51,53; see Fig. S7a in SI. The orbital-dependent incoherence can also be quantified by scattering rate (see Fig. S7b in SI). These observations (i.e., the low-temperature incoherence by JH and orbital-dependent correlations) are key signatures of Hund metal49. For the analysis of the difference between \({e}_{g}^{{\prime} }\) and a1g, see Fig. S7c and S7d in SI.

Fig. 3: Hallmarks of Hund metallicity.
Fig. 3: Hallmarks of Hund metallicity.
Full size image

a Imaginary part of self-energy on the Matsubara frequency axis \({\text{Im}}\Sigma (i{\omega }_{n})\) at T = 150 K and with JH = 0.6, 0.8, and 1.0 eV. b The calculated local spin susceptibility \({\chi }_{\rm spin}^{\rm loc}\) as a function of JH at T = 450 K. c Probability distribution of the Cr-d orbital occupancy, Nd. d The temperature dependence of \({k}_{B}T{\chi }_{\rm spin/orb}^{\rm loc}\). The spin (solid line) and orbital (dashed line) susceptibilities χ within the eg and t2g manifolds are represented with blue and orange colors, respectively. Vertical arrows mark the onset temperatures for spin screening. e The valence histogram within the t2g manifold in terms of the electron number \({N}_{{t}_{2g}}\) and the spin-state \({S}_{z}^{{t}_{2g}}\).

The coexistence of large spin moments and significant charge fluctuations also exemplifies the Hund metal behavior46,50. As shown in Fig. 3b, the local spin susceptibility \({\chi }_{\rm spin}^{\rm loc}\) increases with JH, indicating the formation of large spin moments. This trend is further confirmed by the increased population of high-spin multiplets (Fig. S8a). At the same time, Fig. 3c shows the broad probability distribution of valence states over Nd = 3 (16.9%), 4 (47.7%) and 5 (29.6%), revealing significant charge fluctuations. Its magnitude, estimated by \(\langle {(\Delta {N}_{d})}^{2}\rangle =\langle {N}_{d}^{2}\rangle -{\langle {N}_{d}\rangle }^{2}\approx 0.67\), is comparable to that of iron-based superconductors—prototypical Hund metals83. These fluctuations, which favor high-spin multiplets, impede the screening of local spin moments, thereby promoting Hund metallicity84. For further details, see Fig. S8.

Next we investigate the phenomenon so-called ‘SOS’, another hallmark of Hund metals54,55,56,57. It refers to the large separation between the onset screening temperatures for spin (\({T}_{\rm spin}^{\rm onset}\)) and orbital (\({T}_{\rm orb}^{\rm onset}\)) degrees of freedom; i.e., \({T}_{\rm spin}^{\rm onset}\,\ll\, {T}_{\rm orb}^{\rm onset}\). Following Ref. 56, we define \({T}_{\rm spin/orb}^{\rm onset}\) as the characteristic temperature at which kBTχ begins to deviate from the CW behavior and to decrease (χ: static local susceptibility). This deviation is clearly captured when we separate χ into t2g and eg components, as shown in Fig. 3d. The SOS is well identified for t2g manifold. \({T}_{\rm spin}^{\rm onset}\) is found around 2000 K, much lower than \({T}_{\rm orb}^{\rm onset} > 30000\) K. In contrast, the screening behavior in eg manifold shows a much weaker SOS; much higher \({T}_{\rm spin}^{\rm onset}\) is closer to \({T}_{\rm orb}^{\rm onset} > 30000\) K. This substantial screening of the eg moment is consistent with the Pauli-like susceptibility discussed above.

Whereas these Hund metallic characters are well identified in 1T-CrTe2, notable differences are also found. Figure 3e shows the valence histogram for the t2g manifold. It reveals the predominance of \({t}_{2g}^{2}\) and \({t}_{2g}^{3}\), both favoring high-spin configurations (\(| {S}_{z}^{{t}_{2g}}| =1.0 \,\text {and}\, 1.5\), respectively) in accordance with Hund’s rule. Note that the half-filling \({t}_{2g}^{3}\) configuration has higher probability than the nominally expected \({t}_{2g}^{2}\) while the latter corresponds to the typical filling for Hund metals. As a consequence, the intriguing electronic behavior is observed well distinctive from the typical Hund metals. Figure S9a shows that with increasing JH, the spectral weight around ω ~ 0.6 eV is gradually suppressed. The origin of this redistribution can be understood from the pronounced peak in the t2g self-energy, \(-\text{Im}\Sigma (\omega )\); see Fig. S9b. This behavior is reminiscent of JH-enhanced atomic charge gap observed in half-filled \({t}_{2g}^{3}\) systems47,49. Consequently, charge fluctuations are suppressed by JH (see the inset of Fig. S9a). It contrasts with the JH-dependence of conventional Hund metals, such as those with \({t}_{2g}^{4}\) or \({t}_{2g}^{2}\) configurations, where charge fluctuations typically increase with JH61,85.

Monolayer limit

Finally, we investigate the monolayer limit. The magnetic ground state of monolayer (‘1ML’) is sensitive to the in-plane lattice constant. Previous reports indicate that the smaller or distorted lattice parameter (a ≈ 3.4 – 3.7 Å) gives rise to AFM order86,87,88,89,90,91, whereas the larger (bulk-like; a ≈ 3.8 – 3.9 Å) corresponds to FM22,24,92,93. As our primary focus here is on the fate of bulk ferromagnetism in the 2D limit, we adopt the bulk in-plane lattice parameter. Figure 4b shows the calculated magnetization of 1ML and bulk 1T-CrTe2 (‘bulk’) as a function of temperature. For comparison, we also calculated the monolayer whose structure is fixed to the bulk one. Its result is indicated by ‘1ML-b’ (green crosses). It is clearly noted that the TC is significantly reduced in 1ML (TC = 214 K), ~ 53% of the bulk value. This trend of decreasing TC by reducing the sample thickness agrees well with experiments21,24,25; see Table 2. Although one previous study reports the opposite32, a recent study suggested that sample-dependent variations may account for the observed difference94. Considering the mean-field limitation, we estimated TC ratio between ML and bulk limit, which shows the good agreement: \({T}_{C}^{\rm mono}/{T}_{C}^{\rm bulk}=53 \%\) in our calculation and \({T}_{C}^{\rm thin}/{T}_{C}^{\rm thick}\approx\) 50-67% in experiments24,25.

Fig. 4: Effect of structural deformation on monolayer magnetism.
Fig. 4: Effect of structural deformation on monolayer magnetism.
Full size image

a Structural parameters of hTe (the Te atom height with respect to the Cr layer) and θ (the Cr–Te–Cr bond angle). The major change by the structural relaxation is the Te atom displacements (indicated by small vertical blue arrows), which increase hTe and reduce θ compared to the bulk case (or equivalently 1ML-b). b The calculated magnetization M = μB(〈n↑〉 − 〈n↓〉) as a function of temperature. The dash-dot lines are the fits to U = 5 eV data with \(M(T)={M}_{s}{(1-T/{T}_{C})}^{\beta }\). The fixed JH = 0.85 eV is used.

Table 2 Comparison of structural and magnetic properties for bulk and monolayer
Full size table

Importantly, the calculated TC for ‘1ML-b’ is ~ 94% of bulk value, demonstrating that the decreasing TC in the monolayer limit is mainly attributed to the structural effect rather than the dimensional reduction. The same trend is also found in DFT and DFT+U results of Heisenberg-type exchange couplings (first neighbor J1) and TC estimated from total energy mapping and mean-field estimation, respectively. In Table 2, we summarize our calculation results. It is noted that the Te height and the Cr–Te–Cr bond angle changes by +0.086 Å and −2.1∘, respectively. These changes are effective enough to suppress the intralayer FM couplings by reducing the Cr-d electron hopping through Te anions. In fact, the calculated Slater-Koster hopping parameters95 between nearest-neighbor Cr-d and Te-p are found to be reduced: ∣Vpdσ(π)∣ = 1.148(0.588) eV in 1ML-b and 1.075(0.547) eV in 1ML. The same feature can also be found in the density of states (DOS); see, e.g., the bonding-antibonding splitting and the estimated bandwidth in Fig. S10a. Our results suggest the strain engineering as a useful path to control TC and other magnetic properties22,34,37,38,39,40,96,97,98.

Hund metallic behaviors discussed above persist in the monolayer limit. The characteristic features of orbital differentiation and SOS remain similar to the bulk results, as shown in Fig. S7e–h and Fig. S11, respectively. It demonstrates Hund metallicity and dual nature are robust against the geometric changes in the 1ML structure. To investigate the possible change of Coulomb interaction in the reduced dimension99,100, we repeat the calculation with the larger U and find that the estimated TC does not depend much on U (see Fig. 4b).

One important finding in Table 2 is that, as one approaches the monolayer limit, the moment size Ms increases whereas TC decreases. To understand this intriguing behavior, we first investigate the self-energy. As presented in Fig. S12 in SI, the magnitude of imaginary Σ(iωn) is enlarged in 1ML, indicative of the enhanced correlation. This feature is particularly pronounced for a1g and \({e}_{g}^{{\prime} }\) at high frequency. The same feature has also been observed in the analysis of orbital- and thickness-dependent Z(T), although its numerics likely requires further elaboration. As a Hund metal, 1T-CrTe2 is governed by correlations arising from JH rather than U. Consequently, the enhanced Hund correlation favors higher-spin configurations, giving rise to the enhanced moment Ms for 1ML. The calculated multiplet histogram also shows that the higher spin configurations have more weights in the case of 1ML than 1ML-b (and bulk) as presented in Fig. S13. It is also included in Table 2 in terms of Savg. These analyses provide a natural explanation for the unusual enhancement of spin polarization accompanied by TC reduction in the thinner 1T-CrTe2. Our result is consistent with a recent spin-ARPES study by Zhang et al., which reports an increase in spin polarization as the film thickness is reduced23.

Discussion

The current study suggests a new picture to understand the FM order in 1T-CrTe2. As a DE picture, it is distinct from the previously proposed itinerant Stoner ferromagnetism and RKKY interactions, while it can be closer to superexchange in the sense that anion-mediated hopping is a major factor18,34,37,38,39,40,41,42,43,44,45. Although DE mechanism itself was also considered42,43, the orbital space in these studies was confined to t2g manifold and therefore the mixed-valency of Cr3+(\({t}_{2g}^{3}\)) and Cr4+(\({t}_{2g}^{2}\)) was taken into account. It is clearly different from the self-doped DE picture in which both t2g and eg orbitals are essential as discussed above, and their dual nature provides the indispensable ground for FM order. In fact, FM ordering is suppressed if we define the correlated subspace to be t2g only (see Fig. S14).

It is interesting to compare Hund-metallic 1T-CrTe2 with orbital-selective Mott systems. While the latter has an obvious difference by having well-developed Mott gap for the specific orbital, the inter-orbital interaction also plays a crucial role in OSMP, and JH enhances the incoherence of the metallic (wider-bandwidth) orbital state59,101,102,103. This behavior is observed in 1T-CrTe2 as shown in Fig. S15. It is in fact responsible for the weak magnetic response within the eg sector, reflected in the subtle increase of the low-temperature spin susceptibility (see Fig. 2a). The similarity can also be highlighted from the point of view of the intimacy between DE and OSMP103.

The suggested dual nature can further be corroborated by elaborate experiments. For example, a recent inelastic neutron scattering study successfully captures the dual nature of magnetic excitations and the coexistence of local and itinerant moment in a vdW metallic ferromagnet104. Also, motivated by recent spin-ARPES studies23,105, we present temperature- and spin-dependent spectral function in Fig. S16. It is noted that, upon lowering temperature, the evolution of DMFT spectral function shows some degree of orbital dependence.

In summary, we have performed the detailed investigation into the many-body electronic structure of 1T-CrTe2 using DFT+DMFT, identifying it as a self-doped DE ferromagnet with pronounced Hund metallicity. This picture is built upon our discovery of a dual electronic nature in Cr-d orbitals, where the analysis of local spin susceptibility and self-energy clearly distinguishes itinerant eg electrons from localized t2g moments. The overarching role of JH not only drives the DE mechanism but also establishes 1T-CrTe2 as a Hund metal, with physics analogous to OSMP. Extending to the monolayer limit, we find that while this physical picture remains robust, structural deformation is the dominant factor suppressing the FM coupling and TC, rather than the dimensional reduction itself. Our findings offer a unified perspective connecting DE, Hund physics, and orbital-selective correlations, providing crucial insights for designing and exploring other correlated 2D vdW metallic magnets.

Methods

DFT

We performed density functional theory (DFT) calculations using the all-electron full-potential linearized augmented plane-wave (LAPW) code WIEN2k106. For the exchange-correlation functional, we employed the local density approximation (LDA)107. We used \({R}_{MT}{K}_{\max }=7.0\) where RMT is muffin-tin radius and \({K}_{\max }\) is plane-wave cutoff. The muffin-tin radii of 2.47 and 2.50 a.u. were used for Cr and Te atoms, respectively. We adopted the experimental lattice parameters of a = b = 3.789 Å, and c = 6.095 Å for bulk18. The effect of vdW correction of so-called ‘Grimme D3108’ is found to be less than 0.02% in terms of bond length and angle changes. For monolayers, the in-plane lattice parameters were fixed to the bulk value, and a vacuum layer of 20 Å was taken into account. It is comparable with the experimental reports of Refs. 22,24,92,93. The k-point grids of 16 × 16 × 8 and 16 × 16 × 1 were used for bulk and monolayers, respectively.

To analyze the orbital-dependent electron hopping, we constructed maximally localized Wannier functions (MLWFs) from the non-spin-polarized LDA band structures109,110. For the Wannier construction, we set an energy window from −6.0 to 3.0 eV, which covers the well-isolated five Cr-d and six Te-p band complex from the rest. The nearest-neighbor p-d hopping parameters (Vpdσ and Vpdπ) were extracted by fitting Slater-Koster tight-binding matrix elements95 to the Wannier-based Hamiltonian. To quantify the orbital-dependent bandwidths, the following quantities were estimated with orbital index \(\eta ={e}_{g},{a}_{1g},{e}_{g}^{{\prime} }\):

$${\mu }_{\eta }={\int }_{-\infty }^{\infty }E{D}_{\eta }(E)dE,$$
(1)
$${\sigma }_{\eta }^{2}={\int }_{-\infty }^{\infty }{(E-{\mu }_{\eta })}^{2}{D}_{\eta }(E)dE,$$
(2)

where Dη(E) denotes the calculated LDA density of states (DOS) at the energy E. μη represents the center of mass position of DOS and ση corresponds to the bandwidth.

DFT+U

For DFT+U calculations, we primarily used VASP package111,112. We employed a charge-only exchange-correlation functional within Perdew-Burke-Ernzerhof (PBE) generalized gradient approximation (GGA)113. This choice can be crucial especially for elucidating the JH dependence114,115,116,117,118,119. The functional recipe suggested by Liechtenstein120 was adopted with Hubbard U = 2.9 eV and Hund JH = 0.85 eV. The structural parameters and the internal coordinates were optimized within this scheme. This choice of interaction parameters was the estimation from constrained RPA (cRPA) for SrCrO3, a system that shares the same formal Cr valence of 3d2121. Also, it corresponds to Ueff = U − JH = 2.05 eV that is quite close to Ueff = 2 eV used in previous DFT+U studies for 1T-CrTe2 adopting the Dudarev functional scheme122,123.

To analyze the JH-dependent magnetic stability, we compared the total energies of different magnetic phases, as shown in Fig. S6(a–d); intralayer ferromagnetic (FM), antiferromagnetic (AFM), zig-zag AFM (ZZ) order combined with interlayer FM, AFM order. These DFT+U calculations were performed with the \(2\times \sqrt{3}\times 2\) supercell and the 8 × 10 × 4 k-point grid at a fixed U of 2.9 eV and the crystal structure of the bulk FM phase. We also checked and found that the internal relaxation for each magnetic phase only slightly shifts the phase boundaries and do not change the conclusions.

The DOS in Fig. 1c is obtained from the DFT+U calculation using WIEN2k to maintain the consistency with the other results therein (i.e., LSDA and LDA+DMFT). Since the charge-only DFT combination with +U functional is not available in this code, it is the result from LSDA+U (rather than LDA+U). We found, however, that they are not much different from each other from the comparison to the VASP calculation result conducted within charge-only LDA+U.

DFT+DMFT

To describe the dynamical electronic correlations, we employed DFT plus dynamical mean-field theory (DFT+DMFT)69,70, which treats both itinerant and localized electrons on an equal footing. Our charge self-consistent DFT+DMFT calculations were performed using the EDMFTF package124, interfaced with WIEN2k. For the DFT part of the scheme, we used the LDA functional125. Hybridization-expansion continuous-time quantum Monte Carlo (CTQMC) was adopted for the DMFT quantum impurity solver126. A large hybridization window from −10 to 10 eV was chosen. To obtain one-particle spectral functions on the real-frequency axis, we performed an analytic continuation of the self-energy from the imaginary axis using the maximum entropy method124,127,128,129,130,131,132,133.

In 1T-CrTe2, each Cr atom is under an octahedral environment of six Te anions with trigonal distortion, exhibiting the D3d point group symmetry134. We used a trigonal basis composed of two doublets (eg and \({e}_{g}^{{\prime} }\)) and a singlet (a1g). As also shown in Fig. 1b, t2g (\(={a}_{1g}+{e}_{g}^{{\prime} }\)) states are dominant at the Fermi level (EF) while d-p states also cross EF and finite eg states are available at EF. Including eg orbitals (as well as t2g orbitals) as the correlated subspace in the DFT+DMFT procedure is found to be crucial as discussed in the manuscript and below.

For the local Coulomb interaction, we employed the fully rotationally invariant (‘Full’) form, which includes spin-flip and pair-hopping terms, contrary to another possible choice of so-called density-density (‘den-den’) approximation. It can be crucial for the accurate description of both atomic multiplets and Kondo screening of local moments, which are central to the physics of Hund metals49,135,136. Retaining full rotational invariance is also known to be important for predicting realistic magnetic transition temperatures, which are often significantly overestimated by den-den approximations136,137,138. While the den-den interaction yields a qualitatively similar valence histogram for our system, it fails to capture crucial dynamical effects stemming from spin-flip processes, as exemplified by a spectral feature reminiscent of the non-quasiparticle peak shown in Fig. S2 in SI.

Our analysis explores a range of interaction parameters; however, unless otherwise specified, the results presented are from U = F0 = 5 eV and JH = (F2 + F4)/14 = 0.85 eV. This choice is supported by comparisons between our theoretical results and experimental data (see Section ‘Many-body electronic structure by DFT+DMFT’ and Table 2). We also checked the robustness of our conclusions: variations of up to ± 20% in the interaction parameters did not significantly alter the key physical features, such as the overall FM spectral shape and the magnitude of the magnetic moment, leaving the qualitative picture unchanged. Previous cRPA calculations100,117 of related materials show that the JH change in thin sample limit is negligible whereas that of U can be as large as ~ 45%. For relevant comparative investigations, see Fig. S17 in SI. Furthermore, these parameters are comparable to those used for other 3d metallic chalcogenides48,139. We used the double-counting scheme of Ref. 140. The ferromagnetic transition temperature (TC) was estimated by fitting M(T) data points (Fig. 4b) to the critical scaling formula \(M(T)={M}_{s}{(1-T/{T}_{C})}^{\beta }\). With Cr d-occupancy 〈nσ〉 for spin σ, each M = μB(〈n↑〉 − 〈n↓〉) point is directly obtained from DFT+DMFT calculation.

The DMFT local susceptibilities are defined as:

$${\chi }_{{\rm{s}}{\rm{p}}{\rm{i}}{\rm{n}}/{\rm{o}}{\rm{r}}{\rm{b}}}^{{\rm{l}}{\rm{o}}{\rm{c}}}={\int }_{0}^{\beta }\left\langle {O}_{\mathrm{spin}/\mathrm{orb}}(\tau ){O}_{\mathrm{spin}/\mathrm{orb}}(0)\right\rangle d\tau ,$$
(3)

where β = 1/(kBT) is the inverse temperature (kB: Boltzmann constant), and τ is an imaginary time. For spin susceptibility, \({O}_{\rm spin}=\frac{1}{2}{\sum }_{m}[{n}_{m}^{\uparrow }(\tau )-{n}_{m}^{\downarrow }(\tau )]\). The summation in Ospin runs over orbital index m within t2g, eg, and the whole 3d subspaces; \({\chi }_{\rm spin}^{{t}_{2g}}\), \({\chi }_{\rm spin}^{{e}_{g}}\), and \({\chi }_{\rm spin}^{\rm total}\), respectively. For orbital susceptibility, \({O}_{\rm orb}=\frac{1}{2}{\sum }_{\sigma }[{n}_{e_{g}^{\prime} }^{\sigma }(\tau )/2-{n}_{{a}_{1g}}^{\sigma }(\tau )]\) for \({\chi }_{\rm orb}^{{t}_{2g}}\) and \({O}_{\rm orb}=\frac{1}{2}{\sum }_{\sigma }[{n}_{{d}_{{z}^{2}}}^{\sigma }(\tau )-{n}_{{d}_{{x}^{2}-{y}^{2}}}^{\sigma }(\tau )]\) for \({\chi }_{\rm orb}^{{e}_{g}}\), as defined in the previous studies dealing with spin-orbital separation56,141. The summation runs over the spin degrees of freedom σ = ↑, ↓.