Mechanism of magnetic phase transition in correlated magnetic metal: insight into itinerant ferromagnet Fe_3-δGeTe₂

Recently, van der Walls (vdW) itinerant ferromagnet Fe₃GeTe₂ (FGT) has garnered significant attention. It is a potential candidate for voltage-controlled spintronic devices due to its exceptional exfoliation properties and robust high Curie temperature [17, 18]. In bulk material, the ferromagnetic (FM) Curie temperature $T_{c}$ is around 220 K, and it can even reach room temperature with an ionic gate in its thin flakes [17, 19, 20]. Furthermore, experiments have observed significant electron mass enhancement [20, 21, 22, 23], which is also supported by previous theoretical calculations [24]. Considering its high ferromagnetic transition temperature and strongly correlated effects, the itinerant ferromagnet FGT presents an ideal platform for investigating the dual nature of magnetism and its interplay with other exotic phenomena [25].

Here, we highlight some mysteries that may be related to the issue of magnetism in FGT. First and foremost is the question of how to describe the Fe-$d$ electrons accurately. Although the itinerant Stoner model is commonly accepted [20, 26], recent experiments suggest that the Heisenberg model may be necessary to describe the ferromagnetism in FGT correctly [27, 28, 29, 30]. For instance, the angel-resolved photoemission spectroscopy (ARPES) experiments find that the temperature-dependent evolution towards the ferromagnetic transition is strikingly weak [27, 30]. Secondly, as is commonly discussed in iron-based superconductors [5, 31, 32], it is unclear whether the irons in FGT exhibit Hund physics or Mott physics. Kim et al. suggest that it is a “site-differentiated” Hund metal [33], while another study suggests that it is close to the orbital selective Mott physics [34]. Finally, previous studies have reported heavy fermion behavior in FGT with a large Sommerfeild coefficient ($\gamma_{0}=135\,$mJ/mol K²) [20, 21, 22]. This is striking since $d$-electron heavy fermion systems are rare [35, 36, 37]. While in FGT, heavy fermion behavior appears to be promoted within ferromagnetic order [22]. These controversies encourage us to explore the complex interaction of magnetism with possible Hund, Mott, or Kondo physics in this system.

In this work, we systematically investigate the correlation effects in itinerant ferromagnet Fe_3-δGeTe₂ using density functional theory combined with dynamical mean-field theory (DFT+DMFT). Our findings reveal a detailed evolution of electronic structures during magnetic transition in ferromagnets, characterized by spectral weight transfer between the two spin channels. This process is driven by dynamical correlation effects with frequency dependence self-energy, rather than by static spin splitting with band shift in the mean-field Stoner model. Similarly to Hund metals, the quasiparticles form in itinerant ferromagnet as temperature decreases. Interestingly, the Fe atoms in FGT exhibit two distinct behaviors: one follows Mott physics, while the other exhibits Hund physics. This suggests that FGT may be the material to exhibit both Hund and Mott physics, stemming from the multi-site nature of correlated atoms in its crystal structure. Furthermore, our results indicate that Hund’s coupling promotes the formation of ferromagnetic order, which in turn reduces spin fluctuations and enhances the clarity of the flat bands near the Fermi level. These well-defined flat bands may subsequently hybridize with other itinerant bands, leading to heavy fermion behavior at lower temperatures, akin to what is observed in $f$-electronic heavy fermion materials. Our findings not only naturally reconcile the conflicts between heavy fermion behavior and long-range magnetic order, but also open up a new frontier for discovering new quantum states in itinerant magnets.

In non-magnetic (NM) DFT+U calculations, the Fe-$d$ electrons dominate near the Fermi level in Fig. 1b. However, in ferromagnetic order, the correlation effects strongly localize Fe-$d$ electrons, causing the Fe-bands to be pushed away from the Fermi level. As a result, the Fe-bands in the spin-up channel are almost entirely filled and pushed down to 5 eV below the Fermi level, leaving only itinerant bands with Te character in Fig. 1c. In contrast, the bands in the spin-down channel move up slightly, leaving a small peak of Fe-$d$ density of states (DOS) at the Fermi level. Besides, the partial DOS of Fe1 and Fe2 have similar peak structures, indicating strong hybridization between them. We note that the DOS at the Fermi level in non-magnetic GGA+U calculations is reduced to 2.9 states/eV f.u., much smaller than 13.6 in GGA calculations [21]. Moreover, the calculated band structures in the ferromagnetic state exhibit a large exchange splitting of over 4 eV compared to the non-magnetic state. However, ARPES measurements indicate that upon heating across the Curie temperature, only minor changes in the shape and position of the band dispersions are observed [27, 30]. Hence, the much smaller DOS in DFT+U calculations, combined with the weak temperature-dependent evolution of Fe-$d$ bands in ARPES measurements, suggests to reconsider the application of Stoner model to FGT.

Here, we conclude several significant findings from further inspecting the microscopic processes in DFT+DMFT calculations. The most important observation is the spectral weight transfer in Hubbard bands, rather than the spin splitting suggested by the Stoner model. The colored arrows in Fig. 2b indicate the primary areas where spectral weight transfer occurs. This behavior is significantly different from those observed in DFT and DFT+U calculations. Previous ARPES experiments have strongly indicated this spectral weight transfer behavior [26]. They found that the spectral weight transfer is independent from the details of band structures, with the enhanced DOS at 500 meV and suppressed DOS at 200 meV below the Fermi energy in ferromagnetic state. Furthermore, recent ARPES measurements also show a weak temperature-dependent evolution of band shift towards the Curie temperature, suggesting that local moments may play a crucial role in ferromagnetic order [27, 30]. In short, our proposed mechanism of spectral weight transfer agrees well with ARPES measurements. Additionally, previous DFT+DMFT calculations of Fe and Ni also reveal a weak temperature dependence of the $d$-band shift within the temperature range from 0.6 to 0.9 $T_{c}$ [42].

Secondly, a crossover is observed from high-temperature incoherence states to low-temperature coherence states in FGT, similar to what occurs in Hund metals [5, 10]. It is supported by recent optical experiments, indicating that Hund’s coupling is essential in FGT [28]. In our DFT+DMFT calculations, the formation of ferromagnetic order is more sensitive to Hund’s coupling $J$ than to Coulomb repulsion $U$, highlighting the critical role of Hund’s coupling in establishing ferromagnetic order (See Supplementary Note 2). The incoherent Hubbard bands originate from the significant spin scattering of Fe-$d$ electrons, which can be clearly seen from the self-energy. The imaginary parts of the self-energy in Fig. 2c have significant values at zero frequency, approximately 2 eV for Fe1 and 0.1 eV for Fe2. However, at low temperatures, the imaginary parts of the self-energy approach zero, leading to the quasiparticle bands becoming distinct near the Fermi level, as shown in Fig. 2b. It is important to note that the sharpness observed above the Fermi level for majority-spin states and below the Fermi level for minority-spin states is predominantly influenced by band occupation, as suggested by prior studies on excitations in cobalt [43]. The higher occupation in the spin-up channel results in small upper Hubbard bands, which causes the spectral function to appear sharp above the Fermi level. Conversely, this situation is reversed in the spin-down channel. Furthermore, correlation effects result in the renormalization of the Fe-$d$ bands near the Fermi level, which can be estimated from the real parts of self-energy with quasiparticle weight, $Z=(1-\partial_{\omega}\rm{Re}\Sigma|_{\omega=0})^{-1}$. The real parts of the self-energy are illustrated in Fig. 2e, revealing characteristics indicative of moderate correlation. Table 1 lists the mass enhancements ($m^{*}/m_{\rm LDA}$) of each orbital, calculated as $Z^{-1}$ in DMFT. Our results demonstrate that all the Fe-$d$ orbitals are renormalized in both non-magnetic and ferromagnetic phases.

Thirdly, it is vital to shed light on the differences between FGT and other paramagnetic Hund metals [12, 44]. We focus on the detailed evolution of electronic structures during the magnetic transition. The highest atomic probabilities from the DMFT impurity solver is shown in Fig. 2d. Despite the fact that the average total spin of Fe1 and Fe2 are zero in paramagnetic order, the most significant cases are $S_{z}=\pm 5/2$, and they have the same probabilities. This implies that the Fe atoms have large local moments and fluctuate significantly over time. In the ferromagnetic state, the highest probabilities of Fe1 and Fe2 are primarily composed of only positive values of $S_{z}$ in Fig. 2f, indicating the formation of magnetic order. Compared with paramagnetic Hund metals (See Supplementary Note 3), we recognize that the long-range magnetic order will rapidly suppress the spin fluctuations, making it easier to form quasiparticles in the magnetic phase.

Finally, we recognize that Fe1 and Fe2 atoms exhibit different physical behavior, with Fe1 being described by Mott physics and Fe2 characterized by Hund physics. This difference is evident in the histogram associated with each atomic configuration (See Supplementary Note 4). In Figs. 2d and 2f, Fe1 appear concentrated in a single charge state, $N=5$, typical of the Mott physics [10]. However, the histogram of Fe2 extend over many charge states, a direct signature of Hund physics [5]. Another piece of evidence is the feature of DOS. The partial DOS of Fe1 has a pseudogap at the Fermi level between two broad humps near $\pm$1 eV in Fig. 2a, which is very similar to the Mott system V₂O₃ [32]. Conversely, the partial DOS of Fe2 is characterized by a single broad feature, similar to the Hund metal Sr₂RuO₄ [32]. We realize that the different physics of Fe1 and Fe2 may be due to their local environments, with the occupation of Fe1 and Fe2 being 5.0 and 5.8 in our DFT+DMFT calculations, corresponding to Fe1³⁺ and Fe2²⁺. In multi-orbital correlated systems, Hund’s coupling has a Janus-faced effect depending on the electron occupation [45]. As Fe1 is close to half-filling, an increase in Hund coupling results in a widening of the Mott gap, which is conducive to the emergence of Mott physics. In contrast, for Fe2, which is away from half-filling, the Mott gap decreases with increasing Hund coupling $J$, allowing Hund physics to play a predominant role. The valences of Fe atoms in our calculations are obtained using an exact double-counting scheme in DFT+DMFT calculations [46] and are consistent with previous predictions (See Supplementary Note 5) [17, 19].

Figure 3 shows the temperature-dependent electronic structures in nonstoichiometric Fe_3-δGeTe₂ ($\delta\sim 0.18$). Given that Fe deficiency results in hole doping [48], we implement hole doping simulations by reducing the total number of electrons to decrease computational costs. As expected, the main findings from stoichiometric FGT calculations are retained, including the spectral weight transfer in Hubbard bands, low-temperature quasiparticles with ferromagnetic order, and the coexistence of Hund physics and Mott physics. We observe that the band location remains relatively constant with temperature in Figs. 3a and 3b, which agrees well with recent APRES measurements [27, 30]. Furthermore, the presence of Fe vacancy induces a significant upward shift in the band structures near the Fermi level compared to the stoichiometric results (See Supplementary Note 6). Specifically, as illustrated in Fig. 3c, several circular and hexagonal-shaped Fermi pockets form at the $\Gamma$ point in the nonstoichiometric calculations. Additionally, the elliptical-shaped pockets at the $K$ point closely resemble those observed in experimental results. Moreover, a complex-shaped Fermi pocket appears near the $M$ point in the spin-down channel, which is consistent with experimental findings [22, 27, 30]. At low temperatures, there are quasiparticle flat bands emerging in the spin-down channel near -0.5 eV, which are also observed in ARPES experiments at the same binding energy. These correspondences with experimental data underscore the significant improvements over the stoichiometric results [22, 27]. Figure 3d shows that spectral weight transfer occurs as temperatures decrease. Meanwhile, the DOS has many small sharp peaks due to the complex flat quasiparticle bands formed at low temperatures. The agreement between our hole-doping calculations and ARPES measurements suggests the vital role of Fe2 vacancies in FGT.

Although the peak observed above the Fermi level aligns with the position found in experiments, its height is not sufficiently pronounced. Our calculations indicate that these peaks originate from the hole-like flat quasiparticle bands located at the $\Gamma$ point. Furthermore, as temperature decreases, the quasiparticle bands at the $\Gamma$ point show strong hybridization with other relatively itinerant bands, where the partial DOS of Fe1, Fe2, and Te atoms have similar shapes and peaks positions over a wide range (See Supplementary Note 7). Moreover, at low temperatures, the imaginary parts of the real-frequency hybridization functions exhibit a substantial increase near the Fermi level, particularly for the spin-down channel of the Fe1-$\rm e_{g}^{\prime}$ orbital (See Supplementary Note 8). Besides, these $\Gamma$-centered hole-like bands have a small Fermi velocity in Fig. 3c. Based on this evidence, we conclude that the possible hybridization between these well-defined flat bands and other itinerant bands may be the mechanism responsible for the heavy fermion behavior observed at low temperatures, similar to that seen in $f$-electron heavy fermion materials. Additionally, the crossover temperature for heavy fermion behavior is reported to be between 110 and 150 K based on experimental findings [22]. The large specific heat coefficient has been measured at lower temperatures [20]. And the Kondo peak can be distinctly identified in STM experiments at 6 K [22]. However, in our DFT+DMFT calculations, we have to consider ferromagnetism and the presence of two different Fe atoms. The computations conducted at lower temperatures will exceed our computational capabilities.

In contrast to $f$-electron heavy fermion systems, various mechanisms have been investigated in $d$-electron systems, such as geometrical frustration in LiV₂O₄ [49], symmetry-enforcement in CaCu₃Ir₄O₁₂ [50]. In our study of FGT, Hund’s coupling promotes the formation of ferromagnetic order, which in turn reduces spin fluctuations and enhances the flat bands near the Fermi level. At lower temperatures, these well-defined flat bands may subsequently hybridize with other itinerant bands, leading to heavy fermion behavior. This proposed mechanism may offer a possible explanation of how heavy fermion behavior can be promoted by ferromagnetic order.

As FGT is a layered two-dimensional vdW material, non-local correlation effects may also play a significant role. Particularly in the vicinity of the magnetic transition, long-wavelength spin waves should not be overlooked [42]. A prior study stimulating the Curie temperature in Fe and Ni demonstrates that the single-site nature of the DMFT approach fails to account for the reduction of $T_{c}$ [42]. This overestimation of $T_{c}$ is also evident in our calculations. Moreover, the inadequate mass renormalization observed in our results may also highlight the necessity of incorporating non-local fluctuations, as suggested by previous investigations of iron [51]. Therefore, the non-local correlation effect in FGT warrant further investigation.

In summary, through systematic magnetic DFT+DMFT calculations on FGT, we have uncovered a systematic microscopic description of electronic structures in correlated ferromagnetic metals based on dynamical correlation effects. The Hund’s coupling results in long-range ferromagnetic order accompanied by spectral weight transfer between the two spin channels. This long-range magnetic order suppress spin fluctuation rapidly, leading to the formation of renormalized quasiparticle bands. Our work also show that heavy fermion behavior can be promoted by long-range magnetic order in itinerant magnets, through flat quasiparticle bands around the Fermi level further hybridizing with other itinerant bands at lower temperatures. In addition, the two Fe sites exhibit different physics, with Fe1 characterized by Mott physics and Fe2 by Hund physics, induced by the diversity of their atomic environment. Overall, our research indicates that multi-site itinerant magnets present a new frontier for discovering new quantum states and their competitions.