Abstract
Magnetism in twodimensional (2D) van der Waals (vdW) materials has lately attracted considerable attention from the point of view of both fundamental science and device applications. Obviously, establishing a detailed and solid understanding of their magnetism is the key first step toward various applications. Although Fe_{3}GeTe_{2} is a representative ferromagnetic (FM) metal in this family, many aspects of its magnetic and electronic behaviors still remain elusive. Here, we report our new finding that Fe_{3}GeTe_{2} is a special type of correlated metal known as “Hund metal”. Furthermore, we demonstrate that Hund metallicity in this material is quite unique by exhibiting remarkable site dependence of Hund correlation strength, hereby dubbed “sitedifferentiated Hund metal”. Within this new picture, many of the previous experiments can be clearly understood, including the ones that were seemingly contradictory to one another.
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Introduction
Fe_{3}GeTe_{2} has been a focus of the recent surge of 2D magnetic material research^{1,2,3,4,5,6,7,8,9,10,11,12}. On the one hand, it displays many fascinating properties such as high critical temperature (T_{c} ≃ 220 K)^{6,8}, heavy fermion behavior^{9}, and the relative ease of exfoliation^{5,7}. Its great potential has also been highlighted by recent experiments showing that T_{c} can be increased even up to room temperature^{10,11} and that the large anomalous Hall current is originated from the unique band topology^{12}. On the other hand, its electronic and magnetic properties are far from being clearly understood. The very basic picture for the magnetic moment is still under debate^{9,13,14}. The measured quasiparticle masses are not consistent with each other and exhibit an orderofmagnitude difference depending on experimental probes^{9,12,13,14}. These puzzles are posing a challenge to the current theory of this material.
In this work, we suggest a new physical picture for Fe_{3}GeTe_{2}. We first provide convincing evidence that Fe_{3}GeTe_{2} is a “Hund metal”. The term Hund metal was coined to refer to an intriguing type of correlated (not necessarily magnetic) metals^{15,16,17,18,19} in which Hund coupling J_{H}, rather than Hubbard U, plays the main role in determining the electronic properties^{19}. Hund metals can host various interesting phenomena such as spin freezing^{20,21}, spinorbital separation^{22,23,24,25,26}, orbital differentiation^{27,28}, and unconventional superconductivity^{21,29}. This concept has been providing a compelling view for many of multiorbital systems^{15,19,20,21,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44}, most prominently for ironbased superconductors^{15,19,29,30,32,33,37,45}, ruthenates^{20,21,31,36,40}, and presumably also for recently discovered nickelate superconductors^{46,47}. Our analysis demonstrates that Fe_{3}GeTe_{2} is a new member of this family.
Further, we show that Hund metallicity of this material is distinctive from other known Hund metals by exhibiting the remarkable site dependence of correlation strengths; dubbed “sitedifferentiated” Hund metal (see Fig. 1). This intriguing site dependence is originated from the microscopic details of underlying band structure. The suggested new picture is an important key to understand experiments including the ones that are seemingly contradictory to each other.
Results and discussion
Fe_{3}GeTe_{2} is a Hund metal
Fe_{3}GeTe_{2} is a vdW material with a layered Fe_{3}Ge substructure sandwiched by two Te layers. Five Fe3d levels of both FeI and FeII are split into three groups, namely \({d}_{{{{\rm{{z}}}^{2}}}},{d}_{{{{{\rm{x}}}}}^{2}{{{{\rm{y}}}}}^{2}/{{{\rm{xy}}}}}\) and d_{xz/yz}. More details can be found in Supplementary Note 1 of Supplementary Information (SI).
We firstly demonstrate that all the key features defining Hund metal are well identified in Fe_{3}GeTe_{2}via density functional theory plus dynamical meanfield theory (DFT+DMFT) calculations (see “Methods”). In order to concentrate on the intrinsic nature of correlation (separately from magnetism), we suppress magnetic ordering unless otherwise specified. The result of U = 5 and J_{H} = 0.9 eV will be presented as our main parameter choice. Computation methods, including the parameter dependence and other details, are given in “Methods”.
First, we examine socalled twofaced or “Janus” effect which represents the physics governed by J_{H} distinctively from that of Mott correlation U^{18,19}. The straightforward way is to see the simultaneous increase of both U_{c} (the critical value of U at which the metaltoMottinsulator transition occurs) and “correlation strength” as measured typically by the inverse of quasiparticle weight Z (or equivalently, the quasiparticle scattering rate Γ) as J_{H} increases^{18,19}.
Let us uncover the “first face” of this effect, namely, the U_{c} increase by J_{H}. To avoid the demanding computations scanning the large parameter space, here we simply examine the atomic gap Δ_{at} ≡ E_{N+1} + E_{N−1} − 2E_{N} (E_{N}: the atomic ground state energy of Nelectron subspace). According to the original Mott–Hubbard picture, the gap opening is associated with Δ_{at} > W (W: bandwidth), and the system gets closer to Mott transition as Δ_{at} increases. This analysis, albeit simple, is well justified by the observations that the actual DMFT phase diagrams are consistent with the behavior of Δ_{at} in the large J_{H} regime^{19,42}. We found that Δ_{at} ≈ U − 1.52J_{H} from a fivedegenerateorbital model with 6 electrons, which guarantees that ∂Δ_{at}/∂J_{H} < 0 (see the inset of Fig. 2a). Note that J_{H} drives the system away from being a Mott insulator; hereby the Janus' first face is unveiled.
To see its “second face”, the calculated Γ is presented in Fig. 2a. Here, \({\Gamma }_{l}={Z}_{l}{{{\rm{Im}}}}{\Sigma }_{l}({{{\rm{i}}}}{\omega }_{n}){ }_{{\omega }_{n}\to {0}^{+}}\), where Z_{l} is the quasiparticle weight and Σ_{l}(iω_{n}) the local selfenergy on the imaginary frequency (ω_{n}) axis for a given orbital l (see Supplementary Note 2 of SI for additional discussion). It is clear that the Γ is enhanced as J_{H} increases. Also, the scattering rate is strongly orbitaldependent as depicted by different symbols, and the differences become more pronounced as J_{H} increases. This feature, often called orbital differentiation, has been regarded as one of the key pieces for identifying ironbased superconductors as a Hund metal^{27,28}.
Having evidenced Janus behavior, we now turn to another defining feature of Hund metallicity, namely, spinorbital separation^{22,23,24,25,26}. As discussed in previous works, the electronic behavior of Hund metals follows the localatomic and Landauquasiparticle picture in the high and lowenough energy region, respectively, due to the gradual development of screenings. Deng et al. suggested that the much higher onset screening temperature for the orbital degree of freedom (\({T}_{{{{\rm{onset}}}}}^{{{{\rm{orb}}}}}\)) than that of spin (\({T}_{{{{\rm{onset}}}}}^{{{{\rm{spin}}}}}\)) is a key signature of Hund metals; i.e., \({T}_{{{{\rm{onset}}}}}^{{{{\rm{orb}}}}}\gg {T}_{{{{\rm{onset}}}}}^{{{{\rm{spin}}}}}\)^{26}. Following the literature, we tracked local susceptibilities up to high enough temperature. Figure 2b shows the calculated local spin and orbital susceptibilities multiplied by k_{B}T (k_{B}: Boltzmann constant). The spin susceptibility with no prefactor k_{B}T is presented in Fig. 2c. The onset screening temperatures for spin (below which k_{B}Tχ^{spin} starts to deviate from the Curie law) are indicated by the vertical orange arrows; \({T}_{{{{\rm{onset}}}}}^{{{{\rm{spin}}}}}\approx 3000\) K and 9000 K for FeI and FeII, respectively. Importantly, the onset temperatures for orbital screenings are much higher; \({T}_{{{{\rm{onset}}}}}^{{{{\rm{orb}}}}} > 20000\) K for both FeI and FeII; see the empty symbols. The wellseparated \({T}_{{{{\rm{onset}}}}}^{{{{\rm{spin}}}}}\) and \({T}_{{{{\rm{onset}}}}}^{{{{\rm{orb}}}}}\) provide further evidence that Fe_{3}GeTe_{2}is a Hund metal.
We also investigated the longtime correlators, C^{spin/orb}(τ = 1/2k_{B}T) (τ: imaginary time); see Supplementary Note 3 of SI for its definition and the related discussion. These quantities were recently used as another useful proxy to measure the degree of spinorbital separation^{42}. The results also clearly show that Fe_{3}GeTe_{2}is well classified as a Hund metal.
We finally note that the ferromagnetic transition temperature (T_{c} ≃ 220 K) of Fe_{3}GeTe_{2} is much lower than the characteristic temperature scales of T_{onset} for spin and orbital. It indicates that the Hund metal character is developed well above the ferromagnetic transition temperature and the magnetic order itself is not immediately relevant to Hund metallicity.
Sitedifferentiated Hund metallicity
Hund metallicity in Fe_{3}GeTe_{2}is unique in that its Hund physics is clearly differentiated at two sites. Note that the scattering rate Γ is an orderofmagnitude larger in FeI. Γ of FeI \({d}_{{{{\rm{{x}}}^{2}{y}^{2}/xy}}}\) (orange triangles in Fig. 2a) reaches 0.6 at J_{H} = 1.3 eV, whereas that of FeII \({d}_{{{{\rm{{z}}}^{2}}}}\) only 0.05 (blue circles) for example. The orbitaldependent correlation is also more pronounced in FeI (Fig. 2a).
χ^{spin} also shows remarkable site dependence as presented in Fig. 2c: As the temperature is lowered, χ^{spin} of FeI (orange line) well follows ~1/T behavior in sharp contrast to FeII (blue line) which almost saturates at ~100 K (see inset of Fig. 2c). It implies that, below this temperature, the spin moments of FeII tend to be itinerant. For FeI, on the other hand, we could not find any indication of susceptibility saturation down to the lowest temperature we reached (~100 K), indicative of the persistent local moment character. See also Supplementary Note 3 of SI for further analysis based on longtime correlators.
What is the origin of this sitedependent Hund physics? First of all, we note that the dorbital occupations are almost the same in both sites; the calculated electron occupancy is 6.431 ± 0.001 at U = 0 and 6.18 ± 0.05 at U = 5 eV (−/+ corresponding to FeI/FeII). It is largely attributed to the strong covalency in this chalcogenide material. Although the electron number depends on the chargecounting method, this level of coincidence can effectively exclude the possibility of the sitedifferentiated Hund physics being originated simply from the different charge status or valence. Rather, it indicates the more subtle electronic origins. Second, the calculated density of states (DOS) does not show significant difference between FeI and FeII other than the more pronounced van Hove singularity (vHS) for FeI d_{x2y2/xy} (see Supplementary Note 4 of SI).
Highly useful insights are obtained from the calculated “oneshot” hybridization function Δ(iω_{n}) for each orbital. For its definition and the related figures, refer to Sec. S4 of SI. \({{{\rm{Im}}}}\Delta ({{{\rm{i}}}}{\omega }_{n})\) at high energy (which approximately scales as the square of the bandwidth) is smaller for FeI than FeII (see Supplementary Fig. 4a in SI). It shows that the effective bandwidth for FeI is smaller than FeII, indicative of the stronger correlation at FeI. This feature of smaller bandwidth for FeI is also seen in the integrated DOS analysis (see Supplementary Fig. 4b). It is therefore consistent with the larger Γ (discussed above; see Fig. 2a) and the greater mass of FeI (to be discussed below; see Fig. 3d). Also, in the lowfrequency regime, \({{{\rm{Im}}}}\Delta ({{{\rm{i}}}}{\omega }_{n})\) of FeI d_{x2y2/xy} is smallest and exhibits a downturn as ω_{n} → 0. Note that \({{{\rm{Im}}}}\Delta ({{{\rm{i}}}}{\omega }_{n}){ }_{{\omega }_{n}\to 0} \sim 1/\{(\pi D({E}_{{{{\rm{f}}}}}))\}\). Here, D(E_{f}) is the DOS at the Fermi level E_{f}. Thus, this downturn of \({{{\rm{Im}}}}\Delta ({{{\rm{i}}}}{\omega }_{n})\) is most likely due to the welldeveloped vHS (i.e., the divergence of DOS), which in turn suppresses the lowenergy effective hopping processes^{31}. That is, electron scattering and mass enhancement of FeI d_{x2y2/xy} are further boosted by vHS.
Understanding the experiments
The new picture for Fe_{3}GeTe_{2} as a “sitedifferentiated Hund metal” provides useful insights to understand the experiments. Before discussing those features in detail, let us take a look at the spectral function A(k, ω) which can be directly compared with angleresolved photoemission spectroscopy (ARPES) results, thereby enabling us to check the reliability of our parameter choices for interactions and the correlated electronic structure. Figure 3a shows the excellent agreement between theory and experiment. For example, the pronounced spectral features, namely the bands assigned as γ, ζ and ω in ref. ^{14} are well identified in our A(k, ω). The nearFermilevel states are also well reproduced; see, e.g., α, δ and η bands.
The first experimental quantity we want to revisit is the large quasiparticle mass; m^{*}/m^{b} = γ/γ^{b} ≃ 13 reported from specific heat measurement^{13} (i.e., taken from the measured Sommerfeld coefficient γ and its DFT estimate γ^{b}). This value remains as a puzzle especially because ARPES data was deemed to be in fair agreement with band calculations m^{*}/m^{b} ≃ 1.6^{12,14}. This large discrepancy has been highlighted throughout the literature^{9,13,14}, but the issue remains still unresolved.
Here, we argue that these seemingly contradictory experimental results can be understood as a natural manifestation of the sitedifferentiated Hund physics. In multiorbital systems, the mass enhancement measured by specific heat is mainly determined by the more correlated orbital^{27}. In fact, our calculation shows m^{*}/m^{b} ≈ 7–23 for the more correlated site FeI, which is comparable to the value from specific heat experiment (dashdotted line in Fig. 3d). It is due to that Sommerfeld coefficient γ extracted from specific heat is a sum of each orbital contribution \({\gamma }_{l} \sim {({m}^{* }/{m}^{{{{\rm{b}}}}})}_{l}\); see Supplementary Note 5 in SI for more details. As discussed in ref. ^{27}, it is comparable to the series resistor in a circuit for which total resistance is given by the direct sum, and the total voltage can be approximated by the one applied to the large resistor. Indeed, the site and orbitaldecomposed values (Fig. 3e) clearly show that the FeI contributions dominate the total value γ_{tot}, whereas those of FeII are much smaller. The sum over all orbitals γ_{tot} eventually gives rise to good agreement with experiments.
The ARPES situation is quite different on the other hand. The mass enhancement of a given “band” is a weighted sum of each orbital contribution therein. In fact, the previous study revealed that most bands in this material have both FeI and FeII characters (see, e.g., Supplementary Fig. 3 of ref. ^{14}). As inferred from m^{*}/m^{b} ≈ 2–4 of FeII being comparable to an ARPES estimate (Fig. 3d), we presume that, at least in some bright bands captured by ARPES, the weaker correlated FeII dorbitals as well as the other itinerant Ge and Te states possibly have the larger portion than FeI, which can result in the small m^{*}/m^{b} (see Supplementary Note 6 in SI for the related analysis).
Now let us examine the heavy fermion behavior and the incoherencecoherence crossover observed at T ~ 100 K^{9,48}. Figure 3c presents Γ/k_{B}T as a function of temperature for both paramagnetic (fully filled symbols) and FM phases (partially filled). Here, the coherence temperature T^{*} is defined by Γ/k_{B}T^{*} = 1 (see the yellow region in which Γ≤k_{B}T), namely the temperature below which quasiparticle lifetime exceeds the timescale of thermal fluctuation^{31,42}. We note that the sizable site dependence of correlation makes electron scattering very different at two Fe sites: Whereas Γ/k_{B}T of FeII remains less than 1 even well above the experimental T^{*}, that of FeI is much greater. The larger contribution from FeI, particularly \({d}_{{{{\rm{{x}}}^{2}{y}^{2}/xy}}}\) orbitals, and its significant temperature dependence as T → T^{*} indicate that the experimentally observed incoherencecoherence crossover around T ~ 100 K^{9,48} is mainly attributed to the corresponding orbitals residing in FeI.
For comparison, we also investigate the isostructural material Ni_{3}GeTe_{2} whose γ ~ 9 mJ/mol ⋅ K^{2} is much smaller^{13}. Our calculation reasonably well reproduces the experiment and the mass enhancements m^{*}/m^{b}≤1.5 with no appreciable site differentiation (see Supplementary Note 6 in SI for our calculation results). This contrasting behavior, attributed to the absence of vHS and the different valence (see Supplementary Fig 6c in SI), renders Fe_{3}GeTe_{2} as a unique example of sitedifferentiated Hund metal.
Finally, another important issue for Fe_{3}GeTe_{2} is related to the nature of its magnetic moment. While in the early works it was understood or discussed within itinerant Stoner picture^{9,13}, more recent studies emphasized the local nature of spin. For example, Xu et al. observed the nearly unchanged exchange splitting in their ARPES data even up to T > T_{c}^{14}. The situation is therefore reminiscent of a prototypical FM Hund metal SrRuO_{3} whose exchange splitting persists above T_{c} due to the enhanced local spin character whereas the very lowtemperature behavior is well described within Stoner theory^{49,50}. Note that, near T_{c} ≃ 220 K, χ^{spin} of FeI clearly indicates the local moment behavior (Fig. 2c), and as a consequence, Γ is large. As temperature decreases, on the other hand, Γ gets reduced by the enhanced spin screenings, which eventually result in the longlived quasiparticles well below T ~ 100 K (Fig. 3c). Therefore, our current study provides a unified picture for Fe_{3}GeTe_{2} within which its spin moment can be described as being localized above and itinerant well below T_{c}.
We demonstrated that Fe_{3}GeTe_{2} is a Hund metal with intriguing site dependence originated from the microscopic details in the underlying band structure. The proposed new picture of “sitedifferentiated Hund metal” not just makes this representative metallic vdW ferromagnet an even more exciting material platform, but also provides useful insights to understand the previous experiments including the ones that are seemingly contradictory to each other. Our results hopefully stimulate further experimental and theoretical investigations of the related systems. For example, the direct experimental observations of the site or orbitalselective correlation effects can be an interesting research direction.
Methods
Computation details of DFT+DMFT calculations
The DFT electronic structure of Fe_{3}GeTe_{2} was obtained by using FLAPWMBPT package^{51}, which is based on the fullpotential linearized augmented plane wave plus local orbital method, employing the local density approximation (LDA). We used experimental lattice parameters as reported in ref. ^{5}. On top of this nonspinpolarized DFT band structure, 108 maximally localized Wannier functions were constructed with a wide energy window of [−15:10] eV containing Fes, Fep, Fed, Ges, Gep, Ged, Tes, Tep, and Ted characters, and its tightbinding Hamiltonian was built through the J_{X} interface^{52}. We performed singlesite dynamical meanfield theory (DMFT)^{53,54} calculations for dorbitals of FeI and FeII by employing COMCTQMC^{55} implementation of the hybridizationexpansion continuoustime quantum Monte Carlo (CTQMC) algorithm^{56} as an impurity solver. Namely, we solved two impurity problems, one for FeI and one for FeII, per DMFT selfconsistency loop. Within our DFT+DMFT scheme, electronic selfenergy is momentumindependent (i.e., local in space)^{53}, and is assumed to be diagonal in cubic harmonics basis to avoid a sign problem by omitting the offdiagonal elements of hybridization functions whose values are actually small. The selfenergy on the real frequency axis was obtained from the analytic continuation of imaginary axis data using the maximum entropy method^{57}. For the magnetically ordered phase of Fe_{3}GeTe_{2}, we used EDMFTF package^{58} for DFT+DMFT calculations.
For the double counting (DC) selfenergy, we adopted the nominal DC scheme which reads \({\Sigma }_{{{{\rm{DC}}}}}({n}_{0})=U({n}_{0}\frac{1}{2})\frac{J_{\rm H}}{2}({n}_{0}1)\)^{59}, where n_{0} is the nominal charge. We took n_{0} = 6.0 for both FeI and FeII, which is close to the dorbital occupancy obtained from DFT (n_{DFT} ≃ 6.4). Note also that this DC scheme with n_{0} = 6.0 was used for this material in a previous study^{13}, which yields good agreement of magnetic moment with experimental data. While we mainly present the results obtained from n_{0} = 6.0, we found that varying n_{0} in a range of [5.4:6.6] does not lead to any qualitative changes; e.g., see Supplementary Note 7 of SI for spin and orbital susceptibilities obtained from different Σ_{DC}(n_{0}) values.
The Hubbard U and Hund coupling J_{H} of Fed were chosen to be U ≡ F^{0} = 5.0 eV and J_{H} ≡ (F^{2} + F^{4})/14 = 0.9 eV for both FeI and FeII, which were used in a previous study on Fe_{3}GeTe_{2}^{12}. These values are also comparable to the theoretically estimated values for iron chalcogenide family^{60}. Here, F^{0}, F^{2}, and F^{4} are Slater integrals, and the ratio F^{4}/F^{2} = 0.625 was used for parametrization of onsite Coulomb interaction tensor. For Ni_{3}GeTe_{2} which will be discussed in Supplementary Note 6 of SI, the same computation scheme was used except for the use of a larger nominal charge (n_{0} = 8.0) for the DC selfenergy and a larger Hubbard U of U ≡ F^{0} = 6.0 eV, considering its chemical environment on Ni sites. Note also that, for Ni_{3}GeTe_{2}, we did not consider any magnetic order due to its absence even at very low temperatures^{5,13}. We used experimental lattice parameters of Ni_{3}GeTe_{2}^{5}.
Data availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.
Code availability
All codes used in this work are accessible through their websites.
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Acknowledgements
T.J.K. thanks Hongkee Yoon for the discussion on Fe_{3}GeTe_{2} at the initial stage and comments on the use of J_{X} interface for constructing tightbinding Hamiltonians. S.R. is grateful to Sangkook Choi for the fruitful conversation. This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (Grant Nos. 2021R1A2C1009303 and NRF2018M3D1A1058754), the KAIST Grand Challenge 30 Project (KC30) in 2021 funded by the Ministry of Science and ICT of Korea and KAIST (N11210105), and the National Supercomputing Center with supercomputing resources, including technical support (KSC2020CRE0084).
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T.J.K. performed all the calculations and initiated the project by noticing the site differentiation in Fe_{3}GeTe_{2}. S.R. led the data analysis. M.J.H. supervised the project. All authors participated in the interpretation of the data and wrote the manuscript.
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Kim, T.J., Ryee, S. & Han, M.J. Fe_{3}GeTe_{2}: a sitedifferentiated Hund metal. npj Comput Mater 8, 245 (2022). https://doi.org/10.1038/s4152402200937x
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DOI: https://doi.org/10.1038/s4152402200937x
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