Strong enhancement of magnetic order from bulk to stretched monolayer FeSe as Hund's metals

Despite of the importance of magnetism in possible relation to other key properties in iron-based superconductors, its understanding is still far from complete especially for FeSe systems. On one hand, the origin of the absence of magnetic orders in bulk FeSe is yet to be clarified. On the other hand, it is still not clear how close monolayer FeSe on SrTiO$_3$, with the highest transition temperature among iron-based superconductors, is to a magnetic instability. Here we investigate magnetic properties of bulk and monolayer FeSe using dynamical mean-field theory combined with density-functional theory. We find that suppressed magnetic order in bulk FeSe is associated with the reduction of inter-orbital charge fluctuations, an effect of Hund's coupling, enhanced by a larger crystal field splitting. Meanwhile, spatial isolation of Fe atoms in expanded monolayer FeSe leads into a strong magnetic order, which is completely destroyed by a small electron doping. Our work provides a comprehensive understanding of the magnetic order in iron-based superconductors and other general multi-orbital correlated systems as Hund's metals.


Introduction
Magnetism is one of universal features found in iron-based superconductors (IBS) as superconductivity generally appears in the vicinity of antiferromagnetic (AFM) phase with a specific stripe-type ordering pattern, from which electron pairing mechanisms of the magnetic origin were introduced [1][2][3][4][5][6][7] . Furthermore, nematicity (spontaneous breaking of fourfold rotational symmetry of tetragonal phase), magnetism, and superconductivity in IBS are thought to be closely related [8][9][10][11] . In this context, understanding magnetism can be a starting point to unravel the complex inter-dependence of these properties. In terms of magnetism, FeSe holds a unique position among general IBS as bulk FeSe has no magnetic ordered phase unlike most of other materials [12][13][14] , whose underlying mechanism is still not well understood. FeSe is also of great interest due to the highest superconducting transition temperature among IBS when its monolayer (ML) is on SrTiO 3 substrate 15-18 . Whether or not ML FeSe/SrTiO 3 is close to a magnetic instability is therefore an intriguing question.
Meanwhile, there is a general consensus that the electron correlation should be taken into account to properly understand material properties of this system [19][20][21][22] . Since it is a multi-orbital system in which all five d orbital bands are crossing or near the Fermi energy (E F ), Hund's coupling J H is an indispensable part of interactions as well as the intraorbital Coulomb repulsion U, and IBS in the correlated metallic state are often described as Hund's metals 20,[23][24][25] . In this material state, reduced inter-orbital Coulomb repulsion U ′ = U − 2J H 26 and the tendency to promote parallel spin alignment cooperatively decouple the five d orbital components, which is signaled by the suppression of inter-orbital charge fluctuations. Consequently, coherent and incoherent states can coexist and some orbitals are close to Mott transition while the others are still itinerant. Since this orbital selectivity is known to be enhanced in FeSe [21][22][23]27,28 , its magnetic properties would be better understood in the context of Hund's metal physics.
In this work, a systematic comparative study on the magnetic properties of FeSe in different forms and a reference IBS, LaFeAsO, is performed using a density-functional theory plus dynamical mean-field theory (DFT+DMFT). It is found that the inter-orbital charge fluctuations are greatly reduced between e g and t 2g orbitals for bulk FeSe due to its large crystal field splitting and the resultant strong orbital decoupling induced by the Hund's coupling. Consequently the total charge fluctuation are enhanced leading to a largely re-duced ordered magnetic moment compared with LaFeAsO, consistently with the absence of magnetic order in bulk FeSe in experiments. In contrast, increased fluctuating magnetic moment and suppressed total charge fluctuation due to the increased inter-atomic distance and the reduced dimentionality result in a large ordered magnetic moment in expanded ML FeSe with the lattice constant of that on SrTiO 3 . Thus, the stark contrast of the magnetic order between bulk and ML FeSe is explained in terms of Hund' metal properties within an unified framework. Small electron doping is found to effectively destroy the magnetic order in this system, implying that the superconductivity in ML FeSe/SrTiO 3 is in the vicinity of magnetic order. FeSe is expected to capture most of the essential features of magnetic properties of that on SrTiO 3 as well. Electron doping, another important possible substrate effect, will be also discussed in the later part of this work. Figure 1 displays the imaginary part of the magnetic susceptibility, χ ′′ m , as a function of momentum and frequency, for the three materials in the paramagnetic (PM) phase. The magnetic susceptibility is estimated within DFT+DMFT method from the Bethe-Salpeter equation, using fully momentum and frequency dependent interacting DFT+DMFT oneparticle lattice Green's function and local two-particle vertex function obtained from the DMFT impurity solver 33 . χ ′′ m for LaFeAsO exhibits a typical spin excitation spectrum for IBS, with largest weights at q = (1, 0) near zero frequency indicating the magnetic instability for the stripe-type AFM order and also with high energy excitations near q = (1, 1), as can be seen in previous similar calculations 6,33-35 . Meanwhile, low energy spin fluctuations are much suppressed for bulk FeSe indicating the weakened tendency for the magnetic order in accordance with its absence in experiments. Considerable amount of spectral weights near zero energy are relocated to near 100 meV, implying that some higher frequency processes are involved in the magnetic order suppression. Finally, ML FeSe exhibits the overall increase of spectral weights as well as the recovered dominance of low energy fluctuations over high energy ones compared with bulk FeSe, indicating a much stronger tendency for the magnetic order. In this case, however, strongest low energy excitations are not at q = (1, 0), but slightly shifted from it toward q = (1, 1) suggesting an incommensurate magnetic order.

Results
Besides the spin fluctuation, as the orbital degree of freedom is considered another candidate to drive the nematic order and/or the superconductivity in these materials, we also estimate the orbital susceptibility (see Supplementary Figure 1 and Note 1). Only very weak low energy excitations are found for all the three materials, indicating that DFT+DMFT method does not support the existence of orbital orders in these materials.
Trends in local quantities. We perform a systematic analysis for the trend of local correlations to understand the properties found in the susceptibility results. Ordered magnetic moment S z on a Fe atom is estimated in the stripe-type AFM phase and found to vary from 0.70 to 0.43 and 1.00 µ B for LaFeAsO, bulk and ML FeSe, respectively. We can see that the magnetic order is suppressed and then greatly enhanced for bulk and ML FeSe compared with LaFeAsO as predicted by magnetic susceptibility results in the PM phase in Fig. 1, and also in qualitative agreement with the experimental observation of no magnetic order for bulk FeSe. Usually magnetic order is strong in materials with strong electron correlation, and greatly reduced ordered moment of bulk FeSe is rather puzzling since it is considered to be more correlated than LaFeAsO. Indeed, mass enhancement factor, 1/Z = 1 − ∂Σ(ω) ∂ω | ω=0 , is found to increases considerably for t 2g orbitals, especially d xy as shown in Fig. 2a. Although e g orbitals become less correlated from LaFeAsO to bulk FeSe, the fluctuating magnetic moment ( S 2 1/2 ) which reflects the overall correlation strength, slightly increases in Fig. 2b suggesting that the suppressed magnetic order in bulk FeSe cannot be understood by the overall correlation strength of the material. Meanwhile, mass enhancement increases for all the orbitals for ML FeSe in Fig. 2a along with the fluctuating moment in Fig. 2b defining this material most correlated among the three.
Using the same U ′ s and J ′ s for all the materials in the present study (see Supplementary Note 2), the variation of correlation strength can be attributed mainly to that of the interatomic distance and orbital occupations. In spite of the large reduction of the Fe-anion distance from 2.42 to 2.39Å for LaFeAsO and bulk FeSe, respectively, t 2g orbitals become much more correlated while e g orbitals exhibit the opposite behavior to produce a large orbital differentiation in bulk FeSe. As can be seen in Fig. 2c, it results from the large difference of occupation numbers between t 2g and e g orbitals which are essentially decoupled in a Hund's metal 20,24,25 , indicating a large crystal field splitting in bulk FeSe. We estimate that all five Fe-d orbital levels lie within the range of 0.25 eV for LaFeAsO while the range increases to 0.48 eV for bulk FeSe indeed confirming the enhanced crystal field splitting in bulk FeSe. Noteworthy is that even in bulk FeSe the crystal field splitting is smaller than J value adopted in this work, 0.8 eV, so that the Hund's coupling still plays a major role in the local correlation over all five d orbitals in this material. The overall increase of mass enhancement of ML FeSe can then be related to the elongation of Fe-anion bond to 2.40 A due to the applied strain, considering that its orbital occupations do not change much from those of bulk FeSe. Also, the kinetic energy reduction in a two-dimensional system is expected to further contribute to the stronger overall correlation in ML FeSe, especially for d xz/yz and d z 2 orbitals.
In a Hund's metal, the local charge fluctuation n 2 − n 2 where n is the local density operator on an atom, which quantifies the charge delocalization, can be sizable even in To understand the variation of charge fluctuation over materials, orbital-resolved charge fluctuations defined as , where α and β are orbital indexes, are estimated and listed in Table 1 where every orbital component of the ordered moment is reduced for the latter compared with the former. As mentioned earlier, the enhanced spin flip processes reducing the ordered moment in bulk FeSe can be associated with the 100 meV spin excitations in Fig. 1.
The pronounced suppression of inter-orbital charge fluctuations in bulk FeSe can be attributed to the large difference of its intra-orbital components between e g and t 2g orbitals shown in Table 1, as the inter-orbital fluctuation is expected to be suppressed between orbitals which fluctuate incoherently to each other with very different rates. Since the difference in intra-orbital charge fluctuations among orbitals can be mainly accounted for by that in orbital occupations as mentioned above, their larger difference in bulk FeSe is the direct consequence of the larger crystal field splitting. In short, the suppressed magnetic  are actually below the E F , so in the FS plot in Fig. 3b  as well as previous DFT+DMFT calculations 40,41 . In Fig. 3c and f, χ ′′ m (q, ω = 5 meV) in the PM phase is displayed to figure out how the static magnetic order evolves with doping. In the undoped case, static order is predicted slightly off the stripe-type AFM ordering vector as is already seen in Fig. 1. Despite of significant renormalization of the non-interacting susceptibility χ 0 by the local two-particle vertex to form the fully interacting χ 34 , the FS nesting which features the structure of χ 0 still plays a non-negligible role in stabilizing magnetic ordering 23 . Indeed, one can see that the nesting vectors connecting the hole FSs at Γ and the electron FS at X or Y with same orbital characters in Fig. 3b roughly coincides with the peak positions of χ ′′ m in Fig. 3c. Even though the hole FS is absent in Fig. 3e by the electron doping, actually the hole bands are just below E F as shown in Fig. 3d so that the overall nesting condition is not very different from the undoped case. Consequently, the peak position in χ ′′ m plot in Fig. 3f is almost the same as in Fig. 3c, with only the overall excitation magnitude greatly reduced. The suppressed low energy excitation and tendency for a magnetic order rather result from the local two-particle vertex which includes effects of overall increase of local orbital occupations away from the integer filling by doping, which should suppress the fluctuating moment and enhance charge fluctuations. Zero ordered moment is obtained in the stripe-type AFM calculation for 0.12 e − /Fe doped ML FeSe, in consistence with our χ ′′ m result in the PM phase and also with the suppressed magnetic order by electron doping found experimentally 38 as mentioned above. This large sensitivity of magnetic order on doping therefore results from local correlations, which are well described within the DFT+DMFT method.

Discussion
Our result, that strong magnetic order in strained ML FeSe is destroyed by electron doping on the level where superconductivity is known to appear, implies the close proximity of magnetism to the superconductivity in ML FeSe/SrTiO 3 , imposing a definite constraint on the electron pairing mechanism in this system. Among various pairing scenarios taking into account the absence of hole FS around Γ, our results are most consistent with the "bootstrap" mechanism where electron FSs and "incipient" band (hole band below E F ) have opposite sign gaps (s ± ) 42-44 . This mechanism requires cooperative interplay of attractive q ∼ (0, 0) interaction (e.g., by phonon) and repulsive q ∼ (1, 0) interaction whose existence is identified in our study as the incommensurate spin excitation. Meanwhile, q ∼ (1, 1) interaction connecting separate electron FSs, as required by other scenarios such as "nodeless d" [45][46][47] , sign-preserving "s" 48,49 , and "bonding-antibonding s" 50,51 , is identified from neither spin nor orbital excitations as shown in Fig. 1 and Supplementary Figure 1

Data availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.

Author contributions
C.-Y.M conceived the project, performed calculations, analyzed data and wrote the papaer.