Symmetry-breaking induced surface magnetization in non-magnetic RuO₂

Dai Q. Ho [email protected] Department of Materials Science and Engineering, University of Delaware, Newark, Delaware 19716, United States Faculty of Natural Sciences, Quy Nhon University, Quy Nhon 55113, Vietnam D. Quang To Department of Materials Science and Engineering, University of Delaware, Newark, Delaware 19716, United States Ruiqi Hu Department of Materials Science and Engineering, University of Delaware, Newark, Delaware 19716, United States Garnett W. Bryant [email protected] Nanoscale Device Characterization Division, Joint Quantum Institute, National Institute of Standards and Technology, Gaithersburg, Maryland 20899-8423, United States University of Maryland, College Park, Maryland 20742, USA Anderson Janotti [email protected] Department of Materials Science and Engineering, University of Delaware, Newark, Delaware 19716, United States

Abstract

Altermagnetism is a newly identified phase of magnetism distinct from ferromagnetism and antiferromagnetism. RuO₂ has been considered a prototypical metallic altermagnet with a critical temperature higher than room temperature. Previous interpretations of unusual magnetic properties of RuO₂ relied on the theoretical prediction that local moments on two Ru sublattices, which are connected by four-fold rotational symmetry, are quite significant ( $\sim 1\,\mu_{B}$ ), leading to long-range antiferromagnetic order. However, accumulated experimental data suggest that local moments on Ru in RuO₂ are vanishingly small, indicating the bulk material is likely non-magnetic. This observation is consistent with the delocalized nature of the $4d$ electron of Ru and the strong screening effect in the metallic state. In this work, we show that despite the non-magnetic bulk, RuO₂ surface (110) exhibits spontaneous magnetization, which we attribute to the breaking of local symmetry, resulting in electronic redistribution and magnetic moment enhancement. The emergence of surface magnetism gives rise to interesting spectroscopic phenomena, including spin-polarized surface states, spin-polarized scanning probe microscopy images, and potentially spin-dependent transport effects. These highlight the important role of surface magnetic structures in the otherwise non-magnetic bulk RuO₂.

I Introduction

Altermagnetism (AM) is an emerging phase of magnetism that differs from the well-known ferromagnetism (FM) and antiferromagnetism (AFM) [1]. In altermagnetic materials, spin-splitting phenomena of the electronic band structure occur alternately in reciprocal space directions that are related to each other by rotation or reflection but not translation or inversion, which is determined by the presence of non-magnetic ligands in real space [1]. Among the earliest and most studied candidates for altermagnetic materials is RuO₂, which has become a prototype in this field [1, 2]. Despite various theoretically predicted [2] and experimentally observed phenomena such as the anomalous Hall effect [3], time-reversal symmetry breaking [4], ascribed to the assumption that RuO₂ is an antiferromagnet with a high Néel temperature and a significant local moment (typically assumed $\sim 1\,\mu_{B}$ in theoretical calculations), direct experimental evidence for such a high moment in this $4d$ metallic oxide is lacking. Therefore, the magnetic configuration of RuO₂, especially the presence of local moments and long-range magnetic ordering, is still under active debate. The first neutron scattering experiment on RuO₂ reported a local moment of only $0.05\,\mu_{B}$ [5]. Subsequently, an experiment using the resonant X-ray scattering technique supported the ground state of AFM in RuO₂ [6]. However, recent in-depth investigations strongly suggest that the ground state of RuO₂ is non-magnetic/paramagnetic [7, 8, 9, 10, 11]. For instance, highly local-moment-sensitive muon spin rotation ( $\mu$ SR) and neutron diffraction experiments have shown that the ground state of the bulk RuO₂ is likely non-magnetic, with local moments on the order of $10^{-4}\,\mu_{B}$ [7, 9]. Furthermore, by using broadband infrared spectroscopy combined with first-principles calculations to probe the optical conductivity of RuO₂—its bulk electronic property—researchers concluded that bulk RuO₂ is well described by a non-magnetic model [11]. Very recently, Liu et al. have directly probed the electronic structure of RuO₂ in bulk and thin film samples using angle resolved photoemission spectroscopy (S-ARPES) and found no evidence of spin splitting in the electronic structure of the material [8]. These findings align with the delocalization and weak correlation nature of the Ru $4d$ orbitals, consistent with the metallic property of the compound.

Most of the experiments mentioned above have been concentrated on bulk or thick samples, with less focus on surface properties. However, the surface properties of altermagnets might play a significant role in various phenomena at surfaces and interfaces, or even be the key to explaining some observations [12, 13]. Of many low-index surfaces, RuO₂’s surface and thin films in the crystallographic direction $\langle 110\rangle$ are of particular interest since that is the crystal surface with the lowest surface energy and is naturally cleaved [14, 15]. For example, the RuO₂ (110) surface has been a primary focus in catalysis research thanks to its high electrocatalytic water splitting activity, attributed to the presence of Ru atoms with dangling bonds [14, 16]. However, most studies have ignored the magnetic properties of the surface, except for a couple of theoretical predictions, which highlighted the role of surface magnetism in RuO₂ (110) in catalytic reactions during electrolysis [17, 18]. Interestingly, strained RuO₂ (110) thin films grown on TiO₂ substrates have recently been shown to exhibit superconducting behavior at low temperatures and a metal-to-insulator transition in the ultrathin limit regime [19, 20, 21]. The role of the soft-phonon mode and the enhancement of electron density at the Fermi level ( $E_{F}$ ) have been invoked to explain this superconductivity. However, the possible role of the magnetic structure in supporting the superconductivity of these thin films has been overlooked.

As noted previously, Torun et al. [17] have theoretically predicted that surface magnetization can spontaneously develop on the RuO₂ (110) surface. However, their study was not without limitations. First, the slab they used in their calculations, consisting of only five layers of RuO₂, was probably insufficient to accurately represent both surface and bulk-like regions, potentially affecting the predicted behavior of surface magnetization. In metallic systems such as RuO₂, the electronic wavefunctions of the top and bottom surfaces of the slab used in the calculation can penetrate deep into the bulk and mutually interact if the thickness of the slab is not sufficient, leading to an alteration of electronic properties of the true surface. Second, the origin of the surface magnetism was not comprehensively explored. More importantly, since their work focused on surface catalysis, it did not address how surface magnetization influences electronic and magnetic properties that are critical for spectroscopic and transport phenomena, which is highly relevant for spintronics applications [4, 22].

In this paper, by using first-principles calculations employing Density Functional Theory (DFT), we show that surface magnetism can spontaneously be developed in RuO₂ (110). The spontaneous surface magnetization can be understood from the breaking of symmetry on the surface of RuO₂ (110), leading to a significant reconstruction of the electronic structure of RuO₆ octahedra at the surface layer, i.e., in the band filling of the Ru $4d$ orbital, resulting in the emergence of sizeable local moments. Remarkably, the presence of spontaneous surface magnetization leads to the emergence of spin-polarized surface states and spin-dependent transport effects. These results can be important for understanding the experimental data on RuO₂ obtained using spectroscopic and transport measurements such as spin- and angle-resolved photoemission spectroscopy (S-ARPES) [23], spin-polarized scanning tunneling microscopy/spectroscopy (SP-STM/SP-STS), anomalous Hall measurements, and interfacial spin-dependent phenomena.

II Computational Method

We use first-principles calculations employing Density Functional Theory (DFT) [24, 25] as implemented in the Vienna ab initio Simulation Package (VASP) version 6.4.2 to perform the simulations [26, 27, 28]. The projector augmented wave (PAW) method has been adopted to treat the valence electrons and ion-core interactions [29, 30]. The recommended standard PAW potentials used for Ru and O are Ru_pv (4 $p^{6}$ 4 $d^{7}$ 5 $s^{1}$ ) and O (2 $s^{2}$ 2 $p^{4}$ ), respectively. Bloch wavefunctions of the materials were expanded by a plane-wave basis set with a cutoff energy of up to 600 eV. Due to the metallic nature of RuO₂, the Methfessel-Paxton smearing method (ISMEAR = 1) in combination with a smearing value of 0.2 eV was used for integration over the Brillouin zone (BZ) when optimizing structural parameters. For very accurate total energy calculations, we utilized the tetrahedron method with Blöchl corrections (ISMEAR = $-$ 5) and a Gamma-centered k-point mesh to sample the BZ. Data postprocessing was done with the help from Vaspkit [31].

Bulk RuO₂ was simulated using a tetragonal unit cell with a rutile structure ( $P4_{2}/mnm$ , SG 136). Due to the metallic property of bulk RuO₂, a dense $k$ -point mesh of 12×12×18 was used to sample the bulk BZ. The lattice constant was adopted from a recent experimental determination [32]. RuO₂ films along [110] were built by reorienting the lattice vectors of the primitive cell, i.e., primitive lattice vectors along [001], [ $1\overline{1}0$ ], and [110] become [100], [010], and [001] of the RuO₂ (110) surface unit cell, respectively, thus giving us a nonpolar stoichiometric slab. The symmetry of the bulk unit cell transitions to $Pmmm$ (SG 47) for stoichiometric slabs with an odd number of Ru layers. A $k$ -point mesh of 12×6×1 was used for structural optimization of the surface unit cell, and a denser mesh of 18×9×1 was used for density of state (DOS) calculations. To eliminate spurious image interactions between slabs, a vacuum space of at least 20 Å was added to the slab normal direction. Internal coordinates of the slab were fully relaxed until the Hellman-Feynman force on each atom was smaller than 0.005 eV/Å while keeping the lattice vectors fixed to the experimental bulk values. Due to the recent experimental observation of a non-magnetic ground state in bulk RuO₂ [7, 9], we employed the PBE+ $U$ approach with $U=0$ [33, 34, 35] to describe its electronic structure. Additionally, since spin-orbit coupling does not significantly affect the electronic and magnetic structures of RuO₂, it was neglected in our calculations.

To gain insights into the chemical bonding between atomic pairs contributing to the emergence of spin-polarized states at the surface of RuO₂, we performed a crystal orbital Hamilton population (COHP) analysis [36, 37]. The COHP method decomposes the one-particle band energies into interactions between atomic orbitals of adjacent atoms. It effectively weighs the electronic density of states (DOS) by the corresponding Hamiltonian matrix elements, thus recovering the phase information of calculated wavefunctions that is otherwise missing in the band structure or DOS descriptions of material electronic structures. This enables the identification of bonding, non-bonding, and antibonding characteristics among the pairwise atoms. The stability of these interactions is quantified by the COHP values. A positive COHP value indicates bonding interactions, while a negative value represents antibonding character. Traditionally, COHP analysis was done directly from atom-centered basis set calculations. However, since our calculations were done using the plane-wave basis set, a projection scheme from this basis set to a local orbital basis set was employed using the LOBSTER code [36, 37, 38]. In this projection scheme, the equivalence of the traditional COHP quantity is $-$ pCOHP (negative of projected COHP). In our study, we applied COHP analysis to examine the nearest-neighboring atomic pairs around Ru atoms both on the surface and in bulk-like regions. For the projection, we employed the local basis function as defined in pbeVaspFit2015.

Since spin-polarized scanning tunneling microscopy (SP-STM) technique can be used to recognize spin polarization effects on the surface [39], we simulated spin-resolved STM images employing the Tersoff-Hamann approximation [40]. In this approach, the tunneling current at the simulated probe tip (i.e., at a particular distance from the surface) in an STM experiment is proportional to the local density of states (LDOS) of the integrated electronic states ranging from the Fermi level to a predefined energy level given by a bias voltage $V$ , roughly corresponding to an experimental voltage. The LDOS is given by

n(r,E)=\sum_{\mu}\left|\psi_{\mu}(r)\right|^{2}\delta\left(\varepsilon_{\mu}-E\right),

(1)

and the tunneling current is expressed as

I(r,V)\propto\int_{E_{F}}^{E_{F}+eV}dE\,n(r,E),

(2)

where $n(r,E)$ represents the LDOS at a given position $r$ and energy $E$ . The LDOS can be evaluated from partial charge densities calculated from a pre-converged wavefunction. The partial charge density file from VASP was read by the HIVE-STM program [41], and STM images were generated using the constant-height method (at a height of $\sim 3$ Å above the highest atoms of the slab—the oxygen bridging atoms). For a given bias $V$ , the simulated STM images reflect the contrast in the partial charge densities within the energy range $0<E-E_{F}<V$ (for $V>0$ , positive bias) or $V<E-E_{F}<0$ (for $V<0$ , negative bias).

III Results and Discussion

III.1 Electronic and magnetic properties of bulk RuO₂

RuO₂ crystallizes in the tetragonal rutile structure. Its bulk unit cell consists of two formula units, as shown in Fig. 1(a). The two Ru atoms reside at the center ( $\frac{1}{2},\frac{1}{2},\frac{1}{2}$ ) and corner (0, 0, 0) of the unit cell, and are coordinated octahedrally distorted by the surrounding O atoms, forming two Ru sublattices. RuO₂ belongs to the centrosymmetric tetragonal structure described by space group $P4_{2}/mnm$ with the point group $4/mmm$ . Each Ru resides at the center of a distorted octahedron, with the octahedra being interconvertible through a $C_{4}$ rotational axis aligned along the [001] crystallographic direction.

Refer to caption — Figure 1: Rutile-type crystal structure, electronic structure and chemical bonding analysis of bulk RuO₂. (a) Rutile structure of RuO₂ with two Ru sublattices in red and blue, oxygen in silver color, (b) Bulk Brillouin zone (BZ) shown in black and rotated so that [110] crystallographic direction normal to the (110) surface whose BZ shown in red, (c) spin-polarized electronic band structure of the bulk exhibiting no spin-polarization (Fermi level set to 0), (d) atomic projected density of state for Ru and O in bulk, and (e) the corresponding $-$ pCOHP curve on the same energy scale.

The spin-resolved electronic band structure of the bulk is shown in Fig. 1(c), which does not exhibit spin polarization. In the assumed altermagnetic state, the electronic structure of interest lying along the $\Gamma M$ high-symmetry path of the bulk Brillouin zone (BZ) shows significant spin splitting ( $>1$ eV) as described by theoretical calculations of the band structure with relatively large on-site Hubbard $U$ values ( $U\sim 1.5\div 2$ eV) [2, 5]. However, this spin splitting does not appear in the non-magnetic state. Along this momentum direction, three bands are present near the Fermi level, with an additional flat band lying around $-1.5$ eV below the Fermi level. When considering the electronic structure of the (110) slab, the $\Gamma M$ path of the bulk BZ is of particular importance since bulk band dispersions along this path and parallel to it are projected onto the high-symmetry path $\overline{\Gamma}\overline{Y}$ of the surface BZ, as illustrated in Fig. 1(b).

The density of state plot and chemical bonding analysis, shown in Fig. 1(d) and 1(e), respectively, highlights important characteristics of Ru–O bonds in RuO₂. The DOS plot in Fig. 1(d) clearly shows strong hybridization between the Ru and O orbitals over a wide range of energy. Notably, between 8 and 6 eV below the Fermi level, pronounced peaks in the atomic-projected density of states (PDOS) and $-$ pCOHP (positive—bonding character) with similar contributions from both Ru and O indicate a strong covalent bonding, best described by the $\sigma$ -bonding orbitals formed by Ru $4d$ ( $e_{g}$ ) and O $2p$ components [42]. In addition, approximately from 6 eV to 3 eV below the Fermi level, small positive values of $-$ pCOHP, with a greater contribution of oxygen, suggest a weaker covalent bonding character of the Ru–O bonds due to the $\pi$ -bonding between Ru $4d$ ( $t_{2g}$ ) and O $2p$ . Around $-2$ eV, the contribution of Ru disappears, i.e. Ru PDOS is zero, leaving only O $2p$ states. This indicates that the electrons occupying these states belong to non-bonding orbitals and do not contribute to the Ru–O bonding strength, leading the $-$ pCOHP values to approach zero.

Interestingly, in the vicinity of the Fermi level, i.e., from $-2$ eV to 2 eV, the contribution of Ru $4d$ becomes dominant and populate antibonding crystal orbitals, as indicated by negative $-$ pCOHP values. This implies potential instability when electrons occupy these states. Fortunately, the DOS and $-$ pCOHP peaks lie approximately 0.5 eV below the Fermi level, alleviating the instability due to filling antibonding orbitals [43, 44]. Ultimately, despite the presence of antibonding Ru–O interactions near the Fermi level in the $-$ pCOHP curves, the integrated values of $-$ pCOHP for Ru–O bonds ( $-$ IpCOHP values) are positive (Table S1), indicating a net bonding character for these interactions. Consequently, from a chemical perspective, there is no driving force for electronic reconstruction (i.e., the redistribution of the two spin channels), resulting in a stable non-magnetic ground state rather than an ordered magnetic one.

These characteristics suggest strong mixed covalency and ionicity in Ru–O bonds, which is consistent with the atomic charge states of Ru and O obtained from population analysis based on charge density using the Bader charge method via the atom-in-molecule (AIM) approach [45, 46] or wavefunction-based methods such as Mulliken [47] and Löwdin [48] charges obtained within the projection scheme of the LOBSTER code [49]. The charge states of the atoms determined from these analyzes are lower in magnitude than the formal values of $+4$ for Ru and $-2$ for O, as shown in Table 1.

Table 1: Atomic charge states of Ru and O in bulk RuO₂

Population	Ru	axial-O	equatorial-O
Bader	1.74	-0.87	-0.87
Mulliken	1.38	-0.69	-0.69
Löwdin	1.20	-0.60	-0.60

Despite the non-magnetic character of bulk RuO₂, intriguing phenomena can arise at the (110) surface, leading to the emergence of surface magnetism. This disparity between bulk and surface properties can be attributed to the altered electronic structure and bonding characteristics at the surface, which disrupt the symmetry and electronic interactions present in the bulk. The unique environment at the surface can facilitate localized magnetic moments that do not exist in the bulk material. In the next section, we will explore how these localized states, influenced by surface geometry and reduced coordination, contribute to the overall magnetic behavior of the (110) surface.

III.2 Surface magnetization in RuO₂ (110) surface

III.2.1 The emergence of surface magnetization in RuO₂ (110) surface

To gain insights into the magnetic structure of the RuO₂ (110) surface, we performed calculations using the slab model approach. In isostructural insulating materials such as TiO₂, the (110) surface is electrostatically stable. However, electrostatics are not a concern for RuO₂ since it is a metallic system. To simulate a surface with a bulk-like region well represented by a Ru layer in the middle of the structure, we chose stoichiometric slabs with an odd number of Ru layers. These slabs possess inversion symmetry, eliminating unnecessary complications related to asymmetric systems, such as spurious electric fields across the slab thickness. The slab thickness ranges from three layers (denoted as 3L) up to nineteen layers (19L). Each layer is composed of three atomic planes, with one Ru-containing layer sandwiched between two oxygen-only layers. Each slab has two identical surface layers at the top and bottom of the unit cell [cf. Fig. 2(a)].

The converged, optimized thirteen-layer slab unit cell (13L) is shown in Fig. 2(a). This symmetric structure represents both bulk and surface areas of a semi-infinite bulk, as it is thick enough to decouple the two surfaces, and the middle layer behaves as a bulk-like region [see Fig. S1 in the Supplemental Material (SM) for the dependence of local moments of surface Ru atoms and top-layer magnetization on slab thickness]. Each Ru layer stacked along the [110] direction contains two Ru atoms corresponding to the Ru-center and Ru-corner sublattices of the bulk [cf. Fig. 1(a)]. This arrangement would yield a magnetically compensated structure with oppositely pointed moments within each (110) plane if the ground state of the bulk was antiferromagnetically ordered with the Néel vector along [001], as assumed earlier—similar to the case of the isostructural FeF₂ antiferromagnet [50].

Unlike the magnetic structure in traditional antiferromagnets, where magnetization and spin density vanish, the altermagnetic spin structure of RuO₂ would yield distinct spin densities associated with opposite spin sublattices. That is, real-space spin densities of sublattices connected by the four-fold rotation symmetry around the [001] axis would differ [1]. However, in non-magnetic RuO₂ bulk, the absence of local magnetic moments suggests that the antiferromagnetic order within each Ru plane of the (110) slab and the altermagnetic spin structure are unlikely. This is evident in the spin density visualization for the (110) slab displayed in Fig. 2(a) and the layer dependence of local moments on each Ru site from the surface to the bulk-like region shown in Fig. 2(c) (top and middle panels). Most atoms in the slab are non-magnetic except for those in the outermost layers.

In contrast to the bulk, local moments develop on the surface layers of the slab, decaying rapidly into the bulk. The collinear moments arrange ferrimagnetically within the surface layers, with unequal spin moments: Ru-center ( $\sim$ 0.61 $\mu_{\mathrm{B}}$ ) and Ru-corner ( $\sim$ $-0.17$ $\mu_{\mathrm{B}}$ ) couple collinearly in opposite directions. The majority spin density is located primarily on the Ru-center atom (denoted Ru-6f due to its six-fold coordination) and bridging oxygen (O-br, spin moment $\sim$ 0.19 $\mu_{\mathrm{B}}$ ), while the Ru-corner atom (denoted Ru-5f due to its five-fold coordination) dominates the minority spin density counterpart [cf. Fig. 2(b) for detailed labeling of atomic surface atoms]. Other oxygen ligands of the outermost layers exhibit non-spin-polarized behavior [Fig. 2(a)].

As seen in Fig. 2(a) and evidenced by variations in the local moments for each Ru sublattice across layers, the magnetization density decreases dramatically from the surface layer to the innermost (bulk-like) layer. Local moments on Ru, and consequently the layer magnetization, nearly vanish starting from the layer just beneath the surface [Fig. 2(c), bottom panel]. The emergence of surface-layer magnetism and the rapid decrease in magnetization can also be observed in the layer-projected DOS in Fig. 2(e), where significant spin splitting occurs only in the top layer. This magnetic structure is more stable than the non-magnetic counterpart by approximately 45 meV per unit cell, independent of the slab thickness, thus emphasizing the origin of the magnetic structure due solely to the surface layer (see Fig. S2 in SM for the relative energy of the spin-polarized state compared to the non-spin-polarized counterpart as a function of slab thickness).

In a recent study, Weber et al. [50] demonstrated that surface magnetization can spontaneously develop at the surface of an antiferromagnet with suitable symmetry, including the (110) surface of an AFM with a rutile structure, such as FeF₂, which is isostructural to RuO₂. Accordingly, surface magnetization is intrinsically linked to the magnetic structure of the bulk. However, as shown above, surface magnetization can arise solely from the local breaking of bulk symmetry at the surface of a non-magnetic material. Thus, the presence of surface magnetization alone does not constitute necessary or sufficient conditions to infer the bulk magnetic structure. Bulk measurements are required to identify the magnetic domain conclusively.

III.2.2 The origin of surface magnetization in RuO₂ (110) surface

In this section, we investigate the origin of surface magnetization on RuO₂ (110) surface, which arises from the non-magnetic bulk state. When bulk RuO₂ is transitioned to the (110) surface, the symmetry of the already distorted octahedral around the Ru-corner and Ru-center atoms of the surface layers is further reduced due to surface termination. The Ru-corner (Ru-5f) loses one coordination with one of its axially coordinated oxygen ligands, while the Ru-center (Ru-6f) maintains its coordination number. However, the two bridging oxygen ligands on the surface, which coordinate equatorially to the Ru-6f atom, become covalently unsaturated with only two bonds remaining as compared to three bonds in the bulk, contributing to structural distortions that affect the hybridization between Ru $4d$ and O $2p$ states. These distortions are expected to lift the degeneracy of the Ru $4d$ states further. Without any surface relaxation, i.e., no structural relaxation performed for slabs, the local symmetry of the Ru-5f is $C_{4v}$ (square pyramidal), while the Ru-6f retains its distorted octahedral symmetry as in the bulk. Nevertheless, this configuration is unstable, leading to structural relaxation akin to a Jahn-Teller distortion, which further lowers the local symmetry around the surface Ru atoms. During this relaxation, the Ru–O axial bonds in the Ru-5f-centered square pyramidal shorten, whereas the axial bonds in the Ru-6f-centered octahedra lengthen. Additionally, the equatorial Ru–O bonds centering around the Ru-6f become asymmetrically distorted; those involving the bridging oxygen ligands are significantly shortened, while the others elongate slightly. Despite these structural reconstructions, the total energy of the system remains high, with a considerable DOS at the Fermi level contributing to the occupation of anti-bonding states, as illustrated in Fig. 3(a).

To achieve stability, spin polarization is necessary to redistribute the two spin sublattices, resulting in spontaneous surface magnetization as shown in previous sections. Crystal orbital Hamilton population (COHP) analysis offers valuable insight into the driving forces behind the emergence of magnetization from a chemical bonding perspective [43, 44]. Fig. 3 presents the projected density of states for the atoms of the top surface, along with $-$ pCOHP curves (averaged for the nearest-neighbor Ru–O contacts) for both structurally relaxed non-magnetic and magnetic states. In the non-magnetic state, RuO₂ (110) displays a peak at the Fermi level in both the PDOS and $-$ pCOHP (with negative values) curves, indicating an unstable configuration. This suggests that the non-magnetic surface might typically undergo structural relaxations, such as Peierls or Jahn-Teller instabilities. However, when we compared the relaxed geometrical structure of the non-spin-polarized surface with that of the spin-polarized counterpart, we observed no discernible structural difference. Instead, the surface exhibited pronounced electronic reorganization upon spin-polarization calculation. The surface atoms become magnetized and the electronic structure shows substantial spin-polarization [Fig. 2(e) and 3(a)]. Remarkably, this spin-polarized reorganization shifted the DOS and $-$ pCOHP peaks away from the Fermi level, leading to a drastic decrease in the density of antibonding states at the Fermi level for both spin channels. Ultimately, it is this surface spin-polarization that stabilizes the entire structure.

The development of surface magnetization on RuO₂ (110) can also be understood through the lens of the Stoner criterion, which states that magnetization emerges if the product of the density of states at the Fermi level and the exchange interaction parameter exceeds unity [51]. In the non-magnetic state, the high DOS at the Fermi level suggests instability, driving the surface to electronic redistribution. Instead of structural relaxations like Peierls or Jahn-Teller distortions, the system minimizes its energy through spin polarization. This reorganization substantially lowers the DOS for both spin channels around the Fermi level as described above, stabilizing the surface. The significant reduction in antibonding states and the redistribution of electronic density through spin polarization effectively meets the Stoner criterion, leading to spontaneous surface magnetization.

Electronic reconstruction that results in the emergence of induced magnetic moments and net magnetization in the surface layer can be further seen in the variation of charge states of individual atoms at the surface compared to their bulk counterparts, as detailed in Table S2. Different population methods yield varying values for these charge states, but the trend of electronic reconstruction for the surface atoms remains consistent across these methods. For example, using the charge density-based Bader approach, we find that the charge state of the Ru-5f atom decreases, while that of the Ru-6f atom increases relative to the bulk values. This is primarily due to the absence of one oxygen atom coordination of the Ru-5f (the axial oxygen), which leads to an increased electron density and a corresponding decrease in charge state at the Ru-5f site. In contrast, the Ru-6f retains full coordination with six oxygen ligands, but also specially interacts with two bridging oxygen ligands having higher hole densities due to one unsaturated coordination for each oxygen. This condition facilitates charge transfer from the Ru-6f to these covalently unsaturated ligands, thereby increasing its charge state.

III.2.3 The significance of surface magnetism in RuO₂ (110) surface

Surface magnetism developed on the surface of RuO₂ (110) suggests that we can probe this phenomenon directly, i.e., the spin-splitting effect associated with surface bands should be detectable by surface- and spin-sensitive spectroscopy techniques such as spin-resolved ARPES. Fig. 4 shows the slab band structure with weighted contribution from the top layer atoms (2 Ru and 4 O, cf. Fig. 2(d)) along with projected bulk bands in the light-gray colored background. Interestingly, low energy electronic structures close to the Fermi level exhibit surface states with flatness and significant spin-splitting exchange energy of approximately 0.60 eV, as indicated by black arrows in Fig. 4(a) and 4(b). The occupied flat bands, which lie 0.40 eV below Fermi level, are derived from all magnetic atoms of the surface layer, i.e., Ru-5f, Ru-6f, and O-br, with the strongest contribution from Ru-6f followed by O-br and the lowest contribution from Ru-5f (cf. Fig. S3 in SM). Other non-magnetic atoms of the top layer such as O-3f and O-sub contribute negligibly to these states. The minority spin counterparts of these flat bands are unoccupied (located 0.20 eV above Fermi level) and dominated by the Ru-6f character and a smaller contribution from O-br (cf. Fig. S3 in SM). This is consistent with the highest calculated magnetic moment value for Ru-6f atom amongst others on the surface.

Interestingly, when studying the origin of these surface states along both the [ $1\overline{1}0$ ] direction and the perpendicular in-plane [001] direction, and conditions under which these states exist (dubbed flat band surface state—FBSS), Jovic et al. [52] assigned the existence of FBSS due mainly to bridging oxygen atoms. Consequently, under Ru-rich condition, these atoms are removed and the surface states are expected to disappear. However, our result shows that these surface states are derived from both the Ru and the bridging oxygen atoms of the outermost layers, providing insight into the origin of the flat bands near the Fermi level. It should be noted that the presence of occupied flat bands near the Fermi level has been observed in previous experiments [8, 23, 53, 54]. However, the spin polarization of these bands and the presence of their spin exchange counterpart above the Fermi level have not been reported in the literature. This is likely due to the ARPES technique probing only occupied states. Therefore, unoccupied states-sensitive experiments such as inverse ARPES, scanning tunneling microscopy (STM), or X-ray absorption spectroscopy (XAS) might be required to realize the full spin-polarized spectrum of the flat bands in the proximity of the Fermi level due to the surface magnetism in RuO₂ (110).

As mentioned above, the existence of flat bands lying closely beneath the Fermi level has been observed experimentally, but the origin of these states is still a debate in the literature. Jovic et al. [54] showed that the flat bands have a surface origin. They were considered to be topologically trivial states that connect the projection of nodal lines along the XR directions in the bulk BZ [54]. Very recently, based on spin-resolved ARPES data, people also assigned the flat bands near the Fermi level along the $\Gamma\mathrm{M}$ momentum direction as surface states [10]. However, a recent experiment has also shown the existence of flat bands along that $\Gamma\mathrm{M}$ direction, but the authors argued that it likely has a bulk origin [23]. It is noteworthy that our projected bulk spectrum exhibits a handful of flat bands residing below the Fermi level and overlapping in energy with the flat bands of interest, i.e., those with significant contribution from the outermost layers as described above. This overlap could be the source of confusion in the literature. To reconcile the origin of these flat bands, we calculated the layer-resolved (projected) band structure for each layer of the slab, and the results were shown in Fig. S4 of SM. The layer-projected band structures show a rapid decrease in the contribution of each layer to the flat bands of interest when going from the outermost to the innermost (or the middle, the bulk-like) layer. The middle layer of the slab displays no fingerprint of the pairs of bands being discussed (Fig. S4(f)), suggesting that these bands are derived from the top and bottom layers (the surface layers). In addition, the charge density associated with the flat bands at the $\overline{\Gamma}$ point plotted in Fig. S5 of SM shows a clear outermost-layers-origin of the bands, further corroborating the surface nature of them. The fact that those surface bands are spin-split is clear evidence for the emergence of surface magnetization mostly due to the atoms at the surface. This is consistent with the observation that the developed local moments on atoms decay quickly when going from surface to the bulk region.

Another pair of spin-splitting bands can be observed around $-$ 1.5 eV where the minority spin channel is completely flat while the majority counterpart is slightly dispersive. This contrasts with the corresponding surface-dominated bands of the non-magnetic surface where these bands are also slightly dispersive in the proximity of the $\overline{\Gamma}$ point but without a spin polarization (Fig. 4(c)). Similar to the flat band surface states close to the Fermi level, the spin-polarized states split in energy, i.e., one moving up and the other moving down compared to non-magnetic states upon spin-polarization calculation. In addition, there is another set of surface states sitting at 1.5 eV above the Fermi level. These states are slightly dispersive near the $\overline{\Gamma}$ point and do not show a significant exchange splitting, likely due to the dominating contribution of the small moment Ru-5f atoms to the band character.

Considering that the bulk RuO₂ is non-magnetic and our calculations exclude relativistic effects, the emergence of significant spin exchange splitting associated with surface bands near the Fermi level underscores the unique feature of the spontaneous surface magnetization of RuO₂ (110). This characteristic is likely a key factor in determining the magnetic properties of surface RuO₂. Additionally, the presence of large spin-splitting surface bands, particularly those close to the Fermi level, suggests that the spin polarization of transport at the surface would differ significantly from that in the bulk, which could profoundly influence spin-dependent phenomena in RuO₂. Therefore, we conjecture that surface magnetism can play a vital role in the material’s spin Hall magnetoresistance when coupled with heavy metals such as Pt or Ta, due to the induced exchange interactions and potential spin current manipulation at the surface.

Surface magnetization can also exhibit different characteristics when probed by surface- and spin-sensitive microscopy techniques such as spin-polarized scanning tunneling microscopy (SP-STM) [55]. Since SP-STM is applicable to electrically conductive materials, such as magnetic metals or doped semiconductors, RuO₂’s metallic nature makes it a suitable candidate for the technique. General observation from STM images shown in Fig. 5 is the high intensity at bright spots corresponding to the O-br–derived electronic states due to their positions lying closest to the tip in STM measurement. This has been theoretically simulated and experimentally observed [56, 57]. Notably, although Ru-5f exhibits a high DOS for both spin channels below the Fermi level, the signal is predominantly dominated by bridging oxygen atoms rather than Ru-5f (cf. Fig. 5(a),(c) and Fig. S6(a),(b),(e),(f)). This occurs because the Ru-5f atoms are located one atomic layer further from the STM tip compared to the bridging oxygen atoms, resulting in an exponential decrease in tunneling current with distance. Furthermore, the occupied states of Ru-5f originate from in-plane Ru $4d$ orbitals (Fig. S6(i)), which do not extend significantly out of plane. This interpretation is further supported by spin density visualizations shown in Fig. S6(j),(l), revealing that, under negative bias voltage, the spin density of Ru-6f and O-br covers a greater spatial extent than that of the Ru-5f.

More importantly, our simulated STM images shown in Fig. 5 reveal clear differences in the spin-dependent images. For instance, under negative bias voltage (i.e., probing occupied states), we can see high protrusions at Ru-6f and O-br positions forming horizontal lines in the spin-up images. This is due to the occupied states being dominated by the majority spin (spin-up) channel mostly derived from Ru-6f and O-br as identified by the atomic (site-) projected density of states for surface atoms (Fig. 5(m)) and the charge distribution from partial charge density calculations (Fig. S6(j)). In contrast, when probing the unoccupied states under positive bias voltage, higher protrusions forming horizontal lines in the obtained images were derived from the spin-down channel. The spin-dependent feature of the STM images can be clearly corroborated further by height–profile of the signals scanned over four selected lines indicated in Fig. 5(c). Figures in the left column corresponding to the negative bias voltage images show the dominant contribution of the spin-up density from Ru-6f and O-br positions. In contrast, the figures on the right column, which represent positive bias voltage scans, show the significant effect of the spin-down density from Ru-6f and O-br. This aligns well with the dominant contribution of majority and minority spin density of states in a spin-polarized system.

An intriguing feature emerges when transitioning from negative to positive bias: higher positive biases enhance the visibility of the Ru-5f topography. At +0.5 V bias, the position of Ru-5f becomes discernible in the spin-up images; and at +1.0 V bias, bright vertical columns along [ $1\overline{1}0$ ] appearing periodically along the [001] crystallographic direction are obvious(Fig. S6(c)). This effect arises from the dominant contribution of the Ru-5f spin-up channel over the spin-down channel in unoccupied states as demonstrated in Fig. 5(e). As the bias voltage increases, the position of Ru-5f atoms become more clearly visible due to the higher contribution from the out-of-plane components of its $4d$ orbitals as shown in Fig. S6(i), exhibiting a phenomenon known as contrast reversal. This behavior has previously been observed in studies of rutile transition metal oxide (110) surfaces and helps distinguish stoichiometric surfaces from other structures under varying experimental conditions, further validating our STM image simulations [57].

IV Conclusions

In conclusion, we have shown by first-principles calculations that surface magnetization can be developed from a metallic non-magnetic bulk material. The magnetization is significantly large at the outermost layers of the slab structure and decays quickly as it gets deeper into the bulk region. Symmetry lowering as a result of surface termination induces not only structural relaxation but also significant electronic reorganization at the surface layer, resulting in sizable moments on the atoms forming the surface. This spontaneous surface magnetism proved to exhibit unique properties, including spin-split surface bands near the Fermi level and distinct spin-resolved spectroscopic features. These findings are expected to provide valuable insights into the interpretation of experimental observations in spectroscopy and spin-dependent transport phenomena, highlighting the pivotal role of surface magnetism in such systems.

V Acknowledgement

This work was supported by the NSF through the UD-CHARM University of Delaware Materials Research Science and Engineering Center (MRSEC) Grant No. DMR-2011824. A.J. acknowledges support from the U.S. Department of Energy (Contract No. DE-SC0014388). We also acknowledge the use of Stampede3 at TACC through allocation PHY240154 from the Advanced Cyberinfrastructure Coordination Ecosystem: Services & Support (ACCESS) program, which is supported by National Science Foundation Grants No. 2138259, 2138286, 2138307, 2137603, and 2138296, and the DARWIN computing system at the University of Delaware, which is supported by the NSF Grant No. 1919839.

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