Magnetic and Electronic Properties of Spin-Orbit Coupled Dirac Electrons on a $(001)$ Thin Film of Double Perovskite Sr$_2$FeMoO$_6$

We present an interacting model for the electronic and magnetic behavior of a strained $(001)$ atomic layer of Sr$_2$FeMoO$_6,$ which shows room-temperature ferrimagnetism and magnetoresistance with potential spintronics application in the bulk. We find that the strong spin-orbit coupling in the molybdenum 4$d$ shell gives rise to a robust ferrimagnetic state with an emergent spin-polarized electronic structure consisting of flat bands and four massive or massless Dirac dispersions. Based on the spin-wave theory, we demonstrate that the magnetic order remains intact for a wide range of doping, leading to the possibility of exploring flat band physics, such as Wigner crystallization in electron-doped Sr$_{2-x}$La$_{x}$FeMoO$_6.$

We present an interacting model for the electronic and magnetic behavior of a strained (001) atomic layer of Sr2FeMoO6, which shows room-temperature ferrimagnetism and magnetoresistance with potential spintronics application in the bulk. We find that the strong spin-orbit coupling in the molybdenum 4d shell gives rise to a robust ferrimagnetic state with an emergent spin-polarized electronic structure consisting of flat bands and four massive or massless Dirac dispersions. Based on the spin-wave theory, we demonstrate that the magnetic order remains intact for a wide range of doping, leading to the possibility of exploring flat band physics, such as Wigner crystallization in electron-doped Sr2−xLaxFeMoO6.
PhySH: Monolayer films, Ferrimagnetism, Half-metals The coexistence of a strong spin-orbit coupling (SOC) and low dimensionality gives rise to novel quantum phases of matter [1]. Two-dimensional (2D) systems confined in the atomically thin films can possess rich electronic properties different from the bulk and could host various new correlated phenomena. Especially, the rapid progress in synthesizing atomic-scale slabs, superlattices and heterostructures of correlated transition metal oxides by pulsed laser deposition or molecular beam epitaxy has motivated the exploration of various (perovskite) compounds epitaxially grown on different cubic substrates as potential nano-scale devices [1][2][3][4]. An advantage of the epitaxial growth is that by changing the substrate, one can introduce strain to thin films due to a mismatch of the lattice constants and thereby control the electronic state, which we call strain engineering [5]. By replacing 3d transition metal ions with heavier 4d or 5d ions, one can even control the strength of the SOC. These flexibilities of epitaxially grown atomic-scale layers could pave a way to search for unusual spin-orbit coupled correlated phenomena in 2D systems. In this context, theoretical exploration of possible phases can provide a useful guidance.
In order to explore such collective phenomena, perovskite oxide is one of the best-established platforms [1]. For instance, LaAlO 3 /SrTiO 3 (LAO/STO) interface is known as a good 2D electron gas system [6] and hosts various electronic phenomena, including 2D superconductivity [7] and Rashba SOC effects [8]. Although there have appeared new types of atomic-scale 2D systems like the transition metal dichalcogenides [9], where we can expect a stronger effect of SOC than in graphene [10], perovskite systems are still important playgrounds because the knowledge on their synthesis and chemical properties has been accumulated for a long time.
Specifically, the bulk double perovskite compounds, such as Sr 2 FeMoO 6 (SFMO) and A 2 FeMoO 6 (A = Ca, Ba, Pb), have been investigated intensely as examples of half-metallic ferrimagnets (FiM) with enhanced magnetoresistance at room temperature and as possible spintronics devices based on the high spin polarization of charge carriers [11][12][13][14]. Theoretical studies of the carrier induced FiM in cubic SFMO have been previously discussed within the ab initio [15][16][17][18] and model Hamiltonian approaches [19][20][21] without including the SOC. Since double perovskite compounds have twosublattice structure, the synthesis of high-quality SFMO thin films with completely staggered Fe/Mo sublattices is experimentally challenging. Here, motivated by a recent fabrication of well-ordered thin films of double perovskite SFMO epitaxially grown along the (001) direction on various perovskite substrates that showed a ferrimagnetic ground state [22], we theoretically explore the combined effects of the strong SOC, tetragonal elongation, and carrier doping in a (001) layer of SFMO. For example, a typical perovskite substrate STO has a slightly shorter lattice constant (∼ −1.1%) than SFMO, so STO/SFMO heterostructures [23] would be ideal systems to investigate such effects.
In the insulating compounds such as Ba 2 YMoO 6 , the SOC locally stabilizes the j = 3/2 quartet of Mo 5+ and triggers rich multiorbital physics [24][25][26], unlike in the insulating iridates, where the orbital shape of the lowest energy j = 1/2 state is fixed, and the SOC manifests itself in the anisotropic exchange interactions [27,28]. On the other hand, in SFMO the molybdenum 4d electrons are itinerant, forming a conduction band. In this case, a strong impact of the SOC on the band structure is expected. Indeed, it has recently been proposed that the SOC may stabilize a Chern insulator phase in the (001) and (111) monolayers of double perovskites [29,30] and lead to a topologically nontrivial band structure in BaTiO 3 /Ba 2 FeReO 6 /BaTiO 3 2D quantum wells [31]. It should be noted that the tight-binding model for this system with the SOC and tetragonal compression has been investigated [30] in the free fermion level. However, the effect of carrier doping on the magnetism in the presence of the SOC is still illusive.
In this Letter, we study a (001) layer of pure and doped SFMO based on a minimal microscopic model within a large-S expansion. We find that the strong SOC gives rise to a robust nontrivial magnetic state with an electronic structure consisting of four spin-polarized massive or massless Dirac dispersions as well as flat bands. Based on the spin-wave theory, we demonstrate that such an unusual magnetic state is stable in a large, experimentally relevant range of carrier doping. In the electron-doped Sr 2−x La x FeMoO 6 , we suggest that the extra electrons would occupy a fully polarized flat band.
The model.-In the (001) layer of double perovskite SFMO, Fe and Mo ions form a checkerboard pattern on a square lattice [see Fig. 1(a)], and these metal ions reside inside the oxygen octahedra. In the ionic picture, iron is in the Fe 3+ valence state with five 3d-electrons coupled by Hund's rule into the high-spin state forming a localized S = 5/2 moment. The Mo 5+ ion has a single 4d-electron in the t 2g manifold of degenerate xy, xz, and yz orbitals. The lowest energy coherent charge transfer process takes place when this single electron moves from Mo 5+ to a neighboring Fe 3+ through the hybridization between the same t 2g states along a given bond. For example, on a bond along the x-direction there is a finite overlap either between xy or xz neighboring orbitals with a real amplitude −t.
We consider a tetragonal elongation of the oxygen octahedra due to a substrate-induced compressive strain. The corresponding tetragonal crystal field ∆ T > 0 lifts the threefold t 2g orbital degeneracy by stabilizing an orbital doublet of the axial xz and yz orbitals and by placing the planar xy orbital at a higher energy. It should be noted that for the xy orbital, where the SOC is completely quenched by the tetragonal distortion, we can repeat the analysis of the cubic case without the SOC [20] to discuss the stability against doping. In addition, we include the SOC λ > 0 in Mo 5+ that further lifts the degeneracy of the xz and yz orbitals by stabilizing j z = ±3/2 Kramers doublet of the effective total angular momentum j = s + l = 3/2 quartet [32]. Here, s = 1/2 and l = 1 are spin and effective angular momentum of a t 2g electron, respectively [33]. We will not include the SOC for the Fe 3d-orbitals because it is much weaker than that for the Mo 4d-orbitals. The resulting local energy structure of Mo 5+ is depicted in Fig. 1(c), and the explicit forms of j z = ±3/2 wave functions are given by and are labeled hereafter by fermionic annihilation operators c σ with a pseudospin σ = ↑, ↓.
Here we take the limit of a strong SOC λ and a tetragonal field ∆ T compared to the nearest-neighbor (NN) hopping amplitude |t|. Projecting the t 2g orbitals onto the lowest energy states (1), we have obtained a lowenergy Hamiltonian for a charge transfer between NN Fe and Mo ions [see Fig. 1(d)], as follows.
where i(j) labels Fe(Mo) ions, ij ∈ x(y) refers to each NN bond along the x(y)-direction, σ = ↑, ↓ = ±1 stands for a spin index, 2∆ is a charge transfer gap between Mo 5+ and Fe 3+ , the number operators n j measure carrier density d † i d i and c † j c j , respectively. In the undoped SFMO, there is one carrier n = n (d) + n (c) = 1 per formula unit, ignoring the localized half-filled Fe d-shell. The SOC manifests itself in the spin-dependent hopping in Eq. (2) that explicitly breaks the original SU (2) symmetry. Hereafter, we set t = 1 as the energy scale for simplicity.
When an itinerant electron visits the Fe 3+ ion with a core spin S = 5/2, the resulting total spin S of Fe 2+ could, in principle, take one of the two possible values S = 2 and S = 3. However, the maximum allowed spin quantum number for six electrons in a d-shell is S = 2. The unphysical S = 3 states appear because the local and itinerant spin operators on the Fe site are treated as independent variables. In order to project the enlarged Hilbert space onto the physical one, we supplement the hopping Hamiltonian (2) by a local antiferromagnetic (AF) coupling J → ∞ between the local and itinerant spins [20]. The total Hamiltonian then becomes The sum is taken over every Fe site i, and S i and s i are operators for the local and itinerant spins, respectively. Ferrimagnetic ground state.-The model defined by Eqs. (2) and (3) is one version of canonical double exchange (DE) problems with an infinite exchange coupling between the local and itinerant spins. Similarly to the DE, a maximum kinetic energy gain is achieved when the local moments align ferromagnetically (FM) and, in the present case, antiparallel to the itinerant spins, giving rise to an FiM state.
We consider the FiM order along the tetragonal symmetry z-axis and discuss later its stability within the large-S spin-wave theory. We introduce fermionic operators D ↓(↑) for the carriers on the Fe sites with their spins quantized along the local moments. This representation diagonalizes the spin part of the Hamiltonian Eq. (3) and projects out the fermionic states D ↑ corresponding to the unphysical states with S = 3 (see Ref. [20] for details). The d-operators in Eq. (2) in terms of the new ones are expressed as d xz(yz) where b is a bosonic annihilation operator for a single magnon state. This is created when a spin-down electron moves away from Fe 2+ , which is in the entangled S = 2 state of the local and carrier spins, leaving an Fe 3+ local moment in the S z = S − 1 = 3/2 single magnon state which is tilted away from the fully polarized S = S z = 5/2 state [see Fig. 1(b)]. This representation provides the correct matrix elements of fermionic operators within the eigenstates of the allowed total spin, S = 5/2 and S = 2, in the perturbative level and retains a quantum nature of the local moments.
Inserting the above representation into Eq. (2), we get H = H 0 + H 1 + H 2 , where H 0 is a single-particle part Diagonalizing the noninteracting H 0 part, we get the following expression in the momentum space.
where E k = ∆ 2 + 2(cos 2 k x + cos 2 k y ) and the eigenstates α k↓ , β k↓ , γ k↓ , and c k↑ have been obtained by a unitary transformation [34]. The band structure (4) is composed of four bands and is shown in Fig. 2(a)-(b). The two flat α ↓ and c ↑ bands correspond to a nonbonding state composed of the d xz↓ and d yz↓ orbitals of Fe and a localized j z = 3/2 state of Mo, respectively. The dispersive antibonding β ↓ and bonding γ ↓ bands are made of spin-down states of Mo and Fe. The next-nearestneighbor (NNN) hopping between the same Fe (or Mo) ions, not considered here, might in principle give a finite dispersion to the flat bands. However, the corresponding hopping is between the d xz and d yz orbitals, and is extremely small (∼ few meV) [29]. Moreover, it exactly vanishes when projected onto the complex wave functions of the Mo j z = ±3/2 states due to a destructive quantum interference.
We now discuss the effects of a uniform external magnetic field H on H 0 . This just splits the four bands without hybridization because H 0 conserves the z-component of the real spins. While the external field is uniform, the Zeeman splitting gµ B H of the itinerant electrons on the Fe and Mo ions become staggered due to the difference in the g-factors. As shown in Fig. 3, this would allow us to control the mass of Dirac dispersions and also to dope the flat band just above the Fermi energy.
The flat bands, which are already fully spin-polarized, are different from the unpolarized ones, such as the ones in the (110) thin films of STO [36] or in (metal)organic systems [37,38], supporting the flat-band ferromagnetism [39][40][41].
Therefore, in electron-doped Sr 2−x La x FeMoO 6 or SFMO under a strong magnetic field, where extra electrons occupy the nondispersive Mo band, we anticipate other types of instabilities, such as Wigner crystallization [42] or various types of complex charge ordered patterns, as well as the formation of selftrapped polaronic states of minority spins at the Mo sites. As confirmed in the following part, the FiM state is stable in a wide carrier doping range of the electron doping, and, consequently, the minority-spin flat Mo band can indeed be electron-doped.
Spin-wave spectrum.-We now analyze the stability of the FiM order state postulated above. To this end, we derive a spin-wave excitation spectrum from the magnon Green function G q,ω = 1/[ω − Σ q,ω ] evaluated within the leading order of the large-S expansion. First, we note that in the classical S → ∞ limit, the magnons are localized, and they become dispersive only due to quantum corrections. The corresponding magnon selfenergy corrections (Σ q,ω ∼ 1/S), shown in Fig. 1(c), stem from magnon interactions with propagating transverse and longitudinal particle-hole excitations. Their expressions are quite lengthy and are given in Ref.
[34]. We find that a coherent spin-wave mode emerges below the Stoner continuum with the following dispersion relation in the low-energy limit.
where Γ 1q = cos q x cos q y , Γ 2q = (cos 2 q x + cos 2 q y )/2, J 1 and J 2 are the carrier induced exchange couplings between the NN and NNN local moments, respectively. They depend only on the carrier density and the charge transfer gap ∆ [34]. The spectrum (5) is gapless at q = (π, 0) and at the symmetry-related points, which is very surprising because (i) the model defined by Eqs. (2) and (3) does not host any apparent continuous spin-rotation symmetry, and (ii) the gapless points are away from the FM Bragg point q = (0, 0). Actually, the model has a hidden SU (2) symmetry that can be uncovered by a gauge transformation [34].
For the spectrum (5), the spin stiffness of the FiM ordered state is given by D = 2J 1 S + 4J 2 S. Shown in Fig. 4(a) is the dependence of D on the carrier density n for 0 < n < 1 at various values of the band gap ∆. In the range 1 < n < 2, D remains constant for ∆ = 0. This is because the added electron carriers occupy the unpolarized flat band, and no additional potential or kinetic energy is gained. For ∆ = 0, D becomes very weakly renormalized (∼ few percent [34]) as carriers occupy the minority spin flat band. For comparison, in Fig. 4(b) we plot the spin stiffness obtained at a zero SOC [20]. D vanishes at some critical doping, signaling the instability of the FiM order. Without SOC the FiM ground state cannot be stabilized at n = 1 or n slightly larger than 1.
Thus, a strong SOC extends the stability window of the FiM order to the experimentally accessible electron doping range. The reason behind the extended stability is the SOC-induced electronic band structure reconstruction which transforms a large Fermi surface centered around the q = (0, 0) to four small Fermi pockets around (± π 2 , ± π 2 ), as shown in Fig. 4(c), allowing more kinetic energy gain with an increasing carrier density. Indeed, as seen experimentally [43], electron-doped Sr 2−x La x FeMoO 6 thin films do exhibit the stable FiM order in a wide range of doping as correctly shown here by the model with the SOC.
In conclusion, we have proven the stability of the FiM ground state of SFMO thin films against doping within a perturbative analysis. We discovered that the SOC plays a critical role in the enhancement of the stability and results in a phase with an unusual band structure including Dirac dispersions and flat bands. We anticipate that this gives rise to interesting collective behaviors, such as Wigner crystallization [42].