Magnetically Controllable Two‐Dimensional Spin Transport in a 3D Crystal

2D phases of matter have become a new paradigm in condensed matter physics, bringing in an abundance of novel quantum phenomena with promising device applications. However, realizing such quantum phases has its own challenges, stimulating research into non‐traditional methods to create them. One such attempt is presented here, where the intrinsic crystal anisotropy in a “fractional” perovskite, EuxTaO3 (x = 1/3 − 1/2), leads to the formation of stacked layers of quasi‐2D electron gases, despite being a 3D bulk system. These carriers possess topologically non‐trivial spin textures, indirectly controlled by an external magnetic field via proximity effect, making it an ideal system for spintronics, for which several possible applications are proposed. An anomalous Hall effect with a non‐monotonic dependence on carrier density is shown to exist, signifying a shift in band topology with carrier doping. Furthermore, quantum oscillations in charge conductivity and oscillating thermoelectric properties are examined and proposed as routes to experimentally demonstrate the quasi‐2D behavior.


Introduction
The exploration of 2D states has come to such prominence in condensed matter and materials research because their properties make them desirable, or even essential, for state-of-the-art device applications.2D electron gases (2DEGs) are capable of exceptionally high carrier mobilities, [1][2][3][4] and have also allowed for the observation of fundamental condensed matter phenomena like, for example, the integer and fractional quantum Hall effects, [5][6][7][8][9]  monopole(anti-monopole) form, which arises from a coupling between two component Rashba fields, and the strong SOC of the carriers.As the magnetic ordering of the charge carriers arises indirectly from an exchange interaction with Eu, these spin textures can be externally manipulated, for example, by the magnetic field orientation, without any change in their Fermiology.This, combined with the single orbital character of the conduction states, makes Eu x TaO 3 a promising platform for a broad range of spintronic devices.
We go on to examine possible quantum states arising from this quasi-2D behavior, and propose methods to experimentally investigate them.This includes dimensionally anisotropic quantum oscillations in charge conductivity, observable by Shubnikovde Haas (SdH) transport measurements, [23][24][25] and an intrinsic anomalous Hall effect (AHE) with a non-monotonic dependence on carrier density (n c ).Finally, we demonstrate an oscillatory Seebeck effect [26,27] deviating from typical metallic behavior due to plateaus in the density of states (DOS). [28]This is proposed to be utilizable as a thermoelectric source with robust spin and orbital characterstics.

Crystal Structure of Eu x TaO 3
The structure of an ideal oxide perovskite, ABO 3 , as depicted generally in Figure 1a, has a cubic unit cell formed from a central B site cation octahedrally coordinated by six oxygen ions, with a corner A site cation. [29]This has an O h point group symmetry, splitting d orbital states into two manifolds, e g {d z 2 , d x 2 −y 2 } and t 2g {d xy , d xz , d yz }. [30] The crystal structure of Eu x TaO 3 (shown in Figure 1b) is similar to that of a double perovskite, [31][32][33] which has a unit cell composed of two formula units of ABO 3 along the crystalline c-axis.However, Eu x TaO 3 alternates between an unoccupied A site and an Eu A site with partial occupancy 2x, [32,34,35] hence we name this structure a fractional perovskite.The partial Eu site occupancy results in slight distortions of the two TaO 3 octahedra along the a and b axes, causing oxygen ions at (0, b∕2, c∕2 ± c∕4) to shift more significantly along c compared to those at (a∕2, 0, c∕2 ± c∕4).This anisotropic distortion lowers the symmetry of each octahedron to D 2h , giving rise to an orthorhombic crystal structure with a Pmmm point group symmetry. [32,34,35]While these orthorhombic distortions are small enough that the system can be well approximated by an idealized tetragonal structure (as discussed later in our toy-model analysis), we have fully considered them in our ab initio calculations, employing the orthorhombic structural parameters reported by Sirotinkin et al. [35] with lattice parameters, a = b = 3.8771 Å, c = 7.7960 Å.It should be noted that some other X-ray diffraction measurements [32] have reported minor mismatches in lattice parameters a and b but these have little effect on the key features of the system (see Section I, Supporting Information for a comparison between electron structure calculations with and without a ≠ b distortion).The ionic state of Eu can vary between 3+ and 2+ depending on the value of x, resulting in a net 1+ cationic state that, when added to the 5+ state of Ta, provides the six electrons required by the existing three O 2− anions.Eu x TaO 3 has been experimentally realized in the x = 1/3 stoichiometry where Eu ions have a 3+ charge. [32,34,35]However, with modern molecular beam epitaxy techniques, we predict the Eu 2+ , x = 0.5, stoichiometry is feasible (a detailed analysis of the structural stability of x = 0.5 compound can be found in Section II, Supporting Information).
The alternating pattern of Eu ions leads to two distinct types of charge transfer in Eu x TaO 3 .The O anion in the plane with the Eu is capable of charge transfer with both the Eu and Ta cations, whereas the other O can only receive charge from Ta.As the Ta cations are more important to the O in the latter case for charge transfer, the Ta ions are brought in toward the plane lacking Eu, displacing them ≈0.07 Å along c in the case of Eu 0.5 TaO 3 .This breaks the local inversion symmetry of each Ta site, inducing opposite polarizations, P i ∥ ±z, which overall results in an anti-ferroelectric order.Furthermore, this modifies the crystal potential, V(r), resulting in a TCF that splits bands along orbital characters that are otherwise degenerate in an ideal cubic system.

Energy Splittings of t 2g Conduction Bands
The conduction bands of perovskites are typically of d-t 2g orbital character, and for the simple cubic structure, there are two limiting cases for the band splittings in the TRS breaking regime: The SOC dominates (Figure 1c), which acts as a perturbation of the form SOC ∝ L ⋅ Ŝ, where Ŝ and L are the spin and orbital angular momentum operators, respectively, typically bringing the j = 3∕2 states down in energy.Or an exchange interaction dominates (Figure 1d), spin polarizing the bands, with some orbital splitting from the SOC.
Given the large SOC term of tantalum (≈1 eV), Eu x TaO 3 is closer to the former case.However, the TCF and the 'super-cell' unit cell affords a much richer band diagram, Figure 1e.First, the fractional perovskite structure leads to a Brillouin zone folding (BZF) along k z , Figure 1f, which doubles the number of Ta 5d-t 2g states, hybridizes bands along site index and opens a gap, Δ (BZF) , between the two bands at the BZ boundary.
The largest splitting, Δ (T) , occurs due to the TCF.As the anisotropy is along the z-axis, it is the in-plane d xy orbitals that are separated from the d xz , d yz orbitals.From considering charge transfer, the d xy orbitals must be lower in energy, as these 'cover' more of the lone O anions, better facilitating the ionic bonding in Eu x TaO 3 , i.e., Δ xy > 0. The SOC leads to off-diagonal coupling between orbital and spins states.However, as the d xy states are well isolated from the {d xz , d yz } states, they are not much perturbed by an 'intrinsic' SOC that derives from mixing of atomic orbital character.The {d xz , d yz } states are hybridized to form states of unquenched orbital angular momentum and so are gapped by such a SOC.
Last, the states are spin polarized by a Ruderman-Kittel-Kasuya-Yosida (RKKY) exchange interaction between the Eu 4f and Ta 5d electrons, [25,36,37] where the Eu 4f moments are forcibly aligned by an external magnetic field, B, which breaks TRS.Realization of such an FFM phase has already been demonstrated in many Eu-based compounds under the application of a moderate magnetic field (typically ≈2 − 3 T). [25,38,39]This is due to the localized nature of Eu 4f states, making any direct exchange interaction between them extremely fragile.This situation is expected to be even more extreme in the case of Eu x TaO 3 , as Eu ions are separated substantially in the c direction.To examine this, we have calculated the total energy for Eu 0.5 TaO 3 in three different magnetic configurations: a ferromagnetic phase (FM), an A-type antiferromagnetic (A-AFM) phase, where all Eu moments are parallel within each Eu layer but antiparallel between the adjacent layers, and a C-type antiferromagnetic phase (C-AFM), where all neighboring Eu moments align antiparallel to each other, forming alternating ferromagnetic chains in a staggered pattern.These calculations reveal that the FM and A-AFM phases are essentially isoenergetic, whereas the C-AFM phase is less energetically favorable by 10 meV.Thus, one can assume that this system is intrinsically paramagnetic under ambient conditions.Applying an external magnetic field enables these highly localized magnetic moments to align along the direction of B, forming the aforementioned FFM phase.
The RKKY-induced spin polarization of the Ta 5d-t 2g state has a second-order perturbative form, meaning that its strength depends on the energetic proximity of the t 2g states with the 4f ones.As the d xy states are significantly lower in energy than the {d xz , d yz } manifold one can see that Δ where ,  ∈ {↑, ↓} denote spin along the magnetic axis, z, and  is an amplitude that is dependent on the linear combination of Ta site ±, and the relative sign of  and  due to Hund selection rules.This shows that the interplay of the BZF and RKKY interaction leads to effectively two terms, an on-site (Δ (ex,AA) ) and inter-site (Δ (ex,AB) ) spin splitting.Importantly, the latter reverses the spin ordering for either linear combination of Ta site, producing the ↑, ↓, ↓, ↑ ordering, while the former will then shift one set of spin 'off-center' relative to the other.Putting these together, the four Ta d xy states form the lowest conduction states of Eu x TaO 3 , split along spin eigenvalues that are controllable through their correlation with the local Eu 4f moments forcibly aligned with B. Based on similar FFM phases in Eu compounds [25,38,39] we estimate the strength of the required field to be |B| ≈ 2-3T.Anticipating the 2D-like behavior of these bands, the planar nature of the d xy orbitals is shown schematically in Figure 1h, where in-plane hopping between Ta is greatly favored due to the overlap of the orbitals.

Electronic Structure and Quasi-2D Band Dispersion
Figure 2 shows the calculated electronic structure of FFM Eu x TaO 3 from density functional theory (DFT), with the Eu 4f magnetic moments directed along the [001] axis; see Methods, Section 4, as well as Section III (Supporting Information).Figure 2a,b shows the overall band structures of Eu 1/3 TaO 3 and Eu 0.5 TaO 3 , respectively.As can be seen, the overall features of the two compounds are characteristically the same; they both commonly exhibit effectively dispersionless Eu 4f bands within their bandgap in close proximity to the low energy conduction bands made of Ta d xy orbitals; see Section IV (Supporting Information) for a more detail on orbital projections.Hence, hereafter, we focus our attention specifically on Eu 0.5 TaO 3 , with the understanding that this is broadly applicable to other stoichiometries of Eu x TaO 3 .The energetic proximity of the 4f electrons, ≈0.2-1 eV below the conduction band minimum (CBM), facilitates the RKKY interaction that results in the spin-polarized nature of the t 2g conduction states.Along Γ → X, M the t 2g orbitals behave as they would in a cubic perovskite, with an isotropic parabolic dispersion at low k, with the d xy states highlighted in Figure 2c.However, along Γ → Z the quasi-2D nature of the d xy states in this system is manifest in their minimal dispersion, with Figure 2d showing k z behaves almost like a degeneracy.The importance of the large Δ (T) term is apparent, keeping the other t 2g states (that appear as bonding and anti-bonding bands at 1 − 2 eV and 3 − 4 eV, respectively) well separated from the four d xy states, which is necessary for the quasi-2D behavior.
The lack of dispersion along k z gives rise to elongated Fermi pockets (energy iso-surfaces of individual bands).The inset in Figure 2f shows an example set of Fermi-pockets for n c ≈ 7 × 10 20 cm −3 , where for the larger pockets the elongation is so pronounced that they are almost cylindrical.[25] Similar to its sister perovskites, like SrTiO 3 and EuTiO 3 , Eu x TaO 3 is expected to have high carrier mobility and hence exhibit good quantum oscillations, making this a viable measurement to perform.
Figure 2e shows a calculation of F SDH for the first Fermi-pocket as a function of the angle of the cross-section,  CA , at a typical n c = 4 × 10 18 cm −3 .As  CA is increased, the SdH frequency increases due to the elongation of the band that arises from the lack of k z dispersion.Furthermore, for larger n c (n c = 10 20 cm −3 shown in Figure 2e) the SdH frequency diverges at large  CA as the Fermi-pocket connects across the BZ boundary in the k z direction.Figure 2f shows the SdH frequency at  CA = 0 for the 4 d xy Fermi-pockets as a function of carrier density, n c .At first, only one pocket is populated, but increasing n c systematically introduces new pockets, which will manifest experimentally as a 'beating' in the measured SdH oscillations.Figure 2g shows an example charge density plot for n c ≈ 5 × 10 19 cm −3 (Fermi energy, E F ≈ 0.1 eV), demonstrating that the real space manifestation of this quasi-2D dispersion is a confinement of the charge carriers to the xy planes the Eu ions lie in.The bulk fractional perovskite structure behaves analogously to a material formed by vertically stacking Eu 0.5 TaO 3 layers, with no overlap between the low energy conduction states of each layer.The charge density is enhanced near the Eu to aid the RKKY interaction with the Eu 4f moments.As such, one can see that the system has realized spin-polarized quasi-2DEGs despite the relative simplicity of the 3D crystal.

Topological Spin Textures
While the conduction bands are effectively spin-polarized along the magnetic axis due to the exchange interaction and FFM order, it is worth considering the effect of SOC to form a complete picture of the spin texture.For the low-lying conduction bands, primarily of d xy orbital character, there is minimal inter-orbital mixing with the other t 2g orbitals.As such, the SOC will be driven by spatial variation in crystal potential, [40] i.e., As discussed previously, the TCF in Eu x TaO 3 arises from a local inversion symmetry breaking on each Ta site i, see Figure 1b, that induces a polarization P i ‖∇V i (r)‖ ± z.This leads to a planar Rashba SOC on each Ta site [41] B(SOC as depicted schematically in Figure 3a.Furthermore, due to the anti-ferroelectric order the sites have opposite chirality, B(SOC) Remarkably, the total Eu 0.5 TaO 3 system does not exhibit Rashba-like spin textures, but the inter-site coupling modifies the effective SOC felt by the energy eigenstates to an in-plane monopole/anti-monopole form modulated along k z , Figure 3b,c.Where for clarity, we restate that the spin textures are still predominately collinear along the magnetic axis due to the RKKY interaction.
We To make this more concrete, we construct a minimal 4 × 4 tight-binding model for the low energy d xy states of the complete unit cell of Eu x TaO 3 , making the approximation of an idealized D 4h symmetry.In k-space the full four band Hamiltonian is given by: where, Explicit values for parameters used to approximate DFT bands of FFM Eu 0.5 TaO 3 are given in the methods, Section 4. The model shows a canting of the site pseudo-spin degree of freedom along k z , which is ultimately responsible for the non-vanishing ⟨ Ŝx ⟩, ⟨ Ŝy ⟩ at individual k-points by a non-trivial coupling of the Rashba fields.For low k x ∕k y , it can be shown (see Section V, Supporting Information for derivation) that the effective SOC felt by the energy eigenstates is: where, A(k z ), Figure 3b, captures the k z modulation of the spin texture as a result of the site canting, showing it is anti-symmetric about k z = 0. Due to their inherent chirality, these spin textures have a nonzero contribution to the Berry curvature, [42] Ω: where ,  are spins of opposite direction.One can see that the sign of ⟨ Ŝz ⟩ determines the sign of Ω z contributions from the inplane spin texture in a manner equivalent to the original Rashba , that result from ∇V i .b) Amplitude of monopole SOC term as a fraction of original Rashba term, 2A(k z )∕Δ (SOC) .c) Fermi-pocket projected planar spin textures for d xy bands calculated from Wannier interpolated DFT, color coding shows outward/inward texture as red/blue, respectively.For |+, ↑⟩ , |−, ↓⟩ , |+, ↓⟩ , |−, ↑⟩ the range of in-plane spin, m, takes the values 0.27, 0.28, 0.022, 0.002 ℏ/2 respectively.The out-of-plane component is removed to show monopole/anti-monopole-like texture, as the exchange interaction dominates the spin texture.
fields, [43] giving a pattern of negative, positive, positive, and negative, Ω z contributions.

Anomalous Hall Effect
Given the FFM ordering and the above-mentioned spin textures in the system, combined with the expected good carrier mobility, we anticipate that a large intrinsic AHE could be measured experimentally.Due to the symmetry of the system, we are only interested in the in-plane intrinsic AHE described by the off-diagonal conduction element [44] where f E F ,0 is the Fermi-Dirac distribution for E F , at temperature T = 0, and the integral is across the entire BZ.
Figure 4a shows, in the k z = 0 plane, the in-plane band dispersion and slices of Ω z calculated for Eu 0.5 TaO 3 at selected E F from Wannier functions (See Section 2).This allows one to see the correspondence between the evolution of the band Fermiology and Ω z , where, interestingly, the shifting competition between positive and negative contributions does not match the Rashba-like picture from equation.8.For low E F , Ω z is entirely positive over the whole BZ, as seen for E 1 , E 2 , E 3 .Once E F passes a critical doping concentration and accesses |−, ↑⟩ a net negative Ω z contribution begins to creep in for certain regions of the BZ, E 4 .This negative contribution becomes more intense as E F is further increased, becoming concentrated in the region between |−, ↓⟩ and |−, ↑⟩, E 5 .
This discrepancy between the Ω z expected from the spin textures and Ω z calculated from Wannier interpolation remains unresolved.We have examined the possible role of gapped nodal surfaces formed with the {d xz , d yz } manifold, and/or the presence Color coding shows the two regimes where  AHE xy goes from decreasing to increasing with n c .For the Ω z slices taken in (c), the corresponding E F ∕n c is marked on for reference.
of orbit-momentum locking terms as extra contributions to Ω z as a means to solve this.However, our efforts as of yet have not been able to fully capture the Berry curvature of the DFT calculations, such that this should be a key focus of future research on this system.
Figure 4b shows a schematic setup of an AHE measurement for Eu x TaO 3 , where the magnetization is directed along 001.One can view the measured AHE as arising from the net sum of the contributions from each planar layer of charge carriers due to the aforementioned confinement of charge density.A current along x generates a net voltage along y due to the off-diagonal  AHE xy , with carriers of different Ω z sign having opposite transport velocities.Hence such a shifting Ω z can be expected to produce a non-trivial AHE.
The calculated intrinsic  AHE xy is shown in Figure 4c, demonstrating a non-monotonic dependence on n c (E F ).That is, at some n c there is a turning point in  AHE xy where it goes from decreasing to increasing with n c , due to the shifting competition between positive and negative Ω z contributions.[47][48] This non-monotonic AHE then allows for the possibility of a tunable AHE for device applications, by way of carrier doping.

Thermoelectric Properties
Another transport property worth considering is the Seebeck effect in FFM Eu x TaO 3 . [26,27,49]If a thermal gradient, ∇T, is applied along the material, for example, by applying hot and cold contacts as depicted in Figure 5a, a drift of electrons against the direction of the thermal gradient is expected.Once the system reaches a steady state, this thermal gradient results in an electric field E i = S ij (∇T) j , where S ij are the Seebeck coefficients.This can be measured experimentally as a Seebeck voltage, V s .
Figure 5b shows the calculated in-plane Seebeck coefficient in Eu 0.5 TaO 3 , S xx = S yy , as a function of both temperature, T, and chemical potential, .Due to the isotropic nature of the d xy bands in this plane, and the lack of k z dispersion, this is in good agreement with the Mott formula [50][51][52] S ∝ T∕.
Figure 5c shows the calculated out-of-plane Seebeck coefficient, S zzz , with a direct comparison to the conduction band density of states (DOS), g(E).Interestingly, S zz deviates from normal metallic behavior, exhibiting oscillations with respect to .This matches previous experimental work by Pallechi et al. [28] where an oscillating Seebeck effect in a LaAlO 3 /SrTiO 3 device was shown to arise from localized states at the interface.
The Seebeck coefficients, S ij , are calculated directly as follows: [53] S ij (, T) where f ′ ,T (E) is the first derivative of the Fermi-Dirac distribution function at , and T, and  ij (E) (E) is the transport function, where  i is the group velocity (∇ k E(k)) i ,  is the characteristic lifetime, and  is the Dirac delta function.
To connect these oscillations in S zz to the DOS we construct a model DOS as a series of step functions, where B i are constants that determine the size of the DOS steps, and E i determines their location.These step like increases in the DOS correspond to the introduction of each successive quasi-2D band at E i , which have an effectively constant DOS contribution.Due to the minimal dispersion along k z one can make the approximation for the transport function, where A is a constant that will cancel out.
yz are vanishing due to symmetry and hence S zz =  zz ∕ zz .One can then derive  zz ,  zz as convolutions, providing an analytic expression for S zz , for the model DOS, that is shown in Figure 5d, reproducing the oscillations found in the Wannier calculations for high T. From this, one can see that the measurement of S zz provides another good experimental signature of the 2D-like behavior of the conduction bands by its direct relation to the DOS and the narrow nature of the bands, and the lack of k z dispersion required for Equation 13 to hold true.Furthermore, such a set-up has useful applications in spintronics.For low enough n c , a single conduction band of only a spin-up character is accessed, such that the thermal drift of electrons is spin polarized.Taking this further, one can also consider manipulation of B, exploiting the possibility of off diagonal conduction terms in the thermal drift of the spin-polarized electrons through the Nernst effect, the thermal analog of the Hall effect. [54,55]

Spintronic Applications
To exploit Eu x TaO 3 for spintronic applications, we require easy control of the spin degree of freedom for the lowest band without drastically altering the electronic band structure.As discussed above, the magnetic ordering arises from an RKKY exchange between Eu 4f and Ta 5d electrons.Due to the localized nature of the 4f electrons, their spin is easily controlled by the orientation of B, which in turn manipulates the direction of spin for the conduction states through the RKKY exchange.This is a comparable situation to the FFM phase of the cubic perovskite EuTiO 3 . [25,56]owever, in EuTiO 3 B is also responsible for breaking the cubic symmetry to induce directional anisotropy, and so the conduction band Fermiologies are strongly dependent on field orientation.Eu x TaO 3 has an intrinsic anisotropy due to the fractional perovskite structure, affording it the distinct advantage that manipulation of B does not disturb the quasi-2D behavior of the d xy states (refer to Section VI, Supporting Information for band structures at different magnetic field orientations).Furthermore, while in EuTiO 3 the conduction bands are a mixture of t 2g bands, the four lowest Eu x TaO 3 conduction bands are dominated by one orbital character, d xy .This, combined with their quasi-2D dispersion, and spin polarization, greatly reduces scattering in the system and allows for a single spin conduction channel with large mobility.
Thus, the unique properties of Eu x TaO 3 can be utilized for designing devices for spintronics applications.One such device we propose is a spin polarizer, schematically depicted in Figure 6a.An ohmic contact injects non-spin-polarized electrons into a Eu x TaO 3 device where the doping concentration is sufficiently small that only the first band, and hence a single spin polarization, is accessed.[59] While B does not polarize electrons in the ohmic contact, it aligns spin of electrons traversing Eu x TaO 3 , thus generating a controllable spin polarized current due to the half metallic phase.Moreover, since both the crystal structure and the crystal lattice parameter of Eu x TaO 3 are compatible with the majority of oxide perovskite compounds, by integrating the material with perovskite oxides Eu x TaO 3 can be used as an effective source for injecting a spin-polarized current to probe spin-dependent phenomena in a plethora of materials.
Another possible device is a psuedo-spin valve as depicted in Figure 6b.Here, Eu x TaO 3 is sandwiched by FM contacts, with a thin non-magnetic (NM) layer in between.[62] In such a device the transmission of the channel can be gradually tuned by adjusted the angle  between B and the magnetization of the FM contacts.
Eu x TaO 3 enables a realization of a type of spin transistor depicted in Figure 6c.A current in an L-shaped device made of Eu x TaO 3 flows from contact A to contact C (blue arrow) by application of potential difference.The application of an external magnetic field can then gradually change the resistance of sections AB and BC in a counterphase manner.B applied in-plane ( = ∕2) generates an in-plane spin-polarized current.When B and the current flow direction across a section of the device are (anti-)parallel, the section of the device will have a (high) low resistance due to the anisotropic magneto-resistance (AMR) effect. [63,64]In addition, the dependence of the planar hall resistance of AD, and CE, on B will also be  out of phase with each other. [64,65]

Conclusion
We have shown that in Eu x TaO 3 the low energy carriers are quasi-2D due to the TCF induced by the fractional perovskite structure.Furthermore, in the FFM, TRS breaking regime, they have a spin polarization induced indirectly through an exchange interaction with the Eu 4f magnetic centers, allowing the spin to be easily controlled by an external magnetic field without disturbing their electronic properties.The TCF also results in additional in-plane spin texture with an (anti-)monopole form, which arises from a subtle coupling between two Rashba fields.This makes the system an ideal platform to realize quasi-2DEGs for applications in cutting-edge spintronic devices.Several device applications were proposed, including a spin polarizer, spin transistor, and interfacial spin valve.
In considering transport phenomena, we showed this quasi-2D behavior leads to divergent SdH oscillations and an oscillating Seebeck effect, both of which would be good experimental signatures for said behavior.Furthermore, an intrinsic AHE with a non-monotonic dependence on n c was demonstrated in the Eu 0.5 TaO 3 stoichiometry, giving the possibility of a tunable AHE.The origin of this AHE is still not fully explained and should be a key area of future research.Our findings serve as an example  of a broader idea that bulk 3D crystals can exhibit large confinement of charge carriers, due to inherent crystal anisotropy, realizing quasi-2DEGs.

Experimental Section
Density Functional Theory Calculations: Electronic structure calculations were performed within DFT using the Perdew-Burke-Ernzerhof exchange-correlation functional [66] and projector augmented wave pseudopotentials, as implemented in the VASP program. [67,68]Relativistic effects, including spin-orbit coupling, were fully included.The energy cut-off for the plane wave basis set was chosen to be 400 eV.The strong correlation effects arising from the localized Eu 4f orbital were treated within the Dudarev et al.'s DFT+U approach [69] by adding an additional onsite Hubbard U term and exchange correction J to this orbital, amounting to an effective U eff = U − J = 7 eV.[72] To account for the partial occupancy of the Eu site in Eu 1/3 TaO 3 , two separate DFT calculations were performed using an adapted version of the experimental structural parameters taken from Ref. [35].In one calculation, the Eu site was completely removed from the lattice, i.e., x = 0, while in the other, the occupancy of the Eu site was set to one, i.e., x = 0.5.In both cases, the corresponding Brillouin zone was sampled by a 15 × 15 × 7 k-mesh.The respective DFT Hamiltonians were then downfolded into two 70-band tight-binding models  0 and  0.5 , using maximally localized Wannier functions (MLWFs) [73] with Eu 4f, Ta 5d and O 2p orbitals as the projection centers.Finally, the Hamiltonian of Eu 1/3 TaO 3 ,  1∕3 , was constructed through a linear interpolation as  1∕3 = 1∕3 0 + 2∕3 0.5 (also, see Section III, Supporting Information ).
To calculate the Berry curvature and anomalous Hall conductivity, we constructed a smaller 12-band tight-binding Hamiltonian, describing the lowest 12 conduction bands of the x = 0.5 system using MLWFs with Ta 5d orbitals taken as the projection centers.
Analytical Tight-Binding Model: The Parameters for the analytical model used in Section 5, to reproduce the dispersion and spin textures of Eu 0.5 TaO 3 , are detailed in Table 1.
Thermoelectric Calculations: Calculation of the thermoelectric properties was performed using semi-classical Boltzmann theory [26,27] as implemented in the BoltzTraP2 program. [53,74]This was performed over a 100 × 100 × 50 k-mesh, with a 30 point temperature mesh, using the 12 × 12 Wannier tight binding model.

Figure 1 .
Figure 1.Schematic overview of Eu x TaO 3 system.a) General cubic perovskite crystal structure ABO 3 .b) Eu x TaO3 crystal structure, with 'local' polarization of each half unit cell, depicted schematically.c,d) Band diagrams of d-t 2g conduction orbitals in oxide cubic perovskites showing competition between SOC and exchange terms.c) SOC dominates.d) Exchange dominates.e) Band diagram of both sets of Ta t 2g orbitals in Eu x TaO 3 , with RKKY exchange interaction between t 2g and Eu 4f states.Yellow highlights region of d xy bands, pink highlights d xz , d yz .f) Schematic of BZ folding without RKKY.g) BZ folding with RKKY.h) Real-space visualization of Ta d xy orbitals in Eu x TaO 3 crystal, with spin direction added, to show collinear spin behavior.

1 √ 2 (
. Such an exchange splitting normally leads to either the spin-up or down mediator bands being lowered in energy.However, the interplay between the BZF and RKKY leads to an atypical ordering, ↑, ↓, ↓, ↑, for the low energy d xy states, with an additional shift 'offcenter' for the two spin-down bands as shown in Figure1g.At Γ, our d xy eigenstates are linear combinations of each Ta site |±⟩ = |1⟩ ± |2⟩), such that the RKKY exchange acts as a perturbation with the form:

Figure 2 .
Figure 2. DFT calculated electronic structure of Eu x TaO 3 .a,b) Spin-projected band structure along high symmetry lines for: a) Eu 1/3 TaO 3 b) Eu 0.5 TaO 3 .Inset in (a) shows schematic BZ with high symmetry lines marked in red.c-g) focuses on calculations for Eu 0.5 TaO 3 .c) Close-up of d xy band-structure for in plane dispersion.d) 3D visualization of low energy bands to highlight quasi-2D nature.Spin-up is shown as red/orange tones, and spin-down is shown as blue tones.e) The cross-sectional area/SdH frequency of the first Fermi-pocket as a function of polar angle,  CA , at two carrier concentrations, n c = 4 × 10 18 cm −3 , 1 × 10 20 cm −3 .f) Cross-sectional area/SdH frequency of d xy Fermi-pockets at  CA = 0 as a function of n c , spin-up shown as red/orange tones, spin-down shown as blue tones.Inset shows Fermi-pockets for all d xy bands at n c = 7 × 10 20 cm −3 .Pockets are shifted to be centered on the origin, as the upper two bands have minima away from the Γ point.Pockets 1-3 are cut along the 'top' and 'bottom' as they connect across the BZ boundary.g) Charge density plot of the low energy d xy carriers, with n c ≈ 5 × 10 19 cm −3 (E F ≈ 0.1 eV).

Figure 3 .
Figure 3. Rashba fields and spin texture in Eu 0.5 TaO 3 conduction bands.a) Schematic of k-space SO fields, B (SOC) i

Figure 4 .
Figure 4. Berry curvature, and AHE in Eu x TaO 3 .a) Combination of low energy band-structure and Ω z calculated in the k z = 0 plane.In the positive k y half, the planar dispersion of d xy bands is shown, with spin up shown as red/orange tones and spin down as blue tones.Ω z is shown at five example energy cuts, with positive(negative) Ω z shown as red(blue), and with the 2D Fermi surfaces of each band at these energies shown as black lines.b) Schematic of AHE in the bulk system, where one can view the total AHE as arising from the sum of each layer of Eu x TaO 3 .Electron carriers are color coded on Ω z sign, which have opposite transverse velocities.c)  AHE xy calculated as a function of carrier concentration, n c , for Eu 0.5 TaO 3 .

Figure 5 .
Figure 5. Thermoelectric properties of Eu x TaO 3 .a) Schematic of a thermal device for spin-polarized Seebeck effect and Nernst effect.Hot and cold contacts added to either end of the Eu x TaO 3 device cause a temperature gradient ∇T.V N is the resultant Nernst voltage, and V s the resultant Seebeck voltage from thermal diffusion.b-c) S i of Eu 0.5 TaO 3 calculated from Wannier interpolated DFT as a function of , with temperature color coding.b) i = xx.c) i = zz.DOS added for reference with oscillations in S zz .d) Analytic expression for S zz , Equation 14, with toy model DOS added for reference.

Figure 6 .
Figure 6.Spintronic device applications for Eu x TaO 3 .a) Spin polarizer, generic non-spin polarized current enters material and is spin-polarized along B. b) Inter-facial spin valve, Eu x TaO 3 has itinerant FM contacts at either end, with a thin non-magnetic layer separating them.As transport across the FM-NM-Eu x TaO 3 junction leads to spin scattering, as shown in the inset, adjusting the polar angle of the applied field,, and thus adjusting the relative angle between the magnetizations, manipulates the resistance across the device.c) Spin transistor, current flows around the device from A to C and the orientation of B is parameterized by an azimuthal angle  where  = 0∕ =  2 gives B parallel to the current direction from A to B/B to C.

Table 1 .
Values of analytic model parameters used to recreate DFT bands and spin textures.