Spin to charge conversion at Rashba-split SrTiO$_3$ interfaces from resonant tunneling

Spin-charge interconversion is a very active direction in spintronics. Yet, the complex behaviour of some of the most promising systems such as SrTiO$_3$ (STO) interfaces is not fully understood. Here, on the basis of a 6-band $\boldsymbol{k.p}$ method combined with spin-resolved scattering theory, we give a theoretical demonstration of transverse spin-charge interconversion physics in STO Rashba interfaces. Calculations involve injection of spin current from a ferromagnetic contact by resonant tunneling into the native Rashba-split resonant levels of the STO triangular quantum well. We compute an asymmetric tunneling electronic transmission yielding a transverse charge current flowing in plane, with a dependence with gate voltage in a very good agreement with existing experimental data.

However, the fundamentals and understanding SCC phenomena in 2DEGs oxide and heterostructures as provided in spin-pumping experiments is still in its infancy owing to the particular geometry measurements. In spin-pumping experiments performed on these systems 3,5 , a pure spin current is generated in a top magnetic contact of a tunnel devices, spin-current propagating in the confined STO layer constituting a quantum well (QW) before giving rise to spin-charge conversion via assisted spin-orbit interactions (SOI). This makes largely not suitable the application of a conventional theory and modelling based on the linear Kubo's approach applied to the STO host matrix; unlike the issue of the reciprocal charge-to-spin conversion as required e. g. for the spin-torque problem.
In this article, we propose a modelling of SCC in the k.p framework giving rise to equivalent IREE phenomena in a STO electron gas (2DEG) confined in an oxide triangular quantum well (TQW) considering the specific structure of a NiFe/LAO/STO magnetic tunnel junction. We expose new theoretical insights on these phenomena from a quantum resonant tunneling point of view taking into account the symmetry breaking properties of Kramer's pair conjugates in STO QWs. We combine a k.p method and a scattering approach to describe spin-orbit assisted transport and SCC [24][25][26][27] and demonstrate the occurrence of a lateral charge current with a specific gate dependence in very good agreement with experi-ments. We explain recent experimental data 3,5 and go beyond existing tight-binding (TB) models 5,28-30 for tunnel structures, demonstrating the robustness of our method.
The paper is organized as follow. Section II focuses on the k.p model description and the calculations of the electronic band structure of STO in the presence of a Rashba surface potential. We compare our modelling to recent tight-binding (TB) results involving also the orbital structure. Section III is devoted to the calcualtion details of the resonant tunneling through Rashba states within STO quantum wells (QWs) together with their dependence on the gate voltage or electric field along the confinement direction (direction normal to interfaces). Section IV introduces the asymmetry of the tunneling transmission and discusses the associated spin-charge conversion responsible for the transverse charge current.

II. STO BAND STRUCTURE WITH RASHBA INTERACTIONS.
A. Modelling.
We start by describing the k.p electronic band structure of the STO host material. TB 5,28-32 as well as first principle theory 33 have been extensively used to model the Rashba properties of STO 2DEGs. Nevertheless, although TB may correctly describe the Rashba energy splitting, it does not explicity deal with tunnelling structures. In addition, theories of SCC in oxide systems were limited, up to now, to in-plane charge current 17,18,30 whereas the spin-pumping technique, used in STO 3,5 as in semiconductor-based junctions 34 , implies a tunneling current normal to the layers. To this end, we follow the k.p approach of In the basis set of the order: 3 2 , 3 2 , , the k.p Hamiltonian (Eq. [1]) can be re-written as: with k ± = k x ± ik y . We chose L = 0.6164 eVÅ 2 , M = 9.73 eVÅ 2 , N = −1.615 eVÅ 2 , ∆ SO = 28.5 meV, ∆ T = 2.1 meV, obtained from a fitting procedure to Density Functional Theory (DFT) 35,37 . One may possibly add an additional Bychkov-Rashba extra-term in the SrTiO 3 layer of the form: H R = α R (σ ×p) .ẑ as proposed in Ref. 40 where α R is the Rashba ,ẑ is unit vector along z,p is the momentum operator andσ the Pauli matrices, one obtains the different Fermi surfaces for both the majority ↑ and minority spin ↓ channels.
B. STO Band structure involving Rashba potentials.
We now give a rapid description of the STO spin-resolved band structure in the presence of Rashba interactions where we fix now α R = 15 meV.Å. We compare our results to the ones of recent tight-binding (TB) treatment 30 . Without magnetism, the resulting electronic bands of STO subject to an additional Rashba interaction are displayed on Fig. 1. Such additional Rashba term introduced is equivalent, in spirit, to the had-hoc Rashba hopping surface term proposed in TB 28,30 . The Fermi energy F is chosen here to lie 0.04 eV above the bottom of the conduction band. The different bands and Fermi surfaces corresponds to respective ↑ (Fig. 1a, b and c) and ↓ spin channels (Fig. 1d, e and f), calculated from the above Hamiltonian. Figs  spin-channels in the tunneling process depending on the electron elastic energy.
Unlike the aforementioned calculations of the STO band structure II B involving an explicit Rashba term (Ĥ R = 0), the peculiarity of our tunneling approach will be the appearance of equivalent features from the native triangular potential withĤ R = 0, assumption that we will consider henceforth. We will also simplify the self-consistent potential in STO considering a triangular form of the confined potential 35,36,41,42  The wavefunction of the system that we have to solve is a solution of the Schrödinger equation: that we have to solve without considering additional interface Rashba potential. At the vicinity of a given Rashba quantized level (or resonance n) at energy σ n in the TQW, the band-selected transmission coefficient T (n,σ) k (n here is the band index and σ denotes explicitly the spin) are of the form 45 : where Γ σ L k and Γ R k are respectively the spin-dependent energy broadening from the coupling to the FM and the unpolarized energy broadening towards the STO reservoir. Γ σ L is sensitive to the spin eigenvalue of n (respectively parallel or antiparallel to x) whereas Γ R is not. Γ σ L will differ between the two split Rashba states for k alongẑ ×m in agreement with the symmetry rule required for the IEE process. Γ R generally larger Γ L depicts the electron lifetime τ n = Γ L out of the 2DEG into STO. The voltage-integrated tunneling current on a given resonant level isΓ It results that the electronic transmission across a quantized Rashba state in STO and its hierarchy relative to their spin orientation may depends on the relative ratio Γ R Γ L so that, unlike conventional tunneling, a resonant transmission may invert the apparent sign of the Rashba interactions depending on the electron energy.
The calculation of the band-to-band selected transmission coefficients T   subbands (he1, he2, le1, he3, he4,...) at V g = 0 along [010] and [110]. One observes the appearance of an energy splitting for the first three subbands (he1, he2, le1) relative to the two spins. This indicates a Rashba splitting without the use of any additional surface Rashba terms. Fig. 2c displays the quantized wavefunction for a strict normal incidence (k = 0) for the first five levels and showing a strong hybridization between the he and so components; the le band remains pure. Indeed, a particular he-so mixture leads to a pure d xy character of smaller mass near Γ and able to minimize the quantization energy for the first two levels. The third level (le1) retains a pure d zx character 30 . Generally, at a finite wave vector k = 0, each quantized state will be a mixture of he, le and so subbands leading to a nonparabolic dispersion, especially at the vicinity of the anticrossing Lifshitz point.
Here, the Rashba spin splitting reaches its largest value where the cubic Rashba spin-orbit term dominates the linear contribution close to Γ 37 . Action of a gate-voltage: We now focus on the action of a gate V g = 0 leading to a modulation of F in STO. We model it by considering a change of the band offset ∆ B2 between STO and the barrier with F = ∆ B2 +∆ B3 w 2 . Fig. 3 displays the resonance n for the first three levels (he1, he2, le1) and different k = 0.01Å −1 (Fig. 3a), 0.05Å −1 (Fig. 3c), 0.1Å −1 (Fig. 3e)  extracted from our method and compared to the analytical theory given above with a good matching. On Fig. 3(b,d,f), the effective Rashba parameter α   The results obtained for a non-magnetic contact (∆ exc = 0) are displayed Figs. 4(a-f) showing the resonance transmission at each Rashba quantized levels R in agreement with our previous modelling. In each case, one may observe that the transmission map obey a perfect C4 v cubic symmetry shape for electrons tunneling in the Brillouin zone as expected from the structure lattice structure (C4v symmetry). For F = 1 meV/Å (Fig. 4a), tunneling involves a single he1 spin-split Rashba resonance and giving rise to the same transmission at equivalent points on the cubic Fermi surface. For F = 1.5 meV/Å (Fig. 4b), tunneling involves two Rashba resonance he1 and he2 Rashba-split bands. For F = 3 meV/Å (Fig. 4c), , he-so3 and he-so4 bands.
One observes now that, involving non-zero magnetism in the injector (∆ exc = 0.1 eV), the transmission does not obey a perfect cubic symmetry shape but differs for k = ±k y in the directionẑ ×m as expected (see e.g Fig. 5d). The transmission along ±k x remains symmetric for fixed k y . Concomitantly to a tunneling spin-current along z, a different transmission along +k y and −k y has to be associated with a transverse charge flow along y describing an IREE. In the case I the transmission is larger along +k y for the first he1 band which defines α R > 0. For the case II, the inner Rashba band with smaller k F gives an opposite sign to the transmission asymmetry and this should be linked to α (2) R < 0 as previously mentioned. Note that in this tunneling geometry, the inner Rashba band gives an overall larger conduction owing to the reduced incidence (smaller k F ) and then to an overall negative IRE effect. In the case III, the first Rashba band gives a standard Fermi 'cigar' shape to the resonant transmission however assigned to a very small selected ±k y asymmetry.
In the case III, additional Rashba split band at higher energy (le1, he3-so, he4-so) of a positive Rashba signature yields an overall positive IRE (J +ky > J −ky ). These conclusions are compatible with the description of Fig. 3 in particular in terms of the Rashba coupling sign. Such transmission difference vs. ±k y is characterized by the asymmetry parameter A as: Chirality effects of the same origin have been discussed in terms of tunneling anomalous Hall effects or TAHE in semiconductors 27,38 or in superconducting materials 39 . In this paper, we generalize these phenomena to resonant tunneling in oxide based systems.
A non zero A gives rise to a transverse charge current J c flowing in the QW plane with by our k.p method and θ k 1 the spin-current injection angle fromẑ. Note however that A ∝ P is proportional to the spin-current polarization injected from the contact, and then proportional to the spin-density (or spin accumulation P) injected in the STO TQW.
This gives the connection we are searching for between the transverse charge current and the spin-polarization in the STO, equivalent to IREE. In the present case, calculations give an an out-of-equilibrium spin-polarization P ≈ 0.6in the STO QW textcolorred(not shown) with the chosen parameters. Whereas the charge current J c remains constant within the tunneling structure, J s , of vectorial nature, may vary in the STO TQW owing to the Rashba field inducing a local spin-precession (the Rashba interaction does not commute with the spin operator). Fig. 6b displays A vs. F for an electron beam incoming with incident energy ε = 0 (kinetic energy = 50 meV) from the band bottom. Upon the increase of F above 1 meV/Å , A starts with small positive value of about +2% corresponding to a weak Rashba splitting of the first quantized level (I in Fig. 5a). Then, A decreases down to a large negative due to an opposite spin textures of the second level for F > 1.2 meV/Å. In this region, the asymmetry reaches a maximum absolute value of −20% (II) at the largest Rashba splitting at the vicinity of the Lifshitz point where the current spin-polarization within the TQW approaches unity. Then, when F is further increased, A changes its sign to become positive again (III) when the third and upper levels get involved in the tunneling.
This evolution of A perfectly reproduces in shape the trend of spin to charge conversion length, λ IREE , vs. bias V g reported in experiments of Vaz et al. 5 for ALOx/STO (Fig. 6a).