Single spin-polarised Fermi surface in SrTiO$_3$ thin films

The 2D electron gas (2DEG) formed at the surface of SrTiO$_3$(001) has attracted great interest because of its fascinating physical properties and promise as a novel electronic platform, but up to now has eluded a stable way to tune its properties. Using angle-resolved photoemission spectroscopy with and without spin resolution we here show that the band filling can be controlled by growing thin SrTiO$_3$ films on SrTiO$_3$(001) substrates with different Nb doping levels. This results in a single spin-polarised 2D Fermi surface in a superconducting system, which can be used as platform for Majorana physics. Based on our results it can furthermore be concluded that the 2DEG does not extend more than 3 unit cells into the film and that its properties are determined by the dielectric response of the system.

debate (14 , 15 ), but the lifting of the degeneracy for the d xy -derived states is clear both in experiment and theory.
The presence of both Rashba and Zeeman interactions leads to a spin gap ∆ at the surface Brillouin zone (SBZ) centre of STO (Fig. 1A), promising a wide range of functionalities depending on where the chemical potential is placed. An interesting scenario occurs when the chemical potential is placed inside the Zeeman gap, in which case the system would resemble the situation required to form Majorana fermions (16 ). Especially given the presence of superconductivity in both bulk STO (17 ) and STO interfaces (2 , 3 ), a whole new realm of physics opens up based on the interplay of magnetism, spin-orbit interaction, and superconductivity in a single material.
As indicated above, much of the functionality relies on being able to shift the chemical potential while not altering other properties. In semiconductors and oxides a common approach is to use a gate voltage, but for the LAO/STO interface this is known to also change the magnitude of the Rashba-type splitting (11 ). Furthermore, it is unclear whether such an approach causes any shift in the surface 2DEG of STO(001). Remarkably, apart from the 2DEG generated by Al deposition (18 ), all published ARPES studies on the surface 2DEG show an almost identical band filling irrespective of the bulk doping level and whether the sample is prepared by cleaving (8 , 9 ) or in-situ annealing (10 ), and of the amount of oxygen vacancies. This indicates that the origin of the 2DEG lies beyond a simple band bending picture, in which other ingredients such as structural distortions may also play a role. It should be noted that this behaviour is in stark contrast to anatase TiO 2 (19 ) or TiO 2 enriched surfaces of SrTiO 3 (20 ) in which the band filling can be tuned by the amount of oxygen vacancies.
In this work we follow a different and stable approach to tailor the chemical potential, namely the homoepitaxial growth of thin SrTiO 3 films on TiO 2 -terminated Nb-doped SrTiO 3 (001) substrates. Our main finding is that for films from 3 to at least 20 unit cells (u.c.) on 0.5 wt.% Nb-doped substrates the band filling is such that the Fermi level is exactly in the Zeeman gap at the SBZ centre, thus resulting in a single spin-polarised 2D Fermi surface (Fig. 1B). Further, we show that it is possible to change the position of the Fermi level by varying the amount of Nb dopants in the substrate. The similarities in the 2DEG found in films with different thicknesses indicate that the difference between single crystals and the films is not due to finite size effects, but due to a distinct dielectric response of the system. This in turn is expected to change the properties of the polaronic excitations and thus the band characteristics.
The ARPES data for the 3, 5, and 20 u.c. STO films on 0.5 wt% Nb-doped STO substrate shows marked differences to the data typically obtained for STO single crystals (8 -10 ), exhibiting only the d x y-derived band for all photon energies used (Fig. S1A) (19 , 20 ). It is also worth noting the intense incoherent spectral weight at the center of the SBZ that appears below 150 meV, also often observed for the 2DEG on STO crystals (8 -10 ). The fact that no differences are observed as a function of film thicknesses rules out the influence of finite size effects and constrains the spatial extension of this 2DEG from the top TiO 2 layer to 2 u.c. or less into the film. The 2D curvature data shows the shape of the d xy band that follows a free-electronlike dispersion. The parabola plotted in Figs. 2 F and I, with a band bottom of 80 meV and an effective mass of m * =0.74m e matches well the observed dispersion of the three samples.
The small increase in effective mass (m * =0.65m e for the 2DEG on STO single crystals (10 )) points to an altered bond angle, likely due to surface relaxation. Due to the absence of the heavy d xz /d yz bands, it cannot be confirmed whether the splitting between the latter and the d xy band has changed.
Apart from the small change in bond angle, the most striking difference of the 2DEG found on our STO/Nb:STO films when compared to the universal 2DEG found in cleaved and annealed STO crystals (8 -10 ) is the large reduction of band filling from 230 to 80 meV, which corresponds to a downwards shift of the Fermi level. As will be explained below, this shift can be associated with a change in the dielectric properties of the system. Optical and spectroscopic measurements have shown that the dielectric response of SrTiO 3 changes from single crystals to thin films (22 , 23 ). In addition, the in-gap states observed in our thin films are different from the ones observed in the bulk counterpart (Figs. 2M). For single crystal STO (dashed line) two in-gap states are observed whereas for the STO films the state at 2.5 eV binding energy has disappeared. Both in-gap states are known to originate from defects (24 , 25 ). Although the exact nature of the defects is still under debate, it is commonly accepted that they cause structural distortions, around which localised electronic states are formed, which can be regarded as small polarons. The different in-gap states clearly indicate a different defect structure, which results in an altered dielectric response of our PLD grown STO films when compared to STO crystals.
Of importance to the 2DEG characteristics is how the change in dielectric properties will influence the system's response to polar instabilities, given that undoped STO is an incipient ferroelectric material (26 ). In fact, it was shown that at the surface of cleaved KTaO 3 (001), metallization arises as a response to polar instabilities (27 ). Additionally, polarons have been proposed to drive the surface reconstruction in TiO 2 (28 ). More recently, the superconducting transition temperature in a FeSe monolayer grown on STO was shown to be raised after photoexcitation of the SrTiO 3 substrate, which was attributed to metastable polar distortions at the FeSe/SrTiO 3 interface (29 ). Further, the mechanism underlying the writing of conductive paths in the LAO/STO interface by a metallic tip is attributed to a local polar distortion at the interface (30 ). All of the aforementioned experimental evidences suggest that the properties of the 2DEG are intimately connected to the dielectric response of the surface layer.
Extrapolating from the point-like defects responsible for small polarons, the breaking of translation symmetry at the surface of the sample can be considered as a 2D defect, which will trigger polar lattice distortions. Charges originating from photoexcitation or from oxygen vacancies act in order to minimize the surface free energy, thus screening the polar instability and giving rise to the 2DEG. The 2DEG can then be regarded as a "frozen polaron", i.e. charges coupled to static polar lattice distortions. The polar response to the defects and the efficiency of the screening, and thus the details of the 2DEG, depend on the dielectric response of the system, which is different in the films compared to crystals.
Regardless of the exact origin of the change in band filling observed in our films, it result in only a single band crossing the Fermi level (Figs. 2). Considering the spin texture measured for the 2DEG on the surface of bulk STO (13 ), applying a rigid upwards energy shift would lead to a single spin-polarized Fermi surface. However, given the general differences of thin films, a change in spin splitting cannot be excluded and requires experimental verification.
In order to access this often elusive degree of freedom, we employed spin-resolved ARPES to study 10 u.c. STO/Nb:STO films, well within the previously studied 3-20 u.c. range. To be able to utilise the full potential of this material, it has to be possible to manipulate its properties by external means. In order to simulate the the change in carrier density by gating of the STO substrate, we have grown films on STO wafers where the amount of Nb dopants is changed by one order of magnitude; i.e. 10 u.c. STO film grown on a TiO 2 terminated, 0.05 wt% Nb-doped STO(001) substrate. Similarly, this film also hosts a purely 2D state (Fig. S4, A and B Fig. 3D, also measured at the Fermi level. The data clearly shows two bands that reverse sign at zone center and are oppositely polarised, suggesting that the inner band is the Rashba pair of the outer one. Thus, changing the substrate doping causes a change in band topology because the Fermi level moves above the Zeeman gap. That reducing the amount of Nb dopants in the substrate leads to a surface 2DEG with increased band filling also indicates that the 2DEG is directly affected by the dielectric response of STO. In turn, the latter depends on the metallicity of the environment (33 ).
In other words, the charges provided by the Nb dopants alter the boundary conditions and the dielectric response of the films, thus changing the band filling of the 2DEG. Note that this shift is in the opposite direction as would be expected by a simple model where the additional charges from Nb doping change the band filling. Interestingly, the film grown on the 0.5% substrate is such that the chemical potential lies inside the Zeeman gap, leading to a single spin-polarised band (Fig. 1B). In practice, this result means that local gating of the substrate can be used to create wires at whose tips zero bias anomalies should be observable by tunnelling experiments. Alternatively the wires can also be written by illuminating with an intense light source, by local defect doping, or by writing with a conductive tip (30 ). In combination with the superconducting properties of STO, this unifies all the ingredients for the formation of Majorana fermions in a single material without the need of external fields.     Between the main diffraction intensities, low intensity lines are visible (Fig. S1, D to F). It is interesting to note that the RHEED intensity actually increases with the growth of the first overlayer of STO on the substrate, indicating the high quality of the films.

Fermi surface, photon energy and polarisation dependence
The ARPES data for the 3, 5, and 20 u.c. STO films on 0.5 wt% Nb-doped STO substrate shows marked differences to the data typically obtained for STO single crystals [S2-S4]. In the data obtained for the films (Fig. S2, A to C) a circular Fermi surface is observed around the Γ points and there is no signature of the ellipsoidal d xz -and d yz -derived states for any of the three film thicknesses. In the photon energy scans of the 5 u.c. film (Fig. S2, D and E) the d xy -derived state shows a pure 2D character, also observed for the 3 and 20 u.c.
films as noted below. The observed intensity variations are due to the 3p-3d resonance at around hν = 45 eV (k z ≈ 3.8Å −1 ) and Bloch spectral enhancement at bulk Γ points [S5].
Furthermore, surface reconstructions are visible in all the Fermi surfaces with varying clarity.
The bands around the reconstructed Γ points are relatively featureless and (Fig. S2E), apart from a resonant enhancement do not show the same structure as a function of photon energy as the main Γ points. Therefore it appears that the reconstruction is not long range ordered and thus was not considered in the analysis.
The photon energy scans (i.e. dispersion along k z , calculated using an inner potential V 0 = 14.5 eV [S4]) measured with circularly polarised light for 20 u.c. film on 0.5 wt% Nb-doped STO(001) around Γ 00 and Γ 10 (Fig. S3, A and B), confirm the absence of heavy bands (d xz -and d yz -derived). That the 2DEG is d xy -derived can be seen by dependence of the Fermi surface with light polarisarion (linear vertical and horizontal, LV and LH), which in our experimental geometry is a signature of a band with d xy character (Fig. S3, C and D). For the 3 u.c. film there is no sign of the heavy bands at 85 eV (Fig. 2, A, D and G of main text), which is the typical energy at which these states are expected to appear in STO crystals [S4]. Hence an identical, purely d xy -derived 2D state, is observed for the 3, 5 and 20 u.c. films. Similarly, for the the 10 u.c. film on the 0.05% Nb:STO substrate, the photon energy dependence measured around Γ 00 and Γ 10 with circularly polarised light (S4) show no sign of the heavy bands.
Detailed characterisation of film grown on low doped substrate Fig. S5, A to C, shows ARPES data with hν=47 eV, LV-polarised light, obtained for a 10 u.c. STO film grown on a TiO 2 terminated, 0.05 wt% Nb-doped STO(001) substrate.
Despite also hosting a purely 2D d xy -derived state, its Fermi surface (Fig. S5A) Fig. S6, B and C. For both E 1,2 the main spin polarisation signal is along the x-direction, consistent with a Rashba-type splitting, while the measured out-of-plane spin polarisation is most likely due to spin interference during the photoemission process [S8] and |P y | ≤ 0.04. A well-established routine [S9] was used to simultaneously fit the total intensities and the spin polarisations along the sample x-, z-, and y-axis for E 1,2 , represented by green solid lines in Fig. S6, B and C. For E=E 1 (Fig. S6B) two peaks with opposite spin polarisation result from the analysis, while the spectra measured around E = E 2 (Fig. S6C) can only be fitted assuming four peaks with alternating spin polarisation. The fitted peak positions are marked as red dots in Fig. S5B and match well with the high-resolution spectrum.