Introduction

Electronic devices based on conventional semiconductors are facing the limit of evolution due to the huge consumption of energy, generation of excessive heat, monotonic functionality, and inextensible miniaturization. “Oxide-spintronics”, based on a broad spectrum of electrical and magneto-electrical properties originating from the interplay between orbital and spin degrees of freedom combined with the strong correlation nature of electrons in transition-metal oxides, is considered as one of the promising solutions1,2,3,4,5,6,7,8. For the development of high-performance spintronic devices using oxide materials, an essential ingredient is a good spin channel, where the electron’s spin can be manipulated at will within an infinitesimal energy and length scale.

Recently, observable effects of spin-orbit coupling (SOC) have been reported in various material systems, for example, topological insulators9,10, two-dimensional materials11, and some oxide materials such as LaAlO3/SrTiO3 (LAO/STO) heterostructure12,13 and pyrochlore iridates (A2Ir2O7, where A = yttrium and lanthanide element)14,15,16. In the former, a tunable and moderate strength SOC was reported while, in the latter, electrical and magnetic properties were reported to be dominated by the strong SOC. More recently, we found that a heavily Nb-doped SrTiO3 (SrNb0.2Ti0.8O3) epitaxial film grown on STO showed the strong SOC and the high carrier mobility – ideal characteristics for a good spin channel – which resulted in a large linear magnetoresistance (LMR)17. In this study, we have performed a systematic study on the magnetotransport properties of SrNb x Ti1−xO3 films with varying Nb concentration (nNb) in the range of 2~20 at. % to find that the magnetotransport property is dominated by three-dimensional weak antilocalization (WAL)18,19 and that the SOC strength is proportional to nNb. Furthermore, a few intriguing properties are also found, for example, non-monotonic temperature dependence of the SOC and anisotropy of MR in the intermediate nNb region.

Nb-concentration split SrTiO3 (Nb:STO) films were fabricated at 700 °C in 10−5 Torr of oxygen partial pressure using pulsed laser epitaxy (PLE). Laser (248 nm; IPEX 864, Lightmachinery, Nepean, Canada) fluence of 1.5 J/cm2 and repetition rate of 5 Hz was used. SrTiO3 and Sr(Nb0.2Ti0.8)O3 targets were used to systematically modify the Nb:STO concentration, by controlling the ablation ratio between the two targets within a unit cell thickness20. The advantage of using co-ablation of two targets instead of using Nb:STO solid solution target with different nNb was to avoid additional complexity and quality issues due to preparing distinctive targets. The thickness of the Nb:STO thin films was 16 ± 1 nm, as measured by X-ray reflectometry (XRR). The atomic structure and epitaxy relation of the thin films were characterized using high-resolution X-ray diffraction (XRD) (Rigaku, Smartlab) (in the Supplementary Information, Figure S1).

For the measurement of magnetotransport properties, a specimen was cut into 2(width) × 5(length) mm2 and six electrical contacts were formed by indium. After wiring, the sample was inserted in a commercial cryogen-free cryostat (CMag Vari-9, Cryomagnetics Inc.) and the resistance was measured using a Source-Measure Unit (Keithley 2612A) and a nano-voltmeter (Keithley 2182).

For DFT calculation, Vienna Ab initio Simulation Package (VASP)21 with PAW potentials. We employ HSE06 hybrid functional for the exchange-correlation functional and the energy cutoff of the plane-wave basis set to 450 eV for all calculation. For structural optimization, the 4 × 4 × 4 Monkhorst-pack k-point sampling is used for primitive cell of SrTiO3 and SrNbO3 in cubic perovskite structure. The calculated lattice parameter of SrTiO3 and SrNbO3 are 3.883 Å and 4.017 Å, respectively, which are consistent with experimental values22,23 with error less than 1%. We use the lattice parameter of 3.905 Å for the strained SrNbO3 that corresponds to the experimental lattice of SrTiO3. To calculate the density of states (DOS) of unstrained and strained SrNbO3, we use 6 × 6 × 6 Monkhorst-pack k-point sampling and tetrahedron method for Brillouin-zone integration. For the calculation of Nb:STO, we use the 130-atom supercell that one Ti atom is substituted by Nb which correspond to the doping concentration of 3.8 at %. For k-point sampling of defect cell, we use Γ-only k-point which corresponds to same k-point density compared to primitive cell. For DOS calculation of Nb:STO, we also use Γ-only k-point and Gaussian smearing for Brillouin-zone integration to reduce the computational cost.

Figure 1(a)~(e) show the magnetoresistance (\(MR(B)=\frac{R(B)-R(0)}{R(0)}\times 100\) (%)) of Nb:STO films (nNb = 2, 5, 10, 15, 20 at. %) as a function of the applied magnetic field (B) at various temperature (T), where B is applied perpendicular to the film surface. A few intriguing features can be found. First, at low temperature (T < 20 K), the Nb:STO films show a sharp and linear MR vs. B curve at low field region (|B| < ~2 T) independent of nNb. This behavior is consistent with the LMR shown in heavily-doped Sr(Nb0.2Ti0.8)O3 epitaxial films presented in our previous report17. Second, a lightly-doped Sr(Nb0.02Ti0.98)O3 film has a peculiar T-dependence of the MR vs. B curve. In detail, in the temperature range from 7.5 to 15 K, MR decreases with increasing the strength of B in the high field region (|B| > 2 T) leading to a high negative MR of about −70 % at 10 K and 9 T. In contrast, the other samples do not show such a negative MR although they all show an anomalous behavior (minimum MR or change of slope) around 10 K as shown in Fig. 1(f). Similar behaviors, including the LMR, were observed in top-gated LaAlO3/SrTiO3 devices, where the sign change of MR was induced by electric field12,24. The authors explained those phenomena in terms of the change of the strength of WAL effect resulting from the change in the Rashba SOC constant by the top-gate bias25. The strength of WAL effect also depends on T, which is given by the sum of spin-orbit, spin-flip, and inelastic scatterings18,19. Therefore, the observed complex T-dependence of the MR vs. B curve can be associated with T-dependence of the WAL effect. In contrast to LaAlO3/SrTiO3 where the conduction is confined in the atomically-thin two dimensional (2D) channel, Nb:STO films are assumed to have three dimensional (3D) transport because their thickness is ~16 nm comparable to the mean free path. The 3D WAL effect is described by Fukuyama-Hoshino (F-H) model26,27,28, where MR is given by the following equations (1) and (2).

$$\frac{\Delta {\rho }_{WAL}(B)}{\rho {(0)}^{2}}=\frac{{e}^{2}}{2{\pi }^{2}h}\sqrt{\frac{2\pi eB}{h}}[\frac{1}{2}{f}_{3}(\frac{B}{{B}_{\varphi }})-\frac{3}{2}{f}_{3}(\frac{B}{{B}_{2}})]$$
(1)
$${f}_{3}(y)=\sum _{n=0}^{\infty }[2{(n+1+\frac{1}{y})}^{\frac{1}{2}}-2{(n+\frac{1}{y})}^{\frac{1}{2}}-{(n+\frac{1}{2}+\frac{1}{y})}^{-\frac{1}{2}}]$$
(2)
Figure 1
figure 1

(a~e) Magnetoresistance (MR) vs. B curves of Nb:STO epitaxial films with varying Nb concentration (nNb) at various temperatures. The color code of all plots is same as that in (c). (f) Temperature dependence of MR at 9 T of Nb:STO films.

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In Eq. (1), e and h are the charge of an electron and Planck’s constant, respectively. And, \({B}_{\varphi }={B}_{i}+2{B}_{S}\) and \({B}_{2}={B}_{i}+\frac{2}{3}{B}_{S}+\frac{4}{3}{B}_{SO}\), where \({B}_{x}=\frac{\hslash }{4eD{\tau }_{x}}\) and τ x s (where x = i, S, and SO) are the characteristic time for the inelastic, spin-flip, and spin-orbit scattering, respectively. And, \(\hslash \) and D are the reduced Planck’s constant (=h/2π) and diffusion coefficient of electron in Nb:STO, respectively.

In Fig. 2(a), T-dependent MR vs. B curves of a Nb:STO film (nNb = 2 at. %) are replotted along with the respective fitting curves, which are given by the following equation (Eq. (3))29,30.

$$MR(B,T)=C\ast R(0,T)\ast \frac{{e}^{2}}{2{\pi }^{2}h}\sqrt{\frac{2\pi eB}{h}}[\frac{1}{2}{f}_{3}(\frac{B}{{B}_{\varphi }(T)})-\frac{3}{2}{f}_{3}(\frac{B}{{B}_{2}(T)})]+k{(\frac{B}{R(0,T)})}^{2}$$
(3)
Figure 2
figure 2

(a) MR vs. B curves of a Nb:STO film (nNb = 2 at. %) measured at various temperatures (symbol) together with fitting curves obtained by Eq. (3) (solid lines, see text). (b) Temperature dependence of the phase-coherence field (B ϕ , black solid square), the spin-orbit field (BSO, red solid circle), and the Rashba SOC constant (αR, green solid triangle). A black dashed line is a linear fit to B ϕ (T) curve for T above 7.5 K in the log-log scale, showing a power-law dependence of B ϕ (T). A green dashed line is a fitting curve to the experimental αR(T) assuming a quadratic dependence of Landé g-factor (see text).

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The first and the second term originate from the WAL and the classical orbital motion, respectively, where the latter is assumed to follow the Kohler’s rule31 with k being a constant independent of T. C is a constant which depends only on the geometry of the device and R(0, T) is the resistance under zero magnetic field at various T. Consequently, two fitting parameters, B ϕ and B2, are used except for the curve at 1.8 K, at which k is also used as a fitting parameter and kept constant for the other temperatures. As shown in Fig. 2(a), it is found that Eq. (3) well reproduces the measured MR(B) curves in the investigated temperature range (1.8~30 K) even though the observed T-dependence is rather complex. The obtained values of B ϕ and BSO are plotted as a function of T in Fig. 2(b), which are in the similar range to those of top-gated LaAlO3/SrTiO3 devices, 0.1 T < BSO < 10 T depending on the top-gate bias and 0.05 T < B ϕ  < 0.4 T at 1.5 K12,24. Above 7.5 K, B ϕ increases with T faster than BSO indicating that the inelastic scattering by phonon is as important as SOC above ~15 K. T-dependence of B ϕ is shown to follow a power-law behavior (B ϕ ~T p) for T > 7.5 K as shown in the linear fit in Fig. 2(b). According to ref.32, p is expected to be 3/2 for electron-electron (e-e) scattering, 2 for inelastic scattering by transverse phonon, and 3 for inelastic scattering for longitudinal phonon, respectively. In Fig. 2(b), p is estimated to be ~2.38 indicating that, above 7.5 K, the phase relaxation is dominated by the phonon scattering with mixed contributions from the transverse and longitudinal phonons.

Below 7.5 K, BSO is much higher than B ϕ , indicating that SOC dominates the scattering in that temperature range. An intriguing feature in Fig. 2(b) is the appearance of a minimum BSO at 7.5 K, which reminds of the aforementioned anomalous behavior of MR around 10 K presented in Fig. 1(f). Therefore, it seems to imply that the T-dependence of BSO is closely related to the “10 K anomaly” of MR.

As a possible scenario to explain the T-dependence of BSO, we consider the T-dependent Rashba constant (αR) originating from the T-dependence of Landé g-factor (g). In detail, assuming that the spin relaxation is dominated by the D’Yakonov-Perel (D-P) mechanism, BSO can be expressed in terms of αR by \({B}_{SO}=\frac{{\alpha }_{R}^{2}{m}^{\ast 2}}{e{\hslash }^{3}}\), where m* is the effective mass of an electron27,33. Figure 2(b) also shows the calculated αR(T) (green triangle symbol) using the above relation and the known value of m*(=7.5m0, m0 = the rest mass of an electron) in Nb:STO, as m* = (7.3~7.7)m0 was previously reported34. In addition, neglecting the effect of the band structure on SOC, αR is expressed in terms of g as \({\alpha }_{R}=g(1-g)\frac{\pi e{\hslash }^{2}\varepsilon }{4{m}^{2}{c}^{2}}\), where ε and c are the electric field generated by the asymmetric structure and the speed of light, respectively35. In previous studies36,37,38,39, it has been shown that g depends on T, implying that both αR and BSO should also depend on T. With the specific form of g(T) depending on the material, the observed αR(T) is well reproduced by an assumption of a quadratic T-dependence of g (g(T) = g0 + g1T + g2T 2) similar to CdTe case36, as shown by the fitting curve (green dashed line in Fig. 2(b)). It implies that the anomalous behavior of MR(T) and αR(T) is associated with the T-dependence of g although the specific form of g(T) of Nb:STO needs to be measured for confirmation. As another possibility, it might be associated with the unknown phase of heavily-doped Nb:STO epitaxial films under strain. In our previous work, it was found that a heavily doped Nb:STO film (nNb = 20 at. %) showed an upturn in resistivity and magnetic moment below 10 K as well as a local minimum in Hall carrier density around 10 K17. Further studies are needed to clarify the origin of the phenomenon.

With a plausible explanation of the MR vs. B behavior using the above model (Eq. (3)), MR vs. B curves of Nb:STO films with varying nNb at 1.8 K are plotted in Fig. 3(a) to investigate the nNb-dependence of SOC. The respective fitting curves are also plotted, providing the information of the dependence of B ϕ and BSO on nNb as shown in Fig. 3(b). For all doping concentrations, it is found that BSO is much higher than B ϕ indicating that SOC is the dominant scattering mechanism in Nb:STO epitaxial films in the low temperature region. Except at nNb = 20 at. %, it is found that BSO linearly increases with nNb consistent with an expectation, since SOC is expected to be higher for a compound composed of elements with the higher atomic number (Z) elements (ZNb = 41 compared to ZTi = 22). As for the origin of a decrease in BSO and a drastic reduction in MR at nNb = 20 at. %, we suggest that a new band might participate in the carrier transport as the Fermi level (EF) increases over a threshold. A few evidences supporting this explanation are presented in the following.

Figure 3
figure 3

(a) MR vs. B curves of a Nb:STO films with varying nNb measured at 1.8 K (symbol) together with fitting curves obtained by Eq. (3) (solid lines, see text). (b) B ϕ (black solid square) and BSO (blue open circle) as a function of nNb.

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In Fig. 4(a)~(e), MR vs. B curves of Nb:STO films at 1.8 K are presented with B oriented out-of-plane (MRperp, 0 deg.) and in-plane (MRpara.,90 deg.), respectively. For the in-plane MR, the current is still applied perpendicular to B. The difference of MR (ΔMR = MRperp-MRpara at 9 T) is plotted as a function of nNb in Fig. 4(f). It is interesting to note that a large anisotropy of MR is found for nNb = 5 at. % and nNb = 10 at. % samples whereas a negligibly small anisotropy is observed for the others. This observation seems to imply a change in the shape of the conduction band depending on nNb. The band structure of nNb = 2, 15, and 20 at. % samples should be symmetric along k x (or k y ) and k z direction in k-space while that of nNb = 5 and 10 at. % samples is expected to be asymmetric. This change in the shape of the conduction band depending on nNb might be associated with the crystal field splitting of Nb d-orbitals. It was known that the conduction band of Nb:STO is composed of degenerate Ti d-orbitals and Nb d-orbitals40,41,42. Since the Nb:STO film epitaxially grown on STO is strained as shown in XRD data (see Figure S1 in the Supplementary Information), d-orbitals of Nb atom are exerted by the crystal field possibly resulting in the splitting of energy levels. Consequently, the orbital-resolved conduction band structure of a Nb:STO film is expected to be similar to that seen in Fig. 5(a). For low nNb, if EF is located within the overlap between degenerate Ti-d bands and Nb-\({d}_{{x}^{2}-{y}^{2}}\) band, the transport should be symmetric along k x (or k y ) and k z direction. Since Nb substituting Ti produces electron carriers43,44, EF increases with nNb. As nNb increases, EF enters the region where the conduction band is contributed mainly from Nb-\({d}_{{x}^{2}-{y}^{2}}\) band resulting in 2D transport nature. As EF increases further, Nb-\({d}_{{z}^{2}}\) starts to contribute to the conduction band again making the conduction along k z direction significant. This picture can naturally explain the change of the MR anisotropy depending on nNb.

Figure 4
figure 4

(a~e) MR vs. B curves of Nb:STO epitaxial films at 1.8 K under the out-of-plane (0 deg., black) and the in-plane (90 deg., red) magnetic field. (f) Difference in MR (ΔMR = MRperp-MRpara) at 9 T as a function of nNb.

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Figure 5
figure 5

(a) A schematic illustration of the orbital-resolved band structure of Nb-doped SrTiO3 epitaxial film grown on SrTiO3. (b) Orbital-resolved band structure of unstrained SrNbO3 film (left panel, a = 4.017 Å) and strained SrNbO3 film with the in-plane lattice constant same as SrTiO3 (right panel, a = 3.905 Å). Dashed-line shows the position of valence band maximum. (c) Orbital-resolved band structure of Sr(NbxTi1−x)O3 for x = 3.8 at % and 15.2 at %. The EF of each model is shown as dashed lines.

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To verify the assumed band structure, we have performed the orbital-resolved band calculation using density functional theory (DFT), which is shown in Fig. 5(b). For simplicity, we have calculated the band structure of SrNbO3, which can provide qualitative but essential information on the change of the band structure on doping Nb in SrTiO3. The left and right panels show the partial density of states (PDOS) for an unstrained SrNbO3 (a = 4.017 Å)22,45 and a strained film epitaxially grown on SrTiO3 (a = 3.905 Å)23, respectively. Comparing with the unstrained SrNbO3, it is found that Nb-\({d}_{{x}^{2}-{y}^{2}}\) band moves to lower energy whereas Nb-\({d}_{{z}^{2}}\) band moves to higher energy consistent with our expectation described above. To compare the relative position of Nb-d in Nb:STO, we also calculate the PDOS of Nb:STO (nNb = 3.8 at. % and 15.2 at %) as shown in Fig. 5(c). It is shown that the position of Nb-d composing the conduction band is right above the position of Ti-d as we assumed in Fig. 5(a). Comparing results of different nNb, it is also shown that EF increases with nNb. Therefore, we believe that the crystal-field splitting of the Nb d-bands in Nb:STO films is associated with the observed change in the anisotropy of MR. As alternative explanation, the interaction of d-electrons under a strong SOC should be addressed because it was reported to be important in material systems including transition metal elements46.

In summary, we have investigated magnetotransport properties of epitaxial Nb:STO thin films to find the effect of Nb on the SOC strength. The magnetotransport properties are well described by the F-H model describing 3D WAL and the estimated SOC strength is found to be proportional to nNb. Furthermore, a few intriguing phenomena are presented, non-monotonic temperature dependence of the SOC strength and anisotropy of MR in the intermediate nNb region, which might be explained in terms of the temperature dependence of the g-factor and the change in band structure with nNb. This result demonstrates that the epitaxial Nb:STO film basically possesses the strong SOC, whose strength can be controlled by nNb. We believe that these results not only provide a material platform to study the interplay between SOC and strong correlation, which is an interesting subject in condensed-matter physics, but also enable us to design advanced spintronic devices and other novel functional devices combining the strong SOC and other versatile functionalities of strongly-correlated oxides.