Momentum-space spin texture induced by strain gradient in nominally centrosymmetric SrIrO3 films

ABSTRACT Spin texture in k-space is a consequence of spin splitting due to strong spin–orbit coupling and inversion symmetry breaking. It underlies fertile spin transport phenomena and is of crucial importance for spintronics. Here, we observe the spin texture in k-space of nominally centrosymmetric SrIrO3 grown on NdGaO3 (110) substrates, using non-linear magnetotransport measurements. We demonstrate that the spin texture is not only induced by the interface, which inherently breaks the inversion symmetry in strong spin–orbit coupled SrIrO3 films, but also originates from the film bulk. Structural analysis reveals that thicker SrIrO3 films exhibit a strain gradient, which could be considered as a continuous change in the lattice constant across different layers and breaks the inversion symmetry throughout the entire SrIrO3 films, giving rise to the spin texture in k-space. First-principles calculations reveal that the strain gradient creates large spin-splitting bands, inducing the spin texture with anisotropy, which is consistent with our experimental observations. Our results offer an efficient method for inducing the spin textures in k-space.


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Supplementary Text Figs.S1 to S19 Tables S1 References (1 to 22)

Materials and Methods
Sample preparation.A series of SrIrO3 with different thickness were grown by pulsed laser deposition (PLD) on NdGaO3 (110) substrates (KrF excimer laser, λ = 248 nm).
The base pressure was better than 2e -6 Pa.The growth was monitored by in-situ reflection high energy electron diffraction (RHEED) and the thickness was determined by counting the number of RHEED oscillations.The laser fluence and repetition rate were set as 2 J/cm 2 and 1 Hz, respectively.The oxygen partial pressure was optimized at 10 Pa and the growth temperature was 670 °C, respectively.To avoid possible degradation of the samples in atmosphere, amorphous STO capping layers were deposited at room temperature after growth and annealing 1 h at 10 Pa oxygen atmosphere.

RSM, XRD and Raman measurements.
Reciprocal space mapping (RSM) and X-ray diffraction (XRD) measurements were performed by using a Rigaku SmartLab (9 kW) X-ray diffractometer with a Ge (400) ×2 crystal monochromator.Raman spectra were collected by using a confocal Raman microscope (Horiba HR Evolution) with a 325 nm laser as the excitation source.The polarizer and analyzer are under parallel polarization geometry.
Device fabrication and Transport measurements.We use standard UV lithography to form a micro-fabricated Hall bar pattern with channel length 20 μm.The electrical transport properties were revealed by a Physical Property Measurement System (PPMS, Quantum Design).For the second-order signal measurements, an AC current Iω=I•sin(ωt) were applied by using Keithley 6221 current source while measuring the transverse AC harmonic Hall voltage and extracting the second harmonic resistance R 2ω from V 2ω by using lock-in amplifier (SR830, Stanford Research) at a frequency of 17.7 Hz.

STEM measurements.
For scanning transmission electron microscopy (STEM) image acquisition, the cross-sectional STEM specimens were thinned to less than ~30 μm first by using mechanical polishing and then by performing argon ion milling.The ion-beam milling was carried out using PIPSTM (Model 691, Gatan Inc.) with the accelerating voltage of 3.5 kV until a hole was made.Low voltage milling was performed with accelerating voltage of 0.3 kV to remove the surface amorphous layer and to minimize damage.High-resolution high-angle annular dark-field (HAADF) images were recorded at 300 kV using an aberration-corrected FEI Titan Themis G2 with the convergence semi-angle for imaging 30 mrad and the collection semi-angles snap 39 to 200 mrad.
First-principles calculations.We carried out the first-principles calculations within the framework of the density functional theory (DFT) using projector augmented wave (PAW) method [1,2], as implemented in the Vienna ab initio simulation package (VASP) [3,4].The generalized gradient approximation (GGA) of Perdew-Burke-Ernzerhof (PBE) type [5] was employed for the exchange-correlation potential.The kinetic energy cutoff for plane wave expansion was set to 500 eV.For SrIrO3 slab, the thickness of the vacuum layers along b directions was set to  20 Å.The Brillouin zone was sampled by Γ-centered Monkhorst-Pack method in the self-consistent process, with a 8×8×8 k-mesh for SrIrO3 bulk and a 8×1×8 k-mesh for SrIrO3 slab.The spin-orbit coupling (SOC) was taken into account in all calculations.

Extra electrical transport data of 200-u.c. SIO
Temperature dependent resistivity (ρ-T) curve shows clear semimetallic behavior with the resistivity around 1 mΩ•cm [Fig.S1(a)], which has the same order of magnitude as previous reports [6].The magnetoresistance (MR) measured between ±2 T at 10 K is positive and has the same range of variation as literature reports [7,8], which is shown in Fig. S1(b).Hall measurements [Fig.S1(c)] shows the n-type carrier character and the absence of hysteresis loop, implying the non-magnetic behavior of SIO.Besides, the charge carrier density and mobility can be extracted and shown in Fig. S1(d), which has the same order of magnitude with previous reports [7,8], further reflecting the semimetallic characters of our SIO sample.

NLMR results with current applied along [010]-direction
To investigate the role of crystal anisotropy on the k-space spin textures, we have grown Thirdly, we also measure the magnetic-field angular dependences of   2ω along three geometries at 10 K, as shown in Fig. S10 (f) -(h) and draw the same conclusion as shown in Fig. 3 (d) -(f).This is because when an electric field E is applied, nonlinear spin current is generated simultaneously at both the longitudinal and transverse directions due to the perpendicularly spin-momentum locking.In other words, the NLMR signals are only relevant to the measurement direction of second-harmonic voltage but irrelevant to current direction.
These results also reveal the spin texture is robust and plays a decisive role on NLMR signals.

Details of structure characterizations and strain gradient
Strain gradient is a fancy type of stress-strain relationship in epitaxial films with fullystrained near substrates and partially-relaxed or fully-relaxed far from substrates.This mechanism usually attracts attentions on flexoelectricity area [9][10][11][12][13] in transition metal oxides, such as HoMnO3 [14].Recently, scientists found that apart from flexoelectricity, strain gradient also affects other physical properties, such as tuning magnetic easy axis [15] and stabilizing room temperature skyrmions [16].
Strain gradient in SIO thin films with varied thickness was evaluated by XRD spectra , as shown in Fig. S15(a).The out-of-plane strain gradient can be estimated by zz,z = −/ based on Williamson-Hall plot, as shown in Fig. S15(b).Here,  is the out-of-plane penetration depth of the strain in SIO/NGO films, which is estimated to be ~100 nm, close to the value of LSMO [16]. is the out-of-plane inhomogeneous strain of each SIO/NGO film determined from the slope of the Williamson-Hall plot as: cos( ) 4 sin( ) where Strain gradient analyzed by RSM has been utilized in different systems [17][18][19].Except for the Williamson-Hall plot method, which can be used to obtain the estimation value of strain gradient quantitively [16], using the method described in Ref. 17 the interface with NGO.This layer is highly stressed, and the crystal lattice parameter a is identical to the one of NGO.In turn, this leads to a strongly elongated c-axis.In addition, a non-elliptical shape of the reflection suggests the presence of structural inhomogeneity along out-of-place direction, i.e., a strain gradient.
and smaller than that of NGO, so the higher thermal coefficient mismatch of SIO and LSAT may eliminate the possibility of strain gradient and cause fully relax.Red spheres show the raw data.Red curve is total fitting curve by using bi-cosine fitting function

3 .
another 24-u.c.SIO film on NGO substrate and fabricated into Hall bar device, as shown in Fig. S9.In this sample, current could be applied along both [100] and [010] direction, simultaneously.Fig. S9 (b) and (c) shows the magnetic-field angular dependences of   2ω rotated in x-z plane with varied temperatures when current is applied along [100] and [010] direction, respectively.With current along both directions, all curves display an obvious cosine angular dependence with a period of 2π, indicating that the existence of out-of-plane spin texture along both directions, which is consistent well with the feature of spin texture of the 24 u.c.SIO film shown in Fig. 2 (c) and the 200 u.c.SIO film at 300 K shown in Fig.We have grown another 200-u.c.SIO film on NGO substrates and fabricated into the Hall bar device, as shown in Fig. S10.In this sample, current could be applied along both [100] and [010] direction, simultaneously.Firstly, we set alternating current along [100] directions and measure the magnetic-field angular dependences of   2ω along x -z plane, as shown in Fig. S10 (b).The distinct peak shifts (as shown in Fig.5 in the main text) demonstrate the reproducibility.Secondly, we set alternating current along [010] direction and measure the magnetic-field angular dependences of   2ω along three geometries, as shown in Fig. S10 (c) -(e).The NLMR signals change sign between 10 K and 50 K, consistent with that data presented in Fig. S2&3 measured with current along [100].Besides, it has the same magnitude of   2ω when rotating φ and ψ, together with the result of nearly zero   2ω when rotating θ, suggesting the spin vector orients perpendicularly to the x-z plane without any warping towards to the OOP direction.These results draw the same conclusion as shown in Fig. 3 (a) -(c) with the current applied along [100], and clarify that for NLMR measurements in SIO, measuring   2ω with current along [010] has the same results as measuring   2ω with current along [100].
θ is Bragg angle, β=βmeasured -βinstrument, with βmeasured being the measured breadth of the diffraction peaks of the thin films and βinstrument estimated from the breadth of the nearby substrate reflection./ is a constant.The fitting curves and strain gradients are exhibited in Fig. S15(b) and Fig. S15(c) respectively, where the slopes of curves give  for thin films with different thickness.
, we made further analysis based on RSM data.As shown in Fig. S17(a), the c/a ratio of SIO film is changed gradually with increasing thickness, indicating a gradual reduction of strain.Besides, the asymmetric (103) reflections of 40 -200 u.c.SIO samples enclose a narrow streak at qx values of NGO substrate [Fig.4 (a) -(e) in the main text], indicating the presence of a pseudomorphic layer of SIO at

Fig. S1 .
Fig. S1.Electrical transport of a 200-u.c.SIO film.a temperature-dependent resistance (R-T) curve.Inset is the STEM image of 200-u.c.film, indicating sharp interface.b magnetoresistance (MR) curve measured at 10 K between ±2 T. Red curve: +2 T to -2 T. Black curve: -2 T to +2 T. c transverse electrical transport (Hall) curves under different temperatures.d carrier density and mobility extracted from c.

Fig. S3 . 31 Fig. S4 .
Fig. S3.Longitudinal second-order resistance   2ω measured with rotating H along y-z plane (defined as ψ).a-c   2ω with varied temperatures (10 K -300 K at 9 T and 1 mA, a   2ω at 10 K is divided by 10), varied currents (0.1 mA -1.0 mA at 10 K and 9 T, b) and varied magnetic fields (0 T -9 T at 10 K and 1 mA, c).Solid lines are fitting results by using cosine function.d Δ  2ω extracted from a. Δ  2ω (1) and Δ  2ω (2) are two series of second-order signals.Purple and cyan dash lines are shown to guide eyes.ef Δ  2ω extracted from b or c, dashed line is linear fitting result.

Fig. S5 .
Fig. S5.Transverse second-order resistance   2ω with rotating H along x-y plane (defined as φ).a-c   2ω with varied temperatures (10 K -300 K at 9 T and 1 mA, a), varied currents (0.1 mA -1.0 mA at 10 K and 9 T, b) and varied magnetic fields (0 T -9 T at 10 K and 1 mA, c).Solid lines are fitting results by using cosine function.d Δ  2ω extracted from a. Δ  2ω (1) and Δ  2ω (2) are two series of second-order signals.Purple and cyan dash lines are shown to guide eyes.e-f, Δ  2ω extracted from b or c, dashed line is linear fitting result.

Fig. S6 .
Fig. S6.Transverse second-order resistance   2ω measured with rotating H along y-z plane (defined as ψ). a   2ω with varied currents (0.1 mA -1.25 mA at 9 T and 10 K). b the fitting results of two series of second harmonic signals using 0.75 mA as an illustration.c-d   2ω (1) and   2ω (2) as a function of current.Dashed lines are linear fitting results.e   2ω with varied temperatures (10 K -300 K at 9 T and 1 mA).f   2ω with varied magnetic fields (0 T -9 T at 10 K and 0.5 mA).Solid lines are fitting results by using cosine function.g Δ  2ω extracted from f. Dashed line is linear fitting result.

Fig. S7 .
Fig. S7.The method to fit the second-order signals along x-z scan, using data at 35 K as an illustration.
•cos(θ+π/2).Purple and cyan curves show two series of second-order signals, respectively.Inset is the illustration of scanning geometry.

Fig
Fig. S8.a, angular dependence of   2ω measured at 0.1 -1.25mA (H = 9 T, T = 10 K) with rotating H along x-z plane (defined as θ), solid lines are fitting results.Phase shifts are marked by red triangle arrows.b-c   2ω (1) and   2ω (2) (extracted from a) as a function of current.d angular dependence of   2ω at 0 -9 T (T = 10 K, I = 1 mA), solid lines are fitting results.e   2ω (extracted from d) as a function of magnetic field.

Fig.S9. a
Fig.S9. a Hall bar device of a 24-u.c.SIO film obtained by standard lithography and etching process.Different current directions are marked.The magnetic-field angular dependences of   2ω along x-z plane with varied temperatures when current is set along b [100] direction and c [010] direction.

Fig.S10. a
Fig.S10.a Hall bar device of a 200-u.c.SIO film obtained by standard lithography and etching process.Different current directions are marked.b The magnetic-field angular dependences of   2ω along x-z plane with varied temperatures when current is along [100] direction (H = 9 T, I = 1 mA).c -e The magneticfield angular dependences of   2ω along c x-y plane, d y-z plane, and e x-z plane with varied temperatures when current is along [010] direction.f -h The magnetic-field angular dependences of   2ω along f x-y plane, g y-z plane, and h x-z plane with varied temperatures when current is along [010] direction.

Fig. S12 .
Fig. S12.Statistical details of Δqx, Δqz and φ. a The definition of Δqx, Δqz and φ.Δqx (Δqz) is defined as the deviation between qx (qz) of SIO film and NGO substrate.φ is defined as the rotating angle of the RSM center of SIO comparative to vertical direction.b The change of Δqx (Δqz) as a function of film thickness (t).c, The change of φ and Δqx as a function of t, they all show the same evolution trend.

Fig. S13 .
Fig. S13. a XRD and b Raman results.Both show clear SIO peak shifts.

Fig. S16 .
Fig. S16.XRD and RSM measurements of SIO grown on STO and LSAT substrates.a XRD (002) peaks of SIO grown on STO substrate with different thickness.b -d RSM measurements along STO (103) direction.e XRD (002) peaks of SIO grown on LSAT substrate with different thickness, indicating fully strain lattice of SIO.f -g RSM measurements along LSAT (103) direction.The comparison of Δqx, Δqz are shown by red dashed lines.Indicating fully relax lattice of SIO.

Fig.S17. a b
Fig.S17.a The c/a ratio of all SIO samples.c/a ratio of bulk strain-free SIO (~ 1) is taken as a reference.b The slicing RSM data along qx and qz axes along the black lines.The interslice separation is 0.03 nm along qx and qz axes.Each slice is indicated with a capital letter.c&d The set of line scans showing the change of the intensity of the reflection within the slice area along qx and qz axes.e&h The line scans of the averaged intensity along qx and qz axes.(black curves).The best fit to the experimental data can be done with triple Gaussian peaks, which are indicated with red, green and blue curves.f&i Peak positions extracted from e and h, showing the same for all slices.g&j Peak intensities extracted from e and h.

Fig. S19. a
Fig. S19. a Crystal structure of the SIO bulk with strain gradient.Δ indicates the continuous change in lattice constant across different layers.The green, blue, and red balls represent Sr, Ir, and O atoms, respectively.The gray oxygen octahedron is represented.b The band structures with SOC of the SIO bulk.Inset Brillouin zone and high-symmetry k-point.c Theorical calculation of spin texture (Sx and Sy) on the ka -kc plane (ka ∥ x, kc ∥ y) at Fermi surface for the SIO slab.The direction and length of red arrow indicates the direction and magnitude of the spin vector, respectively.d-f The distribution of the spin component Sx, Sy, and Sz.Blue and red indicate two different directions, and the shade of color indicates the magnitude.