Magnetization generation and giant nonlinear transport at symmetry-engineered interfaces

Domain: physics
Published: 2026-01-09
Editorial outcome: major_revision

Abstract
Interfaces in heterostructures possess inherent inversion asymmetry and display diverse physical effects, however, the pristine in-plane mirror symmetries of the constituent layers are usually preserved at the interface. On-demand manipulation of these symmetries remains challenging. Here, we demonstrate a strategy to control the in-plane mirror symmetries of interfaces by engineering the crystallographic orientation of heterostructures. We design a workhorse system with a new orientation, i.e., the LaAlO3/SrTiO3 heterostructure with metallic interfaces in the (112)-plane. Such a high index orientation leads to the breaking of all the pristine mirror symmetries except the mirror plane perpendicular to the [1-10] direction (M[1-10]), resulting in the Cs point symmetry with a metallic conduction. Consequently, this interface exhibits a giant nonlinear Hall effect characterized by a large Berry curvature dipole, a circular photogalvanic effect, and current-induced out-of-plane magnetization, all functional at room temperature. The magnitude of the nonlinear Hall effect rivals the Weyl and Dirac systems. Our work establishes a new strategy in exploring emerging electronic properties with nontrivial quantum geometry by designing the interface symmetry.

Introduction
Mirror symmetry plays a fundamental role in physical effects and materials properties, especially the orbital hybridization and spin-orbital coupling1. Thus, it controls the geometric property of the material’s electronic wavefunction, especially the Berry curvature and its dipole2–5, and determines the gyrotropic behaviors characterized by second-order axial tensors, including the circular photogalvanic effect6 and (inverse) spin-galvanic effect7,8. For example, the integral of the Berry curvature dipole (BCD) over the k-space in the two-dimensional system, where BCD represents the first moment of the Berry curvature, persists only if there is no more than one mirror symmetry9. Similarly, only in the same condition, an in-plane charge current at the interface generates magnetization with an out-of-plane moment via the inverse spin-galvanic effect10,11. The latter is critical for the field-free magnetization switching by spin-orbital torque12. Moreover, the same in-plane mirror symmetry breaking induces nonlinear electronic transport effects, such as the nonlinear Hall effect (NLHE)9,13,14 and bulk photovoltaic effect6. Usually, semiconductive and metallic materials possess highly symmetric structures. Therefore, the above stringent symmetry requirement confines the manifestation of these intriguing effects in a very limited range of materials, mainly in several two-dimensional layered materials, such as Weyl semimetal WTe2 13,14 and TaIrTe4 15, corrugated bilayer graphene16, and WSe2/SiP17 and WSe2/BP18 heterostructures. Moreover, the above physical effects in these materials generally function at low temperatures with limited exceptions15,19. It is therefore highly desirable to develop a strategy that can deliberately tailor the mirror symmetry in materials, especially those with metallic conduction and induce the above effects.

Here, we demonstrate exactly such a strategy based on the engineering of the crystallographic orientation of any heterostructure. Although this approach generates any desired mirror symmetry at the interface and is both handy and versatile, it has been overlooked until now. Crystallographic orientation engineering has already been the technique in bulk materials, especially piezoelectrics20, to achieve optimized electromechanical coupling and minimized loss. However, how the crystallographic orientation of heterostructures affects the symmetry of the interfacial region remains elusive, let alone the implications on physical properties. To unravel its hidden potential, we choose as a model system, the well-known LaAlO3/SrTiO3 heterostructure, which has garnered tremendous attention since its discovery21. Canonical low-Miller-index LaAlO3/SrTiO3 interfaces with (001), (110) and (111) orientations, respectively possess C4v, C2v, C3v point symmetry, retaining at least two in-plane mirror symmetries at room temperature. As a consequence of the built-in symmetries, these interfaces can generate current-induced magnetization with only in-plane moment22 and the circular photogalvanic effect (CPGE) only works with inclined illumination23. Also, this interface exhibits a negligible nonlinear Hall effect at room temperature. As recently shown, LaAlO3/SrTiO3 two-dimensional electron gas (2DEG) interface shows sizable nonlinear transport properties at low temperatures due to a lower symmetry induced by a phase transition and ferroelectric instability in SrTiO3 crystal24. It can be anticipated that once the unwanted pristine mirror symmetries are removed, the LaAlO3/SrTiO3 interface may exhibit unprecedented physical effects and functionalities even at room temperature or above. In this regard, we have demonstrated here that forming interfaces on crystal planes of high Miller index (11X) enables the rational breaking of the in-plane mirror planes and emergence of hidden properties. Taking the (112)-LaAlO3/SrTiO3 as a model example, this engineered interface exhibits a suite of intriguing phenomena—giant nonlinear Hall effect (featuring a substantial Berry curvature dipole), circular photogalvanic effect, and current-induced out-of-plane magnetization—all operative at room temperature and above.

Results
We grew a thin LaAlO3 layer on (112)-oriented SrTiO3 substrates using the pulsed laser deposition technique (see Fig. 1a and Methods). The thin film exhibits an ultrasmooth surface. The X-ray diffraction (XRD) analysis revealed a high crystalline quality indicated by the presence of Laue oscillation peaks (Supplementary Note 1). The conductive electron layer confined at the interface of LaAlO3/SrTiO3 heterostructures accompanies a potential gradient along the out-of-plane direction, making the interface of polar nature25–27 (see Supplementary Note 2 for discussion on the interfacial polarity). A distinctive feature of the (112)-orientation is that only the M[1-10] mirror symmetry of SrTiO3 crystal is preserved as it aligns with the out-of-plane direction, i.e., the interface polarity (Fig. 1a, b). In contrast, all the other intrinsic mirror symmetries of pristine SrTiO3, such as M[110] and M[001], intersect with the out-of-plane polarity and are consequently broken at the interface region (see Fig. 1c, d). Thus, the (112)-interface possesses only one mirror plane, resulting in the Cs point symmetry group (see Supplementary Note 3 for further discussion on the mirror symmetry). Note that one can further remove the residual mirror symmetry by using a different orientation (1XY) wherein X≠Y≠1. Detailed mathematical analysis of the mirror symmetry breaking at the (11X)-interface is given in the Methods section.

The atomic arrangement at the engineered interface was characterized by scanning transmission electron microscopy (STEM). The atomic arrangement visualized by STEM is consistent with the model shown in Fig. 1c, d. The mirror-symmetrical arrangement of atoms is clearly shown in the (111)-plane corresponding to the M[1-10] mirror symmetry (Fig. 1a), whereas the absence of mirror symmetry in the (11-1) plane is therein (see Fig. 1f).

To confirm that a metallic interface with a monoclinic Cs symmetry is formed at the (112)-LaAlO3/SrTiO3 heterostructure, we measured the temperature dependence of its sheet resistance Rs and the Hall effect. Sheet resistance Rs along both [110] and [111] in-plane directions show the metallic behavior with Rs decreasing with temperature down to ~20 K (Fig. 2a). The small upturn of Rs is likely due to the enhanced electron scattering at lower temperatures28,29. Notably, Rs shows different magnitudes along two in-plane directions in the temperature range investigated, which is consistent with the in-plane anisotropy of the designed symmetry. The Hall measurements (see details in Supplementary Note 4) reveal that the carrier mobility is as high as ~500 cm2V-1s-1 and the carrier density decreases to ~1×1013 cm-2 at low temperatures, which are similar to the canonical low-Miller-index LaAlO3/SrTiO3 interfaces (inset of Fig. 2a)24.

To get an insight into the mirror symmetry breaking in the designed (112)-interface, we have measured the CPGE on the same samples by illuminating the surface using a circularly polarized 405 nm laser beam with normal incidence (see Methods and Supplementary Note 5). In this particular illumination condition, the CPGE current can be generated only along the direction perpendicular to the mirror plane. Consequently, if there are more than two in-plane mirror symmetries in the sample, there will be no in-plane CPGE current generated in the heterostructure. Figure 2b shows the photocurrent flowing along [1-10] in the (112)-LaAlO3/SrTiO3 heterostructure, where θQWP is the angle between the fast axis of the QWP and the polarization direction of the incident linear polarized light. We extract the CPGE contribution by fitting this curve using17:

I = C sin 2(θQWP − θoff) + L sin 4(θQWP − θoff) + I0 (1)

Where I is the photocurrent, C and L denote the CPGE and linear photogalvanic effect (see Methods) current amplitude, respectively; θoff is the angle between the fast axis of the QWP and the crystallographic direction [110] (see details in Supplementary Note 5). I0 is the background photocurrent independent of the light polarization6. This revealed a sizable CPGE current C[1-10] along the [110] direction with an amplitude of ~0.4 nA (Supplementary Note 5). In contrast, there generates a negligible CPGE current (0.02 nA) along the [111] direction (Fig. 2c), likely due to the misalignment of the electrodes. Therefore, all the above results confirm a conductive layer with Cs symmetry emerging at the interface of two centrosymmetric insulators with (112)-orientation.

The interface with such a low symmetry would inevitably possess various physical effects that are normally absent in the well-studied (001), (110) and (111)-orientations. We focus on three such effects shown by the (112)-interface: the charge-to-magnetization conversion, i.e., the inverse spin-galvanic effect, the nonlinear Hall effect and the quantum geometric feature of the electronic wavefunction at the interface. These effects and properties are both of fundamental interest and technological importance. The peculiarity of the present (112)-interface is that it can be generated by a current flowing with a [1-10] direction, driven by a Cs symmetry, and nonlinear Hall current flowing along the [1-10] direction. The current-induced out-of-plane magnetization in the (112)-interface is probed by the polar magneto-optical Kerr effect (MOKE). The experimental setup is shown in Supplementary Note 6 and discussed in Methods. The MOKE signal, i.e., Kerr rotation θK as a function of the a.c. current amplitude is given in Fig. 2d (see Methods). A sizable MOKE signal appears when the current runs along the [110] direction, of which the magnitude increases almost linearly with the current density. In contrast, only a small background noise signal is shown when the source current is driven along the [111] direction or other orientation interfaces, which is in line with the theory (Supplementary Note 7). A spatial distribution of the out-of-plane magnetization has been probed by mapping the polar MOKE signal across the Hall bar (Fig. 2e). Clearly, the almost uniform out-of-plane magnetization is generated over the Hall bar (also see Fig. 2f for line scan results). This is distinctive from the spin/orbital Hall effect that only generates a polar MOKE signal with the opposite sign at the edges of the bar30. Such a current-induced out-of-plane magnetization can be potentially used to switch magnetization by spin-orbital torque.

In addition to the magnetization that is linearly proportional to current, a nonlinear Hall voltage also develops across the Hall bar of the (112)-oriented conductive interface with a quadratic dependence on the current. To characterize this nonlinear transport, we employed two standard Hall bar devices aligned along the two main in-plane crystallographic axes, i.e., [1-10] and [111] (Fig. S11). With a.c. current Iω injected, both first- and second harmonics of longitudinal Vxx and transversal voltages Vxy of the Hall bar are measured using a lock-in amplifier (see Methods). In the case of the conductive interface with Cs symmetry, the second harmonic Hall voltage V2ωxy increases quadratically with Iωx, reaching a value of 10 μV driven by Iωx with an amplitude of 0.5 μA. The magnitude and phase of V2ωxy are almost independent of the frequency of Iωx, excluding the contribution of capacitive coupling and thermal-related effect (Fig. S11). Moreover, the sign of V2ωxy reverses when the bias direction and the Hall voltage probe switch simultaneously.

In addition, the electronic transport of the (112)-orientation shall be strongly anisotropic. To show this, we characterized the tensorial nature of the transport properties of the Cs-electrongas using a disc-style electrode with an angle interval of 22.5°, which allows us to drive Iωx along various in-plane directions with an angle of θ (Fig. S12). The two-fold angle-dependent in-plane resistance agrees well with the Cs symmetry (see Methods). Meanwhile, the Iωx direction dependence of the nonlinear Hall voltage normalized by the square of the driven voltage (i.e. V2ωxy/(Vωxx)2) shows a one-fold angular θ dependence, which is similar to that of WTe2 31 (Fig. 3b). This dependence can be well fitted by the formula derived by symmetry-based phenomenological analysis (see Methods).

To compare with all other systems, the NLHE coefficient is defined as E2ωxy/(Eωxx)2 with E2ωxy denotes the field induced by the Hall voltage and Eωxx refers to the longitudinal driving electric field. For our particular case, the NLHE exhibit three important features: (i) a remarkably wide temperature range, (ii) a very high value of the NLHE coefficient, and (iii) tunability by crystal orientation. Firstly, the NLHE persists in a wide temperature range from 3 K to 475 K (Fig. 3c). With decreasing temperature from 475 K to about 100 K, the NLHE coefficient keeps decreasing and finally reaches nearly zero. This might be due to the decreased electron density in the conduction band that shifts the Fermi level from the e0 band towards the lower a1g band with a decreased magnitude of BCD as discussed later. Further reducing the temperature to 3 K, the NLHE coefficient bounces back to a value slightly higher than the (111)-oriented system and comparable to Bi2Se3 (Fig. 3c and Fig. S14). The temperature-dependent behaviour differs from twisted graphene32, PbTaSe2 33 and other 2D materials, of which the NLHE shows a monotonic dependence on the temperature (see discussion in Supplementary Notes 10 and 11). The very high temperature range approaching 500 K sets a new remarkable benchmark in both the operational temperature range and the NLHE coefficient value, outperforming all previously reported materials.

As mentioned, all the (11X) crystallographic oriented interfaces shall also show similar mirror symmetry breaking. Therefore, in addition to the (112)-orientation, we have explored LaAlO3/SrTiO3 heterostructures with (113) and (114) orientations (Fig. 3d). All these oriented heterostructures possess conductive interfaces with Cs symmetry and exhibit sizable NLHE with the room temperature coefficient value decreasing from (112) to (114). In contrast, NLHE vanishes in (001)-, (110)- and (111)-oriented heterostructures at room temperature (Supplementary Note 12). It is noted that defects such as oxygen vacancies or zigzag patterns may form differently at particularly high Miller-index interfaces during growth. However, as detailed in Supplementary Note 13, the fundamental origin of both the magnetization generation and the giant nonlinear transport is the designed symmetry breaking imposed by crystallographic-orientation engineering. Defects can influence the transport behavior, but their role is derivative—arising from, and modulated by, the underlying orientation-induced symmetry engineering.

Under the relaxation-time approximation, there are three intrinsic contributions to the second-order NLHE, i.e., the Berry-curvature dipole (BCD) mechanism9, the quantum-metric dipole mechanism34, and the Drude mechanism35. Although all three mechanisms require inversion-symmetry breaking, the last two mechanisms require additional breaking of time-reversal symmetry, which is not the case for our non-magnetic LaAlO3/SrTiO3. Thus, only the BCD can be the intrinsic contribution to the NLHE.

Additionally, extrinsic contributions, including the side jump and skew scattering, also play an important role in the NLHE of the (112)-interface36. Detailed analysis of various contributions by the scaling law is given in Supplementary Note 10. It indicates that the electrons of the (112)-interface possess a substantial Berry curvature dipole in their momentum space.

To gain an insight into the quantum geometrical feature of the conductive (112)-interface, we have performed density functional theory calculations to unravel its characteristics, especially the feature of Berry curvature and BCD (see Methods). A superlattice structure was used to model the (112)-LaAlO3/SrTiO3 interface (see Supplementary Note 14). The space group symmetry of the entire superlattice structure is Pm, corresponding to the Cs point group symmetry, which is consistent with that of the (112)-oriented interface under consideration. Our calculations reveal the formation of 2DEG at the interface of the (112)-LaAlO3/SrTiO3 superstructure. The conductive properties of the 2DEG are predominantly contributed by the Ti-3d (dxy, dxz and dyz) orbitals at the n-type interface (see Supplementary Note 14). The three main bands near the Γ point in the K plane of the 3D band structure is shown in Fig. S20 and a 2D band structure with Berry curvature Ωz, is shown in Fig. 4a.

For the 2DEG at the (112)-oriented interface, the breaking of inversion symmetry (P) leads to a nonzero local Berry curvature (Ωz), and the time reversal symmetry dictates that Ωz(-k) = -Ωz(k). Furthermore, the mirror symmetry My requires Ωz(kx, ky) = -Ωz(kx, -ky). Here, the x, y, and z axes correspond to the [111], [110], and [112] crystallographic directions, respectively. Figure 4b illustrates the k-space distribution of the Ωz, clearly consistent with the symmetry analysis. It is highly nonuniform, with hot-spot areas near Fermi lines. More precisely, the Berry curvature exhibits a dipolar distribution about k=0, as allowed by the Cs symmetry. Such texture can be captured by the BCD, defined as:

Dαz = ∫ d2k/(2π)2 Dαz(k) = ∫ d2k/(2π)2 Σn fn ∂Ωn(k)/∂kα (2)

where fn denotes the Fermi distribution function for the n-th band. The time reversal symmetry dictates that D(-k) = D(k); the Cs point group symmetry further requires that Dxz(kx, ky) = -Dxz(kx, -ky) and Dyz(kx, ky) = Dyz(kx, -ky). Consequently, only Dyz(k) can integrate to a nonzero value over the Brillouin zone. In Fig. 4c, we illustrate the k-space distribution of the Dyz, which is consistent with the symmetry analysis, confirming the validity of our calculation. Similar to the Berry curvature, larger Dyz(k) appears near the Fermi lines. Notably, Dyz reaches a sizable value near the Fermi lines and exhibits symmetries with respect to both kx=0 and ky=0, resulting in a finite BCD integrated over the k-space. Since the BCD is a property of the Fermi surface, it substantially depends on the position of the Fermi level in the band structure, as shown in Fig. 4d. The BCD magnitude only show large values near the Fermi level, where two maxima of opposite signs can be found. In contrast, Dxz remains nearly zero across the entire Fermi level, consistent with mirror symmetry My. Note that the peak value of calculated BCD is at the same order of magnitude as that of Weyl semimetal and topological insulators13,15,37. Note that the role of spin-orbit coupling in the geometric features has been discussed in Supplementary Note 15.

In conclusion, we have demonstrated that the crystallographic orientation engineering is a feasible approach to achieve a designed interface symmetry in heterostructures, which allows the emergence of otherwise forbidden physical effects at room temperature. The (112)-oriented LaAlO3/SrTiO3 conductive interface with Cs symmetry exhibits a giant nonlinear Hall effect driven by the large Berry curvature dipole, current-induced out-of-plane magnetization and circular photogalvanic effect, all generated by breaking the in-plane mirror symmetry. The developed strategy is also applicable to other types of heterostructures, including metal-oxide-semiconductor field-effect transistors, tricolor superlattices and thin films with embedded strain gradient or chemical component gradient. Thus, this work opens a new material design realm that not only can be used to explore a wider range of physical effects in a low-symmetric system, such as superconductivity, opto-spintronics, spintronics, magnetoelectric and ferroelectrics, etc., but also offers the prospects to bridge these research fields that were largely decoupled before.

Methods
Sample and device preparation
To achieve the Ti-rich surface, (112)-SrTiO3 substrates were etched in buffered oxide etchant (BOE) for 11 seconds and further annealed at 1000°C in air for 1 h. LaAlO3 thin films were epitaxially grown on SrTiO3 substrates using pulsed laser deposition (PLD) at 800°C in an oxygen pressure of ~0.01 Pa. The laser fluence was about 1 J/cm2 and the repetition rate was 2 Hz. After film growth, the samples were first cooled down to 500°C in the same growth oxygen pressure, then subjected to 10-min in-situ annealing at ~100 Pa oxygen pressure to compensate the oxygen vacancies in the SrTiO3 bulk. Finally, thin films were cooled down to room temperature at the same ambient oxygen pressure. We note here that different orientation LaAlO3/SrTiO3 thin films were grown in the same PLD condition and have a similar thickness of ~4 nm.

The photolithography technique was used to pattern Hall bar (10 μm in width and 70 μm in length) and disc devices. ~60 nm amorphous AlOx thin films grown by PLD at room temperature at an oxygen pressure of 1 Pa were used as hard masks, leaving the bare area of SrTiO3 substrate for the active-Hall bar device growth. Disc devices were fabricated using the same method.

Scanning transmission electron microscopy
Scanning transmission electron microscopy (STEM) specimens were prepared using Tescan Amber FIBSEM with standard lift out procedures. The cross-section specimens were then prepared along <110> and <111> directions. The specimens are polished using low accelerating voltage after thinning to remove debris and amorphized region. Atomic-resolution STEM images were acquired using a double aberration-corrected JEOL ARM200F working at 200 kV. A probe forming a convergence semi-angle of 23 mrad was used to obtain STEM images. Annular dark field (ADF) images were formed using the signal collected from the semi-angle range 45 to 180 mrad.

Circular photogalvanic effect measurements
A homemade optoelectronic measurement setup equipped with optical instruments microscopy cryostat was used for circular photogalvanic effect studies (Fig. S6 and S7). A polarised 405 nm laser (~37 mW nominal power) was employed to generate linearly polarised light. A quarter wave-plate (QWP) was used to vary the light polarization. The photocurrent was read by a transimpedance amplifier and digitized by a data acquisition system. It was further used to record the photovoltage as a function of the QWP angle.

Magneto-optical Kerr effect studies
The Magneto-optical Kerr effect (MOKE) measurements (Fig. S9) were conducted in a dark environment with the temperature of optical components well controlled by a home-made system (within ± mK fluctuation). A 632 nm HeNe laser (power of ~3.5 mW) linearly polarized by a Glan-Taylor Calcite polarizer was used as an excitation source. An objective lens (×20) focused the laser onto the sample, producing a laser spot of about ~1 μm. After being reflected from the sample, the light was collected by the objective lens and further passed through the half-wave plate and then split by a Wollaston prism into two beams, i.e., reference and signal. For a better common-mode rejection performance, we adjusted the ratio between the power of the reference and signal to ~1.9:1 via the half-wave plate. An auto-balanced detector was employed to analyze the intensity difference between light paths. To further enhance the signal-to-noise ratio, we have employed a.c. source to drive the current and analyze the auto-balanced detector signal by a lock-in amplifier. Each data point shown in Fig. 2d–f was statistically calculated from a ~15 min measurement duration with at least 600 data points. The Kerr rotation angle was calculated following the method reported in the 38. We refer to: (1) the finite MOKE sensitivity, (2) the intrinsic offset/noise and finite resolution of our lock-in amplifier. (3) The electromagnetic cross-talk between different electronic components in our home-designed MOKE system. (4) The He-Ne laser used in our system exhibits a sizable fluctuation of power and polarization.

Electronic transport measurements
LaAlO3/SrTiO3 samples were mounted on the chip carrier using conductive silver paste. The Hall bars and disc bars were connected to the chip carrier with aluminum wires penetrating through the LaAlO3 thin film using the ultrasonic wedge-bonding technique. An a.c. current Iω (ω is typically 17.7 Hz in this study) was sourced along the desired crystallographic direction to perform the nonlinear transport measurement. Lock-in techniques were used to simultaneously measure the first- and second- harmonic voltage drops. A low temperature magnetic measurement system with an insert was used for the transport measurements between ~2 K and 300 K; a microscopy cryostat with temperature controller was employed to measure the nonlinear transport at high temperatures up to 475 K. The temperature-dependent NLHE measured (Vωxx and V2ωxy) continuously by heating the sample at a rate of 3 K/min with the ac current biased.

Mathematical analysis of the symmetry evolution at the orientation-engineered interface
How the interface symmetry evolves by varying heterostructure orientation can be analyzed by matrix calculation. For simplicity, we will focus on mirror symmetry here. Regarding the SrTiO3 crystal of Oh point symmetry, it has three axial mirror M {100} and six diagonal mirror planes M {110} of which the matrix can be respectively given as:

M[100] = [[-1, 0, 0], [0, 1, 0], [0, 0, 1]] M[010] = [[1, 0, 0], [0, -1, 0], [0, 0, 1]] M[001] = [[1, 0, 0], [0, 1, 0], [0, 0, -1]]

M[110] = [[0, -1, 0], [-1, 0, 0], [0, 0, 1]] M[1-10] = [[0, 1, 0], [1, 0, 0], [0, 0, 1]] M[011] = [[1, 0, 0], [0, 0, 1], [0, 1, 0]]

M[01-1] = [[1, 0, 0], [0, 0, -1], [0, -1, 0]] M[101] = [[0, 0, 1], [0, 1, 0], [1, 0, 0]] M[10-1] = [[0, 0, -1], [0, 1, 0], [-1, 0, 0]]

The coordinate of the above matrices is in the default cubic lattice and the subscript of M[xyz] represents the normal direction of the mirror plane. The potential gradient developed at the LaAlO3/SrTiO3 heterostructure interface can be represented by a vector, of which direction is determined by the crystallographic orientation (11X) with X being an integer and X≥2. Thus, the interface polarity can be expressed as [11X]. To check whether a mirror plane persists in (11X)-oriented interface, one can perform the mirror reflection operation on the vector [11X]T. If the [11X]T retains intact, the mirror plane is preserved in the interface; otherwise, it is broken at the interface. Based on such a simple principle, we can see that only the M[1-10] plane survives, since:

M[1-10] [11X]T = [[0, 1, 0], [1, 0, 0], [0, 0, 1]] [1, 1, X]T = [1, 1, X]T (3)

Whereas all the other mirror symmetries are broken. Take M[110] as an example:

M[110] [11X]T = [[0, -1, 0], [-1, 0, 0], [0, 0, 1]] [1, 1, X]T = [-1, -1, X]T ≠ [1, 1, X]T (4)

This principle is also applicable to the rotation symmetry. One can find that all the rotation symmetry operations in the pristine SrTiO3 lattice are broken at the (11X)-LaAlO3/SrTiO3. Thus, the symmetry of studied (112)-LaAlO3/SrTiO3 is reduced to Cs. Furthermore, if one chooses an orientation as (1XY) with X≠Y≠1, all the mirror reflection symmetry and rotation symmetry are broken, leading to the symmetry of C1.

Phenomenological theory of circular photogalvanic effect (CPGE)
To understand the CPGE with regard to the crystallographic directions, the classical phenomenological theory is employed. The general expression of the photogalvanic effect (PGE) current is written as6:

Ii = Ilight (βLijk ej e*k + iβCik ek × e*k) (5)

where Ii is the photocurrent in the i direction, Ilight is the intensity of the light, βLijk and βCik are the third and second order tensors describing the linear and circular PGE effect, respectively. ej and ek is the electrical field of the propagating light, e*k is the conjugate of ek. The tensor elements of both linear photogalvanic effect (LPGE) and circular photogalvanic effect (CPGE) are nonzero only in asymmetric materials. Due to the symmetry nature of the 3rd rank tensor of crystals, the LPGE tensor of Cs point group material can be written in the form of 3×6 tensor with 10 independent nonzero elements:

βLij = [[βL11, βL12, βL13, 0, βL15, 0], [0, 0, 0, βL24, 0, βL26], [βL31, βL32, βL33, 0, βL35, 0]] (6)

The CPGE tensor of Cs point group material has 4 nonzero independent elements and is expressed as:

βCij = [[0, βC12, 0], [βC21, 0, βC23], [0, βC32, 0]] (7)

A mixed linear-circular polarised light was generated by the laser and QWP, and hence, the expressions for LPGE and CPGE are:

IL = Ilight βLijk ej e*k = Ilight [[βL11, βL12, βL13, 0, βL15, 0], [0, 0, 0, βL24, 0, βL26], [βL31, βL32, βL33, 0, βL35, 0]] [sin2(2θQWP), 0, 0, 0, 0, 0]T = Ilight [βL11 sin2(2θQWP), 0, βL31 sin2(2θQWP)]T (8)

IC = Ilight iβCik ek × e*k = Ilight [[0, βC12, 0], [βC21, 0, βC23], [0, βC32, 0]] [0, 0, -sin(2θQWP)]T = Ilight [-βC12 sin(2θQWP), 0, -βC32 sin(2θQWP)]T (9)

(where θQWP is the QWP rotation angle (θQWP=0 is in the direction [111]). The total current comprises both linear and circular PGE currents running along [1-10] and [11-1] is expressed as:

I[1-10] = Ilight [1/2 βL26 sin(4θQWP) - βC23 sin(2θQWP)] (10)

I[11-1] = 1/2 Ilight [βL11 + βL12 - βL11 cos(4θQWP) + βL12 cos(4θQWP)] (11)

Hence, CPG current flows only in the [1-10] direction, while LPG current exists in both [110] and [111] directions.

Phenomenological theory of the charge-to-magnetization conversion
The current-induced nonequilibrium magnetization can be phenomenologically described by inverse spin-galvanic effect, i.e., by the formula11:

Mi = γij Jj (12)

Where Mi is the magnetization in the direction i, γij is the second order axial tensor sharing the same symmetry form with the CPGE tensor and Jj is the electric current density in the direction j. When an in-plane current (Jx, Jy, 0) is applied to the conductive interface, the induced magnetization is thus given as:

Mi = [[0, γ12, 0], [γ21, 0, γ23], [0, γ32, 0]] [Jx, Jy, 0]T = [γ12 Jy, γ21 Jx + γ23 Jy, γ32 Jx]T (13)

Hence, current along x ([111]) only generates an in-plane magnetization My = γ21 Jx, while current along y ([1-10]) generates an in-plane magnetization Mx = γ12 Jy and out-of-plane magnetization Mz = γ32 Jy. Only the out-of-plane magnetization was detected in our polar-MOKE setup.

Phenomenological theory of nonlinear transport properties
To understand the tensorial feature of the nonlinear transport of Cs-2DEG, we performed the phenomenological analysis. Generally, the second-order nonlinear Hall effect current induced by the electrical field E is expressed as:

jNαLHE = χαβγ Eβ Eγ (14)

Where χαβγ is the nonlinear susceptibility tensor, and is expressed as a 3×6 tensor with 10 non-zero independent elements, similar to the linear PGE tensor:

χ = [[d11, d12, d13, 0, d15, 0], [0, 0, 0, d24, 0, d26], [d31, d32, d33, 0, d35, 0]] (15)

When an electrical field is applied to the 2D system Eα = (Ex, Ey, 0), the resultant NLHE current is expressed as:

jNαLHE = [[d11, d12, d13, 0, d15, 0], [0, 0, 0, d24, 0, d26], [d31, d32, d33, 0, d35, 0]] [Ex^2, Ey^2, 0, 0, 0, 2ExEy]T = [d11 Ex^2 + d12 Ey^2, 2d26 ExEy, d31 Ex^2 + d32 Ey^2]T (16)

Considering the general resistance anisotropy of Cs symmetry ρ = [[ρ11, 0, ρ13], [0, ρ22, 0], [ρ13, 0, ρ33]], the NLHE electrical field then is expressed as:

ENαLHE = ρ jNαLHE = [[ρ11, 0, ρ13], [0, ρ22, 0], [ρ13, 0, ρ33]] [d11 Ex^2 + d12 Ey^2, 2d26 ExEy, d31 Ex^2 + d32 Ey^2]T (17)

Where specifically, ρ11, ρ22, and ρ33 refer to the resistivity in the main axis direction [1-10], [11-1] and [112]. In our experiments, an in-plane current is sourced, read as: Jα = J [sinθ, cosθ, 0]T, where θ is the angle away from the mirror line.

Because Eα = ρJα = J [ρ11 sinθ, ρ22 cosθ, 0]T, the nonlinear Hall electrical field orthogonal to the source current is then expressed as:

EN?LHE = J^2 ρ11 sinθ (d12 + 2d26) ρ22^2 sin^2θ + d11 ρ11^2 cos^2θ (18)

Hence, we have

EN?LHE / J^2 = ρ11 sinθ (d12 + 2d26) ρ22^2 sin^2θ + d11 ρ11^2 cos^2θ (19)

V2ωxy = W ρ11 sinθ (d12 + 2d26) ρ22^2 sin^2θ + d11 ρ11^2 cos^2θ (20)

V2ωxy = L/2 (ρ32 sin^2θ + ρ11 cos^2θ) (21)

Where, L and W are the Hall bar length and width. Meanwhile, we have the first-order electrical field parallel and perpendicular to the current direction are expressed as:

E|| = J (ρ11 sinθ × sinθ + ρ22 cosθ × cosθ) = J (ρ11 sin^2θ + ρ22 cos^2θ) (22)

E? = J (ρ11 sinθ × cosθ - ρ22 cosθ × sinθ) = J (ρ11 - ρ22) sinθ cosθ (23)

The resistivities of two-fold angular dependence are expressed as:

ρ|| = ρ11 sin^2θ + ρ22 cos^2θ (24)

ρ? = (ρ11 - ρ22) sinθ cosθ (25)

First-principles calculations
The density functional theory (DFT) modeling of SrTiO3/LaAlO3 interfaces has been extensively studied in the literature39,40. In the structural modeling process, we first performed a full structural optimization of SrTiO3, which was then used as the framework for constructing the superlattice. The resulting (112)-SrTiO3/LaAlO3(5.5/6.5) superlattice structure, as shown in Fig. S20, was constructed with one interface chemically modified by substituting an Al atom with a Ti atom, thereby introducing an n-type chemical intermixing at the interface. The other interface was left unmodified to serve as a control. Subsequently, a full relaxation of both the lattice and ionic positions was carried out, allowing the structure to relax freely along the z-axis while keeping the xy-plane constrained.

The DFT calculations were performed using the Vienna Abinitio Simulation Package (VASP)41, adopting the Perdew-Burke-Ernzerhof (PBE) form of the generalized gradient approximation (GGA) for the exchange-correlation functional42. The kinetic energy cutoff for the plane-wave basis was set to 450 eV. To account for the on-site Coulomb interactions, the GGA+U method was employed43, with U=11 eV and J=0.68 eV for the f orbitals of Sr, and U=5 eV and J=0.64 eV for the d orbitals of Ti. The larger U value for Sr was chosen to prevent the incorrect mixing of its f orbitals near the Fermi level44. Brillouin zone integrations were carried out using the Monkhorst-Pack scheme with a k-point grid of 9×13×3. To facilitate high-precision calculations, the energy convergence criterion for the electronic self-consistency was set to 10-7 eV.

To obtain a high-density sampling of the electronic structure in k-space, the WANNIER90 package was employed to compute the maximally localized Wannier functions (MLWFs) and perform Wannier interpolation45,46. The MLWFs were constructed based on the non-self-consistent field (NSCF) calculation results from VASP, utilizing 360 random initial guess orbitals. Additionally, 40 extra NSCF bands were included to facilitate the disentanglement procedure. The BC Ωγ(k) is calculated as follows:

Ωγ(k) = ϵαβγ Ωαβ(k) = Σn Ωn,αβ(k) = -2Im Σn Σm≠n <ψn(k)|vα|ψm(k)> <ψm(k)|vβ|ψn(k)> / (En(k) - Em(k))^2 (26)

Where, n and m denote the band indices, fn represents the Fermi-Dirac distribution function of the n-th band, vα (α=x,y,z) is the velocity operator, ψn(k) and En(k) are the Bloch wave function and energy of the n-th band at wave vector k respectively. It should be noted that Ωn,γ represents the Berry curvature contribution from the n-th band at a given k-point, which corresponds to the results shown in Fig. 4b. Meanwhile, Ωγ denotes the integration of Ωn,γ over all Fermi-occupied states, corresponding to the results presented in Fig. 4c.

The calculation of the BCD, Dαβ, follows

Dαβ = ∫ d2k Dαβ(k) = ∫ d2k Σn ∂(fn Ωβn) / ∂kα (27)

Here, the Fermi–Dirac distribution function, fn is given by

fn(E) = 1 / (e(E-μ)/kBT + 1) (28)

where μ is the chemical potential, kB is the Boltzmann constant, and T is the absolute temperature. As T approaches zero, -∂fn/∂E converges to δ(E-μ), which is approximated in numerical calculations using a reasonable Gaussian smearing. The above calculation of BCD differs from the experimental data, which is further discussed in Supplementary Note 15.

Data availability
The source data underlying all main manuscript figures are provided with this paper. All other data that support the findings of this study are available from the corresponding authors upon request. Source data are provided with this paper.

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "In this work, based on the crystallographic orientation engineering, the authors successfully fabricated the (112)-oriented\nLaAlO3/SrTiO3 conductive interface with the Cs symmetry. In particular, based on the electrical transport measurements,\nthey demonstrated the coexistence of the intriguing current-induced out-of-plane magnetization, circular photogalvanic and\ngiant nonlinear Hall effect effects. The results undeniably enrich the fundamental understanding of the nonlinear transport\neffect in oxides through crystallographic orientation engineering. Before making a decision on the paper, the authors should\naddress the following major points:\n(1) Is it reliable to confirm the Cs symmetry of the (112)-LaAlO3/SrTiO3 heterostructure from the CPGE currents shown in\nFig. 2(b) and (c)? This is due to the fact that the difference for the CPGE currents between (b) and (c) is not significant.\n(2) The spin-orbit coupling (SOC) effect should play a crucial role for the observed current-induced out-of-plane\nmagnetization and nonlinear Hall effect. It seems to me that the study on the SOC of the (112)-LaAlO3/SrTiO3\nheterostructure is missing. In the revision, I suggest the authors specify the magnitude of SOC from the analysis of\nexperimental data and band structures. Further, it is interesting to compare the SOC parameters for different orientated\nLaAlO3/SrTiO3 heterostructures.\n(3) It is known that the nonlinear Hall effect can also be contributed by the Drude and quantum metric mechanisms. In this\nwork, the authors focus on the contribution from the Berry curvature dipole. The authors should specify how to exclude the\nother possible contributions to the nonlinear Hall conductivity.\n(4) Since the BCD has been calculated as shown in Fig. 4d, I suggest the authors further calculate the nonlinear Hall\nconductivity and compare to experimental data.\n(5) From Fig. 3(c), the nonlinear Hall coefficient for the (112)-LaAlO3/SrTiO3 heterostructure increases monotonously with\ntemperature (starting from 100 K). This is uncommon. It is known that the SOC effect will be weakened by the high\ntemperature. If the nonlinear Hall effect is contributed by the BCD due to SOC, the nonlinear Hall coefficient should\ndecrease with temperature. This should be further elaborated. In addition, why can the extrinsic contributions such as side\njumping and screw scatting be excluded at high temperatures.\nMinor points: In the main text, Fig. 3b should be Fig. 3a, Fig. 3c should be Fig. 3b etc.", "reviewer_2": "This study proposes a strategy for controlling in-plane mirror symmetries at oxide interfaces by engineering the\ncrystallographic orientation of heterostructures. By utilizing a (112)-oriented LaAlO₃/SrTiO₃ interface, the high-index\norientation breaks most mirror symmetries, resulting in Cs point symmetry and metallic conduction. This structural tuning\nenables the emergence of novel phenomena, such as a giant nonlinear Hall effect, circular photogalvanic effect (CPGE),\nand current-induced magnetization—all observable at room temperature. The nonlinear Hall response is comparable to that\nobserved in Weyl and Dirac systems. The experimental data are clear and compelling; however, I am not entirely convinced\nby the authors' interpretation of the observed phenomena. I would like the authors to address the following points:\nFigure 1e appears symmetric with respect to the [1-10] mirror plane. However, considering the effects of lattice mismatch and\nstrain, this symmetry may not be preserved throughout the entire structure. Furthermore, if oxygen octahedral rotations are\npresent, the symmetry would be broken. Additionally, a zigzag pattern is visible at the interface between the black and white\nregions, which is typically indicative of atomic intermixing or facet formation. Therefore, the claim that a [1-10] mirror plane\nexists is not readily acceptable.\nThe observed transport anisotropy may arise from this zigzag pattern, which could also produce a stray magnetic field\nperpendicular to the substrate. This, in turn, may explain the current-induced magnetization.\nThere is no clear description of how electrical contacts were made through the LaAlO3 layer, which is known to have high\nresistance.\nThe mobility should also exhibit anisotropy, but this is not discussed.\nThe definition of theta is unclear, so I hope the authors specify its reference direction. Additionally, the term “LPGE” (linear\nphotogalvanic effect) is not defined. What is its physical origin in this system?\nIn Figure 2b, the fit is poor. At around 160 degrees, the experimental data deviate significantly from the fitted curve.\nIn Figure 2c, the CPGE component appears absent. The red fit line diverges considerably from the experimental data.\nFigure 3f, which is referenced in line 194, appears to be missing from the manuscript.\nThe carrier concentration is not discussed. Can the position of the Fermi level account for the carrier density observed in the\nsample?", "reviewer_3": "In their manuscript titled “Magnetization generation and giant nonlinear transport at symmetry-engineered interfaces via\ncrystallographic orientation”, the authors H.-B. Zhang et al. present a series of experimental results in high-Miller-index\nLaAlO3/SrTiO3(hkl) heterostructures which is argued to derive from the specific nonvanishing Berry curvature and Berry\ncurvature dipole densities hosted in the systems under investigation. The overall set of experiments and symmetry-based\nphenomenological models are consistent with the low Cs point-group of their\nheterointerfaces hosting the 2DEGs.\nHowever, there are a number of experimental details which are missing, some desirable more detailed discussions\nregarding e.g., the peculiar temperature-dependence of their nonlinear Hall effect (NLHE),or with regards to the magnitude of\nthe MOKE and the NLHE at different (11X) interfaces; and possibly a demonstration that the systems under investigation are\ngate-tuneable 2DEGs.\n__________\nComment #1\n#1.1. It would be useful/desireable to present In Fig.1, layer-resolved effective charges, in the ionic limit, of unreconstructed\n(11X)-LAO/STO heterostructures as in, e.g., the seminal work of N. Nakagawa et al. [Nature Materials 5, 204–209 (2006).\nhttps://doi.org/10.1038/nmat1569] in (001)-LAO/STO, or that of G. Herranz et al. [Scientific Reports 2, 758 (2012).\nhttps://doi.org/10.1038/srep00758] on (111) and (110)-LAO/STO interfaces. Given the (111)-LAO/STO interface is polar, I\nwould assume that the (11X) interfaces are as well, but I did not “do the math”.\n#1.2. Problematically, it seems to me that the authors argue the existence of a polar interface - as a result of the potential gradient at the interface - and cite Ref. 30 [G. Singh-Bhalla et al., Nat. Phys. 7, 80-86 (2011).], c.f. lines 83-85.\nWithout getting into the whole debate of the “polar catastrophe scenario”, this is near complete inversion of the roles of polar\ninterfaces and existence of 2DEGs therein. The polar interface pre-exists the 2DEG. And in virtue of the electrostatic\nboundary condition (in a self-concistent Poisson-Schrödinger picture), there is no further\nconfinement potential to host the 2DEG without additional charges: meaning one needs electron donors, besides the band-bending due to energy\nmismatched conduction and valence bands at LAO/STO interfaces. More appropriate References would instead include [W.-\nJ. Son et al., Phys. Rev. B 79, 245411 (2009)] and [P. Delugas et al., Phys. Rev. Lett. 106, 166807 (2011)].\nMoreover, it is my understanding that C. Cazorla & M. Stengel [Phys. Rev. B 85, 075426 (2012)] have since shown that for\nmetal-capped − stoichiometric − LAO/STO systems, the LAO internal polar field is a decreasing function of LAO film’s\nthickness, and depends on the electrostatic boundary conditions; or more precisely on the areal carrier density at both\nM/LAO and LAO/STO interfaces. Such that there is and I quote from them “no intrinsic built-in LAO electric field exists in the\nshort-circuited Pt/LAO/STO capacitor system […]” which “[…] in practice rules out the hypothesis of Zener breakdown, which\nwas used in Ref. [G. Singh-Bhalla et al., Nat. Phys. 7, 80-86 (2011).]”\nHowever if the authors want to argue instead that the pristine interfaces are apolar, and that interfacial polarity arises from\nthe existence of an electrostatic confinement potential, then further discussions are needed.\n#1.3. Line 113 “conventional cases” is not well defined and could be replaced by “canonical low-Miller-index LAO/STO\ninterfaces” which I believe is what is implied. This, Ref. [29] is here maybe not the most relevant reference.\n__________\nComment #2\n#2.1. The authors present in the inset of Fig. 2a, the temperature-dependence of the electronic mobility and carrier density in\ntheir 112-Lao/STO interface. I t would be good to specify if those measurements where conducted in Hall bar devices; as\nwell as disclose the raw data of their Hall effect measurements (in Extended data?), as this is the first ever report of transport\nproperties at this specific interface.\n#2.2. This open the questions of whether the Hall effect purely linear? And at which magnetic field value was the Hall\ncoefficient estimated to calculate the sheet carrier densities?\n#.2.3. Have the authors attempted to perform a gate-tuning of the metallicity of the 2DEG at low T (below 10K), where the\nextremely large dielectric constant of the STO substrate should allow for an efficient electrostatic field-effect in back-gating\ngeometry.\nThis is possibly beyond the scope of the present work, but would represent a substantial addition to the current manuscript.\n#2.4. Obviously the corollary to the latter becomes how does the NLHE evolves as a function of sheet conductance (or back-\ngate voltage)?\n__________\nComment #3\n#3.1. The short discussion (lines 107-109) related to small upturn in Rs “commonly existing in LAO/STO 2DEG systems”\nshould be slightly moderated. In some sense, yes, it is commonly observed, but somewhat not in state-of-the art, clean and\noptimally doped systems [c.f. G. Herranz et al., Phys. Rev. Lett. 98, 216803 (2007)]. The upturn is, without debate, a sign of\nincreased scattering rate of electrons at low T. The underlying (microscopic) mechanism is much more difficult to pinpoint.\nThis is generally difficult due general tendency to preferentially form either cationic (La, Al) or anionic/oxygen vacancies\nduring the growth of the LAO layer, depending on the partial pressure of oxygen.\nI would certainly not put “Kondo” and “commonly” side-by-side. Despite the speculated role of oxygen deficiency clusters\nand their associated magnetism in STO-based systems, this is highly debated in the community; see also the decades old\ncontroversy of magnetism in d^0 (or d^0+delta) electron systems. The authors should provide a couple more references for\nthe upturn when the growth conditions are suboptimal, such as in the early reports by A. Brinkman et al. [Nature Materials 6,\n493–496 (2007). https://doi.org/10.1038/nmat1931], or other tentative explanation based on Kondo-screening [J. Ruhman et\nal., Phys. Rev. B 90, 125123 (2014) https://doi.org/10.1103/PhysRevB.90.125123]\nOn the other hand, in the seminal work of G. Herranz et al. [Scientific Reports 2, 758 (2012).\nhttps://doi.org/10.1038/srep00758], on (111) and (110)-LAO/STO interfaces, the upturn was very pronounced for thick (111)-\nand (110)- interfaces, as opposed to nearly absent for thinner LAO layers. However, in later reports, this can be entirely\nmitigated by finding optimal growth conditions, and this thickness-dependent upturn is no longer seen as an intrinsic\nproperty of the interface.\nThis brings me to the following questions.\n#3.2. What is the thickness of each of the LAO(hkl) layers studied in this work?\n#3.3. In Extended Data Fig.1, the authors present a typical X-Ray diffraction patterm for the (112) interface. How about all the\nother interfaces? Namely the LAO/STO(113), (114), (115), as well as the (110), (111) and (001) interfaces; all of which are\ndiscussed in relation to the NLHE displayed in Fig. 2d.\n__________\nComment #4\nRegarding the magnetooptical Kerr effect (MOKE) data and the circular photogalvanic effect (CPGE) presented in Figure 2.\n#4.1. I understand that, on one hand, the current-induced MOKE signal could be expected to be small. However, on the other\nhand, the sensitivity of (longitudinal) MOKE setups can reach down to a couple of nrad, so the signal reported by the authors\nis rather large in comparison. This brings me to the following: How come the Kerr rotation angle at the lowest probed current\ndensity (as well as throughout the whole [11-1] data set) is as large as 0.05 microrad (or 50 nrad)? Can the authors comment\non the tentative origin of this relatively large offset value?\n#4.2. The data in Fig.2e is presented for a current density of 2 A/m, that is, given the 10 microns cross-section of the Hall bar,\nfor a current amplitude of 20 microAmps. This is 40 times larger than any of the second harmonic I-V characteristic presented\nin Fig. 3a and Extended Data Fig. 4c. For consistency, and in the spirit of transparency, the authors could either display a\nsecondary x-axis (top of panel 1d) with the absolute value of the current.\n#4.3. Line 115 the authors claim “To show that indeed the interface possesses a monoclinic Cs symmetry, we have\nmeasured the CPGE […]”. This is incorrect and should be rephrased. The existence of a CPGE cannot prove per say the Cs\nsymmetry of the system. The CPGE is instead a potentially symmetry-allowed effect, and its anisotropy consistent with Cs\nsymmetry.\n__________\nComment #5\nLast but not least, the nonlinear Hall effect data (Fig. 3).\nThe symmetry-based arguments made in favor of the existence of a Berry curvature dipole-driven NLHE, and the first-\nprinciple calculations associated to it, are not being questioned here. Rather:\n#5.1. In relation to my previous point: what happens to the nonlinear I-V characteristics (Fig. 3a, Ext. Data Fig. 4) when\nsourcing current densities comparable to those used in Fig.2? And very importantly, do those NL I-V present a hysteretic\nbehaviou?\n#5.2. Same question goes for the longitudinal linear I-V characteristic, ideally I(omega) vs. Vxx (omega) (see Extended Data\nFig. 4b) - and I(omega) vs. Vyy(omega) - in the same current density range)?\nIn general, how can the authors rule out heating related effects or other sources of nonlinearities?\n#5.3. In Fig. 2c, how is each data point obtained? Do the authors fit a full nonlinear I-V characteristics at a given temperature,\nor is it being measured at constant current bias upon continuously sweeping the temperature (cooldown? or warming? And\nat which rate?).\n#5.4. Could the authors comment further on the quite peculiar temperature-dependence of the NLHE reported in Fig. 3c? is\nthere a tentative explanation for the strong dip at about 100K, and for a recovery of the effect’s magnitude at lower\ntemperature? Can this be understood in relation to the measured temperature-dependence of the sheet conductance and\nsheet carrier density? The argument regarding the Fermi level\n___________________\n## Methods section:\n# Line 324: annealed in which conditions? In vacuo in-situ? in air? In pure oxygen flow furnace?\n# Are all the (11X), the (001), the (110) and (111) interfaces grown in the same conditions?\n__________________\n### Miscellaneous:\n# In Fig. 1b: if I am not mistaken, the “x”-vector: [11-1] should point towards the bottom of the page, otherwise the arrow\nactually indicates the [-11-1] direction.\n# In Fig. 3a, is the y-axis not going from negative to positive values? Or is this deliberate?\n# Line 34: typo “Wely” instead of “Weyl”.\n# Line 82: “ultrasmooth” could be replaced by “atomically (?) sharp”, though the lack of spatially-resolved element specific\nEDXS (energy dispersive x-ray spectroscopy) may prevent to make that claim.\n# Line 181: “it exists”, “it” should be deleted here.\n# Line 229 and 230: “Fermi lines” (twice) could be replaced with “Fermi level” or “Fermi energy” (and give its corresponding\ndefinition).\n#Line 230: “exhibit symmetries concerning both kx = 0 and ky = 0”: which symmetries? What is meant here by “concerning\"?\nI am confused.\n# Line 250: “are” should be “were” or “have been” (due to the use of before at the end of the sentence).\n# Extended Data Fig. 7, what are the corresponding current densities?\n# On many occasions, articles such as “the” or “a/an” are missing, or on the contrary not necessary. I would urge the authors\nto take the necessary English language revisions.\nIn conclusion, I cannot recommend the present manuscript, as it is, for publication in Nature Communications, but I\nencourage the authors to address my comments and questions. Upon satisfying revisions, I am of the opinion that this work\nwould be of great interest to the broader solid-state physics community with a keen interest on nonlinear transport, and their\nconsequences for applications, in low-symmetry systems.", "reviewer_author_response": "We are very grateful to the Reviewers for their valuable and constructive comments.\nWe have accordingly revised the manuscript and the supporting information. Also, we\nhave changed the formality of this manuscript to suit for Nature communications. We\nbelieve we have positively addressed all comments made by the reviewers. A point-by-\npoint response to the reviewers’ comments and changes made in the revised manuscript\nare given below.\nQuestion 1.1: In this work, based on the crystallographic orientation engineering, the\nauthors successfully fabricated the (112)-oriented LaAlO /SrTiO conductive interface\n3 3\nwith the Cs symmetry. In particular, based on the electrical transport measurements,\nthey demonstrated the coexistence of the intriguing current-induced out-of-plane\nmagnetization, circular photogalvanic and giant nonlinear Hall effect effects. The\nresults undeniably enrich the fundamental understanding of the nonlinear transport\neffect in oxides through crystallographic orientation engineering. Before making a\ndecision on the paper, the authors should address the following major points:\nIs it reliable to confirm the Cs symmetry of the (112)-LaAlO /SrTiO heterostructure\n3 3\nfrom the CPGE currents shown in Fig. 2(b) and (c)? This is due to the fact that the\ndifference for the CPGE currents between (b) and (c) is not significant.\nAnswer 1.1: We appreciate the Reviewer’s recognition of the innovation and\nsignificance of our work. We have addressed all the issues raised by the Reviewer in\nthe response letter to make the message in this manuscript clearer.\nRegarding the circular photogalvanic effect (CPGE), it is indeed a reliable method to\ndetect the symmetry of materials of interest, especially the existence or breaking of\nmirror symmetry. The CPGE only functions in gyrotropic materials and is characterized\nby second order axial tensors. In particular, when light is shining along the\ninterface/surface normal direction, only the current generated by the CPGE could flow\nin the direction normal to the mirror symmetrical plane. In other words, there would be\nno CPGE current if there is more than one mirror plane in the interface/surface. Due to\nsuch peculiarity, the CPGE effect has already been used as a probe to detect the\nsymmetry as well as quantum geometric features in 2D layered materials and\ntopological materials [ref.: Xu et al., Nat. Phys. 14, 900 (2018); Ma et al., Nature 565,\n337 (2019); Duan et al., Nat. Nanotechnol. 18, 867 (2023)].\nIn our work, the CPGE generates a photocurrent of 390 pA in the [011] direction\n(marked as red curves in Fig. 2b and 2c), which confirms the breaking of the mirror\nplane 𝑀 . In contrast, the CPGE current along the [111] in-plane direction is just\n011\nabout 20 pA, which is significantly (~20 times) smaller than that along the [011]. Such\na small CPGE current along the [111] direction is likely due to reasons like electrode\nmisalignment, miscut during the substrate polishing process, etc. These technical issues\nslightly deviate either the current flowing direction from [111] direction or the interface\nfrom the perfect (112) orientation. Therefore, the manifestation of CPGE current and\nits substantial in-plane anisotropy is consistent with the C symmetry at the (112)-\ns\noriented LAO/STO interface, which is further consolidated with the behaviours of the\ninverse spin-galvanic effect and nonlinear Hall effect observed in this system.\nAction 1.1: We have added ’20 pA’ of the CPGE in the direction [111̅] in the\nmanuscript for highlighting the substantial contrast. Also, in order to make the figure\nclearer, we have removed the LPGE and CPGE fitting curves in Fig. 2b and 2c; instead,\nwe have put relevant description in the main text.\nQuestion 1.2: The spin-orbit coupling (SOC) effect should play a crucial role for the\nobserved current-induced out-of-plane magnetization and nonlinear Hall effect. It\nseems to me that the study on the SOC of the (112)-LaAlO /SrTiO heterostructure is\n3 3\nmissing. In the revision, I suggest the authors specify the magnitude of SOC from the\nanalysis of experimental data and band structures. Further, it is interesting to compare\nthe SOC parameters for different orientated LaAlO /SrTiO heterostructures.\n3 3\nAnswer 1.2: The spin–orbit coupling (SOC) effect is important to the anomalous Hall\neffect as reported in the literature [Ohuchi et al., Nat Commun 9, 213 (2018); Dugaev\net al., Phys. Rev. B 71, 224423 (2005)]. However, in the nonlinear Hall effect that\ninduced by BCD, the SOC effect does not play the critical role, such as demonstrated\nin the twisted Bilayer Graphene [Duan et al., Phys. Rev. Lett. 129, 18 (2022); Huang\net al., Phys. Rev. Lett. 131, 066301 (2023)] and WTe [Ma et al., Nature 565, 337\n2\n(2019)]. In our LaAlO /SrTiO interface, we claim here that the SOC effect and the\n3 3\nassociated spin contribution do not play a critical role. In 2023, M. T. Mercaldo et al.\nreported that orbital degrees of freedom combined with crystal field effects can\nintrinsically generate strong Berry curvature (BC) features in the 2D electron gas\nsystem, such as the LaAlO /SrTiO interface, in the absence of SOC [Mercaldo et al.,\n3 3\nnpj Quantum Mater. 8, 12 (2023)]. Later, E. Lesne et al. experimentally studied the\nnonlinear transport and BC features of (111)-oriented LaAlO /SrTiO interface at low\n3 3\ntemperature [Lesne et al., Nat. Mater. 22, 576 (2023)]. Their results show the orbital-\nsourced Berry curvature dipole is two orders of magnitude larger than the spin-sourced\none, indicating a minor role of the SOC effect in the nonlinear Hall effect. Regarding\nthe charge-to-magnetization conversion, Johansson et al. showed theoretically that the\norbital Rashba conversion efficiency should dominate in this system and be at least an\norder of magnitude larger than the spin effect [Johansson et al., Phys. Rev. Res. 3,\n013275 (2021)]. Experimentally, A. E. Hamdi et al. showed that the orbital contribution\ndominates in charge-magnetization conversion process of the (001)-LaAlO /SrTiO\n3 3\ninterface, which is about 16 times the spin effect [Hamdi et al., Nat. Phys. 19, 1855\n(2023)]. Overall, previous research on the LaAlO /SrTiO interface supports the\n3 3\ndominant contribution of orbital degree of freedom and a minor role of the SOC effect.\nTo evaluate the role of SOC in our system, we first analysed the band structures of the\n(112)-LaAlO /SrTiO heterostructure, focusing on the electronic states near the Fermi\n3 3\nlevel that are very related to transport properties and BC effects. For the Berry curvature\ndipole (BCD) under investigation in this work, its behaviour is critically dependent on\nthe electronic structure information within a very narrow energy range near the Fermi\nlevel, which is clearly demonstrated in the formula\n∂E\nn β\nD = ∫[d𝐤]D (𝐤) = ∫[d𝐤]∑ Ω 𝛿(𝐸 −𝜇). (1.1)\nαβ αβ ∂k n 𝑛\nα\nn\nAs shown in Fig. R1.1(a), we have shown the energy bands within the range of ±0.2eV\naround the Fermi level along the Γ−X path. The band splitting depicted here is caused\nby SOC. It can be observed that the band splitting is less than 10 meV across the entire\nplotting range, contributing the maximum band splitting (~10 meV) at an energy 0.13\neV above the Fermi level. In Fig. R1.1(b), we have magnified the region where the\nbands cross the Fermi level, revealing a band splitting of approximately 2 meV. These\nresults indicate that the contribution of SOC to the nonlinear Hall transport behaviour\nin (112)-LaAlO /SrTiO is weak, especially at room temperature.\n3 3\nFig. R1.1 SOC-induced band splitting near the Fermi level in the (112)-LaAlO /SrTiO heterostructure.\n3 3\n(a) band structure along the Γ–X direction near the Fermi level. The red dashed box highlights the region\nof interest. (b) Magnified view of the boxed region in (a), showing a spin-orbit-induced band splitting of\napproximately 2.0 meV near the Fermi energy, as indicated by the red arrow.\nTo elucidate the contribution of the SOC effect on the geometrical feature of the (112)-\ninterface, we have calculated the BC and BCD features without the SOC effect and\ncompared them to those with the SOC effect (as shown in Fig. R1.2). Without the SOC\neffect, the BC and BCD shows spin degenerated features. The integrated BCD without\nthe SOC effect shows a similar magnitude to that taking SOC effect into consideration.\nOn the other hand, the SOC effect lifts the spin degeneracy and induces anti-crossing\nfeatures in the band structure, which modifies the dependence of the BCD on the Fermi\nlevel position. In a word, our calculation indicates the major contribution of the BCD\nin the (112)-LaAlO /SrTiO interface arises from the orbital degree of freedom whereas\n3 3\nthe SOC effect, which activates the spin-contribution, modifies the quantum\ngeometrical feature in a minor way. This is consistent with previous works.\nFig. R1.2 Effect of SOC on Berry curvature and BCD distributions. Berry curvature distribution (a)\nwith SOC and (c) without SOC. BCD distribution (b) with SOC and (d) without SOC.\nIn addition, to address the reviewer’s suggestion regarding orientation dependence, we\nperformed a supplementary calculation on the (111)-LaAlO /SrTiO heterostructure.\n3 3\nThe SOC-induced band splitting near the Fermi level reaches approximately 4.2 meV,\nwhile slightly larger than that the (112) case (see Fig. R1.3a, b).\nFig. R1.3 SOC-induced band splitting near the Fermi level in the (111)-LaAlO /SrTiO heterostructure.\n3 3\n(a) band structure along the Γ–X direction near the Fermi level. The red dashed box highlights the region\nof interest. (b) Magnified view of the boxed region in (a), showing a spin-orbit-induced band splitting of\napproximately 4.2 meV near the Fermi energy, larger than that of (112)-LaAlO /SrTiO .\n3 3\nAction 1.2: We have now put the calculation and discussion related to the SOC effects\non the BCD and Berry curvature in SI.\nQuestion 1.3: It is known that the nonlinear Hall effect can also be contributed by the\nDrude and quantum metric mechanisms. In this work, the authors focus on the\ncontribution from the Berry curvature dipole. The authors should specify how to\nexclude the other possible contributions to the nonlinear Hall conductivity.\nAnswer 1.3: Under the relaxation-time approximation, there are three contributions to\nthe second-order nonlinear Hall effect, i.e., the Berry-curvature dipole mechanism\n[Sodemann et al. Phys. Rev. Lett. 115, 216806 (2015)], the quantum-metric dipole\nmechanism [Gao et al., Phys. Rev. Lett. 112, 166601 (2014)], and the Drude mechanism\n[Wang et al., Phys. Rev. Lett. 127, 277201(2021)]. Although all three mechanisms\nrequire inversion-symmetry breaking, the last two mechanisms require additional\nbreaking of time-reversal symmetry [Wang et al., Phys. Rev. Lett. 127, 277201(2021)].\nSince our system is nonmagnetic and its time reversal symmetry persists, only the\nBerry-curvature dipole mechanism can contribute to the nonlinear Hall effect.\nBeyond the relaxation time approximation, there are also side jumping and screw\nscattering contributions to the nonlinear Hall effect. These extrinsic contributions will\nbe discussed in Question 1.4.\nAction 1.3: We have now put a statement in the main text marked in red saying that the\nquantum metric mechanism Drude mechanism only functions in systems without time\nreversal symmetry, which are not applicable in our LaAlO /SrTiO interface.\n3 3\nQuestion 1.4: (1) Since the BCD has been calculated as shown in Fig. 4d, I suggest the\nauthors further calculate the nonlinear Hall conductivity and compare to experimental\ndata. (2) From Fig. 3(c), the nonlinear Hall coefficient for the (112)-LaAlO /SrTiO\n3 3\nheterostructure increases monotonously with temperature (starting from 100 K). This\nis uncommon. It is known that the SOC effect will be weakened by the high temperature.\nIf the nonlinear Hall effect is contributed by the BCD due to SOC, the nonlinear Hall\ncoefficient should decrease with temperature. This should be further elaborated. In\naddition, why can the extrinsic contributions such as side jumping and screw scatting\nbe excluded at high temperatures.\nAnswer 1.4: Since these comments all relate to the BCD and various contributions to\nthe nonlinear Hall effect, we would like to systematically discuss them here.\nIn addition to the intrinsic contribution of the BCD, extrinsic contributions, including\nside jumping and screw scattering, play an important and even dominant role in the\nnonlinear Hall effect (NLHE). To resolve various contributions, we employ the scaling\nmethod at both low (< 30 K) and high temperature (> 100 K) ranges (Fig. R1.4). Data\nbetween 30K and 100K are challenging to analyze due to the occurrence of phase\ntransition with this range [Lesne et al., Nat. Mater. 22, 576 (2023)]. The two curves in\nFig. R1.4 are obtained by corresponding the NLHE coefficient 𝐸2𝜔⁄(𝐸𝜔)2 and\n𝑥𝑦 𝑥𝑥\nconductivity 𝜎 at each temperature. According to the theory [Du et al., Nat. Commun.\n𝑠\n10 3047 (2019)], the scaling law for NLHE can be written as: 𝐸2𝜔⁄(𝐸𝜔)2 =\n𝑥𝑦 𝑥𝑥\n𝐴 (\n𝜎𝑠)2\n+𝐴\n𝜎𝑠\n+𝐴 , where 𝐴 , 𝐴 , 𝐴 are the scaling parameters, 𝜎 is the residual\n1 2 3 1 2 3 0\n𝜎0 𝜎0\nconductivity that we estimate its value as that measured at 1.55 K. The fitting results of\n𝐴 , 𝐴 , 𝐴 are shown in Table R1.1. Based on the NLHE scaling law, the scaling\n1 2 3\nparameters are expressed as:\n𝐴 = 𝐶𝑠𝑘,2 +(𝐶𝑠𝑘,1 +𝐶𝑠𝑘,1 −𝐶𝑠𝑘,1) (1.2)\n1 00 11 01\n𝐴 = 𝐶𝑠𝑘,1 −2𝐶𝑠𝑘,1 +𝐶 𝑠𝑗 −𝐶 𝑠𝑗 (1.3)\n2 01 11 0 1\n𝐴 = 𝐶 +𝐶 𝑠𝑗 +𝐶𝑠𝑘,1 (1.4)\n3 𝑖𝑛 1 11\nwhere 𝐶 is the intrinsic contribution, 𝐶 𝑠𝑗 the side-jump, 𝐶𝑠𝑘,1 the Gaussian skew-\n𝑖𝑛 𝑖 𝑖𝑗\nscattering, 𝐶𝑠𝑘,2 the non-Gaussian skew-scattering, and 𝑖, 𝑗 = 0 (1) denotes the defects\n(phonon) scattering source. There are seven parameters, i.e., seven different\ncontributions in the above three equations, which is impossible to resolve quantitatively.\nTo get a tentative insight into these contributions, here we need to take a bold but\nreasonable assumption.\nFig. R1.4 Scaling law analysis of the NLHE at temperature (a) lower than 30 K and (b) higher than 100\nK.\nTable. R1.1 Scaling parameters\nA A A\n1 2 3\nT < 30 K 7.56 -42.4 56.9\nT> 100 K 90.5 -112.4 36.6\nSince 𝐶𝑠𝑘,2 only exist in the 𝐴 parameter, the non-Gaussian skew-scattering only\n1\nplays a secondary role and we neglect it [J. Duan et al., PRL 129, 186801 (2022)]. In\nthe temperature range below 30 K, the parameter 𝐴 is almost one order of magnitude\n1\nsmaller than 𝐴 and 𝐴 . This indicates that the screw scattering contributions play a\n2 3\nminor role in the low-temperature region. Also, the phonon density is largely\n𝑠𝑗\nsuppressed at low temperatures; it would be reasonable to assume 𝐶 (i.e., side jump\n0\n𝑠𝑗\ndue to phonons) is much larger the 𝐶 (i.e., side jump due to phonons). Thus, we take\n1\n𝑠𝑗\n𝐴 = −42.4 ≈ 𝐶 . Similarly, 𝐴 = 56.9 ≈ 𝐶 .\n2 0 3 𝑖𝑛\nIn the temperature range above 100 K, the parameter 𝐴 shares a similar magnitude\n1\nwith 𝐴 but with opposite sign, while 𝐴 is about three times smaller. Based on\n2 3\nequations (1.2) and (1.3), the opposite sign between 𝐴 and 𝐴 is most likely due to the\n1 2\nopposite sign of 𝐶𝑠𝑘,1 and 𝐶𝑠𝑘,1 in these two equations. Since 𝐶𝑠𝑘,1(screw scattering\n11 01 00\ndue to defects) only exist in 𝐴 , it may play a minor role here. Moreover, since the side\n1\n𝑠𝑗\njump process is insensitive to the temperature, we assume 𝐶 remains as a constant\n0\nover the whole temperature range. Thus, equations (1.2-1.4) can be approximated as:\n𝐴 = 𝐶𝑠𝑘,1 −𝐶𝑠𝑘,1 = 90.5 (1.5)\n1 11 01\n𝐴 = 𝐶𝑠𝑘,1 −2𝐶𝑠𝑘,1 −𝐶 𝑠𝑗 =-70 (1.6)\n2 01 11 1\n𝐴 = 𝐶 +𝐶 𝑠𝑗 +𝐶𝑠𝑘,1 = 36.6 (1.7)\n3 𝑖𝑛 1 11\nThus, we get 𝐶 = 57, which almost equal to that derived at low temperature range,\n𝑖𝑛\nsupporting above approximation.\nBased on above analysis, we can reach here a qualitative understanding of the origins\nand temperature dependence of the NLHE in the (112)-LaAlO /SrTiO interface. Clear,\n3 3\nboth the intrinsic and extrinsic contributions play important but compensating roles in\nthis system. For example, the extrinsic contribution 𝐶 𝑠𝑗 +𝐶𝑠𝑘,1 (= −20.4) is opposite\n1 11\nto that induced by Berry curvature dipole (𝐶 = 57). The peculiar temperature\n𝑖𝑛\ndependence can be ascribed to the role of the parameter 𝐴 , i.e., the contribution of\n2\nscrew scattering and side jump. This also means that the NLHE effect of the (112)-\ninterface is highly related to its conductivity with an approximated quartic dependence.\nWhen the temperature decreases from 300 K to 100 K, the conductivity and mobility\nincreases, the screw scattering process is enhanced, inducing a negative contribution\nthat compensates the contribution of the intrinsic one. At temperatures below 30 K, the\ndensity of phonons is significantly reduced and thus, the related screw scattering\ncontributions (𝐶𝑠𝑘,1 and 𝐶𝑠𝑘,1) are suppressed. In this circumstance, the side jump and\n01 11\nintrinsic effect dominate. With further decreasing temperature towards 1.55 K, the\nconductivity decreases, the compensating contribution of side jump by defects is\nreduced, leading to an enhanced NLHE.\nWe can also infer the magnitude of the Berry curvature dipole based on the value of\n𝐶 = 57 at low temperature. To this end, we employ the formula 𝐷 =\n𝑖𝑛 𝑦𝑧\n𝑉𝑥 2 𝑦 𝜔𝜎𝑥 3 𝑥𝑊2ℏ2 = 𝐶 𝜎 2ℏ2 , where 𝜏 is the relaxation time, ℏ is the Dirac constant 𝐼𝜔 is\n(𝐼𝜔)2 𝑒3𝜏 𝑖𝑛 𝑥𝑥 𝑒3𝜏 𝑥𝑥\n𝑥𝑥\nthe sourced current,𝑉2𝜔 is the nonlinear Hall voltage, 𝜎 is the (sheet) longitudinal\n𝑥𝑦 𝑥𝑥\nconductivity, 𝑊 is the Hall bar width. We estimate the 𝜏 by calculating 𝜏 =\n𝜇H𝑚∗\n,\n𝑒\nwhere 𝑚∗ effective mass, 𝜇 is the Hall mobility. For convenience, we adopt 𝑚∗ of\nH\n(111)-LaAlO /SrTiO reported, that is 3.49 𝑚 , then we have 𝜏 at 1.55 K about 1.0 ps.\n3 3 𝑒\nUsing the NLHE coefficient obtained, we estimated BCD to be ~440 nm at 1.55 K. The\nBCD magnitude derived from experimental results is about one order of magnitude\nlarger than that of the (111)-LaAlO /SrTiO interface at low temperature (< 30 K).\n3 3\nThere is a gap between the BCD magnitude derived from experimental data and that\nfrom the DFT calculation. In fact, it is common that the nonlinear transport coefficient\nfrom DFT calculations are smaller than experimental result (See Refs [E. Lesne et al.,\nNat. Mater. 22, 576 (2023); M. T. Mercaldo et al., npj Quantum Mater. 8, 12 (2023);\nH. Wang et al., npj Comput. Mater. 5, 119 (2019); Q. Ma et al., Nature 565, 337\n(2019)]).\nIn our case, to perform DFT calculations for the (112) high-index interface, we\nconstructed a superlattice structure containing 60 atoms, that are much larger than the\ntypical unit cells used in common studies, which usually include only a few or a dozen\natoms [J.-S. You et al., Phys. Rev. B 98, 121109(R) (2018); H. Wang et al., npj Comput.\nMater. 5, 119 (2019); E. Wang et al., Phys. Rev. Lett. 132, 266802 (2024)]. While the\nstructure is tractable for conventional DFT tasks such as structural relaxation and band\nanalysis, the BCD calculation is considerably more challenging, as it requires extremely\ndense k-space sampling to achieve convergence due to the sensitivity to fine features\nnear the Fermi level. To overcome this difficulty, we employed Wannier interpolation\nand carried out BCD results based on a 1000×1000 k-point mesh. We note that\nconstructing well-localized Wannier functions for such a large supercell is itself a\nnontrivial task, requiring careful disentanglement and projection procedures.\nMoreover, the 2D electron gas in the interface partially comes from the 3d orbitals in\nSrTiO , for which the correlation effect can be large. This correlation effect cannot be\n3\nwell captured by the DFT calculation. Modelling the band scheme in theory is further\ncomplicated by ill-defined oxygen stoichiometries and high dielectric permittivity\n[Gabel et al., Adv. Electron. Mater. 8, 2101006 (2022)]. Nevertheless, our DFT\ncalculation still represents a state-of-the-art theoretical calculation of this newly\ndeveloped system, offering valuable insight into its band structure, distribution of\ngeometrical features in k-space and Fermi energy dependence of the BCD. We hope\nfuture work would bridge the gap between the DFT and experimental characterization.\nAction 1.4: The above discussion of the BCD estimation based on experimental data\nhas been added to the discussion section of the manuscript. The scaling analysis related\nto the temperature dependence is added to the Supplementary Information.\nQuestion 1.5 Minor points: In the main text, Fig. 3b should be Fig. 3a, Fig. 3c should\nbe Fig. 3b etc.\nAnswer 1.5: We thank the reviewer for pointing out these errors.\nAction 1.5: We have revised these minor errors in the manuscript.\nResponse to Reviewer #2:\nThis study proposes a strategy for controlling in-plane mirror symmetries at oxide\ninterfaces by engineering the crystallographic orientation of heterostructures. By\nutilizing a (112)-oriented LaAlO₃/SrTiO₃ interface, the high-index orientation breaks\nmost mirror symmetries, resulting in Cs point symmetry and metallic conduction. This\nstructural tuning enables the emergence of novel phenomena, such as a giant nonlinear\nHall effect, circular photogalvanic effect (CPGE), and current-induced\nmagnetization—all observable at room temperature. The nonlinear Hall response is\ncomparable to that observed in Weyl and Dirac systems. The experimental data are\nclear and compelling; however, I am not entirely convinced by the authors'\ninterpretation of the observed phenomena. I would like the authors to address the\nfollowing points:\nQuestion 2.1: Figure 1e appears symmetric with respect to the [1-10] mirror plane.\nHowever, considering the effects of lattice mismatch and strain, this symmetry may not\nbe preserved throughout the entire structure. Furthermore, if oxygen octahedral\nrotations are present, the symmetry would be broken. Additionally, a zigzag pattern is\nvisible at the interface between the black and white regions, which is typically\nindicative of atomic intermixing or facet formation. Therefore, the claim that a [1-10] mirror plane\nexists is not readily acceptable. The observed transport anisotropy may arise from this\nzigzag pattern, which could also produce a stray magnetic field\nperpendicular to the substrate. This, in turn, may explain the current-induced\nmagnetization.\nAnswer 2.1: We would like to thank the Reviewer for bringing the interface symmetry\ninto further discussion. We would like to respond to this comment from the following\nfour aspects.\nFirstly, we would like to stress that the peculiarity of the method developed in our work,\ni.e., engineering the crystallographic orientation of interfaces, is to break pristine mirror\nsymmetry of the component lattices, represented by the SrTiO crystal in our work. The\n3\nmain advantage of choosing (112)-orientation as an example in our work is to\ndeliberately retain just one mirror symmetry 𝑀 at the interface. By doing so, not\n[11̅0]\nonly can we obtain various physical effects, i.e., nonlinear Hall effect, generation of\nout-of-plane magnetization and circular photogalvanic effect, but also induce\nsignificant in-plane anisotropy of these effects. Particularly, this anisotropy can be the\nsmoking gun evidence that proves the authenticity of these effects. The last remaining\nmirror plane 𝑀 can be readily lifted by using higher Miller-index orientation, such\n[11̅0]\nas (122), as elucidated in the Method section of the main text.\nSecondly, in an ideal situation, the designed mirror plane 𝑀 at the (112)-interface\n[11̅0]\ncan persist only if the (112)-interface is perfect without any defects. We agree that any\nimperfections of the interface that are inevitable in the experimental process, such as\nmiscut induced by the polishing process, interface interdiffusion, interlayer strain, etc.,\nwould impact the 𝑀 mirror plane to a certain extent. However, these imperfections\n[11̅0]\nhappen in a rather random and uncontrollable manner. Thus, their perturbation on the\nmirror plane and the associated physical effects would be reduced to a negligible level\nonce summed over a large area, such as a Hall bar channel. This is confirmed by three\npieces of evidence in our work. (1) If these experimental imperfections would break the\nin-plane mirror symmetry and significantly modulate interface properties, we would\nhave already observed the nonlinear Hall effect at (001), (110), (111)-oriented interface,\nas these imperfects can also happens in these orientations. Clearly, this is not the case\nas shown in Fig. 3d of the main text. (2) If the 𝑀 is significantly broken as\n[11̅0]\nhappened in 𝑀 and 𝑀 , there would also be substantial nonlinear Hall effect\n[110] [001]\nand current-induced magnetization when the current is flowing along the [111̅] in-\nplane direction. This is again not the case in our work. (3) We have also characterized\nthe interface structure over a 54 nm width by STEM (Fig. R2.2), which shows the\ninterface is flat within a 0.5 nm fluctuation without defects occurring in an ordered\nmanner. Therefore, based on the group theory argument and the behaviours of the\nphysical effect at the (112)-interface, the C point symmetry is the most reasonable one\ns\nto describe its structure.\nThirdly, regarding the zig-zag pattern at the interface, it may happen due to interface\nreconstruction. However, this zig-zag pattern is a co-consequence of crystallographic\nengineering of the interface, rather than the origin of emergent physical effects. In other\nwords, the zig-zag pattern itself cannot induce these effects reported in our work. To\nsupport our claim, we studied (110)-oriented LaAlO /SrTiO interface that has been\n3 3\nreported to show the zig-zag pattern [ref.: A. Annadi et al., Nat Commun. 4, 1838 (2013);\nK. Gopinadhan et al., Adv. Electron. Mater. 1, 1500114 (2015)]. As shown in Fig. R2.2,\nthere are no Kerr signals with current flowing along both in-plane directions in the\n(110)- LaAlO /SrTiO heterostructure, which was prepared with the same conditions as\n3 3\n(112)-one. Moreover, (110)-LaAlO /SrTiO interface exhibits negligible nonlinear Hall\n3 3\nresponse.\nAt last, regarding a stray magnetic field perpendicular to the substrate produced by the\nzigzag pattern, we think the Reviewer refers to the work reported by K. Gopinadhan et\nal. [Adv. Electron. Mater. 1, 1500114 (2015)]. However, as mentioned above, zig-zag\nitself cannot allow the generation of out-of-plane magnetization by a current, evidenced\nby the absence of the MOKE signal in (110)-oriented LaAlO /SrTiO interface.\n3 3\nFurthermore, to the best of our knowledge, there is not yet any report of observation of\nroom temperature magnetization/magnetic field in the LaAlO /SrTiO interface,\n3 3\nirrespective of the orientations. Thus, the only mechanism that can generate out-of-\nplane magnetization over the whole conducting channel by flowing a current would be\nthe inverse spin-galvanic effect.\nFig. R2.1 Cross-sectional HAADF-STEM image of (112)-LaAlO /SrTiO over a 54 nm region.\n3 3\nFig. R2.2 Kerr rotation 𝜃 as a function of current amplitude measured in directions of [001] and [11̅0]\nK\nof sample (110)-LaAlO /SrTiO .\n3 3\nAction 2.1: We have now put the HAADF-STEM image and the MOKE measurement\nin the SI to show the flatness of the sample interfaces. We have also added the above\ndiscussion about the interface imperfection of the mirror plane into the SI marked in\nred.\nQuestion 2.2: There is no clear description of how electrical contacts were made\nthrough the LaAlO layer, which is known to have high resistance.\ntechnique. This technique has been widely accepted in the LaAlO /SrTiO interface\n3 3\ntransport measurement experiments, as shown in the work [Lense et al., Nat. Mater.,\n22, 5 (2023); Chen et al., Science, 372, 6543 (2021)].\nAction 2.2: We have already put the description of electrical contact in the “electronic\ntransport measurements” section of Method (marked in red). “In our experiments, the\nHall bars and disc bars are connected to the chip carrier with aluminium wires\npenetrating through the LaAlO thin film using the ultrasonic wedge-bonding\n3\ntechnique.” No further action is taken here.\nQuestion 2.3: The mobility should also exhibit anisotropy, but this is not discussed.\nAnswer 2.3: We agree with the reviewer that the carrier mobility should have an\nanisotropic distribution as the resistivity/conductivity does. Fig. 3b of the main text\nshows that the conductivity/resistivity of the (112)-oriented LaAlO /SrTiO interface\n3 3\npossesses substantial in-plane anisotropy. As we know, the conductivity 𝜎 is\ndetermined by carrier density 𝑛 and carrier mobility 𝜇, i.e., 𝜎 = 𝑛𝑒𝜇. Since the carrier\ndensity 𝑛 is a scalar quantity in the system, the only origin of the conductivity\nanisotropy is mobility.\nTo confirm the mobility anisotropy and its temperature dependence, we conducted the\nlinear Hall measurements in a wide temperature range. As shown in the Fig. R2.3, the\nmobility in direction [11̅0] is higher than that in the direction [111̅], especially in the\nlow-temperature range, which is consistent with our resistivity result.\nFig. R2.3 (a) Carrier mobility measured in direction [11̅0] and [111̅]. (b) carrier mobility anisotropy\n𝑢[111̅]⁄𝑢[11̅0].\n𝑠 𝑠\nAction 2.3: We have added the anisotropic mobility in SI and related description.\nQuestion 2.4: The definition of theta is unclear, so I hope the authors specify its\nreference direction. Additionally, the term “LPGE” (linear photogalvanic effect) is not\ndefined. What is its physical origin in this system? In Figure 2b, the fit is poor. At\naround 160 degrees, the experimental data deviate significantly from the fitted curve.\nIn Figure 2c, the CPGE component appears absent. The red fit line diverges\nconsiderably from the experimental data.\nAnswer 2.4: These questions are related to the photogalvanic effect, we answer here\ntogether.\nFirst, regarding the linear photogalvanic effect, it refers to the generation of a\nphotocurrent under uniform illumination in a non-centrosymmetric material. In some\nliterature, it is called the bulk photovoltaic effect [Z. Dai & A. M. Rappe, Chem. Phys.\nRev. 4, 011303 (2023)]. The microscopic origin of the photogalvanic effect is the\nasymmetric momentum distribution of electrons and holes in space due to the shift\ncurrent and the asymmetric scattering between photon, phonon and electrons in the non-\ncentrosymmetric crystal [Belinicher et al,. Sov. Phys. Usp. 23, 199 (1980)]. In contrast\nto the circular photogalvanic effect that only functions in the gyrotropic material and is\nsensitive to the mirror symmetry, the linear photogalvanic effect exists in any kind of\nnon-centrosymmetric material.\nSecond, regarding the experiment data and their fittings, we think the reviewer might\nmisunderstand these curves in Fig. 2b and 2c. The experiment data (labelled as Exp. in\nthe figure) comprised of both LPGE and CPGE components are marked as grey dots in\nthe figures. The experimental data is then fitted by the curve labelled as ‘Total fit’ (see\nmethods and Fig. R2.4). The blue (LPGE) and red (CPGE) curves shown in the figure\nare the components derived from the fitting (a similar fitting method is given in [Duan\net al., Nat. Nanotechnol. 18, 867 (2023); Knoche et al., Nat Commun 12, 282 (2021).],\nWe have respectively drawn the experimental data, their fitting and the derived\ncomponents of LPGE and CPGE in Fig. R2.4. Clearly, the fitting on experimental data\nis good.\nThird, regarding the definition 𝜃𝑄𝑊𝑃. 𝜃𝑄𝑊𝑃 is the angle between the nominal fast axis\nof the QWP and the polarization direction of the incident linear polarized light [Nat\nCommun 12, 282 (2021)].\nFig. R2.4 photocurrent as a function of QWP. (a) Experimental data and its total fit. (b) The LPGE\ncomponent of the total fit. (c) The CPGE component of the total fit.\nAction 2.4: We have now added a reference for the definition of the linear\nphotogalvanic effect. We have also added the description of the reference direction of\n𝜃𝑄𝑊𝑃 in the Methods. In addition, in order to make the figure clearer, we have removed\nthe LPGE and CPGE fitting curves in Fig. 2b and 2c; instead, we have put relevant\ndescription in the main text.\nQuestion 2.5: Figure 3f, which is referenced in line 194, appears to be missing from\nthe manuscript.\nAnswer 2.5: We thank the reviewer for pointing out this typo. Figure 3f is Figure 3d.\nAction 2.5: We have now corrected Figure 3f in the sentence to Figure 3d.\nQuestion 2.6: The carrier concentration is not discussed. Can the position of the Fermi\nlevel account for the carrier density observed in the sample?\nAnswer 2.6: The Fermi level position is highly related to the carrier density. The carrier\ndensity of our sample, as presented in Fig. 2a in the main text, decreases with the\ntemperature and saturates in the low temperature limit, with its magnitude in the order\nof 1013 −1014 cm−2 in the whole temperature range. These features are similar to\nother reported LAO/STO systems [Lesne et al., Nat. Mater. 22, 576 (2023); Herranz et\nal., Sci Rep 2, 758 (2012)]. In order to unveil the relationship between the carrier density\nand the Fermi level shift, we conducted the DFT calculations, as shown in Fig. R2.5.\nAt the pristine Fermi energy, the carrier density is approximately 2.6×1014 cm-2. As the\nFermi level is raised by +0.1 eV, the carrier density increases to 5.3×1014 cm-2;\nconversely, lowering the Fermi level by -0.1 eV results in a reduced density of 1.1×1014\ncm-2. Although other effects might also affect the carrier density, the present\ncalculations imply a strong connection between the Fermi level and carrier density. We\nalso would like to address here that the calculated carrier density is based on the ideal\nmodel at zero temperature and might diverge from the real situation, as the\nexperimentally observed carrier density at low temperature (such as 2 K) is usually at\nthe order of 1013 cm-2.\nFig. R2.5 Calculated 2D carrier density versus Fermi level shift in the (112)-LaAlO /SrTiO\n3 3\nheterostructure.\nAction 2.6: we have now put the discussion of the carrier density in the supplementary\ninformation.\nResponse to Reviewer #3:\nIn their manuscript titled “Magnetization generation and giant nonlinear transport at\nsymmetry-engineered interfaces via crystallographic orientation”, the authors H.-B.\nZhang et al. present a series of experimental results in high-Miller-index\nLaAlO /SrTiO (hkl) heterostructures which is argued to derive from the specific\n3 3\nnonvanishing Berry curvature and Berry curvature dipole densities hosted in the\nsystems under investigation. The overall set of experiments and symmetry-based\nphenomenological models are consistent with the low Cs point-group of their\nheterointerfaces hosting the 2DEGs.\nQuestion 3.1: However, there are a number of experimental details which are missing,\nsome desirable more detailed discussions regarding e.g., the peculiar temperature-\ndependence of their nonlinear Hall effect (NLHE), or with regards to the magnitude of\nthe MOKE and the NLHE at different (11X) interfaces; and possibly a demonstration\nthat the systems under investigation are gate-tuneable 2DEGs.\nAnswer 3.1: We appreciate the Reviewer’s recognition of the significance of our work.\nBelow, we will address all the issues raised by the Reviewer in the response letter to\nmake the message in this manuscript clearer.\nQuestion 3.1.1. It would be useful/desirable to present In Fig.1, layer-resolved effective\ncharges, in the ionic limit, of unreconstructed (11X)-LAO/STO heterostructures as in,\ne.g., the seminal work of N. Nakagawa et al. [Nature Materials 5, 204–209 (2006).\nhttps://doi.org/10.1038/nmat1569] in (001)-LAO/STO, or that of G. Herranz et al.\n[Scientific Reports 2, 758 (2012). https://doi.org/10.1038/srep00758] on (111) and\n(110)-LAO/STO interfaces. Given the (111)-LAO/STO interface is polar, I would\nassume that the (11X) interfaces are as well, but I did not “do the math”.\nQuestion 3.1.2. Problematically, it seems to me that the authors argue the existence of\na polar interface - as a result of the potential gradient at the interface - and cite Ref.\n30 [G. Singh-Bhalla et al., Nat. Phys. 7, 80-86 (2011).], c.f. lines 83-85.\nWithout getting into the whole debate of the “polar catastrophe scenario”, this is near\ncomplete inversion of the roles of polar interfaces and existence of 2DEGs therein. The\npolar interface pre-exists the 2DEG. And in virtue of the electrostatic boundary\ncondition (in a self-consistent Poisson-Schrödinger picture), there is no further\nconfinement potential to host the 2DEG without additional charges: meaning one needs\nelectron donors, besides the band-bending due to energy\nmismatched conduction and valence bands at LAO/STO interfaces. More appropriate References would instead\ninclude [W.-J. Son et al., Phys. Rev. B 79, 245411 (2009)] and [P. Delugas et al.,\nPhys. Rev. Lett. 106, 166807 (2011)].\nMoreover, it is my understanding that C. Cazorla & M. Stengel [Phys. Rev. B 85,\n075426 (2012)] have since shown that for metal-capped − stoichiometric − LAO/STO\nsystems, the LAO internal polar field is a decreasing function of LAO film’s\nthickness, and depends on the electrostatic boundary conditions; or more precisely on the areal\ncarrier density at both\nM/LAO and LAO/STO interfaces. Such that there is and I quote from them “no intrinsic built-in LAO electric field exists in the\nshort-circuited Pt/LAO/STO capacitor system […]” which “[…] in practice rules out the hypothesis\nof Zener breakdown, which\nwas used in Ref. [G. Singh-Bhalla et al., Nat. Phys. 7, 80-86 (2011).]” However if the authors want to argue instead that the pristine interfaces are polar, and that interfacial polarity arises from\nthe existence of an electrostatic confinement potential, then further discussions are needed.\nAnswer 3.1.1 and 3.1.2: Since both questions are related to the polarity and formation\nof the conductive interface, we would like to respond here in a comprehensive way.\nFirstly, we would like to thank the Reviewer for bringing the “interface polarity” into\nfurther discussion and for the suggestion to revise the references. The polar nature of\nthe LAO/STO interface is indeed an essential ingredient for all the physical effects\ndiscussed in our manuscript, which deserves further discussion. All in all, we agree\nwith the Reviewer’s comment here.\nIt is worth noting that the “polar conductive interface” studied in our work can be\ndefined as “a conductive interface with electrical carriers experiencing an electrical\npotential with an asymmetrical distribution, i.e., a potential gradient”. This polarity of\nthe conductive interface activates the Rashba spin-orbit coupling and/or Rashba orbital-\norbit coupling, underpinning all the exotic physical effect in our work. In our opinion,\nsuch an asymmetrical potential distribution (i.e. polarity) is not unusual in the\nconductive interface in heterostructures. This is because, by the definition of\n“heterostructure”, these two component layers have different chemical elements and\nthus, chemical potential. In the case of the LaAlO /SrTiO interface, the electrical\n3 3\ncarriers are confined in a narrow interface region of the SrTiO side with no carriers in\n3\nthe insulating LaAlO capping layer and the bulk of the SrTiO substrate. Therefore,\n3 3\nthe distribution of electron density and the electrical potential are asymmetrical, i.e.,\npolar. As pointed out by the Reviewer, this has been theoretically discussed by W.-J.\nSon et al., [Son, et al., PRB 79, 245411 (2009)] and P. Delugas et al., [Delugas et al.,\nPRL 106, 166807 (2011)]. Also, it has been experimentally analyzed by J. Gabel et al.\n[Gabel et al., PRB 108, 045125 (2023)].\nIn this sense, the “polar interface” discussed here is not only different but also more\ngeneral than that related to the “polar catastrophe scenario”. In other words, any\nconductive interface with asymmetrical potential distribution can be treated as a “polar\ninterface”, including but not limited to those induced by the “polar catastrophe”. For\nexample, the conduction layer induced in the field effect transistor can also be classified\nas the polar interface. To support this, we fabricated a (112)-orientated metal-oxide\nfield-effect transistor consisting of a conductive La Sr TiO layer and a SrTiO\n0.3 0.7 3 3\ninsulating layer, as illustrated in Fig. R3.1a. The oxide layers of this structure are\nfabricated at 600 ℃ at a pressure of 0.01 mbar to exclude the introduction of oxygen\nvacancies in the SrTiO substrate or the SrTiO capping layer. The electron-doped\n3 3\nLa Sr TiO layer was about 10 nm and shows metallic conduction. When a voltage\n0.3 0.7 3\nof 0.4 V is applied to the 10 nm-SrTiO gating layer, electrons in the La Sr TiO\n3 0.3 0.7 3\nlayer are attracted to the interface, enhancing the asymmetry of carrier distribution. As\na result, a sizable nonlinear Hall effect has been then observed under this condition,\nwith its magnitude 10 times larger than that without the applied voltage (Fig. R3.1b).\nThis case suffices the argument that a potential gradient is a general cause of the polarity,\nthat are the origin of the physical effects discovered in the manuscript.\nFig. R3.1 (a) The (112)-orientated metal-oxide-semiconductor field effect transistors, La Sr TiO layer\n0.3 0.7 3\nis about 10 nm. (b) demonstrates its corresponding NLHE effects measured in the direction [11̅0] with\nand without a bias.\nRegarding question 3.1.1, we show in Fig. R3.2 the layer-resolved effective charge\ndistribution across the LaAlO /SrTiO interface. It shows a similar profile as the (110)-\n3 3\nLaAlO /SrTiO system reported [Herranz et al. Scientific Reports 2, 758 (2012);\n3 3\nAnnadi et al., Nat Commun. 4, 1838 (2013)]. Based on this simple ionic schematic, the\n‘polar discontinuity’ does not exist. However, interfaces are conductive with a metallic\nbehaviour till low temperatures. Thus, our result suggests that a simple catastrophe\nmodel cannot explain the formation of the 2DEG at the interface.\nFig. R3.2 Layout of ‘polar catastrophe’ model for (112) interfaces between LaAlO and SrTiO .\n3 3\nAction 3.1.1 and 3.1.2: Further discussion of the interface polar nature has been added\nto the main text, marked as red. Fig. R3.1 and R3.2 have been added to the\nSupplementary Information. Regarding the reference, we would like to replace “G.\nSingh-Bhalla et al., Nat. Phys. 7, 80-86 (2011)” with the latest “J. Gabel et al., Phys.\nRev. B 108, 045125 (2023)” and the recommended reference “[W.-J. Son et al., Phys.\nRev. B 79, 245411 (2009); P. Delugas et al., Phys. Rev. Lett. 106, 166807 (2011)].”\nQuestion 3.1.3: Line 113 “conventional cases” is not well defined and could be\nreplaced by “canonical low-Miller-index LAO/STO interfaces” which I believe is what\nis implied. This, Ref. [29] is here maybe not the most relevant reference.\nAnswer 3.1.3: Physical properties of LaAlO /SrTiO with orientations of (001), (011)\n3 3\nand (111) have already been investigated by the community, while (112)-\nLaAlO /SrTiO is a new member. This is the reason why we initially defined (112) as\n3 3\nthe ‘unconventional’ case in the LaAlO /SrTiO family, while others are considered\n3 3\n‘conventional’. We thank the reviewer for pointing this out, and we agree that\n‘canonical low-Miller-index LaAlO /SrTiO interfaces’ is a more proper phrase to\n3 3\ndescribe (011) and (111)-LaAlO /SrTiO interfaces in our work. Regarding the\n3 3\nreference, ref. [29] is a work studying the [111]-LaAlO /SrTiO interface, which fits\n3 3\nthe picture.\nAction 3.1.3: We have now changed the phrase accordingly\nQuestion 3.2\nQuestion 3.2.1. The authors present in the inset of Fig. 2a, the temperature-dependence\nof the electronic mobility and carrier density in their 112-Lao/STO interface. It would\nbe good to specify if those measurements where conducted in Hall bar devices; as\nwell as disclose the raw data of their Hall effect measurements (in Extended data?), as this\nis the first ever report of transport properties at this specific interface.\nQuestion 3.2.2: This open the questions of whether the Hall effect purely linear? And\nat which magnetic field value was the Hall coefficient estimated to calculate the sheet\ncarrier densities?\nAnswer 3.2.1 & 3.2.2: These two questions are related to the Hall measurement.\nRegarding the data shown in the inset of Fig. 2a, they were obtained in a Hall bar device\nwith the magnetic field sweep between 3 T and -3 T, as shown in Fig. R3.3a. The carrier\ndensity and mobility are all derived from the linear fits of these curves.\nTo further check the linear dependence in these ordinary Hall effects, we conducted the\nHall measurement with the magnetic field applied as large as ±8 T. As demonstrated\nin Fig. R3.3b, the linearity of these curves at all temperature ranges is maintained up to\n±8 T. Therefore, the magnetic field of ±3 T is sufficient for the Hall measurement to\ninvestigate the carrier density and mobility.\nFig. R3.3 (a) Hall measurement in the magnetic field up to ± 3 T. (b) Hall measurement in the magnetic\nfield up to ± 8 T.\nAction 3.2.1 & 3.2.2: We have put a statement in the SI to declare that the carrier\nmobility was obtained from a Hall measurement in the application of ± 3 T.\nQuestion 3.2.3: Have the authors attempted to perform a gate-tuning of the metallicity\nof the 2DEG at low T (below 10K), where the extremely large dielectric constant of the\nSTO substrate should allow for an efficient electrostatic field-effect in back-gating\ngeometry. This is possibly beyond the scope of the present work, but would represent a\nsubstantial addition to the current manuscript.\n#3.2.4. Obviously, the corollary to the latter becomes how does the NLHE evolves as\na function of sheet conductance (or back-gate voltage)?\nAnswer 3.2.3 & 3.2.4: Indeed, the gate tunability of various physical effects is a merit\nof the LaAlO /SrTiO interface, especially at low temperature due to the substantially\n3 3\nincreased dielectric permittivity of the SrTiO crystal.\nconductivity and NLHE at 1.55 K. To enable a large field effect, we have thinned the\nthickness of the STO substrate to ~50 μm. When sweeping the back gate voltage from\npositive to negative, the conductivity decreases monotonically by 1 order of magnitude.\nMeanwhile, the NLHE coefficient firstly increases and then decreases with the back-\ngating voltage, showing a variation over 10-fold (see Fig. R3.4). This reveals the highly\ntunability of the nonlinear transport of the symmetry-engineered LaAlO /SrTiO\n3 3\nheterostructure. Nevertheless, elaboration of such a gate dependence is non-trivial and\nchallenging. The field effect on the LaAlO /SrTiO heterostructure affects various\n3 3\naspects of the interfaces, including potential asymmetry, carrier distribution, Fermi\nlevel, orbital occupation, etc. We believe it is beyond the scope of the present\nmanuscript and we would like to study it in detail in a following work.\nFig. R3.4 Gate-tuning of conductivity and NLHE.\nAction 3.2.3 & 3.2.4: We believe current manuscript is already a complete piece of\nwork and we would like to study the gate tunability of this system comprehensively in\na separate work. We prefer to publish Fig. R3.4 separately.\n__________\nQuestion 3.3\nQuestion 3.3.1. The short discussion (lines 107-109) related to a small upturn in Rs\n“commonly existing in LAO/STO 2DEG systems” should be slightly moderated. In\nsome sense, yes, it is commonly observed, but somewhat not in state-of-the-art, clean\nand optimally doped systems [c.f. G. Herranz et al., Phys. Rev. Lett. 98, 216803 (2007)].\nThe upturn is, without debate, a sign of increased scattering rate of electrons at low T.\nThe underlying (microscopic) mechanism is much more difficult to pinpoint. This is\ngenerally difficult due general tendency to preferentially form either cationic (La, Al)\nor anionic/oxygen vacancies during the growth of the LAO layer, depending on the\npartial pressure of oxygen.\nI would certainly not put “Kondo” and “commonly” side-by-side. Despite the\nspeculated role of oxygen deficiency clusters and their associated magnetism in STO-\nbased systems, this is highly debated in the community; see also the decades old\ncontroversy of magnetism in d^0 (or d^0+delta) electron systems. The authors should\nprovide a couple more references for the upturn when the growth conditions are\nsuboptimal, such as in the early reports by A. Brinkman et al. [Nature Materials 6,\n493–496 (2007). https://doi.org/10.1038/nmat1931], or other tentative explanation\nbased on Kondo-screening [J. Ruhman et al., Phys. Rev. B 90, 125123 (2014)\nhttps://doi.org/10.1103/PhysRevB.90.125123]\nOn the other hand, in the seminal work of G. Herranz et al. [Scientific Reports 2, 758\n(2012). https://doi.org/10.1038/srep00758], on (111) and (110)-LAO/STO interfaces,\nthe upturn was very pronounced for thick (111)- and (110)- interfaces, as opposed to\nnearly absent for thinner LAO layers. However, in later reports, this can be entirely\nmitigated by finding optimal growth conditions, and this thickness-dependent upturn is\nno longer seen as an intrinsic property of the interface.\nAnswer 3.3.1: We appreciate the Reviewer’s profound insight into the low-temperature\ntransport behavior of the LaAlO /SrTiO interface. We are very grateful that the\n3 3\nreviewer points out this improper phrase in our manuscript. To avoid misinterpretation\nto the readers, we will remove the ‘Kondo effect’ in the sentence as in this manuscript.\nSince we are mainly focusing on the room temperature properties of the (112)-\nLaAlO /SrTiO heterostructure, further detailed analysis and optimization of this\n3 3\nupturn are beyond the scope of this work.\nAction 3.3.1: We have revised the sentence to ‘The small upturn of 𝑅𝑠 at lower\ntemperatures indicates the enhanced electron scattering at low temperatures.’ In\naddition, we also put the recommended references [Brinkman et al., Nature Materials\n6, 493 (2007); Herranz et al., Phys. Rev. Lett. 98, 216803 (2007)] along with the\nstatement.\nThis brings me to the following questions.\nQuestion 3.3.2. What is the thickness of each of the LAO (hkl) layers studied in this\nwork?\nQuestion 3.3.3. In Extended Data Fig.1, the authors present a typical X-Ray diffraction\npattern for the (112) interface. How about all the other interfaces? Namely the\nLAO/STO (113), (114), (115), as well as the (110), (111) and (001) interfaces; all of\nwhich are discussed in relation to the NLHE displayed in Fig. 2d.\nAnswer 3.3.2 & 3.3.3: These two questions are related to the film thickness and XRD\nresults, we answer these two questions together. The (112)-LaAlO /SrTiO thin film\n3 3\nstudied in the work was grown by PLD with a thickness of ~4 nm, as determined by\nSTEM (see Fig. R2.1). Other (hkl)-LAO/STO thin films (for NLHE studies) were\ngrown in the same condition and thus have similar thickness (~4 nm). Their XRD\nspectroscopies are shown in Fig. R3.5. Clearly, (112), (001), (110), (111)-oriented\nLAO/STO heterostructures all show the interference fringes of the LaAlO peaks,\n3\nindicating high film quality. In contrast, the fringes are absent in the (113) and (114)\noriented heterostructures. This also implies that the (112)-orientation is the optimal\nhigh-miller index orientation to explore. In addition, we also characterized the surface\ntopography of all these orientations as shown in Fig. 3.6. All of them show ultra-smooth\nsurfaces with sub-nanometre roughness.\nFig. R3.5 XRD studies of (hkl)-LaAlO /SrTiO thin films. (a) (001)-LaAlO /SrTiO . (b) (110)-\n3 3 3 3\nLaAlO /SrTiO . (c) (111)-LaAlO /SrTiO . (d) (112)-LaAlO /SrTiO . (e) (113)-LaAlO /SrTiO . (f)\n3 3 3 3 3 3 3 3\n(114)-LaAlO /SrTiO .\n3 3\nFig. R3.6 Topography images of (hkl)-LaAlO3/SrTiO3 thin films. (a) (001)-LaAlO3/SrTiO3 (b) (110)-\nLaAlO3/SrTiO3 (c) (111)-LaAlO3/SrTiO3 (d) (112)-LaAlO3/SrTiO3 (e) (113)-LaAlO3/SrTiO3. (f) (114)-\nLaAlO3/SrTiO3.\nAction 3.3.2 & 3.3.3: We have now put a sentence to elaborate on the thickness of the\nsample in the methods section. Fig. R3.5 and Fig. R3.6 have been added to the SI.\n__________\nQuestion 3.4\nRegarding the magnetooptical Kerr effect (MOKE) data and the circular photogalvanic\neffect (CPGE) presented in Figure 2.\nQuestion 3.4.1. I understand that, on one hand, the current-induced MOKE signal\ncould be expected to be small. However, on the other hand, the sensitivity of\n(longitudinal) MOKE setups can reach down to a couple of nrad, so the signal reported\nby the authors is rather large in comparison. This brings me to the following: How\ncome the Kerr rotation angle at the lowest probed current\ndensity (as well as throughout the whole [11-1] data set) is as large as 0.05 microrad\n(or 50 nrad)? Can the authors comment on the tentative origin of this relatively large\noffset value?\nAnswer 3.4.1: The sensitivity of the longitudinal MOKE in previous works can reach\n~10 nrad [e.g. Choi et al., Nature 619, 52 (2023)], which is about 5 times better than\nour work (~50 nrad). To achieve the state-of-the-art sensitivity, each part of the MOKE\nsystem, especially power and polarization of the laser, and the temperature of optical\ncomponents used in the light path, has to maintain as stable as possible. In our system,\nwe have designed a home-made system to maintain the temperature fluctuation of\noptical components, including the waveplates and the Wollaston prism, within 1 mK.\nHowever, the He-Ne laser used in our system exhibits the sizable fluctuation of power\nand polarization, which are difficult to suppress with our current technique and limited\nfunding. This fluctuation of the laser setup causes noise in our MOKE system and\nreduces its sensitivity. Replacing the He-Ne laser with a sophisticated Ti: sapphire laser\n(which is very expensive) will likely increase the MOKE sensitivity [as used in Choi et\nal., Nature 619, 52 (2023)].\nRegarding the ~50 nrad offset of the MOKE signal shown in Fig. 2d of the main text,\nit probably arises from three factors: (1) the finite extinction ratio (100000:1) of the\nWollaston prism placed before the balanced detector; (2) the intrinsic offset/noise and\nfinite resolution of our lock-in amplifier; (3) finite electromagnetic cross talk between\ndifferent electronic components in our home designed MOKE system.\nNevertheless, our MOKE system with its present sensitivity (~50 nrad) is good enough\nto resolve the MOKE signal in the (112)-interface induced by current. More importantly,\na clear contrast of the MOKE signal between [11̅0] and [111̅] in-plane current\ndirection is easily observed by our system, consolidating our claim and confirm the\ngenuineness of the phenomena. We would like to optimize our MOKE system in the\nfuture once further funding is granted.\nAction 3.4.1: We have added further discussion of the MOKE signal and noise in the\nMethods section and SI, marked as red.\nQuestion 3.4.2. The data in Fig.2e is presented for a current density of 2 A/m, that is,\ngiven the 10 microns cross-section of the Hall bar, for a current amplitude of 20\nmicroAmps. This is 40 times larger than any of the second harmonic I-V characteristic\npresented in Fig. 3a and Extended Data Fig. 4c. For consistency, and in the spirit of\ntransparency, the authors could either display a secondary x-axis (top of panel 2d) with\nthe absolute value of the current.\nAnswer 3.4.2: To our best knowledge, it is more common to use the current density to\ndescribe the current-induced magnetization [Choi et al., Nature 619, 52 (2023)]; Stamm\net al., Phys. Rev. Lett., 119, 087203 (2017); Lyalin et al., Phys. Rev. Lett., 131, 156702\n(2023);]. In order to make it consistent with other reported works, it is rational to choose\nA/m as the unit while not μA. But we agree with the reviewer’s opinion that a secondary\naxis of the unit μA on the top of the figure makes the picture clearer. This question is\nalso related to the different current ranges used in NLHE and MOKE, we will further\ndiscuss this question in Question 3.5.1&3.5.2.\nAction 3.4.2: We have now put a secondary x-axis on the top of the panel.\nQuestion 3.4.3 Line 115 the authors claim “To show that indeed the interface possesses\na monoclinic Cs symmetry, we have measured the CPGE […]”. This is incorrect and\nshould be rephrased. The existence of a CPGE cannot prove per say the Cs symmetry\nof the system. The CPGE is instead a potentially symmetry-allowed effect, and its\nanisotropy consistent with Cs symmetry.\nAnswer 3.4.3: As we mentioned in Answer 1.1, the CPGE only functions in gyrotropic\nmaterials and is characterized by second order axial tensors. As a result, when light is\nshining along the interface/surface normal direction, only the current generated by the\nCPGE could flow in the direction normal to the mirror symmetrical plane. Therefore,\nfor a 2D structure, if it has more than one mirror plane (e.g., C or C ), then it has no\n2v 3v\nCPGE; If it has no mirror plane (C ), then it has the CPGE along all the in-plane\n1\ndirections. Only in the system with one mirror plane can the CPGE exhibit\ncharacteristic directional behaviours, that is, its direction is perpendicular to the only\nmirror plane. Therefore, the existence of the CPGE in only one direction is consistent\nwith the C symmetry.\ns\nAction 3.4.3: To make our statement more rigorous, we have rephrased our sentence of\nLine 115 as:\n“To get an insight into the mirror symmetry breaking in the designed (112)-interface,\nwe have measured the circular photogalvanic effect (CPGE) on the same samples by\nilluminating the surface using a circularly polarized 405 nm laser beam with normal\nincidence.”\n__________\nQuestion 3.5\nLast but not least, the nonlinear Hall effect data (Fig. 3).\nThe symmetry-based arguments made in favor of the existence of a Berry curvature dipole-driven NLHE, and the first-\nprinciple calculations associated to it, are not being questioned here.\nRather:\nQuestion 3.5.1 In relation to my previous point: what happens to the nonlinear I-V\ncharacteristics (Fig. 3a, Ext. Data Fig. 4) when sourcing current densities comparable\nto those used in Fig.2? And very importantly, do those NL I-V present a hysteretic\nbehaviour?\nQuestion 3.5.2 . Same question goes for the longitudinal linear I-V characteristic,\nideally I(omega) vs. Vxx (omega) (see Extended Data Fig. 4b) - and I(omega) vs.\nVyy(omega) - in the same current density range)? In general, how can the authors rule\nout heating related effects or other sources of nonlinearities?\nAnswer 3.5.1 & 3.5.2: These two questions are related to the transport behaviours when\na larger current is sourced, we answer here together.\nThe main reason why using different current magnitudes for NLHE and MOKE\nmeasurement is due to their contrasting detectivity. Both physical effects are\ncharacterized by using an AC driving current and detecting the output voltage signal\nwith a lock-in amplifier. The lock-in amplifier can detect an AC voltage with an RSM\namplitude as low as tens of nV. In the case of the NLHE measurement, the second order\nnonlinear Hall voltage sign can easily reach several V with an input a.c. current of a\nfew hundred nA (see Fig. 3a). However, in the case of the MOKE measurement, an\ninput current of tens of A is required to induce a sensible output signal from the\nbalanced detector. This is common for the MOKE measurement on the current-induced\nmagnetization, such as the spin Hall effect [ Stamm et al., Phys. Rev. Lett., 119, 087203\n(2017); Lyalin et al., Phys. Rev. Lett., 131, 156702 (2023)].\nTo answer the reviewer’s request on the transport properties under larger ac current, we\nconducted the transport measurement with a maximum 20 μA applied (see Fig. R3.7).\nAs is seen from the figures, all three measured voltage drops do not show hysteresis\nbehaviours. 𝑉𝜔 linearly grows with 𝐼𝜔 and 𝑉𝜔 keeps it small value all over the current\n𝑥𝑥 𝑥 𝑥𝑦\nrange due to misalignment. Meanwhile, 𝑉2𝜔 increases quadratically up to 20 μA.\n𝑥𝑦\nFig. R3.7 (a) The 1st order voltage drops 𝑉𝜔 and 𝑉𝜔 and (b) the 2nd order voltage drop 𝑉2𝜔 as a\n𝑥𝑥 𝑥𝑦 𝑥𝑦\nfunction of ac current up to 20 μA.\n(2) Heating-related effects and other sources induced by the nonlinearity can be ruled\nout by two phenomena in our work: a) the significant anisotropic performance of NLHE,\nCPGE and MOKE signals between the two orthogonal directions. b) the above\nphenomena in (001)-, (011)- and (111)- LaAlO /SrTiO are not observable under the\n3 3\nsame measurement condition.\nAction 3.5.3: Fig. R3.7 has been added to the SI.\nQuestion 3.5.3 In Fig. 3c, how is each data point obtained? Do the authors fit a full\nnonlinear I-V characteristics at a given temperature, or is it being measured at constant\ncurrent bias upon continuously sweeping the temperature (cooldown? or warming?\nAnd at which rate?).\nAnswer 3.5.3: To obtain the temperature-dependent NLHE coefficient, we measured\nboth 𝑉𝜔 and 𝑉2𝜔 continuously by heating the sample with the ac current biased (i.e., 1\n𝑥𝑥 𝑥𝑦\nμA). Then the NLHE was calculated by using the formula 𝐸2𝜔/(𝐸𝜔)2, see Fig. 3c for\n𝑥𝑦 𝑥𝑥\nexample. This method has also been adapted to measure the temperature-dependent\nNLHE by other works where lots of data are collected in a wide temperature range [Min\net al., Nat. Commun. 14, 364 (2023); Itahashi et al., Nat. Commun. 13, 1659 (2022)].\nIn order to further confirm the rationality of this method, we also conducted the\nmeasurement of both 𝑉𝜔 and 𝑉2𝜔 upon sweeping the ac current at the fixed\n𝑥𝑥 𝑥𝑦\ntemperature. The plot of 𝑉2𝜔 vs (𝑉𝜔)2 at various temperatures is given in Fig. R3.8.\n𝑥𝑦 𝑥𝑥\nThe NLHE coefficient obtained at 450 K, 410 K, 375 K, 350 K, and 300 K by fitting\nthe linear 𝑉2𝜔 vs (𝑉𝜔)2 curves are 116, 100, 69, 55, 41 μm V−1, respectively. These\n𝑥𝑦 𝑥𝑥\nvalues are close to the data obtained by continuously measuring NLHE.\nFig. R3.8 The temperature-dependent NLHE measured at fixed temperatures.\nAction 3.5.3: We have now put the relative description of measuring temperature-\ndependent NLHE in the Method section.\nQuestion 3.5.4. Could the authors comment further on the quite peculiar temperature-\ndependence of the NLHE reported in Fig. 3c? is there a tentative explanation for the\nstrong dip at about 100 K, and for a recovery of the effect’s magnitude at lower\ntemperature? Can this be understood in relation to the measured temperature-\ndependence of the sheet conductance and sheet carrier density? The argument\nregarding the Fermi level.\nAnswer 3.5.4: To understand the peculiar temperature dependence of the NLHE\nreported in our manuscript, we need to resolve its different contributions.\nIt has been reported that, in addition to the intrinsic contribution of the BCD, extrinsic\ncontributions, including side jumping and screw scattering, play an important and even\ndominant role in the NLHE. To resolve various contributions, we employ the scaling\nmethod at both low (< 30 K) and high temperature (> 100 K) ranges (Fig. R3.9). The\ntwo curves in Fig. R3.9 are obtained by relating the NLHE coefficient 𝐸2𝜔⁄(𝐸𝜔)2 and\n𝑥𝑦 𝑥𝑥\nconductivity𝜎 at each temperature. According to the theory [Du et al., Nat. Commun.\n𝑠\n10 3047 (2019)], the scaling law for NLHE can be written as: 𝐸2𝜔⁄(𝐸𝜔)2 =\n𝑥𝑦 𝑥𝑥\n𝐴 (\n𝜎𝑠)2\n+𝐴\n𝜎𝑠\n+𝐴 , where 𝐴 , 𝐴 , 𝐴 are the scaling parameters, 𝜎 is the\n1 2 3 1 2 3 0\n𝜎0 𝜎0\nconductivity at 1.55 K. The fitting results of 𝐴 , 𝐴 , 𝐴 are shown in Table R3.1. Based\n1 2 3\non the NLHE scaling law, the scaling parameters are expressed as:\n𝐴 = 𝐶𝑠𝑘,2𝜎 +(𝐶𝑠𝑘,1 +𝐶𝑠𝑘,1 −𝐶𝑠𝑘,1) (3.1)\n1 0 00 11 01\n𝐴 = 𝐶𝑠𝑘,1 −2𝐶𝑠𝑘,1 +𝐶 𝑠𝑗 −𝐶 𝑠𝑗 (3.2)\n2 01 11 0 1\n𝐴 = 𝐶 +𝐶 𝑠𝑗 +𝐶𝑠𝑘,1 (3.3)\n3 𝑖𝑛 1 11\nwhere 𝐶 is an intrinsic contribution, 𝐶 𝑠𝑗 the side-jump, 𝐶𝑠𝑘,1 the Gaussian skew-\n𝑖𝑛 𝑖 𝑖𝑗\nscattering, 𝐶𝑠𝑘,2 the non-Gaussian skew-scattering, and 𝑖, 𝑗 = 0 (1) denotes the defects\n(phonon) scattering source. There are seven parameters, i.e., seven different\ncontributions in the above equations, which are impossible to resolve quantitatively. To\nget a tentative insight into these contributions, here we take a bold but almost reasonable\nassumption.\nFig. R3.9 Scaling law analysis of the NLHE at temperature (a) lower than 30 K and (b) higher than 100\nK.\nTable. R3.1 Scaling parameters\nA A A\n1 2 3\nT < 30 K 7.56 -42.4 56.9\nT> 100 K 90.5 -112.4 36.6\nSince 𝐶𝑠𝑘,2 only exist in the 𝐴 parameter, the non-Gaussian skew-scattering only\n1\nplays a secondary role. In the temperature range below 30 K, the parameter 𝐴 is almost\n1\none order of magnitude smaller than 𝐴 and 𝐴 . This indicates that the screw scattering\n2 3\ncontributions play a minor role at the low temperatures. Also, the phonon density is\n𝑠𝑗\nlargely suppressed at low temperature, it would be reasonable to assume 𝐶 (i.e., side\n0\n𝑠𝑗\njump due to defects) is much larger the 𝐶 (i.e., side jump due to phonons). Thus, 𝐴 =\n1 2\n𝑠𝑗\n−42.4 ≈ 𝐶 . Similarly, 𝐴 = 56.9 ≈ 𝐶 .\n0 3 𝑖𝑛\nIn the temperature range above 100 K, the parameter 𝐴 shares a similar magnitude\n1\nwith 𝐴 but with opposite sign, while 𝐴 about three times smaller. Based on equations\n2 3\n(2) and (3), the opposite sign between 𝐴 and 𝐴 is most likely due to the opposite sign\n1 2\nof 𝐶𝑠𝑘,1 and 𝐶𝑠𝑘,1 in these two equations. Since 𝐶𝑠𝑘,1(screw scattering due to defects)\n11 01 00\nonly exist in 𝐴 , it may play a minor role here. Moreover, since the side jump process\n1\n𝑠𝑗\nis insensitive to the temperature, we assume 𝐶 remains as a constant over the whole\n0\ntemperature range. Thus, equations (3.1-3.2) can be approximated as:\n𝐴 = 𝐶𝑠𝑘,1 −𝐶𝑠𝑘,1 = 90.5 (3.4)\n1 11 01\n𝐴 = 𝐶𝑠𝑘,1 −2𝐶𝑠𝑘,1 −𝐶 𝑠𝑗 =-70 (3.5)\n2 01 11 1\n𝐴 = 𝐶 +𝐶 𝑠𝑗 +𝐶𝑠𝑘,1 = 36.6 (3.6)\n3 𝑖𝑛 1 11\nThus, we get 𝐶 = 57, which almost equal to that derived at low temperature range,\n𝑖𝑛\nsupporting the above approximation.\nBased on the above analysis, we can reach here a qualitative understanding of the\norigins and temperature dependence of the NLHE in the (112)-LAO/STO interface.\nClearly, both the intrinsic and extrinsic contributions play important and compensating\nroles in this system. For example, the extrinsic contribution 𝐶 𝑠𝑗 +𝐶𝑠𝑘,1 (= −20.4) is\n1 11\nopposite to that induced by Berry curvature dipole (𝐶 = 57).\n𝑖𝑛\nThe peculiar temperature dependence can be ascribed to the role of the parameter 𝐴 ,\n2\ni.e., contributions of screw scattering and side jump. This also means that the NLHE\neffect of the (112)-interface is highly related to its conductivity with an approximated\nquartic dependence. When the temperature decreases from 300 K to 100 K, the\nconductivity and mobility increase, the screw scattering process is enhanced, inducing\na negative contribution that compensates that of the intrinsic one. At the temperature\nbelow 30 K, the density of phonons is significantly reduced and thus, the related screw\nscattering contributions (𝐶𝑠𝑘,1 and 𝐶𝑠𝑘,1) are suppressed. In this circumstance, the side\n01 11\njump and intrinsic dominate. With further decreasing temperature towards 1.55 K, the\nconductivity decreases, the compensating contribution of side jump by defects is\nreduced, leading to an enhanced NLHE.\nAction 3.5.4: We have added the relevant comments on the behaviour of the\ntemperature-dependent NLHE in the main text and put the scaling law of the NLHE in\nSI.\n__________________\nQuestion 3.6 Methods section:\n(1) Line 324: annealed in which conditions? In vacuo in-situ? in air? In pure oxygen\nflow furnace? (2) Are all the (11X), the (001), the (110) and (111) interfaces grown in\nthe same conditions?\nAnswer 3.6: Substrates are annealed in air at 1000 ºC for 2 hours; The thin film growth\nconditions for all (hkl)-LaAlO /SrTiO thin films are the same.\n3 3\nAction 3.6: We have put a relative description in the Methods section.\n__________________\nQuestion 3.7 Miscellaneous:\nQuestion 3.7.1 In Fig. 3a, is the y-axis not going from negative to positive values?\nOr is this deliberate?\nAnswer 3.7.1: We thank the reviewer for pointing this out. Initially, we used the phase\nto denote the ‘negativity’. However, we notice that it is more common to set the y-axis\nfrom negative to positive, such as in the reference [Kang et al., Nat. Mater. 18, 324\n(2019)].\nAction 3.7.1: We have changed the y-axis.\nQuestion 3.7.2 In Fig. 1b: if I am not mistaken, the “x”-vector: [11-1] should point\ntowards the bottom of the page, otherwise the arrow actually indicates the [-11-1]\ndirection.\nAnswer 3.7.2: We are very grateful that the reviewer points this out. The arrow of the\ndirection [111̅] in the figure indeed shall point downwards.\nAction 3.7.2: We have correspondingly revised the coordinate in the figure.\nQuestion 3.7.3 Line 34: typo “Wely” instead of “Weyl”.\nAnswer 3.7.3: We thank the reviewer for pointing out this typo.\nAction 3.7.3: We have corrected this typo.\nQuestion 3.7.4 Line 82: “ultrasmooth” could be replaced by “atomically (?) sharp”,\nthough the lack of spatially-resolved element specific EDXS (energy dispersive x-ray\nspectroscopy) may prevent making that claim.\nAnswer 3.7.4: In the main text, we describe the surface as ‘ultrasmooth’ but not the\ninterface. Thus, we believe this description is correct as the AFM data clearly show the\nroughness of the sample is only about 90 pm. Since the reviewer mentioned the\ninterface, from the HAADF-STEM image we can tell that the interface between LAO-\nSTO is seemly sharp. However, we understand that it is usually hard to achieve a perfect\nsharp interface where the mixture of atoms might happen [Nakagawa et al., Nature\nMater 5, 204 (2006)].\nAction 3.7.4: No action taken.\nQuestion 3.7.5 Line 181: “it exists”, “it” should be deleted here.\nAnswer 3.7.5: We thank the reviewer point out this grammar problem.\nAction 3.7.5: we have removed ‘it exists’ from the sentence.\nQuestion 3.7.6 Line 229 and 230: “Fermi lines” (twice) could be replaced with “Fermi\nlevel” or “Fermi energy” (and give its corresponding definition).\nAnswer 3.7.6: We believe the term ‘Fermi lines’ is correct here. According to the\ndefinition, in the Brillouin zone (BZ) of a two-dimensional (2D) material, Fermi lines\nrepresent the constant-energy contours (Fermi contours) where the electronic band\nenergy equals the Fermi level. However, Fermi level refers to the scalar energy value.\nIn the content of calculating Berry curvature or Berry curvature dipole distribution over\nthe Brillouin zone, it is correct to use ‘Fermi lines’ [Mercaldo et al., npj Quantum Mater.\n8, 12 (2023); Lesne et al., Nat. Mater. 22, 576 (2023)].\nAction 3.7.6: No action taken.\nQuestion 3.7.7 Line 230: “exhibit symmetries concerning both kx = 0 and ky = 0”:\nwhich symmetries? What is meant here by “concerning\"? I am confused.\nAnswer 3.7.7: Initially, we wanted to express that the D is symmetric about k = 0\nyz x\nand k = 0, as displayed in Fig. 4c. We understand that the word ‘concerning’ is not\ny\nproper here, which might cause confusion as the reviewer mentioned.\nAction 3.7.7: We have changed ‘concerning’ to ‘with respect to’.\nQuestion 3.7.8 Line 250: “are” should be “were” or “have been” (due to the use of\nbefore at the end of the sentence).\nAnswer 3.7.8: We thank the reviewer for pointing this out.\nAction 3.7.8: We have revised the sentence.\nQuestion 3.7.9 Extended Data Fig. 7, what are the corresponding current densities?\nAnswer 3.7.9: The corresponding currents in these measurements are in the range of 1\nμA. The feature of the quadratic dependence of 𝑉2𝜔 on 𝐼𝜔 (or 𝑉𝜔) is clearly shown\n𝑥𝑦 𝑥 𝑥𝑥\nwhen the applied current reaches 1 μA, thus it is rational to choose this range.\nAction 3.7.9: We have put the current amplitude in the figure captions.\nQuestion 3.7.10 On many occasions, articles such as “the” or “a/an” are missing, or\non the contrary not necessary. I would urge the authors to take the necessary English\nlanguage revisions.\nAnswer 3.7.10: We thank the reviewer for carefully checking the grammar of this\nmanuscript.\nAction 3.7.10: We have correspondingly revised the mentioned typos and also checked\nthe manuscript thoroughly.\nQuestion 3.8: In conclusion, I cannot recommend the present manuscript, as it is, for\npublication in Nature Communications, but I encourage the authors to address my\ncomments and questions. Upon satisfying revisions, I am of the opinion that this work\nwould be of great interest to the broader solid-state physics community with a keen\ninterest on nonlinear transport, and their consequences for applications, in low-\nsymmetry systems.\nAnswer 3.8: We appreciate the reviewer’s cognition of our work and thank the\nreviewer for giving us many valuable suggestions. By answering the comments point-\nto-point, we believe our manuscript has now reached the standard of Nature\nCommunications."}, "author_response": ""}, {"round": 2, "reviewer_comments": {"reviewer_1": "I have carefully studied the authors' response letter and revised manuscript, which have addressed my concerns. I would\nlike to recommend the revised manuscript for publication.", "reviewer_2": "I would first like to thank the authors for their efforts in revising the manuscript and for addressing several of the previous\nconcerns. While some points have indeed been clarified, there remain a number of important issues that are still\ninsufficiently resolved:\n1. Role of defects and interface structures: The authors argue that if experimental imperfections were to break the in-plane\nmirror symmetry and significantly alter the interface properties, nonlinear Hall effects should also be observable in the (001),\n(110), and (111) interfaces, which is not the case (as shown in Fig. 3d of the main text). However, one should keep in mind\nthat the nature of defects and interdiffusion strongly depends on the growth orientations. In high-energy surfaces such as\n(112), different kinds of interface reconstructions are likely to occur compared to (001), (110), or (111). Although many effects\nmay indeed average out, not all of them do. Since the manuscript does not discuss the difference in defect formation among\nthese orientations, it is still possible that the observed effects originate from specific microstructural features unique to the\n(112) interface. Changes in oxygen octahedral rotations should also be considered. For example, in systems such as SRO,\nout-of-plane spin components have been predicted as a consequence of octahedral rotations. Therefore, the absence of\nnonlinear Hall effects or current-induced magnetization in (001), (110), and (111) interfaces does not necessarily prove that\nthe observed effects in (112) originate solely from the preservation of the M[1-10] mirror plane.\n2. Interpretation of the zigzag pattern: The discussion of the zigzag pattern at the interface remains ambiguous. The authors\nconclude that such patterns do not generate physical effects, but no clear justification is given. Transport properties and spin\nare very sensitive to microscopic features. Furthermore, they cite Annadi et al. (Nat. Commun. 4, 1838 (2013)) to claim that\nzigzag features are also present in (110) interfaces. However, the zigzag reported in that work extends over only about one\nunit cell and is hardly visible in scanning transmission electron microscopy (STEM) images. In contrast, the zigzag\nmodulation observed in the present (112) samples is much more pronounced, extending over several unit cells with a clear\nperiodic composition modulation, as seen directly in STEM image (Fig. 1e). Indeed, when the contrast in Fig. 1e is\nenhanced, the zigzag pattern becomes very evident and is directly discernible in the STEM images. From this perspective, I\nfind it inappropriate to consider the two cases equivalent. From this point of view, I disagree with the authors' statement:\nBoth images demonstrate a sharp interface between the thin film and the substrate (line 99 on page 4). Such roughness is\nunlikely to average out in a Hall bar geometry and may instead contribute a finite value to the measured transport.\nAdditionally, the anisotropy of this roughness, as seen in Fig. 1e,f, could itself be a possible origin of the observed transport\nanisotropy. I strongly encourage the authors to examine this point more carefully.\n3. Definition of theta_QWP: Although the authors define theta_QWP as the angle between the fast axis of the QWP and the\npolarization of the incident light, this explanation is somewhat too brief. It would be helpful to provide a more explicit\nreference direction (e.g., relative to the crystal axes or the measurement geometry in the case of linearly polarized light) to\navoid ambiguity.\n4. Fitting of the circular photogalvanic effect (CPGE)/ linear photogalvanic effect (LPGE) Data (Figs. 2b, 2c, and R2.4): The\nauthors state that the reviewer \"may have misunderstood\" and that the fits are clearly good. However, I did not\nmisunderstand; my point concerned the significant discrepancies between the experimental data and the total fit, especially\nnear 90 degrees and 150 degrees in Fig. R2.4. In fact, in Wang et al., Phys. Rev. Lett. 128, 187401 (2022), the CPGE in\n(111) LAO/STO interfaces is fitted with excellent agreement across all angles and intensities. The substantial mismatch\nobserved here suggests that additional factors may be at play. Moreover, in the new version of the manuscript, the\nseparation into LPGE and CPGE components has been removed, which makes the text difficult to follow, and Figs. 2b and\n2c appear to be interchanged. These figures need to be carefully re-presented.\nI am also concerned that the sum of the red (CPGE) and blue (LPGE) curves does not reproduce the black \"total fit\" curve in\nFig. 3c (original version). For example, at around 135 degrees, the values of LPGE and the total fit differ, despite CPGE\nbeing nearly zero at this point. Is the fitting procedure correct? Has any angular offset been corrected for? If so, this should\nbe clearly described in the manuscript.", "reviewer_3": "I carefully read the rebuttal of the authors to all three reviewers. I have an overall sense that the authors carefully addressed\nmost of the concerns and took the necessary actions in revising the manuscript and in consequently expanding the content\nof the supplementary information.\nThus, I recommend the publication of the present manuscript \"Magnetization generation and giant nonlinear transport at\nsymmetry-engineered interfaces via crystallographic orientation\" by H.-B Zhang et al., in Nature communications. I believe it\nwill be of great interest to many scientific communities working on nonlinear transport, and their theories, in various materials\nsystems.\nRegarding:\nAction 3.2.3 & 3.2.4: \"We believe current manuscript is already a complete piece of work and we would like to study the gate\ntunability of this system comprehensively in a separate work. We prefer to publish Fig. R3.4 separately.\"\nMy comment: I appreciate the authors' effort and openness to disclose unpublished results, and agree that this is a\nsubstantial addition which warrants a detailed investigation for a follow-up publication.\nRegarding:\n\"To further check the linear dependence in these ordinary Hall effects, we conducted the Hall measurement with the\nmagnetic field applied as large as ±8 T. As demonstrated in Fig. R3.3b, the linearity of these curves at all temperature\nranges is maintained up to ±8 T. Therefore, the magnetic field of ±3 T is sufficient for the Hall measurement to investigate the\ncarrier density and mobility.\"\nMy comment: I respectfully disagree with the statement that the ordinary Hall effect is purely linear in the whole magnetic\nfield range. It appears to me that the Hall effect is in fact slightly non-linear in field at the lowest temperatures. Overall I do not\nthink this affects the conclusions of the manuscript, because this non-linearity is still quite small. However the authors do not\nignore that when multiband conduction and Hall effect is involved, the low-field and high-field regions may yield very\ndifferent Hall coefficients, and that their relation to the carrier density of either or both subbands is not the same. In particular,\nonly the high-field Hall coefficient can be taken as inversely proportional to the total carrier density in the system. I would not\ninsists either on a two-bands fit of the Hall effect alone because it is known to be problematic due to the large number of free-\nparameters in the model. I'm looking forward to a more detailed analysis in future studies of the Hall effect in these systems."}, "author_response": "We would like to thank the Reviewers for their valuable comments and appreciations\nof our work. We have accordingly revised the manuscript and the supporting\ninformation. We believe we have positively addressed all comments made by the\nreviewers. A point-by-point response to the reviewers’ comments and changes made in\nthe revised manuscript are given below.\nResponse to Reviewer #1:\nQuestion 1: I have carefully studied the authors' response letter and revised manuscript,\nwhich have addressed my concerns. I would like to recommend the revised manuscript\nfor publication.\nAnswer 1: We are very grateful that the reviewer gave constructive suggestions to\nmake this manuscript more clear and solid. We also thank the reviewer for the\nrecognition of our work.\nAction 1: No action taken.\nResponse to Reviewer #2:\nI would first like to thank the authors for their efforts in revising the manuscript and for\naddressing several of the previous concerns. While some points have indeed been\nclarified, there remain a number of important issues that are still insufficiently resolved:\nAnswer: We thank the reviewer for giving us important suggestions and pointing out\nsome remaining issues. Below, we address in detail all the raised questions and take the\ncorresponding actions.\nQuestion 2.1: Role of defects and interface structures: The authors argue that if\nexperimental imperfections were to break the in-plane mirror symmetry and\nsignificantly alter the interface properties, nonlinear Hall effects should also be\nobservable in the (001), (110), and (111) interfaces, which is not the case (as shown in\nFig. 3d of the main text). However, one should keep in mind that the nature of defects\nand interdiffusion strongly depends on the growth orientations. In high-energy surfaces\nsuch as (112), different kinds of interface reconstructions are likely to occur compared\nto (001), (110), or (111). Although many effects may indeed average out, not all of\nthem do. Since the manuscript does not discuss the difference in defect formation\namong these orientations, it is still possible that the observed effects originate from\nspecific microstructural features unique to the (112) interface. Changes in oxygen\noctahedral rotations should also be considered. For example, in systems such as SRO,\nout-of-plane spin components have been predicted as a consequence of octahedral\nrotations. Therefore, the absence of nonlinear Hall effects or current-induced\nmagnetization in (001), (110), and (111) interfaces does not necessarily prove that the\nobserved effects in (112) originate solely from the preservation of the M[1-10] mirror\nplane.\nAnswer 2.1:\nWe respectfully disagree with the Reviewer’s interpretation, especially with regard to\nthe concluding statement. The development of nonlinear transport and current-induced\nmagnetization does NOT originate from the ‘preservation’ of the mirror symmetry.\nRather, it is a direct consequence of the ‘breaking’ of the other mirror planes, such as\nM . This interpretation is fully consistent with Curie’s principle, which states that\n{100}\n“asymmetry is what creates a phenomenon”. The deliberate preservation of the 𝑀\n[11̅0]\nmirror plane serves exactly this purpose: to prevent the onset of nonlinear transport\neffect with input current flowing along the mirror direction (i.e., [111̅] in our work)\nThus, breaking of all the other mirror planes except the 𝑀 provides evidence, what\n[11̅0]\nwe term “smoking gun confirmation”, for our central hypothesis, specifically the\ncreation of the in-plane anisotropy and observing the associated effects in the\norientation-engineered interfaces. Such in-plane anisotropy has been systematically and\nrepeatedly validated by our experimental results.\nWhen the final mirror plane 𝑀 is broken (for example, in the (122)-LAO/STO\n[11̅0]\norientation), no mirror plane is retained at the interface. As a result, nonlinear transport\nproperties are symmetry-allowed in all in-plane directions. By contrast, systems with\nmore than two planes with preserved mirror symmetries , such as the (110)-, (100)- and\n(111)-LAO/STO, impose mutual constraints that forbid the emergence of a nonlinear\nHall effect. This has been rigorously established in the Method section with explicit\nmathematic support. Thus, the experimental observations reported here are consistent\nwith the symmetry analysis.\nRegarding the role of defects and interface structures, we would like to discuss it in two\naspects.\n(i) Firstly, we concur with the Reviewer that “nature of defects and interdiffusion\nstrongly depends on the growth orientations”. Importantly, all interfaces reported in this\nwork were grown under the same condition. Therefore, any orientation-dependent\nvariation in defect configuration or interdiffusion, if present, must be attributed solely\nto the crystallographic orientation, precisely as the Reviewer has suggested. In this\nsense, the “special” defects and interdiffusion in the (112)-orientation, as well as their\nrole in the observed effects, if any, should be an intrinsic outcome of our symmetry-\nengineering strategy. In other words, such defect-mediated effects are not extraneous\nbut rather an inseparable component of the mechanism we deliberately exploit.\nSince the polar catastrophe scenario is unlikely to operate in the (112)-orientation,\noxygen vacancies are expected to dominate the interface conduction, as already\ndiscussed in our previous response letter. Consequently, defects indeed play an\nimportant role in nonlinear transport and magnetization generation at the (112)-\ninterface, but in a distinct way as the Reviewer suggests here. In our previous response\nletter ‘Answer 1.4’, we have discussed in detail the role of defects in the nonlinear Hall\neffect (NLHE), interrogating the scaling law by varying the temperature and thus the\nconductivity. These analyses demonstrate that the effects associated with defect\nscattering, such as the side jump, skew scattering, coexist with the intrinsic Berry\ncurvature dipole contribution. The overall transport property is the sum of\nasymmetric scattering and intrinsic properties, both of remain strictly governed\nby the underlying symmetry constraint. This is more clearly seen from the tensor\nproperties of the NLHE and the anisotropic resistive response as a function the applied\ncurrent direction.\n2. Secondly, we address the Reviewer’s point regarding the possibility of “different\nkinds of interface reconstructions” in (112)-oriented interface and whether “such a\nunique interface reconstruction” is the fundamental origin of the observed effects or not.\nThe direct characterization of defects and interface reconstruction in complex oxide\nheterostructures, as well as disentangling their precise role in electronic transport, is\ntechnically demanding and could itself constitute an independent research project.\nNevertheless, recognizing the importance of this issue, we have designed three contrast\nexperiments: (b1) (112)-Pt/SrTiO /La SrTiO field effect transistor grown on SrTiO\n3 0.3 3 3\nsubstrate (also seen in the first response letter ‘Answer 3.1’) and (b2) reduced (112)-\nSrTiO substrate.\nmain conclusion: (1) Nonlinear Hall effect is consistently observed whenever (112)-\noriented interfaces are formed, regardless of the different types/natures of the\ndefects (see La Sr TiO field effect transistor). (2) Defects alone cannot generate\n0.3 0.7 3\nnonlinear transport in the absence of symmetry breaking.\nThese findings strongly support our interpretation that the NLHE originates\nfundamentally from symmetry considerations rather than being solely a by product of\ndefect-driven interface reconstructions.\nWe are now presenting the detailed results of the three contrast experiments.\n(b-1) La Sr TiO field effect transistor: the (112)-oriented La Sr TiO /SrTiO\n0.3 0.7 3 0.3 0.7 3 3\nbilayer were grown under high oxygen pressure and relatively low temperature (i.e.,\n600 ℃ and 0.01 mbar oxygen pressure) to minimize the oxygen vacancy density.\nIn this system, both the SrTiO substrate and SrTiO film are insulating, with the\n3 3\nconduction confined within a 10 nm La SrTiO thin film. As shown in Fig. RR2.1c,\n0.3 3\nthe pristine bilayer already shows a small nonlinear Hall effect at room temperature\nwithout apply a gate field. This residual response most likely arises from inequivalent\nboundary conditions at the bottom interface of La Sr TiO /SrTiO substrate and\n0.3 0.7 3 3\nSrTiO /La Sr TiO topinterface. When a gate voltage of 0.4 V is applied via top\n3 0.3 0.7 3\nSrTiO layer, the NLHE increases significantly. Crucially, the nature and configuration\n3\nof the La Sr TiO transistor are entirely distinct from the LaAlO /SrTiO interface,\n0.3 0.7 3 3 3\nwith the exception of the crystallographic orientation. The persistence and enhancing\nof NLHE in this system therefore unambiguously demonstrate that the phenomenon\noriginates from orientation-induced mirror-symmetry breaking, rather than from any\nparticular defect configurations.\nFig. R2.1 NLHE study in a field effect transistor (112)-Pt/La SrTiO /SrTiO . (a) AFM of\n0.3 3 3\nLa SrTiO /SrTiO (b) Schematic illustration. (c) NLHE without a gate voltage and with a 0.4 gate\n0.3 3 3\nvoltage 𝑉 .\nTG\n(b-2) Conversely, in reduced (112)-SrTiO annealed in vacuum, where oxygen\n3-x\nvacancies are abundant but the overall symmetry remains essentially cubic, no NLHE\nis observed. The (112)-SrTiO substrate was first etched and annealed under the same\n3\ncondition mentioned in the Method section of the main text. Subsequently, an\namorphous Al O layer was patterned on the surface of (112)-STO substrate, leaving\n2 3\nthe active Hall bar area exposed to the ambient conditions. The substrate was then\ntransferred into the PLD chamber and annealed for 10 min in vacuum (oxygen pressure\nof 10−4 mbar) at 800 ℃ to create oxygen vacancies in the Hall bar channel. The sample\nwas finally cooled down to room temperature under the same ambient pressure. In this\nconfiguration, the oxygen vacancies in the STO substrate contribute to the transport.\nHowever, the system experiences only weak symmetry breaking since the surface layer\nis not under the application of an electrical field. One may regard this situation as\nanalogous to the (112)-Pt/La SrTiO /SrTiO situation without gate voltage applied.\n0.3 3 3\nAs a result, this reduced (112)-SrTiO mainly retains the bulk cubic symmetry. Such\n3-x\na system is not expected to undergo any symmetry breaking and therefore does not\nexhibit NLHE, as clearly demonstrated in Fig. RR2.2,\nFig. RR2.2 Nonlinear Hall effect in reduced (112)-SrTiO crystal.\n3-x\n(c) Octahedral rotations:\nIn response to the Reviewer’s example regarding the influence of octahedral rotations\non the spin property of SrRuO , we performed controlled experiments by growing two\n3 3\nconclusion is that despite variation in oxygen vacancy concentration and possible\noctahedral distortions, neither sample exhibits measurable NLHE and MOKE signals.\n(c.1) Sample growth: Sample #1: (112)-SRO/STO with oxygen vacancies intentionally\nintroduced in STO. Nominal 8 u.c. SRO was grown on (112)-STO at 800 ℃ under\noxygen pressure of 10−4 mbar, mimicking the growth condition used for the 2DEG\nformation in (112)-LAO/STO. Sample #2: (112)-SRO/STO with minimised oxygen\nvacancy density. Nominal 8 u.c. SRO is grown on (112)-STO at 600 ℃ under an\noxygen pressure of 0.13 mbar, thereby substantially suppressing the defect formation .\n(c.2) NLHE and MOKE studies of (112)-SRO/STO\nAs shown in Fig. RR2.3, that neither sample displays sizable NLHE or MOKE signal.\nThese results unambiguously rule out octahedral rotation (and/or zigzag, as will be\ndiscussed in Answer 2.2) as the primary mechanism responsible for the nonlinear\ntransport and magneto-optical phenomena reported in our work.\nFig. RR2.3 NLHE and MOKE data acquired on two (112)-SRO/STO samples. (a) NLHE comparison.\n(b) MOKE signal comparison\nIn summary, the fundamental origin of the observed effects is the symmetry breaking\nimposed by crystallographic orientation engineering. While defects may influence the\ntransport properties, their role is secondary and derivative, as their impact itself arises\nfrom the underlying orientation-induced symmetry breaking.\nAction 2.1: We have now added to the SI the results on (112)-SRO/STO. KTO results\nare the subject of a separate work and will not be disclosed at this moment.\nQuestion 2.2. Interpretation of the zigzag pattern: The discussion of the zigzag pattern\nat the interface remains ambiguous. The authors conclude that such patterns do not\ngenerate physical effects, but no clear justification is given. Transport properties and\nspin are very sensitive to microscopic features. Furthermore, they cite Annadi et al.\n(Nat. Commun. 4, 1838 (2013)) to claim that zigzag features are also present in (110)\ninterfaces. However, the zigzag reported in that work extends over only about one\nunit cell and is hardly visible in scanning transmission electron microscopy (STEM) images. In contrast, the zigzag\nmodulation observed in the present (112) samples is much more pronounced, extending over several unit cells with a clear\nperiodic composition modulation, as seen directly in STEM image (Fig. 1e). Indeed, when the contrast in Fig. 1e is\nenhanced, the zigzag pattern becomes very evident and is directly discernible in the STEM images. From this perspective, I\nfind it inappropriate to consider the two cases equivalent. From this point of view, I disagree with the authors' statement:\nBoth images demonstrate a sharp interface between the thin film and the substrate (line 99 on page 4). Such roughness is\nunlikely to average out in a Hall bar geometry and may instead contribute a finite value to the measured transport.\nAdditionally, the anisotropy of this roughness, as seen in Fig. 1e,f, could itself be a possible origin of the observed transport\nanisotropy. I strongly encourage the authors to examine this point more carefully.\nAnswer 2.2: First, we thank the reviewer for the careful inspection of the STEM images\nand for bringing the zigzag modulation of the interface into further discussion. After a\ncareful assessment, we agree with the Reviewer that the zigzag modulations in (110)\nand (112) interfaces are different.\nFor the (110)-oriented SrTiO interface/surface, two in-plane mirror planes exists (i.e.,\n3\n𝑀 and 𝑀 ), which constrains the interface/surface zig-zag pattern to a straight\n[1̅10] [001]\nstep configuration, as illustrated in Fig. RR2.4a. In contrast, the (112)-oriented SrTiO\n3\ninterface/surface retains only one in-plane mirror plane (i.e. 𝑀 ), which might\n[11̅0]\nexplain the more pronounced zigzag pattern observed in the STEM image of (112)-\norientated substrate. This reduced symmetry enforces the zig-zag configuration at the\ninterface/surface as schematically illustrated in Fig. RR2.4b, where even the step edge\nadopts a zig-zag pattern. Notably, such zig-zag pattern will become more evident in\nhigher index orientation, such as the (001)-oriented SrTiO with miscut 4° to [110],\n3\ncorresponding to the Miller index (1 1 20) [see: Tailoring the domain structure of\nepitaxial BiFeO thin films, Current Opinion in Solid State and Materials Science 18\n3\n(2014) 39–45].\nFig. RR2.4 Zig-zag pattern analysis in (a) (110)-orientated, and (b) (112)-orientated surfaces.\nAlthough we did not observe long-range, periodic zig-zag pattern in our (112)-\nLAO/STO sample by AFM, we acknowledge that such local structural modulations (if\npresent) may contribute to the observed effect. However, similar to the case of defects\ndiscussed above, these interfacial zigzag modulations are themselves a consequence of\ncrystallographic orientation and act only in a derivative manner. This interpretation is\nsupported by the control experiments on La Sr TiO transistor, which exhibit clear\n0.7 0.3 3\nnonlinear transport despite the absence of long-range zigzag patterns in AFM\ncharacterization (see Fig. RR2.1a). These results confirm that the zigzag interface\nmodulations are not essential for the manifestation of nonlinear transport effects.\nAction 2.2: To address this critical point, we have added the (112)-SRO/STO data in\nSI to state the effect of zigzag together with the illustration. Also, we have now revised\nthe statement on the ‘interfacial sharpness’. We have added a section in the SI to discuss\nthe interface zig-zag pattern.\nQuestion 2.3. Definition of 𝜃𝑄𝑊𝑃: Although the authors define 𝜃𝑄𝑊𝑃 as the angle\nbetween the fast axis of the QWP and the polarization of the incident light, this\nexplanation is somewhat too brief. It would be helpful to provide a more explicit\nreference direction (e.g., relative to the crystal axes or the measurement geometry in\nthe case of linearly polarized light) to avoid ambiguity.\nAnswer 2.3: We thank the reviewer for this suggestion. We define the angle of the\nquarter-wave plate, 𝜃𝑄𝑊𝑃, as the angle between the fast axis of the QWP and the linear\npolarization direction of the incident beam, which is fixed along the laboratory axis\n𝑥𝐿𝑎𝑏, as shown in Fig. RR2.5. The sample is oriented such that its [112] direction\nparallel to the light propagating direction, i.e., the 𝑧𝐿𝑎𝑏. We note that, the in-plane\ncrystallographic axis of the sample are generally not aligned with the laboratory axis\n𝑥𝐿𝑎𝑏 or 𝑦𝐿𝑎𝑏, resulting in an angular offset, 𝜃𝑂𝑓𝑓. This angle offset 𝜃𝑂𝑓𝑓 occurs as a\nshift in the measured and fitted curves of photocurrent vs QWP angle. To avoid any\npotential ambiguity, we have added a discussion of this effect in the SI.\nFig. RR2.5 Relationship between the QWP angle and particular crystallographic direction.\nAction 2.3: We have now given a more detailed description of the 𝜃𝑄𝑊𝑃 in the\nSupporting Information with this figure presented.\nQuestion 2.4. Fitting of the circular photogalvanic effect (CPGE)/ linear photogalvanic\neffect (LPGE) Data (Figs. 2b, 2c, and R2.4): The authors state that the reviewer \"may\nhave misunderstood\" and that the fits are clearly good. However, I did not\nmisunderstand; my point concerned the significant discrepancies between the\nexperimental data and the total fit, especially near 90 degrees and 150 degrees in Fig.\nR2.4. In fact, in Wang et al., Phys. Rev. Lett. 128, 187401 (2022), the CPGE in\n(111) LAO/STO interfaces is fitted with excellent agreement across all angles and intensities.\nThe substantial mismatch observed here suggests that additional factors may be at play.\nMoreover, in the new version of the manuscript, the separation into LPGE and CPGE\ncomponents has been removed, which makes the text difficult to follow, and Figs. 2b and\n2c appear to be interchanged. These figures need to be carefully re-presented.\nI am also concerned that the sum of the red (CPGE) and blue (LPGE) curves does not\nreproduce the black \"total fit\" curve in Fig. 3c (original version). For example, at around\n135 degrees, the values of LPGE and the total fit differ, despite CPGE being nearly zero\nat this point. Is the fitting procedure correct? Has any angular offset been corrected for?\nIf so, this should be clearly described in the manuscript.\nAnswer 2.4: First, we would like to thank the reviewer for pointing out the mistake.\nFig. 2b and Fig. 2c were in the wrong place.\n(a) Regarding the experiment and fitting quality of the photocurrent.\nWe acknowledge the Reviewer’s comment regarding the fitting of the CPGE data, and\nwe agree that a discrepancy exists between the experimental data and the fitted curves.\nSuch minor deviations may arise from light-polarization-dependent photo-\nthermoelectric effects or other polarization-dependent effects along the optical path that\nare not fully calibrated. Nevertheless, the overall agreement between the experiment\nand fit is sufficiently good to confirm the manifestation of the CPGE under\nperpendicular illumination and resolve the contrast between the two in-plane\ncrystallographic directions. In Table RR2.1 we present the fitting parameters for the\n[11̅0] direction, where the R2 reaches 0.93 (with R2 = 1 representing a perfect fit).\nThis indicates that our fitting is quantitatively reliable.\nTable RR2.1 Fitting parameters of CPGE in direction [11̅0]\nAbs. value Standard error\nCPGE component 0.372 nA 0.0197 nA\nLPGE component 0.392 nA 0.0197 nA\n𝜃𝑂𝑓𝑓 52.88° 0.669°\nR2 0.93 N/A\nThe reviewer cited the paper of Wang et al. [Phys. Rev. Lett. 128, 187401 (2022)],\nwhich indeed show an excellent fit in LAO/STO system compared to our results.\nHowever, it should be noted that even in that study, not all the experimental data exhibit\na perfect agreement with the fits. For example, Fig. 4b of Wang et al., the data at around\n100° and 360° deviate from the fitted curve despite an overall R2 of about 0.97 Similar\ndeviations can also be found in other published papers (see Fig. RR2.7): (a) Fig. 4c of\n[Duan et al., Nat. Nanotechnol. 18, 867 (2023)] (R2 ≈ 0.94) ;(b) Fig. 5b of [Zhu et al.,\nNat. Commun. 13, 7702 (2022)] (R2 ≈ 0.83); (c) Fig. 2a of [Liu et al., Nat. Commun.\n11, 323 (2020)] (R2 ≈ 0.85).\nFig. RR2.6 CPGE results in other papers. (a) Duan et al., Nat. Nanotechnol. 18, 867 (2023) (under gate\nvoltage of -50 V), (b) Zhu et al., Nat. Commun. 13, 7702 (2022), (c) Liu et al., Nat. Commun. 11, 323\n(2020) (the red curve)\nThe main purpose of our CPGE study is to demonstrate the symmetry breaking, and in\nthis respect, the fitting quality is fully adequate. The minor deviations noted above do\nnot compromise the validity of our conclusions regarding CPGE and the associated\nsymmetry considerations.\n(b) Regarding the angular offset & the comparison between the total fit and the\nsum of LPGE and CPGE in direction [11𝟏̅].\nWe thank the reviewer for the very careful examination of our results. Indeed, an error\nwas present in the previous version of the manuscript in the fitting of the CPGE along\n[111̅]. Specifically the offset angle in LPGE and CPGE components was mistakenly\nentered as +51.9° to -51.9°. In Fig. RR2.7 we now present the corrected results for\ncomparison. With this correction, the data are accurate and clearly represented.\nFig. RR2.7 CPGE study in direction [111̅] with fit and the corrected CPGE and LPGE component.\nAction 2.4: we have added a dedicated section in SI discussing to the possible deviation\nof the fitting process. The discussion of the offset angle 𝜃𝑂𝑓𝑓 has also been included in\nthe SI for clarity. Furthermore, to provide a clearer understanding of the relationship\nbetween the individual CPGE and LPGE contributions and the total fit (for the [11̅0]\ndirection), we now present the two fitted components separately in SI.\nResponse to Reviewer #3:\nQuestion 3: I carefully read the rebuttal of the authors to all three reviewers. I have an\noverall sense that the authors carefully addressed most of the concerns and took the\nnecessary actions in revising the manuscript and in consequently expanding the content\nof the supplementary information.\nThus, I recommend the publication of the present manuscript \"Magnetization generation and giant nonlinear transport at\nsymmetry-engineered interfaces via crystallographic orientation\" by H.-B Zhang et al., in Nature communications. I believe it\nwill be of great interest to many scientific communities working on nonlinear transport, and their theories, in various materials\nsystems.\nAnswer 3: We really appreciate the reviewer’s valuable suggestions to make this\nmanuscript better.\nQuestion 3.1: Regarding: Action 3.2.3 & 3.2.4: \"We believe current manuscript is\nalready a complete piece of work and we would like to study the gate tunability of this\nsystem comprehensively in a separate work. We prefer to publish Fig. R3.4 separately.\"\nMy comment: I appreciate the authors' effort and openness to disclose unpublished\nresults, and agree that this is a substantial addition which warrants a detailed\ninvestigation for a follow-up publication.\nQuestion 3.2: Regarding:\n\"To further check the linear dependence in these ordinary Hall effects, we conducted the\nHall measurement with the magnetic field applied as large as ±8 T. As demonstrated\nin Fig. R3.3b, the linearity of these curves at all temperature ranges is maintained up to\n±8 T. Therefore, the magnetic field of ±3 T is sufficient for the Hall measurement to\ninvestigate the carrier density and mobility.\"\nMy comment: I respectfully disagree with the statement that the ordinary Hall effect is\npurely linear in the whole magnetic field range. It appears to me that the Hall effect is\nin fact slightly non-linear in field at the lowest temperatures. Overall I do not think this\naffects the conclusions of the manuscript, because this non-linearity is still quite small.\nHowever, the authors do not ignore that when multiband conduction and Hall effect is\ninvolved, the low-field and high-field regions may yield very different Hall coefficients,\nand that their relation to the carrier density of either or both subbands is not the same.\nIn particular, only the high-field Hall coefficient can be taken as inversely proportional\nto the total carrier density in the system. I would not insist either on a two-bands fit of\nthe Hall effect alone because it is known to be problematic due to the large number of\nfree parameters in the model. I'm looking forward to a more detailed analysis in future\nstudies of the Hall effect in these systems.\nAnswer 3.1 and 3.2: We highly value the reviewer’s rigorous scientific attitude and\nprofound knowledge. We plan to further optimize the growth condition to increase\ninterface quality and have more detailed analysis in future studies of the electronical\ntransport in these systems, including the Hall effect and diffusion of current-induced\nmagnetization, etc.\nAction 3.1 and 3.2: No action taken."}, {"round": 3, "reviewer_comments": {"reviewer_2": "I would like to thank the authors for their efforts in revising the manuscript in response to my comments. Also, I apologize for\nthe wording mistake in my previous Question 2.1, where I wrote 'breaking' instead of 'preservation' of the mirror planes. The\nadditional experiments and discussions, particularly regarding defects and octahedral rotations, convincingly support the\nsymmetry-based interpretation. I am satisfied that my concerns have been fully addressed, and I recommend the manuscript\nfor publication."}, "author_response": "We thank all the reviewer for the valuable suggestion to make this manuscript\nbetter.\nAction: No action taken."}]

Decision letter