Designing spin and orbital sources of Berry curvature at oxide interfaces

Quantum materials can display physical phenomena rooted in the geometry of electronic wavefunctions. The corresponding geometric tensor is characterized by an emergent field known as the Berry curvature (BC). Large BCs typically arise when electronic states with different spin, orbital or sublattice quantum numbers hybridize at finite crystal momentum. In all the materials known to date, the BC is triggered by the hybridization of a single type of quantum number. Here we report the discovery of the first material system having both spin- and orbital-sourced BC: LaAlO3/SrTiO3 interfaces grown along the [111] direction. We independently detect these two sources and probe the BC associated to the spin quantum number through the measurements of an anomalous planar Hall effect. The observation of a nonlinear Hall effect with time-reversal symmetry signals large orbital-mediated BC dipoles. The coexistence of different forms of BC enables the combination of spintronic and optoelectronic functionalities in a single material.


Contents
To derive the electronic properties of the two-dimensional system realized at the [111]−LaAlO 3 /SrTiO 3 interface in the hightemperature trigonal phase, we first discuss the orbital and symmetry character of the electronic levels at the center of the Brillouin zone. Let us first neglect spin-orbit coupling. Due to the trigonal coordination, the three t 2g Ti orbitals are split into an a 1g orbital representing a real one-dimensional irreducible representation of the C 3v point group, and an e ′ g doublet forming a real two-dimensional irreducible representation of the group. The e g Ti orbitals are not affected by the trigonal crystal field as they also form a doublet. Let us now include the effect of spin-orbit coupling breaking the SU(2) spin rotation symmetry. The two spin-orbit coupled states originating from the a 1g orbital have a symmetry-protected degeneracy since they form the Γ 4 two-dimensional irreducible representation of the C 3v double-point group. Spin-orbit coupling instead splits the quartet of states originating from the e ′ g doublets. Specifically, two spin-orbit coupled orbitals are singly degenerate and form the Γ 5 and the Γ 6 one-dimensional irreducible representations of the double point group. The remaining two states are instead degenerate and form a Γ 4 representation. Since the Γ 5 and Γ 6 representations are complex, time-reversal symmetry implies that these states must stick together thus forming a Kramers' doublet. The irreducible two-dimensional Γ 4 representation is instead quaternionic and therefore is already equipped with time-reversal invariance. As a result, we have that all levels at the BZ center correspond to an effective spin- 1 2 Kramers' doublet. The minimal model Hamiltonian close to each of these Kramers' doublets can be derived in a k · p expansion accounting for all symmetry-allowed terms. To do so, we note that in the surface Kramers' doublet basis the time-reversal symmetry can be represented as T = iσ y K with K the complex conjugation. The mirror symmetry is instead represented by M = iσ x . Note that from here onwardsx will indicate the 1 10 direction. In the basis |ψ ↑↓ ⟩ the threefold rotation operator takes the form C 3 = e −iσ z π/3 . Under the operation of C 3 and M , momentum and spin transform as follows, k ± → e ±i2π/3 k ± , σ ± → e ±i2π/3 σ ± , σ z → σ z M : where k ± = k x ± ik y and σ ± = σ x ± iσ y . The Hamiltonian must also be invariant under time reversal which adds the constraint H (k) = T H (−k)T −1 = σ y H * (−k)σ y . At linear order in the momentum k, and including an effective regularizing quadratic term, the minimal two-band Hamiltonian for a Kramers' related pair of bands reads: where σ σ σ is a vector of Pauli matrices, σ 0 is the identity matrix,α R is the "Rashba" spin-orbit coupling strength, while m is the effective electron mass. Note that nothing prevents the effective electron mass to be negative. In this case, electron-like transport can be ensured by adding a term quartic in momentum k 4 /(2m 1 ) σ 0 with m 1 > 0. Overall, this amounts to consider a momentum dependent effective electron mass. The Hamiltonian in Eq. (1) does not capture crystalline anisotropy effects. In addition, the Berry curvature associated to this minimal model is zero, since there is no term proportional to σ z . However, higher order momentum terms change this situation. The first symmetry allowed term accounting for crystalline anisotropy is third order in momentum and takes the form, This warping Hamiltonian is proportional to the Pauli matrix σ z , which is crucial to obtain a non-zero Berry curvature and leads to out-of-plane spin textures. Note that since the full Hamiltonian is invariant under the mirror symmetry M , H w (k) is forced to vanish along the mirror line.
To show the presence of a finite Berry curvature induced by warping, we recall that in a two band model the Berry curvature can be calculated by rewriting the full Hamiltonian as H (k) = d(k) · σ σ σ + k 2 σ 0 /[2m(k)], where d is a momentum dependent vector, which for our specific model has components d = −k y , k x , λ k 3 The d vector is independent of terms ∝ σ 0 , and thus of the momentum dependent effective electron mass. The expression for the Berry curvature is then given by . For our minimal model Hamiltonian in the presence of trigonal symmetry, we have where θ is the polar angle in momentum space. The Berry curvature is well defined in each point except the origin where the bands are degenerate. Note that the constraints set by time reversal symmetry and the three-fold rotational symmetry are satisfied as can be verified upon a closer inspection of Eq. (3). Moreover Ω z ± (k, θ ) vanishes along the mirror lines, in accordance with Eq. (2). The orbital sources of Berry curvature can be instead derived by explicitly considering interorbital mixing terms. To this end, it is convenient to neglect spin-orbit coupling. The effective Hamiltonian for the a 1g and e ′ g doublet in the trigonal crystal field can be derived using symmetry principles. Specifically, any generic 3 × 3 Hamiltonian can be expanded in terms of the nine Gell-Mann matrices Λ i as The invariance of the Hamiltonian requires that the components of the Hamiltonian vector b(k) should have the same behavior as the corresponding Gell-Mann matrices Λ i . This means that they should belong to the same representation of the crystal point group. The representation of the Gell-Mann matrices Λ i and those of the polynomials of k can be found using that the generators of the point group for spinless electrons can be written as With this, the effective Hamiltonian up to linear order in momentum reads where Λ 0 is the identity matrix and the other Gell-Mann matrices are reported in the main text. The last term in the equation above corresponds to the orbital Rashba coupling with strength α OR . The second term gives the crystal field splitting of size ∆ between the a 1g singlet and the e ′ g doublet. As discussed in the main text, a direct computation of the Berry curvature using the method outlined in Ref. 1 yields a vanishing Berry curvature at all momenta. This changes by considering a crystalline symmetry lowering to C s . From the representation of Gell-Mann matrices and polynomials of momentum, the effective Hamiltonian now takes the form In the equation above, the crystal field splitting ∆ m yields an energy separation between the e ′ g states at the center of the BZ. This Hamiltonian is characterized by Berry curvature distributions with dipolar hot spots and singular pinch points. Furthermore, it is naturally equipped with a large non-vanishing Berry curvature dipole. The dispersion of the bands is strongly dependent on the ratio between the characteristic orbital Rashba energy E OR = 2mα 2 OR and the crystal field splittings. In particular, the effective mass of the lowest band is negative close to the Γ point of the BZ whenever the orbital Rashba energy is comparable to the crystal-field splitting ∆. In this regime, an intraband Lifshitz transition occurs when increasing the chemical potential. On the contrary, in the opposite regime of large crystal field splittings, all bands possess an effective electron mass that is positive at all momenta, and consequently intraband Lifshitz transitions do not take place. This is explicitly shown in Fig. S1 where we report the evolution of the bands by continuously increasing the ∆/E OR ratio. Note that similar features are also found within the remaining doublet of bands. In this case the momentum dependence of the effective mass is controlled by the ratio between the crystal field splitting ∆ m and the effective orbital Rashba energy E OR = 2mα 2 m , and can acquire a non-trivial dependence along the k y = 0 line.  In Fig. S1 we also indicate the relevant range of energies in the band structure, which we evaluate using the following argument. To obtain a Berry curvature dipole of order k F , the orbital Rashba parameter has to be comparable to k F /(2m). This is explicitly shown in Fig. 3a of our main manuscript -the Berry curvature dipole has a sweet spot and can exceed 1/k F when 2mα OR ≃ k F . Hence, the orbital Rashba momentum k OR ≃ k F . Considering the finite range of the Berry curvature dipole sweet spot and the additional tuning of the carrier density by gating (see Supplementary Note III), we estimate that the Fermi wavevector k OR < k F < 1.5k OR . This consequently gives an upper and lower bounds for the Fermi energy shown as gray lines in Fig. S1. We note that the possibility of occupying more than one band is in line with the observations presented in Ref. 2. Finally we wish to point out that since the orbital Rashba energy corresponds to the characteristic kinetic energy k 2 F /(2m), which lies in the tens of meV range, all values of the crystal field splitting ∆ considered in Fig. S1 could be realized in practice -x-ray absorption spectroscopy [3] indicates a crystal field splitting of the order 8 meV. In other words, the material could be on the verge of a change in the sign of the effective mass close to zero momentum.
We next show that both the Hamiltonian characterizing the orbital degrees of freedom and the Hamiltonian in Eq. (1) for the spin degree of freedom can be derived from a single model that treats orbital and spin degrees of freedom on an equal footing. In order to make a link between the spin and orbital degrees of freedom one needs to include the atomic spin-orbit coupling.
Then, the starting point for the coupled spin and orbital degrees of freedom is provided by the following model Hamiltonian: where we have included the atomic spin-orbit coupling with the strength λ so and σ i indicate the spin Pauli matrices with i = x, y, z.
We recall that the orbital angular momentum projected on the effective subspace with the three selected orbital configurations has the following representation in terms of the Gell-Mann matrices: L x = Λ 2 , L y = Λ 5 , and L z = Λ 7 . For simplicity, we next consider a C 3v symmetric (i.e. ∆ m = α m = 0) crystalline environment. At the Γ point of the BZ, and explicitly including atomic spin-orbit coupling, the Hamiltonian reduces to: with H Γ tot can be diagonalized and yields the following double degenerate (Kramer pairs) eigenvalues According to the character of the trigonal splitting and the Hund's rule (spin-orbit coupling favors the highest total angular momentum at low filling) λ so and ∆ have negative values. Thus, the following energy relation holds The eigenvectors can be also written in a compact vectorial form as: with In the new spin-orbit coupled basis the Hamiltonian H tot has the following matrix form: with H MN being 2x2 matrices.
Let us now focus on the matrix elements of the block Hamiltonian H AA associated with the lowest energy doublet assuming that the energy splitting among the levels is larger than the strength occurring in the inter-doublet terms (i.e. H CB ,H BA , H CA ). We emphasize that since the energy splitting between the spin-orbit Kramers' doublets is controlled (at fixed values of the crystal fields) by the spin-orbit coupling strength, the intradoublet hybridization channels H BA , H CA , H CB can be safely neglected in the large spin-orbit coupling regime λ so ≫ E OR . In this case, one can directly determine the effective spin Hamiltonian by evaluating the following expectation values: Taking into account the expressions of the eigenvectors v A,i and of the k-dependent terms of the total Hamiltonian, the matrix of the block H AA can be written as: with the effective intra-doublet Rashba interaction given by Note that in the Hamiltonian above the trigonal warping in the spin sector is found by equipping the orbital model Hamiltonian with a symmetry-allowed cubic term ∝ γ(k 3 Then the warping coupling λ has the following expression Therefore, the effective low-energy spin model can be directly linked to the orbital model with its sources of Berry curvature dipole. Two remarks are in order here. First, using a Löwdin procedure one can verify that there exist higher-order corrections, e.g. momentum dependent renormalization of the Rashba coupling α R → α R (1 + δ k 2 ). Second, the bare effective electron mass appearing in Eq. (22) is corrected by intraorbital mixing terms due to the finiteness of the spin-orbit coupling strength. This generally results in a momentum dependent mass that can be negative close to zero momentum in agreement with the features of the spin-orbit free orbital bands discussed above. This is explicitly proven in Fig. S2 where we show the evolution of the spin-orbit coupled bands by increasing λ so . Spin-orbit coupling generally weakens the momentum dependence of the effective mass eventually leading to a positive effective mass at all momenta. Importantly, and as mentioned above, the symmetry-based spin model can be applied without restrictions even in the presence of a strongly momentum dependent mass. Therefore, the spin sources of Berry curvature are correctly captured even for intermediate spin-orbit coupling strenghts.
In this case, the spin-sourced model can be applied up to momenta for which the corresponding energy of the spin bands is comparable to the splitting ∆E Γ between two different Kramers doublets at the center of the BZ. This gives an upper bound for the Fermi wavevector k F = k 2 R /4 + m∆E Γ − k R /2 with k R = 2mα R . Our experimental observations indicate the occurrence of a strong anomalous planar Hall effect at the same carrier densities at which a sizable non-linear Hall effect at zero magnetic field is measured. The characteristic features of the anomalous planar Hall effect can be explained using the spin-sourced model -the orbital-sourced Berry curvature is stiff in response to externally applied in-plane magnetic fields. Instead, the non-linear Hall effect at zero magnetic field cannot be quantitatively captured within this picture, and the orbital sources of Berry curvature are needed. Putting these observations together, we can therefore conclude that both orbital Rashba and spin-orbit coupling are relevant ingredients to describe the physical properties of the material structure. Our experimental tools independently probe the Berry curvature associated with orbital and spin degrees of freedom.

II. Supplementary Note II: Magnetotransport theory Anomalous planar Hall effect
In this section we discuss the Hall effect induced by the spin-sourced Berry curvature in the presence of a planar magnetic field using a semiclassical Boltzmann framework.
We first note that the presence of a planar magnetic field generates a Zeeman coupling term in the low-energy Hamiltonian H B = B(σ x cos φ + σ y sin φ ), that breaks C 3 when present, T and the remaining mirror M x except when the direction of the magnetic field with respect to thex axis φ = 2πm/6 with m ∈ N. In this case one mirror symmetry is preserved. The energy dispersion of the total Hamiltonian is then given by: The planar magnetic field breaks the Kramer's degeneracy at the center of the BZ and completely splits the two bands. However, when a mirror is preserved there is a mirror-symmetry protected Dirac point along the mirror line at (k, θ ) = (|B|/α R , φ + sgn(B)π/2). The condition k F = |B|/α R defines the critical magnetic field strength introduced in the main text. Moreover, a Lifshitz transition (a change in topology in the Fermi surface, in this case going from one closed Fermi line to two) appears since the Zeeman coupling shifts the minima of the two bands. The energy for which the upper band is completely depleted is given by To derive the transport characteristic associated to this low-energy model we assume to apply an electric field E = Ex along thex axis and rotate the magnetic field in the x − y plane. Thus we write the magnetic field as B = B(cos φx + sin φŷ). Generally speaking, the semiclassical equations of motions in the presence of a non-zero Berry curvature reaḋ where D(B, Ω k ) = [1 + (e)(B·Ω k )] −1 , Ω k is the Berry curvature and v k is the group velocity. Solving the Boltzmann equation for the electron distribution function f (k) within the relaxation time approximation allows to compute the charge current J = e (d d k/(2π) d )ṙ f (k)D −1 that accounts for the modified phase space factor D. In linear response theory the charge current obeys the relation J a = σ ab E b , where σ ab are the components of the conductivity tensor and E b the external electric field. Following Ref. 4, the planar Hall conductivity, after discarding higher order contributions, is given by where f eq is the equilibrium Fermi-Dirac distribution. Eq. (28) contains all transverse linear responses in the presence of coplanar electric and magnetic fields in both two-and three-dimensional systems. In three-dimensional topological Dirac and type-I Weyl semimetals, the Berry-curvature induced planar Hall effect stems from the term ∝ B 2 sin θ cos θ (v k ·Ω k ) 2 . The last term instead represents the usual anomalous Hall contribution for magnetic materials. In two-dimensional systems, v k ⊥ Ω k . However, in trigonal crystals a non-vanishing contribution to the last term of Eq. (28) appears once the Zeeman interaction is explicitly taken into account, and exists provided all mirror symmetries are broken, even though the material is non-magnetic. This also implies that, contrary to three-dimensional topological semimetals, the Berry-curvature related planar Hall effect of two-dimensional systems does not explicitly depend on the relative angle between the electric and magnetic field but only on the angle between the planar magnetic field and the principal crystallographic directions. This, in turns, allows to directly observe a purely antisymmetric planar Hall effect when the electric and magnetic fields are perfectly aligned since in this configuration the classical contribution of Eq. (28), containing the v x v y term vanishes.
We have computed the behavior of the anomalous planar Hall conductivity assuming a magnetic field direction φ = π/2. In this configuration all mirror symmetries are broken. Moreover the integral of the Berry curvature is maximum. The contribution to the linear transverse response σ xy is given by the integral of the Berry curvature over the Fermi surfaces of the two energy bands. Since the two bands contribute with opposite curvatures it is sufficient to integrate over the exclusion region of the two bands. Hence the anomalous contribution to the transverse conductivity, at zero temperature is given by where S ± are the Fermi surfaces of the two bands and the last integral contains Ω − since it is the curvature of the outermost band. The magnetic field dependence at constant density can be calculated numerically by varying the Fermi energy as the magnetic field is changed. Indeed, by keeping the area of the surface S + ∩ S − fixed the number of electronic carriers stays constant. To obtain the resistivity ρ xy it is necessary to compute the two longitudinal conductivities σ xx and σ yy . In the relaxation time approximation these are given by, where α = (x, y), ε γ is the energy dispersion of the band γ and f 0 is the equilibrium Fermi-Dirac distribution. The resistivity ρ xy is then obtained by inverting the conductivity tensor, Here we have used that the transverse conductivity is purely antisymmetric, i.e. σ xy (B) = −σ yx (B), since the semiclassical contributions are vanishing. Figure S3 shows the corresponding behavior of the anomalous planar Hall resistivity as a function of the magnetic field strength for different values of the carrier density. The plot has been obtained by writing the low-energy Hamiltonian in dimensionless form measuring energies units of k 2 F /2m, lengths in units of 1/k F and density in units of n 0 = k 2 F /2π. The remaining dimensionless parameters have been fixed to α R = 0.4, and λ = 0.1 Furthermore, we have used the approximate expression for the transverse resistivity ρ xy ≃ σ xy /(σ xx σ yy ), which is accurate since the transverse conductivity is much smaller than the linear in τ longitudinal resistivity. We obtain that the anomalous planar Hall resistivity increases non-monotonically and gets enhanced by decreasing the carrier density. Note that when considering the observed decrease of the carrier density as the sheet conductivity increases (see Supplementary Note VI), this trend is in agreement with the behavior of the antisymmetric Hall resistance reported in the main text.

Nonlinear transverse response with planar magnetic fields
Nonlinear transverse currents have two intrinsic contributions: the first is a semiclassical term that depends on the integral of the electronic velocities, whereas the other has a purely quantum nature stemming from the Berry curvature dipole: where ε αβ is the Levi-Civita antisymmetric tensor, ∂ α = ∂ k α , f 0 is the equilibrium Fermi Dirac distribution and ε k is the energy dispersion. The two conductivities are the proportionality factors between the applied AC electric field (E ω α ) 2 and the second harmonic response j 2ω β . Typically the full response is measured, and it is necessary to decouple the two contributions in order to extract the magnitude of the Berry curvature dipole. This can be done by considering how the two conductivities behave when switching the sign of the magnetic field. The σ sc ααβ is odd in B: this can be seen by sending B → −B and applying the coordinate change k → −k. While the integration measure ∂ k x ∂ k y and the f 0 remain unchanged [ε(k, B) = ε(−k, −B)] the three derivatives bring an overall minus sign. On the other hand the contribution σ

Symmetry constraints on the linear and nonlinear resistivity tensor
We recall that the linear conductivity tensor is defined by the relation We can derive the transformation rule of the conductivity tensor under a generic point group symmetry represented by an orthogonal matrix O by simply noticing that both the current j and the driving electric field E transform as vectors under a generic coordinate change. Therefore, the conductivity tensor transforms as O T σ O. For the point group C s the single mirror symmetry M x implies that the transverse conductivity σ xy ≡ σ yx ≡ 0. In crystals with C 3v point group symmetry instead, the additional threefold rotation symmetry implies that the two longitudinal conductivities along the principal crystallographic directions σ xx ≡ σ yy . Crystalline symmetries also pose constraints on the nonlinear conductivity tensor defined by The transformation rule of the nonlinear conductivity tensor imply that in the presence of a M x mirror symmetry, we have χ xxx = χ xyy = χ yxy = χ yyx = 0. The additional threefold rotation symmetry in the C 3v point group symmetry implies that the non-zero component of the nonlinear conductivity tensor satisfy the relation χ xxy = χ xyx = χ yxx = −χ yyy . A violation of this relation implies that the trigonal symmetry is broken and only a mirror symmetry is present in the system.

Planar magnetoresistance computation
We have computed the planar magnetoresistance MR = [ρ xx,yy (B)/ρ xx,yy (0) − 1] considering a planar magnetic field directed along the [110] direction, thus preserving the mirror symmetry. In this case, the transverse Berry-mediated conductance σ xy vanishes. The magnetoresistance is strongly anisotropic and indeed a qualitative difference exists depending on whether the driving current is collinear or orthogonal to the magnetic field. Specifically in the former case (see Fig. S4) a small positive magnetoresistance starts to develop when the Fermi energy crosses the mirror symmetry-protected Dirac point (orange line). This positive magnetoresistance persists up to the Lifshitz point (red line). After the Lifshitz transition, the magnetoresistance starts decreasing and reaches a negative saturation value as also seen in experiments (see Fig. S5). The negative magnetoresistance can be intuitively understood by considering that after the Lifshitz transition scattering between the two energy bands is suppressed, hence enhancing the magnetoconductance. In the configuration in which the driving electric field and the magnetic field are orthogonal, the magnetoresistance is always negative and the weak-field positive magnetoresistance does not occur [see Fig.  S2(a)]. This is in agreement with the features observed at the LaVO 3 -KTaO 3 interface [5]. Note that for configurations in which the magnetic field is not mirror-preserving, a similar type of behaviour is still expected since the quantum contribution is a lower order correction to the semiclassical one. direction and an orthogonal (a), or collinear (b) driving electric field. Since the magnetic field is mirror-symmetry preserving ρ xx,yy = 1/σ xx,yy . The magnetoresistance has been obtained using the same parameter set as in Fig. S1, and is shown also as a function of the carrier density. The magnetic fieldB is normalized in units of k 2 F /2m and the densitiesñ are in units of k 2 F /2π.  In both cases, the MR is seen to grow negatively above a critical planar magnetic field value. Panels (c) and (d) display the corresponding sheet conductance dependences of the MR at constant magnetic field values. For both Hall bar devices, the MR shows an onset above a given value of sheet conductance followed by a monotonic increase, and even an apparent saturation for the curve corresponding to B = 12 T.

III. Supplementary Note III: Additional magnetotransport measurements
Ordinary Hall effect & estimation of the momentum relaxation time − gate dependence Figure S6(a,b) display the gate-modulated ordinary Hall effect, and longitudinal MR (respectively) acquired in the same device, oriented along [110], presented throughout the manuscript; with B ⊥ the out-of-plane magnetic field. At low doping levels, the two-dimensional electron system (2DES) exhibits a linear Hall effect, while the low-field MR is indicative of a weakantilocalization regime, as reported previously [2,6]. At sheet conductance values σ xx exceeding ≈ 2 mS, ρ xy (B ⊥ ) is found to depart from a purely linear Hall effect. Non-linearities in the ordinary Hall effect response have been discussed extensively in the framework of a multi-carrier conduction, or multi-orbital conduction when considering SrTiO 3 -based 2DES [2,7,8]. Here, in the case of the [111]−LaAlO 3 /SrTiO 3 2DES, the nonlinear Hall component has been attributed to the populating of replica sub-bands of the t 2g -manifold (derived from the Ti-3d bands) in the quantum well [2,9]. A number of authors have further discussed the relevance of the two-band model for the determination of meaningful transport parameters values, i.e., carrier densities and mobilities [2,7,8]. We simply point out, that in the limit of large magnetic fields, the total carrier density in the system, which we denote n 2D , can be related to the slope of ρ xy vs. B ⊥ via: where R H = ∂ ρ xy (B ⊥ )/∂ B ⊥ is the (here, high-field) Hall coefficient, whose negative sign is consistent with electron-like transport. Making use of the Drude's formula, it follows for µ H , the electronic mobility: where σ s = σ xx is the sheet conductance of the considered Hall bar device (along [110]). Figure S6(c) displays both the estimated areal carrier density and electron mobility of the [111]−LaAlO 3 /SrTiO 3 2DES across the whole accessible doping range 0.5 ≤ σ xx ≤ 6 mS. While the mobility is found to increase monotonically, the apparent decrease of n 2D versus σ xx is consistent with previous reports making use of a two-band fitting procedure up to 15 T, and which has been physically mapped by self-consistent tight-binding calculations to the redistribution of sub-bands population under the effect of electronic correlations (see Ref. 2 for details). We estimate the momentum relaxation time τ (el) [see Figure S6(d)], within Drude's model, which in the quasi-d.c. limit (ωτ ≪ 1) is given by:  [10], m e the electron mass. The resulting calculated m * is assumed to be gate-and temperature-independent. A linear interpolation of the measured value of the momentum relaxation time τ vs. σ xx allows to calculate the sheet conductance dependence of the BCD's magnitude D x (shown in Fig. 4d) following equation (1)  Additionally, we also report the temperature dependence of the ordinary Hall effect and longitudinal MR of the 2DES, as shown in Fig. S7(a) and S5(b), respectively, for the Hall bar device oriented along [110]. Following the steps described above for the gate-dependent data set, we similarly extract the temperature dependence of the carrier density and electronic mobility [see Fig. S7 Additional doping-dependent Hall effect measurements and magnetoconductance in the weak antilocalization regime We present in Fig. S8(a,b) the gate-dependent ordinary Hall and longitudinal magnetoconductance data set collected for the Hall bar device oriented along the [112] direction. This allows the experimental estimation of total the carrier density (n 2D ), Hall mobility (µ H ), momentum scattering time (τ el ), as well as inelastic and spin-orbit and scattering times (τ i and τ so , respectively) as a function of the 2DES' sheet conductance σ yy (I ω y ∥ [112]). We refer the reader to the Methods sections of the main manuscript for details regarding the Hikami-Larkin-Nagaoka (HLN) model used to fit [see Eq. (3)] the magnetoconductance curves in the weak-antilocalization regime [11,12]. Estimation of the critical magnetic field and corresponding Zeeman energy. Comparison with the Rashba spin-orbit energy On one hand, we establish a criterion for the determination of the 'critical' magnetic field, B c , at which the transverse magnetoresponse in a planar magnetic field exceeds a given threshold value R c xy [shown in the legends of panels S9(a,c), respectively]. The corresponding effective Zeeman energy, at the in-plane field magnitude B = B c , is given by: where g = 2 is the electron g-factor. In the main manuscript, we keep the criterion R c xy ≡ R xy ≥ 6 Ω for the determination of B c and ∆ c Z , as shown in Extended Data Fig. 5. On the other hand, assuming a D'yakonov-Perel' spin relaxation mechanism [13,14], the Rashba spin-orbit energy is given by [12]: with α R the Rashba spin-orbit coupling (SOC) determined from WAL measurements (see Methods section), and k F the electron wavevector at the Fermi energy.

Out-of-plane misalignment of the planar magnetic field
In order to experimentally estimate the value of the out-of-plane misalignment angle, γ, for the measurements displayed in Fig. 3c,d of the main manuscript, we conduct a low-field analysis of both the ordinary Hall and first-harmonic planar Hall effects as a function of doping levels for the same [110]−oriented Hall bar device.
Figures S10(a) and S8(b) are magnified low-field views of the data set displayed in Fig. S6(a) and Fig. 2c (see main manuscript). We perform linear fits between ±2 T of the field-antisymmetrized out-of-plane ρ xy (B ⊥ ), and field-antisymmetrized in-plane R ω xy (B ∥ ) Hall magnetoresponses. The corresponding slopes are shown in Fig. S10(c) and S10(d), respectively. The common dependence of both quantities as a function of σ xx highlights their common origin. We hence attribute the linear contribution to R xy (B) in Fig. 2c [here denoted: R ω xy (B ∥ )] at low-field to a spurious contribution of the ordinary Hall component due to a small out-of-plane magnetic field component: ∆B ⊥ = B ∥ sin(γ), resulting from an imperfect coplanar alignment of the field with the plane of the 2DES. We denote this out-of-plane misalignment angle γ. The contribution ∆R ω xy to the planar Hall effect, from the conventional Hall effect due to this misalignment, is then expected to take the form: To further support this interpretation, we display in Fig. S11 the calculated quantity γ given by: where the argument is the ratio of the low-field slopes from the in-plane and out-of-plane Hall effects. We indeed find that γ is independent of σ xx , and consistently smaller than 1.5 • . This misalignment amounts to an out-of-plane field component ∆B ⊥ ≤ 25 mT at B ∥ = 1 T. Figure S12 displays the change of longitudinal and transverse planar magnetoresponses, when deliberately imposing a small out-of-plane misalignment of the magnetic field (with tilt angle ∆γ). We define γ = 0 • the angle at which the measurements displayed in Fig. 2 of the manuscript were performed. Prior to any measurement campaign, we tentatively minimize γ by finding the tilt-angle which minimizes the low-field slope of R ω xy (B ∥ ), which is typically performed at low doping levels where the BCD-induced planar Hall contribution is absent. device. For both devices: while the linear planar Hall response is found to be quite insensitive to small ∆γ offsets, the quasi-planar MR shows a drastic change of magnitude upon small tilts of the magnetic field out-of-plane.
As seen in Fig. S12(a,c), we find that the planar longitudinal MR is extremely sensitive to a very small out-of-plane misalignment, changing by a factor of two within only one degree. This explains the relative magnitude discrepancies in the planar MR between different cooldowns and measurements campaign. On the other hand, the planar Hall contribution is found to be relatively robust against small out-of-plane tilts of the magnetic field [see Fig. S12(b,d)]. The low-field slope R yx is found to be proportional to ∆γ, corroborating that it originates from a spurious conventional Hall component.

Angular dependence of planar magnetoresponses
We acquire the full in-plane angular dependence of the longitudinal magnetoresistance (MR) at |B| = 12 T, by sweeping the angle θ between the field and the current direction, in steps of 5 • . Due to our mechanical rotator being limited to a 180 • rotation range, we perform the rotation procedure twice, once for B = +12 T and a second time for B = −12 T, while keeping the sourced current and voltage probe contacts polarities unchanged. We can obtain the full field-symmetrized magnetoresistance at |B| = 12 T, as displayed in Fig. S13, by virtue of Onsager's relation: ρ xx,yy (B) = ρ xx,yy (−B). The field-symmetric longitudinal magnetoresistance is then given by: Whether the bias current is sourced alongŷ ∥ [112] or alongx ∥ [110], the planar MR follows the semiclassical cos(2θ ) dependence, where θ is the relative angle between the current direction (along a principal crystal axis) and the planar magnetic field orientation.
Concomitantly, when measuring the planar transverse magnetoresponse, we observe that the field-symmetric contributions R sym xy(yx) , of semiclassical origin (usually referred to as the 'planar Hall effect'), follows a sin(2θ ) dependence, and goes to zero at θ = 0[ π 2 ], as expected for a nonmagnetic system. However, we find that the total transverse resistance R tot αβ (displayed in Fig. 2f of the manuscript) is dominated by the fieldantisymmetric contribution (see Fig. S14) , R as αβ , dubbed "anomalous planar Hall effect" [4,15], which remains finite whenever the external planar magnetic field is not orthogonal to the mirror line M [110] . Independent of whether the current is sourced along the