Abstract
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 orbitalsourced BC: LaAlO_{3}/SrTiO_{3} 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 timereversal symmetry signals large orbitalmediated BC dipoles. The coexistence of different forms of BC enables the combination of spintronic and optoelectronic functionalities in a single material.
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Main
When moving along closed paths, electrons can accumulate a geometric Berry phase related to the flux of a field, called the Berry curvature (BC), encoding the geometric properties of the electronic wavefunctions. In magnetic materials, the adiabatic motion of electrons around the Fermi surface provides such a Berry phase. It is directly observable since it governs the intrinsic part of the anomalous Hall conductivity^{1,2}. Anomalous Hall effect measurements, therefore, represent a charge transport footprint of the intrinsic geometric structure of electronic wavefunctions. In nonmagnetic materials, the BC field is forced to vanish by symmetry when summed over the occupied electronic states. However, local concentrations of positive and negative BC in momentum space are allowed by acentric crystalline arrangements^{3}. This segregation of BC in different regions of momentum space appears whenever electronic states with different internal quantum numbers are coupled to each other by terms that linearly depend on crystalline momentum k. In these regions, the electronic bands typically resemble the dispersion relations of relativistic Dirac or Weyl fermions. The spin–orbit linearink coupling between different spin states shapes the Dirac cones at the surfaces of threedimensional topological insulators^{4,5} as well as the Weyl cones of topological semimetals^{6}. Couplings between different atomic orbital and sublattice states, instead, give rise to the (gapped) Dirac cones of transitional metal dichalcogenides and graphene. Conceptually speaking, the appearance of BC beyond this Dirac/Weyl paradigm is entirely allowed. The fundamental conditions for the occurrence of BC only involve the crystalline geometry of a material, with no restrictions on the specific properties of its lowenergy electronic excitations. Achieving this challenge is of great interest. First, it could, in principle, result in the coexistence of different mechanisms of BC generation. This could be used, in turn, to endow a singlematerial system with different BCmediated effects, for instance, spin and orbital Hall effects. Second, searching for BCs without Dirac or Weyl cones might allow the design of materials with interplay of correlated and topological physics—an unexplored frontier in condensedmatter physics.
Here we reach these two milestones in the twodimensional electron system (2DES) confined at (111)oriented oxide interfaces, with a hightemperature trigonal crystalline structure. This model system satisfies the crystalline symmetry properties for a nonvanishing BC. The combination of spin–orbit coupling, orbital degrees of freedom associated with the lowenergy t_{2g} electrons, and crystal fields leads to the coexistence of a spinsourced and orbitalsourced BC. The two sources are independently probed using two different charge transport diagnostic tools. The observation of the BCmediated anomalous planar Hall effect (APHE)^{7,8} grants direct access to the spinsourced BC, whereas nonlinear Hall transport measurements in timereversal symmetric conditions^{9,10} detect an orbitalmediated Berry curvature dipole (BCD)—a quantity measured so far only in gapped Dirac systems^{9,10,11,12,13,14,15,16,17,18,19} and threedimensional topological semimetals^{20,21,22,23,24,25}. We identify (111)oriented LaAlO_{3}/SrTiO_{3} heterointerfaces as an ideal material system because their 2DES features manybody correlations and a twodimensional superconducting ground state^{26,27,28,29,30}.
We synthesize (111)oriented LaAlO_{3}/SrTiO_{3} heterostructures by pulsed laser deposition (Methods). The samples are lithographically patterned into Hall bars oriented along the two orthogonal principal inplane crystallographic directions: the \([\bar{1}10]\) and \([\bar{1}\bar{1}2]\) axis (Fig. 1a). The sheet conductance and carrier density of the 2DES are controlled by electrostaticfield effects in a backgate geometry (Fig. 1b). We source an oscillating current (I^{ω}) with frequency ω/2π along each Hall bar, and concomitantly measure the longitudinal response as well as the first or secondharmonic transverse voltages in a conventional lockin detection scheme (Fig. 1a).
The nontrivial geometric properties of the electronic waves in the 2DES derive entirely from the triangular arrangement of the titanium atoms at the (111)oriented LaAlO_{3}/SrTiO_{3} interface (Fig. 1c). Together with the \({{{{\mathcal{M}}}}}_{\bar{1}10}\) mirrorline symmetry, this yields a \({{{{\mathcal{C}}}}}_{3v}\) crystallographic point group symmetry. As a result of this trigonal crystal field and the concomitant presence of spin–orbit coupling, the entire dorbital manifold of Ti atoms located at the centre of the surface Brillouin zone (BZ) is split into five distinct Kramers’ pairs (Supplementary Note I). The energy bands of the pairs are shifted in momentum due to spin–orbit coupling. In their simplest form, they acquire a parabolic dispersion reminiscent of a Rashba 2DES (Fig. 1d). However, the trigonal crystal field brings about a specific hexagonal warping^{31,32} that has a twofold effect. First, for each timereversal related pair of bands, the Fermi lines acquire a hexagonal ‘snowflake’ shape^{33}. Second, and the most important, the spin texture in momentum space acquires a characteristic outofplane component^{34,35}, with alternating meron and antimeron wedges respecting the symmetry properties of the crystal (Fig. 1e). This unique spin–momentum locking enables a nonvanishing local BC entirely generated by spin–orbit coupling (Supplementary Note I). The local BC of the spinsplit bands of each pair cancel each other at the same crystal momentum. However, there is a region of crystal momenta populated by a single spin band. In this region (namely, the annulus between the two Fermi lines of the system), alternating positive and negative regions of nonvanishing BC are present (Fig. 1f).
Apart from the spin channel, an inherently different source of BC exists. In systems with orbital degrees of freedom, the lack of crystal centrosymmetry yields coupling that are linear in k, and mix different atomic orbital states. These orbital Rashba couplings^{36} are independent of the presence of spin–orbit coupling. Precisely as its spin counterpart, the orbital Rashba coupling can generate a finite BC^{37}, but only when all the rotational symmetries are broken (Methods and Supplementary Note I). With a reduced \({{{{\mathcal{C}}}}}_{\mathrm{s}}\) symmetry, lowlying t_{2g} orbitals are split into three nondegenerate levels. The corresponding orbital bands then realize a gapped Rashbalike spectrum with protected crossings along the mirrorsymmetric lines of the twodimensional BZ (Fig. 1g). These characteristics result in the appearance of dipolar BC hotspots and singular pinch points (Fig. 1h). Such orbital sources of BC are fully active at the (111) oxide interfaces owing to the reduced lowtemperature symmetries. The cubictotetragonal structural phase transition^{38,39} occurring at 110 K breaks the threefold rotational symmetry along the [111] direction. In addition, the tetragonal to locally triclinic structural distortions at temperatures below ~70 K together with the ferroelectric instability^{40} below 50 K are expected to strongly enhance the orbital Rashba strength.
The orbitalsourced BC is expected to be very stiff in response to externally applied inplane magnetic fields due to the absence of symmetryprotected orbital degeneracies. In contrast, the spinsourced BC is substantially more susceptible to planar magnetic fields. As shown in Fig. 2a,b, an inplane magnetic field is capable of generating a BC hotspot within the Fermi surface annulus. This BC hotspot corresponds to a fieldinduced avoided level crossing between the two spinsplit bands that occurs whenever the applied magnetic field breaks the residual crystalline mirror symmetry. The momentumintegrated net BC is then nonzero (Supplementary Note II), and yields a transverse Hall conductance satisfying the antisymmetric property σ_{xy}ρ_{yx} = −1, even in the absence of any Lorentz force. This effect, theoretically predicted elsewhere^{7,8} and known as the APHE, is different in nature with respect to the conventional planar Hall effect, which is instead related to the anisotropy in the longitudinal magnetoresistance and thus characterized by a symmetric response, namely, σ_{xy}(B) = σ_{xy}(–B).
Figure 2c shows the transverse (Hall) resistance measured with a current applied along the \([\bar{1}\bar{1}2]\) crystal direction and with collinear current and magnetic field. This ensures a vanishing symmetric planar Hall effect^{7}. At fields well below 4 T, a small signal increasing linearly with the field strength is detected. This feature can be attributed to an outofplane misalignment of the magnetic field smaller than 1.5° (Supplementary Note III). Above a magneticfield threshold instead, a large transverse Hall signal sharply emerges (Extended Data Fig. 3). At even larger fields, this response saturates. Electrostatic gating is found to decrease the magneticfield threshold and promotes a nonmonotonic evolution of the response amplitude (Fig. 2d,e). The experimental features of this Hall response can be captured by considering a single pair of spinsplit bands coupled to the external field by the Zeeman interaction. In this picture, the sudden onset of the transverse response is associated with the appearance of the BC hotspot inside the Fermi surface annulus occurring at a critical magneticfield strength (Supplementary Note II). Magnetoconductance measurements in the weak antilocalization regime (Extended Data Fig. 4) show that the onset of the transverse Hall signal precisely coincides with the appearance of the spinsourced BC hotspot (Extended Data Fig. 5). The nonmonotonic behaviour of the transverse response as a function of electrostatic gating and magneticfield strength can also be ascribed to the BC origin of the Hall response. The angular dependence of the transverse resistance (Fig. 2f) indicates a vanishing transverse linear conductivity when the planar magnetic field is along the \([\bar{1}10]\) direction, due to mirror symmetry \({{{{\mathcal{M}}}}}_{[\bar{1}10]}\). This is independent of whether the driving current is along the \([\bar{1}10]\) or \([\bar{1}\bar{1}2]\) direction. Note that the two angular dependencies are related to each other by a 180° shift, in agreement with the Onsager reciprocity relations^{41}.
The absence of linear conductivity makes this configuration the ideal regime to investigate the presence of nonlinear transverse responses, which are symmetryallowed when the driving current is collinear with the magnetic field (Supplementary Note II). We have, therefore, performed systematic measurements of the secondharmonic (2ω) transverse responses (Fig. 3a,b) by sourcing the a.c. current along the \([\bar{1}10]\) direction. We have subsequently disentangled the fieldantisymmetric \({R}_{{{{{y}}}},{{{\rm{as}}}}}^{2\omega }=\left[{R}_{{{{{y}}}}}^{2\omega }(B)\,\,{R}_{{{{{y}}}}}^{2\omega }(B)\right]/2\) and fieldsymmetric \({R}_{{{{{y,\mathrm{sym}}}}}}^{2\omega }=\left[{R}_{{{{{y}}}}}^{2\omega }(B)\,+\,{R}_{{{{{y}}}}}^{2\omega }(B)\right]/2\) contributions, since they originate from distinct physical effects. In particular, the antisymmetric part contains a semiclassical contribution that only depends on the conventional group velocity of the carriers at the Fermi level (Supplementary Note II). Conversely, the symmetric part originates from the anomalous velocity term of the carriers. It is a purely quantum contribution and can be expressed in terms of a BCD. We observe the following features in Fig. 3a,b. The semiclassical antisymmetric contribution has a sudden onset above a characteristic magnetic field (Fig. 3a) that is sensitive to gating (Fig. 3c). The gate dependence displays a monotonic growth consistent with its physical origin. On the contrary, the symmetric contribution displays the typical nonmonotonous gate and fieldamplitude dependence (Fig. 3b,d) of BCmediated effects. The gate dependence of the nonlinear symmetric contribution obtained by sourcing the current along the \([\bar{1}\bar{1}2]\) direction is instead strongly suppressed and featureless (Fig. 3e). This is consistent with a \([\bar{1}10]\)oriented BCD, which gives a vanishing response in this configuration. We note that the symmetric nonlinear transverse resistance has a characteristic quadratic current–voltage (I^{ω}−V^{2ω}), which—combined with the response at double the driving frequency—establishes its secondorder nature (Fig. 3f).
The fact that only the symmetric contribution persists even in the zerofield limit (Fig. 3a,b) indicates the presence of a finite BCD in the absence of externally applied magnetic fields and thus of a nonlinear Hall effect in timereversal symmetric conditions. To support the existence of a finite BCD with timereversal symmetry, we have individually evaluated the dipole originating from the spinsourced BC and the dipole related to the orbitalsourced BC (Methods). Figure 4a shows that in the entire parameter space of our lowenergy theory model, the spinsourced BCD is two orders of magnitude smaller than the orbitalsourced BCD. The latter exceeds the inverse characteristic Fermi wavenumber \(k_\mathrm{F}^{1}\) ≈ 0.5 nm. Besides the intrinsic contribution to the BCD, the nonlinear Hall response with timereversal symmetry also contains disorderinduced contributions^{10,42} due to nonlinear skew and sidejump scattering. We experimentally access such contributions by measuring the longitudinal signal \({V}_{{{{{yyy}}}}}^{2\omega }\) that is symmetryallowed but does not possess any intrinsic BCD contribution. As displayed in Fig. 4b, the strong difference in amplitude between the longitudinal signal and transverse \({V}_{{{{{yxx}}}}}^{2\omega }\) signal over a large drivingcurrent range proves the absence of threefold rotation symmetry as well as a nonlinear Hall effect completely dominated by the intrinsic BCD. The anisotropy between longitudinal and transverse nonlinear signals also allows us to exclude a leading role played by thermoelectric effects due to Joule heating (Fig. 4b, inset). We further observe that both longitudinal \({V}_{{{{{xxx}}}}}^{2\omega }\) and transverse \({V}_{{{{{xyy}}}}}^{2\omega }\) responses have an amplitude comparable with the longitudinal signal \({V}_{{{{{yyy}}}}}^{2\omega }\), thus suggesting their disorderinduced nature. We point out that the finite amplitudes of \({V}_{{{{{xxx}}}}}^{2\omega }\) and \({V}_{{{{{xyy}}}}}^{2\omega }\) imply \({{{{\mathcal{M}}}}}_{\bar{1}10}\) symmetry breaking (Supplementary Note II). This can be related to the mirrorbreaking arrangements of the oxygen atoms caused by the antiferrodistortive octahedron rotations.^{43} It might also be due to the presence of structural domain patterns appearing at the cubictotetragonal structural transition.
We have systematically verified the occurrence of a sizable nonlinear transverse response over the full range of sheet conductances and concomitantly observed a large difference between the two nonlinear transverseconductivity tensor component χ_{yxx} and χ_{xyy} (Fig. 4c). This further proves the main intrinsic BCD contribution to the nonlinear Hall response. By further evaluating the momentum relaxation time τ (Supplementary Note II), we can estimate the size of the BCD (Methods):
The resulting BCD (Fig. 4d) is two orders of magnitude larger than the dipole observed in systems with massive Dirac fermions, such as bilayer WTe_{2} (refs. ^{11,12}) and—over a finite density range—a factor of two larger than the dipole observed in corrugated bilayer graphene^{13}. We attribute the large magnitude of this effect to the fact that the orbitalsourced BC is naturally equipped with a large dipolar density due to the presence of singular pinch points and hotspots with dipolar arrangements. We also monitored the temperature dependence of transverseconductivity tensor components χ_{yxx} and χ_{xyy} (Fig. 4e) and the corresponding behaviour of BCD D_{x} (Fig. 4f). All these quantities rapidly drop approaching 30 K, that is, the temperature above which the strong polar quantum fluctuations of SrTiO_{3} vanish. This further establishes the orbital Rashba coupling as the physical mechanism behind the orbitalsourced BC.
The pure orbitalbased mechanism of BCD featured here paves the way to the atomicscale design of quantum sources of nonlinear electrodynamics persisting up to room temperature. Oxidebased 2DES could be, for instance, combined with a roomtemperature polar ferroelectric layer, triggering symmetry lowering and thus inducing orbital Rashba coupling by interfacial design. This and other alternative platforms combining a lowsymmetry crystal with orbital degrees of freedom and polar modes, including roomtemperature polar metals^{44} and conducting ferroelectric domain walls, are candidate oxide architectures to perform operations such as rectification^{45} and frequency mixing. Moreover, multiple sources of BC can be implemented for combined optoelectronic and spintronic functionalities in a singlematerial system: photogalvanic currents due to the orbitalsourced BC can be employed to create spin Hall voltages exploiting the spinsourced BC. Our study also establishes a general approach to generate topological charge distributions in strongly correlated materials, opening a vast space for exploration at the intersection between topology and correlations.
Methods
Sample growth
The nineunitcellthick LaAlO_{3} crystalline layer is grown on the TiOrich surface of a (111)oriented SrTiO_{3} substrate, from the ablation of a highpurity (>99.9%) LaAlO_{3} sintered target by pulsed laser deposition using a KrF excimer laser (wavelength, 248 nm). We perform the realtime monitoring of growth by following intensity oscillations, in a layerbylayer growth mode, of the first diffraction spot using reflection highenergy electron diffraction (Extended Data Fig. 7a). This allows us to stop the growth at precisely the critical thickness of nine unit cells of LaAlO_{3} (ref. ^{46}) necessary for the (111)oriented LaAlO_{3}/SrTiO_{3} 2DES to form. The SrTiO_{3}(111) substrate was first heated to 700 °C in an oxygen partial pressure of 6 × 10^{−5} mbar. The LaAlO_{3} layer was grown in those conditions at a laser fluence of 1.2 J cm^{−2} and laser repetition rate of 1 Hz. Following the growth of the LaAlO_{3} layer, the temperature is ramped down to 500 °C before performing onehourlong in situ annealing in a static background pressure of 300 mbar of pure oxygen, to recover the oxygen stoichiometry of the reduced heterostructure. Finally, the sample is cooled down at –20 °C min^{−1}, and kept in the same oxygen environment at zero heating power for at least 45 min.
Device fabrication
The (111)oriented LaAlO_{3}/SrTiO_{3} blanket films were lithographically patterned into two Hall bars (width W = 40 μm; length L = 180 μm), oriented along the two orthogonal crystalaxis directions of \([\bar{1}10]\) and \([\bar{1}\bar{1}2]\). The Hall bars are defined by electronbeam lithography into a poly(methyl methacrylate) resist, which is used as a hard mask for argonion milling (Extended Data Fig. 7c). The dryetching duration is calibrated and timed to be precisely stopped when the LaAlO_{3} layer is fully removed to avoid the creation of an oxygendeficient conducting SrTiO_{3−δ} surface. This leaves an insulating SrTiO_{3} matrix surrounding the protected LaAlO_{3}/SrTiO_{3} areas, which host a geometrically confined 2DES.
Electrical transport measurements
The Hall bars are connected to a chip carrier by an ultrasonic wedgebonding technique in which the aluminium wires form ohmic contacts with the 2DES through the LaAlO_{3} overlayer. The sample is anchored to the chip carrier by homogeneously coating the backside of the SrTiO_{3} substrate with silver paint. A d.c. voltage V_{g} is sourced between the silver backelectrode and the desired Hallbar device to enable electrostaticfieldeffect gating of the 2DES, leveraging the large dielectric permittivity of strontium titanate at low T (~2 × 10^{4} below 10 K)^{47,48}. Nonhysteretic dependence of σ_{xx} (σ_{yy}) on V_{g} is achieved following an initial gateforming procedure^{49}.
Standard fourterminal electrical (magneto)transport measurements were performed at 1.5 K in a liquid helium4 flow cryostat, equipped with a superconducting magnet (maximum magnetic field, B = ±12 T). An a.c. excitation current I^{ω} ∝ I^{ω}sin(ωt), of frequency ω/(2π) = 17.77 Hz, is sourced along the desired crystallographic direction. The sheet resistance, \({R}_{{{{\rm{s}}}}}={\sigma }_{{{{{xx}}}}}^{1}\), of a Hallbar device is related to the firstharmonic longitudinal voltage drop V_{xx} according to R_{s} = (V_{xx}/I_{x})(W/L). When the a.c. current is sourced along \(\hat{{{{\bf{x}}}}}\parallel [\bar{1}10]\) (\(\hat{{{{\bf{y}}}}}\parallel [\bar{1}\bar{1}2]\)), we make use of a standard lockin detection technique to concomitantly measure the firstharmonic longitudinal response V_{xx} (V_{yy}), and either the inphase firstharmonic \({V}_{{{{{xy}}}}}^{\omega }\) (\({V}_{{{{{yx}}}}}^{\omega }\)) or outofphase secondharmonic \({V}_{{{{{yxx}}}}}^{2\omega }\) (\({V}_{{{{{xyy}}}}}^{2\omega }\)) transverse voltages (Fig. 1a). We define the first and secondharmonic transverse resistances as \({R}_{{{{{xy}}}}}^{\omega }={V}_{{{{{xy}}}}}^{\omega }/ {I}_{{{{{x}}}}}^{\omega }\) and \({R}_{{{{{y}}}}}^{2\omega }={V}_{{{{{yxx}}}}}^{2\omega }/ {I}_{{{{{x}}}}}^{\omega }{ }^{2}\), respectively. First and secondharmonic measurements are performed at 10 and 50 μA, respectively.
We systematically decompose both first and secondharmonic magnetoresponses into their fieldsymmetric \({R}_{{{{\rm{sym}}}}}^{(2)\omega }\) and fieldantisymmetric \({R}_{{{{\rm{as}}}}}^{(2)\omega }\) contributions according to
In particular, the firstharmonic transverse resistance is purely field antisymmetric, and hence, we chose the simplified notation of \({R}_{{{{{xy}}}}}\equiv {R}_{{{{{xy}}}},{{{\rm{as}}}}}^{\omega }\).
Estimation of the Rashba spin–orbit energy from magnetoconductance measurements in the weak antilocalization regime
In a 2DES, in the presence of a spin relaxation mechanism induced by an additional spin–orbit interaction, the conductance is subject to weak localization corrections at lower temperatures. Extended Data Fig. 4a shows the gatemodulated magnetoconductance curves of the 2DES, which exhibit a characteristic lowfield weak antilocalization behaviour. The magnetoconductance curves, normalized to the quantum of conductance G_{Q} = e^{2}/(πħ), are fitted using a Hikami–Larkin–Nagaoka model that expresses the change in conductivity Δσ(B_{⟂}) = σ(B_{⟂}) – σ(0) of the 2DES under an external outofplane magnetic field B_{⊥}, in the diffusive regime (with negligible Zeeman splitting), as follows^{50,51}:
where Ψ is the digamma function; ħ = h/(2π) is the reduced Planck constant; \({B}_{{{{\rm{i}}}},{{{\rm{so}}}}}=\hslash /\left(4eD{\tau }_{{{{\rm{i}}}},{{{\rm{so}}}}}\right)\) are the effective fields related to the inelastic and spin–orbit relaxation times (τ_{i} and τ_{so}, respectively); and D = πħ^{2}σ(0)/(e^{2}m*) is the diffusion constant. The last term in equation (3), proportional to \({B}_{\perp }^{2}\), contains A_{K}, the socalled Kohler coefficient, which accounts for orbital magnetoconductance.
Hence, from the fit to the weak antilocalization magnetoconductance curves, the effective Rashba spin–orbit coupling α_{R} can be calculated as
based on a D’yakonov–Perel’ spin relaxation mechanism^{51}. A summary of the dependence of the extracted parameters on the 2DES’ sheet conductance is plotted in Extended Data Fig. 5b. The spin–orbit energy Δ_{so} can then be estimated according to
where, in two dimensions, the Fermi wavevector is given by \({k}_{{{{\rm{F}}}}}=\sqrt{2\uppi {n}_{2{{{\rm{D}}}}}}\), assuming a circular Fermi surface. The sheet carrier density n_{2D} is experimentally obtained for each doping value from the (ordinary) Hall effect (Supplementary Note III), measured concomitantly with the magnetoconductance traces.
Spinsourced and orbitalsourced BCD calculations
We first estimate the BCD due to spin sources in timereversal symmetry condition as a function of carrier density considering the lowenergy Hamiltonian for a single Kramers’related pair of bands (Supplementary Note I):
where the momentumdependent mass can be negative close to the Γ point (Supplementary Note I). Although this model Hamiltonian is equipped with a finite BC, its dipole is forced to vanish by the threefold rotation symmetry (Supplementary Note I). We capture the rotation symmetry breaking of the lowtemperature structure at the leading order by assuming inequivalent coefficients for the spin–orbit coupling terms linear in momentum. In other words, we make the substitution α_{R}(σ_{x}k_{y} – σ_{y}k_{x})→v_{y}k_{y}σ_{x} – v_{x}k_{x}σ_{y}. Since the dipole is a pseudovector, the residual mirror symmetry \({{{{\mathcal{M}}}}}_{x}\) forces it to be directed along the \(\hat{{{{\bf{x}}}}}\) direction. In the relaxationtime approximation, it is given by
where Ω_{z} is the BC of our twoband model that we write in a dimensionless form by measuring energies in units of \({k}_{{{{\rm{F}}}}}^{2}/2m({k}_{{{{\rm{F}}}}})\), lengths in units of 1/k_{F} and densities in units of \({n}_{0}={k}_{{{{\rm{F}}}}}^{2}/2\uppi\). Here k_{F} is a reference Fermi wavevector. For simplicity, we have considered a positive momentumindependent effective mass. For the BCD shown in Fig. 4a, the remaining parameters have been chosen as v_{x} = 0.4, v_{y} = (1.2, 1.4, 1.6) × v_{x} and λ = 0.1. Moreover, we account for orbital degeneracy by tripling the dipole of a single Kramers’ pair. This gives an upper bound for the spinsourced BCD.
We have also evaluated the BCD due to orbital sources considering the lowenergy Hamiltonian for spin–orbitfree t_{2g} electrons derived from symmetry principles (Supplementary Note I) and reading
where we introduced the Gell–Mann matrices as
and Λ_{0} is the identity matrix. In the Hamiltonian above, Δ is the splitting between the a_{1g} singlet and \({e}_{\mathrm{g}}^{{\prime} }\) doublet resulting from the t_{2g} orbitals in a trigonal crystal field. Here Δ_{m} is the additional splitting between the doublet caused by rotational symmetry breaking. Finally, α_{OR} and α_{m} are the strengths of the orbital Rashba coupling. Note that in the presence of threefold rotation symmetry, α_{m} ≡ 0, in which case the BC is forced to vanish. For simplicity, we have evaluated the BC for the \({{{{\mathcal{C}}}}}_{\mathrm{s}}\) point groupsymmetric case assuming α_{m} ≡ α_{OR}. In our continuum SU(3) model, the BC can be computed using the method outlined elsewhere^{52}. We have subsequently computed the corresponding dipole measuring, as before, energies in units of \({k}_{{{{\rm{F}}}}}^{2}/2m\), lengths in units of 1/k_{F} and densities in units of \({n}_{0}={k}_{{{{\rm{F}}}}}^{2}/2\uppi\). The dimensionless orbital Rashba coupling has been varied between α_{OR} = 1 and α_{OR} = 2, whereas we have fixed Δ = –0.1 and Δ_{m} = 0.005. The value of the crystal field splitting Δ is consistent with the amplitude determined by Xray absorption spectroscopy^{53} of the order 8 meV, and therefore, it is almost one order of magnitude smaller than our energy unit of ~40 meV for a reference \({k}_{\mathrm{F}}^{1}\) ≃ 0.5 nm and effective mass m ≃ 3m_{e} (Supplementary Note III). The calculated dipole (Fig. 4a) has been finally multiplied by two to account for spin degeneracy. As shown in Supplementary Note I, we remark that the model Hamiltonian for the spin sources of BC (equation (6)) and the model Hamiltonian for the orbital sources (equation (8)) derive from a single sixband model where orbital and spin degrees of freedom are treated on an equal footing.
Estimation of BCD magnitude from nonlinear Hall measurements
The nonlinear current density is mathematically given by \({j}_{\alpha }^{2\omega }={\chi }_{\alpha \beta \gamma }\,{E}_{\beta }\,{E}_{\gamma }\), where χ_{αβγ} is the nonlinear transverseconductivity tensor. When an a.c. current density \({I}_{{{{{x}}}}}^{\omega }/W={\sigma }_{{{{{xx}}}}}{E}_{{{{{x}}}}}^{\omega }\) is sourced along \(\hat{{{{\bf{x}}}}}\), the secondharmonic transverse current density developing along \(\hat{{{{\bf{y}}}}}\) is related to the BCD D according to^{9}
where τ is the momentum relaxation time and e is the elementary charge. Due to the mirror symmetry \({{{{\mathcal{M}}}}}_{{{{{x}}}}}\equiv {{{{\mathcal{M}}}}}_{[\bar{1}10]}\), the dipole is found to point along \(\hat{{{{\bf{x}}}}}\); in the quasid.c. limit, that is, (ωτ) ≪ 1, the BCD expression reduces to
which is the explicit expression for equation (1), in terms of experimentally measurable quantities only, and where
are the measured nonlinear transverseconductivity tensor elements shown in Fig. 4c,e.
Data availability
The data that support the findings of this study are available via Zenodo at https://doi.org/10.5281/zenodo.7575479.
Code availability
The code that support the findings of the study is available from the corresponding authors on reasonable request.
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Acknowledgements
This work was supported by the Swiss State Secretariat for Education, Research and Innovation (SERI) under contract no. MB22.00071, the Gordon and Betty Moore Foundation (grant no. 332 GBMF10451 to A.D.C.), the European Research Council (ERC) (grant no. 677458), by the project Quantox of QuantERA ERANET Cofund in Quantum Technologies, and by the Netherlands Organisation for Scientific Research (NWO/OCW) as part of the VIDI (project 68047543 to C.O. and project 016.Vidi.189.061 to A.D.C.), the ENWGROOT (project TOPCORE) program (to A.D.C. and U.F.) and Frontiers of Nanoscience program (NanoFront). E.L. acknowledges funding from the EU Horizon 2020 research and innovation programme under the Marie SkłodowskaCurie grant agreement no. 707404. We acknowledge A. M. Monteiro, L. Hendl, J. R. Hortensius, P. Bruneel and M. Gabay for valuable discussions.
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A.D.C. proposed and supervised the experiments. C.O. proposed the theory models and supervised their analysis. E.L. grew the crystalline LaAlO_{3} thin films by pulsed laser deposition and performed the structural characterizations. E.L. and Y.G.S. lithographically patterned the samples, performed the magnetotransport experiments and analysed the experimental data with help from T.C.v.T. and U.F. R.B. and M.T.M. performed the BC and semiclassical transport calculations with help from M.C., C.N. and C.O. C.O., E.L., R.B. and A.D.C. wrote the manuscript, with input from all the authors.
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Extended data
Extended Data Fig. 1 Hall effect in a planar magnetic field.
Experimentally measured fieldantisymmetric transverse magnetoresistance \({R}_{{{{\rm{yx}}}}}={V}_{{{{\rm{yx}}}}}^{\omega }/{I}_{{{{\rm{y}}}}}^{\omega }\) at T = 1.5 K, with \({I}_{{{{\rm{x}}}}}^{\omega }\) along \([\bar{1}10]\perp B\) (see inset schematic), for varying sheet conductance values σ_{xx} (indicated by the inset colored scale bar), and tuned via electrostatic field effect in a backgate geometry.
Extended Data Fig. 2 Nonlinear Hall response in a planar magnetic field.
Experimentally measured fieldsymmetric nonlinear transverse magnetoresponse \({R}_{{{{\rm{x}}}}}^{2\omega }\) at T = 1.5 K with \({I}_{{{{\rm{y}}}}}^{\omega }\) along \([\bar{1}\bar{1}2]\perp B\) (see inset schematic), for varying sheet conductance values σ_{yy} (indicated by the inset colored scale bar). The full scale ordinate axis is chosen to be the same as for panel b of Fig. 3, for better comparison, and highlights the comparatively small nonlinear Hall response when the current is sourced along the \({{{{\mathcal{M}}}}}_{\bar{1}10}\) mirror line corresponding to a symmetry demanded zero BCD.
Extended Data Fig. 3 Temperature dependence of the Hall and longitudinal magnetoresistances in a planar magnetic field.
Temperature dependent planar Hall resistance (a,c) and longitudinal MR (b,d) in the linear response regime for various strength of the inplane magnetic field (and at fixed gatevoltage), with \({I}_{{{{\rm{y}}}}}^{\omega }\) along \([\bar{1}\bar{1}2]\parallel B\) (ab) and \({I}_{{{{\rm{x}}}}}^{\omega }\) along \([\bar{1}10]\perp B\) (cd). Both R_{yx} and R_{xy}, as well as the corresponding longitudinal MR, asymptotically go to zero as the temperature increases toward strontium titanate’s nonpolar tetragonal phase (above ≈ 30 K).
Extended Data Fig. 4 Weak antilocalization regime and Rashba spinorbit coupling.
a, Gatemodulated magnetoconductance curves (normalized to the quantum of conductance G_{Q}) for \({I}_{{{{\rm{x}}}}}^{\omega }\) along \([\bar{1}10]\) (see also Supplementary Note III for WAL measurements along \([\bar{1}\bar{1}2]\)). HikamiLarkinNagaoka fits (solid red lines) are performed following Eq. (3). B_{⊥}, the outofplane magnetic field. (c) Left axis: Experimentally estimated momentum, inelastic and spinorbit relaxation times, τ_{el} (named τ throughout the manuscript), τ_{i} and τ_{so}, respectively. Right axis: Strength of the Rashba spinorbit coupling α_{R} versus sheet conductance σ_{xx}.
Extended Data Fig. 5 Effective Zeeman energy and Rashba spinorbit energy of the 2DES.
ab, Right axis: Sheet conductance dependence of the Rashba spinorbit energy Δ_{so}/2 = α_{R}k_{F} and effective Zeeman energy \({\Delta }_{{{{\rm{Z}}}}}^{{{{\rm{c}}}}}=g{\mu }_{{{{\rm{B}}}}}{B}^{{{{\rm{c}}}}}/2\) at the critical inplane field B^{c} (left axis), for \({I}_{{{{\rm{x}}}}}^{\omega }\) along \([\bar{1}10]\) (panel a) and for \({I}_{{{{\rm{y}}}}}^{\omega }\) along \([\bar{1}\bar{1}2]\) (panel b). See Supplementary Note III for details regarding the determination of B^{c}.
Extended Data Fig. 6 Current bias dependent nonlinear transverse signal in a planar magnetic field.
a,b, Fieldantisymmetric, \({R}_{{{{\rm{y}}}},{{{\rm{as}}}}}^{2\omega }\), and fieldsymmetric \({R}_{{{{\rm{y,sym}}}}}^{2\omega }\) second harmonic transverse resistance responses, for various magnitudes of the excitation a.c. current along \([\bar{1}10]\) (∥B, see inset schematic).
Extended Data Fig. 7 Growth, structural characterization and device fabrication of (111)oriented LaAlO_{3}/SrTiO_{3} 2DES.
a, In situ realtime RHEED monitoring of the layerbylayer PLD growth of a 9 u.c. LaAlO_{3} film on a (111)oriented SrTiO_{3} substrate. Insets: RHEED patterns acquired before (left) and after the film growth (right), highlighting the high crystalline quality of the epitaxial LaAlO_{3} film. The vertical arrow marks the end of the growth (at 245 s). b, Atomic force microscopy image of a 9 u.c. thick LaAlO_{3} film on SrTiO_{3}(111). The film’s topography reproduces the characteristic atomically sharp stepsandterraces structure of the substrate’s vicinal surface. Horizontal scale bar: 1 μm. c, Optical micrograph of a patterned PPMA resist hard mask on a 9 u.c. blanket film of LaAlO_{3}/SrTiO_{3} (before the Ar ion milling step) which ultimately defines the Hall bar device area where the 2DES subsists. Prepatterned gold markers are used to signify the position of the Hall bars’ contact pads, as the optical contrast between the etched area (bare substrate) and the crystalline LaAlO_{3}/SrTiO_{3} devices is extremely low. Scale bar: 200 μm.
Supplementary information
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Supplementary Notes I–III and Figs. 1–14.
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Lesne, E., Saǧlam, Y.G., Battilomo, R. et al. Designing spin and orbital sources of Berry curvature at oxide interfaces. Nat. Mater. 22, 576–582 (2023). https://doi.org/10.1038/s41563023014980
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DOI: https://doi.org/10.1038/s41563023014980
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