Main

Recent experiments on the LAO/STO interface have seen tantalizing magnetic signals4,5,6,7,8, often persisting up to high temperatures ∼ 100 K. A large magnetization of 0.3– 0.4μB per interface Ti was observed by torque measurements5 in an external field. In contrast, scanning superconducting quantum interference device (SQUID) experiments6 found an inhomogeneous state with a dense set of local moments with no net magnetization, only isolated micron-scale ferromagnetic patches. Our goal is to reconcile these seemingly contradictory observations and to gain insight into the itinerant versus local moment nature of the magnetism, the exchange mechanism and the ordered state.

LAO and STO are both band insulators, but the TiO2 layers at the interface are n-doped when LAO is terminated by a LaO+ layer. The polar catastrophe1 arising from a stack of charged LaO+ and AlO2− layers grown on STO is averted by the transfer of 0.5 electrons per interface Ti. Oxygen vacancies are also known to provide additional electrons at the interface1.

What is the fate of these electrons at the interface? Transport data suggest that only a small fraction of the electrons (5–10% of the 0.5 electrons per Ti) are mobile2,9,10,11. Interestingly, magneto-transport studies show a large, gate-tunable Rashba spin–orbit coupling (SOC) for these conduction electrons, arising from broken inversion at the interface12. Most of the electrons (comparable to 0.5 electrons per Ti) seem to behave like local moments in the magnetic measurements5,6 discussed above.

We propose a microscopic model of electrons in Ti t2g states at the LAO/STO interface that leads to the following results: local moments form in the top TiO2 layer owing to correlations, with interfacial splitting of t2g degeneracy playing a critical role; conduction electrons mediate ferromagnetic double-exchange interactions between the moments; Rashba SOC for the conduction electrons leads to a Dzyaloshinskii–Moriya interaction and a compass anisotropy term with a definite ratio of their strengths; the zero-field ground state is a long-wavelength spiral with a SOC-dependent pitch; the spiral transforms into a ferromagnetic state in an external field H.

We provide a semi-quantitative understanding of the torque results for the magnetization M(H), and reconcile some of the key differences between the torque and SQUID measurements. We also point to a new magnetoelastic coupling effect that can be important for polar interfaces. Our model naturally explains the coexistence of magnetism and superconductivity at very low temperatures5,6,8. Finally, we make a number of specific predictions that can be tested experimentally.

Symmetry-based considerations

Many of our conclusions regarding magnetism can be understood qualitatively on the basis of symmetry alone. Although symmetry dictates the form of the interactions, the microscopic analysis shows how local moments Sr arise on a square lattice at the interface, and provides quantitative insights into the parameter dependence of exchange couplings. The isotropic Heisenberg exchange is , with and lattice spacing b. Its sign is not determined by symmetry, but our microscopic analysis leads to a ferromagnetic J>0. Inversion symmetry breaking at the interface implies two SOC terms: the Dzyaloshinskii–Moriya interaction with , and the compass anisotropy .

One can see, quite generally, that the ground state of such a model can be a long-wavelength spiral that looks locally ferromagnetic, thus minimizing the J term, with pitch determined by the Dzyaloshinskii–Moriya term. Our microscopic results for D,A,A′ unequivocally predict such a spiral state for H = 0, which turns into an in-plane ferromagnet for H≳1 T. These observations permit us to understand the torque data in a field5 semi-quantitatively, and to see why over most of the sample the H = 0 scanning SQUID measurements find no net moment6.

Electronic structure

Both density functional theory13,14 (DFT) and spectroscopy15,16,17, show that, in the top TiO2 layer dx y states have lower energy than dx z,dy z (Fig. 1a). In addition to z confinement raising the x z,y z levels, the mismatch of in-plane LAO and STO lattice parameters leads to an out-of-plane distortion that lowers dx y. The dx y orbitals delocalize primarily in the x–y plane, and similarly for dx z and dy z. DFT (refs 18, 19) and photoemission20 find the in-plane hopping between dx y orbitals t≃0.3 eV, whereas the out-of-plane t′≃t/30. The resulting x z (y z) bands are quasi-one-dimensional (1D), dispersing primarily along x (y), and confined along z.

Figure 1: Electronic orbitals and charge localization.
figure 1

a, Schematic of Ti t2g energy levels. At the interface dx y is lower than the dy z and dx z orbitals. V conf(z) is the confining potential. b, Phase diagram of the single-band extended Hubbard model at quarter-filling. A COI is obtained for moderate values of on-site Hubbard U and next-neighbour Coulomb V, above the (solid red) phase boundary. Coupling to the breathing-mode phonon further stabilizes the COI. The dashed black line corresponds to an energy gain Eph = 0.25t due to lattice distortion (see Methods).

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Local moments

Coulomb interactions, disorder and coupling to phonons can all play a role in localizing charge. Nevertheless, interactions are essential for forming local moments. Experiments11,21 show that the quasi-1D x z,y z bands, which should be the most sensitive to disorder, contribute to transport, ruling out Anderson localization. We thus make a simple model where we treat correlations first and can then add the effects of phonons and disorder.

The key to understanding correlation effects is the splitting of t2g degeneracy at the interface that leads to a quarter-filled x y band (0.5 electrons per Ti) in the top TiO2 layer. A modest on-site Hubbard U and next-neighbour Coulomb V then lead to a charge-ordered insulator (COI; Fig. 1b). We obtain a simple analytical result for the metal–insulator transition using a slave-rotor approach22. Coupling to the breathing-mode phonon further stabilizes the COI; (see Methods). DFT calculations need a rather large U≃8 eV to stabilize a COI (ref. 23). We see from Fig. 1 that, for realistic values24 of U = 4 eV (U/t≃13) and V ≃0.5–1 eV (V/t≃1.5–3), we are deep in the chequerboard COI state.

Exchange

Consider then the problem of local moments in dx y orbitals on a chequerboard lattice (Fig. 2a) interacting through Hund’s coupling JH with a small density nc of conduction electrons in x z, y z bands. Given JH≃1 eV, it is much more reasonable to work in the non-perturbative double-exchange limit JH≫t than in the perturbative RKKY limit JH≪t. We treat the S = 1/2 moments classically. This is justified a posteriori because of the small quantum fluctuations in the ferromagnetic and long-wavelength spiral states.

Figure 2: Spiral ground state and its evolution with field.
figure 2

a, Local moments in the COI with ferromagnetic exchange J and J′. Anisotropy and Dzyaloshinskii–Moriya terms couple the same spins as J. b, The zero-field spiral ground state, whose evolution with with θH = 15° is shown next. c, The H dependence of the spiral Q normalized by its H = 0 value Q0 for SOC λ/t = π/25. d,e In-plane (Mx; d) and out-of-plane (Mz; e) components of the magnetization obtained from T = 0 variational calculations and T = 0.25J Monte Carlo (MC) simulations. The H axis is scaled by Hpeak at which Mx has a maximum. Panels f–h show the Monte Carlo spin structure factor I(q). f, At H = 0 the peaks of I(q) lie on a white circle, the locus of Q0 corresponding to degenerate variational ground states. g, For 0<H<Hpeak, the I(q) shows a net ferromagnetic moment in addition to spiral peaks. h, A ferromagnetic state is obtained for H>Hpeak.

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We find with

where r denotes positions of occupied sites in the COI (see Methods). is obtained by interchanging and replacing D→−D in . The term couples nearest neighbours along the diagonal (Fig. 2a). The spins have normalization Sr2 = 1. The form of and , as well as J≫J′, arises from the two-sublattice structure of the COI and the quasi-1D nature of x z, y z bands. We find J≃nct/4 and J′≃nct′/4 for low carrier density nc (the number of electrons per Ti in each band).

The new aspect here is the Rashba SOC for the conduction electrons, where k is their momentum, are Pauli matrices and a is the Ti–Ti distance. The SOC strength λ is determined by the electric field at the interface. Experiments12 find a gate-voltage-tunable λ≃1.3–6.3 meV (λ/t≃0.004–0.022). SOC then gives rise to the Dzyaloshinskii–Moriya and compass terms in through the double-exchange mechanism. For λ/t≪1, we find D≃ncλ, A≃2ncλ2/t and A′ = 0. Thus, D/J∼λ/t and A/J∼(λ/t)2 with A J/D2 = 1/2 independent of parameters.

Spiral ground state

We have examined the ground-state properties of in an external field H using a variety of techniques. Using a variational approach we find that, for A J/D2<1, the H = 0 ground state is

a coplanar spiral with spins lying in a plane perpendicular to the interface. Here ; see Methods. For A J/D2>1, the ground state is an in-plane ferromagnet with . For all A J/D2, the ground state is infinitely degenerate (arbitrary 0≤φ<2π), even though has symmetry. This peculiar degeneracy of compass anisotropy25 is expected to be broken by order-by-disorder.

The microscopic value A J/D2 = 1/2 implies a zero-field spiral variational ground state. We have verified the stability of the spiral at H = 0 using two independent, unbiased calculations (see Supplementary Information): finite-temperature Monte Carlo and T = 0 conjugate gradient energy minimization. In Fig. 2b we show an example of spiral ordering in real space with a wavelength of a few lattice spacings. For realistic values of λ/t∼0.02, our variational results give a pitch of (2π/Q0)∼600 Å.

In Fig. 2c–h, we show the evolution of the spiral state as a function of H applied in the x–z plane at an angle θH = 15° with the z axis, (the geometry used in the torque experiments5). In the variational calculation (Fig. 2c) the spiral state () develops into a fully magnetized state above a field Hpeak, which corresponds to a maximum in the in-plane component Mx of M = 〈Sr〉. We plot in Fig. 2d,eMx and Mz as functions of H, where we also see excellent agreement between variational and Monte Carlo results. For the Monte Carlo simulations we used larger SOC λ/t = π/25 that led to smaller pitch spirals. Below we use realistic SOC λ values to study long-wavelength spirals within a variational approach.

The evolution from spiral to ferromagnet as a function of field is best seen in the spin structure factor Iq;|〈Sq〉|2 shown in Fig. 2f–h, where Sq is the Fourier transform of Sr. The spiral at H = 0 shows two peaks in Iq at (Fig. 2f), which lie on the white circle, the locus of Q0 corresponding to the degenerate states in equation (2). At intermediate fields, a uniform component develops along with peaks (Fig. 2g), whereas for H>Hpeak one obtains a ferromagnet (Fig. 2h).

Torque and magnetization

The torque is non-zero only in the presence of spin-space anisotropy, otherwise M would simply align with H. In Fig. 3a, we compare our variational results with the magnetization Mτ≡τ/H versus H derived from torque magnetometry5, where Mτ is the component of M perpendicular to H. (The experimental data have been shown after background subtraction; see Supplementary Information.)

Figure 3: Torque magnetometry.
figure 3

a, Magetization Mτ = τ/H (black symbols) from torque experiments5 at T = 300 mK with θH = 15°; (see Supplementary Information for background subtraction). Theoretical Mτ(H) (red curve) computed with microscopic value A J/D2 = 1/2 with A≃0.3 T chosen to match Hpeak. Better agreement (blue curve) with experiment is obtained by the phenomenological choice A J/D2 = 0.8 with A≃0.7T. The inset shows a schematic of two regimes in torque: an in-plane net magnetization M for small H evolving into a out-of-plane M for large H. b, Hpeak scales with (λ/t)2 for fixed nc, as predicted by the microscopic model. c, Mτ(H) curves, with different SOC that can be tuned by gating, collapse onto a single curve versus H/Hpeak.

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The M(H) curve shows no hysteresis5, because the spiral state has no spontaneous magnetization at H = 0. The in-plane component of the field () induces a uniform magnetization in the spiral state (Fig. 2). This leads to an increasing Mτ for H<Hpeak in Fig. 3a, where Hpeak, at which Mτ is a maximum, depends on θH. For H>Hpeak, the ground state is a ferromagnet (Fig. 2) with the compass term giving rise to a large easy-plane anisotropy A≃0.15–0.3 T (with λ≃0.016–0.022 and nc = 0.05) that tries to keep M in the plane. For large enough H the field dominates over the anisotropy, pulls M out of the plane and Mτ decreases. These two regimes are shown schematically in Fig. 3a inset.

In Fig. 3a we show two M(H) curves that differ in their choice of A J/D2. The red curve with A J/D2 = 1/2 has too rapid a high-field drop in Mτ(H). (Here A = 2ncλ2/t = 0.3 is tuned to match Hpeak and λ/t = 0.022, nc = 0.05.) The blue curve with A J/D2 = 0.8 (with λ/t = 0.028) leads to better agreement with experiment. Such a phenomenological choice of A J/D2 amounts to changing the anisotropy while keeping Dzyaloshinskii–Moriya fixed. Although A and D were both determined by Rashba SOC in our microscopic theory, dipolar interactions and atomic SOC also contribute to A (see Supplementary Information).

The peak position Hpeak;ncλ2/t in our microscopic model and Mτ(H) for different λ collapse onto a single curve when plotted versus H/Hpeak (see Fig. 3b,c). In practice, other sources of anisotropy might modify this scaling. Nevertheless, both nc and λ are gate tunable and we expect Hpeak to vary substantially with bias.

Scanning SQUID experiments

We now discuss how we can reconcile the torque5 and scanning SQUID (ref. 6) results. At H = 0 most of the sample has spiral order and hence no net magnetization, consistent with ref. 6. A detailed modelling of the observed inhomogeneity is beyond the scope of this paper, nevertheless we can offer a plausible picture based on our theory. The energy gain of spiral over ferromagnet is quite small: Δε≃(D2/J−A)/2 = ncλ2/t≃0.1–0.2 K. Therefore, small terms ignored up to now might well upset the balance in favour of the ferromagnet. We find that magnetoelastic coupling to polar distortions, a new aspect of these interfaces, can reverse the balance between spiral and ferromagnetic states and stabilize ferromagnetic patches (see Methods). These ferromagnetic patches can then point in any in-plane direction, consistent with compass anisotropy.

Discussion

We comment briefly on how our theory, which builds on insights from DFT (refs 13, 18, 19, 23), differs from other approaches. It is hard to obtain the large exchange or net moment seen in LAO/STO from itinerant models26, which may be relevant for GdTiO3/SrTiO3. On symmetry grounds, these itinerant systems will also have a SOC-induced spiral ground state similar to the one discussed here. Our model differs from ref. 27 in several respects. We are in the non-perturbative double-exchange limit and not the RKKY regime. In our model, x z,y z carriers mediate exchange. The x y electrons in lower layers have essentially no interaction with moments in the x y orbitals of the top TiO2 layer. This results from the small overlap t′ between x y orbitals along as well as the level mismatch due to confinement. Conventional superconductivity arises in x y states in lower layers and is decoupled from the local moments.

Conclusions

Our theory has several testable predictions. Local moment formation is associated with (π,π) charge order at the interface, although disorder will probably disrupt the order. The zero-field state is predicted to be a coplanar spiral state with several striking properties, including a wave vector that scales linearly with Rashba SOC and hence is gate tunable. The evolution from a spiral to a ferromagnetic state in an external field has characteristic signatures in the spin structure factor. The easy-plane anisotropy strength is also dominated by SOC and exhibits substantial gate voltage dependence that can be tested in torque magnetometry experiments. The ferromagnetic exchange J = nct/4≃40 K (for nc = 0.05 and t = 0.3 eV) should also be tunable by changing nc, the density of carriers. A theoretical analysis of the finite temperature properties of our model is left for future investigations.

Methods

Charge-ordering Mott transition.

We consider the quarter-filled, extended Hubbard model on a 2D square lattice with onsite U and nearest-neighbour Coulomb V. Using slave-rotor22 mean field theory for U and Hartree–Fock for V, we obtain an analytical result for the transition from metal to charge ordered insulator (COI). The chequerboard COI is stable for U>8〈t〉 when U<4V, and for (U−2V)V/U>〈t〉 when U>4V. Here 〈t〉≃0.66t is the kinetic energy of occupied states. Slave-rotor mean field theory has been found to be in excellent semi-quantitative agreement with dynamical mean field theory for several problems. In fact, our 2D square lattice results are quite similar to dynamical mean field theory of the Wigner–Mott transition on a Bethe lattice28.

The COI is stabilized by coupling to the breathing-mode phonon, which has the same (π,π) wave vector as the charge order (see Supplementary Information). The only change above is that , where Eph is the energy gained though lattice distortion. Typical values of Eph≃0.05–0.10 eV (ref. 29).

Double exchange.

The local moments Sr on the sites of a 2D chequerboard lattice are described by their orientation (θr,φr). In the large JH limit, a conduction electron on the local-moment sublattice (a) has its spin parallel to Sr. Thus, we write the spin-full fermion operators , with orbital index α = (x z),(y z), in terms of spinless fermions ar α, through . Here Fr↑ = cos(θr/2) and F r ↓ =sin( θ r /2) e − i ϕ r .

Both spin projections σ = ↑,↓ are allowed on empty sublattice (b) sites, for which we use a common (global) quantization axis. The kinetic energy is then given by

where and .

The Rashba terms and lead to the SOC Hamiltonian

We can rewrite this in terms of the spinless a fermions as

where and we define λ(x z)↑ = −λ(x z)↓ = λ and λ(y z)↑ = λ(y z)↓ = i λ.

To obtain the parameters J, J′, D and A of of equation (1) starting from the microscopic , we match the energies of several low-lying configurations computed for both and . See Supplementary Information for details.

Variational calculation.

We study the ground-state properties of using a variational ansatz Sr = M+R(Q)cos(Q · r)+I(Q)sin(Q · r) with . The normalization Sr2 = 1 is satisfied by the constraints M · R = M · I = R · I = 0; R2 = I2 and M2+R2 = 1. We numerically minimize the energy per spin ε to obtain optimal values for M, R, I and Q as a function of H, and calculate the torque.

At zero field, we find a spiral ground state of equation (2) with Q = Q0 and M = 0 for A J/D2<1 and an in-plane ferromagnet for A J/D2>1. We can see this analytically using the ansatz Sr = Rcos(Q · r)+Isin(Q · r). We write and with θ∈[0,π] and φ∈[0,2π). We first minimize with respect to Q (for Q a≪1), find the optimal and the energy ε(Q)≃−2(J+J′)−A+(A−D2/J)(cos2θ)/2, which is independent of φ. For A J/D2<1, the optimal choice of cosθ = 1 leads to the spiral of equation (2) with Q = Q0 and an energy gain Δε = (D2/J−A)/2 relative to the ferromagnetic state. For our microscopic model Δε = ncλ2/t.

Polar distortions.

Displacements of oxygen ions that bridge Ti–O–Ti bonds30 at the interface affect the local electric field and hence the Rashba SOC λ(u) = λ−λ1u. Here λ1>0 because the local electric field decreases if negatively charged oxygen ions move up (u>0). The resulting coupling to magnetism can be modelled by A→A(u) = 2ncλ2(u)/t and D→D(u) = ncλ(u) in the energy ε(Q), in addition to the cost of distortion. The A and D terms in gain energy by increasing λ(u) with u<0. However, this costs an electrostatic energy −q E u and an elastic energy K u2/2, where is the electric field acting on oxygen ions at the interface and K is the elastic constant. We minimize the total energy with respect to u to find that energy gain of the spiral is reduced: Δε≃(ncλ2/t)[1−2q E λ1/K λ]. With suitable choice of parameters, one can make Δε<0 and stabilize the ferromagnetic state over the spiral.