Abstract
The electronic properties of the polar interface between insulating oxides is a subject of great interest^{1,2,3}. An exciting development is the observation of robust magnetism^{4,5,6,7,8} at the interface of two nonmagnetic materials, LaAlO_{3} (LAO) and SrTiO_{3} (STO). Here we present a microscopic theory for the formation and interaction of local moments that depends on essential features of the LAO/STO interface. We show that correlationinduced moments arise owing to interfacial splitting of orbital degeneracy. We find that conduction electrons with a gatetunable Rashba spin–orbit coupling mediate ferromagnetic exchange with a twist. We predict that the zerofield ground state is a longwavelength spiral. Its evolution in an external field accounts semiquantitatively for torque magnetometry data^{5} and describes qualitative aspects of the scanning superconducting quantum interference device measurements^{6}. We make several testable predictions for future experiments.
Main
Recent experiments on the LAO/STO interface have seen tantalizing magnetic signals^{4,5,6,7,8}, often persisting up to high temperatures ∼ 100 K. A large magnetization of 0.3– 0.4μ_{B} per interface Ti was observed by torque measurements^{5} in an external field. In contrast, scanning superconducting quantum interference device (SQUID) experiments^{6} found an inhomogeneous state with a dense set of local moments with no net magnetization, only isolated micronscale 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 TiO_{2} layers at the interface are ndoped when LAO is terminated by a LaO^{+} layer. The polar catastrophe^{1} arising from a stack of charged LaO^{+} and AlO_{2}^{−} 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 interface^{1}.
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 mobile^{2,9,10,11}. Interestingly, magnetotransport studies show a large, gatetunable Rashba spin–orbit coupling (SOC) for these conduction electrons, arising from broken inversion at the interface^{12}. Most of the electrons (comparable to 0.5 electrons per Ti) seem to behave like local moments in the magnetic measurements^{5,6} discussed above.
We propose a microscopic model of electrons in Ti t_{2g} states at the LAO/STO interface that leads to the following results: local moments form in the top TiO_{2} layer owing to correlations, with interfacial splitting of t_{2g} degeneracy playing a critical role; conduction electrons mediate ferromagnetic doubleexchange 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 zerofield ground state is a longwavelength spiral with a SOCdependent pitch; the spiral transforms into a ferromagnetic state in an external field H.
We provide a semiquantitative 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 temperatures^{5,6,8}. Finally, we make a number of specific predictions that can be tested experimentally.
Symmetrybased 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 S_{r} 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 longwavelength 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 inplane ferromagnet for H≳1 T. These observations permit us to understand the torque data in a field^{5} semiquantitatively, and to see why over most of the sample the H = 0 scanning SQUID measurements find no net moment^{6}.
Electronic structure
Both density functional theory^{13,14} (DFT) and spectroscopy^{15,16,17}, show that, in the top TiO_{2} layer d_{xy} states have lower energy than d_{xz},d_{yz} (Fig. 1a). In addition to z confinement raising the xz,yz levels, the mismatch of inplane LAO and STO lattice parameters leads to an outofplane distortion that lowers d_{xy}. The d_{xy} orbitals delocalize primarily in the x–y plane, and similarly for d_{xz} and d_{yz}. DFT (refs 18, 19) and photoemission^{20} find the inplane hopping between d_{xy} orbitals t≃0.3 eV, whereas the outofplane t′≃t/30. The resulting xz (yz) bands are quasionedimensional (1D), dispersing primarily along x (y), and confined along z.
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. Experiments^{11,21} show that the quasi1D xz,yz 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 t_{2g} degeneracy at the interface that leads to a quarterfilled xy band (0.5 electrons per Ti) in the top TiO_{2} layer. A modest onsite Hubbard U and nextneighbour Coulomb V then lead to a chargeordered insulator (COI; Fig. 1b). We obtain a simple analytical result for the metal–insulator transition using a slaverotor approach^{22}. Coupling to the breathingmode 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 values^{24} 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 d_{xy} orbitals on a chequerboard lattice (Fig. 2a) interacting through Hund’s coupling J_{H} with a small density n_{c} of conduction electrons in xz, yz bands. Given J_{H}≃1 eV, it is much more reasonable to work in the nonperturbative doubleexchange limit J_{H}≫t than in the perturbative RKKY limit J_{H}≪t. We treat the S = 1/2 moments classically. This is justified a posteriori because of the small quantum fluctuations in the ferromagnetic and longwavelength spiral states.
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 S_{r}^{2} = 1. The form of and , as well as J≫J′, arises from the twosublattice structure of the COI and the quasi1D nature of xz, yz bands. We find J≃n_{c}t/4 and J′≃n_{c}t′/4 for low carrier density n_{c} (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. Experiments^{12} find a gatevoltagetunable 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 doubleexchange mechanism. For λ/t≪1, we find D≃n_{c}λ, A≃2n_{c}λ^{2}/t and A′ = 0. Thus, D/J∼λ/t and A/J∼(λ/t)^{2} with AJ/D^{2} = 1/2 independent of parameters.
Spiral ground state
We have examined the groundstate properties of in an external field H using a variety of techniques. Using a variational approach we find that, for AJ/D^{2}<1, the H = 0 ground state is a coplanar spiral with spins lying in a plane perpendicular to the interface. Here ; see Methods. For AJ/D^{2}>1, the ground state is an inplane ferromagnet with . For all AJ/D^{2}, the ground state is infinitely degenerate (arbitrary 0≤φ<2π), even though has symmetry. This peculiar degeneracy of compass anisotropy^{25} is expected to be broken by orderbydisorder.
The microscopic value AJ/D^{2} = 1/2 implies a zerofield spiral variational ground state. We have verified the stability of the spiral at H = 0 using two independent, unbiased calculations (see Supplementary Information): finitetemperature 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/Q_{0})∼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 θ_{H} = 15° with the z axis, (the geometry used in the torque experiments^{5}). In the variational calculation (Fig. 2c) the spiral state () develops into a fully magnetized state above a field H_{peak}, which corresponds to a maximum in the inplane component M_{x} of M = 〈S_{r}〉. We plot in Fig. 2d,e M_{x} and M_{z} 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 longwavelength spirals within a variational approach.
The evolution from spiral to ferromagnet as a function of field is best seen in the spin structure factor I_{q};〈S_{q}〉^{2} shown in Fig. 2f–h, where S_{q} is the Fourier transform of S_{r}. The spiral at H = 0 shows two peaks in I_{q} at (Fig. 2f), which lie on the white circle, the locus of Q_{0} corresponding to the degenerate states in equation (2). At intermediate fields, a uniform component develops along with peaks (Fig. 2g), whereas for H>H_{peak} one obtains a ferromagnet (Fig. 2h).
Torque and magnetization
The torque is nonzero only in the presence of spinspace anisotropy, otherwise M would simply align with H. In Fig. 3a, we compare our variational results with the magnetization M_{τ}≡τ/H versus H derived from torque magnetometry^{5}, where M_{τ} is the component of M perpendicular to H. (The experimental data have been shown after background subtraction; see Supplementary Information.)
The M(H) curve shows no hysteresis^{5}, because the spiral state has no spontaneous magnetization at H = 0. The inplane component of the field () induces a uniform magnetization in the spiral state (Fig. 2). This leads to an increasing M_{τ} for H<H_{peak} in Fig. 3a, where H_{peak}, at which M_{τ} is a maximum, depends on θ_{H}. For H>H_{peak}, the ground state is a ferromagnet (Fig. 2) with the compass term giving rise to a large easyplane anisotropy A≃0.15–0.3 T (with λ≃0.016–0.022 and n_{c} = 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 M_{τ} 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 AJ/D^{2}. The red curve with AJ/D^{2} = 1/2 has too rapid a highfield drop in M_{τ}(H). (Here A = 2n_{c}λ^{2}/t = 0.3 is tuned to match H_{peak} and λ/t = 0.022, n_{c} = 0.05.) The blue curve with AJ/D^{2} = 0.8 (with λ/t = 0.028) leads to better agreement with experiment. Such a phenomenological choice of AJ/D^{2} 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 H_{peak};n_{c}λ^{2}/t in our microscopic model and M_{τ}(H) for different λ collapse onto a single curve when plotted versus H/H_{peak} (see Fig. 3b,c). In practice, other sources of anisotropy might modify this scaling. Nevertheless, both n_{c} and λ are gate tunable and we expect H_{peak} to vary substantially with bias.
Scanning SQUID experiments
We now discuss how we can reconcile the torque^{5} 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≃(D^{2}/J−A)/2 = n_{c}λ^{2}/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 inplane 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 models^{26}, which may be relevant for GdTiO_{3}/SrTiO_{3}. On symmetry grounds, these itinerant systems will also have a SOCinduced spiral ground state similar to the one discussed here. Our model differs from ref. 27 in several respects. We are in the nonperturbative doubleexchange limit and not the RKKY regime. In our model, xz,yz carriers mediate exchange. The xy electrons in lower layers have essentially no interaction with moments in the xy orbitals of the top TiO_{2} layer. This results from the small overlap t′ between xy orbitals along as well as the level mismatch due to confinement. Conventional superconductivity arises in xy 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 zerofield 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 easyplane 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 = n_{c}t/4≃40 K (for n_{c} = 0.05 and t = 0.3 eV) should also be tunable by changing n_{c}, the density of carriers. A theoretical analysis of the finite temperature properties of our model is left for future investigations.
Methods
Chargeordering Mott transition.
We consider the quarterfilled, extended Hubbard model on a 2D square lattice with onsite U and nearestneighbour Coulomb V. Using slaverotor^{22} 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. Slaverotor mean field theory has been found to be in excellent semiquantitative 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 lattice^{28}.
The COI is stabilized by coupling to the breathingmode phonon, which has the same (π,π) wave vector as the charge order (see Supplementary Information). The only change above is that , where E_{ph} is the energy gained though lattice distortion. Typical values of E_{ph}≃0.05–0.10 eV (ref. 29).
Double exchange.
The local moments S_{r} on the sites of a 2D chequerboard lattice are described by their orientation (θ_{r},φ_{r}). In the large J_{H} limit, a conduction electron on the localmoment sublattice (a) has its spin parallel to S_{r}. Thus, we write the spinfull fermion operators , with orbital index α = (xz),(yz), in terms of spinless fermions a_{rα}, through . Here F_{r↑} = cos(θ_{r}/2) and ${F}_{{r}_{\downarrow}}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}({\theta}_{r}/2){\text{e}}^{i\varphi r}$.
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_{(xz)↑} = −λ_{(xz)↓} = λ and λ_{(yz)↑} = λ_{(yz)↓} = iλ.
To obtain the parameters J, J′, D and A of of equation (1) starting from the microscopic , we match the energies of several lowlying configurations computed for both and . See Supplementary Information for details.
Variational calculation.
We study the groundstate properties of using a variational ansatz S_{r} = M+R(Q)cos(Q·r)+I(Q)sin(Q·r) with . The normalization S_{r}^{2} = 1 is satisfied by the constraints M·R = M·I = R·I = 0; R^{2} = I^{2} and M^{2}+R^{2} = 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 = Q_{0} and M = 0 for AJ/D^{2}<1 and an inplane ferromagnet for AJ/D^{2}>1. We can see this analytically using the ansatz S_{r} = 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 Qa≪1), find the optimal and the energy ε(Q)≃−2(J+J′)−A+(A−D^{2}/J)(cos^{2}θ)/2, which is independent of φ. For AJ/D^{2}<1, the optimal choice of cosθ = 1 leads to the spiral of equation (2) with Q = Q_{0} and an energy gain Δε = (D^{2}/J−A)/2 relative to the ferromagnetic state. For our microscopic model Δε = n_{c}λ^{2}/t.
Polar distortions.
Displacements of oxygen ions that bridge Ti–O–Ti bonds^{30} at the interface affect the local electric field and hence the Rashba SOC λ(u) = λ−λ_{1}u. Here λ_{1}>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) = 2n_{c}λ^{2}(u)/t and D→D(u) = n_{c}λ(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 −qEu and an elastic energy Ku^{2}/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≃(n_{c}λ^{2}/t)[1−2qEλ_{1}/Kλ]. With suitable choice of parameters, one can make Δε<0 and stabilize the ferromagnetic state over the spiral.
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Acknowledgements
We acknowledge stimulating conversations with L. Li, L. Balents, W. Cole, J. Mannhart, K. Moler, W. Pickett, S. Satpathy and N. Trivedi, and the support of DOEBES DESC0005035 (S.B.), NSFDMR1006532 (O.E.) and NSF MRSEC DMR0820414 (M.R.).
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Department of Physics, The Ohio State University, Columbus, Ohio 43210, USA
 Sumilan Banerjee
 , Onur Erten
 & Mohit Randeria
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S.B., O.E. and M.R. contributed to the theoretical research described in the paper and the writing of the manuscript.
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