Emergent nanoscale superparamagnetism at oxide interfaces

Atomically sharp oxide heterostructures exhibit a range of novel physical phenomena that are absent in the parent compounds. A prominent example is the appearance of highly conducting and superconducting states at the interface between LaAlO3 and SrTiO3. Here we report an emergent phenomenon at the LaMnO3/SrTiO3 interface where an antiferromagnetic Mott insulator abruptly transforms into a nanoscale inhomogeneous magnetic state. Upon increasing the thickness of LaMnO3, our scanning nanoSQUID-on-tip microscopy shows spontaneous formation of isolated magnetic nanoislands, which display thermally activated moment reversals in response to an in-plane magnetic field. The observed superparamagnetic state manifests the emergence of thermodynamic electronic phase separation in which metallic ferromagnetic islands nucleate in an insulating antiferromagnetic matrix. We derive a model that captures the sharp onset and the thickness dependence of the magnetization. Our model suggests that a nearby superparamagnetic–ferromagnetic transition can be gate tuned, holding potential for applications in magnetic storage and spintronics.

Supplementary Note 1: Additional B z (x, y) and ∆B z (x, y) images We explored several different regions of the samples with no qualitative differences. Supplementary Figure 2 shows a large area B z (x, y) scan of 10 × 10 µm 2 of the N = 12 u.c. sample after ZFC, demonstrating the relative uniformity of the magnetic features. Supplementary Figure 3 shows examples of the differential ∆B z (x, y) images in various samples. All the samples with N > N c = 5 show clear dipole-like features of SPM reversal events. For the N = 200 u.c. sample, our maximal µ 0 H = 250 mT was insufficient to reach H c in order to study SPM reversals.

Supplementary Note 2: Global magnetization measurements
Global magnetization measurements of the samples were done using a Quantum Design magnetic properties measurement system (MPMS) vibrating sample magnetometer. Supplementary Figures  1a,b show the magnetic hysteresis M (H) loops for LMO samples of different thickness N . The 'STO' curve refers to a bare STO substrate that went through the same process, not including PLD. The finite hysteretic signal of the bare STO may either arise from an artifact such as residual magnetic field of the magnetometer's superconducting magnet [1] or from silver paint contamination of the substrate [2].
The N = 4 and 5 u.c. samples show a very small change in magnetization relative to the bare STO, while a substantial difference is observed upon increasing the thickness by a single u.c. to N = 6, as shown in Supplementary Figure 1c. The saturation magnetic moment M s (as well as the coercive field) increases monotonically with N > N c = 5. Supplementary Figure 1c also presents M s per Mn atom, which shows a sharp jump at N = 6 and a non-monotonic behavior at larger thicknesses. The magnetization per Mn atom is always smaller than the expected 4µ B indicating that only a fraction of the Mn atoms are in the FM state.
The temperature dependence of the in-plane magnetic properties of N = 12 u.c. sample are shown in Supplementary Figure 5, revealing the onset of magnetism below T c =100K. Field cooling (FC) was done using a cooling field µ 0 H = 1 T and a measurement field of µ 0 H = 0.1 T was applied during the warm-up process. Zero field cooling (ZFC) measurements were done during warm-up in the presence of the indicated measurement field values. As shown in Supplementary Figure 5a, ZFC curves display a maximum at T max which decreases with H as summarized in Supplementary Figure  5b. In addition, magnetic hysteresis loops ( Supplementary Figures 5c and 5d) acquired at different temperatures show that the coercive field µ 0 H c decreases with increasing temperature, down to 10 mT at 55 K. The behavior of T max and the hysteresis loops point to a possible existence of a blocking temperature T B 80 K [3][4][5][6][7].
In order to evaluate the existence of AFM ordering, Supplementary Figure 10a shows the highfield magnetic moment M vs. T in N = 12 u.c. sample measured in 1 T field. The critical temperature T c = 82 K is estimated from peak position in the derivative of the magnetization dM/dT as shown in Supplementary Figure 10b. The inverse of the magnetic susceptibility H/M in Supplementary Figure 10c shows a clear Currie-Weiss behavior with extrapolated Θ = 61 K. The fact that Θ < T c is a clear indication of AFM interactions in the sample [8].

Supplementary Note 3: In-plane anisotropy
By applying H at different angles, we find a significant in-plane magnetic anisotropy of the SPM islands. For H oriented close (θ = 7 • ) to the [100] STO direction (x-axis), the angular distribution of the SPM magnetization reversals is peaked at θ = 0, as shown in Supplementary Figure 4a and illustrated by the ∆B z (x, y) image in Supplementary Figure 4d. For H at 52 • , most of the events are still oriented around θ = 0 • (Supplementary Figures 4b,e). However, few events appear at angles close to θ = 90 • . When H is at 97 • (Supplementary Figures 4c,f), the angular distribution shows a broad maximum around the y-axis ([010] STO). The in-plane magnetization thus shows fourfold anisotropy with fourfold easy axes along the LMO crystallographic directions that are locked to the underlying STO crystal structure. The observed differences in the anisotropy barrier for the two orthogonal directions is caused apparently by symmetry breaking at the cubic-to-tetragonal transition of STO at T < 105 K, leading to domain structure [9].

Supplementary Note 4: Theoretical model for magnetism in LMO/STO heterostructure Phase separation in bulk LMO
We first give a simple theoretical description of the phase separation phenomena and formation of ferromagnetic (FM) islands in bulk LMO following Ref. 10. As discussed in the main text, the 'A-type' antiferromagnetic (AFM) state of undoped LMO consists of FM planes that are aligned antiferromagnetically [11][12][13][14]. The AFM state can be described by the Hamiltonian where i is the position of the Mn 3+ ions on a simple cubic lattice with spacing a = 0.39 nm,μ denotes the directions in the FM planes andν the out-of-plane AFM direction. S i and s i are the core spin (S = 3/2) and e g electron spin, respectively, coupled via Hund's coupling J H . We work with J F = J AF = J > 0 and in the limit J H → ∞. The AFM in LMO is slightly canted, leading to a small magnetic moment ∼ 0.2 µ B per u.c. due to Dzyaloshinskii-Moriya exchange [12,15]. We incorporate this by assuming a background magnetic moment of ∼ 0.2 µ B per u.c. while estimating the saturation magnetization of the sample. LMO in the bulk can be doped by injecting excess e g electrons or holes chemically, e.g. by doping with Ce or Sr, respectively. Alternatively oxygen excess, e.g. induced during the growth of LMO thin film, could give rise to similar effect. The kinetic energy of the carriers in doped LMO is described by the 'double exchange' model [14] Here θ i is the polar angle of the core spin and t is the hopping amplitude of the carriers (a i ). The above term prefers the core spins to align ferromagnetically (θ i = θ j ), and thereby tends to induce metallicity. We take t = 0.3 eV and J = 0.1t [14] for our calculations. The competition of FM double exchange with the AFM superexchange is believed to be at the root of the nanoscale phase separation in doped manganites [14,16,17]. In the PS state, the longrange Coulomb interaction between non-uniform excess charge distributions plays a crucial role in determining the typical scale of the phenomenon.
We consider the phase separation in bulk LMO doped with x electrons per Mn. In the low doping regime of interest here, we assume that the excess electrons segregate to form a periodic arrangement of spherical FM islands or puddles of radius R with a density ρ (per site) within an undoped AF background. The periodic arrangement is defined by a cubic u.c. of volume (4π/3)R 3 /p, where p is the FM volume fraction that determines the average spacing ∼ R/p 1/3 between FM islands. The u.c. have a neutralizing uniform positive charge density xe corresponding to the dopants and the charge neutrality condition implies ρ = x/p. The energy of the phase separated state can be written as, E PS = E Kin + E Mag + E Coulomb , where E Kin corresponds to the kinetic energy of the electrons confined within the FM metallic island and can be easily estimated [10]; E Mag = −(2J F + J AF )S 2 + 2J AF S 2 p is the magnetic exchange energy of the phase separated state. We take J F = J AF = 0.1t, with t = 0.3 eV [14]. The Coulomb energy cost is entirely due to the charging energy of each spherical u.c., as there is no inter-island interaction in this approximation, and could be obtained as where V = e 2 / a is the strength of Coulomb interaction that is estimated by using static dielectric constant ≈ 100 for doped manganites at low temperature and low-frequency [18]. We obtain the optimal size of the FM island by minimizing E PS with respect to R and p. As evident in Supplementary Figure 7a, the phase separated state has lower energy than FM for 0 < ∼ x < ∼ x c ≈ 0.1 . As shown in the inset, the FM volume fraction p → 1 as x → x c and whole system becomes a uniform FM beyond x = x c . Supplementary Figure 7b shows the radius R as function of doping; R increases with x and diverges approaching the transition to the uniform FM. The diameter (2R) of the FM island is between 4 − 20 nm. The inset of Supplementary Figure 7b implies that the charge density ρ within the FM island varies weakly as a function of x and stays close to the critical density x c ≈ 0.1.

Charge reconstruction in LMO/STO heterostructures
As discussed in the main text, due to the polar nature, LMO/STO heterostructure can undergo an electronic reconstruction as in LAO/STO [19]. As a result, the heterostructure consists of an electrondoped layer within the LMO near the interface and a hole doped layer at the top surface [20]. We estimate the charge density qe (per 2D u.c.) of doped LMO layers using q(N ) = 0.5(1 − N c /N ) (Fig. 5e), where we take the critical thickness N c = 5 in conformity with experiment. This simple form can be obtained in the intrinsic polar catastrophe scenario [20]; however, here we treat it as an empirical formula. Since our model is electron-hole symmetric, from here on, we only refer to the electron-doped layer.

Charge distribution in LMO: Schrödinger-Poisson calculation
The excess charges are confined close to the surface and interface due to electrostatics. However, they can lower their kinetic energy by delocalizing in the z-direction. We self-consistently obtain the spread N e of the electrons from the interface along the z-direction by performing a Schrödinger-Poisson calculation, assuming a single hole-doped layer with charge +qe per 2D u.c. as a boundary condition at the top surface. This gives us an estimate of the layer-resolved charge distribution n(l), l being the layer index, and the effective single-particle potential V eff (l) that confines the electrons near the interface. The electric field (in the z-direction) between layers l and l +1 is E(l, l +1) = E pol +E S +E H , where E pol = 2πe/˜ a 2 is the electric field due to alternating polar LaO + and MnO 2 − sublayers, E S = −2πqe/˜ a 2 the field due to the hole-doped layer at the surface and is the electric field due to the Hartree potential for the charge distribution {n(l)}. Here˜ 18 [18] is the low temperature dielectric constant of bulk undoped LMO. The potential V eff (l) is obtained by summing over the fields from the interface to the l-th layer. The kinetic energy is given by H 0 = k,ll ll (k)a † kl a kl , where the energy dispersion ll (k) contains the z-direction hopping t and the 2D dispersion in the xy-plane, 0 (k) = −2t(cos k x a + cos k y a) ≈ −4t + ta 2 k 2 , with k = (k x , k y ). We work with spinless Fermions, as appropriate for the double exchange model (see below) assuming a uniform FM phase for the doped layers. The Hamiltonian H 0 + V eff is diagonalized starting with an initial charge distribution {n(l)} and n(l) is obtained self-consistently via n(l) = kλ n F (ε λ (k)) |ψ λl (k)| 2 , where ε λ (k) and ψ λl (k) are the eigenvalues and eigenfunctions, respectively, and n F is the Fermi function. The Fermi energy is determined by the charge neutrality constraint N l=1 n(l) = q. The results for n(l) and V eff (l) are shown in Supplementary Figure 9. Since the charge density profile decays exponentially with the number of layers, to determine the number of doped layers we used a cut-off of n = 0.005. Below we show that the doped layers lead to a phase-separated (PS) state exhibiting superparamagnetism.

Phase separation in LMO/STO heterostructures
As in the case of bulk LMO, we estimate various contributions to the energy of the PS state as a function of the FM area fraction p a and the radius R of the islands in the 2D case of the LMO/STO heterostructure.
Kinetic energy: We estimate the kinetic energy E kin (R, p a ) of the electrons within the FM island subjected to the effective confining potential V eff (l). The kinetic energy of the electrons confined within an area πR 2 in the xy plane is obtained from H 0 = n,ll ll (n)a † nl a nl , where n = (n x , n y ); n x , n y being positive integers and ll (n) contains z-direction hopping t and 2D particle-in-a-box energy levels 0 (n) ≈ −4t + ta 2 π(n 2 x + n 2 y )/R 2 for a box of linear dimension √ πR. By diagonalizing H 0 + V eff , we obtain the kinetic energy of the electrons E kin (R, p a ) as a function of R and the FM fraction p a .
Magnetic energy: The formation of FM islands, while reducing the kinetic energy, leads to loss of magnetic exchange energy, which essentially limits the FM area fraction p a . As shown in Supplementary Figure 6, there are three possible A-type AFM arrangements for the LMO/STO structure. If the spin configurations of Supplementary Figures 6a 6b are realized, then one expects to see a large magnetic signal from different AFM domains in the SOT scans for odd number of LMO layers for N ≤ N c , in contrast to our observations (Fig. 1). Also, the configuration of Supplementary Figure 6b is highly unlikely as our SOT measurements find that the SPM islands have in-plane magnetic moment. Therefore, for our calculations, we consider the spin configuration of Supplementary Figure 6c. In principle, the AFM configuration in LMO/STO heterostructure for N ≤ N c could be different from the A-type AFM in the bulk, e.g. G-type or C-type. However, the qualitative fact that we obtain an inhomogeneous SPM state for all N ≤ 200 will not change if we take G-type of C-type AFM states as FM tendencies will be even more suppressed.
For N > N c , N e layers get doped with electrons. If these layers host FM islands in an AFM matrix with a FM area fraction p a , then the magnetic energy of the N e layers is given by E mag (p a ) = −(3N e − 2p a N e − 1)JS 2 . As in the case of bulk LMO, the competition between kinetic double exchange and magnetic superexchange gives rise to a PS state with p a < 1. However, as the excess charges segregate within the FM regions, it costs a lot of Coulomb energy to form a large FM region. This essentially limits the size of the FM islands.
Coulomb energy: To obtain the Coulomb energy cost, we approximate the hole-doped layer at the surface as a uniformly charged 2D plane with surface charge density σ 0 = qe/a 2 and the electron doped layer at the interface as a square lattice of 2D disks, with radius R and surface charge density σ f = −σ 0 /p a , having average spacing (π/p a ) 1/2 R. The Coulomb energy is obtained from E Coulomb = (π/ L 2 ) dk z k |ρ(k)| 2 /k 2 , where k = (k , k z ), L 2 is the area of the system, and ρ(k) is the Fourier transform of the 3D charge density. For FM area fraction p a < 1, the Coulomb energy (per 2D u.c.) contribution from the non-uniform part of the charge distribution is obtained as where g = 2 √ π(g 1x + g 2ŷ ), g 1 , g 2 being integers, and J 1 (x) the Bessel function, and V = e 2 / PS a is determined by the dielectric constant PS in the PS state. Since PS is not known, we take for our calculation PS = ≈ 100, the value for doped LMO [18]. However, our results do not change qualitatively over a range of PS values.

Numerical Results
Summing over E mag (p a ), E kin (R, p a ), and E Coulomb (R, p a ), we obtain the energy E PS (p a , R) of the PS state and minimize it to obtain the optimal diameter D and area fraction p a of the FM islands, as shown in figures. 5f and 5g. The magnetic moment m (Fig. 5f) of the FM islands is obtained from their volume πR 2 N e a assuming 4µ B per Mn atom. The total magnetic moment M of the sample (Fig. 1g) is calculated by summing the magnetic moments m of the electron-and hole-doped layers over the 5×5 mm 2 area of the sample, as well as the background contribution of 0.2µ B per Mn for the (N − 2N e ) + 2(1 − p a )N e undoped AFM part of the LMO layers. Energies of the SPM and FM states are compared in Supplementary Figure 8a. We find the SPM state to be stabilized over uniform FM, i.e., p a < 1, for all thicknesses 6 ≤ N ≤ 200, in conformity with our SOT measurements. The charge density inside each FM island varies weakly with N for N > 6 and stays around 0.17 (Supplementary Figure 8b). Figure 5f shows that the size of the FM islands is on the nm scale, giving rise to the SPM behavior. The calculated moments and diameters of the FM islands are in good agreement with corresponding typical values, D 19 nm and m 1.5 × 10 4 µ B , found experimentally (Fig. 3j). However, in reality, disorder can give rise to a distribution of these quantities, as seen in figure. 3j. The quantities D, m, and p a show non-monotonic dependence on N , peaking at N 12 ( Fig. 5f,g). Around this thickness, a transition from insulating SPM to the metallic FM state could be induced by increasing the carrier concentration at the interface by an external gate voltage.

Supplementary Note 5. SQUID Characteristics
The scanning SOT microscopy technique, including the Pb SOT fabrication and characterization, is described in Refs. 21, 22 and 23. Supplementary Figure 11 shows the measured quantum interference pattern I c (H ⊥ ) of the Pb SOT used to investigate the 8 u.c. sample, which is typical for our devices. It had an effective diameter of 114 nm (204 mT modulation period), 66 µA critical current at zero field, and white flux noise (at frequencies above a few hundred Hz) of 200 nΦ 0 Hz −0.5 . A different SOT of ∼ 100 nm diameter was used for each sample to study the local B z (x, y), as summarized in Supplementary Table 1. Since the 4 and 5 u.c. samples produced a very weak signal, a larger SOT of 229 nm was used for both samples.
SOTs are sensitive only to the out-of-plane component of the magnetic field B z and can operate in the presence of elevated in-plane and out-of-plane fields. The field sensitivity of a SOT arises from the field dependence of its I c (H ⊥ ) and is maximal around the regions of large |dI c /dH|. Therefore, the SOTs usually have poor sensitivity at H ⊥ = 0, as seen from Supplementary Figure 11. Using a vector magnet, we have applied a constant H ⊥ to bias the SOT to a sensitive region and then imaged the local B z (x, y) at various values of H up to our highest field µ 0 H = 250 mT. The presence of H ⊥ did not cause any observable effect on B z (x, y) because of the in-plane magnetization of LMO with large anisotropy. The values of the applied H ⊥ for the various samples are listed in Supplementary  Table 1 along with the estimated scanning height h of the SOT above the sample surface. For 6 to 24 u.c. samples, we have a more accurate evaluation of h, obtained from the best fit to ∆B z (x, y), as demonstrated in Fig. 2d and described in method.