Unconventional electron states in δ-doped SmTiO3

The Mott-insulating distorted perovskite SmTiO3, doped with a single SrO layer in a quantum-well architecture is studied by the combination of density functional theory with dynamical mean-field theory. A rich correlated electronic structure in line with recent experimental investigations is revealed by the given realistic many-body approach to a large-unit-cell oxide heterostructure. Coexistence of conducting and Mott-insulating TiO2 layers prone to magnetic order gives rise to multi-orbital electronic transport beyond standard Fermi-liquid theory. First hints towards a pseudogap opening due to electron-electron scattering within a background of ferromagnetic and antiferromagnetic fluctuations are detected.


STRUCTURAL DETAILS FOR THE δ-DOPED ARCHITECTURE
The δ-doped SmTiO 3 architecture is realized by a superlattice based on a 100-atom unit-cell. It consists of 10(9) TiO 2 (SmO) layers, separated by a single SrO monolayer. Each TiO 2 layer is build from two possibly symmetryinequivalent Ti ions, to allow for potential intra-layer spin and/or charge ordering. Together with the five-layer resolution distant from the doping layer, there are thus 10 inequivalent Ti single-site DMFT problems in our realistic modelling. The original lattice parameters [1] are brought in the directional form of the experimental works, i.e. Ref. [2,3], but without lowering the P bnm symmetry. The original c-axis is parallel to the doping layer and the original a, b-axes are respectively inclined. The plane-wave energy cutoff is set to E cut = 11 Ryd and a 5 × 5 × 3 kpoint grid is used. With fixed lattice parameters, all atomic positions in the supercell are structurally relaxed within DFT(GGA) until the maximum individual atomic force settles below 5 mRyd/a.u.. The lattice distortion introduced by the SrO layer is well captured by the structural relaxations. No relaxation of the lattice paramteres is performed. A change of lattice parameters is expected to be very small due to the structural similarity and does not invoke changes of the key physics discussed in the given work. The obtained crystal structure is used for the PM, A-AFM as well as the pre-converged electronic structure studies.

INFLUENCE OF THE LOCAL-INTERACTION STRENGTH
The chosen local Coulomb interactions are well established for bulk titanates [4]. A first-principles computation of these parameters for the large δ-doped architecture, including their layer dependence, is currently numerically unfeasible. In order to still examine the influence of a smaller/larger local Coulomb interaction, we additionally studied the DFT+DMFT electronic structure for U = 3.5 eV and U = 6.5 eV. The Hund's exchange is even less sensitive and therefore remains fixed at J H = 0.64 eV. Figure 2 exhibits the layer-dependent Ti orbital filling with U . As expected, for the smaller value of U , the orbital polarization is much weaker for the layers beyond Ti1. Also the Mott state is not reached with distance from the doping layer up to Ti5, thus U = 3.5 eV appears too small to account for the correct δ-doping physics. On the other hand, for U = 6.5 eV the orbital polarization for Ti3-5 is even increased. Moreover, the second TiO 2 layer with Ti2 is also strongly orbital polarized for this larger U . The additionally shown Ti2 local spectral function in Fig. 2 reveals that the second layer is about to enter a Mott-insulating state. Hence a substantially larger, somewhat unphysical, Hubbard U destroys the revealed two-layer dichotomy.

DETAILS ON THE SELF-ENERGY FITTING
The low-frequency characteristics of the local electronic self-energy within the two TiO 2 layers closest to the SrO doping layer, i.e. Ti1 and Ti2, is analyzed in some detail in the main text. This allows to extract important information on the (non)-Fermi-liquid quality connected to states close to the Fermi energy. To asses this quality, we focus on two features of the self-energy, namely the intercept C 0 = lim ω→0 Σ and the formZ(ω) = 1 − ∂Σ ∂ω ω→0 −1 . For a Fermi liquid (FL) at low temperature, the intercept C 0 approaches zero andZ(ω) = Z is the constant quasiparticle (QP) weight. Significant deviations from these features signals a non-Fermi liquid (NFL) regime.
We here provide further information on the self-energy fitting procedure in the δ-doped paramagnetic phase to enable such a discrimination, as well as on the grade of those fits. The relevant object in this context is the imaginary part of the Matsubara self-energy. In correct mathematical terms, this function reads Im Σ(iω n ), with the fermionic Matsubara frequencies defined as ω n = (2n + 1)π T . From a sole fitting-function point of view, we however refer in the following to Im Σ(ω n ). Moreover we discuss the fitting parameters as dimensionless, imagining proper normalization. Figure 3 shows different resolutions of the imaginary part of the multi-orbital self-energy with respect to the frequencies ω n . The quantum-Monte-Carlo data is well converged at T = 48 K to facilitate our low-frequency examination. In order to check the influence on the frequency cutoff n c (i.e. all frequencies with n ≤ n c are used for the fit), different values for n c are chosen. Two different stages of fitting procdures are performed and results are here exemplified for the dominant Ti |2 state in the first (Ti1) and second (Ti2) layer next to SrO (see Tab. II). First, motivated by assuming Fermi-liquid theory holds, a polynomial fit of simplistic first order (n c ≤ 4) and of 5th order (n c ≥ 6) is processed. Intercept and Z are therefrom easily extracted from the zeroth and first-order terms, respectively. A good quality of the fitting shall focus on the very low-frequency region, but takes care of the ω n evolution. Hence the sole linear fitting for very small n c as well as the higher-order fit for rather large n c ∼ 16 − 32 are less suited to the problem. But the results show that a proper fitting leads to robust values for C 0 and Z with e.g. an error ∆Z ∼ 0.03.
Questioning Fermi-liquid behavior asks for an exponential fitting function with an exponent α (cf. Tab. II). Only for α = 1 a constant QP weight and thus Fermi-liquid behavior is recovered. However the fitting is more exclusively restricted to small n c . A large n c results in a too small intercept C 0 , since the fitting function tries to account for the natural bending of Im Σ(ω n ) at larger ω n by shifting C 0 towards zero. Still, the quality of our numerical data is high enough to reveal the resilient FL-to-NFL crossover from the first to the second TiO 2 layer. In the second layer, the exponent α manifestly deviates from unity and the intercept C 0 from zero. Hence electron-electron scattering is beyond FL theory and does not easily vanish at the Fermi energy. Of course the previous FL statements only fully hold for T → 0, but our temperature is already well below a possible QP coherency scale and the contrasting behavior  in the seemingly FL-like first TiO 2 layer is obvious.