Deep-learning Hamiltonian reveals twist-tunable flat bands and nonlinear photocurrents in SrTiO3 moire bilayers

Technical Summary: Deep-learning Hamiltonian reveals twist-tunable flat bands and nonlinear photocurrents in SrTiO$_3$ moiré bilayers

Problem Statement
The extension of moiré physics to complex oxides offers new avenues for manipulating electronic states; however, the large supercells required for oxide moiré systems make systematic first-principles calculations computationally prohibitive. While twist-angle control has successfully generated flat bands and correlated states in van der Waals materials like graphene and transition-metal dichalcogenides, applying similar engineering to perovskite oxides such as SrTiO$_3$ is challenging. SrTiO$_3$ features strongly coupled lattice, orbital, charge, and spin degrees of freedom, making it difficult to reduce to simple continuum or tight-binding models. Furthermore, the computational cost of Density Functional Theory (DFT) scales steeply with system size, hindering a systematic description of twist-angle-dependent electronic structures and optical responses in bilayer SrTiO$_3$.

Methodology
To overcome the computational bottleneck, the authors combine DFT with the E(3)-equivariant deep-learning Hamiltonian framework, DeepH-E3.

• Training Data Generation: Using the OpenMX package, the authors performed DFT calculations on untwisted bilayer SrTiO$_3$ supercells with various interlayer sliding configurations (623 total configurations). These structures sample the local stacking environments found in moiré superlattices.
• Model Training: The DeepH-E3 model, an E(3)-equivariant graph neural network, was trained to predict Hamiltonian matrix elements directly from the atomic structures, incorporating spin-orbit coupling and preserving spatial symmetries. The model was validated against DFT reference data, achieving a mean absolute error (MAE) of 0.26 meV for Hamiltonian elements and excellent agreement in band structures.
• Application: The trained model was applied to commensurate twisted bilayer SrTiO$_3$ systems with twist angles ranging from $53.13^\circ$ down to $8.80^\circ$ (corresponding to supercells with up to 2,380 atoms).
• Response Calculations: Using the predicted Hamiltonians interfaced with the HopTB package, the authors calculated the dielectric response, second-harmonic generation (SHG), shift current, and intrinsic spin Hall conductivity (SHC).

Key Contributions and Results

1. Twist-Tunable Flat Bands:
   The study reveals a systematic evolution of the valence band maximum (VBM) as the twist angle decreases.

   • In the large-angle regime ($\theta > 20^\circ$), bands remain relatively dispersive with weak folding.
   • As the angle decreases to the intermediate and small regimes ($\theta \lesssim 15^\circ$), the VBM bands flatten significantly.
   • At the smallest angles studied (e.g., $8.80^\circ$), nearly dispersionless flat bands emerge, with the reported hole mass increasing by more than an order of magnitude compared to the untwisted bilayer.
   • Unlike graphene-based systems where flat bands arise from interlayer hopping interference, the flattening in SrTiO$_3$ is attributed to stacking-dependent electrostatic and orbital hybridization effects. The bandwidth and effective mass show non-monotonic variations, indicating sensitivity to the discrete commensurate stacking geometry rather than twist angle alone.

2. Differential Impact on Optical and Transport Responses:
   The authors evaluated how twisting affects various physical responses relative to the untwisted bilayer:

   • Linear Dielectric Response: The dielectric function (both real and imaginary parts) remains largely unchanged across the calculated angle range. The optical gap and main absorption features vary only weakly, suggesting the linear optical structure is robust against moiré modulation in this regime.
   • Spin Hall Conductivity (SHC): The SHC, dominated by the intrinsic spin Berry curvature, remains nearly invariant with twist angle. The peak positions and amplitudes show only minor non-monotonic variations, indicating the integrated spin Berry curvature is insensitive to the moiré perturbation in the large-to-intermediate angle range.
   • Nonlinear Optical Responses (SHG and Shift Current): These responses are strongly affected by twisting.
     • Second-Harmonic Generation (SHG): Twisting breaks interfacial inversion symmetry, leading to a strong enhancement of SHG compared to the weak response of the untwisted bilayer.
     • Shift Current: The shift current (a second-order nonlinear photocurrent) exhibits a clear twist-angle dependence. While the untwisted bilayer shows a weak response, the twisted structures show pronounced enhancement. The peak amplitude of the shift current increases significantly as the twist angle decreases from $53.13^\circ$ to $22.62^\circ$.

Significance and Claims
The paper claims that twist engineering serves as an effective control mechanism for the electronic and optoelectronic properties of oxide moiré systems, specifically bilayer SrTiO$_3$. The primary findings are:

• Electronic Structure: Twist angle can systematically tune the low-energy electronic structure, inducing nearly flat valence bands in the small-angle limit.
• Optoelectronic Control: While linear responses and spin transport are relatively insensitive to twisting in the studied range, nonlinear optical responses are highly tunable. Specifically, the shift current shows the clearest dependence on twist angle, offering a new handle for controlling nonlinear photocurrents in oxide systems.
• Methodological Validation: The work demonstrates the capability of deep-learning Hamiltonian approaches (DeepH-E3) to perform large-scale electronic structure calculations in complex, strongly coupled oxide materials where traditional DFT is impractical.

The authors conclude that while the full treatment of all coupled responses in twisted oxides remains an open direction, this study successfully establishes the feasibility of using machine learning to explore twist-engineered phenomena in complex oxides.