Quantifying angular momentum of coherently driven circular phonons

Technical Summary: Quantifying Angular Momentum of Coherently Driven Circular Phonons

Problem and Context
The connection between magnetization and angular momentum, historically established by the Einstein-de Haas effect, has recently been revisited in the context of ultrafast laser physics. Theoretical frameworks, specifically "magnetophononics," propose that circular ionic motions driven by terahertz (THz) fields carry angular momentum capable of inducing internal magnetic fields. While recent experiments in nonmagnetic strontium titanate (SrTiO₃, STO) have reported THz-induced magnetic fields and even ferromagnetic switching, these findings contrast sharply with first-principles predictions, which suggest such effects should be orders of magnitude weaker. A critical gap in the field has been the lack of direct, quantitative measurement of atomic trajectories and the specific angular momentum carried by individual ions during and after THz excitation. Previous studies have explored chiral phonons in various materials, but a time-resolved quantification of the angular momentum transfer in solids remained elusive.

Methodology
To address this, the authors employed ultrafast x-ray diffraction (UXRD) at the Bernina beamline of the SwissFEL Free Electron Laser to resolve time-dependent ionic trajectories in a free-standing 80 nm STO film. The experiment was conducted in the paraelectric cubic phase at 320 K.

• Excitation: The sample was driven by broadband, single-cycle THz pulses (1.5–4 THz) generated via optical rectification in a DSTMS crystal. The polarization was manipulated using a Y-cut quartz crystal acting as a quarter-waveplate to produce circular left (CL) and circular right (CR) polarizations, alongside linear vertical (LV) and horizontal (LH) controls. Peak electric fields inside the film were approximately 180 kV cm⁻¹, remaining below the threshold for anharmonic phenomena like phonon up-conversion.
• Detection: 50 fs x-ray pulses (8.3 keV) were used to probe the structural dynamics. The experimental geometry allowed for the simultaneous recording of the (203) and (023) Bragg diffraction peaks. These peaks are sensitive to atomic motions along the orthogonal [100] and [010] in-plane directions, respectively.
• Analysis: The time-dependent changes in diffracted intensity were used to reconstruct the collective ionic displacements. By modeling the frequency-dependent polarization and utilizing density functional theory (DFT) to determine phonon eigenvectors, the authors extracted the mass-normalized mode amplitudes ($Q(t)$). These amplitudes were combined with the eigenvectors to calculate the real-space trajectories of all ions (Sr, Ti, and O) within the unit cell. The phonon angular momentum ($\mathbf{L}$) was then computed as the sum of the cross products of position and velocity vectors for each atom, weighted by their masses.

Key Results
The study successfully reconstructed the full time-resolved trajectories of ions in STO with sub-picometer precision.

1. Circular Trajectories: Under circular THz excitation, the oxygen, strontium, and titanium ions exhibited circular motions with opposite phases but the same sense of rotation (clockwise for CL, counter-clockwise for CR). The oxygen ions reached displacement amplitudes of approximately 4 pm, while strontium ions reached 1 pm.
2. Angular Momentum Quantification: The analysis revealed a significant imbalance in the contribution to the total phonon angular momentum. Despite having a much smaller mass than strontium or titanium, the three oxygen ions contribute approximately 90% of the total phonon angular momentum. This dominance is attributed to their larger number and larger trajectory radii.
3. Temporal Dynamics: The angular momentum peaks approximately 250 fs after excitation, coinciding with the peak polar mode amplitude, and decays rapidly within less than 1 ps as the momentum transfers to other modes (likely acoustic). The sign of the angular momentum reverses with the helicity of the THz excitation.
4. Mechanism for Induced Magnetism: The authors identify that the imbalance between the angular momentum of the negatively charged oxygen ions and the positively charged metal ions provides a clear physical mechanism for the breaking of time-reversal symmetry and the generation of the net magnetic moment observed in optical experiments.

Significance and Claims
The paper claims to provide the first quantitative measurement of circular ionic motions and their associated angular momentum in a solid. By establishing a general methodology to reconstruct atomic trajectories and compute angular momentum from ultrafast x-ray diffraction data, the work offers a direct experimental validation of the mechanisms underlying magnetophononics.

The authors assert that these findings resolve the debate regarding the origin of THz-induced magnetism in STO by demonstrating that the specific distribution of angular momentum among ions (dominated by oxygen) creates the necessary conditions for induced magnetic fields. Furthermore, the paper posits that this methodology serves as a general toolkit for investigating angular momentum transfer in quantum materials, potentially enabling new strategies for controlling topological phonon transport and phonon-driven magnetism. The work does not propose new applications but rather establishes a foundational measurement technique for understanding chiral phonon dynamics in real time.