Barium titanate's electro-optic roll-off traces to collective titanium motion, not the soft phonon frequency alone
Synopsis
Barium titanate's electro-optic coefficient falls markedly as modulation frequency rises, limiting modulator bandwidth. Using machine-learned potentials trained on PBE and HSE data and Green–Kubo molecular dynamics for the fixed-cell ionic coefficient, the work finds that nonharmonic damping and atomic displacement correlations spanning many unit cells, rather than phonon frequency itself, set both the low-frequency amplitude and the attenuation frequency. The HSE potential gives about 590 pm/V and about 137 GHz, against experimental values of 480 pm/V and 72.6 GHz.
Interpretation
The microscopic origin of barium titanate's frequency-dependent electro-optic attenuation is traced to collective correlations of atomic displacements rather than to the frequency of an individual phonon mode. A harmonic phonon picture cannot describe this attenuation, since even the softest optical phonon sits at several THz, far above the experimentally observed attenuation frequency. For tetragonal barium titanate at 300 K, Green–Kubo calculations with an HSE machine-learned potential give a low-frequency coefficient of about 590 pm/V and an attenuation frequency of about 137 GHz, against experimental values of 480 pm/V and 72.6 GHz.
The calculation takes the fixed-cell ionic electro-optic coefficient directly from molecular-dynamics trajectories, retaining anharmonic dynamics and all displacement cross correlations. The Green–Kubo form writes the response as a displacement correlation function weighted by Raman tensors and Born effective charges, whose zero-frequency limit recovers the static coefficient and whose time evolution sets the frequency dependence. With the PBE potential, Green–Kubo magnitudes at 10 GHz are 177.0 and 212.7 pm/V for 83 and 123 supercells, bracketing the SSCHA mode-sum static reference of 186.2 pm/V.
Changing the short-range interaction or the lateral supercell size separately tunes correlation strength and correlation range, and both shift the low-frequency amplitude and the attenuation frequency together. At fixed 8×8×4 geometry, interpolating the short-range interaction from PBE to HSE and extrapolating to λ=2 raises the nearest-neighbor correlation from 0.406 to 0.633, the 10 GHz coefficient from 196 to 1635 pm/V, and lowers the attenuation frequency from 1211.59 to 20.41 GHz. With HSE interactions fixed and lateral size grown from 6 to 24, the low-frequency coefficient rises from 328 to 2465 pm/V and the attenuation frequency falls from 313.73 to 47.38 GHz, while the nearest-neighbor correlation stays between 0.498 and 0.505.
Perspective
The scope is fixed-cell, single-domain simulation of tetragonal barium titanate at 300 K without domain walls; the largest supercell holds 243 cells and 69,120 atoms, and the authors note that further size and sampling tests are computationally demanding even with the NEP force field.
Finite-size dependence remains in the HSE results, with fitted attenuation frequencies of 169.81, 148.82, 158.67 and 137.15 GHz for 123, 163, 203 and 243 cells; the 20.41 GHz value at λ=2 lies slightly below the fit range and is treated as an extrapolated estimate. In the overdamped regime, phonon frequency and friction cannot be determined independently from the molecular-dynamics spectra.