Fig. 4: Screening of the LO–TO splitting and polar optical scattering.
a Decomposition of the optical phonon bands shows the LO–TO splitting. b Calculated phonon dispersion in ZrNiSn parent sample. The insets illustrate the phonon vibration modes for the two longitudinal optical phonons (a propagation vector along c axis is used here to define longitudinal and transverse phonon vibrations). The LO–TO splitting mainly occurs for the high-frequency optical branches of LO1 and TO1,2. c Schematic illustration of the screening effect on the LO–TO splitting (following the Eqs. (2) and (4) in Methods) with different Thomas-Fermi screening lengths, rTF (in units of the lattice constant). With enhanced screening or smaller rTF as traced by the arrow, the LO frequency around the Brillouin zone center is suppressed. The Born effective charge, \({\cal{Z}}_a\), is supposed to be insensitive to the presence of free carriers, for this purpose, which is an approximation. d Comparison of the neutron-weighted phonon DOSs between ZrNiSn and ZrNiSn0.875Sb0.125 calculated with first principles. e Variation of the LO–TO splitting (or polarization field) with n. The dash curve is fitted using the Eqs. (2) and (4) in Methods including screening. The solid curve is the n-dependent mobility limited by polar optical phonon scattering in ZrNiSn1−xSbx compounds40. With increasing screening, the LO–TO splitting and polarization field are reduced, and the polar optical phonon scattering is reduced. f The polar coupling constant for some selected TE semiconductors as listed in Fig. 1. The value for the half-Heusler compounds is calculated based on the parameters as shown in Supplementary Table 1, while the data for other compounds is from literature27.
