First‐Principles Calculations of Phonons and Thermodynamic Properties of Zr(Hf)S2‐Based Nanotubes

Comparative hybrid density functional calculations on the structure, stability, and phonon frequencies of monolayers and single‐walled nanotubes are performed for Zr(Hf)S 2 disulfides. The first‐principles calculations of HfS 2 ‐based nanotubes are made for the first time. The symmetry analysis of infrared and Raman active vibrational modes in ZrS 2 and HfS 2 nanotubes is made using the induced representations of the isogonal point groups of line groups. It is shown that the number of infrared and Raman active modes is constant for NTs with the same chirality type. The correlation of the phonon modes of the nanotubes of relatively large diameters with those of monolayer is analyzed. The thermodynamic functions of monolayers and nanotubes with various chirality and diameters are calculated on the basis of the obtained phonon frequencies. It is established that the phonon contribution to the nanotube strain energy is small, but may be important for an accurate estimate of the stability of the nanotubes of small diameters. The calculated results show that the thermal contributions to Helmholtz free energy are positive; thereby they slightly reduce the stability of ZrS 2 and HfS 2 nanotubes at elevated temperatures. © 2019 Wiley Periodicals, Inc.