Fermi surface geometry and momentum dependent electron-phonon coupling drive the charge density wave in quasi-1D ZrTe$3$

TL;DR

This work addresses the origin of the charge density wave in quasi-1D ZrTe3 by combining ab initio electronic structure, lattice dynamics, and electron-phonon coupling analyses. Using DFT+U for Te 5p orbitals and DFPT, the authors show that an accurate Fermi surface and a localized soft phonon at the CDW wavevector emerge only when Te correlations are included. They demonstrate that the momentum dependence of electron-phonon coupling, rather than nesting alone, dominates the CDW driving force, with the coupling matrix elements amplifying the susceptibility near the CDW wavevector. The study thus identifies a cooperative mechanism where Fermi surface geometry and momentum-dependent electron-phonon interactions jointly determine the instability, providing insights applicable to other quasi-1D systems such as trichalcogenides and Peierls-like chains.

Abstract

ZrTe$_3$ is a prototypical quasi-one-dimensional compound undergoing a charge density wave transition via a very sharp Kohn anomaly in phonon momentum space. While Fermi surface geometry has long been considered the primary driver of the instability, a full understanding of the lattice dynamics and electron-phonon role has remained elusive. Our first principles calculations in the high-symmetry phase show that the Fermi surface is correctly reproduced only when the Hubbard interaction on the Te $5p$ orbitals is included, which in turn is essential for the appearance of a soft harmonic phonon mode at the CDW wavevector. Analyzing the mode and momentum dependence of the electron-phonon coupling, we find that its variations with phonon momentum dominate over electronic effects. These results identify unambiguously the CDW origin in ZrTe$_3$ as a cooperative effect of Fermi surface geometry and momentum-dependent electron-phonon coupling, with the latter playing the leading role. The mechanisms revealed in our work are directly relevant to other quasi-1D systems, including trichalcogenides and compounds hosting Peierls-like chains.

Figures (5)

• Figure 1: Crystal structure of ZrTe$_3$ viewed along the b-axis. The unit cell is outlined with black lines, while shaded areas inside indicate the Zr-centered prisms along b. Te(2) and Te(3) atoms forming the secondary chain along a are highlighted. Alternating intraprismatic ($\Delta_{\rm Intra}$) and interprismatic ($\Delta_{\rm Inter}$) distances along this dimerized chain are listed in Table \ref{['tab:Te_distances']}.
• Figure 2: Fermi surface of ZrTe$_3$ at $k_z$ = 0. The background color map shows the experimental ARPES intensity from Ref. Yokoya2005. Black arrows, drawn in that study, indicate the q$_x$ component of the CDW vector. Cyan lines represent the theoretical FS from Ref. Stowe1998. Our calculated FS cross sections are overlaid for comparison: (a) PBE (red), (b) PBE+SOC (purple), and (c) PBE+U applied to the Te atoms (dark blue).
• Figure 3: Quantities related to the non-interacting static electronic susceptibility at different levels of theory (PBE: red; PBE+SOC: purple; PBE+U: blue). (a) Nesting function along ($h$, 0, 1/3) r.l.u for $h \in [0, \; 0.10]$. (b) Real part of the non-interacting susceptibility in the same momentum range. Note that different y-axis scales are used in panel (b) for PBE/PBE+SOC (left, red--purple) and for PBE+U (right, blue).
• Figure 4: Harmonic phonon spectra of ZrTe$_3$ calculated with PBE (red) and PBE+U (blue). Only acoustic modes are shown.
• Figure 5: Harmonic phonon spectra and electron-phonon interaction of calculated with PBE+U. (a) Acoustic phonon spectra along ($h$, 0, 1/3) r.l.u. for $h \in [0,\; 0.10]$. (b) Phonon linewidths (full width at half maximum) from electron--phonon coupling for the same modes and momentum range. (c) Ratio between the electron--phonon linewidths and the nesting function, calculated both with the same electronic parameters.