Importance of Non-Adiabatic Effects on Kohn Anomalies in 1D metals

TL;DR

This paper investigates non-adiabatic effects on Kohn anomalies in 1D metals by developing a minimal 1D model with a parabolic band and a single phonon mode to study how $\omega_Q$, $m^*$, and EPC strength $|g_Q|$ renormalize phonons. A threshold $g_0$, depending only on $\omega_Q$ and $m^*$, is derived to predict low-temperature instabilities, and a self-consistent treatment including a phonon linewidth $\gamma_Q$ yields finite renormalized frequencies and a criterion for instability when $|g_Q|>g_0$. These predictions are validated against first-principles DFPT/EPW calculations for a (3,3) carbon nanotube and boron/strained gold monoatomic chains, reproducing the observed trends: strong renormalization or imaginary frequencies for modes with $|g_{KQ}|$ exceeding $g_0$ and subdued effects otherwise. The work highlights the pivotal role of non-adiabatic EPC in 1D Kohn anomalies and provides a practical stability criterion that can guide experiments and future extensions to higher dimensions.

Abstract

Kohn anomalies are kinks or dips in phonon dispersions which are pronounced in low-dimensional materials. We investigate the effects of non-adiabatic phonon self-energy on Kohn anomalies in one-dimensional metals by developing a model that analyzes how the adiabatic phonon frequency, electron effective mass, and electron-phonon coupling strength influence phonon mode renormalization. We introduce an electron-phonon coupling strength threshold for low-temperature system instability, providing experimentalists with a tool to predict them. Finally, we validate the predictions of our model against first-principles calculations on a 4 Å-diameter carbon nanotube.

Figures (12)

• Figure 1: Normalized phonon frequency $\Omega_Q$ as a function of the normalized EPC matrix element $|\hat{g}_Q|$, defined in Eq. \ref{['eq:ghat']} at 0 K. Various values are reported for $\omega_Q$: 10, 50, 100, 150, and 200 meV in blue, red, green, purple, and orange respectively with $m^*=1$. The black dashed line is the reference circular arc. The inset shows the deviation from this perfectly circular curve. The color code is the same and the shaded area account for different $m^*$ ranging between 0.2 and 5.0.
• Figure 2: The (a) electronic and (b) phonon bandstructures of the (3,3) carbon nanotube. The EPC matrix elements for $A_1(T)$ (c), $A_1(L)$ (e) and RBM (g). The dotted line is $g_0$ computed with Eq. \ref{['eq:g0']}. The renormalized phonon frequency for the three modes (d), (f) and (h). The solid line is computed with Eq. \ref{['eq:O_renorm_tot']} while the dashed line is the DFPT reference.
• Figure S1: Mean absolute error of the difference between Eq. \ref{['eqSI:fd_diff1']} and Eq. \ref{['eqsI:num-tanh']}, across the Brillouin Zone as a function of temperature for $m^*=1$.
• Figure S2: Solution of Eq. \ref{['eq:O_renorm_tot']} in the main manuscript as a function of $|g_{\rm Q}|$ considering the real part only at $T=0$ K. Therefore, here, $\gamma_{\rm Q}=0$.
• Figure S3: (a) Integral of the real part of the susceptibility at $T$ = 0 K as a function of the renormalized frequency $\Omega_Q$. (b) $\gamma_{\rm Q}$ at $T$ = 0 K as a function of the renormalized frequency $\Omega_Q$. In both panels, the lines are computed numerically, while the dots are the analytical results for $\Omega_Q = 0$ through Eqs. \ref{['eqSI:rechiana']} and \ref{['eqSI:gammana']}, respectively.