Role of ionic quantum-anharmonic fluctuations on the bond length alternation and giant piezoelectricity of conjugated polymers

Abstract

Functionalized conjugated polymers are promising materials for electromechanical applications due to predicted giant piezoelectricity, arising from anomalously large dynamical effective charges and an enhanced response in the proximity of the dimerization phase transition. In this work, we assess the impact of quantum ionic fluctuations on piezoelectricity using the stochastic self-consistent harmonic approximation with a Rice-Mele diatomic chain model, parametrized to reproduce hybrid-functional first-principles calculations of prototypical carbyne. The model's accuracy is validated against first-principles calculations both with and without quantum-anharmonic effects. We find that ionic fluctuations strongly impact the structural properties, with the boundary of the dimerization phase transition shifted by $34\%$. Despite quantum fluctuations in the bond length reaching magnitudes comparable to the average, the strong piezoelectric response persists. The topological enhancement of the effective charges remains robust and is even enhanced by about $\sim20\%$ thanks to a quantum-induced shrinking of the electronic gap. The piezoelectric coefficient remains dominated by the internal relaxation and retains a morphotropic-like character, reaching maximum values near the renormalized boundary, with quantum anharmonicity mainly shifting the optimal enhancement window.

Figures (10)

• Figure 1: Comparison of ab initio values and model prediction of Born effective charge $Z^*$(a) and piezoelectric coefficient $c_\mathrm{piezo}$(b) evaluated at different He-C distances, shown on the horizontal axis at the top. To each value of the distance, it corresponds a fitted value of $\Delta$ in the model, shown on the horizontal axis at the bottom. In the inset, we illustrate the decorated carbyne system displaying two simulation cells.
• Figure 2: Energy profile of carbyne as a function of the bond-length difference BLA with and without quantum-anharmonic effects (dots and lines, respectively). Fully ab initio results obtained with PBE0 functional and SSCHA are shown in black. Model results, obtained using parameters of table \ref{['tab:fitted_parameters']}, are shown in red.
• Figure 3: Panel (a): temperature dependence of the BLA for carbyne ($\Delta=0$ in the model), computed with the inclusion of QAE. We observe an hysteresis cycle which suggests that the polyyne-to-cumulene transition is of first order, in contrast with the common Landau-Peierls picture of a second order transition, and consistently with the findings of Refromanin2021dominant. Panel (b): free energy differences $\delta F=F_\mathrm{cooling}-F_\mathrm{heating}$ between the minimized free energies of the optimal structures obtained in the cooling cycle and those obtained in the heating cycle, shown in the range of temperatures where the hysteresis is found. From a linear fit of $\delta F$ with respect to $T$, we deduce a $T\mathrm{_{C}^{QAE}}\simeq4300\;\mathrm{K}$ above which the undimerized structure with no BLA becomes favorable.
• Figure 4: Effects of quantum-anharmonicity on the manifestation of BLA in the model at $T=0\;\mathrm{K}$. Panel (a): for three representative values of the onsite energy $\Delta=0\;\mathrm{eV}<\Delta_\mathrm{c}^\mathrm{QAE}$, $\Delta=0.47\;\mathrm{eV}\simeq\Delta_\mathrm{c}^\mathrm{QAE}$ and $\Delta=0.50\;\mathrm{eV}>\Delta_\mathrm{c}^\mathrm{QAE}$, we display the histograms with the values of the bond length differences $l_1-l_2$ computed along the supercell configurations as described in the text. Panel (b): effects of quantum anharmonicity on the displacive phase transition guided by the onsite energy $\Delta$. The black line correspond to the values obtained neglecting the ionic fluctuations, whose effects are included in the values indicated by the triangles. To test for the presence of meta-stable states, we performed free energy minimizations for increasing and decreasing values of the onsite energy $\Delta$, as described in the text. We observe how the inclusion of QAE shifts the phase boundary between the lower-symmetric and the higher-symmetric phases. Panel (c): We obtain a value of $\mathrm{\Delta_{c}^{QAE}}\simeq0.47\,\mathrm{eV}$ from a linear fit of the free energy differences $\delta F=F_{\Delta\;\mathrm{dec.}}-F_{\Delta\;\mathrm{inc.}}$ between the free energy of the minimized configurations of the $\Delta$-decreasing cycle and those of the $\Delta$-increasing one, computed in the range of values of $\Delta$ where the two phases coexist.
• Figure 5: Structural phase diagram of the model with respect to the temperature and the onsite energy. For different pairs of $T$ and $\Delta$, we display with different colors whether a BLA is predicted or not. The green area represents the region for which the system manifests a BLA and hence non-zero effective charges and a piezoelectric response, whereas for values of $T$ and $\Delta$ in the pink region, the system presents no BLA and thus no piezoelectricity. With light-blue we indicate the region where the inclusion of QAE suppresses the BLA. The black line correspond to the values of $T\mathrm{_{C}^{QAE}}(\Delta)$ obtained with the inclusion of QAE as described in the text, representing the quantum-anharmonic-corrected phase boundary. The vertical dashed red line is in correspondence of the critical value of the onsite energy obtained at $T=0\;K$ without QAE.