Role of high-order anharmonicity and off-diagonal terms in thermal conductivity: A case study of multiphase ${\mathrm{CsPbBr}}_3$

We investigate the influence of three- and four-phonon scattering, perturbative anharmonic phonon renormalization, and off-diagonal terms of coherent phonons on the thermal conductivity of ${\mathrm{CsPbBr}}_3$ phase change perovskite, by using advanced implementations and first-principles simulations. Our study spans a wide temperature range covering the entire structural spectrum. Notably, we demonstrate that the interactions between acoustic and optical phonons result in contrasting trends of phonon frequency shifts for the high-lying optical phonons in orthorhombic and cubic ${\mathrm{CsPbBr}}_3$ as temperature varies. Our findings highlight the significance of wavelike tunneling of coherent phonons in ultralow and glasslike thermal conductivity in halide perovskites.

Figure 1

(a)–(c) Crystal structures of ${\mathrm{CsPbBr}}_3$ in the orthorhombic, tetragonal, and cubic phases, respectively. The green, brown, and yellow colors represent cesium (Cs), bromine (Br), and lead (Pb) atoms. (d) The first Brillouin zone of the cubic phase with high symmetry points $\mathrm{\Gamma }, X, M$, and $R$ indicated by red dots. (e) Lattice thermal conductivity ${\kappa }_L$ of ${\mathrm{CsPbBr}}_3$ includes diagonal and off-diagonal contributions in three phases using self-consistent phonon approximation. P1, P2, and P3 represent orthorhombic, tetragonal, and cubic phases. ${\kappa }_p$ [Eq. (1)] is the standard Peierls contribution and ${\kappa }_c$ [Eq. (4)] is the coherent contribution from the off-diagonal Wigner distribution elements. 3ph indicates only three-phonon scattering is included and 3,4ph means both three-phonon and four-phonon scattering are considered. For comparison, the unified first-principles theory [8] and the quasiparticle nonlinear theory (QP-NL) [9] are plotted for reference. Expt.${}^L$, Expt.${}^M$, and Expt.${}^S$ refer to experiments on single crystalline nanowires [21] with cross sections of $800\times 380 {\mathrm{nm}}^2, 320\times 390 {\mathrm{nm}}^2$, and $300\times 160 {\mathrm{nm}}^2$, respectively.

Figure 2

Renormalized phonon dispersions for (a) orthorhombic, (b) tetragonal, and (c) cubic phases at different temperatures, respectively. HA is the harmonic approximation. Panels (d), (e), and (f) are the frequency-resolved ${\kappa }_L$ (dashed lines) and cumulative ${\kappa }_L$ (solid lines) using 3ph (the upper line) and 3,4ph (the lower line) methods at different temperatures, respectively.

Figure 3

Scattering strength $I_{\lambda {\lambda }_1}$ of the 4ph interaction matrix elements between the highest zone-center optical phonon mode and all other remaining phonons in three various phases according to Eq. (2) and Eq. (3) for the (a) orthorhombic at 100 K, (b) tetragonal at 350 K, and (c) cubic phase at 500 K, respectively.

Figure 4

(a) Phonon scattering phase space and (b) scattering rates between 3ph and 4ph for cubic ${\mathrm{CsPbBr}}_3$ at different temperatures. (c) 3ph and 4ph scattering with channels resolution, including the splitting ($\lambda \to {\lambda }_1+{\lambda }_2, \lambda \to {\lambda }_1+{\lambda }_2+{\lambda }_3$), combination ($\lambda +{\lambda }_1\to {\lambda }_2, \lambda +{\lambda }_1+{\lambda }_2\to {\lambda }_3$), and redistribution ($\lambda +{\lambda }_1\to {\lambda }_2+{\lambda }_3$) processes at 500 K. (d) $v^2$, where $v$ is phonon group velocity.