Thermal transport and phonon focusing in complex molecular crystals: Ab initio study of polythiophene

Thermally conductive molecular crystals are of fundamental interest because they are unlike typical complex crystals, which conduct heat poorly owing to their large phonon scattering phase space. While molecular crystals with high thermal conductivity in the range of tens of ${\mathrm{Wm}}^{-1}{\mathrm{K}}^{-1}$ have been known experimentally for decades, their intrinsic upper limits for thermal conductivity are unclear. Ab initio methods that have been successfully applied to simple crystals have proved difficult to adapt to molecular crystals due to quantum nuclear motion and their complex primitive cells. Here, we report the thermal transport properties of crystalline polythiophene with 28 atoms per primitive cell using an ab initio approach that rigorously includes finite-temperature anharmonicity and quantum nuclear effects. The calculated room temperature thermal conductivity is $198{\mathrm{Wm}}^{-1}{\mathrm{K}}^{-1}$ along the chain axis, a high value that arises from exceptional phonon focusing along the chain for both acoustic and optical branches for nearly all wave vectors and despite short lifetimes in the picosecond range. Our finding, along with other recent ab initio studies of polyethylene, suggests that the intrinsic upper bounds for the chain axis thermal conductivity of polymer crystals may exceed $100{\mathrm{Wm}}^{-1}{\mathrm{K}}^{-1}$.

Figure 1

The structure and dispersion of polythiophene (PT). (a) Structure of a PT crystal with chains extending in the $z$ axis. (b) View of the $xy$ plane of the unit cell. There are two chains in each unit cell. Chains from the adjacent cell are shown. (c) Single-polymer chain of PT showing that the length of unit cell in the chain axis is two thiophene chemical repeat units. (d) Calculated phonon dispersion of PT, showing numerous branches. The dark blue and light blue branches correspond to the acoustic-like and optic modes, respectively, as seen in Fig. 2. Inset: The phonon dispersion along symmetry directions from $\mathrm{\Gamma }$. The branches for which isoenergy contours are plotted later are highlighted in orange [Fig. 4] and red [Fig. 4].

Figure 2

Thermal conductivity vs temperature and spectral thermal conductivity. (a) Thermal conductivity along the chain (blue solid line) and along the $y$ axis perpendicular to the chains (orange dashed line) vs temperature using force constants obtained at 300 K. The polymer chains are oriented along the $z$ axis (blue line). The chain-axis thermal conductivity is $198{\mathrm{Wm}}^{-1}{\mathrm{K}}^{-1}$ at room temperature. (b) Spectral thermal conductivity vs phonon frequency along the chain axis. The contributions from acoustic-like modes and optic modes are highlighted in dark blue and light blue, respectively.

Figure 3

Phonon lifetimes and group velocities of PT and Si. (a) Phonon lifetimes vs frequency for PT (blue dots) and Si (orange triangles). (b) Thermal conductivity accumulation function vs lifetime. The thermal conductivity of PT is due almost entirely to modes with lifetimes less than 50 ps, in contrast to Si which has more than half of its thermal conductivity from modes with lifetimes longer than 100 ps. (c) Phonon group velocities along the chain axis ($z$) vs frequency of PT (blue dots) and Si (orange triangles). (d) Thermal conductivity accumulation function vs group velocity along the chain axis. Nearly two thirds of the heat in PT is carried by modes with group velocity larger than the maximum group velocity in Si.

Figure 4

Isoenergy contours in PT. (a) The isoenergy contours of an acoustic branch with an anisotropy in the maximum group velocity ratio of 7.2 in PT for the $k_x=0$ plane. The lines are contours of equal frequency. The arrows show the direction of the group velocity vector. The contours are similar to those Ref. [20] corresponding to the quasitransverse acoustic branch. (b) The isoenergy contours of an optical branch with a much larger anisotropy ratio of 22.7. The group velocity of nearly all the modes are pointed along the chain axis. We observe a similar effect in other planes in the Brillouin zone for this branch and for other optical branches with a large anisotropy ratio.

Figure 5

The spectral thermal conductivity vs frequency in the case where the group velocities are parallel to their wave vector (blue) compared to the actual case (orange). Phonon focusing underlies the high thermal conductivity of PT, particularly for modes below 5 THz.