  The lattice thermal conductivity of silicon clathrate framework Si$_{23}$ and
two Zintl clathrates, [Si$_{19}$P$_{4}$]Cl$_{4}$ and
Na$_{4}$[Al$_{4}$Si$_{19}$], is investigated by using an iterative solution of
the linearized Boltzmann transport equation (BTE) in conjuntion with ab initio
lattice dynamical techniques. At 300 K, the lattice thermal conductivities for
Si$_{23}$, [Si$_{19}$P$_{4}$]Cl$_{4}$, and Na$_{4}$[Al$_{4}$Si$_{19}$] were
found to be 43 W/(m K), 25 W/(m K), and 2 W/(m K), respectively. In the case of
Na$_{4}$[Al$_{4}$Si$_{19}$], the order-of-magnitude reduction in the lattice
thermal conductivity was found to be mostly due to relaxation times and group
velocities differing from Si$_{23}$ and [Si$_{19}$P$_{4}$]Cl$_{4}$. The
difference in the relaxation times and group velocities arises primarily due to
the phonon spectrum at low frequencies, resulting eventually from the
differences in the second-order interatomic force constants (IFCs). The
obtained third-order IFCs were rather similar for all materials considered
here.
