  The lattice thermal conductivity ($\kappa$) of newly synthesized
two-dimensional (2D) MoSi$_2$N$_4$ family and its associated abnormality is
anatomized by $ab$ $initio$ phonon Boltzmann transport calculations. $\kappa$
of MoSi$_2$N$_4$ and WSi$_2$N$_4$ is found over 400 Wm$^{-1}$K$^{-1}$ at 300 K.
$\kappa$ of MoSi$_2$Z$_4$ (Z=N,P,As) obeys Slack's rule of thumb, decreasing by
one order of magnitude from Z=N to Z=As with 46 Wm$^{-1}$K$^{-1}$. However, in
MSi$_2$N$_4$ (M=Mo,Cr,W,Ti,Zr,Hf), the variation of $\kappa$ with respect to M
is anomalous, i.e. deviating from Slack's classic rule. For M in the same
group, $\kappa$ of MSi$_2$N$_4$ is insensitive to the average atomic mass,
Debye temperature, phonon group velocity, and bond strength owing to the
similar phonon structure and scattering rates. MSi$_2$N$_4$ with heavy
group-VIB M even possesses a three to four times higher $\kappa$ than that with
light group-IVB M, due to its much stronger M-N and exterior Si-N bonds and
thus one order of magnitude lower phonon scattering rates. Nevertheless, this
abnormality could be traced to an interplay of certain basic vibrational
properties including the bunching strength and flatness of acoustic branches
and their nearby optical branches, which lie outside of the conventional
guidelines by Slack. This work predicts high $\kappa$ of 2D MSi$_2$Z$_4$ for
thermal management and provides microscopic insight into deciphering the
anomalous $\kappa$ of layered 2D structures.

Materials Science