Coexistence of size-dependent and size-independent thermal conductivities in phosphorene

Thermal conductivity of single layer phosphorene is investigated by combining density functional calculations and the Peierls-Boltzmann transport equation. Differing from isotropic and divergent thermal conductivities in two-dimensional graphene and ${\mathrm{MoS}}_2$, a compelling coexistence of size-dependent and size-independent thermal conductivities is discovered for single layer black phosphorus (BP) along zigzag (ZZ) and armchair (AM) directions, respectively. Additionally, thermal conductivities in single layer phosphorene are found to be highly anisotropic because of orientation dependent group velocities and phonon relaxation times; e.g., thermal conductivities at 300 K are 83.5 and 24.3 W/m K along ZZ and AM directions, respectively, for single layer BP with a size of $10\mu \text{m}$.

Figure 1

Top and side view of single layer BP. Red arrows represent lattice vectors of the primitive cell along ZZ and AM directions. Heavy and light colors denote phosphorus atoms in different plane.

Figure 2

Phonon dispersion of single layer BP.

Figure 3

Thermal conductivity ($\kappa$) as a function of (a) the size of q-point mesh ($N\times N\times 1$). (b) Thermal conductivity along the ZZ direction contributed from ZA, TA, LA, and optical phonon branches as a function of N.

Figure 4

Dependence of phonon relaxation time on frequency for (a,b) ZA, (c,d) TA, and (e,f) LA phonons. Top and bottom panels denote the wave vector $\mathbf{q}$ approaching the Brillouin-zone center along ZZ ($\Gamma -Y$) and AM ($\Gamma -X$) directions, respectively.

Figure 5

Scaling exponent of phonon relaxation time ($\tau$) on frequency ($\omega$), $n$, as a function of the orientation of wave vector $\mathbf{q}$ approaching the Brillouin-zone center, namely, $\tau \sim {\omega }^{-n}$. Exponent $n$ for (a) ZA, (b) TA, and (c) LA phonons are plotted in left, central, and right panels, respectively. The angles of 0 and ${90}^{\circ }$ correspond to ZZ ($\Gamma -Y$) and AM ($\Gamma -X$) directions in the Brillouin zone, respectively. The solid lines in cyan denote $n=2$, while magenta solid lines are plotted to guide the eye.

Figure 6

Thermal conductivity ($\kappa$) as a function of length (a) and temperature (b). Open squares in panel (b) are experimental thermal conductivities for bulk BP [44].

Figure 7

Omega resolved (a) thermal conductivities [$\kappa \left(\omega \right)$] and (b) average group velocities [$V\left(\omega \right)$] along ZZ and AM directions.