Phonon-limited carrier mobility in monolayer black phosphorus

We calculate an electron-phonon scattering and intrinsic transport properties of black phosphorus monolayer using tight-binding and Boltzmann treatments as a function of temperature, carrier density, and electric field. The low-field mobility shows weak dependence on density and, at room temperature, falls in the range of 300–1000 ${\mathrm{cm}}^2$/Vs in the armchair direction and 50–$120{\mathrm{cm}}^2$/Vs in the zigzag direction with an anisotropy due to the effective mass difference. At high fields, drift velocity is linear with field up to 1–2 V/$\mu \mathrm{m}$ reaching values of ${10}^7$ cm/s in the armchair direction, unless self-heating effects are included.

Figure 1

(a) Black phosphorus monolayer crystal structure showing tight-binding hopping integrals. (b) Brillouin zone in the reciprocal space. (c) Phonon dispersion according to our model (blue solid curves) fitted to the DFT calculations in Ref. [15] (red dashed curves). (d) Phonon contributions to the weighted average scattering rate in Eq. (4) at $T=300$ K and $n={10}^{13}{\mathrm{cm}}^{-2}$ for the armchair direction (red) and zigzag direction (green). The two peaks at 320 and 465 ${\mathrm{cm}}^{-1}$ correspond to the out-of-plane and in-plane vibrations near the $\mathrm{\Gamma }$ point, correspondingly.

Figure 2

Band structure using tight-binding model with $t_1=-1.15$ eV and $t_2=3.07$ eV (blue solid curves) fitted to the $GW$ calculations from Ref. [14] (red solid curves).

Figure 3

Calculated temperature dependence of the low-field mobility in black phosphorus along (a) the armchair $x$ and (b) the zigzag $y$ directions at $n={10}^{12},5\times {10}^{12}$, and ${10}^{13}{\mathrm{cm}}^{-2}$ in red, blue, and green solid curves, correspondingly. The black dashed curves show $1/T^2$ scaling to guide the eye. (c) Inverse mobility at $n=5\times {10}^{12}{\mathrm{cm}}^{-2}$ for $x$ and $y$ directions along with the best fit to Eq. (7), shown by the black dashed curves.

Figure 4

Density dependence of mobility at $T=300$ K are shown along the armchair $x$ (red) and zigzag $y$ (blue) directions, correspondingly.

Figure 5

Field dependence of drift velocity along the armchair $x$ (a) and zigzag $y$ (b) directions. Isothermal curves for $T=300$ K are shown for $n={10}^{12}{\mathrm{cm}}^{-2}$ (red) and $n={10}^{13}{\mathrm{cm}}^{-2}$ (blue). In (a) we show isothermal curve for $T=750$ K (cyan) and calculations including self-heating effect on 300 nm ${\mathrm{SiO}}_2$ at $n={10}^{13}{\mathrm{cm}}^{-2}$ (green). The black dashed curves are best fits to Eq. (8).

Figure 6

Monolayer black phosphorus DFT band gap dependence on strain applied in the perpendicular to the plane direction (001). This strain direction keeps the in-plane P-P bond length $r_1$ constant. The solid line corresponds to the best fit to Eq. (A1) with $g_2=2.46$ eV/Å.

Figure 7

Monolayer black phosphorus DFT band gap dependence on strain applied in the in-plane direction (110). This strain direction keeps the in-plane bond angle constant. The solid line corresponds to the best fit to Eq. (A2) with $g_1=1.53$ eV/Å for a fixed $g_2=2.46$ eV/Å.