Ab initio investigation on exciton-phonon interaction and exciton mobility in black phosphorene

Black phosphorene (BP) is an emerging two-dimensional material that has potentially wide applications in nano- and optoelectronics. Despite the strong excitonic effects observed in few-layer BP, there is still a lack of understanding regarding exciton-phonon (ex-ph) interaction and exciton mobility. In this work, using $\mathit{GW}$ plus real-time Bethe-Salpeter equation approach, it is revealed that the out-of-plane acoustic phonon mode $\mathit{ZA}$, and the optical phonon modes $A_g^1$ and $A_g^2$ couple strongly with the exciton. Among them, the $A_g^2$ mode has a uniquely significant influence on the oscillator strength. Driven by these ex-ph interactions, the exciton becomes more localized accompanied with strengthened binding and the absorption spectrum undergoes a notable renormalization at 300 K. In the framework of deformation potential theory, the exciton mobility of monolayer BP (MLBP) is found to be largely suppressed compared to the free-charge carriers, showing quasi-one-dimensional character. This work provides rich knowledge of ex-ph interaction and exciton mobility in MLBP at the ab initio level, offering insights for the understanding and controlling the exciton behavior in BP-based devices.

Figure 1

(a) Atomic structure of MLBP. (b) $\mathit{GW}$ (thick lines) and DFT bands (thin lines) of MLBP. (c) Polarized absorption spectra of MLBP in equilibrium from $\mathit{GW}$-RPA without e-h interaction (dashed lines) and $\mathit{GW}$-BSE with e-h interaction (solid lines) under AR direction (top) and ZZ direction (bottom) polarized light.

Figure 2

(a) Time evolution of KS electronic-state energies of MLBP. (b) Top: FT spectra of evolution of VBM and CBM. Bottom: comparison between sum of FT spectra of CBM and VBM and FT spectrum of band gap. (c) Time evolution of band gap and energy of $A$ exciton. (d) FT spectra of evolution of band gap and $A$ exciton. (e) Corresponding frequencies and vibrational schematic diagrams of phonon modes appearing in FT spectra.

Figure 3

(a) Time evolution of binding energy and ${|A_{\mathrm{CBM}-\mathrm{VBM}}|}^2$ of $A$ exciton. (b) FT spectra of $A$-exciton binding energy and ${|A_{\mathrm{CBM}-\mathrm{VBM}}|}^2$. Amplitudes of two FT spectra have been normalized to same level. (c), (d) Modulus squared of the real-space $A$-exciton wave function at 0 K (c) and averaged over AIMD trajectory at 300 K (d). Rectangle frame corresponds to size of $18\times 18\times 1$ supercell. Two dotted ellipses in (c) and (d) both refer to outline of $A$ exciton at zero temperature. And, isosurfaces in (c) and (d) are plotted using same value.

Figure 4

(a), (b) Time evolution of square of transition dipole moment from VBM to CBM ${|\langle CBM\left|\vec{r}\right|VBM\rangle |}^2$ and corresponding FT spectrum. (c), (d) Time evolution of oscillator strength of $A$ exciton and corresponding FT spectrum. $Y$-axis coordinates in (a) and (c) denote relative change of square of transient transition dipole or exciton oscillator strength compared to result of equilibrium structure. (e) VBM orbital distribution variations with $A_g^2, A_g^1$, and $B_{2\mathrm{g}}$ phonon mode excitation. To display variation clearly, we introduce considerably large displacement ($\delta =\pm 0.3\AA$ within primitive cell). $\delta =0$ refers to equilibrium structure. $\delta >0$ and $\delta <0$ suggest atomic displacements aligned to two opposite phonon vibrational directions. (f) Six different snapshots of absorption spectra along AIMD trajectory. Red lines denote exciton oscillator strengths. (g) Top: Averaged absorption spectrum of AIMD ensemble at 300 K. Bottom: Absorption spectrum at zero temperature. Shaded areas are fitted Lorentz functions of first peak.

Figure 5

$\mathit{GW}$ band gap and energy of $A$ exciton under different lattice deformation along AR direction (dots and solid lines) and ZZ direction (crosses and dashed lines) calculated by $GW+\mathrm{BSE}$. Positive values of deformation correspond to tensile strain while negative values correspond to compressive strain. Dots (or crosses) represent calculated values and solid (or dashed) lines are linearly fitted curves.