Strain-controlled fundamental gap and structure of bulk black phosphorus

We study theoretically the structural and electronic response of layered bulk black phosphorus to in-layer strain. Ab initio density functional theory (DFT) calculations reveal that the strain energy and interlayer spacing display a strong anisotropy with respect to the uniaxial strain direction. To correctly describe the dependence of the fundamental band gap on strain, we used the computationally more involved GW quasiparticle approach that is free of parameters and is superior to DFT studies, which are known to underestimate gap energies. We find that the band gap depends sensitively on the in-layer strain and even vanishes at compressive strain values exceeding $\approx 2$%, thus suggesting a possible application of black P in strain-controlled infrared devices.

Figure 1

(a) Ball-and-stick model of the structure of bulk black phosphorus in top and side views. (b) Fractional change of the interlayer distance $a_3$ as a function of the in-layer strain $\epsilon$ along the ${\vec{a}}_1$ and ${\vec{a}}_2$ directions. (c) Dependence of the strain energy $\mathrm{\Delta }E$ per unit cell on the in-layer strain along the ${\vec{a}}_1$ and ${\vec{a}}_2$ directions.

Figure 2

Electronic band structure (left panels) and density of states (right panels) of bulk black phosphorus (a) without strain, (b) when stretched by 1% along the ${\vec{a}}_1$ direction, and (c) when stretched by 2% along the ${\vec{a}}_1$ direction. GW results are shown by the solid black lines. DFT-PBE results, which underestimate the band gap, are shown by the dashed red lines.

Figure 3

(a) Correlation between quasiparticle (GW) energies $E_{\mathrm{qp}}$ and Kohn-Sham energy values $E_{\mathrm{PBE}}$, obtained using the DFT-PBE functional, for states along high symmetry lines in the Brillouin zone. The straight lines in the valence and conduction band regions are drawn to guide the eye. The results represent bulk black phosphorus subject to strain values $\epsilon \left({\vec{a}}_1\right)=1\%$ and $\epsilon \left({\vec{a}}_2\right)=0$. The Fermi level is at zero energy. (b) Dependence of the quasiparticle (GW) electronic band gap $E_g$ on the in-layer strain applied along the ${\vec{a}}_1$ and ${\vec{a}}_2$ directions.