Analytical study on strain tunable electronic structure and optical transitions in armchair black phosphorene nanoribbons.

We theoretically investigate the electronic structure and optical absorption spectrum of armchair-edged black phosphorene nanoribbons (APNRs) with and without uniaxial strain based on the tight-binding Hamiltonian and Kubo formula. We analytically obtain the energy spectrum and wavefunction, and reveal the band gap scaling law as $1/(N+1)^2$ for APNRs in the presence and absence of uniaxial strain, where $N$ is the number of armchair dimer across the ribbon. We find the band gap of APNRs linearly increases (decreases) with increasing in-plane uniaxial tensile (compressive) strain $\varepsilon_{x/y}$, but shows contrary dependence on the out-of-plane uniaxial strain $\varepsilon_{z}$. The effective mass versus strain exhibits the same behavior to that of band gap but with nonlinear dependence. Under an incident light linearly-polarized along the ribbon, we demonstrate that the inter-band optical transitions obey the selection rule $\Delta n=n-n^{\prime}$=0, but the intra-band transitions are forbidden for both pristine and strained APNRs originating from the orthogonality between the transverse wavefunctions of the sublattices belonging to different subbands. Importantly, the transverse electric field or impurities can release the optical selection rules by breaking the wavefunction orthogonality, which results in that the optical transitions between any subbands are all possible. Our findings provide further understanding on the electronic and optical properties of APNRs, which may pave the way for designing optoelectronic devices based on phosphorene.