Strain-engineered widely tunable perfect absorption angle in black phosphorus from first principles

Using the density functional theory of electronic structure, we compute the anisotropic dielectric response of bulk black phosphorus subject to strain. Employing the obtained permittivity tensor, we solve Maxwell's equations and study the electromagnetic response of a layered structure comprising a film of black phosphorus stacked on a metallic substrate. Our results reveal that a small compressive or tensile strain, $\approx 4\%$, exerted either perpendicular or in the plane to the black phosphorus growth direction, efficiently controls the epsilon-near-zero response and allows perfect absorption tuning from low angle of the incident beam $\theta =0^{\circ }$ to high values $\theta \approx {90}^{\circ }$ while switching the energy flow direction. Incorporating the spatially inhomogeneous strain model, we also find that for certain thicknesses of the black phosphorus, near-perfect absorption can be achieved through controlled variations of the in-plane strain. These findings can serve as guidelines for designing largely tunable perfect electromagnetic wave absorber devices.

Figure 1

Schematic configuration involving a slab of bulk black phosphorus of thickness $d$ adjacent to a metallic substrate. The black phosphorus layer is exposed to a transverse magnetic field from the vacuum region where the incident electric field is polarized in the $x-z$ plane, and the magnetic field is polarized along $y$. The incident wave vector ${\mathit{k}}_0$ makes an angle $\theta$ with the $z$ axis. The tensile or compressive strains are applied in the $x-y$ plane and along the $z$ direction (note that the arrows marking applied strain can be reversed corresponding to compressive and tensile strain, respectively). The crystallography principal directions are set as follows: $x\equiv a, y\equiv b$, and $z\equiv c$.

Figure 2

Top row: (a),(b) Computed frequencies ${\omega }_p$ from DFT that lead to an ENZ response in bulk black phosphorus as a function of strain ${\epsilon }_{zz}$. The system is unstrained in the plane of the sample (${\epsilon }_{\parallel }=1.0$). Bottom row: (c),(d) The same as a function of strain ${\epsilon }_{\parallel }$ where the system is unstrained in the direction normal to the plane of the sample (${\epsilon }_{zz}=1.0$).

Figure 3

Directional control of energy flow through strain: Two-dimensional maps illustrating the behavior of the time-averaged Poynting vector for a plane wave incident on a black phosphorus film with metallic substrate. The incident angle corresponds to $\theta =34.1^{\circ }$, and the frequency is set at $20.7\mathrm{eV}$. In (a) the strain parameters correspond to ${\epsilon }_{\parallel }=1.04$, leading to perfect absorption, while for (b) the system is unstrained (${\epsilon }_{\parallel }=1$). For both cases there is no strain in the $z$ direction (${\epsilon }_{zz}=1$). The film thickness is $d=5.42\times {10}^{-3}\mu \mathrm{m}$. The interface separating the black phosphorus from the vacuum region is located at $z=0$ and the vacuum region corresponds to $z<0$.

Figure 4

(a) Absorptance as a function of the incident angle $\theta$ for three in-plane strains ${\epsilon }_{\parallel }$, as shown in the legend. The frequency is set at $\omega =22.6\mathrm{eV}$. (b),(c) The corresponding real and imaginary parts of the permittivity component ${\varepsilon }_{1z}$ (obtained by DFT) are shown as a function of the incident wave frequency.

Figure 5

Color maps representing the absorptance as a function of the angle of incidence and thickness of a black phosphorus film with metallic substrate. Six different strains are considered as labeled. Top row: The parameters used correspond to the strains and frequencies calculated in Figs. 2 and 2 where there is only out-of-plane strain (${\epsilon }_{\parallel }=1$). Bottom row: Only in-plane strain is present (${\epsilon }_{zz}=1$) and the frequency is set to $\omega =22.6\mathrm{eV}$.

Figure 6

Inhomogeneous strain in the black phosphorous is accounted for by using a linear $z$-dependent model. The spatially inhomogeneous strain develops along the $z$ direction and it varies linearly (red lines) along the strain direction from a maximum (horizontal dashed line) at locations $z=0$ and $z=d$ to zero strain at points marked by the vertical small indicators at $z=d/6,d/3,d/2,2d/3,5d/6$. The pertaining curves to these five linear models are marked by 1,2,3,4,5. The arrows portray the direction of applied strain that propagates linearly inside the BP region confined between $z=0$ and $z=d$.

Figure 7

The permittivity of BP as a function of compressive strain ${\epsilon }_{zz}$ from 0 to $-10\%$. The panels (a) and (b) show the real and imaginary parts of ${\varepsilon }_{1z}$ whereas the panels (c) and (d) are the real and imaginary part of ${\varepsilon }_{1x}$. The in-plane strain spans the strain-free, compressive, and tensile strain regimes; ${\epsilon }_{\left|\right|}=1.0,0.94,0.90,1.06,1.10$.

Figure 8

Absorptance vs the incident angle $\theta$ of electromagnetic wave. The frequency of the electromagnetic wave is set to $\omega =22.6$ eV. The in-plane strain increases from the leftmost column to the rightmost column, i.e., ${\epsilon }_{\left|\right|}=-10\%,-6.0\%,0.0\%,+6.0\%,+10\%$ in (a),(f), (b),(g), (c),(h), (d),(i), and (e),(j), respectively. In the top row panels, three different thicknesses of BP are set; $\mathit{d}=0.5\mu \mathrm{m},0.3\mu \mathrm{m},0.1\mu \mathrm{m}$, when model 1 is implemented. In the bottom row panels, the thickness of BP is set to $\mathit{d}=0.3\mu$ and the five different models (marked by 1,2,3,4,5) are implemented.

Figure 9

Electromagnetic modes for the system in Fig. 1 with the same parameter values as used in Fig. 5. The red curve depicts the fast-wave ($k_{0x}/k_0<1$) waveguide modes, while the green curves correspond to the impedance matched modes (overlapping with the red curve). As the thickness increases, eventually radiative leaky waves emerge (blue curve). The angle $\theta$ is determined from the propagation constants $k_{0x}$ via $\theta =arcsin(k_{0x}/k_0)$. For reference, the black region is extracted from the high absorptance ($>95\%$) data of Fig. 5.

Figure 10

Electronic band structures and total densities of states for bulk black phosphorus. The strain parameters are as follows: (a) ${\epsilon }_{\parallel }=1.0$, (b) ${\epsilon }_{\parallel }=0.9$, and (c) ${\epsilon }_{\parallel }=1.1$ for in-plane strain and (d) ${\epsilon }_{zz}=0.9$ and (e) ${\epsilon }_{zz}=1.1$ for out-of-plane strain. In (a)–(c) the system is unstrained along $z$ (${\epsilon }_{zz}=1$), and there is no in-plane strain in (d) and (e) (${\epsilon }_{\parallel }=1.0$). The Fermi energy is located at the energy $E=0$. The axes labels of the top row panels are the same as for the bottom row panels.

Figure 11

Band structure and permittivity components for unstrained BP using PBE and GLLBsc functionals. (a) The band structure is shown for PBE and GLLBsc by the blue and red curves, respectively. (b),(c) Plots of the permittivity tensor components (${\varepsilon }_{1x,1y,1z}$) as a function frequency. The solid and dashed curves show PBE and GLLBsc results, respectively. The real (Re) and imaginary (Im) parts are displayed by the blue and red colors, respectively.

Figure 12

Absorptance ($A$) vs the incident angle $\theta$ for three in-plane strain values ${\epsilon }_{\left|\right|}=1.1,1.0,0.9$. The frequency of the incident wave is set at $\omega =22.6$ eV. The corresponding permittivity components of the dashed and solid lines are obtained using the GLLBsc and PBE functionals, respectively.