Anomalous magneto-optical response of black phosphorus thin films

We theoretically investigate the Landau levels (LLs) and magneto-optical conductivity (MOC) of black phosphorus thin films under a perpendicular magnetic field based on an effective $\mathit{k}·\mathit{p}$ Hamiltonian and linear-response theory. We obtain the analytical expression for LLs, which agrees well with the numerical calculations, and find that the LLs sublinearly depend on the magnetic field and LL index. By using the Kubo formula, we evaluate the longitudinal and Hall optical conductivities as functions of the photon energy and the magnetic field. The analytical optical transition matrix elements reveal unusual selection rules for the interband (intraband) optical transitions between the LLs $\mathrm{\Delta }n=0,\pm 2$ $\left(\pm 1,\pm 3\right)$. The MOC shows strongly anisotropic behaviors of the band structure. For the interband transition, the MOC for linearly polarized light along the armchair direction is three orders of magnitude larger than that along the zigzag direction. Interestingly, we find a beating pattern in the interband MOC due to the interference among the three kinds of optical transitions. For the intraband transition, the MOC can be used to determine the band parameters such as the effective masses and the interband coupling at zero magnetic field. Our results about the MOC can also be applied to the monolayer black phosphorus.

Figure 1

(a) Schematic illustration of magneto-optical setup on BP TF structure. The wavy lines represent linearly polarized light. The lattice structure of the BP TF is shown on the right. (b) Band structure for 10-nm BP thin film obtained from self-consistent calculations. (c) The probability distribution and band profile for the lowest sub-band along the $z$ direction for 10-nm BP TFs obtained from self-consistent calculation. The carrier density (hole doped) used in the self-consistent calculation is $3.0\times {10}^{12}{\mathrm{cm}}^{-2}$. (d) Schematic illustration of the dominated interband transitions. The interband transitions mainly occur when LL index changes $\mathrm{\Delta }n=0,\pm 2$ [Eq. (18)].

Figure 2

LLs as a function of the magnetic field for the lowest sub-bands. (a) and (b) represent for conduction and valence band, respectively. The red solid (blue dashed) lines represent the result obtained from numerical (perturbed) calculation. The green dash-dotted lines are eigenvalues of $H_0$ given in Eq. (5). The number of basis function used in the calculation is 200 to get convergent numerical results. (c) LL spacings of valence band (in unit of $E_{0,v}-E_{v,1})$ as a function of LL index. (d) Derivative of LL $\left(d{E}_{v,n}/dB\right)$ in valence band as a function of magnetic field.

Figure 3

(a) The real part of the longitudinal magneto-optical conductivity and (b) imaginary part of the Hall magneto-optical conductivity (in units of ${\sigma }_0=2e^2/h)$ as a function of the photon energy under the magnetic field $B=10$ T and level broadening factor $\mathrm{\Gamma }=0.15$ meV.

Figure 4

The real part of longitudinal magneto-optical conductivity for linearly polarized light along the $x$ direction as a function of magnetic field under photon energy $\hslash \omega =0.8$ eV and Landau-level broadening factor $\mathrm{\Gamma }=0.5$ meV. (a)–(c) represent the results for the lowest three sub-bands and (d) donates the total magneto-optical conductivity.

Figure 5

The real part of longitudinal magneto-optical conductivity for linearly polarized light along the $y$ direction as a function of magnetic field under photon energy $\hslash \omega =0.8$ eV and Landau-level broadening factor $\mathrm{\Gamma }=0.5$ meV. (a)–(c) represent the results for the lowest three sub-bands and (d) donates the total magneto-optical conductivity.

Figure 6

The imaginary part of Hall magneto-optical conductivity for linearly polarized light as a function of magnetic field under photon energy $\hslash \omega =0.8$ eV and Landau-level broadening factor $\mathrm{\Gamma }=0.5$ meV. (a)–(c) represent the results for the lowest three sub-bands and (d) donates the total magneto-optical conductivity.

Figure 7

The real part of the longitudinal optical conductivity for intraband transition as a function of photon energy under magnetic field $B=20$ T and level broadening factor $\mathrm{\Gamma }=0.1$ meV, for (a) $|n+1\rangle \to |n\rangle$ and (b) $|n+3\rangle \to |n\rangle$ transitions, respectively.