Effective $g$ factor in black phosphorus thin films

We theoretically investigate the effective $g$ factor in the black phosphorus (BP) thin films (TFs) based on a multiband $\mathbf{k}·\mathbf{p}$ theory. We demonstrate that the effective single particle $g$ factor in pristine BP TF is anisotropic arising from its anisotropic band structure with $g_{xx}^{*}\approx g_{yy}^{*}\approx 2.0$ and $g_{zz}^{*}$ sensitively depending on the interband coupling and the band gap. The $g_{zz}^{*}$ approaches 2.0 with increasing hole doping density and gate electric field since both of them minish the interband coupling by reducing the overlap integral between the electron and hole wave functions. We also estimate the exchange interaction enhancement on the effective single particle $g$ factor by using the screened Hartree-Fock approximation. The exchange interaction enhanced $g$ factor ($g_{\text{ex}}$) shows maxima (minima) at odd (even) filling factors. The effective $g$ factor ($g^{*}$) oscillates with the increase of magnetic field and sensitively depends on the Landau level broadening as well as the gate electric field since both of them affect the interband coupling and the electron-electron interactions.

Figure 1

Schematic illustration of experimental setup on black phosphorus (BP) thin film (TF) structure, which is sandwiched by the hexagonal boron nitrogen (h-BN) TFs to avoid degeneration. The graphite back gate induces a strong electric field in the BP TF.

Figure 2

Band profile and electron density for states in the conduction (blue lines) and valence (red lines) band obtained from self-consistent calculations for hole doping density $n_h=5\times {10}^{12}{\mathrm{cm}}^{-2}$ with gate electric field (a) $E_z=0$ and (b) $E_z=0.5$ MV/cm. The dielectric function $\varepsilon =10$ and temperature $T=30$ K. Notably, the density probabilities for electron and hole ground states are both positive and plotted schematically in the figure to describe the spatial distribution of electron and hole states in the BP thin film without and with external electric fields.

Figure 3

(a) The $z$ component of $g$ factor ($g_{zz}^{*}$ ) in pristine BP TFs as a function of the number of layers; (b) the $g_{zz}^{*}$ versus the hole doping concentration $n_h$ under different BP thickness; (c) the $g_{zz}^{*}$ versus the gate induced electric field $E_z$ under hole doping concentration $n_h=3\times {10}^{12}{\mathrm{cm}}^{-2}$ with different thickness. The blue solid (red dash-dotted) line represent the results for 7.5 (10) nm BP TF.

Figure 4

(a) The exchange interaction enhancement on the $g$ factor $g_{\text{ex}}$ and (b) the effective $g$ factor $g^{*}$ as a function of filling factor $\nu$ (hole doping concentration $n_h=eB/h$) under magnetic field $B=30$ T for different LL broadenings width $\mathrm{\Gamma }={\mathrm{\Gamma }}_0\sqrt{B}$ meV with and without electric field $E_z$ (in unit of MV/cm).

Figure 5

(a) The exchange interaction enhancement on the $g$ factor $g_{\text{ex}}$ and (b) the effective $g$ factor $g^{*}$ as a function of magnetic field $B$ when the hole doping density is $2.0\times {10}^{12}{\mathrm{cm}}^{-2}$ for different LL broadenings $\mathrm{\Gamma }={\mathrm{\Gamma }}_0\sqrt{B}$ meV with and without electric field $E_z$ (in unit of MV/cm).