Tunable quantum interference effect on magnetoconductivity in few-layer black phosphorus

In this Rapid Communication, we develop a systematic weak localization and antilocalization theory fully considering the anisotropy and Berry phase of the system, and apply it to various phases of few-layer black phosphorus (BP), which has a highly anisotropic electronic structure with an electronic gap size tunable even to a negative value. The derivation of a Cooperon ansatz for the Bethe-Salpeter equation in a general anisotropic system is presented, revealing the existence of various quantum interference effects in different phases of few-layer BP, including a crossover from weak localization to antilocalization. We also predict that the magnetoconductivity at the semi-Dirac transition point will exhibit a nontrivial power-law dependence on the magnetic field, while following the conventional logarithmic field dependence of two-dimensional systems in the insulator and Dirac semimetal phases. Notably, the ratio between the magnetoconductivity and Boltzmann conductivity turns out to be independent of the direction, even in strongly anisotropic systems. Finally, we discuss the tunability of the quantum corrections of few-layer BP in terms of the symmetry class of the system.

Figure 1

Electronic structure of few-layer BP in the (a) insulator phase, (b) semi-Dirac transition point (SDTP), and (c) Dirac semimetal (DSM) phase. (d) The Fermi surfaces of the DSM phase. At a sufficiently low Fermi energy (lower panel), a momentum state has time-reversed counterparts both within the node and in the opposite node. Thus, the quantum interference effect is contributed by two types of scatterings: intranode (red dashed arrow) and internode (blue dashed arrow) scatterings. As the Fermi energy increases (upper panel), the Fermi surface is distorted and the time-reversal symmetry around a node is broken. Thus, the quantum interference effect via intranode scattering is suppressed.

Figure 2

Feynman diagrams describing the corrections to the dc conductivity. (a) The current-current correlation function supplemented with the ladder vertex correction gives results equivalent to the Boltzmann transport theory. (b) The ladder vertex correction satisfies the self-consistent Dyson's equation. (c) The quantum correction to the dc conductivity is mostly contributed by a bare Hikami box and two dressed Hikami boxes. (d) The Cooperon operator obeys the self-consistent Bethe-Salpeter equation.

Figure 3

Ratio of the field-induced change in the magnetoconductivity to the Boltzmann conductivity for various phase coherence lengths with the mean free path ${\ell }_e=10\mathrm{nm}$. The ratios for the WL correction in (a) the insulator phase and (b) SDTP are plotted in units of ${\alpha }_1=n_{\mathrm{imp}}{V}_{\mathrm{imp}}^2/{\hslash }^2{v}_y^2$, where $V_{\mathrm{imp}}$ denotes the strength of the impurity potential. As for the DSM phase, (c) intranode scattering induces the WAL effect, (d) while internode scattering leads to the WL effect. Both corrections are plotted in units of ${\alpha }_2=n_{\mathrm{imp}}{V}_{\mathrm{imp}}^2/{\hslash }^2{v}_x{v}_y$. As for the field dependence, the black dashed lines denote the $\pm lnB$ dependence, while the red dashed line and blue dashed line in (b) represent the $B^{\nu }$ dependence with $\nu \approx 0.31$ and $B^{2/3}$ dependence, respectively. The exponent $\nu$ is not universal but depends on the system parameter.