Model for charge carrier spectra in topological semimetals of the TaAs family

Abstract

We propose a four-band model describing the electron energy spectra near the Weyl points in the topological semimetals of the TaAs family (TaAs, TaP, NbAs, NbP). This model takes into account the fact that these Weyl points result from the band-contact lines which would exist in the mirror-reflection planes of these materials if the spin-orbit interaction were absent in them. Within this model, we obtain conditions for the existence of the Weyl points, determine their positions in the Brillouin zone, and derive the explicit formula for dispersion of the bands along the straight line connecting the two close Weyl points with opposite topological charges. Using NbP as an example, the values of the parameters defining the model spectrum are found. The obtained results show that for the semimetals of the TaAs family, the charge-carriers spectrum in the vicinity of the two close Weyl points can be analyzed without complex band-structure calculations.

Figures (9)

• Figure 1: Cross section (hexagon) of the first Brillouin zone of NbP by the mirror-reflection plane $p_y=0$, and the band-contact rings in this plane. The red rhombi depict the projections of the Weyl points onto this plane. The upper right ring (enclosed by the dashed square) is also shown in an enlarged scale. For this ring, the W1 and W2 Weyl points are marked, near which the energy bands were calculated in Ref. lee (see subsequent Figs. \ref{['fig2']}, \ref{['fig3']}, \ref{['fig5']}-\ref{['fig8']}, and Tables \ref{['tab1']}, \ref{['tab3']}). For these Weyl points, the angle $\theta$ between the $p_z$ axis and the $p_{\|}$ direction is indicated.
• Figure 2: Dispersion of the four bands along the $p_y$ direction ($k_y=p_y/\hbar$) at $p_x=p_x^W$, $p_z=p_z^W$, Eq. (\ref{['17']}). The point ($p_x^W$,$p_z^W$) in the mirror-reflection plane $p_y=0$ is the projection of the two Weyl points onto this plane. The bands are plotted for $m_3=m_5=0$ and the values of the other parameters presented in Table \ref{['tab2']}. (Interestingly, the same bands are obtained if the values of $m_2$ and $m_4/m_6$ are replaced by $m_2=10$ meV and $m_4/m_6=0.578$.) The red cycles are the data of Fig. 8a in Ref. lee for the W1 point in NbP (see Fig. \ref{['fig1']}) ; $\varepsilon_i$ mark the energy bands. All the energies are measured from the Fermi level (red dashed line).
• Figure 3: Dispersion of the bands along the $p_x$ axis ($k_x=p_x/\hbar$) at $p_z=2\pi\hbar/c$, $p_y=p_y^{W1}$ for the parameters from Table \ref{['tab2']}. The solid lines are plotted with Eqs. (\ref{['5']})--(\ref{['7']}). The $p_x$ is measured from $p_x^{W1}$. The red cycles are the data of Fig. 8b in Ref. lee for the W1 point in NbP (see Fig. \ref{['fig1']}).
• Figure 4: Dispersion of the bands along the $p_x$ axis at $p_z=2\pi\hbar/c$, $p_y=0$ (i.e., along the $Z-S$ axis of the Brillouin zone) for $\tilde{m}$, $\bar{\varepsilon}_0$, $t_2$ from Table \ref{['tab2']} but with $m_2=10$ meV and $m_4/m_6=0.578$ (the other parameters $m_1=9$ meV, $a=-0.54\times 10^5$ m/s, $a'=2.46\times 10^5$ m/s, are found from a fit like in Fig. \ref{['fig3']}). The lines are plotted with Eqs. (\ref{['5']})--(\ref{['7']}). The $p_x$ is measured from $p_x^{W1}$. Note that the additional Weyl point is visible in the plane $p_y=0$.
• Figure 5: Dispersion of the two crossing bands along the $p_z$ direction ($k_z=p_z/\hbar$) in NbP at $p_x=p_x^{W1}$, $p_y=p_y^{W1}$, Eq. (\ref{['25']}). The black solid lines show the bands for the parameters presented in Table \ref{['tab2']} and for $R=0.2(2\pi\hbar/a)$, $d^2\varepsilon_0/dp_{\|}^2\approx 0.37/m$, and $v_{\|,\|}=0.79\times 10^4$ m/s (which corresponds to $v_{\|}=0.95\times 10^4$ m/s) where $a=3.334$ Å, and $m$ is the free-electron mass. Note that the black solid lines remain practically unchanged if $v_{\|,\|}< 10^4$ m/s. The red cycles are the data of Fig. 8c in Ref. lee. These data were obtained for the W1 point marked in Fig. \ref{['fig1']}. All the energies are measured from the Fermi level (red dashed line).