Ab initio investigation of magnetic transport properties by Wannier interpolation

We present an efficient ab initio approach for the study of magnetic transport properties based on the Boltzmann equation with the Wannier interpolation scheme. Within the relaxation-time approximation, band-resolved electric conductivity under a finite magnetic field is obtained and the historical motion of the electron wave packet in reciprocal space is determined. As a typical application of this method, we have calculated the electric conductivities of ${\text{MgB}}_2$ under finite magnetic fields. Multiband characters for the individual bands are revealed, and the field dependence of the conductivity tensor is studied systematically with the field orientated parallel and normal to the $c$ axis, respectively. The obtained historical motion is employed to simulate directly the cyclotron motion in the extremal orbit and determine the corresponding effective mass. Moreover, this approach is further exploited to calculate the Hall coefficient in the low-field limit without the complicated computation for the second $\mathbf{k}$ derivative of the band.

Figure 1

(Color online) The calculated Fermi surface of ${\text{MgB}}_2$ and the extremal orbits labeled for the magnetic field along the $c$ axis.

Figure 2

(Color online) ${\sigma }_{xx}/\tau$ as a function of $B\tau$ of the four bands of ${\text{MgB}}_2$ with the magnetic field along the $c$ axis.

Figure 3

(Color online) Historical track of the electron wave packet in reciprocal space. The initial positions are (a) $\mathbf{k}\left(0.065,0.0,0.0\right)$ in orbit 1 and (b) $\mathbf{k}\left(0.444,0.0,0.0\right)$ in orbit 3, respectively. $\mathbf{k}$ points are expressed as $\mathbf{k}\left(k_1,k_2,k_3\right)$ in the unit of reciprocal lattice vectors. There are 4000 points on each curve. $k_3$ is fixed in the $\Gamma$ plane during the simulation. The abscissa ${\omega }_{e}t$ is related to $Bt$ as ${\omega }_{e}t=\frac{e}{m_e}Bt$ where $m_e$ is the mass of a free electron.

Figure 4

(Color online) The partial Hall conductivity ${\sigma }_{xy}^{\left(n\right)}\left(k_z\right)/{\tau }^{\left(n\right)}$ contributed by orbits between $-k_z$ to $k_z$ of the individual bands as a function of $k_z$.

Figure 5

(Color online) $B\tau$ dependences of (a) ${\sigma }_{xx}/\tau$, (b) ${\sigma }_{yy}/\tau$, and (c) ${\sigma }_{zz}/\tau$ of the four bands of ${\text{MgB}}_2$. The magnetic field is applied along the $x$ axis in the $ab$ plane.

Figure 6

The band structure of fcc Pd. The Fermi level is fixed at 0 eV.