Magnetoresistance of a three-dimensional Dirac gas

We study the transversal magnetoconductivity and magnetoresistance of a massive Dirac fermion gas. This can be used as a simple model for gapped Dirac materials. In the zero-mass limit, the case of gapless Dirac semimetals is also studied. In the case of Weyl semimetals, to reproduce the nonsaturating linear magnetoresistance seen in experiments, the use of screened charged impurities is inevitable. In this paper, these are included using the first Born approximation for the self-energy. The screening wave number is calculated using the random phase approximation with the polarization function taking into account the electron-electron interaction. The Hall conductivity is calculated analytically in the case of no impurities and is shown to be perfectly inversely proportional to the magnetic field. Thus the magnetic field dependence of the magnetoresistance is mainly determined by ${\sigma }_{xx}$. We show that in the extreme quantum limit at very high magnetic fields the gapped Dirac materials are expected to have ${\sigma }_{xx}\propto B^{-3}$ leading to ${\varrho }_{xx}\propto B^{-1}$, in contrast with the gapless case where ${\sigma }_{xx}\propto B^{-1}$ and ${\varrho }_{xx}\propto B$. At lower fields, we find that the effect of the mass term is negligible and in the region of the Shubnikov-de Haas oscillations the two systems behave almost identically. We suggest a phenomenological scattering rate that is able to reproduce the linear behavior at the oscillating region. We show that in the case of the scattering rate calculated using the Born approximation, the strength of the relative permittivity and the density of impurities affects the magnetic field dependence of the conductivity significantly.

Figure 1

Landau levels (5) of the Dirac Hamiltonian. The degeneracy of each level is 2 except the $n=0$ levels. $\mathrm{\Delta }=0.5/{\ell }_B$ and ${\ell }_B=\sqrt{1/eB}$ are used.

Figure 2

Feynman diagram for the first-order Born approximation of the self-energy [see Eq. (21)].

Figure 3

Feynman diagram for the RPA of the impurity potential. The double dashed lines are the effective impurity potentials, the single dashed lines are the bare impurity potentials, and the wavy line represents the electron-electron interaction.

Figure 4

The bubble diagram used to calculate the screening wave number in Eq. (27).

Figure 5

Density of states calculated from Eq. (36). The mass term is chosen as ${\ell }_B\mathrm{\Delta }=1$.

Figure 6

Magnetic field dependence of the chemical potential for two mass terms ${\mathrm{\Delta }}_0=0$ and ${\mathrm{\Delta }}_1=2/{\ell }_{1\mathrm{T}}\approx 50\mathrm{meV}$ and for three temperatures $T_0=0\mathrm{K}$, $T_1=30\mathrm{K}$, and $T_2=100\mathrm{K}$. The carrier density is fixed at $n_e={10}^{18}{\mathrm{cm}}^{-3}$. (a) is for the small magnetic fields with oscillating behavior and (b) is for the large magnetic field limit. The vertical lines show the magnetic fields where a new Landau level crosses the chemical potential as calculated from Eq. (40).

Figure 7

Magnetic field dependence of the screening wave number for several cases as in Fig. 6.

Figure 8

Scattering rate calculated from Eq. (46a) with indices $n=0$ and $\lambda =1$ as a function of magnetic field at zero temperature. The screening wave number is calculated using Eq. (41). The density of charge carriers is $n_e={10}^{18}{\mathrm{cm}}^{-3}$. ${\mathrm{\Delta }}_0=0$ and ${\mathrm{\Delta }}_1=2/{\ell }_{1\mathrm{T}}$.

Figure 9

Scattering rate calculated from Eq. (46a) with indices $n=0$ and $\lambda =1$ as a function of magnetic field. The density of charge carriers is $n_e={10}^{18}{\mathrm{cm}}^{-3}$. Different types of screening wave numbers are used: a constant, a linear in magnetic field, and the one calculated from Eq. (42). $\mathrm{\Delta }=0$ and $T=0\mathrm{K}$.

Figure 10

Hall conductivity ${\sigma }_{xy}$ calculated from Eq. (49) as a function of magnetic field at $T=30\mathrm{K}$. The density of charge carriers is $n_e={10}^{18}{\mathrm{cm}}^{-3}$. ${\sigma }_0/{\ell }_{1\mathrm{T}}\approx 15{\mathrm{\Omega }}^{-1}{\mathrm{cm}}^{-1}$.

Figure 11

Transverse diagonal conductivity ${\sigma }_{xx}$ calculated from Eq. (52) as a function of magnetic field at different temperatures. $\mathrm{\Delta }=0$ (top plot) and $\mathrm{\Delta }=2/{\ell }_{1\mathrm{T}}$ (bottom plot). The scattering rate is chosen phenomenologically based on the numerical results in Fig. 8. The inset shows the scattering rates used (no Landau level dependence is assumed). $T_0=0\mathrm{K}$, $T_1=50\mathrm{K}$, the density of charge carriers is $n_e={10}^{18}{\mathrm{cm}}^{-3}$ and ${\sigma }_0/{\ell }_{1\mathrm{T}}\approx 15{\mathrm{\Omega }}^{-1}{\mathrm{cm}}^{-1}$.

Figure 12

Transverse diagonal conductivity ${\sigma }_{xx}$ calculated from Eq. (52) as a function of magnetic field at zero temperature. $\mathrm{\Delta }=0$ (top) and $\mathrm{\Delta }=2/{\ell }_{1\mathrm{T}}$ (bottom). The scattering rate is calculated using Eq. (46a) using screening wave numbers calculated through Eq. (41). The inset figure shows the scattering rates used ($n=1$ Landau level). The density of charge carriers is $n_e={10}^{18}{\mathrm{cm}}^{-3}$ and ${\sigma }_0/{\ell }_{1\mathrm{T}}\approx 15{\mathrm{\Omega }}^{-1}{\mathrm{cm}}^{-1}$.

Figure 13

Magnetoresistance ${\varrho }_{xx}$ calculated from Eq. (56) as a function of magnetic field. $\mathrm{\Delta }=0$ (top) and $\mathrm{\Delta }=2/{\ell }_{1\mathrm{T}}$ (bottom). The resistivity calculated from the phenomenological result (red line) and the resistivity calculated microscopically using the first Born approximation with $\varepsilon =1$ (blue). The density of charge carriers is $n_e={10}^{18}{\mathrm{cm}}^{-3}$ and ${\sigma }_0/{\ell }_{1\mathrm{T}}\approx 15{\mathrm{\Omega }}^{-1}{\mathrm{cm}}^{-1}$.