Quantum Interference Theory of Magnetoresistance in Dirac Materials

Magnetoresistance in many samples of Dirac semimetals and topological insulators displays nonmonotonic behavior over a wide range of magnetic fields. Here a formula of magnetoconductivity is presented for massless and massive Dirac fermions in Dirac materials due to quantum interference of Dirac fermions in scalar impurity scattering potentials. It reveals a striking crossover from positive to negative magnetoresistivity, uncovering strong competition between weak localization and weak antilocalization in multiple Cooperon channels at different chemical potentials, effective masses, and finite temperatures. This work sheds light on the important role of strong coupling of the conduction and valence bands in the quantum interference transport in topological nontrivial and trivial Dirac materials.

Figure 1

The dimensionless Cooperon gap $z_i$ and the summarized dimensionless weighting factor ${\mathcal{F}}_i$ as a function of the chemical potential $\mu$ for (a) topological insulator ($mb>0$), (b) trivial insulator ($mb<0$), and (c) Dirac semimetal ($mb=0$). and the mass term(a) $m{v}^2=-0.001\mathrm{eV}$, (b) 0.001 eV, and (c) 0. ${\mathcal{F}}_{\mathrm{tot}}$ is defined as ${\mathcal{F}}_{\mathrm{tot}}=\sum _i{\mathcal{F}}_i$. The model parameters are fixed for all calculations in this Letter to be $b{\hslash }^2=-18\mathrm{eV}{\AA }^2$, $\hslash v=1\mathrm{eV}\AA$.

Figure 2

(a) The quantum correction to conductivity in units of $e^2/(2\pi h{\ell }_e)$ as a function of chemical potential $\mu$ for several different Dirac mass $m$. The magnetoconductivity in (b) topological nontrivial phase ($m{v}^2=-0.02\mathrm{eV}$), (c) topological trivial phase ($m{v}^2=0.02\mathrm{eV}$), and (d) Dirac semimetal ($m{v}^2=0$) for several different chemical potentials. Insets: Enlarged view for small magnetic field denoted by the pink square region; the green lines and the yellow lines represent the weak field asymptotic $\alpha \zeta (\frac{1}{2},\frac{1}{2})(e^2/4\pi h{\ell }_B)$ with $\alpha =1$ and 2, respectively. The coherent length ${\ell }_{\varphi }=100{\ell }_e$ is much larger than the mean free path ${\ell }_e=20\mathrm{nm}$.

Figure 3

(a) The relative longitudinal MR of a ${\mathrm{Cd}}_2{\mathrm{As}}_3$ sample. The measured data (open squares) are extracted from Fig. 2(b) in Ref. [15] and the solid lines are fitted by using Eq. (4) at different temperature $T$. (b) The temperature dependence of the fitted coherent length ${\ell }_{\varphi }$ (open squares). The red straight line indicates the temperature-dependent coherent length ${\ell }_{\varphi }\propto T^{-0.75}$ arising from electron-electron interaction as predicted in Ref. [34]. (c) The relative MR as a function of temperature at several magnetic field strengths. The calculation parameters for the solid lines are ${\rho }_0=20.2\mathrm{m}\mathrm{\Omega }\mathrm{cm}$, ${\ell }_e=8.5\mathrm{nm}$, ${\eta }^2=0.268$, and ${\ell }_{\varphi }=472(T/2\mathrm{K}{)}^{-3/4}\mathrm{nm}$.