Anomalous Temperature Dependence of Quantum Correction to the Conductivity of Magnetic Topological Insulators

Quantum transport in magnetic topological insulators reveals a strong interplay between magnetism and topology of electronic band structures. A recent experiment on magnetically doped topological insulator ${\mathrm{Bi}}_2{\mathrm{Se}}_3$ thin films showed the anomalous temperature dependence of the magnetoconductivity while their field dependence presents a clear signature of weak antilocalization [Tkac et al., Phys. Rev. Lett. 123, 036406 (2019)]. Here, we demonstrate that the tiny mass of the surface electrons induced by the bulk magnetization leads to a temperature-dependent correction to the $\pi$ Berry phase and generates a decoherence mechanism to the phase coherence length of the surface electrons. As a consequence, the quantum correction to conductivity can exhibit nonmonotonic behavior by decreasing the temperature. This effect is attributed to the close relation of the Berry phase and quantum interference of the topological surface electrons in quantum topological materials.

Figure 1

Magnetoconductivity as a function of the temperature at different magnetic field strengths for two Mn-doped ${\mathrm{Bi}}_2{\mathrm{Se}}_3$ thin films of Mn-doped concentration (a) $x_{\mathrm{Mn}}=0\%$, (b) $x_{\mathrm{Mn}}=4\%$, and (c) $x_{\mathrm{Mn}}=8\%$. The open squares are the experimental data extracted from Ref. [26]. The solid red lines are the fitting results at different magnetic fields $B$ by using the formula in Eq. (6).

Figure 2

Schematic diagram of the band structure and spin orientation for (a) massless and (b) massive Dirac fermions. The spin vectors at a certain Fermi energy are depicted by the red arrows. (c) and (d) show the corresponding Berry phase as the solid angle traced out the spin vectors on the Bloch sphere for (a) and (b), respectively. (e) and (f) show the trajectory of backscattering (solid line) and corresponding time-reversal trajectory (dashed line) for massless and massive Dirac fermions, respectively. The black arrow represent the momentum direction, and the red arrow denotes the spin orientation.

Figure 3

(a) The Cooperon gap ${\ell }_i^{-2}$ in units of square of the mean free path ${\ell }_e^{-2}$ and (b) the weighting factors as functions of spin polarization $\eta$, where $t_{0,\pm }$ and $s$ represent the WL and WAL channels, respectively. The weighting factors for $t_{+}$ and $t_{-}$ channels are equal.

Figure 4

Zero-field conductivity correction and slope ${\kappa }_{\mathrm{qi}}^{(n)}$ as a function of the ratio of the mean free path to the coherence length ${\ell }_e/{\ell }_{\varphi }$ for (a) WAL of spin polarization $\eta \sim 0$ and (b) WL of $\eta \sim 1$. Magnetoconductivity at different values of ${\ell }_e/{\ell }_{\varphi }$ for (c) $\eta =0.01$ and (d) $\eta =0.9$. The calculation parameter ${\ell }_e=10\mathrm{nm}$.

Figure 5

(a) Magnetoconductivity at different temperatures for the Cr-doped ${\mathrm{Bi}}_2{\mathrm{Se}}_3$ thin film of $x=0.23$. The open squares are the experimental data extracted from Fig. 2(j) of Ref. [31]. The solid red lines are the fitting results. (b) The temperature dependence of the fitted phase coherence length ${\ell }_{\varphi }$ (open squares). The red line indicates ${\ell }_{\varphi }\propto T^{-1.14}$.