Microscopic mechanism and quantitative theory of the weak-antilocalization-to-weak-localization crossover in magnetically doped topological-insulator thin films (Cr-doped Bi2Se3)

In three-quintuple-layer Cr-doped Bi2Se3 films, the low-field quantum correction to the surface conductance crosses over from weak antilocalization to weak localization as the magnetic-dopant concentration and perpendicular magnetization increase and as temperature decreases, including a nonmonotonic magnetoconductance at intermediate doping, reentrant weak antilocalization on warming, and a low-temperature resistivity enhancement of more than two orders of magnitude. The 2012 experiment interpreted this phenomenologically through a Berry phase shifted from $\pi$ toward $2\pi$ by a time-reversal-symmetry-breaking gap at the Dirac point, but its authors acknowledged the analysis as oversimplified and its temperature dependence as theoretically untreated. The open problem is to determine the microscopic magnetic and electronic structure of the magnetically doped regime and construct a controlled microscopic theory that quantitatively connects the sample's magnetization, its field- and temperature-dependent gap landscape, and the observed magnetoconductance lineshapes, including the mechanism of the dilute-magnetic-semiconductor-like weak localization and the large low-temperature resistivity enhancement. Partial theoretical progress exists for spatially uniform gaps, but the inhomogeneity-coupled quantitative account and the insulating-state mechanism remain open.

Background

A three-dimensional topological insulator (TI) is a bulk insulator whose surfaces host a single cone of massless Dirac fermions with spin-momentum locking, protected by time-reversal symmetry (TRS). Weak localization (WL) is the quantum-interference correction to the conductivity of an ordinary two-dimensional disordered conductor: closed time-reversed diffusive loops interfere constructively, enhancing the return probability and suppressing the conductance; a perpendicular magnetic field destroys this interference, giving a positive low-field magnetoconductance. For spin-helical Dirac fermions the two time-reversed loops accumulate a Berry phase of $\pi$ (the geometric phase picked up when the locked spin traces the Fermi circle), the interference becomes destructive, and the correction reverses sign: this is weak antilocalization (WAL), visible as a sharp negative cusp in the low-field magnetoconductance $\Delta\sigma(B)=\sigma(B)-\sigma(0)$. The standard Hikami-Larkin-Nagaoka (HLN) formula describes the WAL magnetoconductance of a single diffusive channel with spin-orbit coupling. In the symmetry-class language of disordered electron systems, the TI surface without TRS breaking is in the symplectic class (WAL); TRS breaking by magnetization moves it toward the unitary class, while a fully spin-polarized (massive) Dirac cone behaves as an orthogonal-class conductor (WL). Magnetic doping of a TI is expected to open an exchange gap $\Delta$ at the Dirac point and is the route to predicted topological magnetoelectric effects, including the quantized anomalous Hall effect. A dilute magnetic semiconductor (DMS) is a nonmagnetic semiconductor doped with a small fraction of magnetic ions; the crossover of the doped TI toward DMS-like behavior is the central phenomenon here.

Liu et al. [Phys. Rev. Lett. 108, 036805 (2012)] grew three-quintuple-layer (QL; one QL is the natural Bi-Se trilayer stacking unit, about 1 nm) films of Bi$_{2-x}$Cr$_x$Se$_3$ ($x=0$ to $0.23$) on sapphire by molecular beam epitaxy (MBE), so that transport is dominated by the topological surface states. Angle-resolved photoemission spectroscopy (ARPES) showed the Dirac-point gap widening with Cr content and the surface-state spectral weight fading at $x=0.23$. Magnetotransport showed a systematic WAL-to-WL crossover with increasing $x$, with low temperature and with strong perpendicular field; the anomalously nonlinear Hall resistance of the $x=0.23$ film resembles the anomalous Hall effect (AHE) of ferromagnets, indicating partial out-of-plane magnetization; and the low-temperature sheet resistance grows by more than two orders of magnitude relative to undoped Bi$_2$Se$_3$. At intermediate doping ($x=0.07$) the 1.5 K magnetoconductance is nonmonotonic in field, and in heavily doped films WAL reenters on warming but is driven back to WL by a strong field. The authors interpreted the crossover via the Berry phase $\delta\varphi=\pi(1+\Delta/2E_F)$ derived by Lu, Shi, and Shen [Phys. Rev. Lett. 107, 076801 (2011)], which interpolates from $\pi$ (massless, WAL) to $2\pi$ (fully polarized, WL) as the magnetic gap $\Delta$ grows relative to the Fermi energy $E_F$, and they fitted all magnetoconductance curves with the two-term WL-plus-WAL magnetoconductance formula of that theory. They explicitly acknowledged that this analysis is oversimplified — neglecting hybridization between the top and bottom surfaces, possible Kondo-like electron-impurity interaction, and electron-electron interactions — and that the temperature evolution of the magnetoconductance in magnetically doped TIs had not been treated theoretically.

Problem Statement

For magnetically doped topological-insulator thin films — paradigmatically the three-quintuple-layer MBE-grown Bi$_{2-x}$Cr$_x$Se$_3$ films ($x \le 0.23$) of Liu et al. (2012) and the related Mn- and Cr-doped bismuth-chalcogenide films — establish a validated microscopic account of the weak-antilocalization-to-weak-localization crossover, comprising three jointly linked results, all of which are required:

(1) The microscopic magnetic and electronic structure of the intermediate-to-heavily doped regime: determine experimentally which of the candidate structures (an illustrative, not exhaustive, set) the crossover regime realizes — a uniform TRS-broken phase with a homogeneous exchange gap at the Dirac point, aggregated superparamagnetic dopant clusters with a chemical-potential-dependent and spatially inhomogeneous gap, or nanoscale near-surface ferromagnetic regions — and quantify the gap magnitude, its spatial correlation, and its dependence on doping, carrier density, field, and temperature. The competition of distinct scattering channels is a separate, dynamical aspect rather than a candidate structure. This experimental determination is bounded by the published data of Liu et al. (2012) and the related films, without new measurements.

(2) A controlled microscopic theory that quantitatively connects the sample's magnetization (including its temperature and field dependence and its anomalous-Hall signature), the resulting Dirac-point spectral gap and its inhomogeneity, and the measured magnetoconductance lineshapes — reproducing, within the same parameter set, the WAL-to-WL crossover with increasing dopant concentration, the nonmonotonic intermediate-doping magnetoconductance, the reentrant WAL on warming with WL recovered at strong field, and the temperature-dependent quantum corrections — going beyond the acknowledged oversimplifications of the 2012 analysis (surface-surface hybridization, Kondo-like electron-impurity interaction, electron-electron interactions, and uniform-gap assumptions).

(3) The mechanism of the dilute-magnetic-semiconductor-like weak localization and of the low-temperature resistivity enhancement of more than two orders of magnitude at heavy doping — candidate mechanisms include magnetic-polaron formation, nanoscale ferromagnetic phases, impurity-band or Kondo-type physics, and strong-disorder localization (an illustrative, not exhaustive, list) — and its quantitative role in the crossover.

These three results serve one overarching objective: a quantitative, experimentally validated microscopic picture of how breaking time-reversal symmetry with magnetic dopants converts the topological weak antilocalization of the surface Dirac fermions into weak localization in this material family. Phenomenological two-term (WL plus WAL) magnetoconductance fits alone, or theories restricted to a spatially uniform gap with scalar disorder, do not by themselves resolve the problem.

References

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