Competition between Weak Localization and Antilocalization in Topological Surface States

Abstract

A magnetoconductivity formula is presented for the surface states of a magnetically doped topological insulator. It reveals a competing effect of weak localization and weak antilocalization in quantum transport when an energy gap is opened at the Dirac point by magnetic doping. It is found that, while random magnetic scattering always drives the system from the symplectic to the unitary class, the gap could induce a crossover from weak antilocalization to weak localization, tunable by the Fermi energy or the gap. This crossover presents a unique feature characterizing the surface states of a topological insulator with the gap opened at the Dirac point in the quantum diffusion regime.PACS numbers: 73.25.+i, 03.65.Vf, Topological surface states, composed of an odd number of massless Dirac cones, are peculiar to threedimensional (3D) topological insulators [1][2][3]. Electrons in these states have a helical spin structure in momentum space, and acquire a π Berry's phase after completing a closed trajectory adiabatically around the Fermi surface. The π Berry phase could lead to the absence of backscattering[4], weak antilocalization [5], and the absence of Anderson localization [6,7]. In the quantum diffusion regime (mean free path ≪ system size ∼ phase coherent length), an electron maintains its phase coherence after being scattered by static centers for many times. As a result, the destructive interference due to the π Berry phase can give a quantum enhancement to the classical electronic conductivity, leading to weak antilocalization (WAL) [8,9]. Applying a magnetic field tends to break the destructive interference, giving rise to negative magnetoconductivity (MC), a key signature of WAL. WAL is expected in systems with symplectic symmetry. Much effort has been devoted to observing WAL in graphene [5,[10][11][12][13]. However, graphene has two valleys of gapless Dirac cones with opposite chiralities, and the intervalley scattering will inevitably suppress WAL [5,[10][11][12][13]. In contrast, the surface states of recently discovered topological insulators Bi 2 Te 3 and Bi 2 Se 3 have only one helical Dirac cone [14][15][16], and WAL is intrinsic to them. Many observations of WAL in Bi 2 Te 3 and Bi 2 Se 3 have been reported recently [17][18][19][20][21][22]. In particular, there is great interest in the effect of magnetic doping, which is considered to be an efficient way to open an energy gap in the Dirac cone by breaking time reversal symmetry (TRS) [23][24][25]. This gap is expected to give rise to many interesting phenomena, such as Majorana fermion [26], topological magnetoelectric effect [27] and quantized anomalous Hall effect [28]. These developments call for a thorough theoretical investigation on WAL in topological insulators, in particular, in the presence of magnetic doping.

Interplay between ferromagnetism, surface states, and quantum corrections in a magnetically doped topological insulator

“…Since ARPES measurements show that the Fermi level in these samples is located well in the bulk conduction band states, the magnetoconductance data probe a complicated interplay between states that include the pure TI surface, confined two dimensional (2D) states created by band bending and three dimensional (3D) bulk states. Despite this complex situation, the onset of time reversal symmetry breaking ferromagnetism qualitatively yields the magnetoconductance behavior predicted for 2D Dirac cone TI surface transport 18 . We note that a recent study 19 of ultrathin (3 QL) samples of Cr-doped Bi 2 Se 3 showed a similar cross-over from weak anti-localization to weak localization, but in a regime where the samples are paramagnetic and where a surface state gap is already developed at a temperature (∼ 100 K) much higher than that implied by transport measurements ( 10 K).…”

“…We now discuss an interpretation of these data based upon diagrammatic calculations 18,28 for quantum corrections to transport in Bi 2 Se 3 . In samples such as the ones studied here, the bulk states dominate the conductivity because of the large Fermi energy: a simple estimate using the surface state energy dispersion and a Fermi energy 300 meV above the Dirac point shows that the surface carrier density (∼ 1.3 × 10 13 cm −2 ) is about an order of magnitude smaller than that of the bulk in our samples.…”

Nonuniversal weak antilocalization effect in cubic topological Kondo insulators

“…In addition, as it has been shown recently, the opening of the gap in the Dirac bands may actually change the sign in the quantum correction to conductivity from weak-antilocalization to weak localization. 25 Perhaps the fact that this crossover has not been observed 31 suggests that at least one of the Dirac bands remains ungapped.…”

“…The magnitude of the spin-orbit interaction for the surface electrons in topological insulators can be indirectly probed by studying the quantum interference correction to conductivity: [24][25][26][27][28] upon decrease in temperature, δT < 0, increase in conductivity (δσ > 0) would signal weak anti-localization effect as opposed to weak localization corresponding to decrease in conductivity (δσ < 0). The sign of the correction to conductivity is determined by the ratio of the spin-orbit scattering length, l SO , to the dephasing length l φ : for weak spin-orbit coupling, l SO l φ (here l is a mean-free path) correction to conductivity is negative, while in the opposite limit of strong spin-orbit coupling l SO l φ and the interference correction to conductivity is positive.…”

“…25,29 Application of an external magnetic field perpendicular to conducting surface destroys quantum interference processes leading to positive or negative magnetoconductivity -another signature of weak localization or weak anti-localization.…”