† Corresponding author. E-mail:
Project supported by the National Basic Research Program of China (Grant Nos. 2012CB921703 and 2015CB921102), the National Natural Science Foundation of China (Grant Nos. 61425015, 11374337, and 91121003), and the Strategic Priority Research Program of the Chinese Academy of Sciences (Grant No. XDB070202).
The recent discovery of three-dimensional (3D) topological insulators (TIs) has provided a fertile ground for obtaining further insights into electron localization in condensed matter systems. In the past few years, a tremendous amount of research effort has been devoted to investigate electron transport properties of 3D TIs and their low dimensional structures in a wide range of disorder strength, covering transport regimes from weak antilocalization to strong localization. The knowledge gained from these studies not only offers sensitive means to probe the surface states of 3D TIs but also forms a basis for exploring novel topological phases. In this article, we briefly review the main experimental progress in the study of the localization in 3D TIs, with a focus on the latest results on ultrathin TI films. Some new transport data will also be presented in order to complement those reported previously in the literature.
Study of electron transport under the influence of disorder is at the heart of condensed matter physics. In a seminal paper published in 1958, Anderson pointed out that the prorogation of waves would be absent in a random media with sufficiently strong disorder.[1] In the late-1970s, application of scaling analysis to the localization problem provided another major breakthrough in understanding the interplay between disorder and metal–insulator transition.[2] An interesting outcome of the scaling theory is that no truly metallic state could exist in a two-dimensional (2D) electron system no matter how weak the disorder is.[3] Even at the clean limit, interference between the time-reversed pairs of electron’s self-intersected trajectories leads to enhancement of backscattering probability, which is manifested as a lnT-type decrease in conductivity as temperature approaches to zero.[3–5] Such a quantum correction to conductivity is referred to as weak localization (WL), a precursor to the Anderson localization that occurs at strong disorder. Many predictions on the 2D electron transport have been confirmed in experiment.[5,6] Kravchenko et al., however, reported a possible metallic ground state in a low density Si-based 2D electron system, in which Coulomb interactions between electrons overwhelm electron’s kinetic energy.[7,8] Despite a lot of attention drawn to this subject by this surprising observation, whether or not a truly metallic state can exist in 2D is still a controversial issue.[9,10] This may be attributed to the formidable challenges in theoretical treatments of strong electron–electron interactions, as well as the experimental inability in reaching the absolute zero-temperature.
The recently discovered three-dimensional (3D) topological insulators (TIs) provide an interesting alternative to obtain truly metallic state in 2D.[11,12] The surface of a 3D TI hosts gapless Dirac particles with spin directions locked perpendicularly to their momenta. Such spin-helical surface states are believed to be immune to localization.[11,12] In the single-particle picture, this can be understood as a consequence of the Berry phase π associated with the spin-momentum locking, which leads to suppression of backscattering. This effect is opposite to the WL and thus termed as weak antilocalization (WAL). The topological protection against localization can survive even if the electron–electron interactions are present, according to a number of theoretical studies.[13–16] The WAL effect has been observed in numerous magnetotransport measurements, in which the experimental hallmark is positive, cusp-shaped magnetoresistance that is strongly temperature-dependent at low temperatures.[17–22] The connection between the positive magnetoresistance and the Dirac surface states is, however, not always straightforward due to a number of complications, such as the magnetoresistance of bulk carriers, the surface-bulk and inter-surface couplings, and the possible coexistence of topologically trivial surface states.[23] Nevertheless, in the past several years there have been significant advances in suppressing the bulk conductivity by doping[22,24] or alloying[25–28] various TI compounds or by electrostatic gating.[17,18,20,21,29–31] The magnetoresistance due to the WAL has become a sensitive probe to the surface transport and various couplings to the surface states.[17,18,20–22,30–33]
In addition to the WAL, it is also possible to realize other interesting transport regimes by utilizing the hybridization between the top and the bottom surface states in ultrathin films of 3D TIs. These include the WL, the quantum spin Hall effect, topologically trivial insulating states, and so on.[34–37] Evidence for the surface hybridization gaps has been reported from angular resolved photoemission spectroscopy (ARPES)[38] and scanning tunneling spectroscopy.[39] Transition from the WAL to the strong localization has been observed.[40] Clear experimental evidence for crossover from the WAL to the WL, however, still remains elusive, even though negative magnetoresistances have been reported by several groups.[32,33,40,41]
In this article, we give a short review of the latest experimental progress in the study of electron antilocalization and localization in 3D TIs and related structures. We do not intend to exhaustively describe all of the exciting results reported by many groups worldwide because of the limitation in space. We would rather focus on the experimental results reported in the last few years, which provide new insights into the electron localization in this fascinating class of materials. We hope that further efforts can be made to narrow the gaps between theory and experiment in this intriguing subfield of quantum transport. Readers may refer to Refs. [42]–[44] for more detailed summaries of the electron transport results prior to 2013 and Refs. [11], [12], [42], and [45] for more comprehensive reviews of TIs.
At low temperatures and in weak magnetic fields, magnetoconductivities in 3D TIs are often related to the WAL, which is a counterpart of the WL occurring in ordinary electron systems with negligible spin–orbit coupling.[5] Both the WAL and the WL require the transport to be in diffusive regime, in which the sample’s size L, dephasing length lϕ and mean free path le satisfy L ≫ lϕ ≫ le at low temperatures. In addition, the diffusive regime also requires kFle ≫ 1, which corresponds to a sheet resistivity (or resistance per square in 2D) ρxx = (kFle)−1(h/e2) ≪ h/e2. Elastic scatterings by the disorder potential lead to quantum corrections to the electron conductivity. For the WAL (WL), the corrections in resistivity are negative (positive), which can be attributed to the destructive (constructive) interferences between electron waves propagating along the time-reversed pairs of self-crossing loops, as illustrated in Fig.
![](cpb_25_11_117201/cpb160595ieqn2.gif)
![]() | Fig. 1. Diffusive transport and weak antilocalization (WAL). (a) Illustration of a time-reversed pair of electron trajectories on a self-crossing loop. Destructive and constructive interferences between them lead to weak antilocalization and weak localization (WL), respectively. (b) Sketch of the band diagram of a topological insulator with Fermi energy EF located in the bulk conduction band. When EF is in the shaded regions, scatterings between the bulk and the surface states make the sample behave like a single-channel system in the phase coherent transport. (c) Extracted α values from fitting the low field magnetoconductivity data to the simplified HLN equation [Eq. ( |
Early transport studies of TI thin films were mostly focused on Bi2Se3 and Bi2Te3, in which the Fermi level is not located inside the bulk band gap due to various defects [see Fig.
For an ideal 3D TI thin film, in which the bulk is insulating and the thickness is sufficiently large so that hybridization between the top and the bottom surfaces is negligible, the transport takes place in two independent channels. If the dephasing fields of these two surfaces can be tuned equal to each other, one would expect α = 1. Electrostatic gating is a necessary tool to obtain such smoking-gun evidence for the surface WAL effect. Many methods have been proven effective in depleting excessive bulk carriers,[55] such as back-gating with SrTiO3[17] or SiO2[18] and top-gating with Al2Ox grown with atomic layer deposition[32] or liquid electrolytes.[56] In 2011, Chen et al. and Checkelsky et al. independently reported observation of the crossover from α ≈ 1/2 to α ≈ 1 in back-gated Bi2Se3 thin films (or exfoliated thin plates).[18,20] It is noteworthy, however, that tuning Bi2Se3 samples into the decoupled two-channel transport is very difficult because of high densities of bulk electrons. Despite a lot of work worldwide, there have been only a handful of reports of α ≈ 1 in Bi2Se3, and one of them has relied on doping in the bulk or on the top surface.[33] More recently, successful syntheses of ternary compound (Bi,Sb)2Te3 and quarterly compound (Bi,Sb)2(Se,Te)3 have made the surface-dominated transport much easier to access.[26–29] High quality (Bi,Sb)2Te3 films have been grown with molecular beam epitaxy onto SrTiO3 and InP substrates.[30,57] These films are particularly suitable for the study of the localization-related physics because the samples sizes can be readily made much larger than electron’s dephasing length. In contrast, nanoplates exfoliated from (Bi,Sb)2(Se,Te)3 single crystals are usually on the micrometer scale, and in some samples universal conductance fluctuations become pronounced.[58,59] A large range of tuning of the chemical potential can be obtained in gated (Bi,Sb)2Te3 and (Bi,Sb)2(Se,Te)3 structures on a regular basis, and many groups have reported characteristics of ambipolar transport and large α values.[27,30–32,60] Figure
In the Anderson (strong) localization regime, electron wavefunctions are localized around certain positions in the form of |ψ(r)| ≈ exp(−|r − r0|/ξ), where the localization length ξ satisfies ξ < lϕ.[2,61] Theories have predicted surface states of truly 3D TIs are immune to the strong localization.[13–16] In ultrathin TI films, the hybridization effect between the top and the bottom surfaces can open an energy gap near the Dirac point, resulting in a ground state of either a quantum spin Hall insulator or a topological trivial semiconductor.[34–36] This makes observation of the strong localization in ultrathin TI films possible if one can tune the chemical potential into or close to the hybridization gap and makes the bulk insulating. However, as pointed out by Skinner et al., the bulk resistivity is very difficult to remove completely, even in well-engineered, highly compensated TI samples, such as (Bi,Sb)2Te3, and the best situation that one could get is the variable range hopping transport involving poorly screened electron and hole puddles.[62]
Even though ARPES measurements have provided evidence for sizable surface hybridization gaps (e.g., as large as 0.2 eV for a few nm thick Bi2Se3 films) as early as 2009,[38] clear observation of the strong localization has not been reported until recently.[40] This was demonstrated with ultrathin (Bi, Sb)2Te3 thin films. With a combination of precise control of thickness and gate-voltage tuning of chemical potential, the transport can be varied from the well-defined diffusive regime to the variable range hopping regime. As shown in Fig.
Similar positive magnetoconductivities have also been observed in conventional 2D electron systems in the hopping regime.[63–65] For instance, Hsu et al. reported a crossover of magnetoconductivity from negative to positive values when the resistivity of Ag–Ge thin films exceeds πh/e2(∼ 80 kΩ).[65] In the hopping regime, electron transport is dominated by quantum interferences between different paths for forward scatterings because of the localized nature of electron wavefunctions.[66,67] The phase differences between various trajectories are modified by an applied magnetic field. The positive magnetoconductivity can, hence, be regarded as a delocalization effect of the magnetic field. This effect is independent of the strength of SOC.[65–67] Quantitatively, one expects Δσ ∝ B2 in sufficiently low fields and saturates at a sufficiently high magnetic field.[64–67] Figures
![]() | Fig. 3. Transition from weak antilocalization (WAL) to strong localization in ultrathin TI thin films. (a) In the diffusive transport regime, the zero-field conductivity σ has a logarithmic dependence on temperature, and the magnetoconductivity shows the WAL behavior described by Eq. ( |
![]() | Fig. 4. Positive magnetoconductivity in the strong localization regime. (a) Gate voltage dependence of zero-field sheet resistivity ρxx of a 2 nm thick BST film. (b) Magnetoconductivity curves of the same sample for three representative ρxx values, which are marked as points A–C in panel (a). (c) Magnetoconductivity curves of the 2 nm BST thin film in the variable range hopping regime at selected temperatures. The sheet resistivity is ρxx = 2.3 MΩ at 1.6 K. Panels (a)–(c) are adapted from Ref. [40]. (d) Magnetoconductivity curves of a 3 nm thick Bi2Se3 sample for a set of gate voltages. The inset shows the gate voltage dependence of ρxx. |
In addition to a possible transition to the quantum spin Hall state, the hybridization of the surface states in ultrathin 3D TI films may also lead to other interesting physics. For instance, Lu et al. pointed out that the spin texture of massive surface states [see Fig.
![]() | Fig. 5. Sketch of the gapped helical surface states of TIs and the predicted crossover from the WAL to the WL when the ratio Δ/E is varied from 0 to 0.999. (b) Prefactor α obtained from the fit to the simplified HLN equation [see Eq. ( |
Several groups have reported magnetotransport results of ultrathin TI films, in which the magnitude of magnetoconductivity is much smaller than that of the standard WAL effect observed in thicker films.[32,33,40,41] In most cases, the low field magnetoconductivity remains negative and can be fitted with the simplified HLN equation [see Eq. (
We have shown the important role of disorder in the phase coherent transport in 3D TI thin films. When the disorder is weak (kFle ≫ 1), the transport is diffusive and the magnetoconductivity can be well characterized by the WAL effect of coupled or decoupled multiple channels existing on the top and the bottom surfaces and possibly in the bulk. The advances in improving sample quality and gating techniques have allowed for surface-dominating transport detected with the magnetoconductivity in the WAL regime. In ultrathin TI films, much stronger disorder (kFle ≪ 1) can be realized, and the transport can be driven into the strong (Anderson) localization regime, in which variable range hopping and positive magnetoconductivity are observed. In the crossover regime between the WAL and the strong localization, a gradual suppression of the negative magnetoconductivity takes place as the sheet resistivity increases. The suppression of WAL is argued as a consequence of the disorder-related effect during the transition from the WAL to strong localization, instead of the crossover from the WAL to the WL. It should be noted, however, that the experimental results reviewed in this article do not exclude the existence of the Berry phase effect predicted by Lu et al. for the massive Dirac fermions due to surface hybridization.[37] Further experimental effort in improving the quality of ultrathin TI films is needed to bring the transport into the well-defined diffusive regime in order to confirm the predicted crossover from the WAL to the WL.
In this article, we have limited our discussions to the transport in perpendicular magnetic fields. The phase coherent transport in parallel or tilted magnetic fields can also give valuable information of the TI samples, which are difficult to acquire with other techniques.[73,74] Additional work is also needed for understanding the complicated results in those circumstances. As mentioned in the Introduction, this article is not intended to cover all of the important results in this rapidly developing field. For instance, manifestations of weak antilocalization and Aharonov–Bohm-like effect in TI nanoribbons are not discussed.[75–77] Readers may find many previous reviews valuable in obtaining a more comprehensive knowledge of this exciting research area.[11,12,43–45,78]
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