Cite this Article
Song Hua-Ding, Sheng Dian, Wang An-Qi, Li Jin-Guang, Yu Da-Peng, Liao Zhi-Min. Topological transport in Dirac electronic systems: A concise review . Chinese Physics B, 2017, 26(3): 037301  
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Topological transport in Dirac electronic systems: A concise review
Song Hua-Ding1, Sheng Dian1, Wang An-Qi2, Li Jin-Guang1, Yu Da-Peng1, 3, 4, Liao Zhi-Min1, 3, †
State Key Laboratory for Mesoscopic Physics, School of Physics, Peking University, Beijing 100871, China
Academy for Advanced Interdisciplinary Studies, Peking University, Beijing 100871, China
Collaborative Innovation Center of Quantum Matter, Beijing 100871, China
Department of Physics, South University of Science and Technology of China, Shenzhen 518055, China

 

† Corresponding author. E-mail: liaozm@pku.edu.cn

Abstract

Various novel physical properties have emerged in Dirac electronic systems, especially the topological characters protected by symmetry. Current studies on these systems have been greatly promoted by the intuitive concepts of Berry phase and Berry curvature, which provide precise definitions of the topological phases. In this topical review, transport properties of topological insulator ( , topological Dirac semimetal ( ), and topological insulator-graphene heterojunction are presented and discussed. Perspectives about transport properties of two-dimensional topological nontrivial systems, including topological edge transport, topological valley transport, and topological Weyl semimetals, are provided.

PACS: 73.50.-h;73.20.Fz;71.27.+a;79.60.Jv
Keyword:transport property;topological insulator;topological semimetal;graphene heterostructure
1. Introduction

Inspired by the notion of “topological order”, the exploration for novel classifications of matter phases becomes an intriguing theme in condensed-matter physics. Over three decades, the discovery of the quantum Hall effect (QHE) enlightened investigations on topological phases of matters.[1–6] Assisted by the powerful concepts of Berry phase and Berry curvature, precise definitions of topological phases are achieved. Berry phase is the phase generation during an adiabatic evolution process with steady external parameters producing a loop in the parameter space.[1,2] The Berry phase can be written as the integral of the Berry curvature.[2] Moreover, the integrals of the Berry curvature over closed surfaces are always employed to identify topological nontrivial phases, known as quantized Chern numbers.[2] Hence, theoretical predictions for nontrivial matters emerged endlessly for more than a decade, and novel quantum systems were continuously revealed. Such topological nontrivial systems exhibit fascinating properties, especially in transport experiments.

The QHE was discovered by Klaus von Klitzing et al., where the Hall conductivity of a two-dimensional (2D) system was found to be exactly quantized under a strong magnetic field.[7–9] Thouless et al. firstly employed topological considerations to explain the QHE.[5] From their milestone work, researchers began to realize that the electronic topological structure of quantum materials may be directly related to the transport behaviors. Various materials with novel topological phase have already been revealed. Graphene, the first 2D material containing Dirac fermions, has been proved to possess a Berry phase of π, leading to half-integer QHE observed in transport measurements.[10–16] Topological insulators (TIs) like and , where strong spin-orbit coupling creates topological insulating electronic phases and non-zero integer Chern numbers, are very potential platforms for spintronics and next-generation information devices.[17–33] Gapless surface states protected by time-reversal symmetry exist on the surface of these materials, and less-dissipative transport can be achieved by the surface states, which will lead to an information highway in the future. Similar to the graphene system, the surface states of TIs also behave as 2D Dirac fermions with π Berry phase.[18–21] Thus, Hall conductivity of the surface states is half-integer quantized as well. However, the parallel connection of the top and bottom surfaces may necessarily carry an integer QHE in real measurements.[23] Rui Yu et al. proposed that quantum anomalous Hall effect (QAHE) can be realized in a magnetized TI system,[25] which was confirmed by Cui-Zu Chang et al.[26,27] Recently, topological semimetals became another focus of condensed-matter physics. The topological Dirac semimetals (for example, Bi and ) are natural three-dimensional counterpart of graphene, with bulk linear-dispersion Dirac fermions and Fermi arcs surface state.[34–39] The unique chiral character of the Dirac fermions in topological Dirac semimetals generates abundant physical entities, such as chiral anomaly induced charge pumping behavior.[39] Novel transport properties, like anomalous negative magnetoresistance and surface state assisted Aharonov–Bohm oscillations,[40,41] were successively discovered.

In this topical review, we generally introduce the experimental investigations about transport behaviors in topological nontrivial systems, mainly based on our own works. We have reported abundant phenomena originating from surface states in topological insulators, topological Dirac semimetal, and monolayer graphene. In Section 2, we demonstrate the enhanced photothermoelectric effect by TI helical surface states[42] and the quantum oscillations from TI nanoribbon sidewalls.[43] In Section 3, our investigations on the topological Dirac semimetal ), including negative magnetoresistance induced by chiral anomaly,[40] Aharonov–Bohm oscillations in nanowires, and two-carrier transport in nanoplates are summarized.[41,44] In Section 4, we exhibit the novel tunneling behavior in the vertical heterojunction formed by graphene and TI.[45] In Section 5, we will review other significant theoretical and experimental progresses on the topological edge state transport, the valley transport, and the topological Weyl semimetals. Finally, we give perspectives and conclusions in Section 6.

2. Transport properties of topological insulator
2.1. Enhanced photothermoelectric effect of 2D topological surface states

Here we review the photothermoelectric effect in topological insulator nanoribbons. , as a strong three-dimensional topological insulator, has a bulk energy gap of ∼ 0.3 eV and helically conducting chiral surface states.[22,24] The 2D surface states could be excited to be spin-polarized by means of circularly polarized light. Due to spin-momentum locking in topological insulators, the spin-polarized surface states have oriented motions, which could be accelerated by the temperature gradient, resulting in an enhanced photothermoelectric effect.[46,47]

The nanoribbons were synthesized via the chemical vapor deposition (CVD) method. To better characterize the crystalline quality and study the layered microstructure of the nanoribbons, the as-synthesized nanoribbons were prepared to be cross-sectional on a substrate (Fig. 1(a)). Measured from the cross section along the [100] zone axis, the diffraction pattern indicates that the nanoribbon is single crystalline (Fig. 1(b)) with the basic unit of quintuple layer of –Se–Bi–Se–Bi–Se– sequentially (Fig. 1(c)). The ARPES spectrum acquired at room temperature confirms the existence of TI surface states (Fig. 1(d)).

Fig. 1. (color online) (a) Cross-sectional HAADF-STEM image of a nanoribbon. (b) Diffraction pattern acquired from the cross-section along the [100] zone axis. (c) High resolution HAADF-STEM image of the nanoribbon surface, where the quintuple layer layer is indicated. (Bi represented by red dots; Se represented by green dots.) (d) Room-temperature ARPES spectrum for a nanoribbon.[42]

Individual nanoribbons were fabricated to contact with Cr/Au electrodes on a Si substrate with 300 nm layer via electron beam lithography techniques, as shown in Fig. 2(a). To study the surface state enhanced photothermoelectric effect in the nanoribbons, the voltage response to circularly polarized light illumination was measured. Right-circular polarization (RCP), linear polarization (LP), and left-circular polarization (LCP) of the incident laser were achieved via tuning the λ/4 waveplate. The incident laser was along the y–z plane, where θ, the relative orientation of the incident light and the z axis, was chosen to be 0° and 30° (Fig. 2(b)). Considering that the most notable voltage response was detected when illuminating near the electrodes, the incident laser in our experiment was illuminated near the measurement electrodes. At θ = 0°, the voltage vs. time curves of RCP, LP, LCP show a little difference, as a result of that the vertically incident laser cannot induce in-plane surface states to be spin polarized, thus no additional surface state related photocurrent can be excited.

Fig. 2. (color online) (a) Optical image of a typical nanoribbon PTE device. (b) Sketch of the measurement set up. (c) The photo-voltage response measured when the laser illumination on was near the positive electrode. (d) The photo-voltage response measured when the laser illumination on was near the ground electrode. (e) The physical mechanism for the light polarization-selective transition.[42]

At θ = 30°, it is found that the RCP and LCP light have different effects on the photothermoelectric output. When the laser spot is near the electrode connected to the voltmeter positive terminal (the positive electrode), the RCP light induces an obviously enhanced voltage signal, the voltages induced by LCP and LP are similar (Fig. 2(c)). However, when the laser is illuminated near the voltmeter ground terminal (the negative electrode), the LCP induced photothermoelectric effect is notable, while the voltage response of RCP is almost the same as that of LP (Fig. 2(d)).

The phenomena above are dominated by the light polarization selective rule according to the helical character of the carriers (Fig. 2(e)). We define the electrons in the Dirac cone with a positive slope as spin-up, while those in the branch with a negative slope correspond to spin-down states. The quantum numbers of the orbital angular momentum of each branch are +1 and −1, respectively.[48–51] A photon can selectively excite the surface states to the conduction band states (l = 0) by transferring angular momentum.[48,51] If spin-polarized surface electrons have the same direction as the electrons driven by the temperature gradient, the generated voltage would be enhanced. Conversely, if the additional surface electrons have the opposite motion direction with the electrons driven by the temperature gradient, there would be little impact on the total generated voltage.

When the laser illumination on is near the positive electrode, the σ– photons (RCP) only excite the surface electrons with to conduction bands (l = 0), while the hopping of electrons is forbidden, resulting in additional spin-up surface states. Considering the spin-momentum locking in , the spin-up surface electrons have a translational direction along the +x axis,[52] which is parallel to the direction of the temperature gradient. Naturally, additional surface electrons would be accelerated and enhance the photothermoelectric effect. Contrarily, the LCP would hardly enhance photovoltage as a result of the anti-parallel relationship between the momentum of spin-down surface electrons and the temperature gradient. A similar analysis can be made for the laser illumination position near the ground electrode, in this case, the LCP would induce significant enhancement of the photothermoelectric effect.

2.2. Side-wall surface states associated quantum oscillations

Three-dimensional topological insulators have been demonstrated to possess rich physical connotations.[17–24] is always considered as a perfect platform to explore the novel properties of 2D Dirac fermions on its surface. However, even containing a large bulk energy gap, the conductance contributed from the bulk is still notable due to Se vacancies in making identification of the surface states by transport measurements a great challenge. In this section we review our investigations on prominent SdH oscillations originating from the surface Dirac fermions in the sidewalls of nanoplates. Importantly, the SdH oscillations from the sidewalls appear with a dramatically weakened magnetoresistance background, offering a direct path to detect the 2D surface states when the coexistence of bulk and surface conductions is inevitable.

The nanoplates were prepared by the vapor-liquid-solid mechanism, using gold as the catalyst. Electrodes of Cr/Au (10 nm/130 nm) were deposited by electron beam evaporation for transport measurements (Fig. 3(b)). When applying an in-plane magnetic field, obvious SdH oscillations can still be observed on a smoothed magnetoresistance background (Fig. 3(c)), the oscillation features are gradually smeared out by increasing temperature (over 20 K). Such in-plane field induced SdH oscillations have rarely been reported before, and apparently, this phenomenon does not originate from the top/bottom surfaces. These results indicate that the most possible origination is the surface states on two sidewalls.

Fig. 3. (color online) (a) Schematic of the in-plane magnetic field referring to the sidewall (Se: green spheres; Bi: purple spheres).(b) Optical image of a typical nanoplate device and a schematic of the orientation of the applied magnetic field. The length between the two inner longitudinal voltage leads, and the width of the two Hall voltage leads of the nanoplate device are 1.48 μm and 2.26 μm, respectively. (c) Plot of longitudinal resistance versus under in-plane magnetic field at different temperatures. (d) Landau-level fan diagram for the Shubnikov–de Haas oscillation in at the lowest temperature 1.3 K.[43]

Considering the Onsager semiclassical quantization relation, a linear relationship can be acquired: , where indexes the N-th minimum in conductance, K is a constant, and is related to the Berry phase by . Based on such a relation, the Landau level fan diagram can be established (Fig. 3(d)). The extrapolated Landau index at extreme field limit ( ) is about 0.5, acquired from the Landau level fan diagram, indicating an additional Berry phase π in the system. This is direct evidence of novel 2D Dirac fermions. We substantially demonstrated that the oscillation behavior derived from Landau level formation in 2D Dirac fermions. Since the applied in-plane field cannot induce such formation in the top/bottom surface, the sidewalls are the only possible origin for the quantum oscillation we observed.

Angle dependent magneto-transport measurements show that the positive magnetoresistance background, which is the contribution from three-dimensional bulk transport, remarkably decreases as the out-of-plane field component gradually disappears (Fig. 4(a)). To avoid the complexity that top/bottom and sidewalls simultaneously take part in transport, we measured the transversal resistance (2 K), which should not be affected by the sidewall conductance (Fig. 4(b)). Clear oscillation features have been observed (Fig. 4(c)). We derived the Landau level fan diagrams from the results in Fig. 4(b). We found that the derived slopes can be well fitted by 1/cos θ (Fig. 4(d)). Therefore we can confirm that the observed SdH oscillations avoid influences from three-dimensional bulk transport, and the oscillations under in-plane field should originate from the sidewalls. Moreover, such sidewall associated SdH oscillations successfully avoid disturbances from bulk transport, providing an easy method to detect surface states in topological insulators.

Fig. 4. (color online) (a) Longitudinal resistance and (b) Hall resistance as a function of the applied magnetic field B under different orientations described by the angle θ at a temperature of 2 K. (c) Oscillations of the Hall resistance at 2 K and at different angle θ. (d) Fitted slope of the Landau level fan diagram derived from as a function of the angle θ; the solid line gives a good fitting with 1/cos θ.[43]
3. Transport properties of topological Dirac semimetal
3.1. Giant negative magnetoresistance induced by the chiral anomaly

is one kind of the so-called three-dimensional Dirac semimetals, which are also described as three-dimensional analogues of graphene. These materials are unusual quantum materials containing massless Dirac fermions.[19] By breaking the time-reversal symmetry or spatial inversion symmetry, the Dirac semimetal is believed to transform into a Weyl semimetal with chiral anomaly.[34–39,53] In single crystal nanoplates, we find large negative magnetoresistance induced by chiral anomaly. Moreover, the negative magnetoresistance is closely linked with the carrier density, which is tunable by gate voltage and temperature. Our finding is helpful to understand Weyl fermions in Dirac semimetals.

The nanoplates were prepared by the chemical vapor deposition method. The exposed surface of the nanoplate samples was the (112) plane, identified by TEM results (Fig. 5(a)). The nanoplates possess a thickness of several hundred nanometers (Fig. 5(b)). The samples were then fabricated to contact with Au electrodes on an oxide Si substrate, serving as the back gate. Four-probe measurements were accepted in the experiments (Fig. 5(c) inset). The temperature dependence of resistivity of the samples demonstrates semiconducting-like behavior, which is different from the metallic behavior in bulk crystal (Fig. 5(c)). This is mainly attributed to the deviation of carrier density. The sample has a low carrier density and its Fermi level is close to Dirac points, where the holes in valence bands can be easily thermally activated to conduction bands in high temperature. So the resistivity of increases as the temperature decreases due to the reduced thermal activation. After decreasing to a critical low temperature, the thermal energy is not enough for holes to activate across the gap, leading to the metallic behavior. Unlike previous nanowire, the bulks usually maintain high carrier density and the Fermi level is always above Dirac points, where the metallic p–T mechanism is dominant. As shown in Fig. 5(d), nanoplates exhibit a notable negative MR with in-plane B, even at room temperature. The negative MR also emerges in nanowire samples with . The negative MR is mainly induced by the chiral anomaly effect. , as a three-dimensional Dirac semimetal, its Dirac point is composed of two overlapping Weyl nodes with opposite chiralities (left handed and right handed) (Fig. 6(a)), which would be separated in the magnetic field.[54,55] Applying an extra parallel electric field, the two kinds of Weyl fermions would have unequal chemical potentials ( ) (Fig. 6(b)). In such a situation, the continuity equation of the Weyl nodes is

(1)
In addition to the typical band dispersion contribution, an anomalous term due to the Berry curvature emerges, the extra term is always transverse to the electric field, leading to a Hall current.[2] For the system with inversion symmetry breaking, the Berry curvatures in different valleys have opposite signs under the protection of time-reversal symmetry, thus non-zero valley conductivity can exist in the transverse direction, a net valley degree accumulation is expected on both transverse sides. This phenomenon is called the “valley Hall effect”. To experimentally realize the VHE, suppressed valley relaxation must be achieved. The spontaneous inversion symmetry breaking and strong spin–orbit coupling make an ideal platform to realize VHE.[90] Mak et al. firstly reported the experimental observation of VHE in monolayer .[91] Circularly polarized light was used to create a chemical potential imbalance in the two valleys of , inverting the valley signal to an observable electric signal; an anomalous Hall signal induced by VHE was substantially detected.

5.2.2. The valley transport in graphene

Theoretical analysis has shown that the graphene with inversion symmetry breaking can also exhibit the VHE.[92] However, pristine graphene (single layer or bilayer) is under inversion symmetry protection. The experimental investigations on the valley transport in a graphene system have been achieved recently, with the help of improvement on device fabrication.

In 2014, Gorbachev et al. reported a large non-local signal detected in single layer graphene with underlying BN substrate.[93] The inversion symmetry of graphene is believed to be broken as the consequence of A/B sublattice global asymmetry induced by the substrate potential. The inversion symmetry breaking leads to a non-zero Berry curvature distribution in the Brillouin zone. When the Fermi level is tuned to the energy level where non-zero Berry curvature exists, a large non-local signal can be detected from the transport measurements. The anomalously obvious non-local signal is due to the valley Hall effect and the inverse VHE, the latter is the phenomenon where pure valley current gives rise to charge current.

The novel work of Gorbachev et al. not only firstly reported the experimental realization of graphene valley transport, but also provided an efficient method to detect the VHE. In 2015, the valley transport in bilayer graphene was discovered by Sui et al. and Shimazaki et al. The two groups both utilized a perpendicular electric field tuned by a gate voltage to break the inversion symmetry of graphene, a strong non-local signal was detected, which was attributed to the VHE and the inverse VHE.[94,95] Their investigations show that electrically driven inversion symmetry breaking is achievable, shedding light on future valleytronics. Furthermore, the AB/BA stacking domain wall in bilayer graphene has also been reported to support a non-diffusional valley current.[96] The bilayer graphene system with broken inversion symmetry may possess a non-zero “valley” Chern number, leading to a topological valley edge state existing along the domain wall.

5.3. Topological Weyl semimetals

Based on the quantum field theory, the fermions of our universe are divided into three kinds: the Dirac fermions, the Weyl fermions, and the Majorana fermions.[97] Interestingly, the condensed-matter systems host quasiparticles possessing the characters of these high-energy physics concepts. The most typical example of Dirac semimetals is graphene, which has a Dirac point protected by time-reversal symmetry and inversion symmetry.[10–16] Extending the 2D linear energy–momentum dispersion relation to 3D momentum space, the 3D Dirac semimetals were recently discovered. Moreover, the Dirac point in 3D Dirac semimetals is composed of two Weyl nodes. The topological Weyl semimetals are the systems where low-energy excitations obey the Weyl equations.[57,98–100] Compared to the Dirac points (four-fold degeneracy), the Weyl nodes are only two-fold degenerate. Thus the Weyl node can only exist in systems with spontaneous symmetry-breaking, because in the systems possessing both the time-reversal and inversion symmetries, the band crossings are always four-fold degenerate.

The topology of the Weyl nodes is also defined by the Berry curvature,[98–100] an analogy of the magnetic flux in momentum space. Generally speaking, the Weyl nodes can be considered as the “magnetic monopoles” in momentum space, which are the origins or ends of the Berry curvature. The chirality (+1 or −1) of a Weyl node is similar to the “magnetic charge”, reflecting the Berry curvature distribution embracing the Weyl node. The Weyl nodes always come in pairs because the Chern number of the Weyl semimetal must be zero, otherwise it becomes a topological insulator.[2]

There are a large number of unusual physical features associated with Weyl semimetal, including open Fermi arcs in the surface and various exotic transport phenomena induced by the chiral anomaly. The open Fermi arcs connect the projections of the bulk Weyl nodes with opposite chiralities onto the surface, and the Fermi arcs can be understood as the surface states of Weyl semimetals.[98] Under the presence of external electric and magnetic fields, the chiral charge of one Weyl node is not conserved, which is the so-called chiral anomaly. The charge pumping effect between Weyl nodes with opposite chiralities gives rise to large negative magnetoresistance.[40,54–57,97,98]

It is worth noting that the Weyl semimetals discussed above have zero density of state at the zero energy, and they obey the Lorentz invariance in quantum field theory. This kind of Weyl semimetals have been predicted to exist in several symmetry-breaking materials, including TaAs, NbP, and other systems.[99,100] By the ARPES technique, the existence of Weyl fermions in TaAs, NbAs has been experimentally confirmed,[101–105] and chiral-anomaly-induced negative magnetoresistance was also detected in TaAs recently.[106]

Recently, a new type of Weyl semimetal has been predicted in the system.[107] In this type of Weyl semimetal, non-zero density of state at zero energy exists, and such system seems to break through the limit of Lorentz invariance, which is referred as a type-2 Weyl semimetal. The remarkable distinction between the Fermi surfaces of the two types of Weyl semimetals leads to notable differences in the thermodynamics and magneto-transport. The type-2 Weyl semimetal exhibits the chiral anomaly effect with strong anisotropy.

6. Conclusion and perspectives

Topological nontrivial systems provide splendid platforms for uncovering novel physics in mesoscopic scale, and also for practical applications. As a powerful method of physical characterization, transport measurements make significant contributions in exploring these miraculous systems. This topical review summarizes our experimental investigations on transport behaviors in topological insulator ( , topological Dirac semimetal ( ), and TI-graphene heterojunction. We demonstrate fantastic quantum effects of these systems, exhibiting their great potential in realizing next-generation electronic and spintronic devices. We also review significant breakthroughs from other groups, to depict a general status of current research on topological transport. Although great triumphs have been achieved, future investigations are still in eagerly awaited.

The past ten years have witnessed great achievements on the graphene system. Moreover, this intriguing 2D material still possesses unpredictable academic and application prospects. Researchers begin to enlarge the functionalization of graphene using graphene-based heterostructures by the proximity effect. Other than the inversion symmetry breaking and enhancement of spin–orbit coupling discussed above, a large exchange field has been introduced into graphene by a magnetic insulator EuS.[108] We believe experimental realizations of QSHE and QAHE in graphene-based heterostructures will come true in the near future. The physical contents and application possibilities are obviously diversified by investigations on the graphene-based heterostructures.

The proximity effect also shows extraordinary talents in other two-dimensional systems. The superconducting proximity effect is believed to induce Majorana fermions in a topological nontrivial system,[109] and transform the hybrid system into a “topological superconductor”. The controllable realization of such fantastic fermions will start the revolution of quantum computing. These features lead the superconducting proximity effect to become a research focus in recent years. Researchers have successfully detected the signature of Majorana fermions in topological superconductor nanowires,[110,111] and we notice that a recent report has provided evidence of the Majorana zero mode in the topological superconductor.[112] The superconductivity in Dirac semimetal was also achieved by a point contact.[113] The topological superconductor systems will remain intriguing for a long time, and there is still an inestimable amount of physical content for us to discover.

Many other novel 2D materials are waiting to be revealed. The black phosphorus, which has both band gap and high mobility, was discovered in recent years.[114–117] Researchers have inquired into its application prospect,[114,115] and the QHE has been realized in black phosphorus.[116,117] Other new systems, like type-2 Weyl semimetal, are explored continuously. The variety of 2D materials affords us incessant inspirations, and the combination of 2D materials and other systems (for example, 2D material with superconductors or ferromagnets) will actually maximize such variety.

On the other hand, the improvement of device fabrication techniques provides substantial foundation for intensive works. The MBE method has been developed to grow high-quality 2D materials and their Van der Waals heterostructures. The interference of impurity potential can be greatly suppressed by the device encapsulation technology (for example, BN encapsulation),[118] leading to an ultraclean device environment. By the help of technological advances, more and more physical phenomena have been discovered experimentally.

We hope the readers of Chinese Physics B may acquire inspirations from our review, and our investigations over these years may shed light on future explorations.

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