† Corresponding author. E-mail:
Project supported by the National Basic Research Program of China (Grant Nos. 2012CB821402 and 2015CB921401), the National Natural Science Foundation of China, the Program for Professor of Special Appointment (Eastern Scholar) at Shanghai Institutions of Higher Learning, and STCSM of China (Grant No. 15XD1500200).
The discovery of the three-dimensional Dirac semimetals have expanded the family of topological materials, and attracted massive attentions in recent few years. In this short review, we briefly overview the quantum transport properties of a well-studied three-dimensional Dirac semimetal, Cd3As2. These unusual transport phenomena include the unexpected ultra-high charge mobility, large linear magnetoresistivity, remarkable Shubnikov–de Hass oscillations, and the evolution of the nontrivial Berry’s phase. These quantum transport properties not only reflect the novel electronic structure of Dirac semimetals, but also give the possibilities for their future device applications.
Two-dimensional (2D) Dirac electron systems, represented by graphene and the surface state of three-dimensional (3D) topological insulators (TIs), exhibit exotic quantum phenomena and have been considered as one of the central topics in modern condensed matter physics.[1–4] Recently, 3D Dirac semimetal (DSM) has been proposed as another new class of topological material,[5–8] and caught plenty of attention quickly. This new kind of topological material can be regarded as a 3D analog of graphene because it possesses 3D gapless Dirac points (at Fermi level) in the bulk Brillouin zone, whose low-energy physics can be effectively described as four-component Dirac fermions,
Theoretically, several materials have been proposed as candidates for DSM, such as BiO2, A3Bi (A = Na, K, Rb), Cd3As2, and some distorted spinels.[5–7,14] Many unusual phenomena were also predicted, such as the giant diamagnetism which diverges logarithmically when the Fermi energy approaches the 3D Dirac point;[5,15] quantum linear magnetoresistance even up to room temperature and large negative magnetoresistance due to the novel bulk band structure;[5,6,16–21] the quantum spin Hall effect in its quantum-well structure;[5,22] and possible topological superconductivity when doping carriers.[6,12,23,24] Soon after these theoretical predictions, the angle-resolved photoemission spectroscopy (ARPES) experiments were performed on Na3Bi and Cd3As2 single crystals and two bulk 3D Dirac points were observed on the opposite sides of the Brillouin zone center point Γ along kz.[25–29] Scanning tunneling microscopy (STM) experiments also support the existence of 3D Dirac points in Cd3As2 single crystals.[30] Later, the distinct surface states of both two compounds were revealed by ARPES measuremens.[11,31] More recently, the point contact experiments have shown a hint of topological superconductivity under the tip in both Cd3As2 single crystals and polycrystals.[32,33] Pressure-induced superconductivity was also observed in Cd3As2 by high-pressure resistance measurements.[34] The magneto-transport measurements on Cd3As2 nano-structures have discovered a large negative magnetoresistance, which was attributed to the chiral anomaly.[35,36] Moreover, due to the large surface-to-volume ratio in nano-structures, the Aharonov–Bohm oscillations have been detected in individual single-crystalline Cd3As2 nanowires.[37]
The quantum transport measurement is an important tool to detect the unusual band structure of 3D DSMs. Following those theoretical predictions, a lot of quantum transport studies were carried out on Na3Bi and Cd3As2 single crystals, and many interesting transport phenomena were observed.[38–47] Considering that Cd3As2 is far more stable in air than Na3Bi, the Cd3As2 single crystal is more suitable for transport measurements. In this short review, we will summarize these quantum transport properties of Cd3As2 single crystals.
The standard method of growing Cd3As2 single crystals is using Cd flux with the starting composition Cd: As = 8: 3, as described in Refs. [38] and [48]. The typical size of as-grown Cd3As2 single crystals is several millimeters. This method usually gives Cd3As2 single crystals with large natural (112) surface. It is also convenient to employ vapor transport method to grow Cd3As2 single crystals with large natural (001) surface.[46,49,50] At ambient pressure, Cd3As2 has a distorted superstructure of the antifluorite structure type with a tetragonal unit cell of a = 12.633(3) Å and c = 25.427(7) Å in the centrosymmetric I41/acd space group,[48] which is shown in Fig.
Figure
Cd3As2 is a well-known compound which has been studied for decades.[52–54] The typical temperature dependence of the longitudinal resistivity of Cd3As2 single crystals at zero magnetic field shows a metallic behavior, with the carrier concentration ne varying from ∼ 1016 to ∼ 1018 cm−3. Considering the different qualitity of Cd3As2 single crystals, the residual resistivity ρ0 could vary from ∼10 nΩ·cm to 10 μΩ·cm with the residual resistivity ratio (RRR) ranging from ∼ 5 to ∼ 1000 (Table
![]() | Table 1. The parameters of transport properties, including residual resistivity ρ0, residual resistivity ratio (RRR), carrier concentration ne, mobility μ, cyclotron mass m*, Fermi velocity vF, and Fermi vector kF. . |
![]() | Fig. 2. (a) Curves of the resistivity versus temperature measured along the needle axis in five “Set A” and two “Set B” samples (semilog scale).[39] (b) Measured mobility versus the zero field conductivity for Set A samples (A4, A5, A6, and A8).[39] |
The first magnetotransport study of Cd3As2 single crystals after the theoretical prediction was reported by He et al.[38] The single crystals used in that work were grown in Cd flux, with ρ0 = 28.2 μΩ cm and RRR ≈ 5.7. Figure
![]() | Fig. 3. The longitudinal magnetoresistance (MR) of Cd3As2 single crystal at various temperatures with the field perpendicular to the (112) plane.[38] The MR presents remarkable oscillations at low temperatures and unusual linear behavior even near room temperature. |
At 280 K, the MR reaches as high as 200% at 14.5 T and shows a quite linear behavior above 3 T, which is consistent with the theoretical prediction.[6] This unusual linear MR has been observed by other transport works as well.[39–41] Usually there are two scenarios to explain the linear MR: a) the quantum limit scenario[16,17] and b) the classical disorder scenario.[55–57] For the former one, in the presence of linear energy dispersion, linear MR could be achieved when all of the carriers occupy the lowest Landau level (quantum limit);[16,17] for the latter one, linear MR arises because the local current density acquires spatial fluctuations in both magnitude and direction, as a result of the heterogeneity or microstructure caused by nonhomogeneous carrier and mobility distribution.[55–57] Both scenarios have been used to successfully explain the linear MR observed in some materials.[55–60]
Wang et al. proposed that Cd3As2 can support sizable quantum linear MR even up to room temperature at the quantum limit due to its linear energy dispersion.[6] However, this requirement is actually not satisfied in the field range, since there is clearly more than one Landau levels occupied, as will be seen in Fig.
![]() | Fig. 4. (a) The oscillatory component ΔRxx, extracted from Rxx by subtracting a smooth background, as a function of 1/B at various temperatures for Cd3As2 single crystal.[38] (b) Fast Fourier transform (FFT) of the oscillatory component ΔRxx at T = 1.5 K shows only one single frequency. (c) Temperature dependence of the relative amplitude of ΔRxx for the sixth Landau level. The solid line is a fit to the Lifshitz–Kosevich formula.[38] |
Quantum oscillation is a powerful tool to study the electronic properties of materials. It stems from the quantization of Landau levels. In a semiclassical approximation, this results in electrons executing cyclotron orbits confined to quantized Landau tubes. The Landau levels will pass over the Fermi surface one after another as the applied magnetic field is increased, which leads to the density of states oscillating with oscillatory behavior being periodic in inverse magnetic fields. This means that many quantities will exhibit quantum oscillations, since they are functions of the density of states. Various parameters such as electronic spectrum, scattering mechanism, geometry of Fermi surface, etc., can be extracted from the shape, period, and phase of quantum oscillations.[61] Therefore, this technique is widely used to study the electronic properties of materials. For resistance and magnetization, their oscillations are named as Shubnikov–de Hass (SdH) and de Hass–van Alphen (dHvA) oscillations, respectively.
He et al. investigated the properties of Fermi surface of Cd3As2 by analysing the SdH oscillations at low temperatures.[38] Figure
The SdH oscillation amplitude in Cd3As2 can be well described by the Lifshitz–Kosevich formula for a 3D system[61–64]
![](cpb_25_11_117105/cpb161861ieqn2.gif)
Berry’s phase is a geometrical phase factor, acquired when quantum mechanical systems adiabatically evolute on a closed path in parameter space.[66,67] A nontrivial or nonzero Berry’s phase could result in plenty of emergent phenomena, such as the anomalous and quantum Hall effects,[68,69] and topological insulating and superconducting phases.[3] A distinguished feature of Dirac fermions is that they carry the nontrivial π Berry’s phase because the electron orbits enclose a single Dirac point.[70–74] This nontrivial π Berry’s phase can be experimentally accessed by analyzing the SdH oscillations, which has been successfully detected in graphene,[73,74] elemental bismuth,[75] the bulk Rashba semiconductor BiTeI,[64] and bulk SrMnBi2.[76] However, situations have become complicated in topological insulators because of the large Zeeman energy effects and bulk conduction.[77–79]
In general, any closed cyclotron orbit is quantized under an applied magnetic field, and can be described by the Lifshitz–Onsager quantization rule
He et al. have firstly reported that the nontrivial π Berry’s phase exists in Cd3As2 by means of analysing the SdH oscillations.[38] The Landau index plot of Cd3As2 single crystals is presented in Fig.
![]() | Fig. 5. Landau index n plotted against 1/B for Cd3As2 single crystals.[38] The closed circles denote the integer index (ΔRxx valley), and the open circles indicate the half integer index (ΔRxx peak). Two straight-line fittings extrapolate to 0.58 ± 0.01 and 0.56 ± 0.03, respectively, which is a strong evidence for the nontrivial π Berry’s phase of 3D Dirac fermions in Cd3As2.[38] |
![]() | Fig. 6. The existence of the nontrivial Berry’s phase in Cd3As2 has been verified by several other measurements of SdH[41–44,46] and dHvA[43] oscillations. For example, the Landau plots in panels (a) and (b) come from the SdH oscillations measured by Narayanan et al.[41] and Zhao et al.,[42] respectively; (c) and (d) come from the dHvA oscillations measured by Pariari et al.[43] |
One important thing is that once the rotational symmetry of the crystal structure is broken, the 3D Dirac points in Cd3As2 are no longer preserved and a mass term will be introduced.[5,6,30] In this situation, a Dirac gap will open at the Dirac points, which means that the nontrivial Berry’s phase is no longer π constantly and will turn into a function of the induced mass term.[81] Therefore, if the gap is wide enough, the nontrivial Berry’s phase will reduce to zero.[46,81] Actually, a magnetic field whose orientation deviates from the kz direction would break the rotational symmetry in Cd3As2, and induce a small gap.[30] In our electric transport measurements, the external fields are fixed along the (112) direction, and this leads to a gap opening at the Dirac points.[30] However, according to Jeon et al., this Dirac gap is only about 8 meV (at 14.5 T in our measurements).[30] Such a small gap could not affect the value of Berry’s phase significantly.[30,81] That is why the nontrivial Berry’s phase is still detected in most of quantum oscillation experiments with magnetic field along the (112) direction.[38,41,44,46,81] In this sense, it will be interesting to investigate the evolution of Berry’s phase with the magnetic fields and the tilt angles.
Cao et al. discovered that the value of the Berry’s phase changes with increasing magnetic field.[44] They found that fitting higher magnetic field regimes would give a lower value of Berry’s phase when the orientation of the field is perpendicular to (112) direction.[44] Furthermore, the Berry’s phase also manifests an angular dependence at different magnetic field regimes.[44] Their results are reproduced in Fig.
![]() | Fig. 7. (a) Schematic sketch of the magnetoresistance measurements in Ref. [44]. The Berry’s phase was fitted in the regimes of (b) 5–7 T (corresponding to ninth to eleventh Landau levels); (c) 7–10 T (seventh to ninth Landau levels); and (d) 10–15 T (fifth to seventh Landau levels).[44] They proposed that these changes of the Berry’s phase suggest a field-induced topological phase transition.[44] |
![]() | Fig. 8. (a) Angular dependence of the transverse magnetoresistance with magnetic field rotated in the (010) plane.[46] (b) Angular dependence of the values of the Berry’s phase with magnetic field rotated in the (010) plane.[46] (c) Angular dependence of the transverse magnetoresistance with magnetic field rotated in the (100) plane.[46] (d) Angular dependence of the values of the Berry’s phase with magnetic field rotated in the (100) plane.[46] |
Xiang et al. investigated the angular dependence of the Berry’s phase thoroughly, and discovered that it can be continuously tuned by the orientation of the magnetic field.[46] In their measurements, the current is along the [010] direction and the external field rotates from [100] to [001] direction.[46] They found that the value of the Berry’s phase clearly deviates from 0 when the magnetic field is tilted away by 60 degrees from the initial [100] direction,[46] and finally reaches ∼π at [001] direction.[46] Meanwhile, when the external field rotates in the ab plane, similar result is also obtained.[46] These results are reproduced in Fig.
In summary, we have briefly reviewed the quantum transport properties of the three-dimensional Dirac semimetal Cd3As2 single crystals, including the unexpected ultra-high charge mobility, large linear magnetoresistivity, remarkable SdH oscillations, nontrivial Berry’s phase, and the evolution of the nontrivial Berry’s phase. These unusual transport results are related to the novel electronic properties of Cd3As2. Moreover, because its Fermi surfaces only consist of 3D Dirac fermions, Cd3As2 single crystals or its other structures may have a huge potential for future device applications.
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