Quantum transport properties of the three-dimensional Dirac semimetal 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. 1(a). Note that in the first theoretical paper predicting Cd₃As₂ to be DSM,^[6] one of the two crystal structures from early experiment^[51] is non-centrosymmetric and the four-fold Dirac nodes are protected by proper uniaxial rotational symmetry instead of inversion symmetry. Ali et al. reinvestigated the crystal structure by single-crystal x-ray diffraction and pointed this out.^[48] They calculated the electronic structure using the experimentally determined centrosymmetric structure and found that the band structure is similar to the noncentrosymmetric one.^[48] The overall structure of Cd₃As₂ in Fig. 1(a) can be viewed as a 2 × 2 × 4 superstructure stacked from the small cubic sub-cell shown in Fig. 1(b),^[48] in which the cubic Cd lattice with two vacancies resides in a face-centered cubic As lattice.

Figure 1(c) shows the calculated electronic structure of Cd₃As₂ from the crystal structure shown in Figs. 1(a) and 1(b). The bulk conduction band (contributed by Cd 5s states) and valence band (contributed by As 4p states) cross at a discrete point (i.e., Dirac point) along Γ–Z direction at the Fermi level. In the presence of C₄ rotational symmetry and time-reversal symmetry, the 3D Dirac point in Cd₃As₂ is 4-fold degenerate at the Fermi level.^[48] The bulk Brillouin zone of Cd₃As₂ is plotted in Fig. 1(d), in which two Dirac points locate on the opposite sides of the Brillouin zone center point Γ along the k_z direction. Moreover, according to Fig. 1(c), there are no other additional trivial bands crossing the Fermi level, which means the Fermi surface of Cd₃As₂ only consists of 3D Dirac points. Therefore, the transport properties are only dominated by 3D Dirac fermions. As mentioned above, Cd₃As₂ is very stable in air, unlike Na₃Bi. Therefore, Cd₃As₂ is an ideal system to investigate the quantum transport properties of DSM.

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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 n_e 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 1). The high charge mobility (∼ 10⁴ cm²/Vs) in Cd₃As₂ has been known from early transport studies.^[53,54] Surprisingly, Liang et al. discovered an unexpected ultra-high charge mobility (as high as 9 × 10⁶ cm²/Vs at 5 K) in some Cd₃As₂ single crystals, which is shown in Fig. 2.^[39] Actually, they found that there are two kinds of single crystals which were named as "Set A" (single domain) and "Set B" (multidomain) samples. The ultra-high charge mobility exists in "Set A" samples with very low residual resistivity (21 nΩ·cm) and very high RRR (4100) (Table 1). They proposed a protection mechanism that strongly suppress backscattering in zero magnetic field,^[39] and this protection results in the ultra-high mobility. Later, this kind of Cd₃As₂ single crystal with ultra-high mobility was also investigated by Zhao et al.^[42]

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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 3 shows the longitudinal magnetoresistance (MR) at various temperatures, with the magnetic field perpendicular to the (112) plane.^[38] The MR is defined by MR = (R_xx(B) − R_xx(0 T))/R_xx(0 T) × 100%. Firstly, one can see that the MR shows no sign of saturation with magnetic field up to 14.5 T even near room temperature (280 K). Secondly, with decreasing temperature, the MR increases dramatically and oscillates remarkably. At 1.5 K, a large MR up to ∼ 1600% was observed in a field of 14.5 T and the oscillations can be tracked down to a field as low as B ≈ 2 T which reflects the high mobility in Cd₃As₂.

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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. 4(a). Similar situations have also occurred in other transport works.^[39–41] Liang et al. and Feng et al. suggested that the linear MR in Cd₃As₂ single crystals may relate to the field-induced change to the Fermi surface,^[39,40] since the Dirac–Fermi surface would split into two Weyl–Fermi surfaces due to time-reversal symmetry breaking under magnetic fields.^[40] However, Narayanan et al. performed magnetotransport measurements of Cd₃As₂ single crystals with linear MR in magnetic fields up to 65 T (approaching the quantum limit) and found no additional frequencies (only spin splitting due to the large g factors) or changes in scattering.^[4] Therefore, they proposed that the unconventional linear MR is likely caused by disorder effects, as it scales with the high mobility rather than directly linked to Fermi surface changes.^[41] Note that quantum linear MR has been observed in Cd₃As₂ single crystals at low temperature by Zhao et al., in which the linear MR appeared after reaching the quantum limit (about 43 T in their samples).^[42]

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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 4(a) shows the oscillatory component ΔR_xx versus 1/B at various temperatures after subtracting a smooth background. One can see that the oscillations are periodic in 1/B, as expected from the successive emptying of the Landau levels when the magnetic field is increased. Fast Fourier transform (FFT) method is employed to analyze the frequency of the oscillations, and the result is shown in Fig. 4(b), which consists of only one single frequency F = 58.3 T.^[38] Therefore, one can directly obtain the Fermi surface area A_F normal to the applied field via the Onsager relation F = (Φ₀/2π²)A_F, where Φ₀ = 2.07 × 10⁻¹⁵ T·m² is the flux quantum. The frequency F = 58.3 T corresponds to A_F = 5.6 × 10⁻³ Å⁻².^[38] By assuming a circular cross section, a very small Fermi momentum k_F ≈ 0.042 Å⁻¹ is estimated.^[38] This result is in good agreement with the ARPES experiment, which gives k_F ≈ 0.04 Å⁻¹.^[29] Note that there are two 3D Dirac points along k_z direction in Brillouin zone, i.e., two Fermi pockets. If the k_F is larger than the distance between the high-symmetry Γ point and Dirac point, these two Fermi pockets will touch each other and the Lifshitz transition (i.e., the Fermi level reaches the valence band top) occurs. Therefore, it is necessary to check whether the Lifshitz transition happens or not. From an ARPES study on a Cd₃As₂ single crystal, the distance between Γ point and Dirac point is about 0.15 Å⁻¹.^[27] According to the relation we can estimate the critical carrier concentration where the Lifshitz transition happens is about 2.4 ×10²⁰ cm⁻³. Such a value is much larger than our k_F and other works' (Table 1), which means that Cd₃As₂ samples with carrier concentration n_e being the order of ∼ 10¹⁸ cm⁻³ are not close to the Lifshitz transition. It is worth mentioning that Zhao et al. have observed double period oscillations with aligning the magnetic field at certain directions, and they attribute the anomalous oscillation behavior to the Lifshitz transition in their Cd₃As₂ samples.^[42]

The SdH oscillation amplitude in Cd₃As₂ can be well described by the Lifshitz–Kosevich formula for a 3D system^[61–64]

where ω_c is the cyclotron frequency and T_D is the Dingle temperature. R_T is the thermal damping factor and R_D is the Dingle damping factor. Figure 3(c) shows the temperature dependence of the normalized oscillation amplitude at 1/B = 0.0928 T⁻¹, which corresponds to the 6th Landau level. Fitting the data by employing Eq. (2), ħω_c ≈ 24.6 meV for n = 6 can be obtained.^[38] For Dirac system, cyclotron frequency ω_c follows the square root dependence on magnetic field.^[6,65] By employing and v_F = ħk_F/m*, rather small cyclotron effective mass m* ≈ 0.044m₀ and very large Fermi velocity v_F ≈ 1.1 × 10⁶ m/s are obtained.^[38] These values evaluated by transport measurements (Table 1) are consistent with the ARPES and STM results.^[27–30] This very large Fermi velocity may explain the unusual high mobility in Cd₃As₂.

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

where A_F is the extremal cross-sectional area of the Fermi surface related to the Landau level n; 2πβ is the Berry's phase, and δ is an additional phase shift resulting from the curvature of the Fermi surface in the third direction, taking the value δ = 0 and ±1/8 for the 2D and 3D, respectively.^[64,65,80] Therefore, to obtain the value of Berry's phase, one can plot the index field 1/B versus the Landau index and track the intercept in the limit B → ∞. For the trivial parabolic dispersion in conventional metals, the Berry's phase should be zero; for Dirac systems with linear dispersion, there should be a nontrivial π Berry's phase because the zeroth Landau level is pinned to the Dirac point.^[71–74]

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. 5.^[38] The data points from two Cd₃As₂ single crystals fall into two straight lines, and the linear extrapolation gives the intercepts 0.58 ± 0.01 and 0.56 ± 0.03, respectively.^[38] This is a strong evidence for the existence of Dirac fermions in Cd₃As₂ from the bulk transport measurements. The slight deviations from 1/2 indicate the additional phase shifts δ ≈ 0.08 and 0.06, which may result from the 3D nature of the Fermi surface of Cd₃As₂.^[39–41,64] The existence of the nontrivial Berry's phase has been verified by some other measurements of SdH^[41–44,46] and dHvA^[43] oscillations, which are presented in Fig. 6. Therefore, complementary to surface-sensitive ARPES^[27–29] and STM^[30] experiments, the bulk transport measurement results confirm the 3D DSM phase in Cd₃As₂.

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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 k_z 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. 7, which are consistent with the theoretical prediction.^[5,6,30] They proposed that these changes of the Berry's phase suggest a possible topological phase transition.^[44]

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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. 8. They proposed that this shift of ∼π of the Berry's phase may also hint at a possible topological phase transition induced by a magnetic field tilted away from the high-symmetry direction.^[46]