Anisotropic Fermi Surface and Quantum Limit Transport in High Mobility 3D Dirac Semimetal Cd3As2

The three-dimensional (3D) topological Dirac semimetal is a new topological phase of matter, viewed as the 3D analogy of graphene with a linear dispersion in the 3D momentum space. Here, we report the angular dependent magnetotransport in Cd3As2 single crystal and clearly show how the Fermi surface evolves when tilting the magnetic field. Remarkably, when the magnetic field lies in [112] and [44-1] axis, only single oscillation period features present, however, the system shows double period oscillations when the field is applied along [1-10] direction. Moreover, tilting the magnetic field at certain direction also gives double period oscillations. We attribute the anomalous oscillation behavior to the sophisticated geometry of Fermi surface and illustrate a complete 3D Fermi surfaces with two nested anisotropic ellipsoids around the Dirac point. Additionally, a sub-millimeter mean free path at 6 K is observed in Cd3As2 crystal, indicating a large ballistic transport region in this material. Tracking the magnetoresistance oscillations to 60 T, we reach the quantum limit (n = 1 Landau Level) at about 43 T. These results improve the knowledge of the Dirac semimetal material Cd3As2, and also pave the way for proposing new electronic applications based on 3D Dirac materials.

Motivated by theoretical predications [1,2], to date, Na 3 Bi [3] and Cd 3 As 2 [4,5,10,21] have been identified to be 3D topological Dirac semimetal by angle resolved photoemission spectroscopy (ARPES). More recently, scanning tunneling microscopy (STM) experiments have revealed quasiparticle interference and the extended Dirac-like dispersion in Cd 3 As 2 [6]. This unusual band dispersion also makes the electrons around the Fermi surface behave many unusual transport phenomena such as strong quantum oscillations [8], ultrahigh mobility [9] and large magnetoresistance [9,12]. However, most magnetotransport measurements have been limited to study the quantum oscillations in one direction or by rotating the magnetic field only in certain plane. The scarcity of the complete 3D Fermi surface analysis hinders an in-depth understanding of the physical properties in Cd 3 As 2 single crystal. Therefore, it is highly desirable to demonstrate the transport property study at different magnetic field direction as well as the angular dependent magneotransport to reveal the complicated 3D Fermi surface in Cd 3 As 2 system. Further, the high magnetic field transport measurements to reveal the physics in quantum limit of Cd 3 As 2 are also highly pursued.
Here, we present a systematic study of the magnetotransport in Cd 3 As 2 single crystal and firstly extend our study to the angular dependence of magnetoresistance in three independent directions as well as the high magnetic field experiments (up to 60 T). The SdH oscillations signified a previously unknown Fermi surface with two nested ellipsoids, leading to a good understanding on its 3D Dirac nature and also providing a platform to explore exotic physical phenomena.

II. RESULTS
A. Sample structure Single crystals of Cd 3 As 2 are synthesized from a Cd-rich melt with the stoichiometry Cd 8 As 3 in the evacuated quartz ampoule [22]. The Cd 3 As 2 single crystal is examined by a FEI TITAN Cs-corrected cross sectional scanning transmission electron microscopy (STEM) operating at 200kV. Fig. 1(a) shows the atomic layer by layer high angle annular dark field scanning transmission electron microscopy (HAADF-STEM) image which manifests a high quality single crystal nature of Cd 3 As 2 sample. An optical image of the measured sample is shown in the inset of Fig. 1(d). The crystal is needle-like and grows preferentially along the [1ī0] direction (the length direction). The width direction is along [44ī] and the largest facet of the crystal is (112) plane.
Standard four probe method is used to measure the Cd 3 As 2 samples on its (112) plane. Two indium 3 or silver paste current electrodes (I+ and I-) are pressed on both ends and across the entire width of the sample, so that the current can homogeneously go through the sample in the length direction [1ī0]. The other two indium or silver paste electrodes were pressed on the crystal as voltage probes. Angular dependence of magnetoresistance was measured by rotating the sample in (1ī0) plane and (112) plane, characterized by the angle and illustrated in Fig. 1(b)respectively. The transport measurements were carried out in a PPMS-16T system (Quantum Design) and pulsed high magnetic field at Wuhan National High Magnetic Field Center.
B. Ultrahigh mobility in Cd 3 As 2 single crystal More than ten samples have been studied. Data presented here are from three typical samples: Sample 1, 2 and 3. Fig. 1(c) shows the resistivity of sample 1 as a function of temperature (T). The resistivity decreases almost linearly when T decreases from 300 K to about 6 K and then tends to be saturated. It displays a perfect linear metallic property with a large residual resistivity ratio RRR = 2120 (the resistivity at room temperature over the resistivity at 6 K). As shown in the insert panel of Fig. 1(c), the resistivity is quite low (about 11.6 n cm   at 6 K). Similar results have been reported before [9], suggesting the behavior is related to its long transport lifetime. More interestingly, when T is lower than 6 K, the resistivity oscillates near zero resistivity within the resolution of our measurement instrument (about 4.5 n cm   ), which could be understood in terms of the quantum ballistic transport. Estimated from the Shubnikov-de Haas (SdH) oscillations of sample 1, the carrier density is about 18 3 5.86 10 cm  

(details shown in Supplemental
Material [23]). It is worth noting that the carrier density estimation from SdH might be two to ten times smaller than the carrier density in Hall measurement [9]. Therefore, the mobility is conservatively estimated to be 9.19×10 6 ~ 4.60×10 7 cm 2 /Vs and the mean free path 0 l is about 0.25 ~ 1.25 mm at 6 K ,which are consistent with the previous transport study [9]. The high mobility and long mean free path may shed new light on realizing various applications on future functional devices. Additionally, the linear metallic T behavior is also observed in other samples, such as sample 2 shown in Fig. 1 In order to verify and analyze the SdH oscillations, we fit the entire oscillatory component with the standard Lifshitz−Kosevich (LK) theory for a 3D system [24][25][26] cos 2 sinh where m * is the cyclotron effective mass of the carriers and T D is the Dingle temperature. is the phase related to the berry phase while is a phase shift determined by the dimensionality, taking the value ±1/8for 3D system [25,27]. Considering the two frequencies obtained in B [1ī0] direction, we use two independent parameters F 1 and F 2 to fit the oscillation. As shown by the red solid line in Fig. 2 direction is plotted in Fig. 3(a). The maxima of  are assigned to be the integer indices (solid circles) while the minima of are plotted by open circles in the diagram as half integer indices [25]. A linear extrapolation of the index plot gives the intercept value close to 0.3. Considering the study before [25], in a D system, the intercept of the index plot should be ±1/8 (+ for holes and -for electrons). In our system, the intercept is 0.3, deviating from ±1/8, indicating the Fermi surface is an anisotropic ellipsoid instead of spherical with perfect Berry phase. Similarly, the LL fan diagram in B [44ī] direction is shown in Fig. 3(b). As discussed previously, the intercept value obtained is about 0.2, different from 0.3 yielded in B [112] direction, also consistent with the assumption of the anisotropic Fermi surface in Cd 3 As 2 system. Fig. 3(c) displays the magnetoresistance behavior measured in pulsed high magnetic field normal to the (112) plane up to 54 T at various temperatures. The larger and more obvious SdH oscillations in high field provide an opportunity to study the further physics in the quantum limit. However, limited by the measurement resolution, the small oscillations in the lower field are hard to distinguish from noise.
A similar plot based on the maxima and minima of versus the index n is shown in Fig. 3(d). The intercept shifts to 0.38 instead of 0.3 deduced from the relatively low magnetic field. It suggests that the high magnetic field is needed to fix the intercept more reliable. It is worth noting that we reach the quantum limit in sample 3 and more details are shown in section D.
To study the 3D Fermi surface of Cd 3 As 2 systematically, the angular dependence of SdH oscillations are important to show the evolution of Fermi surface changing in different directions. Fig. 4(a) shows the angular dependent oscillations after removing the polynomial background for the transverse rotation (B⊥I) by varying magnetic field angle tilting from perpendicular field (B [112] ) to parallel field (B [44ī] ). is the angle between magnetic field and [112] axis. Fig. 4 Umklapp relation [28] can be satisfied and the Umklapp electron-phonon scattering processes plays dominant role on the resistivity at low temperature in Cd 3 As 2, which leads to R ~ T· N ph with N ph denoting the number of phonons that satisfies the Umklapp relation [28]. Moreover, due to the ultra-large unit cell of Cd 3 As 2 , low energy optical phonon modes might exist in the system. Therefore, the Umklapp processes and optical phonon modes can lead to almost linear R-T relation down to very low temperature, and the observed linear R-T behavior in Fig. 1(c) and (d) deviates from the ordinary Bloch-Grüneisen Law with R~T 5 and the electron-electron interaction induced R~T 2 law [28].
D. Quantum limit of Cd 3 As 2 single crystal: SdH oscillations with Zeeman splitting and linear magnetoresistance in the quantum limit We extend the SdH measurements to 60 T to explore the quantum limit. According to the theory before [29], a quantum linear magnetoresistance would be expected to occur in the quantum limit  Fig. 6(b), with the intercept is about 0.11. When approaching the quantum limit of n = 1, the maxima (or minima) points shows small deviations from the fitting line, which originates from the anomalous oscillation in 3D Dirac system around the Quantum limit [30].

III. CONCLUSIONS
In summary, we firstly report pronounced SdH oscillations measured along three different directions in high quality Cd 3 As 2 single crystals, and extend our study to the angular dependent magnetotransport and in the high magnetic field up to the quantum limit. By analyzing the SdH oscillations in different magnetic field orientations, we obtain a complete 3D Fermi surface with two nested ellipsoids. Furthermore, we present the changing of the angular-dependent oscillation periods are essentially due to the complicated nested 3D Fermi surface. In addition, the sub-millimeter scale mean free path and ballistic transport region as well as the quantum limit have been demonstrated in Cd 3 As 2 single crystal. These results intensify the previous knowledge of the Dirac semimetal material Cd 3 As 2 , offer a better understanding of existing 3D Dirac semimetals, and reveal the potential of application in topological electronic devices.