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Supplemental Materials

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

Yanfei Zhao1,2 , Haiwen Liu1,2, Chenglong Zhang1,2, Huichao Wang1,2, Junfeng Wang3, Ziquan Lin3, Ying Xing1,2, Hong Lu1,2, Jun Liu4, Yong Wang4, Scott M. Brombosz5, Zhili Xiao5, Shuang Jia1,2, ‡, X. C. Xie1,2, † and Jian Wang1,2,*

1International Center for Quantum Materials, School of Physics, Peking University, Beijing 100871, China

2Collaborative Innovation Center of Quantum Matter, Beijing 100871, China

3Wuhan National High Magnetic Field Center, Huazhong University of Science and Technology, Wuhan 430074, China

4Center of Electron Microscopy, State Key Laboratory of Silicon Materials, Department of Materials Science and Engineering, Zhejiang University, Hangzhou, 310027, China

5Materials Science Division, Argonne National Laboratory, Argonne, Illinois 60439, USA

Correspondence authors

*jianwangphysics@pku. (J.W.)

†xcxie@pku. (X.C.X.)

‡gwljiashuang@pku. (S.J.).

Contents:

I. Sample growth and XRD results

II. Quantum oscillations in Sample 1

III. SdH oscillations analysis in B[112] and B[44ī] directions for sample 2

IV. FFT analysis at varied temperatures in different directions for sample 2

V. FFT analysis of the angular dependence of SdH oscillations for sample 2

VI. Hall Measurements of Sample 2

VII. Magnetotransport measured in Pulsed High Magnetic Field of Sample 2

VIII. The anisotropic Fermi surface and estimation of carrier density

I. Sample growth and XRD results

Single crystals of Cd3As2 were grown from a Cd-rich melt with the ratio of Cd:As = 85:15. High purity elements (99.99%) were sealed in an evacuated quartz ampoule. The ampoule was heated up to 825oC and kept there for 48 hours. Then it was cooled down to 425oC at a rate of 6 oC/hr. The remaining liquid was decanted at this temperature. The single crystals of Cd3As2 crystallized in a needle-like form with metallic luster. The typical dimensions of our measuremented sample are ～ 0.9 mm × 0.2 mm × 0.3 mm (length × width × thickness).

X-ray diffraction (XRD) patterns were collected on a Bruker AXS Smart diffractometer with a Mo Kα x-ray source (λ = 0.71073 Å) and an APEX II CCD detector. Face indexing was performed utilizing the Bruker Apex2 software (v2012.4). ShelXTL (v. 2008/1) was utilized to determine the space group and unit cell dimensions. CrystalMaker for Windows (v 9.1.4) was utilized to visualize crystallographic planes and calculate their relative angles.

Nice diffraction patterns are shown in the inset of Fig. 1(a) in the main text. We were able to identify the Miller indices for each face of the crystal. The collection of 1979 frames produced 23816 reflections. The derived lattice parameters are a = b = 12.6207(5) and c = 25.3756(14) which matches the literature report to within 0.2%. The initial refinement also was able to identify the space group as I41/acd, which also matches the literature reports. Since the x-ray beam dimensions (0.5 mm diameter) are comparable to the crystal size, we conclude that our samples are single crystals.

II. Quantum oscillations in Sample 1

[pic]

Supplementary Figure S1: Quantum oscillations from 4 T to 15 T in Sample 1 at B[112] direction. (a) After subtracting the nonoscillating background deduced by fitting a fourth-order polynomial, the oscillatory component Δρ as a function of 1/B at T = 2 K. Compared with the oscillations in sample 2, the amplitude of the oscillation here is smaller. (b) FFT analysis obtain a single frequency F = 54.30T, similar as the frequency measured in sample 2.

III. SdH oscillations analysis in B[112] and B[44ī] directions for sample 2

[pic]

Supplementary Figure S2: SdH oscillations analysis in B[112] and B[44ī] directions for sample 2. (a)-(b) The frequency of the SdH oscillations was extracted from the fast Fourier transform (FFT) analysis from 2 T to 15 T. Single frequency was observed in both B[112] and B[44ī] directions. BF[112] = 51 T and BF[44ī] = 41 T. (c)-(d) shows the fitting to the entire oscillatory component with the standard Lifshitz − Kosevich (LK) theory for a 3D system in B[112] and B[44ī] directions from 4 T to 15 T respectively. The red solid line is the reasonable close fit to the observed oscillations (the black solid line) by the LK expression with a single frequency. (e)-(f) Plotted is the relative amplitude of oscillation Δρ/ρ2K as a function of temperature for the Landau Level located at n = 7 marked in the maintext Fig. 2 in B[112] and B[44ī] direction, respectively. By fitting the curve, we subtract the cyclotron effective mass m*[112] = 0.043 me and m*[44ī] = 0.036 me (me is the free electron mass).

IV. FFT analysis at varied temperatures in different directions for sample 2

[pic]

Supplementary Figure S3: FFT analysis at varied temperatures in different directions. (a) B // [112] axis. (b) B // [44[pic]] axis. (c) B // [1[pic]0] axis.

The frequency of FFT analysis is independent of the temperature as well as the disorders in the sample. The FFT analysis in three different directions at various temperatures is shown in Fig. S3. In the perpendicular case when B is parallel to [112] axis (Fig. S3(a)), there is only one dominating peak appeared at 51 T. With increasing the temperature, the amplitude of the peak is gradually decreased. As for the parallel case with B // [44[pic]] (Fig. S3(b)), only single peak is observed at 41 T, and shows little temperature dependence too. These observations are consistent with previous experimental results. When B is parallel to [1[pic]0] axis, two periods are shown in Fig. S3(c). When changing the temperature, the position of the two periods kept almost unchanged. The truncation range of FFT analysis for B[112] and B[44ī] direction is from 2 T to 15 T, and the truncation range of FFT analysis for B[1ī0] is from 2.5 T to 15 T. The number of oscillation periods is independent of the truncation range.

V. FFT analysis of the angular dependence of SdH oscillations

[pic]

Supplementary Figure S4: FFT analysis of angular dependent SdH oscillations for various magnetic field angles from (a) B[112] (θ ’ 0 deg, B in the [112] direction) to (g) B[44ī] (θ ’ 90 deg, B in the [44ī] direction).

[pic]

Supplementary Figure S5: FFT analysis of angular dependent SdH oscillations for various magnetic field angles from (a) B[44ī] (ϕ ’ 0 deg, B in the [44ī] direction) to B[1ī0] (ϕ ’ 90 deg, B in the [1ī0] direction).

The truncation range of FFT analysis is from 2 T to 15 T (the oscillations are totally suppressed below 2 T). The FFT analysis corresponding to different magnetic field directions are shown in Fig. S4 and Fig. S5. The frequencies of FFT analysis show a little differences from that obtained under truncation 4 T -- 15 T. However, it is worth noting that the numbers of the periods are unchanged when analyzing the different range of data. Thus, the Fermi surface with nested ellipsoids shape is unchanged. As for the frequencies obtained from Fig. S5 and frequency vs angle behavior shown in Fig. S6, due to the almost isotropic Fermi surface in the a-b plane, when magnetic field is twisted from [44ī] to [1ī0] (with dihedral angle [pic] [pic] respect to the a-b plane), the period changes a little during the rotation. Moreover, when magnetic field rotates between [112] and [44ī], the first frequency F1 shows the largest frequency around 30 deg before leaving the nested region.

[pic]

Supplementary Figure S6: Frequency vs. Angle plots when magentic field is twisted from [112] to [44ī] (a) and from [44ī] to [1ī0] (b). F1 and F2 stand for the two different FFT frequencies.

VI. Hall Measurements of Sample 2

[pic]

Supplementary Figure S7：Hall measurements in Cd3As2 sample at different temperatures

From the Hall measurements in our Cd3As2 sample at different temperatures shown in Figure S4, the carriers are electrons, indicating that the Fermi energy lies in the conduction band instead of valance band.

VII. Magnetotransport measured in Pulsed High Magnetic Field of Sample 2

[pic]

Supplementary Figure S8: Magnetotransport measured in Pulsed High Magnetic Field. Figure displays the magnetoresistance behavior measured in pulsed high magnetic field normal to the (112) plane up to 54 T at various temperatures.

VIII. The anisotropic Fermi surface and estimation of carrier density

We calculate the Fermi momentum using Weyl Hamiltonian: [pic] based on the SdH oscillation data. Due to the C4 symmetry in X-Y plane and anisotropic SdH oscillation period in [112] and [44ī] direction, the velocity satisfies [pic] and [pic]. Based on the SdH period SF[112] = 51 T and SF[44ī] = 41 T for B[112] and B[44ī] direction, we calculate the momentum [pic] = 0.033 [pic] and [pic]= 0.073 [pic], and thus find Fermi velocity vx = vy = 2.2* vz. For the B[1ī0] direction case, the small period F1 = 44 T and [pic]= 0.034 [pic] give out the half major axis value k1 = 0.039[pic] for red (or blue) ellipse in Fig. 5(d) of the main text. From the analysis of two periods behavior in B[1ī0] direction, the Dirac point location is deduced to be (0, 0,[pic]k0). Considering kz = 0.073 [pic] and k1 = 0.039 [pic], one can estimate the location of Dirac point K (K’) = (0, 0,[pic]k0), with k0 = 0.056 [pic]. Moreover, we give out the orientation range of the nested ellipses region as shown by purple dashed line in Fig. 5(d) in the maintext. From kz = 0.073 [pic] and k1 = 0.039 [pic], we calculate the orientation range is about [-[pic],[pic]], which is qualitatively consistent with the angle range with double periods features. The little deviation in the orientation range may attribute to the deformation of energy bands in the overlapping region. This deformation of energy bands around the Lifshitz saddle point might influence the Fermi velocity. From the momentum [pic] = 0.033 [pic] and [pic]= 0.073 [pic], we estimated the carrier density from two Dirac pockets to [pic]. We consider that the Fermi surface geometry of sample 1 is similar to sample 2. Thus, based on SF[112] = 54.3 T in sample 1, the carrier density is estimated to [pic].

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