Topological Phase Transition in Single Crystals of (Cd1−xZnx)3As2

Single crystals of (Cd1−xZnx)3As2 were synthesized from high-temperature solutions and characterized in terms of their structural and electrical properties. Based on the measurements of resistivity and Hall signals, we revealed a chemical-doping-controlled transition from a three-dimensional Dirac semimetal to a semiconductor with a critical point xc ~ 0.38. We observed structural transitions from a body-center tetragonal phase to a primitive tetragonal phase then back to a body-center tetragonal phase in the solid solutions as well, which are irrelevant to the topological phase transition. This continuously tunable system controlled by chemical doping provides a platform for investigating the topological quantum phase transition of three-dimensional Dirac electrons.

The electrical properties of these four compounds are distinct in several aspects. Zn 3 As 2 , Zn 3 P 2 and Cd 3 P 2 are semiconductors with low carrier mobility and the direct band gaps being 1.0 eV, 1.5 eV and 0.5 eV respectively [13][14][15] . Both Zn 3 P 2 and Zn 3 As 2 are p-type, while Cd 3 P 2 and Cd 3 As 2 are n-type [13][14][15] . On the other hand, previous studies on the optical properties of Cd 3 As 2 suggested that it was a semiconductor with a narrow band gap around 0.1 eV 13 . The mobility for Cd 3 As 2 was reported as high as 1.5 × 10 4 cm 2 /Vs at room temperature 13 . For comparing, the hole mobility for Zn 3 As 2 is only 10 cm 2 /Vs at room temperature 13 . Cd 3 As 2 was believed to manifest an inverted band structure due to the spin-orbital coupling (SOC) 16,17 while the other three had normal band structures.
Recent studies on Cd 3 As 2 have revealed the topological aspect of its electrical properties [18][19][20][21][22][23][24][25][26] . Band structure calculation predicted that Cd 3 As 2 was a three-dimensional (3D) Dirac semimetal with the band inversion 18 . The energy dispersion of the Dirac electron is protected by the rotational symmetry along the crystallographic c axis in the tetragonal unit cell. The 3D Dirac cones of Cd 3 As 2 have been observed in angle-resolved photoemission spectroscopy (ARPES) [19][20][21] . The Dirac-like band dispersion and the inverted band ordering was probed by the Landau level spectroscopy and quasiparticle interference in scanning tunneling microscopy (STM) 22 . Based on the electrical transport measurements, two experimental groups found an ultrahigh mobility of the Dirac electrons 23,24 . A strongly sample-dependent, large linear magnetoresistance (MR) was observed in Cd 3 As 2 at low temperatures 23 . The nonsaturating linear MR in n-type Cd 3 As 2 up to 65 T was believed to result from its mobility fluctuations 26 . The anisotropic Fermi surface with two ellipsoids of Dirac electrons along the c axis was revealed from the angular dependent measurements of SdH oscillations 24 .
Noticing the opposite band orderings in Cd 3 As 2 and the other three members in the family, we expect a band inversion transition in pseudo-binary compounds of (Cd 1−x Zn x ) 3 As 2 and Cd 3 (As 1−x P x ) 2 . A band inversion transition due to the change of the SOC strength has been observed in solid solutions of semiconductors and semimetals such as Hg 1−x Cd x Te and Pb 1−x Sn x Se 27,28 . Recent studies on the solid solutions of TlBiSe 2−x S x and Bi 2−x In x Se 3 have confirmed the existence of a quantum phase transition tuned by chemical doping from topological insulators to trivial band insulators 29,30 . A topological phase transition from a 3D Dirac semimetal to a trivial semiconductor was predicted in Na 3 Bi 1−x Sb x and Cd 3 (As 1−x P x ) 2 by first-principle calculations as well 31 .
The changes of the structures and physical properties of polycrystalline (Cd 1−x Zn x ) 3 As 2 and Cd 3 (As 1−x P x ) 2 were studied before [32][33][34][35][36] . (Cd 1−x Zn x ) 3 As 2 crystallize in a primitive tetragonal structure 32 . The majorities undergo a crossover from n-to p-type when x increases in (Cd 1−x Zn x ) 3 As 2 33,34 , while the band gap increases linearly with the proportion of Zn, according to the magneto-optical measurements 35 . However the change of the topological properties of the electronic structure has not been addressed.  In this study, we report the single-crystalline (Cd 1−x Zn x ) 3 As 2 obtained from high-temperature solution growth. Powder X-ray Diffraction (XRD) measurements revealed structural transitions from a body-center tetragonal phase for Cd 3 As 2 , to a primitive tetragonal phase for 0.07 ≤ x < 0.52, and then back to a body-center tetragonal phase for x > 0.52. The electrical resistivity and Hall measurements revealed a metal-insulator transition at a critical point x c ~ 0.38. The analysis of the MR demonstrated a transition from a 3D Dirac semimetal to a trivial direct-gap semiconductor via the modulation of the SOC strength.

Results
The low temperature phases of Cd 3 As 2 and Zn 3 As 2 were reported as α″ (P4 2 /nmc), α′ (P4 2 /nbc) and α (I4 1 cd) at different temperatures [5][6][7]12 , which evolve from a high-temperature β ( m Fm3 ) phase 6,8,12 . The β phase belongs to an antifluorite structure in which an arsenic atom is coordinated by six cationic and two vacancies randomly distributed in corners of a cube. In the low-temperature phases, two cationic atoms are missed along a diagonal of one face in the distorted cube [5][6][7]12 . At room temperature, both Cd 3 As 2 and Zn 3 As 2 were reported to crystallize in a body-center tetragonal phase 6,7,12,[19][20][21][22][23][24][25][26] . Recently single-crystal XRD measurements suggested that the crystals of Cd 3 As 2 form in the structure of I4 1 /acd instead of I4 1 /cd at room temperature 9 . Considering that our powder XRD cannot distinguish these two structures, we prefer to believe the result in ref. 9 and take I4 1 /acd to be α-phase hereafter. The unit cell of I4 1 /acd phase is made of the unit cells of P4 2 /nmc phase associated with the lattice constants = a 2 a I P and c I = 2c P (The subscripts I and P present the body-center and primitive space group respectively) ( Fig. 1(a)-(d)).
Our powder XRD measurements revealed that (Cd 1−x Zn x ) 3 As 2 have different crystal structures for different x at room temperature. Figure 2(a) shows that the crystals of Cd 3 As 2 grown from flux are α phase, and their XRD pattern exhibits the characteristic peaks of (231) I , (233) I and (237) I of the body-center tetragonal phase for 25° < 2θ < 36°. This result is consistent with what was previously reported 7,19 . Once a small amount of Zn is added (x = 0.07), the XRD pattern is distinct from that of Cd 3 As 2 . The characteristic peaks of (231) I , (233) I and (237) I disappear, while the (032) P peak of the P4 2 /nmc group occurs ( Fig. 2(a)). This peak remains resolvable until the doping level reaches x = 0.38. For 0.38 < x < 0.46, the structure reenters the body center tetragonal structure I4 1 / acd accompanied by (240) I and (244) I peaks which are exceedingly weak in the pattern of Cd 3 As 2 7, 36, 37 ( Fig. 2(b)). The peaks of (440) I with the strongest intensity stand at 40.0° and 43.1° for α − Cd 3 As 2 and Zn 3 As 2 respectively ( Fig. 3(a)). The peak of (040) P is the counterpart of (440) I in P4 2 /nmc group. Figure 3(a) shows that the peaks of (040) P and (440) I shift gradually when x changes from 0 to 1. Although the volume of the unit cell changes about 20% from x = 0 to 1, the peak shape does not change significantly in the solid solutions, indicating homogeneous chemical distributions in the crystals. The lattice constants for the samples in the I4 1 /acd group were presented in the view of the P4 2 /nmc group as = a 2 a I P and c I = 2c P . Figure 3(b) shows that a P and c P change in a precisely linear relation with respect to x, albeit the structural transitions. This result is similar as what is reported for polycrystalline (Cd 1−x Zn x ) 3 As 2 36 . Temperature dependent resistivity of (Cd 1−x Zn x ) 3 As 2 shows a clear change from a metallic to semiconducting profile when x increases from 0~0.31 to 0.38~0.58 (Fig. 4(a)). The resistivity of Cd 3 As 2 is close to what was reported in ref. 22 with the residual resistivity ratio (RRR = ρ (T=300K) /ρ (T=2K) ) being 10. The values of RRR keep almost invariant when x increases up to 0.31. For x = 0.38, the ρ(T) decreases with decreasing temperatures above 200 K, and then increases below this temperature. This complicated behavior indicates that the sample is likely a very narrow bandgap semiconductor for x = 0.38. The values of RRR then dramatically decrease for x ≥ 0.38, being 0.58 for x = 0.38 and 7.7 × 10 −6 for x = 0.58 (see more details in the final phase diagram). The changes of the RRR for different x indicate a process of band gap opening for x ≥ 0.38. Such metal-semiconductor transition point is close to that reported for polycrystals 36 . Figure 4(b) shows the MR for the samples for x ≤ 0.46 at 2 K. When x ≤ 0.31, the values of the MR are comparably large as that for Cd 3 As 2 25,38 . The values of the MR decline significantly for the semiconducting samples for x ≥ 0.38. Recent studies of Cd 3 As 2 reported linear MR at low temperatures 25,26 . In our experiments, the MR of (Cd 1−x Zn x ) 3 As 2 follows the power law of MR ∝ H α where α varies from 0.9 to 1.5 for different samples.
Here we observed the precious studies reported the SdH oscillations for (Cd 1−x Zn x ) 3 As 2 when x ≤ 0.1 and x = 0.2 39,40 . Conspicuous SdH oscillations occur in the field dependent resistivity for all the samples for x ≤ 0.38 here at low temperatures, while the oscillations were not observed for the samples for x ≥ 0.46. This observation is consistent with a metal-insulator transition with a critical point x c ~ 0.38. The part of resistivity with oscillations versus the reciprocal of the magnetic field is presented in Fig. 5. For x < 0.29, only one frequency was observed for each sample (Fig. 6(a)). The frequencies of the oscillations show a clear trend of a decline with respect to x up to 0.29 ( Fig. 6(a)). For 0.29 ≤ x ≤ 0.38, the frequencies show more significant sample difference in a same batch. Some samples show single frequencies from 15 T to 30 T, while the second and third frequencies as large as 70 T As shown in Fig. 5, the temperature dependent amplitudes of the SdH oscillations were fitted by the Lifshitz-Kosevich formula [41][42][43][44] : where k B is the Boltzmann's constant; ω c is the cyclotron frequency; T D is the Dingle temperature and A(T) is the thermal damping factor which helps to fit the energy gap ω c . For the samples with multi-frequencies, their main frequencies were analyzed. For large x, the amplitudes of the oscillations are damped less significantly by the temperatures, which indicates a smaller cyclotron effective mass (Fig. 5).
The parameters of the SdH oscillations for different x are listed in , and the cyclotron  Table 1.
effective mass m * = eB/ω c are listed as well. Figure 6 shows that m * changes in a similar manner as S F1 with respect to x. In order to better understand the metal-insulator transition, we measured Hall resistivity in (Cd 1−x Zn x ) 3 As 2 at 2 K. The field dependent Hall resistivity of the samples for x ≤ 0.31 shows a linear negative profile with SdH oscillations on the background. The negative linear-field-dependent ρ yx (H) indicates that the carriers in the samples for x ≤ 0.31 simply originate from an electron band. The carrier density decreases nearly linearly with increasing x from 0 to 0.31 ( Fig. 7(b)). These results are consistent with the observation of the decreasing SdH oscillation frequencies with respect to x 36,39,40 . ρ yx (H) becomes smaller and nonlinear for 0.38 ≤ x < 0.59. This nonlinear feature is clear for x = 0.46 (inset of Fig. 7(a)). In this range, the samples manifest a semiconducting ρ(T) profile while multi-frequencies were observed in the SdH oscillations in their MR. The Hall signals and the resistivity indicate two types of carriers. For x ≥ 0.59, the Hall signals become large and positively field-dependent, which indicate p-type semiconductors consistent with low carrier concentrations. The change of the carrier density n H and mobility μ c with respect to x is summarized in Fig. 7(b,c). Meanwhile we used the standard Bloch-Boltzmann transport to estimate the mobilities which read μ m = 1/B max as the minimum on the σ xy curves in Fig. 7(c). The mobility undergoes a similar linear decline with the respect to x.

Discussion
Our measurements show that the change of the electrical properties of (Cd 1−x Zn x ) 3 As 2 has no observable correlation with the structural transitions. This result is not unexpected according to previous band structural calculation. Both α′′ (P4 2 /nmc) and α (I4 1 /acd) phases of Cd 3 As 2 manifest similar simple band structures near the Fermi surface. Only two Dirac cones protected by rotational symmetry cross the Fermi level (E F ) along the high symmetric line Γ − Z in the Brillouin zones 18 . Therefore a structural transition cannot influence the bands near the E F .
Previous studies of the calculation and experiments showed that the negative gap is about − . − . 0 3eV 0 7eV for Cd 3 As 2 18, 35 , while the direct gap is 1.0 eV for Zn 3 As 2 13 . With a semimetal and a semiconductor as two terminals, the band inversion transition should accompany a metal-semiconductor transition at a certain x. If we assume that the band gap of (Cd 1−x Zn x ) 3 As 2 changes linearly with respect to x 35 , the critical point of the band inversion transition is estimated to occur in the range of 0.23 ≤ x ≤ 0.41. This estimation is consistent with our experimental results. The critical point can also be estimated by considering the change of the SOC strength in (Cd 1−x Zn x ) 3 As 2 . Here we assume that the band inversion is solely induced by the change of the SOC strength, which is proportional to Z 4 /n 3 in case of the hydrogenic wavefunctions in a Coulomb field where Z is the nuclear charge and n is the principal quantum number 45 . Then the critical point is estimated as .  The band structure calculation for Cd 3 (As 1−x P x ) 2 revealed a topological phase transition from a Dirac semimetal to a trivial semiconductor induced by the change of the SOC strength 31 . When x increases, the two Dirac points along the k z axis gradually move closer and then merge at the Γ point under the protection of the crystal symmetry. A direct band gap is opened beyond the critical point. The process of the band inversion transition in (Cd 1−x Zn x ) 3 As 2 should be similar as that in Cd 3 (As 1−x P x ) 2 . Despite of large chemical replacement in the crystals,   the Dirac cones are robust, which is supported by the vanishing disorder self-energy around the crossing points 46 . Further investigation such as ARPES measurements for (Cd 1−x Zn x ) 3 As 2 will help to reveal the details of this topological phase transition. Cd 3 As 2 is always n-type due to As vacancies, while Zn 3 As 2 is p-type because extra Zn vacancies serve as electron acceptors 13,38 . Since both two types of carriers come from element vacancies, an n to p transition is expected in (Cd 1−x Zn x ) 3 As 2 . With increasing x, the zinc doping will suppress the chemical potential, which crosses a small Fermi surface near the Dirac cones. The decrease of S F1 with increasing x is a comprehensive result of the change of the band structure and chemical potential.

Parameters
For 0.29 ≤ x ≤ 0.38, we found strongly sample-dependent frequencies of SdH oscillations. For a very narrow bandgap semiconductor or semimetal, any small change of the carrier concentrations will affect the chemical potential dramatically near the band touching. The strong sample-dependence and the complicated SdH oscillations in this regime are not unexpected.
Summary. Single crystals of (Cd 1−x Zn x ) 3 As 2 were synthesized from high-temperature solutions. Based on the analysis of the electrical properties, we realized a transition from a 3D Dirac semimetal to a semiconductor with the critical point . x 0 38 c in these solid solutions (Fig. 8). The structural transitions do not affect the electrical properties in this system. The topological aspect of this metal-insulator transition needs experimental exploration in the future.

Methods
Single crystalline (Cd 1−x Zn x ) 3 As 2 samples were grown from high temperature solutions with the initial concentration of starting elements being (Cd 1−y Zn y ) 9 As 1 . The mixtures were sealed in evacuated quartz ampoules, and then kept at a high temperature between 800 °C and 1100 °C for two days, and then slowly cooled down to 425 °C with a rate of −5 °C/hour. After staying at 425 °C for one day, the ampoules were centrifuged to separate crystals from flux. The single crystals of Cd 3 As 2 were mainly 3D bulks with triangular facets, but several needle-like crystals were found in the growth as well. The shapes of the crystals were similar as that described in refs 20, 21. When zinc was added to the solutions, the sizes of the 3D crystals decreased, while some flake-like crystals with smooth or mesa-landscape-like surfaces appeared. For x ≥ 0.58, the flake-like crystals were dominant and no 3D crystals appeared in the growth. XRD measurements revealed that both the triangular facets ( Fig. 1(f)) and the large surface of the flake-like crystals (Fig. 1(e)) were either the (011) P face of the P4 2 /nmc structure or the (112) I face of the I4 1 /acd structure, which were the counterparts of each other (Fig. 2(c,d)). In a same batch of growth, the crystals with different morphologies did not show larger difference of physical properties than those with the same morphologies.
In order to determine the zinc concentration x, we measured the Energy Dispersive X-ray Spectrum (EDX) of the samples in an FEI Nova NanoSEM 430 spectrometer. The samples with no residual cadmium were selected in the measurements and their EDX spectrum was observed through an overall area scanning. The linear relation of the measured zinc concentrations x and the initial y (0 ≤ y ≤ 0.1) is shown in Fig. 3(c). By weighing the mass of the crystals yielded in every growth, we found that all the initial stoichiometric zinc was compounded in the crystals.
The powder XRD data was collected from a Rigaku MiniFlex 600 diffractometer and then refined by a Rietica Rietveld program. As shown in Fig. 3(b), the lattice constants and the volume of the unit cell change linearly with x in accordance with a Vegards law. This result is same with what observed in polycrystalline (Cd 1−x Zn x ) 3 As 2 36,47 . Figure 8. Phase diagram for (Cd 1−x Zn x ) 3 As 2 . With the concentration of Zn increasing, the samples transform from a topological Dirac semimetal to a semiconductor. The upper sketches of the band structure illustrate this transition. The structure transforms from I4 1 /acd to P4 2 /nmc then back to I4 1 /acd with vertical dashed lines serving as rough boundaries. The background color presents the gradual change of resistivity at 2 K as x increases (inset of Fig. 4(a)). The changes of S F1 and RRR are plotted in the diagram as well.
Scientific RepoRts | 7: 3148 | DOI:10.1038/s41598-017-03559-2 Therefore we selected x determined by EDX measurements as the nominal zinc concentrations, which were estimated to have less than ± 1% difference between the real zinc concentrations. More details of the XRD experiments are discussed in the Result part.
Single crystals were polished to the bars with length ~1.0 mm, width ~0.4 mm and thickness ~0.3 mm for electrical transport measurements. The crystallographic orientation of the bars was same as those chosen in the recent experiments [21][22][23][24][25] , i.e., the current was perpendicular to the [112] I direction in I4 1 /acd (or [011] P direction in P4 2 /nmc), and the magnetic field was along the [112] I direction in I4 1 /acd (or [011] P direction in P4 2 /nmc). The electrical resistance and Hall voltage were measured via a four-point contact method in Quantum Design Physical Property Measurement System (PPMS-9).