Transport evidence of mass-less Dirac fermions in (Cd1−x−yZnxMny)3As2 (x + y = 0.4)

Charge carriers parameters on a 2D-layer surface for (Cd1−x−yZnxMny)3As2 (y = 0.08) (the concentration n 2 D = 1.9 × 1012 cm–2, the effective value of the 2D-layer d 2 D = n 2 D / n 3 D = 14.5 nm, the wave vector k F = 0.1 nm–1, the charge carriers relaxation time due to dispersion τ D = 1.8 × 10–13 s, the velocity of charge carriers on Fermi surface v F = ℏ k F / m c = 2.65 × 105 m s−1, the mean free path l F = v F τ D = 47.7 nm) were determined. It was found that the dependence of the cyclotron mass mс(0)/m0 on Fermi wave vector kF for (Cd1−x−yZnxMny)3As2 (y = 0.08) is in compliance with a theoretical linear dependence, that describes mass-less Dirac fermions.

3 =14.5 nm, the wave vector k F =0.1 nm -1 , the charge carriers relaxation time due to dispersion t D =1.8×10 -13 s, the velocity of charge carriers on Fermi surface =  v k m F F c =2.6 5 ×10 5 m s −1 , the mean free path t = l v F F D =47.7 nm) were determined. It was found that the dependence of the cyclotron mass m с (0)/m 0 on Fermi wave vector k F for (Cd 1−x−y Zn x Mn y ) 3 As 2 (y=0.08) is in compliance with a theoretical linear dependence, that describes mass-less Dirac fermions.

Introduction
Among topological Dirac and Weyl semimetal (TDSs and TWSs) materials Cd 3 As 2 has been treated as ideal because of its ultrahigh mobility and chemical stability in air. It allows considering Cd 3 As 2 as a promising candidate for finding new topological phases [1][2][3][4]. The existence of nontrivial topological characteristics of 3D and 2D electronic states are of wide interest [5][6][7].
Earlier we discussed the results of studying Shubnikov-de Haas (SdH) oscillations in (Cd 1−x−y Zn x Mn y ) 3 As 2 (CZMA) compound (х+y =0.4) [8]. SdH effect was investigated in a temperature range T=4.2÷300 K and in a transverse magnetic field B=0÷25 T. The values of the cyclotron mass m c , the effective g-factor g * and Dingle temperature T D were determined. For a sample with a composition y=0.04, x=0.36 a strong dependence of the cyclotron mass on a magnetic field was observed. Our results of Fast Fourier Transform (FFT) analysis based on studying Shubnikov-de Haas oscillations indicate the presence of topological properties. For other composition (y=0.08, x=0.32) the magnitude of the phase shift was β= 0.44 being close to β=0.5, which also suggests that single CZMA crystals with y=0.08 demonstrate properties of Dirac semimetals and indicates the presence of Berry phase and 3D Dirac fermions in Cd 3 As 2 single crystals [8,9]. Magnetic field dependences of resistivity have been recently measured at various orientations between a magnetic field vector and electrical current,  J directed along (100) crystal plane. Magnetoresistance dependences ( ) r r D B demonstrate unusual features in (Cd 1−x−y Zn x Mn y ) 3 As 2 (х+y =0.4; y=0.04) single crystal at different orientation. An asymmetry and parity violation of magnetoresistance of magnetic diluted Dirac-Weyl semimetal (Cd 0.6 Zn 0.36 Mn 0.04 ) 3 As 2 was established [10].
The purpose of this investigation was to continue the study of transport properties of solid solutions diluted a magnetic semiconductor (Cd 1−x−y Zn x Mn y ) 3 As 2 (х+y =0.4) containing Mn (y=0.04 and 0.08). Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.

Experimental details
A modified Bridgeman method was used to obtain single crystals of CZMA. All the samples had tetragonal crystal structure (s. g. P4 2 /nmc). Well-resolved single-period SdH oscillations were observed well in the all investigated CZMA (х+y=0.4) specimens at temperatures between T=4.2 and 50 K (figure 1, see [8]).
It has been recently found that the cyclotron mass is independent on a magnetic field, B, for CZMA monocrystals (y=0.08 and y=0.04) (figure 2, [8]). And an anomalous dependence of the cyclotron mass on a magnetic field was observed that obeys a linear law: Our further studies of CZMA (x+y=0.4) were prolonged on the basis of the results obtained in [8].  [8].
The concentration of charge carriers, n ,   The wave vector can be determined if the density of charge carriers is known, that can be expressed as: where g-the factor of degeneration of Landau bands. In our spin-filtered densities case we apply degeneration factor as g=25 [15]. As a result, it was found that for CZMA (y=0.04) the wave vector is k F =0.1 nm -1 .
According to Lifshitz-Kosevich theory [14] the temperature dependence of SdH oscillation amplitude can be expressed as: where T D and DE N are adjustable parameters, and H corresponds to a magnetic field at the minimum (maximum) of longitudinal magnetoresistance. The value DE N is an energy gap between N and (N+1) Landau band:

R H T k T E H k T E H k T E H
where m c -is an effective cyclotron mass. The parameter T D is Dingle temperature where t D -a is relaxation time for charge carriers due to diffraction, for samples y=0.04 and y=0.08 t D =1.9×10 -13 s, t D =1.8×10-13 s, respectively.
From the values k , F m c and t D calculated for the samples y=0.04 the velocity on Fermi surface .7 nm were calculated. In the table 2 effective 2D-mobility and Hall 3D-mobility are presented. A linear dispersion law is an important feature of quantum transport ( figure 3). This kind of dependence was also observed for Dirac fermions in graphene [16,17]. The dispersion law for the carriers (electrons): where v F -Fermi velocity, k-a wave vector. The relation with effective mass: From the data in figure 3 it can be seen that the values obtained experimentally [11,18,19] and the values obtained for CZMA (y=0.04) (marked with symbols) are in a good accordance with the theoretical linear dependence, that describes mass-less Dirac fermions (the continuous line).
In agreement with [8] rising Mn concentration leads to changes in transport properties of diluted magnetic semiconductor (Cd 1−x−y Zn x Mn y ) 3 As 2 (x+y=0.4). The results of SdH oscillation investigations in y=0.04 samples showed the absence of a phase shift β and evidence of Berry phase. Thus, (Cd 0.6 Zn 0.36 Mn 0.04 ) 3 As 2 samples are not topological insulators but they demonstrate an anomalous dependence of charge carriers' cyclotron mass on a magnetic field. Thus, we have shown the presence of a relation between manganese concentration and topological properties in CZMA diluted magnetic semiconductor and the presence of mass-less Dirac fermions in (Cd 1−x−y Zn x Mn y ) 3 As 2 (x+y=0.4).