Our samples are nonstoichiometric single crystals with deficiencies in Sr and Mn. The sample providing the transport data has the specific composition of which is obtained by averaging four energy-dispersive x-ray spectroscopy (EDX) results. Despite the deficiency of the Sr and Mn elements, the bulk crystal exhibits a good crystalline quality as confirmed by the sharp diffraction peaks from the x-ray diffraction (XRD) spectrum (Fig. 1(b)). All the peak positions match well with those of the stoichiometric coordinate,^[12] and the layer normal is along the a-axis. Data for additional samples are available in the supplementary materials.

Fig. 1.

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Fig. 1. (color online) Characterization of . (a) A schematic drawing of crystal structure. Single crystal SrMnSb2 is composed of two formula layers (SrSb and MnSb), stacking along the axis. (b) X-ray diffraction pattern of on the (00l) plane. The XRD pattern peaks of our sample match well with the stoichiometric coordinate. (c) The temperature dependence of in-plane resistivity. A constant current was applied within the bc plane. The inset shows the geometry of Hall bar. (d) The carrier density and mobility at different temperatures. The mobility of the sample exceeds at 2.5 K.

Our samples were measured in a Hall bar geometry as illustrated in the inset of Fig. 1(c). A constant current was applied in the bc plane with a magnetic field along the a-axis (the out-of-plane direction). Figure 1(c) displays the temperature dependence of the resistivity (R–T curve) at zero field, which exhibits a metallic feature. Figure 1(d) shows the carrier density and mobility at different temperatures. Mobility up to at 2.5 K was obtained from the Hall effect measurement. In the whole temperature range, the Hall signal shows positive slopes, indicating a hole-dominated transport behavior which is consistent with the previous report.^[12] Notably, the electron carriers were reported to be dominant in SrMnSb₂ elsewhere.^[15] The striking difference in the carrier type implies that SrMnSb₂ is a sensitive system that stringently relies on composition and growth details.

Typical magnetoresistance () is shown in Fig. 2(a). To explore the conduction channels’ dimensionality, MR was measured at different angles. With the perpendicular geometry (the field direction is parallel to the a axis), the MR ratio almost reaches 200% at 2.5 K, which exceeds the previously reported giant MR in Dirac semimetals CaMnBi₂ (105%)^[5] and SrMnBi₂ (%).^[2] Moreover, clear SdH oscillations start to appear around 4 T. Subtracting the parabolic MR background resolves the evident SdH oscillations that persist beyond 20 K. Only one main frequency was observed in fast Fourier transform (FFT) spectrum at each angle in the field range of 0–9 T. The oscillation frequency is associated with the external cross-sectional area of the Fermi surface . For a 2D Fermi surface, the cross section has an angular dependence following . The angle is defined as the angle between the magnetic field and the normal of the bc plane. As displayed in Fig. 2(b), the angular dependence of the frequency follows the cosine law (the red dashed curve). The quasi-cosine behavior of the frequency indicates a highly anisotropic Fermi surface. Similar oscillation analysis was carried out for the temperature-dependent MR in Fig. 2(c). For the oscillation at 2.5 K, the Fourier spectrum reveals a single frequency of F = 74.6 T (the inset of Fig. 2(c)). Oscillations at other temperatures also reveal the same frequency since there was no apparent peak shift. Deducing from the Onsager relation ( is the flux quantum), the frequency of 74.6 T corresponds to the external cross-sectional area of the Fermi surface . This area is just slightly larger than that of the nonstoichiometric sample which hosts the relativistic fermions.^[12]

Fig. 2.

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Fig. 2. (color online) SdH oscillations at low fields. (a) The magneto-resistance ratio at different angles. The angle of magnetic field is defined as the angle between the magnetic field and the normal of the bc plane (zero degree means that the magnetic field is perpendicular to the bc plane). The maximum MR ratio is approximately 200%. Clear SdH oscillations persist beyond . (b) The angular dependence of the oscillation frequency, following the cosine law . The red dashed curve indicates the cosine fitting. (c) SdH oscillations at different temperatures. were obtained by subtracting a parabolic background. Inset: Fourier transform of the oscillations at 2.5 K, revealing a single frequency F = 74.6 T. (d) The Lifshitz–Kosevich formula fitting to the oscillation amplitudes at different temperatures. The fit yields the cyclotron effective mass , where is the bare electron mass. (e) Dingle plot. The oscillation amplitude at 2.5 K decays exponentially. The Dinge temperature fitting yields the Dingle temperature of 15 K. The quantum lifetime obtained is s. (f) Landau index plot. Inset: integers correspond to the valleys in the conductance spectrum. All the data align well on a straight line. The red dashed line is the linear fit using . The intercept corresponds to a trivial Berry phase either for 2D or 3D system.

We performed the fitting of the temperature-dependent oscillation amplitudes to the Lifshits–Kosevich formula:^[16]

Identifying the oscillation amplitudes at different temperatures at one specific magnetic field strength and fitting the set of data to the formula enable the extraction of the effective mass. The fitting yields an effective mass , where m₀ is the bare electron mass. This value is larger than the conventional Dirac semimetal system, such as Cd₃As₂ (0.044m₀),^[17] but smaller than those of Dirac cousins SrMnBi₂ (0.29m₀)^[2] and CaMnBi₂ (0.32m₀).^[5]

The expression of Dingle temperature is , where is the quantum lifetime of carries due to scattering. It can be obtained from the slope of the linear fitting of , . As shown in Fig. 2(e), the Dingle temperature is found to be 15 K at T = 2.5 K. Then we extract the carrier lifetime s. The quantum lifetime (also known as the single-particle lifetime) and the transport lifetime are two important time scales. It is widely accepted^[18] that the ratio is an indication of the dominant scattering mechanism since the quantum lifetime obtained from SdH oscillations measures the collision events at all directions while obtained from the conductivity ( counts the collision events in only particular direction.^[18] In our trivial SrMnSb₂ system, is s, giving , which is in the conventional case that the transport lifetime is regarded as the same order as the quantum lifetime. Commonly, larger ratio is expected in topological system, since the large ratio between the two time scales is usually associated with the suppression of backscattering that originates from the chiral nature in topological semimetal systems.^[19] Besides the 2DEG systems with a factor of 10–100,^[18,20–22] the large was reported in Weyl semimetal TaAs^[23] and in topological semimetal LaSb with an extraordinary large value reaching .^[24]

The corresponding mobility obtained from the quantum lifetime is cm²/Vs, which is significantly larger than that of SrMnBi₂.^[2] It is remarkable that the sample’s physical properties are comparable with its Dirac cousins, requiring us to attain more information on sample’s relativistic properties. Therefore, we calculated the Berry phase accumulated along cyclotron orbits.

In the Lifshitz–Kosevich (LK) formula,^[16] is proportional to , where F, B, and stand for frequency, magnetic field, and phase shift, respectively. It is widely believed that in a Dirac system, the linear dispersion band contributes an extra π Berry phase, which leads to a different phase shifts in 2D (1/8) and 3D (0) systems.^[25] To avoid the obstacles in carrying out a complete fitting for the oscillation pattern by the LK formula, an alternative approach is through the Landau fan diagram. In the Landau fan diagram in which the inverse magnetic field is set as x-axis and integer n as y-axis, the position of peaks and valleys will fall on a straight line. Then F and can be extracted from the linear fit using .^[25] It is worth to mention that due to the complexity of the Landau level quantization in 3D semimetal systems, the assignment of the integer index is different from the cases of the topological insulator and graphene.^[25] We follow the rule that is suggested by Wang^[25] to index the extrema, in which the Landau level integers are assigned to the peaks in resistivity oscillation. The slope of the linear fitting corresponds to the oscillation frequency. The value T is consistent with the FFT result. A phase shift (Fig. 2(f)) was obtained, which corresponds to a trivial Berry phase.

The trivial Berry phase was suggested by the theoretical calculation.^[14] As illustrated in Fig. 1(a), the crystal structure of SrMnSb₂ is similar to its cousin material SrMnBi₂, consisting of a periodic arrangement of Mn–Sb and Sr–Sb layers along the a axis, but with an orthorhombic distortion. The distortion in Sb layer dictates the symmetry and is responsible for the absence of the Dirac fermion. Practically, the composition of our sample has a small deviation from the stoichiometry, which is Sr_(0.88±0.02)Mn_(0.91±0.06)Sb₂, while the crystal structure remains unchanged compared to the stoichiometric sample as evidenced by the XRD pattern. Thus, we consider that the chemical deficiency is not likely to significantly modify the crystal symmetry and introduce Dirac points in our sample’s band structure.

To investigate the transport behavior when approaching the quantum limit, we applied a pulsed magnetic field up to 60 T. We observed the developing of quantized plateaus in . Each plateau position is aligned with the minima of as shown in Fig. 3(a). We interpreted the quantized feature as the multilayered quantum Hall effect arising from the stacked 2D conduction channels. Due to the weak interlayer interaction between unit layers and the strong 2D confinement revealed in the transport, the bulk sample behaves like a stacked system forming by many parallel 2D channels. The bulk quantum Hall effect has been reported in highly-doped topological insulator Bi₂Se₃,^[26] Dirac semimetal ZrTe₅,^[27] η-M₄O₁₁,^[28] and Bechgaard salts.^[29–31] The quantized Hall resistance takes value , where Z is the number of stacked quantum layers, and N is the Landau index. Figure 3(b) displays scaled with Z as a function of N. The sample thickness is around , the value of lattice parameter c calculated from XRD data is Å (the uncertainty comes from the standard deviation across the measured peaks), and each unit layer contains two van der Waals layers Sr–Sb and Mn–Sb. The inset shows that the scaled falls on a straight line. The linear fit gives a slope of , which indicates a deviation of 3% to the value of . The error could come from the estimation of sample thickness and the deficiency-leading non-ideal current paths. The highly linear relation between the value and the index shows an excellent quantization coming from the multilayered quantum Hall effect.

Fig. 3.

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Fig. 3. (color online) Pulsed high field experiments. (a) Magnetoresistance and Hall resistance as functions of the perpendicular field along the a axis. The data is presented at 4.2 K. (b) Nearly quantized plateau in divided by the number Z as a function of 1/B. Adjacent LLs are separated by . Inset: normalized of each Landau level aligns in a straight line. Red dashed line is the linear fitting, yielding a slope of with 3% deviation from the standard value . (c) Oscillatory beating patterns were developed in sample 1 and sample 2. The first node position is around 10 T. (d) Peak splitting in the FFT transform spectrum of high field SdH oscillations of sample 1. The two frequencies are 74.6 T and 88.2 T respectively, indicating two Fermi surfaces differing 18% in cross-section areas. (e) Conductivity oscillation in 1/B scale at 4.2 K with . Pulsed field of 60 T drove sample to the second Landau level. Pronounced Zeeman splitting was observed as labeled on the plot. An oscillatory beating pattern can be observed. The red dashed line is for guiding eyes. (f) Angular dependence of the Zeeman splitting spacing. Red and navy arrows show the splitting spacing change of the and Landau levels.

Meanwhile, the field-dependent resistance evolves into a beating pattern with the first beating node positioned at 10 T. The beating pattern indicates the existence of two frequencies. Accordingly, the Fourier transform analysis of the resistance oscillation engenders a double-peak spectrum (Fig. 3(d)). The first frequency 74.6 T corresponds to the frequency appeared at low fields. The other frequency is 88.2 T. The two frequencies stand for two differing by 18%, which means that there are two sets of Landau levels with comparable carrier density and cross-section area.

With the magnetic field strength up to 60 T, we can probe the transport channels with relatively low mobility. It is possible that an additional channel with lower mobility contributes to the beating feature. According to the SdH oscillations, the Fermi level lies around 138 meV below the top of the valence band where a small pocket located at Y could be involved in the transport. However, the difference between the cross-section areas of pocket at Y and is remarkable and tends to rule out the two-band origin of the beating oscillation.

Zeeman splitting only produces two identical sets of sublevels along the field direction, so that the interference between the two sets of sublevels will not give rise to the beating pattern. On the other hand, a large spin orbit interaction often causes electronic band structure splitting and can manifest itself as a beating behavior in the SdH oscillations.^[32–34] There are two types of spin–orbit interaction: Rashba type and Dresselhaus type. Rashba terms appear in the systems lacking of interface inversion symmetry, while Dresselhaus effect is considered in bulk inversion asymmetry crystals.^[35] The distorted Sb atom net intrinsically breaks the inversion symmetry in SrMnSb₂ along the a crystallographic axis. Meanwhile, Sb atom is relatively heavy, thus sizable spin–orbit coupling should be expected in the sample.^[35] Besides, Rashba terms usually play significant roles in 2D electron gas systems^[32,33,36] and also manifest in topological insulator surface states.^[37] For a trivial parabolic band, the Fermi energy is proportional to carrier density. The 18% splitting in density can be translated into 18% splitting in the Fermi energy, corresponding to the SOI splitting on the order of 20 meV.^[34] Although the dominated SOI type requires more detailed work to determine, the presence of splitting in SdH oscillation frequency is a direct manifestation of the SOI splitting in the Fermi surface.

In the presence of the strong SOI, spins with different orientations are mixed in the Zeeman sublevels, and give rise to variations in g factor from the pure Zeeman splitting. It is noticeable that the magnetic field above 22 T lifts the Zeeman splitting as shown in Fig. 3(e), where pronounced splitting peaks and valleys were resolved in the conductance oscillation curves. It is known that Zeeman splitting is subjected to the total external magnetic field and it should be independent of the field direction. However, we observe the angular-dependent oscillations in the high field regime, as displayed in Fig. 3(f). The splitting spacing in our sample shows an obviously angular dependence. The g factor calculated at 10 degree is 5.0 which differs from the value at 40 degree (3.5) by 30%. An angular-dependent Zeeman splitting can be developed when SOI is involved by which the external field can induce exchange interaction. Any exchange splitting can be decomposed into the orbital independent and dependent part.^[38] The former one corresponds to the isotropic Zeeman splitting part, while the latter one is highly sensitive to the field’s direction, giving rise to a highly anisotropic g factor.^[39] The anisotropic g factor is discovered in many 2D hole systems^[40] as well as in the interface of heterostructrues,^[41–44] where Rashba-type SOI usually dominates. Similar g factor variation is also discovered in materials that exhibit highly anisotropic Fermi surface, such as ZrTe₅.^[27] The anisotropic g factor provides another unambiguous evidence of the SOI in SrMnSb₂.