Dirac material is a group of compounds whose low-energy excitation behaves as massless Dirac particles.^[1] In recent years, a variety of Dirac materials have been discovered, such as graphene,^[2] three-dimensional (3D) topological insulators Bi_1−xSb_x, Bi₂Se₃, and Bi₂Te₃,^[3,4] and Dirac semimetals Na₃Bi,^[5] Cd₃As₂.^[6–9] The energy spectrum of Dirac materials exhibits linear behavior and can be described by relativistic Dirac equation. More interestingly, when the time reversal symmetry or space inversion symmetry is broken, Dirac semimetals (DSM) evolve into Weyl semimetals (WSM), whose extraordinary properties such as Fermi arc and chiral anomaly have attracted great attention.^[10–24] One interesting consequence of the linear energy dispersion is quantum transport phenomena. For Dirac materials, a moderate magnetic field can compel all carriers occupy the lowest Landau level and lead to the quantum linear magnetoresistance (MR).^[25,26]

In Dirac materials, there is a class of ternary 112 type compounds such as CaMnBi₂,^[27–30] SrMnBi₂,^[31–34] EuMnBi₂,^[35,36] and LaAgBi₂,^[37] which have been researched deeply. More recently, time-reversal symmetry breaking Weyl state has been suggested to exist in YbMnBi₂,^[38] and Sr_1−yMn_1−zSb₂^[39] is also identified as a promising candidate of WSM. Among these materials, the Bi/Sb square net is a common feature and has been considered as the platform that hosts Dirac/Weyl fermions. Therefore, it is of considerable interest to explore new materials which have similar structure and further study their physical properties.

In this work, we have synthesized the single crystals of BaMnBi₂ and investigated the transport properties. Hall resistivity shows carriers in BaMnBi₂ are electron type. Magnetic property measurement indicates the magnetic structure of BaMnBi₂ may be slightly different from that of SrMnBi₂ or CaMnBi₂. Angle-dependent MR implies the anisotropic quasi-two-dimensional Fermi surface in BaMnBi₂. The in-plane MR displays a crossover from semiclassical quadratic field dependence to linear field dependence with the increase of magnetic field. The linear MR indicates the possible existence of Dirac fermions in this material. It is worth doing ARPES experiments to check whether BaMnBi₂ is a Weyl semimetal.

Single crystals of BaMnBi₂ were grown from Bi flux. The mixtures of Ba, Mn and Bi were placed in a crucible and sealed in a quartz tube with a ratio of Ba:Mn:Bi=1:1:6. The quartz tube was heated to 1180 °C in 60 h, held there for 30 h, and cooled to 370 °C at a rate of 3 °C/h, and then the excess Bi-flux was removed by centrifuging. Elemental analysis was performed using energy dispersive x-ray spectroscopy (EDS, Oxford X-Max 50). The determined atomic proportion was consistent with the composition of BaMnBi₂ within instrumental error. Single crystal x-ray diffraction (XRD) pattern was collected from a Bruker D8 Advance x-ray diffractometer using Cu K_α radiation. Resistivity measurements were performed on a Quantum Design physical property measurement system (QD PPMS-14T) and the magnetic properties were measured with vibrating sample magnetometer (VSM) option.

BaMnBi₂ is isostructural with SrMnBi₂. As shown in Fig. 1(a), the crystal structure of BaMnBi₂ is comprised of alternating MnBi and BaBi layers.^[40] In the MnBi layer, each Mn atom is surrounded by four Bi atoms, which form the MnBi₄ tetrahedra. In the BaBi layer, Bi atoms are separated and form a square net (highlighted by red atoms). Figure 1(c) shows the x-ray diffraction pattern of a BaMnBi₂ single crystal on the (00l) plane. An additional small peak at about 27.2° also has been observed. It may come from the residual Bi flux on the surface. The inset is an optical image of a representative crystal.

Fig. 1.

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	Fig. 1. (a) The crystal structure of BaMnBi2. (b) The definition of polar angle θ and azimuthal angle φ. (c) Single crystal x-ray diffraction pattern of a BaMnBi2 crystal, showing only the (00l) reflections. The inset is an image of a typical single crystal with a scale of 4 mm.

Both the in-plane resistivity ρ_xx(T) and out-plane resistivity ρ_z(T) exhibit a simple metallic behavior as shown in Fig. 2(a). The out-plane resistivity is nearly 50 times the in-plane resistivity in a range of 2.5 K–300 K. Such a significant anisotropy suggests quasi-2D electronic band structure in BaMnBi₂.

Fig. 2.

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Fig. 2. (a) Temperature dependence of in-plane resistivity ρxx(T) and out-plane resistivity ρz(T). (b) Hall resistivity ρxy versus magnetic field H with the temperature T from 5 K to 150 K. (c) and (d) Temperature dependence of carrier concentration and mobility, respectively. (e) Magnetization versus temperature with the magnetic field H = 1 T applied parallel to ab plane (up) and c axis (down) under zero field cooling (ZFC) (red) and field cooling (FC) (violet) conditions. The dashed lines denote two anomalous temperatures.

Figure 2(b) plots the magnetic field dependence of Hall resistivity ρ_xy(H) measured at various temperatures. The negative slope of ρ_xy suggests that the dominant charge carriers in BaMnBi₂ are electrons. Single band model has been employed to analyze the Hall effect data. The carrier concentration and carrier mobility are shown in Figs. 2(c) and 2(d) respectively. The carrier concentration is given by n = 1/eR_H, and the Hall coefficient R_H is obtained by R_H = ρ_xy(8 T)/H. At 5 K, the carrier concentration is 1.1×10¹⁹ cm⁻³. We calculate the carrier mobility by μ = 1/enρ_xx(0 T). With temperature reduced, the mobility becomes larger and reaches 1500 cm²/V·s at 5 K.

Figure 2(e) presents the temperature dependence of magnetization of BaMnBi₂ measured under zero field cooling (ZFC) and field cooling (FC) conditions with an applied field H = 1 T parallel to ab plane and c axis respectively. Similar to CaMnBi₂ and SrMnBi₂,^[33,34] there are two anomalous temperatures in BaMnBi₂. T₁ = T_N = 288 K is the temperature of antiferromagnetic transition, below T₁ the magnetization exhibits strong anisotropy, the linear temperature dependence of magnetization with positive slope above T₁ indicates strong antiferromagnetic correlations.^[41] There is a ZFC–FC splitting for magnetic field parallel to ab plane when T < T₂ = 270 K. This is quite different from the behavior in CaMnBi₂ or SrMnBi₂, in which the ZFC–FC splitting occurs when the field is parallel to c axis. The difference implies that the magnetic structure of BaMnBi₂ may be slightly different from that of CaMnBi₂ or SrMnBi₂.

Fig. 3.

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Fig. 3. (a) Polar angle θ dependence of in-plane resistivity ρxx at H = 10, 6, 2 T. The temperature and azimuthal angle are fixed at T = 2.5 K and φ = 90° respectively. The red curve is the fitting of cosine function. (b) Magnetic field H dependence of in-plane resistivity ρxx with different polar angle θ at 2.5 K. (c) In-plane magnetoresistance MR versus magnetic field H with the temperature from 5 K to 150 K. (d) The field derivative of the MR as a function of magnetic field H. The straight lines in low field regions are linear fitting results using the relationship MR = A2H2. Lines in high field regions are fitting results using MR = A1H + O(H2). The intersection of fitting lines in low field and high field is defined as critical point H*. (e) Temperature dependence of critical point H*. The solid curve is the fit to the equation

The carriers of a metal in magnetic field are subject to the Lorentz force. The Lorentz force affects the carriers’ momentum components in the plane perpendicular to the field and MR is partially determined by the mobility in this plane. The carriers in a quasi-2D material will only be affected by the magnetic field component H|cosθ|. Figure 3(a) shows the angular-dependent in-plane MR measured in different magnetic fields. When θ = 0°, 180° (field parallel to c axis), MR has the maximum value. With the increase of the polar angle θ, MR decreases gradually and reaches the minimum value when θ = 90°. The |cosθ| function behavior of the whole curve indicates the existence of a quasi-2D Fermi surface in BaMnBi₂. The small deviation implies the existence of 3D electronic transport in BaMnBi₂.

Figure 3(c) shows the in-plane MR as a function of magnetic field at different temperatures. The MR exhibits quantum linear behavior in high field and semiclassical quadratic dependence in low field.^[42] The transition from semiclassical to quantum can be seen more clearly from the field derivative of the MR [Fig. 3(d)]. Initially, dMR/dH is proportional to H, which indicates a semiclassical H²-dependent MR. With the increase of magnetic field, the field crosses a critical point H* and dMR/dH nearly becomes a constant. This implies that MR follows linear field dependence plus a small quadratic term in high field, namely MR = A₁H+O(H²). The linear MR suggests the existence of Dirac fermions in BaMnBi₂.^[43–46]

A sufficiently strong perpendicular magnetic field can cause the complete quantization of orbit of Dirac fermions and the quantized Landau level can be described as where v_F is the Fermi velocity and n is the Landau index. So the energy splitting between the first and lowest Landau level is At a specific temperature T, with the increase of field H, the energy splitting becomes larger than the Fermi energy E_F and the thermal fluctuations k_BT. As a result, all carriers will occupy the lowest Landau level and reach the quantum limit. With these two relationships combined, we can get the temperature dependence of the critical magnetic field ^[47] However, the condition for absolute quantum limit is n < (eH/ħ)^3/2, where n is the carrier concentration. Substituting the electron concentration n(5 K) = 1.1 × 10¹⁹ cm⁻³ obtained from the Hall measurement into the expression, we get the value H > 32.5 T. Although the system does not enter the absolute quantum limit in our situation, the linear quantum MR also could appear at lower field with more than one Landau level filled, which also has been observed in previous research on InSb.^[45] In addition, considering the relationship between T and H* can be well fitted by the equation as shown in Fig. 3(e), it is suggested that the linear MR in BaMnBi₂ originates from Dirac fermions.

In summary, single crystals of BaMnBi₂ have been grown. Resistivity, Hall resistivity, magnetic property and magnetoresistance have been measured and analyzed. Compared with CaMnBi₂ and SrMnBi₂, the different behavior of magnetization brings the possibly different magnetic structure of BaMnBi₂. Quasi-2D electronic transport is also observed in angle-dependent MR. The crossover from semiclassical parabolic field-dependent MR in low field to linear field-dependent MR in high field can be explained by combining semiclassical and quantum magnetoresistance. Our results clearly indicate the existence of Dirac fermions in BaMnBi₂.

Note added When the paper was being finalized, we noticed one similar related work on BaMnBi₂,^[48] where the Shubnikov–de Hass oscillation is also observed under high magnetic field. However, the angle-dependent resistivity in their work is different from ours and deviates greatly from the |cosθ| relationship, which may be caused by misaligned leads on the samples besides the contribution from 3D Fermi surface.