Abstract
Layered compounds AMnBi_{2} (A = Ca, Sr, Ba, or rare earth element) have been established as Dirac materials. Dirac electrons generated by the twodimensional (2D) Bi square net in these materials are normally massive due to the presence of a spinorbital coupling (SOC) induced gap at Dirac nodes. Here we report that the Sb square net in an isostructural compound BaMnSb_{2} can host nearly massless Dirac fermions. We observed strong Shubnikovde Haas (SdH) oscillations in this material. From the analyses of the SdH oscillations, we find key signatures of Dirac fermions, including light effective mass (~0.052m_{0}; m_{0}, mass of free electron), high quantum mobility (1280 cm^{2}V^{−1}S^{−1}) and a π Berry phase accumulated along cyclotron orbit. Compared with AMnBi_{2}, BaMnSb_{2} also exhibits much more significant quasi twodimensional (2D) electronic structure, with the outofplane transport showing nonmetallic conduction below 120 K and the ratio of the outofplane and inplane resistivity reaching ~670. Additionally, BaMnSb_{2} also exhibits a Gtype antiferromagnetic order below 283 K. The combination of nearly massless Dirac fermions on quasi2D planes with a magnetic order makes BaMnSb_{2} an intriguing platform for seeking novel exotic phenomena of massless Dirac electrons.
Introduction
Threedimensional topological semimetals, including Dirac semimetals (DSMs)^{1,2,3,4,5,6}, Weyl semimetals (WSMs)^{7,8,9,10,11,12,13,14,15} and Dirac nodalline semimetals^{16,17,18,19,20,21} represent new quantum states of matter and have stimulated intensive studies. These materials possess bulk relativistic quasiparticles with linear energymomentum dispersion. DSMs feature linear band crossings at discrete Dirac nodes. In WSMs, the Weyl nodes with opposite chirality appear in pairs and each pair of Weyl nodes can be viewed as evolving from the splitting of Dirac node due to the lifted spin degeneracy arising from either broken spatial inversion symmetry or broken time reversal symmetry (TRS). The linear band dispersions in these materials are topologically protected by crystal symmetry and lead to many distinct physical properties such as large linear magnetoresistance and high bulk carrier mobility^{22}. WSMs also show exotic surface “Fermi arc” connecting a pair of Weyl nodes of the opposite chirality^{7,8}. These exotic properties of topological semimetals have potential applications in technology.
AMnBi_{2} (A = alkali earth/rare earth metal) is one of the established Dirac semimetals^{23,24,25,26,27,28,29,30,31,32,33}. These materials share common structure characteristics, consisting of alternately stacked MnBi4 tetrahedral layers and ABiA sandwich layers^{23,26,27,29,31,33}. In an ABiA sandwich layer, Bi atoms form a square net and harbor Dirac fermions, with coincident (e.g. SrMnBi_{2}^{34}) or staggered (e.g. CaMnBi_{2}^{35}) stacking of A atoms above and below the Bi plane. In the Mn centered edge sharing MnBi4 tetrahedral layers, antiferromagnetic (AFM) order usually develops near room temperature^{36,37} and such layers are expected to be less conducting^{23,27}. Dirac fermions in AMnBi_{2} have been found to interplay with magnetism, leading to novel exotic properties. This has been demonstrated in YbMnBi_{2}^{15} and EuMnBi_{2}^{31}. Evidence for Weyl state has been observed in YbMnBi_{2}, which has been proposed to be caused by the TRS breaking due to the ferromagnetic (FM) component of a canted AFM state^{15}. In EuMnBi_{2}, halfinteger bulk quantum Hall Effect (QHE) occurs due to the magnetic order induced twodimensional (2D) confinement of Dirac fermions^{31}.
One disadvantage of AMnBi_{2} as Dirac semimetals is that the strong spin orbit coupling (SOC) due to heavy Bi atoms opens gap at Dirac nodes^{23,38}, leading to massive Dirac electrons. For instance, in SrMnBi_{2}, the SOCinduced gap at the Dirac node is about 40 meV^{23} and the effective mass of Dirac fermions estimated from the analyses of Shubnikovde Haas (SdH) oscillations is ~0.29m_{0} (m_{0}, the mass of free electron)^{23}, much heavier than the Dirac fermions in the 3D gapless DSM Cd_{3}As_{2} where m^{*} ~0.02–0.05m_{0}^{39,40,41,42}. Therefore, one possible route to realize massless Dirac fermions in AMnBi_{2}type material is to replace Bi with other lighter main group elements such as Sb and Sn, whose SOC effect is much weaker. Under this motivation, we previously studied SrMnSb_{2}^{43} and found the 2D Sb layer can indeed harbor much lighter relativistic fermions with m* ~0.14m_{0}. Moreover, this material shows FM properties: the Mn sublattice develops a FM order for 304 K < T < 565 K, but a canted AFM state with a FM component for T < 304 K. Coupling between ferromagnetism and quantum transport properties of relativistic fermions has also been observed^{43}. These interesting results further motivated us to investigate the isostructural compound BaMnSb_{2}, the material studied in this work.
BaMnSb_{2} crystalizes in a tetragonal structure with the space group of I4/mmm^{34}, similar to the structure of SrMnBi_{2} but different from the orthorhombic structure of SrMnSb_{2}^{43}. The orthorhombic distortion in SrMnSb_{2} leads Sb atoms in the SrSbSr sandwich layer to form zigzag chains. However, in BaMnSb_{2}, as Ba has larger ionic radius than Sr, the stronger interaction between Ba atoms and Sb atoms suppresses orthorhombic distortion^{38} and lead to a Sb square net lattice (Fig. 1a), which is an analogue of the Bi square net in SrMnBi_{2}. In addition, the Ba layers are coincidently stacked along the Sb layer in BaMnSb_{2}, which is also distinct from the staggeredly arranged Sr atoms in SrMnSb_{2}^{38}. First principle calculations have predicted that BaMnSb_{2} exhibits Dirac fermion behavior and its SOC induced gap near the Dirac node is as small as ~20 meV, about half of the gap in SrMnBi_{2}^{38}.
In this paper, we will show BaMnSb_{2} indeed is a Dirac material with the Dirac node being very close to the Fermi level. Through the analyses of SdH oscillations, we find this material hosts nearly massless Dirac fermions (m^{*} ~0.052m_{0}). As compared with AMnBi_{2} and SrMnSb_{2}, BaMnSb_{2} possesses the smallest Fermi surface (FS) with the most significant 2D character. Additionally, BaMnSb_{2} also exhibits a Gtype AFM order below 283 K. These findings suggest that BaMnSb_{2} is a promising candidate for seeking novel exotic phenomena of massless Dirac fermions.
Results and Discussions
The BaMnSb_{2} single crystals used in this study were synthesized using selfflux method (see Method). The composition measured by an energy dispersive Xray spectrometer (EDS) can be expressed as Ba_{1y}Mn_{1z}Sb_{2}, with y < 0.05 and 0.05 < z < 0.12. The nonstoichiometry of Sr and Mn was also found in SrMnSb_{2} where the actual composition measured by EDS can be described by Sr_{1y}Mn_{1z}Sb_{2} (y, z < 0.1)^{43}. The neutron diffraction experiment on a piece of single crystal with the measured composition of Ba_{0.96}Mn_{0.94}Sb_{2} confirms the tetragonal structure with the space group I4/mmm reported by Cordier & Schafer^{34}. The lattice parameters and atomic positions extracted from the structural confinement are summarized in Table 1. We note that the lattice constant c = 23.85 Å obtained from our structural refinement is smaller than the previouslyreported value of 24.34 Å^{34}, which may be attributed to the deficiencies of Ba and Mn in our sample.
From magnetic susceptibility measurements on BaMnSb_{2} single crystals, we have observed signatures of antiferromagnetism. As shown in Fig. 1b, at temperatures below 285 K, the susceptibility χ exhibits a striking decrease for the magnetic field applied along the caxis (B//c), but a clear upturn for inplane field (B//ab). As the temperature is lowered below 50 K, χ displays sharp upturns for both field orientations. Similar features have been observed in AMnBi_{2} (A = Ca, Sr and Ba)^{23,25,26,32,33,36}. Neutron scattering studies on CaMnBi_{2} and SrMnBi_{2} have demonstrated that those magnetic anomalies probed in susceptibility originate from an AFM order formed on the Mnsublattice^{36}. Given the similar behavior in susceptibility between BaMnSb_{2} and AMnBi_{2}, we can reasonably attribute the magnetic transition at 285 K seen in BaMnSb_{2} to an AFM transition. We note BaMnSb_{2} exhibits distinct magnetic anisotropy from AMnBi_{2}. As seen in Fig. 1b, χ of B//c (χ_{c}) is much larger than χ of B//ab (χ_{ab}) at temperatures both above and below T_{N}. However, in AMnBi_{2}, χ_{ab} is almost equal to χ_{c} for T ≥ T_{N}, but > χ_{c} for T < T_{N}^{23,26,32}. Additionally, the isothermal magnetization M(H) of BaMnSb_{2} (Fig. 1c) also looks different from that of AMnBi_{2}. We have observed weak FM polarization behavior (Fig. 1c), in contrast with the nearly linear field dependence of M for BaMnBi_{2}^{32}. These discrepancies imply that BaMnSb_{2} and AMnBi_{2} may not share an identical magnetic structure.
Magnetic structures of CaMnBi_{2}, SrMnBi_{2} and YbMnBi_{2} have been determined from neutron scattering experiment^{36,37}. Although these materials show similar Néeltype AFM coupling within the plane and the moments are oriented along the caxis, their interlayer coupling is different. CaMnBi_{2} (space group P4/nmm) and YbMnBi_{2} (P4/nmm) features FM interlayer coupling, whereas SrMnBi_{2} (I4/mmm) is characterized by interlayer AFM coupling^{36,37}. The magnetic structure of BaMnSb_{2} determined from our neutron scattering experiments is similar to that of SrMnBi_{2}, i.e. both the interlayer and intralayer couplings between two nearest moments are AFM, which is also called the nearest neighbor Gtype AFM order. The Néel temperature probed in the neutron scattering experiment is ~283 K, as shown in Fig. 1d which shows the Bragg peak at (101) (dominated by the magnetic scattering, see the inset) as well as the temperature dependence of the (101) Bragg peak intensity. The ordered moment of Mn at 4 K estimated from the magnetic structure refinement at 4 K is 3.950(85) μ_{B}/Mn, comparable to the ordered moments probed in CaMnBi_{2}, SrMnBi_{2}, YbMnBi_{2} and Sr_{1y}Mn_{1z}Sb_{2}^{36,37,43}. As indicated above, the isothermal magnetization of BaMnSb_{2} exhibits weak FM polarization (Fig. 1c). This feature, together with the upturn of χ_{ab} below T_{N}, sharp upturns of χ_{ab} and χ_{c} below 50 K and the small irreversibility of χ_{ab} between field cooling (FC) and zerofieldcooling (ZFC) measurements, is reminiscent of a canted AFM state with a FM component. However, moment canting is generally not expected for a tetragonal structure for symmetry considerations. If the weak ferromagnetism turns out to be intrinsic for BaMnSb_{2}, the possible origin may be associated with its actual nonstoichiometric composition Ba_{1y}Mn_{1z}Sb_{2} as mentioned above. In our previous studies on orthorhombic SrMnSb_{2}^{43}, we have demonstrated a FM component arising from a canted AFM state; the saturated FM moment sensitively depends on Sr and Mn deficiencies, ranging from 0.6 μ_{B}/Mn to 0.005μ_{B}/Mn. With this in mind, we can speculate that Ba and Mn deficiencies possibly lead to local orthorhombic distortion, thus resulting in local canted AFM states. However, we have to point out that small FM components cannot be resolved in neutron scattering experiments. Hence it is not surprising to see the absence of FM response in our neutron scattering experiment.
We have also characterized the electronic transport properties of BaMnSb_{2} single crystals. In Fig. 1e we present both inplane (ρ_{in}) and outofplane (ρ_{out}) resistivity as a function of temperature, from which we found several signatures distinct from that of (Ca/Sr/Ba)MnBi_{2}^{26,30,32,33}. First, BaMnSb_{2} shows much stronger electronic anisotropy than (Ca/Sr/Ba)MnBi_{2}, which is manifested in its larger ρ_{out}/ρ_{in} ratio. The ρ_{out}/ρ_{in} ratio at 2 K ranges from 15 to 100 for (Ca/Sr/Ba)MnBi_{2}^{26,30,32,33}, but rises to 670 for BaMnSb_{2}, which is comparable to the value of ρ_{out}/ρ_{in} (~609) seen in SrMnSb_{2}^{43}. Such a large electronic anisotropy of BaMnSb_{2} suggests its electronic structure is quasi2D like, which is further confirmed in our measurements of angular dependence of SdH oscillation frequency as shown below. Second, unlike (Ca/Sr/Ba)MnBi_{2} whose ρ_{out}(T) always exhibits a hump due to a crossover from hightemperature incoherent to lowtemperature coherent conduction^{26,30,32,33}, BaMnSb_{2} displays an opposite behavior in ρ_{out}(T) (Fig. 1e); a crossover from hightemperature metallic conduction to lowtemperature localization is observed, which leads to a broad minimum in ρ_{out}(T) around 120 K. The temperature dependence of inplane resistivity ρ_{in}(T) of BaMnSb_{2} also differs from that of (Ca/Sr/Ba)MnBi_{2}. (Ca/Sr/Ba)MnBi_{2} features a quadratic temperature dependence for ρ_{in} in low temperature range^{23,26,32,33}, while ρ_{in} of BaMnSb_{2} exhibits localization behavior below 80 K but crossovers to metallic behavior below 11 K. These differences imply the transport mechanism in BaMnSb_{2} is somewhat different from that in (Ca/Sr/Ba)MnBi_{2}. We note that in the temperature region where the localization behavior occurs, both ρ_{out}(T) and ρ_{in}(T) follow a logarithmic temperature dependence, as denoted by the dashed lines in Fig. 1f which presents ρ_{out}(T) and ρ_{in}(T) on the logT scale. This observation is reminiscent of Kondo effect. Given that we have an AFM lattice formed from local moments of Mn ions, the presence of Kondo effect is possible in principle. But, this naturally leads to a question why such an effect occurs only to BaMnSb_{2}, but not to (Ca/Sr/Ba)MnBi_{2} and SrMnSb_{2} with similar AFM lattices. Clear understanding of this issue requires further studies, but one possible interpretation is that the Kondo effect depends on the dimensionality of electronic structure and may be enhanced in BaMnSb_{2} due to its highly 2D electronic structure. Moreover, we would like to point out the localization behavior seen in BaMnSb_{2} cannot be attributed to disorder induced localization since in Sr_{1y}Mn_{1z}Sb_{2} with the level of disorders being comparable or higher than that of BaMnSb_{2}, no localization behavior is observed^{43}.
Like (Ca/Sr/Ba)MnBi_{2} and SrMnSb_{2}, BaMnSb_{2} also exhibits quantum transport properties as revealed by our magnetotransport measurements, In Fig. 2a,d, we present the field dependences of inplane (ρ_{in}) and outofplane (ρ_{out}) resistivity measured at various temperatures for BaMnSb_{2}, respectively. Strong SdH oscillations, which sustain up to above 40 K, are observed in both ρ_{in}(B) and ρ_{out}(B). In Fig. 2b,e we present the oscillatory components of ρ_{in} and ρ_{out}, respectively. From Fast Fourier Transformation (FFT) analyses of Δρ_{in} and Δρ_{out} (see the insets to Fig. 2c,f), we find that the SdH oscillations of ρ_{in} consists of a single frequency (~22T), whereas the oscillations of ρ_{out} include two frequencies (i.e. F_{α}~25T and F_{β} ~35T). Such a difference is likely caused by the nonstoichiometric composition. As mentioned above, the actual composition of our BaMnSb_{2} crystals involves Ba and Mn nonstoichiometry, which could lead to slight modification for electronic structure in different samples. To verify this speculation, we have measured many samples and find that their oscillation frequencies indeed show variation, ranging from 20T to 35T. Given that the quantum oscillation frequency is directly linked to the extremal Fermi surface crosssection area A_{F} by the Onsager relation F = (Φ_{0}/2π^{2})A_{F}, a small oscillation frequency is generally expected for topological semimetals with the Dirac node being near the Fermi level. We note that the quantum oscillation frequency of 22T probed in our BaMnSb_{2} crystals is the smallest as compared with AMnBi_{2} and SrMnSb_{2}, implying that if BaMnSb_{2} turns out to be a Dirac material, its Dirac band crossing points must be very close to the Fermi level.
Evidence for Dirac fermions in BaMnSb_{2} has been obtained from the further analyses of the SdH oscillations. As shown in Fig. 2c,f, the effective cyclotron mass m^{*} can be extracted from the fit of the temperature dependence of the normalized FFT peak amplitude to the thermal damping factor of LifshitzKosevich (LK) equation^{44}, i.e. , where ρ_{0} is the zero field resistivity and α = (2π^{2}k_{B}m_{0})/(ħe). μ is the ratio of effective mass of cyclotron motion to the free electron mass. is the average inverse field for FFT analysis. We did the FFT within the field 3T–31T range for ρ_{in} and the 5T–31T range for ρ_{out}, with being 5.47T and 8.61T respectively. As seen in Fig. 2c,f, the best fits yield m* = 0.052m_{0} and 0.058m_{0}, respectively, for the SdH oscillations of ρ_{in} and ρ_{out}. For ρ_{out}, the fit was performed for the component with the oscillation frequency of 35T, whose FFT peak can be clearly resolved. The effective mass of m* = 0.052m_{0} and 0.058m_{0} seen in BaMnSb_{2} is much smaller than that of other known AMnBi_{2}^{23,26,32,37} and SrMnSb_{2}^{43}, but comparable to that of gapless Dirac semimetal Cd_{3}As_{2}^{39,40,41,42}. Detailed comparisons of m* as well as other parameters derived from SdH oscillations are shown in Table 2.
To further verify if the nearly massless electrons in BaMnSb_{2} is of topological nature of Dirac fermions, we extracted the Berry phase accumulated along cyclotron orbit from the analyses of SdH oscillations. Berry phase should be zero for a nonrelativistic system with parabolic band dispersion, while a finite value up to π is expected for Dirac fermions^{45,46}. We present the Landau level (LL) fan diagram constructed from the SdH oscillations of ρ_{in} for BaMnSb_{2} in Fig. 3a,b, where integer LL indices are assigned to the maxima of ρ_{in}. Our definition of LL index is based on the customary practice that integer LL indices are assigned to the minima of conductivity^{46,47}. Inplane conductivity σ_{xx} can be converted from the longitudinal resistivity ρ_{xx} and the transverse (Hall) resistivity ρ_{xy} using σ_{xx} = ρ_{xx}/(ρ_{xx}^{2} + ρ_{xy}^{2}). Since our measured ρ_{xy} (Fig. 3d) is about 1/3–1/4 of ρ_{xx} (Fig. 2a) for B < 9 T, σ_{xx} ≈1/ρ_{xx}, which justifies our definition of LL index. As seen in Fig. 3b, the intercept on the LL index axis obtained from the extrapolation of the linear fit in the fan diagram is 0.53, very close to the expected value of 0.5 for a 2D Dirac system with a π Berry phase. The oscillation frequency derived from the fit is 21.8T, nearly the same as the frequency obtained from the FFT analyses of the SdH oscillations of ρ_{in} (see the inset to Fig. 2c), suggesting that our linear fit in the fan diagram is reliable^{46}. The Berry phase derived from the above fan diagram analyses clearly indicates that the nearly massless electrons probed in the SdH oscillations are Dirac fermions.
Dirac Fermions are usually characterized by high quantum mobility, as seen in Cd_{3}As_{2}^{22}. This is also seen in BaMnSb_{2}. The quantum mobility is directly related with the quantum relaxation time τ_{q} by μ_{q} = eτ_{q}/m*. τ_{q} characterizes quantum life time, the time scale over which a quasiparticle stays in a certain eigenstate. τ_{q} can be found from the field damping of quantum oscillation amplitude, i.e., . T_{D} is the Dingle temperature and is linked with τ_{q} by T_{D} = ħ/(2πk_{B}τ_{q}). With m^{*} being the known parameter, τ_{q} at T = 2 K can be extracted through the linear fit of ln ([B* sinh (αTμ/B)/αTμ]*Δρ/ρ_{0}) against 1/B. As shown in Fig. 3c, we have obtained τ_{q} = 3.8 × 10^{−14} s, from which the quantum mobility μ_{q}(=eτ_{q}/m^{*}) is estimated to be 1280 cm^{2}V^{−1}s^{−1}, much higher than that of SrMnBi_{2} (250 cm^{2}V^{−1}s^{−1} ^{23}) or SrMnSb_{2} (~570 cm^{2}V^{−1}s^{−1} ^{43}) (see Table 2). In general, the transport mobility is one or two orders of magnitude higher than quantum mobility, since the transport mobility is sensitive only to large angle scattering of carriers, while the quantum mobility is sensitive to both small and large angle scatterings. However, this was not observed in BaMnSb_{2}. Using the Hall coefficient R_{H} data extracted from Hall resistivity data shown in Fig. 3d, the transport mobility μ_{tr}(=R_{H}/ρ_{xx}) at 1.8 K is estimated to be ~1300 cm^{2}V^{−1}S^{−1} for BaMnSb_{2}, much less than that of SrMnSb_{2} (μ_{tr} ~12500 cm^{2}V^{−1}S^{−1}) at low temperature^{43}. The low μ_{tr} in BaMnSb_{2} may be associated with the transport localization behavior seen in ρ_{out} and ρ_{in}.
The Dirac fermion behavior probed in our experiments for BaMnSb_{2} is in good agreement with the prediction by first principle calculations that BaMnSb_{2} is a Dirac material at ambient pressure^{38}. Next, we will make more detailed comparisons between the predicted electronic band structure and our experimental observations. First, the Dirac bands are predicted to be generated by the Sb square net plane; thus the Fermi surface formed by Dirac bands is expected to be highly 2D, which is supported by our observations. As shown in Fig. 4a,d, systematic evolutions of SdH oscillation patterns for ρ_{in} and ρ_{out} are clearly observed as the magnetic field is rotated from the outofplane to the inplane direction (see the insets to Fig. 4b,e for the experiment setup). The oscillation frequency F(θ) extracted from the FFT for ρ_{in} measurements can be fitted to F(θ) = F(θ = 0°)/cosθ (Fig. 4c), suggesting that the Fermi surface responsible for SdH oscillations in BaMnSb_{2} is indeed 2D. However, in ρ_{out}(B,θ) measurements, we observed two frequency branches. As shown in Fig. 4f, the higher frequency branch also follows F(θ) = F(0°)/cosθ, while the lower frequency branch shows a weak angular dependence, suggesting the sample used for ρ_{in} measurements has slightly different morphology in its Fermi surface from the sample used for ρ_{out} measurements, which presumably originates from slightly different nonstoichiometric compositions. The 2D Fermi surface also explains the aforementioned large electronic anisotropy manifested in the large ρ_{out}/ρ_{in} ratio (~670). Second, the first principle calculations also predicted that the linear Dirac bands crossing occurs near the middle of ΓM, with the crossing point (i.e. the Dirac node) being right above the Fermi level, which implies the Fermi pocket hosting Dirac Fermions should be a hole pocket and small. In addition to the hole pocket enclosing the Dirac nodes, small electron pockets with quasilinear band dispersion are also predicted to exist at X and Y points. The quantum transport properties of Dirac fermion revealed in our experiments provide strong support to these predictions. As shown in Fig. 3d, our measured Hall resistivity exhibits linear field dependence with positive slopes at all temperatures as well as remarkable SdH oscillations below 50 K. These features prove that holes are dominant carriers and responsible for the SdH oscillations. From the SdH oscillation frequency of 22T of ρ_{in}, the extremal crosssection area of the Fermi surface is estimated to be ~0.2 nm^{−2}, about 0.1% of the total area of the first Brillouin zone, indicating an extremely small Fermi surface, the smallest as compared with AMnBi_{2} and SrMnSb_{2} (see Table 2). Third, the SOCinduced gap at the Dirac node in BaMnSb_{2} was predicted to be half of that in SrMnBi_{2} due to the weaker SOC of the Sb square net as mentioned above, which should result in lighter Dirac Fermions in BaMnSb_{2}. Our observations of small cyclotron Frequency and effective mass of the Dirac fermions in BaMnSb_{2} are in line with these predicted results. Furthermore, the Dirac cone in BaMnSb_{2} was predicted to be anisotropic^{38}, similar to that of SrMnBi_{2} and CaMnBi_{2}^{23,27,28}. ARPES experiments are called in to verify it.
The signatures of Dirac fermions in BaMnSb_{2} imply that materials including 2D Sb square net planes can harbor Dirac electrons. We note many such candidate materials indeed exist, e.g. ReAgSb_{2} (Re = rare earth), for which small mass quasiparticles have been found^{48,49,50}. Recent studies on LaAgSb_{2} have shown its small mass quasiparticles indeed originate from the Diraccone like band structure formed by Sb 5 P_{x,y} orbitals^{51}. Band structure calculations^{52} predicted that the Dirac cone in LaAgSb_{2} can host nearly massless Dirac fermions with m^{*} ~0.06m_{0} and has a very small Fermi surface with a quantum oscillation frequency of ~20T. However, these predictions were not seen in experiments^{48,52}; the smallest m* measured in experiments is 0.16m_{0} and the least quantum oscillation frequency is 72T^{48}. Surprisingly, the Dirac electron behavior observed in BaMnSb_{2} is very close to that predicted for LaAgSb_{2}. Note that the electronic structure of BaMnSb_{2} is much simpler than that of ReAgSb_{2}. BaMnSb_{2} exhibits only a single frequency quantum oscillations (~22T) and its transport properties can almost be described by a singleband model, whereas LaAgSb_{2} possesses a much complicated band structure, showing four frequencies in quantum oscillations^{48,49}.
Given that BaMnSb_{2} exhibits a Gtype AFM order with possible FM components due to Ba and Mn nonstoichiometry as discussed above, a natural question is whether its FM component can be tuned by changing Ba and Mn nonstoichiometry and coupled to quantum transport properties. In our previous studies on Sr_{1y}Mn_{1z}Sb_{2}, we have shown that its saturated FM moment M_{s} can be tuned from 0.6μ_{B}/Mn to 0.005μ_{B}/Mn by changing y and z^{43}. The samples with heavier Sr deficiencies have larger M_{s} than the samples with heavier Mn deficiencies. The M_{s} (~0.04 μ_{B}/Mn at 7T, see Fig. 1c) probed in BaMnSb_{2} seems comparable to that of type B samples of our previously reported Sr_{1y}Mn_{1z}Sb_{2} where M_{s} ~0.04–0.06 μ_{B}/Mn. Although we have examined many samples, all measured samples show comparable M_{s}. Therefore, it is difficult to find samples with a wide range of M_{s}, which would allow us to examine the coupling between Dirac electron behavior and ferromagnetism as we did for Sr_{1y}Mn_{1z}Sb_{2}^{43}.
Conclusion
In summary, we have demonstrated that in BaMnSb_{2} the Sb square net layers with coincident stacking of Ba atoms can host nearly massless Dirac fermions due to the weaker SOC effect of Sb, in contrast with massive Dirac fermions hosted by the Bi square net planes in AMnBi_{2}. Compared with AMnBi_{2}, BaMnSb_{2} displays much more significant 2Dlike electronic band structure, with the outofplane transport showing noncoherent conduction below 120 K and the ρ_{out}/ρ_{in} ratio reaching ~670. Its quantum transport properties can be almost described by a single band model, consistent with its simple electronic band structure predicted by first principle calculations. In addition, BaMnSb_{2} also exhibits a Gtype AFM order below 283 K and the Ba and Mn nonstoichiometries might cause a weak FM component. These findings establish BaMnSb_{2} as a promising platform for seeking novel exotic properties of massless Dirac fermions in low dimensions.
Methods
Single crystal growth and characterization
Single crystals of BaMnSb_{2} were synthesized using selfflux method with a stoichiometric ratio of Ba pieces, Mn and Sb powder. The starting materials were mixed in a small crucible, sealed into a quartz tube under Argon atmosphere and heated up to 1050 °C in one day. The temperature was maintained at 1050 °C for two days. After that, it was first cooled down to 1000 °C at a fast rate, 50 °C/h and then followed by a slowly cooling down to 450 °C at a rate 3 °C/h. Subsequently the furnace was turned off for fast cooling. Platelike crystals with lateral dimensions of several millimeters (see the inset to Fig. 1b) can easily be obtained from the final product. The compositions of the crystals were measured using an energy dispersive Xray spectrometer (EDS). The measured composition can be expressed as Ba_{1y}Mn_{1z}Sb_{2}, with y < 0.05 and 0.05 < z < 0.12. The structure of the single crystals was characterized by an Xray diffractometer.
Magnetization and magnetotransport measurements
The magnetization data were taken by a 7T SQUID magnetometer (Quantum Design). The magnetotransport properties were measured using standard four and five probe method for longitudinal and Hall resistivity, respectively, in a Physics Property Measurement System (PPMS, Quantum Design) and the 31T resistive magnet at National High Magnetic Field Laboratory (NHMFL) in Tallahassee.
Neutron Scattering
Singlecrystal neutron diffraction was performed at the HB3A Fourcircle Diffractometer equipped with a 2D detector at the High Flux Isotope Reactor(HFIR) at ORNL. Neutron wavelength of 1.546 Å was used from a bent perfect Si220 monochromator^{53}. The Rietveld refinement was performed using FullProf^{54}.
Additional Information
How to cite this article: Liu, J. et al. Nearly massless Dirac fermions hosted by Sb square net in BaMnSb_{2}. Sci. Rep. 6, 30525; doi: 10.1038/srep30525 (2016).
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Acknowledgements
The work at Tulane is supported by the U.S. Department of Energy under EPSCoR Grant No. DESC0012432 with additional support from the Louisiana Board of Regents (support for a graduate student, materials, travel to NHMFL). The work at NHMFL is supported by National Science Foundation Cooperative Agreement No. DMR1157490 and the State of Florida (high field measurements). The work at ORNL HFIR was sponsored by the Scientific User Facilities Division, Office of Science, Basic Energy Sciences, U.S. Department of Energy.
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The single crystals used in this study were synthesized and characterized by J.L., Y.Z. and A.C. The magnetotransport measurements in PPMS were carried out by J.L., D.J.A., S.M.A.R., L.S. and Z.M. The high field measurements at NHMFL were conducted by J.L., D.G., J.H., S.M.A.R., I.C. and Z.M. J.L. and Z.M. analyzed the data and wrote the manuscript. H.C. performed neutron scattering experiments and data analyses. All authors read and commented on the manuscript. The project was supervised by Z.M.
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Liu, J., Hu, J., Cao, H. et al. Nearly massless Dirac fermions hosted by Sb square net in BaMnSb_{2}. Sci Rep 6, 30525 (2016). https://doi.org/10.1038/srep30525
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