Robust magnetism and crystal structure in Dirac semimetal EuMnBi₂ under high pressure

Dirac materials, characterized by low-energy fermionic excitations behaving as massless Dirac particles, find potential application in electronic systems [1–3]. For exploring novel topological materials, magnetic Dirac materials with a layered structure comprising an insulating block layer and conducting Dirac/Weyl fermion layer appear to be promising for the possibility of manipulating the fermions through control of magnetic order under external stimuli such as magnetic field [4], chemical substitution [5], or pressure [6]. The layered material AMnX₂ (A: alkaline earth and rare-earth ions, X: Sb, Bi) [7–20] could implement the block-layer concept for Dirac/Weyl semimetals as AMnX₂ consists of an alternating arrangement of the X⁻ square net layer hosting quasi 2D Dirac fermions [21], and the insulating A²⁺-Mn²⁺-X³⁻ block layer [5].

Among the AMnX₂ materials, the initial focus turned toward SrMnBi₂ [6–8] for its manifestation of spin-degenerate quasi-2D Dirac bands observable along the Γ-M line [22]. In SrMnBi₂, the antiferromagnetic (AFM) order within the Mn layer seems to exert minimal impact on the Dirac fermion, while in EuMnBi₂, with the substitution of nonmagnetic Sr by magnetic Eu, high-field transport measurements indicate a robust influence of the AFM order from the Eu layers [23], on quantum transport and the Dirac band due to strong exchange interactions [4].

EuMnBi₂ crystallizes in a tetragonal structure (space group I4/mmm) consisting of a MnBi layer with edge-sharing MnBi₄ tetrahedrons and a Bi square net layer sandwiched by Eu atoms [13, 24]. The Eu layers demonstrate AFM ordering below the Néel temperature, T $_\textrm{N}$ = 22 K, characterized by a ferromagnetic arrangement of Eu moments within the ab plane and a sequential alignment along the c-axis as up-up-down-down [13, 14, 24]. EuMnBi₂ stands out as a distinctive system within the AMnBi₂ family, where the transport characteristics and the Fermi surface are significantly impacted by the magnetism from the Eu sublattice. It was found that the interlayer conductivity, along with the extent of spin splitting, significantly relies on the magnetic arrangement of the Eu layers and this arrangement can be regulated using an external magnetic field [4]. The Quantum Hall effect was detected in the system with the field-induced reorientation of the Eu moment [14]. This reorientation led to a decrease in the interlayer coupling, effectively constraining the Dirac fermions within the individual 2D Bi layer. Additionally, the Fermi energy in EuMnBi₂ can be regulated through the partial substitution of Gd³⁺ for Eu²⁺ within the block layer [5]. These studies demonstrate the strong tunability of this system.

These findings motivate us to use pressure as a control parameter and investigate the impact on the magnetic state, which provides the foundation for the magnetic control of electronic structure. Earlier electronic transport study indicates that the application of pressure initially enhances the T $_\textrm{N}$ of Eu ions up to 4 GPa, beyond which the magnetic order is suppressed [25]. Under further compression to 6 GPa, another low-temperature transition of a different nature was observed. It was proposed that this transition is linked to temperature-induced valence transition from Eu²⁺ towards Eu³⁺. Furthermore, x-ray absorption spectroscopy (XAS) studies up to 28 GPa provide evidence for divalent toward trivalent transition above 7 GPa without structural transformation. The valence transition is concurrent with Fermi surface modification suggested by Hall measurements. However, the sample used in this study contains a significant amount of Bi metal and an unidentified impurity phase. This necessitates further careful investigation into the magnetic and electronic states of this important material.

In this work, we have carefully analyzed the magnetism of Eu sublattice and crystal structure using a pure EuMnBi₂ sample under quasihydrostatic pressure using time-domain synchrotron Mössbauer spectroscopy (SMS) in ¹⁵¹Eu and synchrotron x-ray diffraction (XRD) techniques up to 33.1 GPa and 31.7 GPa, respectively. Our experiments show that the magnetic ordering temperature (T_o ) rises monotonically with increasing pressure and the magnetic order survives up to 33.1 GPa, contrary to the previous findings [25]. In addition, the tetragonal crystal lattice remains stable under compression up to at least 31.7 GPa.

Single crystals of EuMnBi₂ were grown by the Bi self-flux method. Reactants included elemental Eu pieces (Alfa Aesar, sublimed dendritic, 99.9$\%$), Mn pieces (Alfa Aesar, 99.9%), and Bi chunks (Alfa Aesar, 99.99%). The three metals were mixed in the ratio of Eu/Mn/Bi = 1:1:15, placed in an alumina crucible, and subsequently sealed in an evacuated silica tube ($\lt$10⁻⁵ torr) with a sample mass of ∼3 g. The sample was heated up to 1050 ^∘C at a rate of 60 ^∘C h⁻¹ and kept at 1050 ^∘C for 10 hours. Then the sample was cooled to 850 ^∘C at a rate of 100 ^∘C h⁻¹, and finally, slowly cooled to 400 ^∘C at a rate of 3 ^∘C h⁻¹. The samples were then centrifuged to remove the excess Bi flux. Plate-like single crystals with a typical size of ∼3 $\,\times\,$ 3 $\,\times\,$ 1 mm³ were obtained. The crystal structure was determined using an XtaLAB Synergy-S single crystal x-ray diffractometer, equipped with Mo K $_{\alpha}$ radiation (λ = 0.710 73 Å) with an ω of 2.0^∘ per scan and an exposure time of 10 s per frame. A SHELXTL package with the direct methods and full-matrix least-squares on the F² model was used to determine the crystal structure [26, 27].

The magnetic state of EuMnBi₂ was probed under high pressure using ¹⁵¹Eu SMS. Mössbauer spectroscopy under high-pressure sample environment is more feasible at the third-generation synchrotron sources compared to the conventional method using radioactive sources, due to the enhanced brilliance and compact size of the x-ray beams. Therefore, SMS is frequently employed in high-pressure magnetic research [28–32]. In conventional Mössbauer spectroscopy using a radioactive source, the Doppler effect modulates the source energy for energy-dependent absorption, whereas in time-domain SMS experiments, synchrotron radiation with an energy linewidth of ∼1 meV is used to coherently excite nuclear transitions and probe the hyperfine interactions from the reemitted radiation in time domain. Particularly, the 7/2 $\rightarrow$ 5/2 M1 nuclear transition at 21.54 keV in ¹⁵¹Eu with half life of 9.5 ns is well suited for synchrotron radiation studies [33]. The substantial 48% natural abundance of the ¹⁵¹Eu isotope allows for the use of samples with natural isotope abundance. The experiments were conducted at the Beamline 3ID of the Advanced Photon Source (APS), Argonne National Laboratory (ANL). The time-domain data was collected using 24-bunch timing mode with 153 ns intervals between the consecutive electron bunches. A miniature diamond anvil cell (DAC) and a compatible helium flow cryostat were utilized to achieve high pressure and low temperature [34]. A single crystal sample was loaded in the DAC. Pressures were determined in situ based on the shift of the ruby R1 fluorescence line [35]. Neon gas served as a quasihydrostatic pressure-transmitting medium. CONUSS program was used to analyze the SMS data and derive hyperfine parameters such as magnetic hyperfine field (B $_\textrm{hf}$) and electric quadrupole interaction (Δ) [36].

To investigate the crystal structure under high pressures, angle-dispersive synchrotron XRD experiments were carried out at room temperature at the 13BM-C beamline (PX²) of the APS, ANL [37]. X-rays with a wavelength of 0.434 Å were employed. To generate pressures up to 32 GPa, a BX-90 DAC equipped with Boehler-Almax anvils of 400 µm diameter culet size was used. A Re gasket, preindented to 76 µm with a 210 µm hole drilled using electrical discharge machining, was positioned between the anvils to create a sample chamber. A powder sample was loaded in the chamber with two ruby balls serving as an in situ pressure marker [35]. Neon gas was used as a pressure-transmitting medium. The two-dimensional diffraction images were integrated using the DIOPTAS software [38]. High-pressure XRD data were fitted using Le Bail method in GSAS-II [39].

The substantial difference in isomer shift values between the Eu²⁺ and Eu³⁺ valence states renders it a valuable parameter for providing information on the valence state of Eu ions [29, 40]. The isomer shift (δ) of ¹⁵¹Eu in EuMnBi₂ at 0.6 GPa and room temperature was determined from the SMS spectrum taken together with a trivalent EuF₃ (δ:0 mm s⁻¹) as a reference sample. Figure 1 shows the time-domain spectrum as well as the simulated Mössbauer spectrum in the energy domain based on the parameters obtained from fitting the experimental data. δ is derived to be −11.3 mm s⁻¹, a typical value for Eu²⁺. This value is also comparable to the reported −11.4 mm s⁻¹ (relative to the ¹⁵¹SmF₃ source) in a sister compound EuMnSb₂ [41].

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Representative SMS spectra in ¹⁵¹Eu from EuMnBi₂ at different temperatures and pressures are displayed in figure 2. B $_\textrm{hf}$ and Δ, along with the effective sample thickness, served as fitting variables. For 4f⁷ ions, the magnetic hyperfine field arises due to contributions from core electron polarization, conduction electron polarization by ions, and neighboring atoms [28, 42]. In the presence of magnetic order, Zeeman splitting of nuclear levels leads to quantum oscillations in the SMS spectrum, with additional beating frequency from quadrupole interaction. At constant pressure, the oscillation frequency decreases as the temperature rises due to a reduction in the B $_\textrm{hf}$ strength. The oscillations disappear when the sample is warmed up to the paramagnetic phase. At 1.6 GPa and 13.2 K B $_\textrm{hf}$ is 25.98(3) T, consistent with the magnetic hyperfine field observed in other divalent Eu-based compounds [41, 43, 44]. At 1.6 GPa T_o lies between 23 K and 37 K. Above T_o no sizable quadrupole interaction is resolved. At pressure higher than 8.6 GPa, T_o increases to a temperature above 35 K. At the lowest temperature (∼13 K), the oscillation frequency increases as pressure is raised from 1.6 GPa to 33.1 GPa, attributed to increasing B $_\textrm{hf}$ which leads to nuclear Zeeman splitting in larger energy range. This is also demonstrated in the simulated energy-domain spectra (see figure 2(b)) based on the fitting parameters to the experiment data. The B $_\textrm{hf}$ values at various temperatures and pressures are summarized in figure 3. At around 13 K, B $_\textrm{hf}$ increases monotonically with pressure.

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Table 1 tabulates the derived hyperfine parameters. The main determinant of Δ is the electric field gradient (EFG) at the nucleus, which arises from lattice charge distribution and incompletely filled electronic shells. In the case of Eu²⁺, the half-filled 4f shell conforms to Hund's rule, rendering the isotropic electronic ground state (⁸ S $_{7/2}$) and yielding no electronic contribution to Δ due to zero total angular momentum [45]. Consequently, Δ values predominantly mirror the EFG originating from the lattice effects. Note that in the magnetic phase, a small Δ value is needed to fit the SMS data.

Table 1. List of magnetic hyperfine field ($B_\textrm{hf}$) and electric quadrupole interaction (Δ) of ¹⁵¹Eu in EuMnBi₂ at high pressures and various temperatures. The error bars are generated from the fitting uncertainties in CONUSS.

P (GPa)	T (K)	$B_\textrm{hf}$ (T)	Δ (mm/s)
	13.2	25.98(3)	4.0(1)
	15.9	24.29(3)	3.7(1)
	18	23.42(2)	3.8(2)
1.6	21	21.57(1)	3.3(1)
	23	18.80(5)	3.0(2)
	25	15.36(4)	2.6(1)
	37	0	1.2(1)
	13.2	26.67(4)	4.3(1)
	16	25.33(5)	3.7(2)
3	19	23.58(4)	3.0(2)
	25	18.56(6)	3.3(2)
	50	0	0
	13	27.72(4)	3.8(2)
6.1	18	26.37(5)	4.2(2)
	30	7.92(3)	4.3(7)
	13	28.30(5)	4.3(2)
	20	26.52(4)	3.6(2)
8.6	30	22.0(5)	3.9(2)
	35	18.04(5)	4.1(1)
	75	0	0
	13	30.07(6)	6.0(2)
	25	27.34(1)	5.5(3)
	35	23.50(6)	5.1(2)
13	42	18.48(8)	2.1(2)
	50	1.36(3)	0
	75	0	0
	120	0	0
	12.9	32.62(5)	5.2(3)
20.5	25	31.01(3)	5.3(2)
	35	29.03(4)	5.3(1)
25.8	12.9	34.06(6)	4.8(3)
30.7	12.8	35.30(3)	4.7(4)
33.1	12.9	36.51(7)	2.7(9)

The tetragonal crystal structure (I4/mmm) of EuMnBi₂ was confirmed at ambient conditions from single crystal XRD with Mo radiation (λ = 0.71073 Å). The lattice parameters agree well with the reported values in [13]. The atomic occupancies were checked carefully and no vacancies were observed from the XRD data. The detailed crystal structure information is tabulated in table 2.

Table 2. Single crystal XRD refinement and atomic coordinates and equivalent isotropic displacement parameters for EuMnBi₂ at ambient conditions. U_eq is defined as one-third of the trace of the orthogonalized U_ij tensor (Ų).

Space group	I4/mmm
a (Å)	4.5510(7)
c (Å)	22.560(6)
V (A3)	467.1(2)
θ range (∘)	3.613–25.102
# of reflections; $R_\mathrm{int}$	1072; 0.1429
# of independent reflections	161
# of parameters	13
R1; $\omega R_2$ (I $\gt$2δ(I))	0.1428; 0.3045
Goodness of fit (GOF)	1.085

Atom	Wyck.	x	y	z	Occ.	Ueq
Eu	4e	0	0	0.1124(2)	1	0.036(3)
Mn	4d	0	0.5	0.25	1	0.032(2)
Bi1	4c	0	0.5	0	1	0.035(3)
Bi2	4e	0	0	0.3282(3)	1	0.032(3)

The crystal structure of EuMnBi₂ was investigated under pressures up to 31.7 GPa using synchrotron XRD. As shown in figure 4(a), the XRD data can be indexed with the tetragonal space group I4/mmm up to at least 31.7 GPa, indicating the stable tetragonal lattice in this pressure range. The XRD data also confirms the purity of the sample. The lattice parameters and volume under pressure are displayed in figures (b)–(e). Analyzing the pressure-volume data with the third-order Birch–Murnaghan equation of state yields the bulk modulus B₀ = 36.7(3) GPa, its pressure derivative B $^{\prime}_0$ = 4.32(5), and zero-pressure volume V₀ = 457.87(6) ų. The B₀ value is slightly lower than the reported value of 41.8 GPa [25]. This small discrepancy may originate from the less hydrostatic pressure medium used in the earlier study.

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Our high-pressure SMS measurements show that the magnetic ordering temperature in EuMnBi₂ lies above 25 K at 3 GPa, in agreement with the earlier report that T_N increases continuously to 31 K at 3.9 GPa. This initial increase may be attributed to the enhanced Ruderman–Kittel–Kasuya–Yosida exchange interaction, primarily the intralayer exchange coupling due to the smaller Eu–Eu distance within the ab-plane than along the c-axis. At higher pressure, our SMS data show that at 20.5 GPa the T_o lies higher than 35 K and the magnetic order persists up to 33.1 GPa, contrary to the report from Susilo et al that T_N is suppressed above 4 GPa based on electrical resistivity measurements [25]. It is noted that Mössbauer effect is a sensitive probe for detecting magnetic transition, while in resistivity measurements other phenomena may obscure the signature of magnetic order and it becomes challenging to interpret the data on magnetic state with full confidence.

In Eu-based magnetic intermetallic compounds, pressure typically has the effect of reducing the magnetic transition temperature and/or magnetic moment [46]. This is because a valence transition from magnetic Eu²⁺ (J = 7/2) to nonmagnetic Eu³⁺ (J = 0) is anticipated under high pressure, given that the ionic radius of Eu³⁺ is smaller than that of Eu²⁺. This valence transition has been observed in many compounds such as parent and doped EuFe₂As₂ [43, 47], Eu(Fe$_{1-x}$Ni_x )As₂ [29], and EuRh₂Si₂ [48]. Similarly, in EuMnBi₂ high-pressure XAS measurements at Eu L₃ edge indicate a large increase in Eu mean valence by ∼0.4 at 27 GPa [25]. It is surprising to maintain the magnetic order up to 33.1 GPa with such a significant valence increase. To understand the robust magnetism in this important material, further theoretical investigations are desired.

In this study, we investigated the magnetic properties and crystal structure of EuMnBi₂ under quasihydrostatic pressure conditions. Using time-domain SMS in ¹⁵¹Eu, we have found that the magnetic order of Eu sublattice persists up to 33.1 GPa with T_o higher than 35 K at 20.5 GPa. These results are in sharp contrast to the previous claim of suppression of magnetic order above 4 GPa based on electrical resistivity studies. Our synchrotron XRD experiment on a high-quality EuMnBi₂ sample shows that the ambient tetragonal structure remains stable up to pressure as high as 31.7 GPa. Our findings highlight that high pressure serves as an effective approach to enhance the magnetic exchange interaction. The experimental data provide foundation for further experimental as well as theoretical studies.

This work was supported by the National Science Foundation (NSF) Award No. OIA-2033131. G C J acknowledges the financial support from the Alabama Graduate Research Scholars Program (GRSP) funded through the Alabama Commission for Higher Education and administered by the Alabama EPSCoR. W X was supported by U.S. DOE-BES under Contract DE-SC0023648. This research used resources of the Advanced Photon Source, a U.S. DOE Office of Science user facility operated for the DOE Office of Science by Argonne National Laboratory under Contract No. DE-AC02-06CH11357. Uses of the COMPRES-GSECARS gas loading system and the PX² program were supported by COMPRES under NSF Grant EAR-1661511 and GSECARS through NSF Grant EAR-1634415. B L acknowledges the support from COMPRES. We are grateful to Haozhe Wang's analysis of the single crystal diffraction data.

The data cannot be made publicly available upon publication because they are not available in a format that is sufficiently accessible or reusable by other researchers. The data that support the findings of this study are available upon reasonable request from the authors.