Neutron diffraction determination of the AF order

Figure 1a shows neutron diffraction data for the (h0l) reciprocal-lattice plane taken below T_N ≈ 18 K at T = 6 K, and Fig. 1b shows data along the (00l) direction taken above and below T_N at 30 and 6 K. Nuclear Bragg peaks occur in Fig. 1b at l = 2n positions for both temperatures whereas magnetic Bragg peaks only appear in the 6 K data. Two AF propagation vectors define the locations of the magnetic Bragg peaks: (1) peaks at l = 2n ± τ_1z positions correspond to τ₁ = (0, 0, τ_1z) with τ_1z = 0.303(1); (2) nuclear-forbidden peaks at l = 2n + 1 positions in Fig. 1b correspond to τ₂ = (0, 0, 1). [τ₂ = (0, 0, 1) is equivalent to τ₂ = (0, 0, 0) for P6₃/mmc. We use τ₂ = (0, 0, 1) to facilitate presentation of the diffraction data.] Within the (h0l) reciprocal-lattice plane, magnetic Bragg peaks matching τ₂ overlap with nuclear Bragg peaks with odd values of l when the conditions h = 3n + 1 or h = 3n + 2 (n = integer) are satisfied. The predicted A-type order would be consistent with magnetic Bragg peaks appearing only at positions corresponding to τ₂. However, the existence of additional magnetic Bragg peaks at τ₁ reveals the presence of more complex helical or spin-density-wave type (itinerant) AF order. Our measurements cannot distinguish between these two types of order, but our DFT calculations reveal that the Eu 4f bands are located well below the Fermi level which is more supportive of local-moment helical order.

Using magnetic symmetry analysis and single-crystal refinements to our data from the TRIAX triple-axis spectrometer and the CORELLI time-of-flight spectrometer, we discover that EuIn₂As₂ has the complex broken-helix AF order illustrated in Fig. 1d and Supplementary Fig. 5. It consists of ferromagnetically aligned layers with μ⊥c that are helically stacked along c. We find that the orthorhombic MSG $$ C{2}^{\prime }{2}^{\prime }{2}_1$$ (No. 20.33) describes the symmetry, but for continuity we refer to directions with respect to the hexagonal chemical unit cell. We find μ = 6.0(3)μ_B/Eu at T = 6 K, which is smaller than the expected value of 7.0μ_B/Eu and the saturation moment of μ_sat = 7.00(6)μ_B shown below and in Supplementary Fig. 11. We note that neutron diffraction studies of other Eu containing compounds report less than 7.0μ_B/Eu^19,20, and, as shown below, the ordered moment is still increasing with decreasing temperature below 6 K. Details of the corrections performed to account for the strong thermal neutron absorption of Eu are given in the “Methods” section and Supplementary Note 1.

To analyze the broken-helix order, we assume a tripling of the chemical unit cell since $$ {\tau }_z=0.303\left(1\right)\approx \frac{1}{3}$$ and the relatively small incommensurability results in maximum disagreement only at the 33rd Eu layer. Referring to Fig. 1d, we designate the two Eu crystallographic sites in $$ C{2}^{\prime }{2}^{\prime }{2}_1$$ as red and blue. This in turn indicates red and blue ferromagnetically aligned layers, and allows for defining the helix turn angles ϕ_rr and ϕ_rb between successive red–red and red–blue layers, respectively. The MSG symmetry dictates that ϕ_rr + 2ϕ_rb = 180^∘ and constrains magnetic moments in the blue layers to lie along $$ \left[\overline{1}10\right]$$. Our refinements find that ϕ_rb = 127(3)^∘ at T = 6 K, and more details of the refinements are given in Supplementary Figs. 1–9 and Supplementary Tables 1–3.

We simulate in Fig. 2a the integrated intensities for representative magnetic Bragg peaks for different values of ϕ_rb using the determined MSG and μ⊥c. Magnetic Bragg peaks corresponding to τ₂ disappear as ϕ_rb → 60^∘ (pure 60^∘-helix order), whereas magnetic Bragg peaks matching τ₁ disappear as ϕ_rb → 180^∘ (A-type order). We find that the pure 60^∘-helix order can be described by the higher-symmetry MSG $$ P{6}_1{2}^{\prime }{2}^{\prime }$$ (No. 178.159) and that the A-type order can be described by MSG $$ C{m}^{\prime }{c}^{\prime }m$$ (No. 63.462). The darker shaded area in Fig. 2a indicates the experimental value of ϕ_rb at T = 6 K.

Fig. 2

Details of the broken-helix magnetic order.

a Simulations of the magnetic neutron diffraction intensity for the (2, 0, 1) Bragg peak and the average of the (2, 0, 2) ± τ₁ Bragg peaks as functions of the broken-helix turn angles ϕ_rb and ϕ_rr. b Temperature dependencies of the integrated intensity of the (2, 0, 1) Bragg peak and the average integrated intensity for the (2, 0, 2) ± τ₁ magnetic Bragg peaks measured on the CORELLI spectrometer and scaled to 1 at T = 6 K. The inset displays the helix turn angles ϕ in the pure 60^∘-helix phase (T_N2 < T ≤ T_N1) and ϕ_rb for the broken-helix phase (T ≤ T_N2). c Temperature dependence of τ₁ = (0, 0, τ_1z) from the average of the positions of the (0, 0, 6) ± τ₁ magnetic Bragg peaks. In b, ϕ(T) is calculated from the temperature evolution of τ_1z. ϕ_rb(T) is calculated from the temperature dependence of the ratio of the integrated intensity of the (0, 0, 5) magnetic Bragg peak to that of the (006) − τ₁ magnetic Bragg peak. These data are from measurements made on the TRIAX spectrometer and are shown in Supplementary Fig. 9.

Figure 2b, c reveals the temperature evolution of the AF order. Figure 2b displays the measured integrated intensities of the (2, 0, 2) ± τ₁ and (2, 0, 1) magnetic Bragg peaks scaled to 1 at T = 6 K. The curves suggest that μ continues to increase with decreasing temperature below 6 K. Two transitions are evident: the incommensurate magnetic Bragg peaks corresponding to τ₁ with $$ {\tau }_{1z}\approx \frac{1}{3}$$ emerge at T_N1 = 17.6(2) K and commensurate magnetic Bragg peaks corresponding to τ₂ appear at T_N2 = 16.2(1) K. These transitions also appear in the magnetic susceptibility, resistance, and ¹⁵¹Eu Mössbauer data shown in Supplementary Fig. 10. Figure 2c displays the temperature dependence of τ₁ which indicates that pure 60^∘-helix order first emerges at T_N1, and τ_1z decreases upon cooling until T_N2 where broken-helix order emerges. This sequence follows the rightmost path in the group-subgroup chart in Supplementary Fig. 18 which traces second-order magnetic transitions from the paramagnetic state to pure 60^∘-helix order to broken-helix order.

The inset to Fig. 2b shows the evolution of the helix turn angle ϕ below T_N1. ϕ ≈ 60^∘ at T_N1 and slightly decreases upon cooling for T_N2 < T < T_N1. The small change from 60^∘ reflects the change in τ_1z. For T < T_N2, the broken-helix order emerges and the inset to Fig. 2b plots ϕ = ϕ_rb, which is calculated from data shown in Supplementary Fig. 9. ϕ_rb rapidly increases immediately below T_N2 and approaches ϕ_rb = 130^∘ as T → 0 K. The temperature evolution of ϕ_rb below T_N2, while maintaining τ_1z = 0.303(1), agrees with the calculations in Fig. 2a showing that broken-helix order persists over a range of ϕ_rb.

Density functional theory calculations and symmetry analyses

Having established and described the emergence of complex helical order, we now discuss and compare the impacts of broken-helix, pure 60^∘-helix, and A-type order on the electronic-band structure and topology. First, similar to previous reports¹⁸, our DFT calculations and symmetry analyses find that A-type order creates an AXI state with inverted bulk bands near Γ and an electronic-band gap of ≈100 meV. Looking at the phase diagram in Fig. 2a, we realize that by adiabatic continuation the magnetic symmetry can be lowered away from the pure 60^∘-helix or A-type ordered states by varying ϕ_rb while maintaining the constraints that ϕ_rr + 2ϕ_rb = 180^∘ and ordered moments in the blue layers lie along $$ \left[\overline{1}10\right]$$. For example, starting from A-type order, reducing ϕ_rb from 180^∘ immediately breaks the mirror and $$ \mathcal{I}$$ symmetries present in MSG $$ C{m}^{\prime }{c}^{\prime }m$$.

Using this adiabatic continuation approach, we find via DFT calculations that a full band gap persists for any value of ϕ_rb. Figure 3a shows a diagram of the Brillouin zone and Fig. 3b shows the calculated electronic bands for ϕ_rb = 130^∘. Figure 3c focuses on the gap region near Γ where band inversion is highlighted by green bars denoting As 4p_z orbital character, and Fig. 3d demonstrates that the inverted gap remains for pure 60^∘-helix order. Supplementary Figure 15 shows that this result holds for other values of ϕ_rb spanning 180^∘ ≤ ϕ_rb ≤ 60^∘. Thus, the inverted gap remains across all three magnetic orders connected by the internal parameter ϕ_rb and indicates that the topological phase is robust to changes to ϕ_rb. In contrast, we find that bands cross the Fermi level for the case of ferromagnetic order as shown in Supplementary Fig. 15. This opens the possibility of controlling the band structure by applying a magnetic field strong enough to align μ along a single direction. This point is discussed further below.

Fig. 3

Bulk and surface electronic-band structures for various magnetic ground states.

a Hexagonal Brillouin zone and the projected (110) and (001) surface-Brillouin zones with high-symmetry points (capital letters) and reciprocal-lattice axes (b₁ = a^*, b₂ = b^*, and b₃ = c^*) indicated. b Bulk band structure along high-symmetry paths for broken-helix order with ϕ_rb = 130^∘. E_F is the Fermi energy. c, d Views of the minimal gap region along the K-Γ-M direction for ϕ_rb = 130^∘ (broken-helix order) (c) and ϕ_rb = 60^∘ (pure 60^∘-helix order) (d) . The top valence band according to simple band filling is indicated in blue and the rest of the bands are colored red. Green vertical bars show As 4p_z orbital character and indicate band inversion near the minimal gap. e–h Surface bands for the (110) (e, f) and (001) (g, h) projected surfaces along high-symmetry lines for ϕ_rb = 130^∘ (e–g). A Dirac cone exists on each surface, but the presence of a gap and whether or not the cone is pinned to a time-reversal-invariant momentum (TRIM) depends on symmetry: a gapless Dirac cone unpinned from a TRIM occurs for the (110) surface preserving the $$ 2^{\prime }$$ symmetry axis. On the other hand, the (001) surface (g) has a gapped Dirac cone at $$ \overline{\mathrm{\Gamma }}$$ due to the absence of a $$ 2^{\prime }$$ axis and any other symmetry that reverses an odd number of space–time coordinates. The surfaces with gapped Dirac cones support half-integer quantum-anomalous-Hall-type conductivity. The non-magnetic calculation for the (001) surface (h) has a gapless Dirac cone pinned to $$ \overline{\mathrm{\Gamma }}$$. Note that panels e and f are zoomed in around the bottom of the band gap to show the details of the surface Dirac cone. E_F is inside the gap and the surface Dirac point is at E_F − 0.061 eV, very close to the top of the valence band projection. More details of the surface band structure are found in Supplementary Fig. 15g.

Sample synthesis

Single crystals of EuIn₂As₂ were grown using a flux method similar to ref. ¹⁵ and found to be single phase via powder X-ray diffraction. An initial composition of Eu:In:As = 3:36:9 was weighed out and packed in fritted alumina crucibles²⁴, followed by sealing in a fused silica tube. The prepared ampoule was first heated up to 300 ^∘C over 2 h and held there for an hour, then heated up to 580 ^∘C over 3 h and held there for 2 h, and finally heated up to 900 ^∘C over 10 h followed by a 2-h dwell. The intermediate dwells were to ensure maximum dissolving and incorporation of the volatile elements into the melt. The final dwell was followed by a 48-h cool down to 770 ^∘C, at which point excess flux was decanted using a centrifuge. This process yielded plate-like crystals of EuIn₂As₂ with masses of a few milligrams.

Magnetization experiments

Magnetization measurements were made on single-crystal samples down to T = 2 K under applied magnetic fields up to μ₀H = 5 T using a Quantum Design, Inc. Superconducting Quantum Interference Device.

Neutron diffraction experiments

Neutron diffraction measurements were made on the TRIAX triple-axis spectrometer at the University of Missouri Research Reactor, and on the time-of-flight (TOF) CORELLI spectrometer at the Spallation Neutron Source at Oak Ridge National laboratory²⁵. Single-crystal samples with masses of 4.5 and 11.6 mg were used on TRIAX and CORELLI, respectively. The samples were attached to Al sample mounts and cooled down to T = 6 K using closed-cycle He refrigerators. The integrated intensities of recorded Bragg peaks were corrected for the effects of neutron absorption using the MAG2POL²⁶ software according to the procedure described in ref. ²⁷. The samples were flat rectangular plates with thicknesses of 0.1 mm along [001]. The TRIAX sample has an approximate width of 3 mm along [100] and approximate length of 2 mm along [010]. The CORELLI sample has an approximate width of 3 mm along [110] and approximate length of 3 mm along $$ \left[1\overline{1}0\right]$$.

The TRIAX experiments were performed in elastic mode with incident and final neutron energies of E_i,f = 14.7 and 30.5 meV. Söller slit collimators with divergences of $$ 60^{\prime }$$-$$ 60^{\prime }$$-$$ 80^{\prime }$$-$$ 80^{\prime }$$ were inserted before the pyrolytic graphite (PG) (002) monochromator, between the monochromator and sample, between the sample and PG (002) analyzer, and between the analyzer and detector, respectively. PG filters were inserted before and after the sample to eliminate higher-order beam harmonics. The sample’s (h0l) reciprocal-lattice plane was aligned within the horizontal scattering plane. In order to collect data for purely nuclear and magnetic Bragg peaks, a series of (h0l) reflections were recorded at T = 30 and 6 K, i.e. significantly above T_N1 and below T_N2, respectively. Each Bragg peak was fitted with a Gaussian lineshape in order to determine its integrated intensity (area), center, and full-width at half-maximum. The magnetic signals of the low-temperature τ₂ Bragg peaks were then extracted by subtracting the integrated intensity of the 30 K nuclear contribution when appropriate. A Lorentz factor ($$ L=\frac{1}{\sin 2\theta }$$) was also applied. Refinements were made with FULLPROF²⁸. Throughout this work, error bars represent one standard deviation.

CORELLI experiments were performed at several temperatures above and below T_N1 and T_N2 with the (h0l) plane set horizontal. However, the spectrometer employs 2-D position-sensitive detectors which allow for recording some Bragg peaks above and below this plane. The TOF measurement technique employs a white neutron beam, and the sample was rotated around its vertical axis at discrete intervals in order to cover a large swath of reciprocal space. Individual Bragg peaks were obtained by performing cuts through the 2-D maps recorded by the instrument, returning intensity versus Q profiles of the peaks. Integrated intensities were in turn calculated by adding up all of the data points located in the peak profile that were above background. To account for the pulsed beam’s non-linear distribution of wavelengths, a normalization to vanadium scattering was applied. CORELLI data used for the structural and magnetic refinements presented in Supplementary Note 1 are restricted to a bandwidth of incoming neutron energies of 45–55 meV in order to accurately correct for neutron absorption.

Electronic-band structure calculations

Band structures using DFT with spin–orbit coupling were calculated with the Perdew–Burke–Ernzerhof exchange-correlation functional, a plane-wave basis set, and the projected-augmented-wave method as implemented in VASP^29,30. To account for the half-filled strongly-localized Eu 4f orbitals, a Hubbard-like³¹U value of 5.0 eV is used. For different helical magnetic structure with τ ≈ (0, 0, 1/3), i.e. the hexagonal unit cell is tripled along the c-axis with atoms fixed in their bulk positions. A Monkhorst-Pack 12 × 12 × 1 k-point mesh with a Gaussian smearing of 0.05 eV including the Γ point and a kinetic-energy cutoff of 250 eV have been used. To search for possible band gap closing points in the full Brillouin Zone, a tight-binding model based on maximally localized Wannier functions³² was constructed to reproduce closely the bulk band structure including spin–orbit coupling in the range of E_F ± 1 eV with Eu sdf, In sp, and As p orbitals as implemented in WannierTools³³.