Single crystal growth, chemical defects, magnetic and transport properties of antiferromagnetic topological insulators (Ge_1-δ-xMn_x)₂Bi₂Te₅ ($x\leq 0.47$, $0.11\leq\delta\leq 0.20$)

The discovery of magnetic topological insulators (TIs) marks an important breakthrough in condensed matter physics in the past decade. When magnetism is introduced in TIs and breaks the time-reversal symmetry that protects the gapless Dirac surface states, a gapped surface state and dissipationless quantized edge conduction may appear. Therefore magnetic TIs can host a set of emergent phenomena such as quantum anomalous Hall effect, Axion insulating state and quantum magnetoelectric effectTokura et al. (2019); He et al. (2018); Liu et al. (2016); Wang et al. (2015). Among the magnetic TIs, MnBi_2nTe_3n+1 family with alternating [MnBi₂Te₄] septuple layer (SL), and $(n-1)$[Bi₂Te₃] quintuple layer (QL) is the first family that hosts intrinsic magnetism rather than introduced by doping Lee et al. (2013); Rienks et al. (2019); Zhang et al. (2019); Li et al. (2019); Otrokov et al. (2019a); Gong et al. (2019); Lee et al. (2019); Yan et al. (2019); Zeugner et al. (2019); Otrokov et al. (2019b); Aliev et al. (2019); Hu et al. (2020a); Ding et al. (2020); Wu et al. (2019); Shi et al. (2019); Tian et al. (2020); Yan et al. (2020); Gordon et al. (2019); Deng et al. (2020a). MnBi_2nTe_3n+1 goes from an A-type antiferromagnetic (AFM) state ($n\leq 3$) to a ferromagnetic state ($n\geq 4$), with magnetic moment pointing out-of-plane. The van der Waals nature makes it easy to exfoliate a bulk crystal into a thin-film device, in which quantized anomalous Hall conductance Liu et al. (2020); Deng et al. (2020b) and electric-field tuned Layer Hall effect Gao et al. (2021) are experimentally achieved in odd-layer and even-layer MnBi₂Te₄ devices, respectively.

The discovery of MnBi_2nTe_3n+1 was inspired by the existence of the nonmagnetic XBi_2nTe_3n+1 (X = Ge, Sn, Pb) series which have already been synthesized for decades Shelimova et al. (2004). XBi_2nTe_3n+1 are previously known thermoelectric materials, and recently attracted research interest due to their non-trivial band topology Neupane et al. (2012); Okamoto et al. (2012); Kuroda et al. (2012). When nonmagnetic X atoms are replaced by Mn, the quasi-metastable MnBi_2nTe_3n+1 compounds can be made in a very narrow temperature region Hu et al. (2020b). To the XTe-rich end of the XTe-Bi₂Te₃ phase diagram, besides XBi_2nTe_3n+1, thicker layered structures with more X in one building block exist. For example, X₂Bi₂Te₅ (X = Ge, Sn, Pb), abbreviated as the 225 phase, is made of nonuple layers (NL) while X₃Bi₂Te₆ (X = Ge, Sn, Pb) phase consists of undecuple layersShelimova et al. (2004); Kuropatwa and Kleinke (2012); Chatterjee and Biswas (2015); Matsunaga et al. (2007). The NL of X₂Bi₂Te₅ can be seen as inserting an additional XTe layer into XBi₂Te₄ SL, as shown in Fig. 1(a). Given the close structural correspondence between MnBi₂Te₄ and XBi₂Te₄, one may suspect Mn₂Bi₂Te₅ (Mn225) and Mn₃Bi₂Te₆ to exist, being potential candidates of intrinsic magnetic topological insulators. Indeed theoretical calculation has indicated Mn225 to be an intrinsic magnetic topological insulator that could host dynamical axion field Li et al. (2020); Zhang et al. (2020); Eremeev et al. (2022); Li et al. (2023); Tang et al. (2023). However, the successful growth of pure Mn225 phase is very challenging, hindering the investigation of its intrinsic physical properties Yan et al. (2022); Cao et al. (2021). For example, only a few layers of Mn225 phase were found embedded inside the MnBi₂Te₄ pieces in chemical vapor transport (CVT) growths while the Mn225 single crystals obtained via the self-flux growth might show significant contamination from the MnBi₂Te₄ phase.

In this paper, we report the growth, crystal and magnetic structures, as well as the transport and thermodynamic properties of high quality (Ge_1-δ-xMn_x)₂Bi₂Te₅ ($x\leq 0.47$, $0.11\leq\delta\leq 0.20$) single crystals. While our attempt to grow pure Mn225 single crystals is not successful using both the CVT and flux growth methods, pure (Ge_1-δ)₂Bi₂Te₅ (Ge225) single crystals were made by the flux method using Te as the self flux while (Ge_1-δ-xMn_x)₂Bi₂Te₅ (GeMn225) with $x\leq 0.47$ can be grown by the CVT method Hu et al. (2021a); Yan et al. (2022). The wavelength-dispersive X- ray spectroscopy (WDS) measurements as well as the refinements of the powder X-ray diffraction (PXRD) and single-crystal neutron diffraction data indicate the presence of significant Ge vacancies of $0.11\leq\delta\leq 0.20$, leading to holes dominating the electrical transport. We find that GeMn225 shows a $T_{N}=10.8$ K at $x=0.47$ with a spin flop transition at 2.0 T when $H\parallel c$. Our neutron analysis of the $x=0.47$ compound suggests negligible amount of Mn_Bi antisite formation and a bilayer A-type AFM with a refined Mn moment of 3.0(3) $\mu_{B}$ at 5 K.

Ge225 single crystals were grown using the self-flux method with Te as the flux. Ge chunks, Bi chunks and Te chunks were mixed at the ratio of Ge : Bi : Te = 2 : 2 : 8 in an alumina crucible and sealed in an evacuated quartz tube. The ampule was first heated to 1000 ^∘C overnight, then quickly cooled to 600 ^∘C before it was slowly cooled to 520 ^∘C in 3 days. At last, single crystals were separated from the flux by a centrifuge. Large and shiny mm-sized single crystals were obtained using this method.

Our flux-growth trials of the Mn-doping series using Te self flux did not yield the 225 phase. However, our CVT growth trials using MnI₂ as the transport agent resulted in high quality GeMn225 single crystals. Mn pieces, Ge chunks, Bi chunks, Te chunks and I₂ pieces were mixed at the ratio given in Table I, loaded and sealed in a quartz tube under vacuum. The tube was placed vertically in a box furnace and slowly heated to 1000 ^∘C overnight. It was then moved to a horizontal tube furnace where the low-temperature and high-temperature were set to be 520^∘C and 540^∘C, with the starting material on the high-temperature end. The cold-end temperature was selected as 520 ^∘C since it was the synthesis temperature reported for pure Ge225 in solid-state reaction Kuropatwa and Kleinke (2012). After two weeks, GeMn225 single crystals were taken out and rinsed with distilled water to remove the iodide impurities.

To identify the pieces of the 225 phase, (0 0 $L$) reflections were collected on both the top and bottom surfaces of single crystals using a PANalytical Empyrean diffractometer equipped with Cu K$\alpha$ radiation. Following this, we performed PXRD for further impurity checking and structural refinement. Then WDS measurements were conducted to obtain the elemental analysis of the samples, specifically the Mn level $x$. Magnetization data were collected in a Quantum Design (QD) Magnetic Properties Measurement System (MPMS). Specific heat and electrical transport measurements were made inside a QD DynaCool Physical Properties Measurement System (PPMS). Electrical contacts were made to the sample using Dupont 4922N silver paste to attach Pt wires in a six-probe configuration. To eliminate unwanted contributions from mixed transport channels, electrical resistivity ($\rho_{xx}$) and Hall ($\rho_{yx}$) data were collected while sweeping the magnetic field from -9 T to 9 T. The data were then symmetrized to obtain $\rho_{xx}(H)$ using $\rho_{xx}(H)=(\rho_{xx}(H)+\rho_{xx}(-H))/2$ and antisymmetrized to get $\rho_{yx}(H)$ using $\rho_{yx}(H)=(\rho_{yx}(H)-\rho_{yx}(-H))/2$. The magnetoresistance is defined as MR $=(\rho_{xx}(H)-\rho_{xx}(0))/\rho_{xx}(0)$. In our measurement geometry, the positive slope of $\rho_{yx}(H)$ suggests hole carriers dominate the transport. Single-crystal neutron diffraction was performed for the $x$ = 0.47 sample at 5 K and 0 T on the HB-3A DEMAND single-crystal neutron diffractometer located at Oak Ridge National LaboratoryChakoumakos et al. (2011). Both the neutron and X-ray diffraction data were refined using the Fullprof suit Rodríguez-Carvajal (1993).

The presence of vacancies has profound impact in the transport properties of the 225 compounds. Research on Mn-Bi-Te systems indicates that defect-free compounds are charge-neutral, with carriers in actual samples being contributed by various defects Hu et al. (2021b). Mainly, electron carriers are contributed by Bi_Mn/Ge and Te vacancies, whereas hole carriers are contributed by (Mn/Ge)_Bi and cation vacancies. This can be seen in the following defect chemistry for native Ge225:

Indeed one Ge_Bi produces one hole while one Bi_Ge creates one electron. In the presence of Ge vacancies, we can write

Therefore, the hole carrier density can be estimated by calculating $2\delta/A$, where $A$ represents the unit cell volume with cm³ as the unit. The carrier densities calculated through this defect analysis are denoted as $p_{2}$ and are summarized in Table I. As observed, the correspondence between $p_{1}$ and $p_{2}$ is reasonably good, especially for the $x=0$ and 0.33 samples.

We may tentatively estimate the saturation field of the $x=0.47$ sample by assuming linear field dependence of $M$ above 14 T. The interpolation is shown in Fig. 5 (a). When the magnetization at 2 K reaches 4.5 $\mu_{B}$/Mn, the saturation field is estimated to be around 30 T. For a uniaxial antiferromagnet, long-range order requires either interlayer coupling, or uniaxial magnetic anisotropy. Due to the bilayer nature of 225, there exist two interlayer exchange couplings, as depicted in Fig. 5(b): one is $J_{c1}$, representing the interlayer AFM coupling within each NL per Mn, the other is $J_{c2}$, denoting the interlayer coupling between adjacent NL per Mn. We can write down the full Hamiltonian in the ordered state, per Mn, asQian et al. (2022):

where $g$ is the Lande factor, $\mathbf{S}_{i}$ represents the Mn spin under investigation, $\mathbf{S}_{i+1}$ is the Mn in the same NL as $\mathbf{S}_{i}$ while $\mathbf{S}_{i-1}$ is the Mn in the adjacent NL, $K$ is the magnetic anisotropy parameter per Mn and $S=5/2$. Since $\mathbf{S}_{i+1}$ and $\mathbf{S}_{i-1}$ represent identical spin, we can combine two exchange coupling as $J_{c}=1/2(J_{c1}+J_{c2})$. The relationship between $J_{c}$ and $K$ is then,

Where $H_{sf}$ and $H_{s}$ is the spin flop field and saturation field. Using critical fields obtained above, we can get $SK$ = 8.0 $\mu$eV and $SJ_{c}$ = 1.8 meV. Table IV summarizes $SJ_{c}$, $K$, Mn-Mn distances and bond angles in Mn-Bi-Te family for comparison. Since $J_{c1}\gg J_{c2}$ owing to the much shorter superexchange path of $J_{c1}$ compared to $J_{c2}$, $J_{c1}$ can be approximated as $J_{c}$. $J_{c1}$ of (Mn_0.47Ge_0.41)₂Bi₂Te₅ is much larger than that of the MnBi_2nTe_3n+1 series. This is reasonable, given the much shorter Mn-Mn nearest-neighbor interlayer distance of 4.39 $\rm{\AA}$ in GeMn225 ($L_{2}$) compared to 13.86 $\rm{\AA}$ in MnBi₂Te₄ ($L_{1}$) or other Mn-Bi-Te compounds. Meanwhile, owing to a similar exchange path, $J_{c1}$ should be comparable to the coupling between the primary Mn site and the Mn_Bi antisite in MnBi₂Te₄. Indeed, the latter is responsible for the high full saturation field in MnBi₂Te₄ Lai et al. (2021). A much smaller magnetic anisotropy is obtained for (Mn_0.47Ge_0.41)₂Bi₂Te₅, compared to (Mn_0.6Pb_0.4)Bi₂Te₄, despite both have similar Mn occupancy and ordering temperature. This can be understood qualitatively by the bond angle analysis. As depicted in Table IV, both the Te-Mn-Te ($\alpha$) and Te-Mn-$z$ ($\beta$) angles exhibit a significant decrease in (Mn_0.47Ge_0.41)₂Bi₂Te₅ compared to MnBi_2nTe_3n+1. When the bond angles decrease, the ligand-field splitting will also become smaller due to a less overlap of wavefunctions, leading to a reduced magnetic anisotropy Huisman et al. (1971); Yan et al. (2021). This also explains why $SK$ remains similar across the MnBi_2nTe_3n+1 series (refer to Table IV), as the lattice environment of Mn remains consistent in these compounds.

When GeMn225 is exfoliated into even-NL or odd-NL thin flakes, both the inversion symmetry $\mathcal{P}$ and time reversal symmetry $\mathcal{T}$ are broken, while the combined $\mathcal{PT}$ symmetry is preserved. This symmetry condition is the same as the even-layer MnBi₂Te₄ device, where the Layer Hall effect Gao et al. (2021) and quantum metric nonlinear Hall effect Gao et al. (2023) are discovered. Therefore, the bilayer A-type AFM and the non-trivial band topology nature of GeMn225 Li et al. (2023); Zhang et al. (2020) make it an excellent system for probing these emergent phenomena, eliminating the need to differentiate between even-NL or odd-NL devices.

In summary, we have grown high-quality single crystals of (Ge_1-δ-xMn_x)₂Bi₂Te₅ with the doping level $x$ up to 0.47. Elemental analysis and diffraction techniques not only suggest Ge/Bi mixing, but also reveal the presence of significant Ge vacancies of $0.11\leq\delta\leq 0.20$, being responsible for the holes dominating the charge transport. As $x$ increases, long-range AFM order with the easy axis along $c$ emerges at 6.0 K for the $x=0.33$ sample and at 10.8 K for the $x=0.47$ sample. Spin-flop transitions observed at 0.7 T for $x=0.33$ and 2.0 T for $x=0.47$. Our refinement of the neutron diffraction data of the $x=0.47$ sample suggests a bilayer A-type AFM structure with the ordered moment of 3.0(3) $\mu_{B}$/Mn at 5 K. Our analysis of the magnetization data reveals a much stronger interlayer AFM exchange interaction and a much reduced uniaxial magnetic anisotropy when contrasted with MnBi₂Te₄. We argue the former arises from the shorter superexchange path and the latter to be linked to the smaller ligand-field splitting in (Ge_1-δ-xMn_x)₂Bi₂Te₅. Our study illustrates that this series of materials always exhibit broken $\mathcal{P}$ and broken $\mathcal{T}$ symmetries yet preserved $\mathcal{PT}$ symmetry upon exfoliation into thin flakes, providing a platform to explore the Layer Hall effect and quantum metric nonlinear Hall effect.

We thank Randy Dumas at Quantum Design for high field magnetization measurements. Work at UCLA was supported by the U.S. Department of Energy (DOE), Office of Science, Office of Basic Energy Sciences under Award Number DE-SC0021117. E. F. and H.C. acknowledges the support from U.S. DOE BES Early Career Award KC0402010 under contract No. DE-AC05-00OR22725. This research used resources of the Advanced Light Source, which is a DOE Office of Science User Facility under contract No. DE-AC02-05CH11231.