- Understanding the quantum spin Hall effect
- New topological phases in mercury compounds
- Developing quantum spin Hall materials: mercury compounds and grey tin
- Predicting new quantum spin Hall systems in atomically-thick 2D materials
- Magnetic impurities in topological semiconductors: superexchange vs. Van Vleck mechanism
- Quantum and topological phase transitions

*Unlike in the quantum Hall effect and quantum anomalous Hall effect, the quantization precision in the quantum spin Hall effect depends on unavoidable background impurities and defects. However, doping with magnetic ions restores the quantization accuracy. In specific systems, topologically trivial edge states and spontaneous formation of excitonic liquid can play a role.*

Inspired by experimental data accumulated in Wuerzburg and by CETRERA/MagTop collaboration [1,2] for HgTe and (Hg,Mn)Te topological quantum wells, MagTop’s researchers addressed the origin of a short protection length in quantum spin Hall effect from various angles. Results of Refs. [3,4] demonstrated quantitatively and with no adjustable parameters that the presence of paramagnetic acceptor dopants explain a variety of hitherto puzzling properties of two-dimensional topological semiconductors at the topological phase transition and in the regime of the quantum spin Hall effect. In particular, the theory elucidates the origin of short topological length (see Figure) and provides methods for its enhancement by negative U centers and bound magnetic polarons in Mn-doped samples.

Tight-binding model for CdTe/HgTe/CdTe or CdTe/HgS/CdTe quantum wells and InAs/GaSb heterostructures show that all these materials support additional unprotected edge states that may lead to backscattering between helical states in the topological regime [5]. While these states are far from the gap in CdTe/HgTe/CdTe, they can be relevant in other systems. Furthermore, a relatively strong coupling between electron and hole bilayers in InAs/GaSb heterostructures may result in a spontaneous symmetry breaking (formation of exciton condensate) that will also have a destructive effect on the quantum spin Hall effect [6].

[1] I. Yahniuk, S. S. Krishtopenko, G. Grabecki, B. Jouault, C. Consejo, W. Desrat, M. Majewicz, A. M. Kadykov, K. E. Spirin, V. I. Gavrilenko, N. N. Mikhailov, S. A. Dvoretsky, D. B. But, F. Teppe, J. Wróbel, G. Cywiński, S. Kret, T. Dietl, W. Knap,* Magneto-transport in inverted HgTe quantum wells,* npj Quantum Materials 4, 13 (2019).

[2] I. Yahniuk, A. Kazakov, B. Jouault, S. S. Krishtopenko, S. Kret, G. Grabecki, G. Cywiński, N. N. Mikhailov, S. A. Dvoretskii, J. Przybytek,V. I. Gavrilenko, F. Teppe, T. Dietl, W. Knap, *HgTe quantum wells for QHE metrology under soft cryomagnetic conditions: permanent magnets and liquid 4He temperatures*, arXiv:2111.07581 (2021) [retracted from a journal to do not publish with Russian institutions].

[3] T. Dietl, *Effects of charge dopants in quantum spin Hall materials,* Phys. Rev. Lett. 130, 086202 (2023).

[4] T. Dietl, *Quantitative theory of backscattering in topological HgTe and (Hg,Mn)Te quantum wells: Acceptor states, Kondo effect, precessional dephasing, and bound magnetic polaron*, Phys. Rev. B 107, 085421 (2023) (Editors’ Suggestion).

[5] N. M. Nguyen, G. Cuono, R. Islam, C. Autieri, T. Hyart, W. Brzezicki, *Unprotected edge modes in quantum spin Hall insulator candidate materials*, Phys. Rev. B 107, 045138 (2023).

[6] T. Paul, V. Fernandez Becerra, T. Hyart, *Interplay of quantum spin Hall effect and spontaneous time-reversal symmetry breaking in electron-hole bilayers I: Transport properties*, Phys. Rev. B 106, 235420 (2022).

*Since the beginning of the era of topological materials, HgTe has been one of the most attractive compounds due to a large band inversion that makes it a robust topological system [**B. A. Bernevig et al. **Science, 1757 (2006)**; M. Koenig, **Science 318, 766 (2007)**]. **MagTop’s computational team engineered new topological phases of two kinds of HgTe-based superlattices, one preserving time-reversal symmetry and another violating it by magnetic ions.*

Using *ab initio* computations, we investigate how topological phases evolve as a function of hydrostatic pressure and uniaxial strain in non-magnetic HgTe/HgSe superlattices. Our analysis unveils the presence of isoenergetic nodal lines, which could host strain-induced three-dimensional flat bands at the Fermi level when fabricated, for instance, as core-shell nanowires. Additionally, we found to harbor a rich phase diagram with a plethora of new topological phases as an ideal Weyl semimetal phase (see Figure); a sequentiall transition to a Dirac semimetal, to a nodal-line semimetal, and finally to a topological insulator [1].

Regarding the topological phases with the breaking of the inversion symmetry, we study theoretically the interplay between magnetism and topology in HgTe/MnTe superlattices. An axion insulator phase is observed for the antiferromagnetic order with the out-of-plane Néel vector direction below a critical thickness of MnTe. Defining T as the time-reversal symmetry, this axion insulator phase is protected by a magnetic twofold rotational symmetry C_{2}·T (Figure). The axion insulator phase evolves into a trivial insulator as we increase the thickness of the trivial MnTe layers. By switching the Néel vector direction into the ab plane, the system realizes a strong antiferromagnetic topological insulator evolving into a weak antiferromagnetic topological insulator with the increasing of the trivial MnTe layers [2].

[1] __R. Islam__, B. Ghosh, __G. Cuono, A. Lau, W. Brzezicki__, A. Bansil, A. Agarwal, B. Singh, __T. Dietl, and C. Autieri__ *Topological states in superlattices of HgTe class of materials* *for engineering three-dimensional flat bands.* Phys. Rev. Research **4**, 023114 (2022).

[2] __R. Islam__ , S. Mardanya, __A. Lau, G. Cuono__, T.-R. Chang, B. Singh, C. M. Canali, __T. Dietl, and C. Autieri.__ *Engineering axion insulator and other topological phases in superlattices without inversion symmetry.* Phys. Rev. B **107**, 125102 (2023).

**Invited talk:** __C. Autieri__ “25th International Conference on the Electronic Properties of Two-Dimensional Systems (EP2DS-25) and 21st International Conference on Modulated Semiconductor Structures (MSS-21)” to be held in Grenoble, France, from 9 to 14th July 2023. Title of the talk: “Topological phases of the HgTe-based system”. See also the website: https://ep2ds25mss21.sciencesconf.org/

*Experimental and theoretical studies carried at University of Wuerzburg, MagTop, and elsewhere call for significant progress in the development of 3D, 2D, and 1 D systems of non-magnetic and magnetic mercury-based chalcogenides and related topological materials, such as grey tin, in order to find new topological phases (such as axion insulator) and explore the interplay between Kondo, Luttinger, and magnetic polaron effects in a topological setting. MagTop’s MBE Group has launched an extensive programme for the growth and characterization of grey tin and, in collaboration with the University of Rzeszów, is striving to develop its own MBE growth technology for mercury compounds.*

In the Hg-MBE chamber in Rzeszów, three new cells (for Cd, Mn and Cr) as well as kSA Bandit precise substrate temperature control system have been installed. In parallel, the MBE Group is producing Hg-based gated devices (from wafers grown abroad) using MagTop’s equipment and expertise in electron beam lithography (EBL), etching, atomic layer deposition (ALD) of oxides, and subsequent metal deposition. The devices produced are than used in experiments conducted by MagTop in collaboration with CENTERA [1].

The appealing counterpart of HgTe is grey tin α-Sn, which also has the inverted band structure, but without the inversion symmetry breaking. The **application of in-plane compressive strain to epitaxial layers of α-Sn results in the emergence of a Dirac semimetal** (DSM) which turns into a Weyl semimetal (WSM) in a magnetic field, as it breaks the time-reversal. MagTop has carried out a comprehensive investigation of **α-Sn **layers of various thicknesses grown by the MBE technique on CdTe/GaAs (001) hybrid substrates, which induce a compressive in-plane strain that is sufficiently large to induce the DSM phase. Remarkably, **we observed a non-saturating negative longitudinal magnetoresistance (NLMR) for all studied samples when the magnetic field is applied parallel to the current direction **(Figure). After ruling out alternative mechanisms such as current jetting and weak localization, **we attributed this feature to a chiral anomaly**, which is a characteristic signature of the WSM phase. Notably, **this study represents the first detailed investigation of NLMR in α-Sn**. The non-trivial nature of our samples is supported by the extraction of a π Berry phase from Shubnikov-de Haas oscillations.

[1] I. Yahniuk, __A. Kazakov__, B. Jouault, S. S. Krishtopenko, S. Kret, G. Grabecki, G. Cywiński, N. N. Mikhailov, S. A. Dvoretskii, J. Przybytek,V. I. Gavrilenko, F. Teppe, __T. Dietl__, W. Knap, *HgTe quantum wells for QHE metrology under soft cryomagnetic conditions: permanent magnets and liquid ^{4}He temperatures, *arXiv:2111.07581 (2021) [retracted from a journal to do not publish with Russian institutions].

[2] J. Polaczyński, A. Kazakov, B. Turowski, R. Rudniewski, Z. Muhammad, W. Zaleszczyk, T. Wojciechowski, T. Wojtowicz, V.V. Volobuev, P. Dłużewski, M. Aleszkiewicz, B. Kurowska, N. Olszowska, M. Rosmus, G. Krizman, J. Bermejo-Ortiz, L.-A. de Vaulchier, Y. Guldner,

*In quantum spin Hall (QSH) materials studied so far the longest topological protection length is of the order of a few micrometers and the operation temperature is too low for potential applications. MagTop’s researchers have theoretically demonstrated a large topological gap and a strong sensitivy to an electric field of MoSi _{2}N_{4} and related 2D systems.*

In 2020, a new 2D material class including MoSi_{2}N_{4} was synthesized. MagTop’s researchers in a broad international collaboration have investigated the electronic, structural, and magnetic properties of these new materials by *ab initio* methods [1-4]. New topological phases, characterized by the topological invariant Z_{2 }= 1 and hosting the QSH effect, were found together with and an electric field-driven topological phase transition (see Figure). These compounds have exceptional properties such as a large inverted band gap making robust the topological properties. At the same time, a small electric field can switch the topological phase faster than in any other QSH compound. The appearance of both large band gap and fast electric switching makes these materials promising candidates for designing room-temperature fast topological field-effect transistors [5,6].

*These studies were included in the Ph.D. dissertations of R. Islam (now at the University of Alabama, USA) and G. Hussain.*

[1] __R. Islam__, B. Ghosh, __C. Autieri__, S. Chowdhury, A. Bansil, A. Agarwal, B. Singh, *Tunable spin polarization and electronic structure of bottom-up synthesized MoSi _{2}N_{4}*

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*A series of works carried out by MagTop/IFPAN collaboration shows why superexchange, rather than interband Van Vleck/Bloembergen-Rowlad mechanism, dominates exchange coupling between localized spins in topological materials.*

A consensus emerged [He Ke, Yayu Wang, and Qi-Kun Xue, *Annu. Rev. Cond. Mat. Phys. 9, 329 (2018)*; Y. Tokura, K. Yasuda, and A. Tsukazaki,

MagTop/IFPAN collaboration has called this insight into question. As a first step a theoretical formalism was developed that takes on equal footing various possible mechanisms which can lead to spin-spin interactions in semiconductors doped with transition metals [1]. Subsequently, *ab initio* computations and *kp* modelling served to test our understanding of *sp-d* band splitting in the whole Brillouin zone generated by spin-polarized Mn ions in topological HgTe and non-topological CdTe [2]. The theory explained the origin of decades-long difficulties in the understanding of band splittings at the L-points of the Brillouin zone in Mn-doped II-VI semiconductor compounds. In the key paper [3], the theoretical analysis demonstrated that the interband susceptibility (accounting for the VV-BR mechanism) is dominated by a large self-interaction term which does not contribute to the strength of the spin-spin interaction between *different* magnetic ions and, thus, is irrelevant as far as magnetic ordering of the system is concerned (see, Figure). Quantitative computations of the coupling magnitude for Mn pairs, *J*_{i}, in non-topological CdTe and topological HgTe were carried and showed that the superexchange dominates in both systems. The theory explains why there is no ferromagnetic interaction in Fe-doped topological materials [4] but appears in Cr-doped HgTe and related systems [5].

[1] C. Śliwa, __T. Dietl__, *Thermodynamic perturbation theory for non-interacting quantum particles with application to spin-spin interactions in solids*, Phys. Rev. B **98**, 035105 (2018).

[2] __C. Autieri__, C. Śliwa, __R. Islam, G. Cuono, T. Dietl__, *Momentum-resolved spin splitting in Mn-doped trivial CdTe and topological HgTe semiconductors*, Phys. Rev. B **103**, 115209 (2021).

[3] C. Śliwa, __C. Autieri__, J. A. Majewski, __T. Dietl__, *Superexchange dominates in magnetic topological insulators, *Phys. Rev. B **104**, L220404 (2021) [Editors’ Suggestion].

[4] __Y. Satake__, J. Shiogai, __G. P. Mazur__, S. Kimura, S. Awaji, K. Fujiwara, T. Nojima, K. Nomura, S. Souma, T. Sato, __T. Dietl__, A. Tsukazaki, *Magnetic-field-induced topological phase transition in Fe-doped (Bi,Sb) _{2}Se_{3} heterostructures*, Phys. Rev. Materials

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*One of the central questions in the field of condensed matter physics is understanding the microscopic mechanisms governing phase transformations. MagTop’s theoretical teams addressed conditions for the appearance of quantum or topological phases in selected systems, including (i) a subtle interplay between itinerant ferromagnetic and antiferromagnetic correlations in various strongly correlated metals under pressure; (ii) the formation of the quantum anomalous Hall phase for electrons experiencing the Rashba interaction, Zeeman splitting, and ionic potential], and (iii) the effect of many-body interactions on the Chern insulator clarifying a misleading notion of the first-order topological transition*.

In order to explain pressure-induced phase transitions between ferromagnetic (FM) and antiferromagnetic (AF) ordering in metallic magnets [e.g., U. S. Kaluarachchi *et al.* *Nat. Comm.* 8, 546 (2017)]) we considered theoretically two possibilities: (i) a competition between characteristic energy scales ubiquitous for *d*-electron magnets [1]; (ii) the presence of crystallographically inequivalent ligand atoms applicable to specific uranium-based magnets [2]. Additionally, we established that development of the AF component in ferromagnetic La_{5}Co_{2}Ge_{3} under pressure has structural rather than electronic origin [3].

When it comes to topological 2D systems, we demonstrated that a quantum anomalous Hall phase emerges for electrons experiencing the ionic potential, Rashba interaction, and Zeeman splitting [4]. We also explored topological properties of the system in the presence of many-body interactions local in the momentum space (see Figure). Interestingly, our investigations of the exact many-body wave function of the quantum anomalous Hall phase shed a new light on the widely explored phenomenon termed „first order topological phase transition”, describing the absence of a spectral gap closing at the topological phase transition in the presence of strong interactions. We showed that the absence of spectral gap closing is related to the many-body nature of the wave function and a two-particle nature of the first excited state. Moreover, on the example of Z_{2} topological Mott insulating phases realized in a version of Anderson lattice model we have demonstrated [5] that the border between two distinct, topological phases also features absence of spectral gap closing, suggesting universality of such observation when topology intertwines with Mott physics. Instead, we find that in such systems topological phase transitions are signalled by kinks or singularities at selected time reversal invariant momenta.

[1] __M. M. Wysokiński__, *Mechanism for transitions between ferromagnetic and antiferromagnetic orders in d-electron metallic magnets*, *Sci. Rep. 9, 19461 (2019)*

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