First-principles discovery of novel quantum physics and materials:
From theory to experiment

Let us first discuss the physical requirements of QAH insulators [14, 15, 16]. The topological Chern number is defined as an integral of Berry curvature over the 2D momentum space. For TRI systems, the Berry curvature is an odd function of the momentum, implying a vanishing $C$. It is thus necessary to break TRS for obtaining a nonzero $C$. For that purpose, one may apply magnetic or Coriolis fields or introduce a spontaneous magnetization into materials. The latter way that preserves the lattice translational symmetry and enables the use of Bloch theorem is relevant for the discussion of QAH insulators. Physically, the nonzero $\sigma^{A}_{xy}$ is generated either by a non-vanishing magnetization in combination with SOC or by a non-collinear antiferromagnetic (AFM) structure with a scalar spin chirality [22, 12, 23, 24]. For simple magnetic materials that typically have collinear spins and include one type of magnetic atoms, the resulting QAH insulators should simultaneously have a 2D ferromagnetic (FM) order, an insulating bulk gap opened by the SOC, and a nontrivial band topology.

Finding high-temperature FM semiconductor is a well-known challenging task [25]. For comparison, seeking high-temperature QAH insulators is even more challenging (illustrated in Fig. 1A), since the QAH insulators belong to a special kind of FM semiconductors satisfying much more strict material requirements: they are crystallized in 2D and have topological electronic structure with band gap induced by SOC. One fundamental problem invoked by the Mermin-Wagner theorem is that the FM order in 2D is easily destroyed by thermal fluctuations and maybe not experimentally realizable [26]. Recent works found that the 2D ferromagnetism can be stabilized by magnetic anisotropy and survive at finite temperatures [27]. A few 2D FM semiconductors have been obtained experimentally mainly from van der Waals (vdW) layered materials, such as CrI₃ and Cr₂Ge₂Te₆ [28, 29]. Nevertheless, they all have low Curie temperatures (below the nitrogen temperature), and more importantly none of them are topologically nontrivial.

From the material point view, a high-temperature QAH insulator should meet both criteria of strong ferromagnetism and large SOC-induced topological gap (Fig. 1A). While metallic systems composed of light $3d$ elements are preferred by the former criterion, insulating materials comprised by heavy elements are demanded by the latter [30]. The conflicting material requirements make the pursuit challenging. Previous works have tried to construct magnetic topological materials via two compromising ways: introducing magnetism extrinsically into TIs either by applying magnetic doping or alloying (Fig. 1B) or by fabricating magnetic-topological heterostructures (Fig. 1C). The major drawback of the resulting material systems is that the extrinsic magnetic effects are usually weak, easily affected by structural disorders or defects, and very difficult to control in experiment. Although the QAH effect was originally realized in the first class of materials, one can hardly improve the working temperature limited by the strong disorder effects [13]. Many groups have attempted to find the QAH states in the second class of materials, but no successful results was reported. One promising direction is to find the next-generation QAH materials—intrinsic magnetic TIs (Fig. 1D), which are stoichiometric magnetic compounds inherently possessing both intrinsic magnetism and topological electronic structures [31, 32]. However, despite long-time systematic experimental search, no vdW layered materials has been found to fit the requirements of intrinsic magnetic TIs (before the discovery of candidate material MnBi₂Te₄ in 2019). In this context, first-principles material predictions become indispensable for the development of this field.

In the following, we will present two strategies to transform the existing vdW layered materials into intrinsic magnetic TIs (including QAH insulators) as illustrated in Fig. 2. Advantages of vdW material systems (e.g., easy to prepare thin-film samples and construct heterostructures) thus could be inherited by magnetic TIs. The first strategy is to intercalate a magnetic layer into a vdW layered TI, namely magnetic layer intercalation (Fig. 2A) [31, 32]. For instance, the intercalation of a bilayer Mn-Te into each quintuple-layer Bi₂Te₃ creates a vdW-coupled septuple-layer compound MnBi₂Te₄. MnBi₂Te₄ is an intrinsic magnetic TI that beautifully combines the ferromagnetism induced by Mn and the nontrivial band topology contributed by Bi-Te. The other strategy is to decorate Li into vdW layered materials containing magnetic elements (Fig. 2B). The Li decoration alters the electron occupation of magnetic elements, which might dramatically change the magnetic and topological properties of the mother material. As an example, we will demonstrate that a monolayer LiFeSe, created by decorating Li on the superconducting material FeSe monolayer, is chemically and physically distinct from FeSe. The new material is a large-gap QAH insulator with extremely robust 2D ferromagnetism, which could serve as a promising material candidate to realize high-temperature QAH effects [30].

Previous studies of QAH states predominately focused on magnetically doped TIs. In contrast, we find that the magnetic layer intercalation can generate a series of intrinsic magnetic TIs—the MnBi₂Te₄-family materials, which provide a useful material platform to study magnetic topological physics, including the QAH effect, topological magnetoelectric effects and topological superconductivity [31].

The existing intrinsic QAH insulators, including MnBi₂Te₄ films [64] and twisted bilayer graphene [65], have working temperatures far below the room temperature, which is limited by the low magnetic transition temperature originated from the weak magnetic coupling. It is thus preferred to utilize stronger exchange coupling mechanisms for obtaining 2D ferromagnets with high Curie temperature $T_{C}$. In nature, a few transition metals in their bulk are well known to have very high $T_{C}$. For instance, $T_{C}$ of Fe, Co, and Ni are 1043, 1400 and 627 K, respectively. Materials including 2D layers of 3$d$ transition metals are thus of particular interest. Experimentally, a state-of-the-art gating technique has been developed to inject a large amount of lithium (Li) ions into layered materials for tuning magnetic properties [66]. This motivates us to apply the strategy of Li decoration to change magnetic properties of existing vdW layered materials and search for high-temperature QAH insulators.

Based on first-principles calculations, we predicted that the Li-decorated iron-based superconductor materials, LiFe$X$ ($X=$ S, Se, Te), are ideal candidate materials to realize high-temperature QAH effects [30]. The FeSe-family materials are high-temperature superconductors having vdW layered structures. A full adsorption of Li atoms on monolayer FeSe leads to the formation of LiFeSe (Fig. 6A), which is dynamically and thermally stable as supported by phonon calculations and molecular dynamic simulations. Though sharing the same crystal symmetry as FeSe (space group $P4/nmm$), this new 2D material has unexpectedly extraordinary material properties as shown below.

The monolayer LiFeSe is an extremely robust 2D ferromagnet, whose FM exchange coupling strength is stronger than any other known 2D materials, giving $T_{C}$ far beyond room temperature [30]. Due to the small Fe-Fe distance (2.6 $\AA$), the 3$d$ orbitals of neighboring Fe atoms overlap significantly with each other. The electron injection from Li to Fe increases the 3$d$ orbital occupation and thus stabilizes a high-spin state. As a result, the electron hopping from one Fe to the other is allowed (forbidden) by the Pauli exclusion principle if their spins are aligned parallelly (anti-parallelly), leading to strong FM kinetic exchange [67, 68]. Interestingly, the lattice parameters and magnetic moments of Fe atoms are essentially the same as in monolayer Fe that has been experimentally fabricated in graphene pores [69]. Whereas the monolayer Fe is unstable on its own, it can stably exist within the sandwich structure of LiFeSe.

In the absence of SOC, LiFeSe is polarized by the strong ferromagnetism to be a half Dirac semimetal, which displays a large insulating band gap for spin up and a Dirac-like linear band crossing for spin down (Fig. 6C). The two pairs of Dirac points, whose band degeneracies are protected by the $M_{x}$ and $M_{y}$ mirror symmetries, appear in the Brillouin zone along $\Gamma$-$Y$ and $\Gamma$-$X$, respectively. The predicted electronic structure is ensured by symmetry and insensitive to computational methods, as confirmed by various advanced approaches including density functional theory (DFT) in combination with dynamical mean-field theory (DMFT), namely DFT+DMFT. The inclusion of SOC in combination with an out-of-plane ferromangetism breaks all the mirror symmetries, opening large Dirac gaps of $\sim$35 meV (Fig. 6D,E). As each Dirac gap contributes a Berry phase of $\pi$ by occupied electrons, the whole system has a Berry phase of $4\pi$ in total, corresponding to $C=2$. The nonzero topological Chern number is further verified by calculations anomalous Hall conductance (Fig. 6E) and edge states. Conclusively, monolayer LiFeSe is a large-gap QAH insulator with room-temperature 2D ferromagnetism, which is promising for realizing high-temperature QAH effects.

Moreover, the 3D LiOH-LiFeSe, possibly obtainable from the existing material LiOH-FeSe by Li injection, is predicted to be a 3D QAH insulator [70]. This new topological state have macroscopic number of chiral conduction channels, which may find important applications in low-dissipation electronics. By gating-controlled space-modulated Li injection, heterostructures between a high-temperature superconductor and a QAH insulator could be constructed in the FeSe-related materials (Fig. 6B). Such kind of systems, if realized experimentally, could open great opportunities to explore chiral topological superconductivity and Majorana fermion related physics and applications at high temperatures [8, 9].

Finally let us discuss possible experimental evidences. Recent works discovered that the injection of Li into (Li,Fe)OH-FeSe and (Li,Fe)OH-FeS can induce a superconducting to ferromagnetic insulating transition, leading to the emergence of a high-$T_{C}$ ferromagnetism and strong anomalous Hall effects in the insulating state [71, 72]. The observed effects were tentatively attributed to be caused by the Fe atoms substituted from (Li,Fe)OH. However, the major physical properties can be reproduced by first-principles calculations excluding the substituted Fe [30], which supports our theoretical prediction. Despite the preliminary works, the striking prediction of high-temperature QAH states in LiFeSe-family materials is awaiting for experimental proof.

As a brief summary, we have predicted four types of novel 3D magnetic topological states in realistic materials, including 3D AFM TI, FM WSM and FM high-order TI in MnBi₂Te₄-family materials and their heterostructures, as well as 3D QAH insulator in LiOH-LiFeSe. By reducing the physical dimensions from 3D to 2D, quantum confinement of these topological materials all can generate QAH states. This offers four different physical scenarios to realize QAH insulators with unusual properties of high Chern number, “pseudo-3D” and high working temperature. The findings greatly enrich the whole family of QAH materials and facilitate the study of QAH-related physics and applications.

TI materials can be classified into two types in 2D and 3D by their physical dimensions, which give gapless topological Dirac fermions on their edges and surfaces, respectively [1, 2]. Both kinds of topological boundary states are protected against backscattering by TRS. Differently, elastic scatterings at any angle other than 180 degrees are allowed for the surface states of 3D TIs but forbidden for the edge states of 2D TIs, which renders 2D TIs more preferable for low-dissipation electronics. However, in contrast to the extensive studies of 3D TIs, much less research progresses have been achieved on 2D TIs or QSH insulators, despite the fact that the early theoretical and experimental discoveries of TIs were made in 2D material systems of graphene and HgTe/CdTe quantum well [77, 78, 4].

Compared to the 3D counterpart, the theory of 2D TIs looks simpler, but the experimental realization is turned out be much more complicated. The key issue is on the materials, which get easily affected by environment and difficult to fabricate and characterize when thinned down to the 2D limit. The proposal of finding QSH effect in graphene is theoretically beautiful but limited by the extremely weak SOC effects of the material, which can be hardly realized by experiment [77]. The QSH effect was first experimentally discovered in HgTe/CdTe quantum well and further observed in InAs/GaSb quantum well at very low temperatures [78, 4, 79, 80]. These quantum well systems have kept as the only experimentally available 2D TI materials, before the predictive discovery of stanene- and WTe₂-related materials [74, 81]. Model material systems with simple structures and large QSH gaps are obviously desired for promoting fundamental research and practical applications.

Based on first-principles calculations, we predicted that 2D thin films of $\alpha$-Sn are ideal candidate materials of large-gap QSH insulators (Fig. 7) [74]. The thinnest $\alpha$-Sn (111) film has a buckled honeycomb lattice, which corresponds to the tin (Sn) counterpart of graphene, namely stanene. The “magic” honeycomb structure of stanene introduces QSH states near the K/K’ valleys by the $p_{z}$ orbitals, similar as found in graphene [77, 82]. Such kind of QSH states, however, could only exist in ideal experimental conditions, because the partially occupied $p_{z}$ orbitals of stanene are chemically active and easily affected by substrates and adsorbates. Importantly, if fully saturating the $p_{z}$ orbitals with chemical groups $X$ ($X=$ -F, -Cl, -Br, -I, -OH, etc.), large band gaps are opened at K/K’, the material becomes chemically stable, and a topological band inversion between the bonding $s$ orbitals and the anti-bonding $p_{xy}$ orbitals of Sn occurs at the Brillouin zone center ($\Gamma$ point), giving rise to a new kind of QSH states. Caused by the strong SOC effects of Sn $p_{xy}$ orbitals, these QSH states have sizable band gaps $\sim$0.3 eV, suitable for room-temperature uses. Moreover, their properties are highly tunable, for instance, by varying chemical groups or applying lattice strains, which could potentially applied to design topological electronic devices [83]. Furthermore, we predicted that the large-gap QSH states can survive in stanene samples coupled strongly with substrates (Fig. 8A) [84] or in $\alpha$-Sn films of different crystallographic orientations and film thickness [85], which facilitates the experimental realization.

Our collaboration with experimentalists leads to fruitful progresses on the study of stanene. In 2015, we for the first time fabricated monolayer Sn on Bi₂Te₃(111) by MBE and identified its atomic and electronic structures by STM and ARPES, respectively, which confirmed the existence of stanene structure [86]. However, the stanene sample is metallic on the TI substrate. After long-time intensive experimental efforts and guided by theory, we successfully obtained the first insulating stanene sample on PbTe(111) (Fig. 8D) [87]. Unfortunately, a careful comparison between experiment and theory indicates that the insulating sample is topologically trivial, due to the small lattice constant constrained by the substrate. One can either enlarge the lattice constant or increase the layer numbers to achieve a topological band inversion. For an ultraflat stanene with a large lattice constant fabricated on Cu(111), our experiments indeed observed a band gap of $\sim$0.3 eV opened by the SOC and the existence of metallic edge states within the gap (Fig. 8B) [88]. Moreover, we experimentally studied the quantum size effects in few-layer stanene and unexpectedly discovered robust superconductivity in the material (Fig. 8E) [89]. The finding is somewhat surprising, considering that the bulk $\alpha$-Sn is non-superconducting. Owning to the coexistence of topological and superconducting states, $\alpha$-Sn films become promising material candidates to explore topological superconductivity and Majorana fermions. Furthermore, we theoretically predicted a new type of Ising superconductivity in stanene [76] and the novel quantum state was soon found by our experimental collaborators (Fig. 8F) [90], as to be discussed below. The interplay of theory and experiment continually gives us surprises in this research direction.

A new 2D semiconductor Sn₂Bi, which is composed of a unique honeycomb lattice of Bi atoms connected with each other by bonding with a triangular net work of Sn atoms beneath (Fig. 8C), has been discovered on Si(111) by our theoretical and experimental works [73]. The original motivation is to grow monolayer Sn on Si(111)-$\sqrt{3}\times\sqrt{3}$-Bi. The STM experiments observed a high-quality single-layer honeycomb structure with an insulating gap of $\sim$0.8 eV. By considering thousands possible structures, first-principles calculations identified this stable phase of Sn₂Bi, which can well reproduce all the experimental results of STM and ARPES, thus conclusively confirmed the predicted phase.

The non-layered 2D material Sn₂Bi has unusual chemical stoichiometry and atomic structure, which cannot find a counterpart in bulk materials [73]. The electronic structure is also quite special, showing strong SOC effects and the coexistence of dispersive valence bands and flat conduction bands. The high electron-hole asymmetry is caused by the unique atomic structure. Specifically, the Bi atoms do not bond directly with each other but indirectly via Sn atoms. The Bi-Sn orbital hybridization is strong in valence bands, leading to nearly free hole excitations; whereas the orbital hybridization becomes weak in conductance bands, giving rise to strongly localized electron excitations. The special electronic properties might be useful for exploring strong-correlation physics and device applications.

In an unpublished work, we found unusual defect physics in Sn₂Bi that can realize quinary charge states in solitary defects [75]. The realization of high-charge in-gap defect states in semiconductors is demanded by charge-based quantum devices but is a considerably challenging task. This is limited by the fact that the defect Coulomb charing energy increases with the number of defect charges, which could push the defect levels outside the band gap and thus forbid high-charge defect states. One might suggest to use larger band gaps to keep charged defect states in-gap. However, materials with larger insulating gaps usually have more localized defect charge distributions and thus stronger Coulomb repulsions. Therefore, the pursuit is challenged by the conflicting requirements of delocalization for low charging energy and localization implied by large band gap. A compromise can be achieved in Sn₂Bi, where the band gap is sizable and defect charge states are unusually delocalized, benefiting from the network structure composed of “metallic” elements with similar electronegativities. Thus, an ultralow defect Coulomb charing energy is realized in a large-gap system, making high-charge in-gap defect states feasible. The finding suggests a new route to realize high-charge defect states for developing charge-based quantum devices.

The discovery of TIs sheds new lights on the research of thermoelectrics, since many TIs, like (Bi,Sb)₂Te₃, actually are excellent thermoelectric (TE) materials [6, 91, 92, 93, 94, 95]. In fact, TI and TE materials share similar material traits of heavy elements and narrow band gaps [91, 92]. So that TIs can have strong SOC effects to induce topological band inversions, whereas TE materials can have low lattice thermal conductivities as well as large power factors. How to utilize topological states to improve TE performance, however, is an important open question.

We predicted that TE effects become strongly size dependent in TIs, and the optimization of geometric sizes could significantly enhance the TE performance (Fig. 9) [6]. Using stanene as an example, we showed that the TE performance is greatly improved by optimizing the contribution of edge sates. Specifically, by introducing defect and disorders in the material and by reducing (increasing) the materials width (length), the phonon contribution get depressed, whereas the edge-state contribution is almost unaffected. This effectively decouples contributions of electrons and phonons, thus realizing the concept of “phonon-glass electron-crystal” (Fig. 9B). The way of improving TE performance by geometric optimization is efficient and easy to control, which generally applies to topological materials in 2D, including QSH and QAH insulators [93, 94, 95].

Ising superconductors belong to a novel kind of superconductors that are protected against external magnetic fields by the SOC effects and can display an upper critical field $B_{c2}$ exceeding the classical Pauli limit $B_{p}=1.86T_{C}$ [96, 97] ($B_{p}$ and the zero-field critical temperature $T_{C}$ are in units of Tesla and Kelvin, resepectively) [98, 99, 100], showing promise for potential applications in high-speed maglev, high-field magnet, high-energy collider, etc. In the traditional physical scenario of Ising superconductivity, the inversion symmetry must be broken and an out-of-plane mirror reflection symmetry is required. Thus few candidate materials have been found, which are limited to the monolayers of transition metal dichalcogenides in the 2H phase [98, 99, 100].

A common knowledge is that the SOC-induced spin splitting is zero if both inversion symmetry and TRS are preserved, implying zero SOC fields. Guided by this analysis, previous works on Ising superconductivity only considered inversion asymmetric materials. We found an important loophole in the symmetry analysis that is applicable to single-orbital systems only. Realistic materials usually have orbital degeneracies protected by crystalline symmetries, which should be described by the physics of multiple orbitals.

Based on the above observation, we predicted a new type of Ising supercondcuting pairing mechanism that does not require inversion symmetry breaking, called type-II Ising superconductivity [76]. The key physics can be well demonstrated in stanene that is inversion symmetric and has degenerate $p_{xy}$ orbitals, as ensured by the $C_{3\rm{v}}$ symmetry. The $p_{xy}$-related Bloch states have an four-fold degeneracy at $\Gamma$ when excluding the SOC. They are split into two spin-degenerate copies by the SOC, including the $|j_{z}=\pm 3/2\rangle$ (i.e., $|p_{x+iy},\uparrow\rangle$ and $|p_{x-iy},\downarrow\rangle$) and $|j_{z}=\pm 1/2\rangle$ (i.e., $|p_{x+iy},\downarrow\rangle$ and $|p_{x-iy},\uparrow\rangle$) states. Importantly, the SOC results in a spin-orbital locking, which gives opposite Zeeman-like fields for opposing orbitals $p_{x\pm iy}$ (Fig. 10A). As the circular orbital motions are in-plane, the generated Zeeman-like fields are along the out-of-plane direction. Such kind of effective SOC-fields are extremely large ($\sim$10³ Tesla in stanene), which greatly suppresses the Zeeman splitting caused by external in-plane magnetic fields, leading to an enhanced $B_{c2}$ significantly beyond $B_{p}$. By high-throughput first-principles calculations, we predicted many candidate materials for experimental studies (Fig. 10B). This work greatly enriches the research on new physics and materials of Ising superconductivity.

As recent experimental progresses, our collaborators studied the influence of magnetic fields on superconductivity in few-layer stanene, and observed that the in-plane upper critical field indeed exceeds the Pauli limit, which provides a smoking-gun evidence for the existence of type-II Ising superconductivity [90]. The theoretical prediction has also been experimentally confirmed in other material systems [101], demonstrating the generality of Ising superconductivity physics.