Switchable quantum anomalous and spin Hall effects in honeycomb magnet EuCd₂As₂

In order to realize magnetic topological quantum states, the materials are not only magnetic that breaks the time-reversal symmetry $\mathcal{T}$, but also host strong SOC effect that causes band inversion. With exotic magnetic and thermoelectric properties, layered rare-earth pnictides have attracted great attention, and remarkably, multiple magnetic topological quantum states have been recently observed in Eu-based pnictides [41–44]. We promote EuCd₂As₂ as a prospective material candidate with unique magnetic and topological properties. Its magnetic ground states can be engineered by temperature and pressure [45–47], and in particular, both the Dirac semimetal and Weyl semimetal are observed experimentally under AFM and FM phases of bulk EuCd₂As₂ respectively [48, 49]. Bulk EuCd₂As₂ has a layered structure with space group $P\bar{3}m1$ (no. 164), which can be visualized as a stacking of QLs, with Cd–As layers separated by Eu layers to form a sandwich structure, along the z direction [50]. It has been demonstrated that the intralayer Eu atoms in bulk EuCd₂As₂ are coupled by FM ordering, while a AFM ordering for Eu atoms in neighbor QLs. As shown in figures 1(a) and (b), the EuCd₂As₂ QLs form a hexagonal lattice and the optimized lattice constant is a = 4.48 Å. The calculated exfoliation energy of EuCd₂As₂ is 33 meV Å⁻², comparable to that of MoS₂ (26 meV Å⁻²) [51], indicating that the EuCd₂As₂ QLs are experimentally feasible. To reveal the magnetic ground state, a $\sqrt{3}\times 1$ supercell is considered. Similar to the bulk configuration, intralayer Eu for 2D EuCd₂As₂ QLs couple ferromagnetically with magnetic moments of 6.9 μ_B on each Eu with an electronic configuration of 4f⁷. In the absence of SOC, as illustrated in figure 1(d), the spin-up channel is gapless while a gap appears in the spin-down channel. Taking into account the SOC, a gap about 72 meV opens up for the spin-up channel, leading to the QAHE distinguished by an integer Chern number $\mathcal{C}$, which can be obtained by $\mathcal{C}=\frac{1}{2\pi }{\int }_{\text{BZ}}{\Omega}(\mathbf{k}){\mathrm{d}}^{2}k$ and Ω(k) is the so-called Berry curvature of all occupied states [52, 53],

$$\begin{equation}{\Omega}(\mathbf{k})=\sum\limits _{n< {E}_{\text{F}}}\sum\limits _{m\ne n}\,2\,\mathrm{I}\mathrm{m}\frac{\langle {\psi }_{n\mathbf{k}}\vert {\upsilon }_{x}\vert {\psi }_{m\mathbf{k}}\rangle \langle {\psi }_{m\mathbf{k}}\vert {\upsilon }_{y}\vert {\psi }_{n\mathbf{k}}\rangle }{{({\varepsilon }_{m\mathbf{k}}-{\varepsilon }_{n\mathbf{k}})}^{2}},\end{equation} \tag{ 1 }$$

where m, n are the band indices, ψ_{m/n k} and ɛ_{m/n k} are Bloch wavefunctions and corresponding eigenenergies for band m/n, respectively, and υ_x/y are the velocity operators. Due to the intrinsic EuCd₂As₂ QLs exhibit FM ordering along out-of-plane direction, only the z-component of Berry curvature is involved in the calculation. Moreover, the QAHE is further explicitly confirmed by the emergence of one chiral edge state in semi-infinite EuCd₂As₂ nanoribbons terminated by Eu, Cd and As atoms, in direct agreement with the value of $\mathcal{C}=1$, as illustrated in figure 1(e).

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There is no doubt that the appearance of intrinsic QAHE in 2D EuCd₂As₂ QLs provides a concrete material platform for exploring the exotic magnetic topological states and topological phase transitions, as well as their external control, which is critical for not only the fundamental investigations but also practical applications in topological spintronics. It is well known that the strain engineering is a feasible way to regulate and control the electronic, magnetic, and even the topological properties of materials, such as the phase transition between topological crystalline insulator and topological insulator in 2D TlSe [54], and/or the phase transition between normal insulator and AFM topological insulator in 2D EuCd₂Sb₂ QLs [55]. Moreover, it is interesting to note that 2D materials have great mechanical tunability and the strain engineering is much more readily controlled in experiment than that of 3D bulks [56, 57]. Indeed, the strain-induced topological phase transition has been confirmed experimentally such as in 2D 1T'–WTe₂ monolayer [58]. Here, we focus on the strain-induced topological phase transition in EuCd₂As₂ QLs and reveal that the obtained QSHE is robust against the magnetism. The magnitude of strain is described by (a − a₀)/a₀ × 100%, where a and a₀ represent the lattice parameters of the strained and unstrained EuCd₂As₂ QLs, respectively.

To get preliminary insight into the engineered topological phase transition, we present in figure 2 the global energy gaps (marked by the squares) of EuCd₂As₂ QLs with SOC versus the strain variation. As expected, the energy gaps can be effectively modified by the biaxial strain. As shown in figure 2(a), the global energy gap of EuCd₂As₂ QLs with QAHE decreases under a small compressive strain while that increases under the tensile strain, reaching as much as 79.4 meV for the 1% tensile strain. With further increase of the tensile strain, a decreasing of the global energy gaps can be caused similar to the behavior of compressive strain. A closure of the global energy gaps occurs for both the compressive and tensile strains, around respectively −1% and 2%, and remarkably, an global energy gap reopens (see also figure 2(d)). As generally accepted, a process of the energy gap closing and reopening indicates the preliminary insight of a topological phase transition. To show the topological phase transitions explicitly, we calculate the Chern number $\mathcal{C}$ and the so-called spin Chern number ${\mathcal{C}}_{S}=({\mathcal{C}}_{+}-{\mathcal{C}}_{-})/2$, where ${\mathcal{C}}_{+}$ and ${\mathcal{C}}_{-}$ are Chern numbers of all occupied bands for spin-up and spin-down manifolds, respectively [59–61]. Eigenvalues of the projected spin operator P_α , given by P_α = Pσ_α P (α = x and z for in-plane and out-of-plane magnetizations, respectively), with P is the projector operator of the occupied band eigenstates and σ_α represent Pauli matrices [59–61], are computed based on the constructed MLWFs to distinguish the spin-up and spin-down manifolds. Indeed the spectrum of P_z is gapped out, and, with the well distinguished spin-up and spin-down channels, the ${\mathcal{C}}_{+}$ and ${\mathcal{C}}_{-}$ of given subsets for both the compressive and tensile strains can be obtained, which are ${\mathcal{C}}_{\pm }=0$ and ${\mathcal{C}}_{\pm }=\pm 1$, respectively. Therefore, the total Chern number $\mathcal{C}={\mathcal{C}}_{+}+{\mathcal{C}}_{-}=0$, meaning that the QAHE is no longer conserved, but interestingly, an integer spin Chern number is obtained with a value of ${\mathcal{C}}_{S}=1$ for the tensile strain, i.e., the intriguing FM QSHE appears in EuCd₂As₂ QLs.

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To better understand the exotic topological phase transitions, after including SOC, the evolution of band structures for both the spin-up and spin-down subbands, which are distinguished by the matrix elements of the Pauli matrix σ_α , ⟨ψ_{m k} |σ_α |ψ_{n k} ⟩, on the basis of MLWFs, versus the strain variation is investigated. Figure 2 presents the energy gaps at the Γ point as a function of the strain, where the negative values indicate the inverted energy gaps. The band inversion takes place for spin-up manifold while the energy gap of spin-down manifold is normal at the equilibrium lattice parameters, leading to the Chern numbers ${\mathcal{C}}_{+}=1$ and ${\mathcal{C}}_{-}=0$, and hence the total Chern number $\mathcal{C}=1$ as discussed above. Under compressive strain, a band gap closing and reopening occurs for the spin-up manifold and the normal energy gaps increase for the spin-down one, the system changes into a normal insulator. On the other hand, a band gap closing and reopening occurs for the spin-down manifold and the inverted energy gap remains intact for the spin-up one. The band inversion emerges in both spin channels which carry nonzero Chern numbers with opposite signs, i.e., ${\mathcal{C}}_{+}=1$ and ${\mathcal{C}}_{-}=-1$, yielding the spin Chern number ${\mathcal{C}}_{S}=1$ and thus the QSHE is obtained. However, as shown in figures 2(a) and 3(b), the EuCd₂As₂ QLs show the semimetallic character with negative global energy gaps.

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In addition to the topological phase transition for the EuCd₂As₂ QLs, a magnetic transition is also obtained even under a tensile strain as small as 1%. The in-plane AFM ordering has the lowest energy with a magnetic anisotropic energy of 0.2 meV under 1% tensile. The in-plane magnetizations have currently emerged as promising candidates for exotic topological phases such as the QAHE and higher-order topological insulators but usually in FM ordering [62, 63]. Most interestingly, we find that the real insulating QSHE can be realized under the in-plane AFM ordering. Figure 2(d) displays the global energy gaps and energy gaps at the Γ point for spin-up and spin-down subbands at different strain values. Unlike to the FM states, spin-up and spin-down bands are doubly degenerate without net magnetic moments in AFM case due to the preservation of combined $\mathcal{P}\mathcal{T}$ symmetry. Across the critical tensile strain, band inversion occurs in both the spin-up and spin-down manifolds. The corresponding orbitally resolved band structures without and with SOC are plotted in figures 3(d) and (f) by taking 5% as an example. In absence of SOC, the system is gapless with one band crossing exactly at the Γ point. Similar to graphene, a band gap opens when the SOC is switched on, revealing the inverted nontrivial gap clearly.

To confirm the nontrivial topology, we calculate the spin Chern number ${\mathcal{C}}_{S}$ and an nonzero spin Chern number ${\mathcal{C}}_{S}=1$ is obtained. In addition, the calculation of spin hall conductivity ${\sigma }_{xy}^{S}$, defined as ${\sigma }_{xy}^{S}=e/{(2\pi )}^{2}{\int }_{\text{BZ}}{{\Omega}}^{S}(\mathbf{k}){\mathrm{d}}^{2}k$ with the spin Berry curvature Ω^S (k) of all occupied bands,

$$\begin{equation}{{\Omega}}_{n}^{S}(\mathbf{k})=-2\,\mathrm{I}\mathrm{m}\sum\limits _{m\ne n}\frac{\langle {\psi }_{n\mathbf{k}}\vert {J}_{x}^{S}\vert {\psi }_{m\mathbf{k}}\rangle \langle {\psi }_{m\mathbf{k}}\vert {\upsilon }_{y}\vert {\psi }_{n\mathbf{k}}\rangle }{{({\varepsilon }_{m\mathbf{k}}-{\varepsilon }_{n\mathbf{k}})}^{2}},\end{equation} \tag{ 2 }$$

is performed [64]. As generally accepted, the nontrivial topology can be distinguished by finite spin Hall conductivity with exotic plateaus in the respective insulating region [22]. As shown in figure 4(a), the plateaus within the energy window of SOC gap, which arises mainly from the spin Berry curvature Ω^S (k) near the Γ point, is clearly visible. We also employ the WCC to confirm the nontrivial topology, the verification of ${\mathbb{Z}}_{2}$ invariant can be obtained by counting the numbers of intersections between any arbitrary horizontal reference line and evolution of the WCCs, where the odd and even numbers mean topological nontrivial and trivial states, respectively [65, 66]. As clearly illustrated in figure 4(b), the number of crossings between WCC and reference horizontal line is odd, suggesting the AFM QSHE with ${\mathbb{Z}}_{2}$ = 1. Moreover, a main advantage of the ground in-plane AFM state is the insulating energy gap, and its magnitude can be tuned via strain, reaching as much as 133 meV for 5% tensile strain.

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Nontrivial topological QSHE occurs also for the strained EuCd₂As₂ QLs under in-plane FM and out-of-plane AFM configurations. As illustrated in figures 2(b) and (c), although the system is topologically trivial at the equilibrium lattice constants, band inversion can emerge in both the spin-up and spin-down channels under tensile strain, similar to the above discussed configurations. Figure 3 presents the band structures without and with SOC under the 5% tensile strain. One can see clearly the band crossings at Γ for both the FM and AFM configurations in the absence of SOC, and interestingly, a SOC gap is opened around the Fermi level. The calculated spin Chern number of all occupied states is ${\mathcal{C}}_{S}=1$, confirming explicitly the QSH state. Therefore, the magnetic QSHE in strained EuCd₂As₂ QLs is highly robust against the magnetic configurations, including both the FM and AFM ordering along in-plane and out-of-plane directions.

Finally, as a further demonstration, we focus on the edge states of strained EuCd₂As₂ QLs, where the existence of metallic edge states is one of the most exotic characters of the nontrivial QSHE, and the integer value of ${\mathcal{C}}_{S}=1$ implies that there is one pair of gapless edge states in the SOC gap. We take 5% tensile strain as an example. The edge-state band structures were calculated by iterative Green's function method based on the MLWFs of a $\sqrt{3}\times 1$ rectangle supercell [67]. Figures 4(c) and (d) display the results of semi-infinite EuCd₂As₂ QLs terminated by Eu, Cd and As atoms with in-plane FM and AFM configurations. It is clearly visible that a pair of nontrivial edge states cross at the Γ point, in direct agreement with the nonzero topological invariant of ${\mathcal{C}}_{S}=1$.