Phys. Status Solidi RRL 2020, 14, 2000206

DOI: 10.1002/pssr.202000206

In the published article, it was reported that 2D FeCl can realize the valley Hall effect, and that the valley polarization comes from the opposite signs of the Berry curvatures around the two valleys (B and B′) on the high-symmetry lines Γ–X and Γ–X′. However, it is now realized that the distribution of the Berry curvatures of FeCl should match with its crystal symmetry. The distribution of the Berry curvatures reported violates this rule because FeCl has C₄ symmetry, which requires the same signs for the Berry curvatures around the valleys B and B′.

After analyzing all calculation processes in detail, an error is found in the step of calculating the maximal localized Wannier functions (MLWFs). Although the energy bands from the MLWFs match with the results of first-principles calculations, the C₄ symmetry is broken for low-energy electronic states and the Berry curvatures. After analyzing the influence of C₄ rotation symmetry to the distributions of the Berry curvatures, it is found that the Chern number must be 2 or −2 for FeCl. To obtain the correct results of the Berry curvatures, a tight-binding model is built with the basis of the five d orbitals of the Fe atom, and this model matches with the crystal symmetry of FeCl.

According to the orbital projected band structures without spin–orbit coupling (SOC) in the published article, the main contribution to the spin-polarized bands around the Fermi level is from the five d orbitals of the Fe atoms. Thus, we constructed a five-band tight-binding Hamiltonian with the basis set of the five spin-down d orbitals. Figure 1a herein shows the band structure from the tight-binding model and first-principles calculation, and they match with each other. To consider the effect of SOC, we add an intrinsic SOC term into this Hamiltonian. The hopping strength of SOC is set to 0.05 eV to match the bandgap (20 meV) of the first-principles result (see Figure 1b). The broken time-reversal symmetry generally results in the nonvanishing Berry curvatures for the electronic states below the Fermi level. With the help of the above tight-binding model, we calculated the Berry curvatures in the first Brillouin zone (see Figure 1c) and the anomalous Hall conductivity (see Figure 1d). It is shown that the Berry curvatures have the same signs around B and B′. By integrating the Berry curvatures near the valleys B and B′, we obtain Chern numbers of C_B and C_B′, and they are approximately equal to −1/2, which indicates VHE cannot happen if only a proper electronic field is applied.^[1]

To induce valley polarization from FeCl, we apply uniaxial strain to break the energy degeneracy of the valleys B and B’. The energy splittings of the different valleys are very important for the applications of FeCl in valleytronics. For 2D FeCl (see Figure 2c herein), the bands of the different valleys (B and B′) have the same energies for unstrained FeCl and the energy degeneracy is protected by the C₄ rotation symmetry and M_x(y) mirror symmetries of the point group D_4h.^{[2, 3]} Valleys B and B′ are related by C₄ symmetry, which leads to similar Berry curvature distributions around B and B′. Meanwhile, due to the existence of ${\mathrm{C}}_4^2$ or M_x(y), the Berry curvatures of the four valleys have same signs in the first Brillouin zone, as shown in Figure 1c. If symmetries are broken by uniaxial strains, the point group becomes D_2h, which only has C₂ symmetry and M_x(y) mirror symmetry. Therefore, the Berry curvatures around valleys B and B′ can be different. It is worth mentioning that the Berry curvature distribution of the two valleys in k_x = 0 or k_y = 0 should remain the same and it is protected by C₂ and Mx(y). We find the size of movement for the energies of valley B and B′ almost linearly dependent on the strain. When −5% strain is applied (see Figure 2b), the energy of valley B′ moves to above the Fermi level (+0.04 eV); at the same time, the energy of B moves to −0.11 eV. If the Fermi level is tuned to these two energies, a valley polarized current would be produced. Moreover, as shown in Figure 2d, the tensile strain can induce the opposite movement on the energies of B and B′. Our results revealed that uniaxial strain can effectively adjust the energies of different valleys and induce valley polarization in 2D FeCl.

物理状态 Solidi RRL 2020 , 14 , 2000206

DOI：10.1002 / pssr.202000206

在已发表的文章中，据报道，二维 FeCl 可以实现谷霍尔效应，谷极化来自高对称线Γ -上两个谷（B 和 B '）周围的 Berry 曲率的相反符号。X和Γ - X ′。然而，现在人们意识到 FeCl 的 Berry 曲率的分布应该与其晶体对称性相匹配。报告的 Berry 曲率分布违反了这一规则，因为 FeCl 具有 C ₄对称性，这要求 B 和 B ' 谷周围的 Berry 曲率具有相同的符号。

在详细分析了所有计算过程后，在计算最大局部万尼尔函数（MLWFs）的步骤中发现了一个错误。尽管 MLWF 的能带与第一性原理计算的结果相匹配，但 C ₄对称性因低能电子态和 Berry 曲率而被破坏。在分析了 C ₄旋转对称性对 Berry 曲率分布的影响后，发现对于 FeCl，陈数必须为 2 或 -2。为了获得正确的 Berry 曲率结果，以 Fe 原子的 5 个 d 轨道为基础建立了紧束缚模型，该模型与 FeCl 的晶体对称性相匹配。

根据已发表文章中没有自旋轨道耦合 (SOC) 的轨道投影带结构，费米能级周围的自旋极化带的主要贡献来自 Fe 原子的五个 d 轨道。因此，我们用五个自旋向下的 d 轨道的基组构建了一个五带紧束缚哈密顿量。数字 图1a显示了来自紧束缚模型和第一性原理计算的能带结构，它们相互匹配。为了考虑 SOC 的影响，我们在这个哈密顿量中添加了一个固有的 SOC 项。SOC 的跳跃强度设置为 0.05 eV，以匹配第一性原理结果的带隙 (20 meV)（见图 1b）。破坏的时间反转对称性通常导致费米能级以下电子态的非零贝里曲率。借助上述紧束缚模型，我们计算了第一布里渊区的贝里曲率（见图 1c）和异常霍尔电导率（见图 1d）。结果表明，Berry 曲率在 B 和 B ' 周围具有相同的符号。通过对 B 和 B ' 谷附近的 Berry 曲率积分，我们得到C 的陈数_B和C _{B '}，它们大约等于-1/2，这表明如果只施加适当的电场，VHE 不会发生。^{[ 1 ]}

为了从 FeCl 引起谷极化，我们应用单轴应变来打破谷 B 和 B ' 的能量简并。不同谷的能量分裂对于 FeCl 在谷电子学中的应用非常重要。对于 2D FeCl（参见 此处的图2c），不同谷（B 和 B '）的带对于无应变的 FeCl 具有相同的能量，并且能量简并受到 C ₄旋转对称性和M _{x ( y )}镜像对称性的保护点群 D _4h。^{[ 2, 3 ]} B 和 B ' 谷与 C ₄相关对称性，这导致 B 和 B ' 周围的 Berry 曲率分布相似。同时，由于存在${\mathrm{C}}_4^{\mathrm{二}}$或M _{x ( y )}，四个谷的 Berry 曲率在第一布里渊区具有相同的符号，如图 1c 所示。如果对称性被单轴应变破坏，则点群变为 D _2h，它只有 C ₂对称性和M _{x ( y )}镜像对称性。因此，谷 B 和 B ' 周围的贝瑞曲率可能不同。值得一提的是，k _x  = 0 或k _y  = 0处的两个谷的 Berry 曲率分布应保持不变，并受到 C ₂和Mx ( y）。我们发现谷 B 和 B ' 能量的运动大小几乎与应变线性相关。当施加-5% 应变时（见图 2b），谷 B ' 的能量移动到费米能级以上（+0.04 eV）；同时，B 的能量移动到 -0.11 eV。如果将费米能级调整到这两种能量，就会产生谷极化电流。此外，如图 2d 所示，拉伸应变可以引起 B 和 B ' 能量的相反运动。我们的结果表明，单轴应变可以有效地调节不同谷的能量并在二维 FeCl 中引起谷极化。