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由于MnBi2Te4电子结构具有对晶格常数的改变相当敏感的特性, 本文采用基于密度泛函理论的第一性原理方法对MnBi2Te4反铁磁块体的电子结构施加等体积应变调控. 研究发现体系能带结构在材料等体积拉伸和压缩作用下变化灵敏, 体系出现绝缘体-金属相变. 特别地, 当施加特定应变后导带和价带在Γ处出现交叉, 体系呈零带隙状态. 在此应变下仍可观察到能带反转的现象, 具有非平庸的能带拓扑性质. 根据不同应变下的电荷密度图, 发现等体积应变会影响体系七倍层层间距, 其中等体积压缩和拉伸应变可分别增大和减小Te原子层间距, 表明等体积压缩有利于降低反铁磁层间耦合. 通过等体积压力应变调控, 掌握了MnBi2Te4的电子结构的变化规律, 这对本征磁性拓扑绝缘体MnBi2Te4的物性研究和实验制备具有重要的指导意义.MnBi2Te4 as an intrinsic magnetic topological insulator has attracted lots of attention. Since the electronic structure of MnBi2Te4 is quite sensitive to the change of lattice constant, here in this work, we use a first-principles method based on density functional theory to implement the isometric strain control of the electronic structure of MnBi2Te4 antiferromagnetic bulk. The so-called isometric strain is to change the lattice constant under the premise that the volume of the crystal remains unchanged. Our results show that the energy band structure of the system changes sensitively under the action of isometric tension and compression strains of the material, and the system has an insulator-metal phase transition. In particular, when a certain strain is applied, the conduction band and the valence band cross at Γ, and the system presents a zero band gap state. Under this strain, the band inversion can still be observed, showing non-trivial energy band topological properties. According to the charge density and local charge density maps under different strains, it is found that the isometric strain will affect the interlayer spacing of the system's seven-fold layers. The isometric compression and tensile strain can increase and reduce the Te atomic layer spacing respectively, indicating that isometric compression is beneficial to reducing the antiferromagnetic interlayer coupling. Through the control of isometric pressure and strain, we can master the change law of the electronic structure of MnBi2Te4, which has important guiding significance for the research of physical properties and experimental preparation of the intrinsic magnetic topological insulator MnBi2Te4.
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图 1 (a) MnBi2Te4反铁磁结构(上下自旋由不同颜色的箭头标出)及其(b)顶部视图; (c) 包含能带计算过程每个高对称点的第一布里渊区
Fig. 1. (a) MnBi2Te4 antiferromagnetic structure (the upper and lower spins are marked by arrows of different colors) and its (b) top view; (c) the first Brillouin zone containing each high symmetry point in the energy band calculation process.
图 2 (a)−(f)等体积拉伸应变作用下的能带结构图
Fig. 2. (a)−(f) The band structure diagram under isometric tensile strains.
图 3 (a)−(f) 等体积压缩应变作用下的能带结构图
Fig. 3. (a)−(f) The band structure diagram under isometric compression strains.
图 4 (a) 体系总能随应变的变化趋势; (b)单胞晶格常数随应变的演变规律; (c)等体积应变对带隙的影响; (d)图(c)虚线框处的局部放大图(CBM和VBM随应变的演变趋势也被给出)
Fig. 4. (a) Variation trend of total energy of the system with strain; (b) evolution regular of unit cell lattice constant with strain; (c) the effect of isometric strain on the band gap; (d) part a enlarged view of the dotted frame in Fig.4 (c). (The evolution trends of the bottom of CBM and VBM with strain are also given)
图 5 不同等体积应变作用后的电荷密度图 (a) –10%; (b) –5%; (c) 无应变体系; (d) 5%; (e) 10%. Mn, Bi, Te原子的位置用不同颜色的球对应标出; (1
$\bar{1}$ 0)和(001)晶面距离原点所在平面分别为0$\times \;{d}$ 及0.41$\times\;{d}$ (对于饱和度: 红色取13%表示电荷增加, 蓝色取7%代表电荷减少)Fig. 5. Charge density diagram after different isometric strains: (a) –10%; (b) –5%; (c) unstrained system; (d) 5%; (e) 10%. The positions of Mn, Bi and Te atoms are correspondingly marked with balls of different colors; The crystal plane (1
$ \bar{1} $ 0) and (001) are 0$\times\;{d}$ and 0.41$\times\;{d}$ respectively. For saturation: red takes 13% means charge increase, blue takes 7% means charge decrease
图 6 最高价带的局部电荷密度随不同等体积应变的演变图(并相应给出ab平面的平均面电荷密度曲线) (图中三维局部电荷密度的isosurface值均取0.0006 e/bohr3, 黄色代表电荷积累, 蓝色表示电荷减少)
Fig. 6. The evolution diagram of the local charge density of the highest valence band with different isometric strains and correspondingly give the average surface charge density curve of the ab plane. (The isosurface values of the three-dimensional local charge density in the figure are all 0.0006 e/bohr3. The yellow color represents the accumulation of charge, while the blue color represents the decrease of the charge)
图 7 (a)七倍层间距结构示意图; (b)七倍层层间距和(c)Te-Te原子间距随应变的变化规律曲线. 图(b)和图(c)中具体距离也相应标出
Fig. 7. (a) Schematic diagram of the structure of the sevenfold interval; Variation of the curve of (b) the interval of the sevenfold interval and (c) Te-Te interatomic distance with strain. The specific distance in Fig. (b) and Fig. (c) is also marked accordingly
图 8 施加2.26%等体积压缩应变时的(a) 总能带结构图, (b) 每类原子的态密度及总态密度, (c) Bi和Te的p轨道能带投影, (d) Bi-p和Te-p轨道分波态密度. 图(c)中含
$\varGamma$ 点费米能级附近的放大图, 由虚线框标出并由箭头指示Fig. 8. When 2.26% isometric compressive strain is applied: (a) Structure diagram of the total energy band; (b) state density and total state density of each type of atom; (c) the p orbital energy band projection of Bi and Te; (d) Bi-p and Te-porbit partial wave density. Fig. (c) contains an enlarged view of the
$\varGamma$ point near the Fermi level, marked by a dotted frame and indicated by an arrow
图 9 –2.26%等体积应变作用后同无应变体系2 × 2 × 2超胞的差分电荷密度 (a)三维图(黄色表电荷增加, 而蓝色表电荷减少); (b) (100)晶面切面二维图(红色和蓝色分别表示电荷增加及减少, 饱和度的值由图中标尺标出). 图(a)取isosurface = 0.008 e/bohr3, 图(b)的切面取原点所在平面(即0 × d)
Fig. 9. Differential charge density of –2.26% isometric strain and unstrained system: (a) Three dimensinal graph (yellow color represents charge accumulation and blue color charge depletion); (b) two dimensional drawing of crystal plane (100). (Red and blue indicate charge increase and decrease respectively. The values of saturation are marked on the scale in the figure). Fig. (a) takes isosurface = 0.008 e/bohr3, the cut plane of Fig. (b) takes the plane of the origin (ie 0
$ \times$ d)表 1 不同等体积应变下体系的晶格常数
Table 1. The lattice constants of the system under different isometric strains.
拉伸应变
$ \eta$/%a = b/Å c/Å 压缩应变
$ \eta$/%a = b/Å c/Å 0 4.360 40.600 0 4.360 40.600 1 4.404 39.793 –1 4.316 41.432 2 4.447 39.027 –2 4.273 42.270 3 4.491 38.266 –3 4.229 43.154 4 4.534 37.544 –4 4.186 44.045 5 4.578 36.825 –5 4.142 44.986 6 4.622 36.134 –6 4.098 45.948 7 4.665 35.462 –7 4.055 46.942 8 4.709 34.808 –8 4.011 47.968 9 4.752 34.172 –9 3.968 49.028 10 4.800 33.554 –10 3.924 50.123 
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