Study of the Electronic Structure, High-Pressure Elastic Property and Phase Stability of Germanium Nitride
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摘要: 采用密度泛函理论框架下的平面波方法结合准谐近似，研究了Ge3N4的β相、w相和γ相在高温高压条件下的力学稳定性、相变点、电子结构和弹性性质。结果表明：γ相的抗剪切能力最强、刚度最大、属于超硬材料，β相具有最小的体弹模量、杨氏模量和G/B值、因此韧性和延性最好；w相和β相都属于延性相；而γ相则呈现出脆性，在30 GPa后转变为延性材料。3种相都属于半导体，成键主要来自于N-2p轨道和Ge-4s、Ge-4p轨道的杂化；它们的结构能保持稳定是源于强烈的共价键；β相、w相和γ相分别属于Γ-A型、Γ-N型间接带隙和直接带隙半导体。研究发现，β→w相变对压强非常敏感，可以认为该相变是因压强的改变而引起的；w→γ相变伴随着晶胞体积的塌缩；同时，还成功地得到了β→w→γ相变的相界。Abstract: Through the plane-wave method in the framework of density functional theory in combination with the quasi-harmonic approximation, the mechanical stability, phase boundary, electronic structures and elastic properties of β-, w- and γ-Ge3N4 are investigated at high temperatures and pressures.The γ phase, which belongs to super-hard material, has the maximum rigidity and shear stress resistance.The β phase has the minimum bulk modulus, Young's modulus and G/B ratio.The β phase has better ductility and tenacity than the other two phases.β- and w-Ge3N4 belong to ductile materials while the γ phase shows a brittle manner.When the pressure exceeds 30 GPa, the γ phase belongs to ductile materials too.β-, w- and γ-Ge3N4 are typical semi-conductors, the bonding interactions are mainly contributed by the Ni-2p, Ge-4s and 4p bands.The β, w and γ phases belong to Γ-A, Γ-N indirect band gap and direct band gap semiconductors, respectively.The β→w phase transition is sensitive to pressure.The w→γ phase transition is accompanied by the shrinkage of volume.Meanwhile, the phase boundaries of β→w→γ transitions are also successfully obtained.
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Key words:
- nitrides /
- density of states /
- band structure /
- elastic modulus
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图 1 (a) 1 100 K时β相和γ相的吉布斯自由能与w相的自由能之差(ΔG); (b) β→w→γ相变的相界；(c) 1 200 K时β相和γ相的吉布斯自由能差(Gβ-Gγ)
Figure 1. (a) Gibbs free energy differences of β and γ phases with respect to the w phase at 1 100K; (b) T-p phase diagram of β-, w- and γ-Ge3N4; (c) Free energy differences Gβ-Gγ at 1 200 K
图 2 3种Ge3N4同质异形体的能带
Figure 2. Band structures of Ge3N4
(a) β phase (b) w phase (c) γ phase
图 4 Ge3N4晶体中N原子的分波态密度
Figure 4. Partial-wave density of states of the N atoms in Ge3N4
图 5 Ge3N4晶体中Ge原子的分波态密度
Figure 5. Partial-wave density of states of the Ge atoms in Ge3N4
表 1 β-Ge3N4的点阵常数a、c, 弹性常数Cij, 多晶体弹模量B, 剪切模量G, 杨氏模量E, B/G比值, 泊松比μ, 弹性各向异性因子A1、A2、A3在不同压强下的值
Table 1. Lattice constants a, c, elastic constants Cij, bulk modulus B, shear modulus G, Young's modulus E, B/G, Poisson ratio μ, anisotropy factors A1, A2 and A3 for β-Ge3N4 under pressures
p/
(GPa)a/
(nm)c/
(nm)C11/
(GPa)C12/
(GPa)C13/
(GPa)C33/
(GPa)C44/
(GPa)B/
(GPa)G/
(GPa)E/
(GPa)$\frac{B}{G}$ μ A1 A2 A3 0 0.811 9 0.310 4 286.2 144.3 120.3 277.8 99.2 180.7 83.9 217.9 2.153 0.298 1.226 1.226 2.407 2 0.808 8 0.309 4 297.5 156.9 130.6 288.8 101.1 190.8 84.4 220.6 2.260 0.307 1.243 1.243 2.460 4 0.805 9 0.308 4 305.1 168.1 137.9 297.5 104.4 199.2 84.8 222.7 2.349 0.313 1.277 1.277 2.565 6 0.803 1 0.307 5 314.3 175.5 145.6 300.7 103.9 206.6 84.9 224.0 2.433 0.319 1.283 1.283 2.568 8 0.800 3 0.306 6 313.1 188.5 152.7 303.4 103.8 212.6 80.9 215.3 2.627 0.331 1.334 1.334 2.806 10 0.797 7 0.305 8 329.1 196.1 161.4 313.7 107.3 222.8 84.5 225.0 2.636 0.332 1.341 1.341 2.706 p/
(GPa)acalc/(nm) aExp/(nm) ccalc/(nm) cExp/(nm) C11，calc/
(GPa)C12, calc/
(GPa)C13, calc/
(GPa)C33, calc/
(GPa)C44, calc/
(GPa)BExp/(nm) Ref.[25] Ref.[13] Ref.[1] Ref.[26] Ref.[25] Ref.[13] Ref.[1] Ref.[26] Ref.[5] Ref.[27] 0 0.782 6 0.789 9 0.803 8 0.803 2 0.299 3 0.301 4 0.307 4 0.307 7 364.3[21] 184.9[25] 111.7[25] 486.3[25] 80.4[25] 218.0 185.0 
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表 2 不同压强下w-Ge3N4的点阵常数a, 弹性常数Cij, 多晶体弹模量B, 剪切模量G, 杨氏模量E, B/G比, 泊松比μ, 各向异性因子A*和维氏硬度HV
Table 2. Pressure dependences of the lattice constant a, elastic constants Cij, bulk modulus B, shear modulus G, Young's modulus E, B/G, Poisson ratio μ, anisotropy factors A* and Vickers hardness HV for w-Ge3N4
p/(GPa) a/(nm) C11/(GPa) C12/(GPa) C44/(GPa) B/(GPa) G/(GPa) E/(GPa) B/G μ A* HV 10 0.675 3 281.6 179.6 160.4 213.6 101.5 262.8 2.104 0.294 0.149 9.49 15 0.670 1 282.4 189.9 163.9 220.7 99.1 258.5 2.227 0.304 0.179 8.53 20 0.664 9 279.1 198.1 169.7 225.1 96.3 252.8 2.337 0.312 0.225 7.72 25 0.660 1 270.1 199.4 174.2 222.9 93.2 245.3 2.391 0.316 0.273 7.23 30 0.654 9 260.1 198.8 177.7 219.2 89.9 237.2 2.438 0.319 0.322 6.79 35 0.649 9 250.4 200.4 181.3 217.1 85.2 226.0 2.548 0.326 0.392 6.01
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表 3 不同压强下γ-Ge3N4的点阵常数a, 弹性常数Cij, 多晶体弹模量B, 剪切模量G, 杨氏模量E, B/G比, 泊松比μ, 弹性各向异性因子A*和维氏硬度HV
Table 3. Pressure dependences of the lattice constant a, elastic constants Cij, bulk modulus B, shear modulus G, Young's modulus E, B/G, Poisson ratio μ, anisotropy factors A* and Vickers hardness HV for γ-Ge3N4
p/(GPa) a/(nm) C11/(GPa) C12/(GPa) C44/(GPa) B/(GPa) G/(GPa) E/(GPa) B/G μ A* HV 0 0.821 2 376.7 145.8 223.0 219.8 168.6 402.7 1.305 0.196 0.058 26.43 0 0.828 8 395.1 165.4 234.5 242.0 176.2 452.4 1.371 0.188 0.059 25.43 20 0.808 1 480.9 225.9 254.1 310.9 192.6 478.9 1.614 0.243 0.056 21.78 25 0.803 8 507.5 245.1 259.6 332.5 197.4 494.3 1.684 0.252 0.054 20.92 30 0.799 9 530.8 264.2 265.1 353.1 201.2 507.2 1.751 0.260 0.055 20.05 35 0.796 3 552.8 282.7 269.7 372.8 204.3 518.2 1.824 0.268 0.056 19.22 40 0.792 9 589.8 320.1 274.5 410.1 206.4 530.2 1.986 0.284 0.059 17.24 45 0.789 6 593.2 329.3 278.1 417.3 208.5 535.3 2.001 0.286 0.061 17.19 50 0.786 5 613.0 341.3 281.8 431.8 210.3 542.7 2.053 0.290 0.062 16.69 55 0.783 6 632.8 360.7 285.4 451.4 211.9 549.6 2.130 0.297 0.064 15.94
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