First-principles study of five isomers of two-dimensional GeSe under in-plane strain

Zuo Bo-Min; Yuan Jian-Mei; Feng Zhi; Mao Yu-Liang

doi:10.7498/aps.68.20182266

Abstract

Using first-principles calculations, we investigate the stability and electronic properties of five isomers of two-dimensional (2D) GeSe monolayer under in-plane strain. Our calculated results show that the five isomers of GeSe monolayer are all stable. It is found that the α-GeSe has a direct band gap, while each of the β-GeSe, γ-GeSe, δ-GeSe and ε-GeSe possesses an indirect band gap. By applying compressive or tensile uniaxial and biaxial strain to the five GeSe isomers, the indirect-to-direct transition in band gap is found. In the α-GeSe, the changes from indirect-to-direct and semiconducting-to-metallic are both found under an applied strain. In the 2D β-GeSe and γ-GeSe, an adjustable range of indirect band gap under strain is found. Moreover, a direct band gap in the δ-GeSe is found separately under the biaxial compression strain of σ_xy = –2% and σ_xy = –4%. By applying a tensile strain of 10% along the armchair direction in ε-GeSe, a transition from an indirect to direct band gap occurs. When the tensile strain is continuously increased to 20%, the band structure of ε-GeSe maintains direct character. This direct band gap can be tuned from 1.21 eV to 1.44 eV. When 10% tensile strain is applied along the biaxial direction, the transition in band gap from indirect-to-direct also occurs. Our results indicate that the direct band gap can be tuned from 0.61 eV to 1.19 eV when the tensile strain is increased from 10% to 19% in ε-GeSe.

Keywords:

Authors and contacts

1.
School of Physics and Optoelectronic, Xiangtan University, Xiangtan 411105, China
2.
School of Mathematics and Computational Science, Xiangtan University, Xiangtan 411105, China

Corresponding author: Yuan Jian-Mei, yuanjm@xtu.edu.cn ; Mao Yu-Liang, ylmao@xtu.edu.cn

Funds: Project supported by the National Natural Science Foundation of China (Grant No. 11471280), the Research Foundation of Education Bureau of Hunan Province, China (Grant No. 17A207), and the Natural Science Foundation of Hunan Province, China (Grant No. 2018JJ2130).

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References

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图 1 单层GeSe的5种同分异构弛豫后结构的俯视图和侧视图(图中a和b表示晶格常数)　(a) α-GeSe; (b) β-GeSe; (c) γ-GeSe; (d) δ-GeSe; (e) ε-GeSe

Figure 1. Top view and side views of relaxed structures of five isomers of GeSe monolayer (in the figures, a and b denote the lattice constants, respectively): (a) α-GeSe; (b) β-GeSe; (c) γ-GeSe; (d) δ-GeSe; (e) ε-GeSe.

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图 2 5种GeSe单层同分异构体的吸收系数

Figure 2. Absorption coefficients for five isomers in GeSe monolayer.

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图 3 带隙随平面内应变的变化图(方块表示沿扶手椅(σ_x)方向的应变, 圆圈表示沿锯齿形(σ_y)方向的应变, 三角形代表双轴(σ_xy)应变, 实心和空心符号分别表示间接和直接带隙)

Figure 3. Variation of band gap along with the applied in-plane strain (The square represents the strain along armchair (σ_x) direction, while the circle represents the strain along zigzag (σ_y) direction, the triangle represents the bi-axial (σ_xy) strain, the solid and hollow symbols denote the indirect and direct band gap, respectively)

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图 A1 单层α-GeSe在应变调控下能带结构　(a) σ = 0; (b) σ_x = 2%; (c) σ_x = 7%; (d) σ_y = –8%; (e) σ_y = –1%; (f) σ_y = 8%; (g) σ_xy = –1%; (h) σ_xy = –6%; (i) σ_xy = 8%

Figure A1. Band structures of α-GeSe monolayer under applied strains: (a) σ = 0; (b) σ_x = 2%; (c) σ_x = 7%; (d) σ_y = –8%; (e) σ_y = –1%; (f) σ_y = 8%; (g) σ_xy = –1%; (h) σ_xy = –6%; (i) σ_xy = 8%.

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图 4 应变调控下ε-GeSe单层的能带结构 (σ_x, σ_y和σ_xy分别表示沿扶手椅形、锯齿形和双轴方向的应变箭头表示导带最小值 (CBM)指向价带最大值 (VBM)的方向)　(a) σ = 0 (图中a表示ε-GeSe的导带最小值, b表示与a相同路径点的价带点, c是ε-GeSe的价带最大值, d是与c的相同路径点的导带点); (b) σ_x = 10%; (c) σ_x = 20%; (d) σ_y = 10%; (e) σ_y = 20%; (f) σ_xy = 10%; (g) σ_xy = 20%

Figure 4. Band structures of ε-GeSe monolayer under applied strains (σ_x, σ_y, σ_xy represent the strains along the armchair, zigzag and biaxial directions, respectively. The arrow represents the direction from the conduction band minimum (CBM) to the valence band maximum (VBM)): (a) σ = 0 (in the figure, a represents the conduction band minimum of ε-GeSe, b represents the valence band point corresponding to the same path point of a, c is the valence band maximum of ε-GeSe, d is the conduction band point corresponding to the same path point of c); (b) σ_x = 10%; (c) σ_x = 20%; (d) σ_y = 10%; (e) σ_y = 20%; (f) σ_xy = 10%; (g) σ_xy = 20%.

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图 A2 ε-GeSe单层施加20%拉伸应变下优化后的结构　(a) 扶手椅方向; (b) 锯齿形方向; (c) 双轴方向

Figure A2. Optimized structures of ε-GeSe monolayer under 20% tensile strain, respectively: (a) Armchair direction; (b) zigzag direction; (c) biaxial direction.

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图 5 单层ε-GeSe部分电荷密度的等值面 ((a), (b), (c), (d)分别对应于图4(a)中a, b, c和d所标注的点)

Figure 5. Isosurfaces of partial charge densities of monolayer ε-GeSe ((a), (b), (c), (d) are corresponding points to a, b, c and d in Fig. 4(a), respectively).

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表 2 5种GeSe同分异构体结构的振动频率计算结果 (E₀代表零点能, E_f/i代表虚拟频率, E_f/i/E₀为虚拟频率占据零点能的比例)

Table 2. Calculated results of vibration frequencies of five isomers of GeSe (E₀ represents zero energy, E_f/i represents virtual frequency, E_f/i/E₀ denotes virtual frequency occupies the proportion of zero energy).

	α-GeSe	β-GeSe	γ-GeSe	δ-GeSe	ε-GeSe
E₀	73.88	39.55	77.92	151.22	179.68
E_f/i/meV	0.037	0.016	0.041	0.077	0.664
E_f/i/E₀	0.05%	0.04%	0.05%	0.05%	0.03%

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表 1 弛豫后单层GeSe 5种同分异构体的结构参数 ( a和b分别是晶格常数, h为GeSe的翘曲高度, E_b是相对于单个原子之间的结合能, E_g为基本带隙)

Table 1. Relaxed structural parameters of five isomers of GeSe monolayer (a and b are the lattice constants, respectively. h is the buckling height of GeSe; E_b is the bind energy per atom; E_g is the fundamental band gap).

	α-GeSe	β-GeSe	γ-GeSe	δ-GeSe	ε-GeSe
a/Å	4.27	3.68	3.67	5.80	6.85
b/Å	3.98	3.68	5.89	5.84	6.43
h/Å	2.52	1.45	1.76	2.52	1.72
\|E_b\|/eV·atom^–1	4.14	4.12	4.11	4.13	4.08
E_g/eV	1.14	2.30	1.77	1.58	1.78

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表 3 ε-GeSe沿扶手椅形(σ_x)、锯齿形(σ_y)和双轴(σ_xy)方向平面应变优化后的几何参数及能隙(负应变表示压缩应变, 正应变表示拉伸应变; d₁, d₂和d₃(如图1(e)所示)分别表示Ge和Se原子之间的距离, θ₁表示θ (Ge-Se-Ge)的键角, θ₂表示θ (Se-Ge-Se)的键角; E_g (eV)表示对应应变下的带隙, (ind.)表示间接带隙，(dir.)表示直接带隙)

Table 3. Summary of the optimized geometric structures and energy gaps for ε-GeSe under in-plane strains along the armchair (σ_x), zigzag (σ_y) and biaxial (σ_xy) directions (Negative values of strain denote compress strains, while positive values denote tensile strains. d₁, d₂ and d₃ (as shown in Fig. 1(e)) represent the distance between Ge and Se atoms, respectively. θ₁ represents the θ (Ge-Se-Ge) bond angle. θ₂ represents the θ (Se-Ge-Se) bond angle. E_g(eV) is the band gap under the corresponding strain，（ind.）is the indirect band gap, (dir.) is the direct band gap)

Strain/%		d₁/Å	d₂/Å	d₃/Å	θ₁	θ₂	E_g/eV
	σ_x	2.68	2.51	2.60	85.6	86.7	1.56 (ind.)
–10	σ_y	2.60	2.58	2.56	98.2	85.1	1.41 (ind.)
	σ_xy	2.60	2.52	2.52	89.6	77.8	0.85 (ind.)
	σ_x	2.67	2.51	2.60	87.4	87.2	1.61 (ind.)
–8	σ_y	2.61	2.57	2.57	97.1	85.8	1.55 (ind.)
	σ_xy	2.60	2.52	2.54	90.2	80.3	1.32 (ind.)
	σ_x	2.67	2.52	2.60	89.1	87.8	1.66 (ind.)
–6	σ_y	2.61	2.56	2.58	96.3	86.6	1.67 (ind.)
	σ_xy	2.61	2.52	2.56	90.8	82.8	1.71 (ind.)
	σ_x	2.66	2.52	2.60	90.9	88.3	1.72 (ind.)
–4	σ_y	2.62	2.55	2.59	95.5	87.5	1.77 (ind.)
	σ_xy	2.62	2.53	2.57	91.7	85.5	1.90 (ind.)
	σ_x	2.65	2.53	2.60	92.4	88.8	1.75 (ind.)
–2	σ_y	2.63	2.55	2.59	94.8	88.4	1.84 (ind.)
	σ_xy	2.64	2.53	2.59	92.7	87.4	1.89 (ind.)
0		2.65	2.54	2.60	94.1	89.3	1.78 (ind.)
	σ_x	2.65	2.55	2.60	95.8	89.7	1.75 (ind.)
2	σ_y	2.67	2.54	2.60	93.3	90.5	1.62 (ind.)
	σ_xy	2.67	2.55	2.60	95.8	91.3	1.62 (ind.)
	σ_x	2.64	2.56	2.60	97.2	90.2	1.66 (ind.)
4	σ_y	2.69	2.54	2.60	93.2	91.6	1.48 (ind.)
	σ_xy	2.68	2.56	2.60	98.0	92.6	1.50 (ind.)
	σ_x	2.63	2.57	2.61	99.0	90.2	1.60 (ind.)
6	σ_y	2.71	2.54	2.60	92.8	92.8	1.35 (ind.)
	σ_xy	2.70	2.57	2.60	100.2	93.7	1.40 (ind.)
	σ_x	2.63	2.59	2.61	100.8	90.4	1.51 (ind.)
8	σ_y	2.74	2.54	2.60	92.8	94.1	1.24 (ind.)
	σ_xy	2.71	2.59	2.60	102.6	94.9	1.30 (ind.)
	σ_x	2.62	2.60	2.61	103.5	89.9	1.44 (dir.)
10	σ_y	2.77	2.54	2.59	93.0	95.5	1.16 (ind.)
	σ_xy	2.73	2.60	2.60	105.1	96.1	1.19 (dir.)
	σ_x	2.62	2.61	2.62	105.5	90.0	1.37 (dir.)
12	σ_y	2.80	2.54	2.59	92.9	96.8	1.09 (ind.)
	σ_xy	2.75	2.61	2.59	107.7	97.6	1.00 (dir.)
	σ_x	2.61	2.62	2.62	108.2	89.4	1.32 (dir.)
14	σ_y	2.84	2.53	2.58	93.2	98.3	1.05 (ind.)
	σ_xy	2.77	2.62	2.58	110.3	99.4	0.84 (dir.)
	σ_x	2.60	2.63	2.63	110.8	89.0	1.28 (dir.)
16	σ_y	2.87	2.53	2.57	93.4	100.0	1.03 (ind.)
	σ_xy	2.78	2.64	2.57	112.6	101.7	0.67 (dir.)
	σ_x	2.60	2.63	2.63	113.1	88.8	1.24 (dir.)
18	σ_y	2.92	2.52	2.56	93.5	102.0	0.98 (ind.)
	σ_xy	2.79	2.65	2.55	115.2	103.7	0.63 (dir.)
	σ_x	2.59	2.64	2.64	115.6	88.3	1.21 (dir.)
20	σ_y	2.96	2.51	2.54	94.1	103.8	0.94 (ind.)
	σ_xy	2.72	2.70	2.54	123.7	108.1	0.99 (ind.)

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表 A1 单层ε-GeSe在施加20%拉伸应变下优化后的结构的振动频率的计算结果

Table A1. Calculated optimized results of vibration frequencies of ε-GeSe monolayer under application of 20% tensile strain.

	σ_x = 20%	σ_y = 20%	σ_xy = 20%
E₀/meV	165.49	156.07	132.56
E_f/i/meV	2.650	0.030	0.005
E_f/i/E₀	1.60%	0.02%	0.03%

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[1]	Zhao Y, Chen Y, Zhang Y H, Liu S F 2017 Mater. Chem. Phys. 189 215 Google Scholar
[2]	Fiori G, Bonaccorso F, Iannaccone G, Palacios T, Neumaier D, Seabaugh A, Colombo L 2014 Nat. Nanotech. 9 768 Google Scholar
[3]	Wang Q H, Kalantar-Zadeh K, Kis A, Coleman J N, Strano M S 2012 Nat. Nanotech. 7 699 Google Scholar
[4]	Ando T, Fowler A B, Stern F 1982 Rev. Mod. Phys. 54 437 Google Scholar
[5]	Schedin F, Geim A K, Morozov S V, Hill E W, Blake P, Katsnelson M I, Novoselov K S 2007 Nat. Mater. 6 652 Google Scholar
[6]	Wang T H, Zhu Y F, Jiang Q 2013 J. Phys. Chem. C 117 12873 Google Scholar
[7]	Liu L, Feng Y P, Shen Z X 2003 Phys. Rev. B 68 104102 Google Scholar
[8]	Balandin A A, Ghosh S, Bao W, Calizo I, Teweldebrhan D, Miao F, Lau C N 2008 Nano Lett. 8 902 Google Scholar
[9]	Mak K F, Lee C, Hone J, Shan J, Heinz T F 2010 Phys. Rev. Lett. 105 136805 Google Scholar
[10]	Splendiani A, Sun L, Zhang Y B, Li T S, Kim J, Chim C Y, Galli G, Wang F 2010 Nano Lett. 10 1271 Google Scholar
[11]	Peng X H, Wei Q, Copple A 2014 Phys. Rev. B 90 085402 Google Scholar
[12]	Liu H, Neal A T, Zhu Z, Luo Z, Xu X F, Tománek D, Ye P D 2014 ACS Nano. 8 4033 Google Scholar
[13]	Jain A, McGaughey A J 2015 Sci. Rep 5 8501 Google Scholar
[14]	Liang L B, Wang J, Lin W Z, Sumpter B G, Meunier V, Pan M 2014 Nano Lett. 14 6400 Google Scholar
[15]	Kou L Z, Chen C F, Smith S C 2015 Phys. Chem. Lett. 6 2794 Google Scholar
[16]	Reich E S 2014 Nature 506 19 Google Scholar
[17]	Singh A K, Hennig R G 2014 Appl. Phys. Lett. 105 042103 Google Scholar
[18]	Huang L, Wu F G, Li J B 2016 J. Chem. Phys. 144 114708 Google Scholar
[19]	Zhang S L, Wang N, Liu S G, Huang S P, Zhou W H, Cai B, Xie M Q, Yang Q, Chen X P, Zeng H B 2016 Nanotechnology 27 274001 Google Scholar
[20]	Wiedemeier H, Georg H, von Schnering G 1978 Z. Krist-Cryst. Mater. 148 295
[21]	Vaughn II D D, Patel R J, Hickner M A, Schaak R E 2010 J. Am. Chem. Soc. 132 15170 Google Scholar
[22]	de Oliveira I S S, Longuinhos R 2016 Phys. Rev. B 94 035440 Google Scholar
[23]	Bosse J L, Grishin I, Gyu Choi Y, Cheong B K, Lee S, Kolosov O V, Huey B D 2014 Appl. Phys. Lett. 104 053109 Google Scholar
[24]	Zhang X Y, Shen J W, Lin S Q, Li J, Chen Z W, Li W, Pei Y Z 2016 J. Materiomics 2 331 Google Scholar
[25]	Xue D J, Liu S C, Dai C M, Chen S Y, He C, Zhao L, Hu J S, Wan L J 2017 J. Am. Chem. Soc. 139 958 Google Scholar
[26]	Liu S C, Mi Y, Xue D J, Chen Y X, He C, Liu X F, Hu J S, Wan L J 2017 Adv. Electron. Mater. 3 1700141 Google Scholar
[27]	Sannyal A, Zhang Z Q, Gao X F, Jang J 2018 Comp. Mater. Sci. 154 204 Google Scholar
[28]	Gomes L C, Carvalho A 2015 Physics 9 2
[29]	Fei R X, Li W B, Li J, Y L 2015 Appl. Phys. Lett. 107 237
[30]	Zhang S L, Liu S G, Huang S P, Cai B, Xie M Q, Qu L H, Zou Y S, Hu Z Y, Xu X C, Zeng H 2015 Sci. China: Mater. 58 929
[31]	Zhao H Q, Mao Y L, Mao X, Shi X, Xu C S, Wang C X, Zhang S M, Zhou D H 2018 Adv. Funct. Mater. 28 1704855 Google Scholar
[32]	von Rohr F O, Ji H, Cevallos F A, Gao T, Ong N P, Cava R J 2017 J. Am. Chem. Soc. 139 2771 Google Scholar
[33]	Gomes L C, Trevisanutto P E, Carvalho A, Rodin A S, Neto A C 2016 Phys. Rev. B 94 155428 Google Scholar
[34]	Mao Y L, Xu C S, Yuan J M, Zhao H Q 2018 Phys. Chem. Chem. Phys. 20 6929 Google Scholar
[35]	Mao Y L, Long L B, Yuan J M, Zhong J X, Zhao H Q 2018 Chem. Phys. Lett. 706 501 Google Scholar
[36]	Ersan F, Arkin H, Aktürk E 2017 RSC Adv. 7 37815 Google Scholar
[37]	Xu Y F, Zhang H, Shao H Z, Ni G, Li J, Lu H L, Zhang R J, Peng B, Zhu Y Y, Zhu H Y, Soukoulis C M 2017 Phys. Rev. B 96 245421 Google Scholar
[38]	Hu Y H, Zhang S L, Sun S F, Xie M Q, Cai B, Zeng H B 2015 Appl. Phys. Lett. 107 122107 Google Scholar
[39]	Ong Z Y, Cai Y Q, Zhang G, Zhang Y W 2014 J. Phys. Chem. C 118 25272 Google Scholar
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Vol. 68, Issue 11 - 2019-06-05

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