Materials Science in Semiconductor Processing 43 (2016) 187–195 Contents lists available at ScienceDirect Materials Science in Semiconductor Processing journal homepage: www.elsevier.com/locate/mssp Mechanical and electronic properties of Si, Ge and their alloys in P42/mnm structure Qingyang Fan a,n, Changchun Chai a, Qun Wei b, Qi Yang a, Peikun Zhou c, Mengjiang Xing d, Yintang Yang a a Key Laboratory of Ministry of Education for Wide Band-Gap Semiconductor Materials and Devices, School of Microelectronics, Xidian University, Xi’an 710071, PR China b School of Physics and Optoelectronic Engineering, Xidian University, Xi’an 710071, PR China c Faculty of Science, University of Paris-Sud, Paris 91400, France d Faculty of Information Engineering & Automation, Kunming University of Science and Technology, Kunming 650051, PR China article info abstract Article history: Structural, mechanical, and electronic properties of Si–Ge alloys in P42/mnm structure were studied Received 12 September 2015 using first-principles calculations by Cambridge Serial Total Energy Package (CASTEP) plane-wave code. Received in revised form The calculations were performed with the local density approximation and generalized gradient ap- 11 December 2015 proximation in the form of Perdew–Burke–Ernzerhof, PBEsol. The calculated excess mixing enthalpy is Accepted 17 December 2015 positive over the entire germanium composition range. The calculated formation enthalpy shows that the Si–Ge alloys are unstable at 0 K; however, the alloys might exist at specified high temperature scale. Keywords: The anisotropic calculations show that Si12 in P42/mnm structure exhibits the greatest anisotropy in Stability U Poisson’s ratio, shear modulus, Young’s modulus and the universal elastic anisotropy index A , but Si8Ge4 Electronic properties has the smallest anisotropy. The electronic structure calculations reveal that Si and Si–Ge alloys in P4 / Mechanical properties 12 2 mnm structure are indirect band gap semiconductors, but Ge in P4 /mnm structure is a direct semi- Si–Ge alloys 12 2 conductor. & 2015 Elsevier Ltd. All rights reserved. 1. Introduction quasi-direct band gaps using crystal structure searches combined with ab initio calculations. These structures can absorb sunlight The group 14 elements silicon and germanium have attracted with different frequencies, providing attractive features for appli- more and more interest and been extensively investigated. These cation in the tandem multijunction photovoltaic modules. Under elements have an s2p2 valence electronic configuration, which applied pressure of around 11 GPa, both silicon and germanium brings similar chemical characteristics but significant differences. transform into the β-Sn structure [18], which has 4þ2 coordina- Pure carbon found on the Earth mainly in graphite and diamond tion and is metallic. Other stable and well-researched phases of forms, which exhibits some of the strongest bonds known in these elements exist at higher pressures [18,19].Si1ÀxGex alloys nature. Pure silicon and germanium also adopt the diamond form also have been studied a lot in recent years due to their applica- under ambient conditions, and they are both with an ideal tetra- tions in both optoelectronics and microelectronics industry [20– hedral coordination. Thereby, both semiconductors have great 24]. The structural stability, dynamical, elastic and thermodynamic important applications in microelectronics industry. In nature, properties of Si–Ge, Si–Sn and Ge–Sn alloys in zinc-blende struc- carbon, silicon and germanium have a lot of allotropes [1–17]. ture were studied using first-principles calculations by Zhang et al. Nguyen et al. [13] found a new low–energy and dynamically stable [21]. The calculated heats of formation and cohesive energies in- 3 distorted sp –hybridized framework structure of silicon and ger- dicate that Ge–Sn has the strongest alloying ability and Si–Ge has manium in the P42/mnm symmetry. The band gap of the Si12 in the highest structural stability. The structure, formation energy, P4 /mnm structure is indirect band gap semiconductor, while Ge 2 12 and thermodynamic properties of Si0.5Ge0.5 alloys in zinc-blende in P42/mnm structure is direct band gap semiconductor. Wang and rhombohedra structures were investigated through first- et al. [14] found six metastable silicon allotropes with direct or principles calculations by Zhu et al. [23]. Bautista-Hernandez et al. [12] found a stable of silicon and germanium in the monoclinic (M n Corresponding author. phase) and orthorhombic structures (Z phase). From these works, E-mail address: [email protected] (Q. Fan). both the M and Z phases happen to be mechanically and http://dx.doi.org/10.1016/j.mssp.2015.12.016 1369-8001/& 2015 Elsevier Ltd. All rights reserved. 188 Q. Fan et al. / Materials Science in Semiconductor Processing 43 (2016) 187–195 Table 1 The calculated lattice parameters (Å) of Si12,Si8Ge4,Si4Ge8,Ge12 in P42/mnm structure (SG: Space Group). SG PBE PBEsol CA-PZ Expermental ac a c a c Si12 P42/mnm 5.374 9.674 5.362 9.674 5.300 9.474 a P42/mnm 5.388 9.629 Si8Ge4 P42/mnm 5.477 9.756 5.481 9.719 5.364 9.525 Si4Ge8 P42/mnm 5.513 10.046 5.504 10.020 5.371 9.731 Ge12 P42/mnm 5.638 10.149 5.624 10.124 5.451 9.791 a P42/mnm 5.652 10.140 Si Fd-3m 5.465 5.466 5.374 5.430b Ge Fd-3m 5.694 5.692 5.578 5.660b a Ref [13]. b Ref [52]. dynamically stable and the energy of these two phases for Si and [31,32] exchange correlation potential. The structural optimiza- Ge are slightly larger than that of Si and Ge in diamond structure. tions were conducted using the Broyden–Fletcher–Goldfarb– Therefore, these phases can be synthetized at room temperature. Shanno (BFGS) minimization [33]. The interactions between the Amrit De and Craig E Pryor [25] calculated the electronic and ionic core and valence electrons were described by the ultrasoft optical properties of C, Si and Ge in the lonsdaleite phase using a pseudo–potential [34]. The valence electron structures of Si and Ge 2 2 2 2 transferable model empirical pseudopotential method with spin– atoms are 3s 3p and 4s 4p , respectively. For Si12,Ge12 in P42/ orbit interactions. Diamond and Si are indirect band gap semi- mnm structure and their alloys, energy cut–off was used with conductors in the lonsdaleite structure, while Ge is transformed 340 eV, 380 eV and 340 eV, respectively. A high-quality k-point À1 into a direct semiconductor with a much smaller band gap. grid of 0.025 Å , which is corresponding to 7 Â 7 Â 4 for Si12,Ge12 Nguyen et al. [13] reported the structural and electronic in P42/mnm structure and their alloys, was used in all calculations. properties of silicon and germanium in P42/mnm structure, but The electronic properties of Si12,Ge12 in P42/mnm structure and the elastic and anisotropic properties were not investigated. In the their alloys are calculated by Heyd–Scuseria–Ernzerhof (HSE06) present work, the elastic and anisotropic properties of silicon and hybrid functional [21]. The self-consistent convergence of the total À6 germanium in P42/mnm structure are also studied. Additionally, energy is 5 Â 10 eV/atom; the maximum force on the atom is À4 we will report the Si–Ge alloys in P42/mnm structure, including 0.01 eV/Å, the maximum ionic displacement within 5 Â 10 Å the stability, elastic, anisotropic and electronic properties. and the maximum stress within 0.02 GPa. 2. Methods of calculation 3. Results and discussion The Si12,Ge12 in P42/mnm structure together with their alloys The Si12,Ge12 in P42/mnm structure and their alloys Si8Ge4 (Si8Ge4 and Si4Ge8) were investigated based on the density func- (Si0.667Ge0.333), Si4Ge8 (Si0.333Ge0.667) are new cagelike distorted tional theory (DFT) [26,27] using the Cambridge Serial Total En- sp3-hybridized framework structures including 12 atoms in a ergy Package (CASTEP) plane-wave code [28]. The calculations conventional cell with the P42/mnm (No. 136) structure in tetra- were performed with the generalized gradient approximation gonal symmetry. The crystal structures of Si12,Ge12 in P42/mnm (GGA) in the form of Perdew–Burke–Ernzerhof (PBE) [29], PBEsol structure together with their alloys are shown in Fig. 1. The Si [30] and local density approximation (LDA) in the form of Ceperley atoms occupy two Wyckoff positions: 4d (0.00000, 0.50000, and Alder data as parameterized by Perdew and Zunger (CA-PZ) 0.25000) and 8j (0.34159, 0.34159, 0.12316) in Si12; the Ge atoms Fig. 1. Unit cell crystal structures of Si12 (Ge12), Si8Ge4 and Si4Ge8 in P42/mnm structure. Q. Fan et al. / Materials Science in Semiconductor Processing 43 (2016) 187–195 189 Table 2 0.30 The calculated elastic constants (GPa) and elastic modulus (GPa) of Si12,Si8Ge4, Si4Ge8,Ge12 in P42/mnm structure. Ω=1.0817 0.25 U C11 C12 C13 C33 C44 C66 B G B/G E v A 0.20 Ω=1.3055-0.4476x Si12 123 48 47 146 49 61 75 48 1.56 119 0.24 0.134 Si8Ge4 111 36 42 130 43 38 66 40 1.65 100 0.25 0.035 Si4Ge8 100 32 37 110 39 44 58 38 1.53 94 0.23 0.056 0.15 Ge12 88 26 32 100 35 40 50 34 1.47 83 0.22 0.054 (eV/pair) Si (Fd-3m) 154 56 79 88 64 1.38 155 0.21 0.223 H 0.10 Exp.a 166 64 80 102 Ge (Fd-3m) 121 49 62 73 50 1.46 122 0.22 0.342 Exp.a 129 48 67 77 0.05 a Ref [53].