Mechanical and Electronic Properties of Si, Ge and Their Alloys in P42

Materials Science in Semiconductor Processing 43 (2016) 187–195

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Materials Science in Semiconductor Processing

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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].Si1xGex 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 1 into a direct semiconductor with a much smaller band gap. grid of 0.025 Å , which is corresponding to 7 7 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 6 germanium in P42/mnm structure are also studied. Additionally, energy is 5 10 eV/atom; the maximum force on the atom is 4 we will report the Si–Ge alloys in P42/mnm structure, including 0.01 eV/Å, the maximum ionic displacement within 5 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]. 0.00 0.0 0.2 0.4 0.6 0.8 1.0 occupy two Wyckoff positions: 4d (0.00000, 0.50000, 0.25000) Composition x and 8j (0.34412, 0.34412, 0.12349) in Ge12. The Si atoms occupy the Wyckoff positions 8j (0.34599, 0.34599, 0.12196) and the Ge 0.30 atoms occupy the Wyckoff positions: 4d (0.00000, 0.50000, 0.25000) in Si8Ge4; the Si atoms occupy the Wyckoff positions: 4d 0.25 Ω=0.9731 (0.00000, 0.50000, 0.25000) and the Ge atoms occupy the Wyckoff positions 8 j (0.34023, 0.34023, 0.12476) in Si4Ge8. The 0.20 bonding lengths of the 4d-site to 8j-site and the 8j-site to 8j-site Ω=1.2772-0.6082x bonds are 2.367 Å, 2.383 Å and 2.408 Å in Si , respectively. The 12 0.15 same bonding lengths in Ge12 are 2.487 Å, 2.486 Å and 2.506 Å, (eV/pair)

respectively. They are both in excellent agreements with Ref [13]. H 0.10 These bonding lengths are slightly elongated from those in the diamond Si structure and diamond Ge structure, which are 2.367 Å and 2.484 Å in our calculation, respectively. The optimized lattice 0.05 parameters of the Si12,Ge12 in P42/mnm structure and their alloys are listed in Table 1.FromTable 1,InTable 1, it can be easily found 0.00 0.00.20.40.60.81.0 that our results are in excellent agreement with Ref [13]. Mean- Composition x wile, the calculated lattice parameters of diamond–Si and diamond–Ge (Space group: Fd–3 m) are in excellent agreement with previous experimental results, indicating our calculations are valid Ω=0.2206 and believable. 0.06

The calculated elastic constants of Si12,Ge12,Si8Ge4 and Si4Ge8 in P42/mnm structure are shown in Table 2. The ﬁrst and foremost, tetragonal symmetry has six independent elastic constants (C , 11 0.04 C33, C44, C66, C12, C13), and these independent elastic constants should obey the following generalized Born’s mechanical stability

criteria of tetragonal symmetry [35,36]: (eV/pair)

H Ω=0.3402-0.2392x

Ciii >=0, 1, 3, 4, 6, ()1 0.02

(−)>CC11 12 0, ()2

0.00 (+CC11 33 −20, C 13 )> ()3 0.0 0.2 0.4 0.6 0.8 1.0 Composition x

[(240.CC11 + 12 )+ C 33 + C 13 ]> ()4 Fig. 2. Mixing enthalpy △H per cation-anion pair as a function of germanium

concentration for Si-Ge alloys in P42/mnm structure calculated using (a) PBE, The independent elastic constants of Si12 and Ge12 in P42/mnm (b) PBEsol and (c) LDA functionals. Black and red curves indicate △H with the x- structure satisfy the above generalized Born’s mechanical stability dependent and x-independent interaction parameters Ω, respectively. criteria of tetragonal symmetry. In other words, the calculated results show that the Si12 and Ge12 in P42/mnm structure are respectively. The mixing enthalpy △H(x) can also be expressed by mechanically stable under ambient conditions. The phonon spec- the following expression: tra of the Si and Ge in P42/mnm structure are shown in Ref [13].

There is no imaginary frequency, which means that the Si12 and ΔΩHx()= x (1. − x ) ()6 Ge in P4 /mnm structure are dynamic stability at ambient 12 2 Ω pressure. In order to investigate the stability of Si–Ge alloys, we where is the interaction parameter which depends on the ma- calculated the phase diagram based on the regular-solution model terials. Ω could be calculated using the Eqs. (5) and (6). The x– [37]. The mixing enthalpy of Si–Ge alloys is expressed as follows: dependent interaction parameter is gained from a linear ﬁt to the Ω values. Using linear x–dependent and average values of the ΔHx( )= ESi (1−xx Ge )−(1. − xESi ) ( )− xEGe ( ) ()5 interaction parameter Ω, the mixing enthalpies of Si1-xGex alloys where E(Si1xGex) is the total energy of Si1xGex alloys, E(Si) is the for PBE, PBEsol and CA-PZ functionals are depicted in Fig. 2.For total energy of Si12 and E(Ge) is the total energy of Ge12, Si1-xGex alloys, the linear ﬁts are Ω¼1.3055–0.4476x eV/pair, 190 Q. Fan et al. / Materials Science in Semiconductor Processing 43 (2016) 187–195

14 12

12 10

10 8

8 6 6 Frequency (THz) Frequency 4 4

2 2

0 0 Z A M G Z R X G Z A M G Z R X G Si Ge Ge 8 4 Si4 8

Fig. 3. The phonon spectra of Si8Ge4,Si4Ge8 at ambient pressure.

Table 3 Ω¼1.2772–0.6082x eV/pair and Ω¼0.3402-0.2392x eV/pair with The density (g/cm3), anisotropic sound velocities (m/s), average sound velocity (m/ PBE, PBEsol and CA-PZ, respectively. The average values of Ω are s) and the Debye temperature (K) for Si12,Si8Ge4,Si4Ge8,Ge12 in P42/mnm struc- 1.0817 eV/pair, 0.9731 eV/pair and 0.2206 eV/pair for x¼0.5. From ture, and Si and Ge in diamond structure. discussion above, we can ﬁnd that a smaller interaction parameter Ω P42/mnm structure Fd-3m structure for CA-PZ compared with that of PBE and PBEsol. The CA-PZ has the largest deviation ΔH with x-dependent Ω from ΔH with Si12 Si8Ge4 Si4Ge8 Ge12 Si Ge average Ω. This can be associated with smaller equilibrium en- ρ 2.003 2.922 3.770 4.483 2.276 5.108 ergies and larger difference between the equilibrium energies of [100] [100]vl 7836 6163 5150 5150 8226 4871

[010]vt1 4946 3836 3216 2794 5892 3467 the constituent binary obtained with CA-PZ. From Fig. 2, it can be

[001]vt2 5520 3606 3416 2987 5892 3467 inferred that the mixing enthalpy of an alloy depends on the in- v [010] [001] l 8540 6670 5402 4723 teraction between atoms of constituents. To ensure the stability of [100]vt1 5520 3606 3416 2987 Si8Ge4 and Si4Ge8, the phonon spectra are calculated at ambient [010]vt2 5520 3606 3416 2987 [110] [110]v 8552 6975 5402 4652 8992 5354 pressure (see Fig. 3). The mixing enthalpies are all positive, and the l ﬁ [110]v 4327 3582 3003 2630 6562 3765 alloys might exist at a speci ed high temperature scale, due to the t1 Δ [001]vt2 4946 3836 3216 2794 4960 3097 entropy H effects considered. The Helmholtz free energy of v [111] [111] l 9233 5505 mixing ΔG can be expressed as 5092 2955 [112]vt1,2 Δ=Δ–GETHΔ ()7 vl 8324 6383 5354 4623 8727 5220

vt 4890 3703 3162 2762 5303 3119 v Moreover, for the regular solution model of alloys, the entropy m 5421 4111 3504 3057 5859 3452 Δ ΘD 566 422 354 304 639 358 of mixing H can be given as ΔHRxlnxxlnx=− [ +(11 − ) ( − )] ( 8 ) 120 Bulk modulus where R represents the gas constant and x is the composition of Ge Shear modulus in Si–Ge alloys. From Eqs. (7) and (8), it can be seen that the Si–Ge Young's modulus alloys in P42/mnm structure can be synthesized in a high tem- 100 perature environment.

Calculated bulk modulus B, shear modulus G for Si12,Ge12 and – 80 Si Ge alloys in P42/mnm structure are also shown in Table 2. From Table 3, it can been seen that our calculation results of Si and Ge in diamond structure are in excellent agreements with the experi- 60 mental or previous theoretical data. Bulk modulus B and shear modulus G are calculated by the Voigt–Reuss–Hill approximation Elastic modulus (GPa) [38–40]. The Young’s modulus E and Poisson’s ratio v are obtained 40 by the following formulas [40]: E¼9BG/(3BþG), v¼(3B-2G)/[2 (3BþG)]. Fig. 4 shows the elastic modulus as a function of com- Si Si Ge Si Ge Ge position x for Si Ge alloys. From Fig. 4, it can be ﬁnd that Bulk 12 8 4 4 8 12 1 x x modulus B, shear modulus G and Young’s modulus E all decrease – Fig. 4. Elastic modulus as a function of germanium concentration for Si Ge alloys with the increasing composition x. From silicon to germanium in in P42/mnm structure. P42/mnm structure, the Young’s modulus of Ge12 is decrease Q. Fan et al. / Materials Science in Semiconductor Processing 43 (2016) 187–195 191

0.3 0.3

0.2 0.2

0.1 0.1

0.0 0.0

-0.1 -0.1

-0.2 -0.2

-0.3 -0.3 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 Poisson's ratio xy plane Poisson's ratio xz plane

75 0.3

50 0.2

0.1 25

0.0 0

-0.1 -25

-0.2 -50

-0.3 -75 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 -75 -50 -25 0 25 50 75 Poisson's ratio yz plane shear modulus (GPa) xy plane

75 75

50 50

25 25

0 0

-25 -25

-50 -50

-75 -75 -75 -50 -25 0 25 50 75 -75 -50 -25 0 25 50 75 shear modulus (GPa) xz plane shear modulus (GPa) yz plane

Fig. 5. 2D representation of Poisson’s ratio in the xy plane (a), xz plane (b) and yz plane (c) for Si–Ge alloys in P42/mnm structure. 2D representation of shear modulus in the xy plane (d), xz plane (e) and yz plane (f) for Si–Ge alloys in P42/mnm structure. The solid and dash dot lines represent the maximal and minimal value of xy, xz and yz planes, respectively. The black, blue, red and cyan lines represent the Si12,Si8Ge4,Si4Ge8 and Ge12, respectively.

30.25% than that of Si12, and bulk modulus is 26.67%, shear mod- (v40.26) [42]. The value of B/G is smaller than 1.75 and the value ulus is 29.17%, respectively. According to Pugh [41], a smaller B/G of v is smaller than 0.26 for Si–Ge alloys in Table 2. These show value (B/Go1.75) for a solid represents a brittle manner but a that Si–Ge alloys are brittle, Ge12 has the most brittleness and larger B/G value (B/G41.75) usually represents a ductile manner. Si8Ge4 has the least brittleness. Moreover, Poisson’s ratio v is consistent with B/G, which refers to The anisotropy of the crystal lattice along different directions, ductile compounds usually with a larger Poisson’s ratio v the atomic arrangement of the periodicity and the degree of 192 Q. Fan et al. / Materials Science in Semiconductor Processing 43 (2016) 187–195

Fig. 6. 3D representation of the Young’s modulus for Si12 (a), Si8Ge4 (b), Si4Ge8 (c), Ge12 (d).

density are not the same. This leads to the different physical and modulus. The 3D figures of the Young’s modulus for Si12,Si8Ge4, chemical properties of crystals in different directions, which are Si4Ge8 and Ge12 are shown in Fig. 6, and the 2D representation of represented as the crystal anisotropy. The anisotropy in poly- the Young’s modulus for Si12,Si8Ge4,Si4Ge8 and Ge12 are shown in crystalline materials can also be influenced to certain texture Fig. 7. The surface in each graph denotes the magnitude of E along patterns often produced during manufacturing of the material. It different directions. The 3D figure appears as a spherical shape for can be defined as a difference, when measured along different an isotropic structure, while the deviation from the spherical axes, in a material’s physical or mechanical properties. The de- shape exhibits the content of anisotropy [45]. From Fig. 6,itis pendence of Young’s modulus on the direction of load can be used obvious that the 3D figures evidently deviate in shape from the as an example. Most materials exhibit the anisotropy behavior. The sphere along the xy plane than then xz plane, as the C33 larger than anisotropy of crystal is different in different directions, such as the C11 for Si12, which may result in the xy plane is more aniso- elastic modulus, hardness, fracture resistance, thermal expansion tropic than the xz plane in Si12. The maximal and minimum values coefficient, thermal conductivity, electrical resistivity, magnetic of Young’s modulus both appear in the xy plane, but only the susceptibility and refractive index. It is well known that the ani- minimum values appear in the xz and yz planes. The ratio of sotropy of elasticity is an important implication in engineering maximal values and the minimum values are 1.337; 1.163; 1.181; science and crystal physics. In this paper, we mainly discuss the 1.173 for Si12,Si8Ge4,Si4Ge8 and Ge12, respectively, which means anisotropy of elastic modulus of materials. The directional de- the Si12 exhibits the largest anisotropy. The elastic anisotropy of a pendence of the anisotropy is calculated by the Elastic Anisotropy crystal can be characterized by many different ways, for example, Measures (ELAM) [43,44] code. The calculated Poisson’s ratio and the universal anisotropic index AU. The universal elastic anisotropy shear modulus along different directions as well as the projections index AU for a crystal with any symmetry can be given by follows U U in different planes are shown in Fig. 5. Fig. 5 shows that the [46–48]: A ¼5 GV/GR þBV/BR–6. The calculated results of A are Poisson’s ratio and shear modulus in xy plane of Si1-xGex alloys listed in Table 2. Table 2 shows that Si has the greatest anisotropy U exhibits to be more anisotropic than on other planes. The Si12 has in A , while Si8Ge4 has the smallest anisotropy. Consequently, Si12 the greatest anisotropy in Poisson’s ratio, Si8Ge4 has the smallest exhibits the greatest anisotropy in Poisson’s ratio, shear modulus, U anisotropy in Poisson’s ratio. The anisotropy of shear modulus of Young’s modulus and A , and Si8Ge4 exhibits the smallest Si12,Si8Ge4,Si4Ge8 and Ge12 are as well as the anisotropy of anisotropy. Poisson’s ratio. The Si12 has the greatest anisotropy in shear The sound velocities are determined by the symmetry of the modulus, while Si8Ge4 has the smallest anisotropy in shear crystal and propagation direction. Using the elastic constants, the Q. Fan et al. / Materials Science in Semiconductor Processing 43 (2016) 187–195 193

Si Si Ge 100 12 100 8 4

E =131 GPa 50 max 50 E =107 GPa max E =98 GPa E =92 GPa 0 min 0 min

-50 -50

-100 -100

-100 -50 0 50 100 -100 -50 0 50 100 Young's modulus (GPa) Young's modulus (GPa)

100 100 Ge Si Ge 12 4 8

50 50 E =88 GPa E =98 GPa max max E =75 GPa E =83 GPa min 0 min 0

-50 -50

-100 -100 -100 -50 0 50 100 -100 -50 0 50 100 Young's modulus (GPa) Young's modulus (GPa)

Fig. 7. 2D representation of the Young’s modulus for Si12 (a), Si8Ge4 (b), Si4Ge8 (c), Ge12 (d). The black, red and blue lines represent the xy, xz and yz planes, respectively. phase velocities of pure transverse and longitudinal modes of the In the principal directions, the acoustic velocities in a tetra-

Si12,Si8Ge4,Si4Ge8 and Ge12 can be calculated by following the gonal symmetry are given by the following expressions: procedure of Brugger [49]. For example, the pure transverse and longitudinal modes can only be found for [001], [110] and [111] []=100vCl 11 /ρ , ()16 directions in a cubic crystal, and the sound propagating modes in other directions are the quasi-transverse or quasi-longitudinal waves. In the principal directions, the acoustic velocities in a Cubic []=010vCt144 /ρ , ()17 symmetry can be expressed by:

[]=100vCl 11 /ρ , ()9 []=001vCt266 /ρ , ()18

[]=[]=010vvCtt1244 001 /ρ , ()10

[]=001vCl 33 /ρ , ()19 []=(++110vCCCl 11 12 2 44 ) /2ρ , ()11

[]=(−)110vCCt11112 /ρ , ()12 []=[]=100vvCtt1266 010 /ρ , ()20

[]=001vCt212 /ρ , ()13

[]=(++110vCCCl 11 12 2 66 ) /2ρ , ()21 []=(+111vCCCl 11 2 12 + 4 44 ) /3ρ , ()14

¯ ¯ [112]=[vvCCCtt1 112]=(− 2 11 12 + 44 ) /3ρ . ()15 []=(−)110vCCt11112 /2ρ , ()22 194 Q. Fan et al. / Materials Science in Semiconductor Processing 43 (2016) 187–195

4 4 3 3 2 2 1 1.85 eV 1 1.90 eV 0 0 Energy (eV) -1 -1 -2 -2 Z A M G Z R X G Z A M G Z R X G (a) Si (b) Si Ge 12 8 4 4 4 3 3 2 2 1 1.65 eV 1 1.60 eV 0 0 Energy (eV) Energy -1 -1 -2 -2 Z A M G Z R X G Z A M G Z R X G (c) Si Ge (d) Ge 4 8 12

Fig. 8. Electronic band structure of Si12 (a), Si8Ge4 (b), Si4Ge8 (c), Ge12 (d).

Table 4 It is well known that the electronic structure determines the The calculated band gap (eV) of for Si12,Si8Ge4,Si4Ge8,Ge12 in P42/mnm structure. fundamental physical and chemical properties of materials. The calculated electronic band structures for Si ,SiGe ,SiGe and PBE PBEsol CA-PZ HSE06 mBJ 12 8 4 4 8 Ge12 in P42/mnm structure are presented in Fig. 8. The bottom of a a – Si12 1.24, 1.28 1.15 1.17 1.85 2.03 the conduction band occurs along the Z A direction and the top of Si8Ge4 1.25 1.17 1.21 1.90 the valence band occurs at A point, which means that Si12 in the Si4Ge8 1.03 0.98 1.02 1.65 P42/mnm structure is a semiconductor with an indirect band gap Ge 0.71, 0.81a 0.68 1.12 1.61 1.46a 12 of 1.85 eV within HSE06. However, both the valence band max- a Ref. [13]. imum and the conduction band minimum locate at the A point for Ge12 in P42/mnm structure, indicating that Ge12 in P42/mnm structure is a direct band gap semiconductors with a band gap of 1.61 eV within HSE06. The calculated results using different

[]=001vCt244 /ρ , ()23 functions of Si12,Si8Ge4,Si4Ge8 and Ge12 in P42/mnm structure are listed in Table 4. The calculated results of Si12 and Ge12 in P42/ ρ – v where is the density of Si Ge alloys; l is the longitudinal sound mnm structure are 1.28 eV and 0.81 eV within GGA by Nguyen v v ﬁ velocity; t1 and t2 refer to the rst transverse mode and the et al., respectively, and our calculated results are in excellent second transverse mode, respectively. The equations of Debye agreements with the results of Nguyen et al. [13]. Compared with temperature, longitudinal sound velocity and transverse and the results of Nguyen et al., the band gap of Si12 within HSE06 is sound velocity are given by Ref [48,50]. The calculated sound vo- slightly smaller within mBJ, but the band gap of Ge12 within HSE06 locity of Si–Ge alloys in P4 /mnm structure together with Si, Ge in 2 is slightly larger within mBJ (see Table 4). For Si8Ge4, the con- diamond structure are shown in Table 3. The density and elastic duction band minimum is at (0.231 0.231 0.5) point along the ZA constants of the Si12,Ge12 in P42/mnm structure are slightly direction (see Fig. 8(b)), while the valence band maximum is lo- smaller than that of Si, Ge in diamond structure. Hence, the sound cated at (0.0 0.389 0.0) point along the XG direction. For Si4Ge8, velocities along different directions (such as [100], [010], [001], the conduction band minimum is at (0.250 0.250 0.5) point along – [110] and [1 10] directions) of Si12,Ge12 in P42/mnm structure are the ZA direction, while the valence band maximum is located at smaller than that of Si, Ge in diamond structure. The Debye tem- A point. In addition, Si8Ge4 and Si4Ge8 are indirect band gap perature of Si12,Ge12 in P42/mnm structure and Si, Ge in diamond semiconductors with band gap of 1.90 and 1.65 eV, respectively. structure are also shown in Table 3. The Debye temperatures of Si, Ge in diamond structure have excellent agreements with other 4. 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