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 inP42/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’an710071, PR Chinab School of Physics and Optoelectronic Engineering, Xidian University, Xi’an 710071, PR Chinac Faculty of Science, University of Paris-Sud, Paris 91400, Franced Faculty of Information Engineering & Automation, Kunming University of Science and Technology, Kunming 650051, PR China

a r t i c l e i n f o

Article history:Received 12 September 2015Received in revised form11 December 2015Accepted 17 December 2015

Keywords:StabilityElectronic propertiesMechanical propertiesSi–Ge alloys

x.doi.org/10.1016/j.mssp.2015.12.01601/& 2015 Elsevier Ltd. All rights reserved.

esponding author.ail address: qyfan_xidian@163.com (Q. Fan).

a b s t r a c t

Structural, mechanical, and electronic properties of Si–Ge alloys in P42/mnm structure were studiedusing first-principles calculations by Cambridge Serial Total Energy Package (CASTEP) plane-wave code.The calculations were performed with the local density approximation and generalized gradient ap-proximation in the form of Perdew–Burke–Ernzerhof, PBEsol. The calculated excess mixing enthalpy ispositive over the entire germanium composition range. The calculated formation enthalpy shows thatthe Si–Ge alloys are unstable at 0 K; however, the alloys might exist at specified high temperature scale.The anisotropic calculations show that Si12 in P42/mnm structure exhibits the greatest anisotropy inPoisson’s ratio, shear modulus, Young’s modulus and the universal elastic anisotropy index AU, but Si8Ge4has the smallest anisotropy. The electronic structure calculations reveal that Si12 and Si–Ge alloys in P42/mnm structure are indirect band gap semiconductors, but Ge12 in P42/mnm structure is a direct semi-conductor.

& 2015 Elsevier Ltd. All rights reserved.

1. Introduction

The group 14 elements silicon and germanium have attractedmore and more interest and been extensively investigated. Theseelements have an s2p2 valence electronic configuration, whichbrings similar chemical characteristics but significant differences.Pure carbon found on the Earth mainly in graphite and diamondforms, which exhibits some of the strongest bonds known innature. Pure silicon and germanium also adopt the diamond formunder ambient conditions, and they are both with an ideal tetra-hedral coordination. Thereby, both semiconductors have greatimportant applications in microelectronics industry. In nature,carbon, silicon and germanium have a lot of allotropes [1–17].Nguyen et al. [13] found a new low–energy and dynamically stabledistorted sp3–hybridized framework structure of silicon and ger-manium in the P42/mnm symmetry. The band gap of the Si12 inP42/mnm structure is indirect band gap semiconductor, while Ge12in P42/mnm structure is direct band gap semiconductor. Wanget al. [14] found six metastable silicon allotropes with direct or

quasi-direct band gaps using crystal structure searches combinedwith ab initio calculations. These structures can absorb sunlightwith different frequencies, providing attractive features for appli-cation in the tandem multijunction photovoltaic modules. Underapplied pressure of around 11 GPa, both silicon and germaniumtransform into the β-Sn structure [18], which has 4þ2 coordina-tion and is metallic. Other stable and well-researched phases ofthese elements exist at higher pressures [18,19]. Si1�xGex alloysalso have been studied a lot in recent years due to their applica-tions in both optoelectronics and microelectronics industry [20–24]. The structural stability, dynamical, elastic and thermodynamicproperties of Si–Ge, Si–Sn and Ge–Sn alloys in zinc-blende struc-ture were studied using first-principles calculations by Zhang et al.[21]. The calculated heats of formation and cohesive energies in-dicate that Ge–Sn has the strongest alloying ability and Si–Ge hasthe highest structural stability. The structure, formation energy,and thermodynamic properties of Si0.5Ge0.5 alloys in zinc-blendeand rhombohedra structures were investigated through first-principles calculations by Zhu et al. [23]. Bautista-Hernandez et al.[12] found a stable of silicon and germanium in the monoclinic (Mphase) and orthorhombic structures (Z phase). From these works,both the M and Z phases happen to be mechanically and

Table 1The calculated lattice parameters (Å) of Si12, Si8Ge4, Si4Ge8, Ge12 in P42/mnm structure (SG: Space Group).

SG PBE PBEsol CA-PZ Expermental

a c a c a cSi12 P42/mnm 5.374 9.674 5.362 9.674 5.300 9.474

P42/mnma 5.388 9.629Si8Ge4 P42/mnm 5.477 9.756 5.481 9.719 5.364 9.525Si4Ge8 P42/mnm 5.513 10.046 5.504 10.020 5.371 9.731Ge12 P42/mnm 5.638 10.149 5.624 10.124 5.451 9.791

P42/mnma 5.652 10.140Si 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].

Q. Fan et al. / Materials Science in Semiconductor Processing 43 (2016) 187–195188

dynamically stable and the energy of these two phases for Si andGe are slightly larger than that of Si and Ge in diamond structure.Therefore, these phases can be synthetized at room temperature.Amrit De and Craig E Pryor [25] calculated the electronic andoptical properties of C, Si and Ge in the lonsdaleite phase using atransferable model empirical pseudopotential method with spin–orbit interactions. Diamond and Si are indirect band gap semi-conductors in the lonsdaleite structure, while Ge is transformedinto a direct semiconductor with a much smaller band gap.

Nguyen et al. [13] reported the structural and electronicproperties of silicon and germanium in P42/mnm structure, butthe elastic and anisotropic properties were not investigated. In thepresent work, the elastic and anisotropic properties of silicon andgermanium in P42/mnm structure are also studied. Additionally,we will report the Si–Ge alloys in P42/mnm structure, includingthe stability, elastic, anisotropic and electronic properties.

2. Methods of calculation

The Si12, Ge12 in P42/mnm structure together with their alloys(Si8Ge4 and Si4Ge8) were investigated based on the density func-tional theory (DFT) [26,27] using the Cambridge Serial Total En-ergy Package (CASTEP) plane-wave code [28]. The calculationswere performed with the generalized gradient approximation(GGA) in the form of Perdew–Burke–Ernzerhof (PBE) [29], PBEsol[30] and local density approximation (LDA) in the form of Ceperleyand Alder data as parameterized by Perdew and Zunger (CA-PZ)

Fig. 1. Unit cell crystal structures of Si12 (Ge12)

[31,32] exchange correlation potential. The structural optimiza-tions were conducted using the Broyden–Fletcher–Goldfarb–Shanno (BFGS) minimization [33]. The interactions between theionic core and valence electrons were described by the ultrasoftpseudo–potential [34]. The valence electron structures of Si and Geatoms are 3s23p2 and 4s24p2, respectively. For Si12, Ge12 in P42/mnm structure and their alloys, energy cut–off was used with340 eV, 380 eV and 340 eV, respectively. A high-quality k-pointgrid of 0.025 Å�1, which is corresponding to 7�7�4 for Si12, Ge12in P42/mnm structure and their alloys, was used in all calculations.The electronic properties of Si12, Ge12 in P42/mnm structure andtheir alloys are calculated by Heyd–Scuseria–Ernzerhof (HSE06)hybrid functional [21]. The self-consistent convergence of the totalenergy is 5�10�6 eV/atom; the maximum force on the atom is0.01 eV/Å, the maximum ionic displacement within 5�10�4 Åand the maximum stress within 0.02 GPa.

3. Results and discussion

The Si12, Ge12 in P42/mnm structure and their alloys Si8Ge4(Si0.667Ge0.333), Si4Ge8 (Si0.333Ge0.667) are new cagelike distortedsp3-hybridized framework structures including 12 atoms in aconventional cell with the P42/mnm (No. 136) structure in tetra-gonal symmetry. The crystal structures of Si12, Ge12 in P42/mnmstructure together with their alloys are shown in Fig. 1. The Siatoms occupy two Wyckoff positions: 4d (0.00000, 0.50000,0.25000) and 8j (0.34159, 0.34159, 0.12316) in Si12; the Ge atoms

, Si8Ge4 and Si4Ge8 in P42/mnm structure.

Table 2The calculated elastic constants (GPa) and elastic modulus (GPa) of Si12, Si8Ge4,Si4Ge8, Ge12 in P42/mnm structure.

C11 C12 C13 C33 C44 C66 B G B/G E v AU

Si12 123 48 47 146 49 61 75 48 1.56 119 0.24 0.134Si8Ge4 111 36 42 130 43 38 66 40 1.65 100 0.25 0.035Si4Ge8 100 32 37 110 39 44 58 38 1.53 94 0.23 0.056Ge12 88 26 32 100 35 40 50 34 1.47 83 0.22 0.054Si (Fd-3m) 154 56 79 88 64 1.38 155 0.21 0.223Exp.a 166 64 80 102Ge (Fd-3m) 121 49 62 73 50 1.46 122 0.22 0.342Exp.a 129 48 67 77

a Ref [53].

0.0 0.2 0.4 0.6 0.8 1.00.00

0.05

0.10

0.15

0.20

0.25

0.30

H (e

V/p

air)

Composition x

Ω=1.3055-0.4476x

Ω=1.0817

0.0 0.2 0.4 0.6 0.8 1.00.00

0.05

0.10

0.15

0.20

0.25

0.30

Composition x

Ω=1.2772-0.6082x

H (e

V/p

air)

Ω=0.9731

0.0 0.2 0.4 0.6 0.8 1.00.00

0.02

0.04

0.06 Ω=0.2206

Ω=0.3402-0.2392x

Composition x

H (e

V/p

air)

Fig. 2. Mixing enthalpy △H per cation-anion pair as a function of germaniumconcentration for Si-Ge alloys in P42/mnm structure calculated using (a) PBE,(b) PBEsol and (c) LDA functionals. Black and red curves indicate △H with the x-dependent and x-independent interaction parameters Ω, respectively.

Q. Fan et al. / Materials Science in Semiconductor Processing 43 (2016) 187–195 189

occupy two Wyckoff positions: 4d (0.00000, 0.50000, 0.25000)and 8j (0.34412, 0.34412, 0.12349) in Ge12. The Si atoms occupythe Wyckoff positions 8j (0.34599, 0.34599, 0.12196) and the Geatoms occupy the Wyckoff positions: 4d (0.00000, 0.50000,0.25000) in Si8Ge4; the Si atoms occupy the Wyckoff positions: 4d(0.00000, 0.50000, 0.25000) and the Ge atoms occupy theWyckoff positions 8 j (0.34023, 0.34023, 0.12476) in Si4Ge8. Thebonding lengths of the 4d-site to 8j-site and the 8j-site to 8j-sitebonds are 2.367 Å, 2.383 Å and 2.408 Å in Si12, respectively. Thesame bonding lengths in Ge12 are 2.487 Å, 2.486 Å and 2.506 Å,respectively. They are both in excellent agreements with Ref [13].These bonding lengths are slightly elongated from those in thediamond Si structure and diamond Ge structure, which are 2.367 Åand 2.484 Å in our calculation, respectively. The optimized latticeparameters of the Si12, Ge12 in P42/mnm structure and their alloysare listed in Table 1. From Table 1, In Table 1, it can be easily foundthat our results are in excellent agreement with Ref [13]. Mean-wile, the calculated lattice parameters of diamond–Si and dia-mond–Ge (Space group: Fd–3 m) are in excellent agreement withprevious experimental results, indicating our calculations are validand believable.

The calculated elastic constants of Si12, Ge12, Si8Ge4 and Si4Ge8in P42/mnm structure are shown in Table 2. The first and foremost,tetragonal symmetry has six independent elastic constants (C11,C33, C44, C66, C12, C13), and these independent elastic constantsshould obey the following generalized Born’s mechanical stabilitycriteria of tetragonal symmetry [35,36]:

> = ( )C i0, 1, 3, 4, 6, 1ii

( − ) > ( )C C 0, 211 12

( + − ) > ( )C C C2 0, 311 33 13

[ ( + ) + + ] > ( )C C C C2 4 0. 411 12 33 13

The independent elastic constants of Si12 and Ge12 in P42/mnmstructure satisfy the above generalized Born’s mechanical stabilitycriteria of tetragonal symmetry. In other words, the calculatedresults show that the Si12 and Ge12 in P42/mnm structure aremechanically stable under ambient conditions. The phonon spec-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 andGe12 in P42/mnm structure are dynamic stability at ambientpressure. In order to investigate the stability of Si–Ge alloys, wecalculated the phase diagram based on the regular-solution model[37]. The mixing enthalpy of Si–Ge alloys is expressed as follows:

Δ ( ) = ( ) − ( − ) ( ) − ( ) ( )−H x E Si Ge x E Si xE Ge1 . 5x x1

where E(Si1�xGex) is the total energy of Si1�xGex alloys, E(Si) is thetotal energy of Si12 and E(Ge) is the total energy of Ge12,

respectively. The mixing enthalpy △H(x) can also be expressed bythe following expression:

Δ Ω( ) = ( − ) ( )H x x x1 . 6

where Ω is the interaction parameter which depends on the ma-terials. Ω could be calculated using the Eqs. (5) and (6). The x–dependent interaction parameter is gained from a linear fit to theΩ values. Using linear x–dependent and average values of theinteraction parameter Ω, the mixing enthalpies of Si1-xGex alloysfor PBE, PBEsol and CA-PZ functionals are depicted in Fig. 2. ForSi1-xGex alloys, the linear fits are Ω¼1.3055–0.4476x eV/pair,

0

2

4

6

8

10

12

14

0

2

4

6

8

10

12

Freq

uenc

y (T

Hz)

8Ge4

Z A M G Z R X GSi Si4Ge8

Z A M G Z R X G

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

Table 3The density (g/cm3), anisotropic sound velocities (m/s), average sound velocity (m/s) and the Debye temperature (K) for Si12, Si8Ge4, Si4Ge8, Ge12 in P42/mnm struc-ture, and Si and Ge in diamond structure.

P42/mnm structure Fd-3m structure

Si12 Si8Ge4 Si4Ge8 Ge12 Si Geρ 2.003 2.922 3.770 4.483 2.276 5.108[100] [100]vl 7836 6163 5150 5150 8226 4871

[010]vt1 4946 3836 3216 2794 5892 3467[001]vt2 5520 3606 3416 2987 5892 3467

[010] [001]vl 8540 6670 5402 4723[100]vt1 5520 3606 3416 2987[010]vt2 5520 3606 3416 2987

[110] [110]vl 8552 6975 5402 4652 8992 5354

[ ]v110 t1 4327 3582 3003 2630 6562 3765

[001]vt2 4946 3836 3216 2794 4960 3097[111] [111]vl 9233 5505

[ ]v112 t1,2 5092 2955

vl 8324 6383 5354 4623 8727 5220vt 4890 3703 3162 2762 5303 3119vm 5421 4111 3504 3057 5859 3452ΘD 566 422 354 304 639 358

40

60

80

100

120

Elas

tic m

odul

us (G

Pa)

Bulk modulus Shear modulus Young's modulus

Si12 Si8Ge4 Si4Ge8 Ge12

Fig. 4. Elastic modulus as a function of germanium concentration for Si–Ge alloysin P42/mnm structure.

Q. Fan et al. / Materials Science in Semiconductor Processing 43 (2016) 187–195190

Ω¼1.2772–0.6082x eV/pair and Ω¼0.3402-0.2392x eV/pair withPBE, PBEsol and CA-PZ, respectively. The average values of Ω are1.0817 eV/pair, 0.9731 eV/pair and 0.2206 eV/pair for x¼0.5. Fromdiscussion above, we can find that a smaller interaction parameterΩ for CA-PZ compared with that of PBE and PBEsol. The CA-PZ hasthe largest deviation ΔH with x-dependent Ω from ΔH withaverage Ω. This can be associated with smaller equilibrium en-ergies and larger difference between the equilibrium energies ofthe constituent binary obtained with CA-PZ. From Fig. 2, it can beinferred that the mixing enthalpy of an alloy depends on the in-teraction between atoms of constituents. To ensure the stability ofSi8Ge4 and Si4Ge8, the phonon spectra are calculated at ambientpressure (see Fig. 3). The mixing enthalpies are all positive, and thealloys might exist at a specified high temperature scale, due to theentropy ΔH effects considered. The Helmholtz free energy ofmixing ΔG can be expressed as

ΔΔ = Δ – ( )G E T H 7

Moreover, for the regular solution model of alloys, the entropyof mixing ΔH can be given as

Δ = − [ + ( − ) ( − )] ( )H R xlnx x ln x1 1 8

where R represents the gas constant and x is the composition of Gein Si–Ge alloys. From Eqs. (7) and (8), it can be seen that the Si–Gealloys in P42/mnm structure can be synthesized in a high tem-perature environment.

Calculated bulk modulus B, shear modulus G for Si12, Ge12 andSi–Ge alloys in P42/mnm structure are also shown in Table 2. FromTable 3, it can been seen that our calculation results of Si and Ge indiamond structure are in excellent agreements with the experi-mental or previous theoretical data. Bulk modulus B and shearmodulus G are calculated by the Voigt–Reuss–Hill approximation[38–40]. The Young’s modulus E and Poisson’s ratio v are obtainedby the following formulas [40]: E¼9BG/(3BþG), v¼(3B-2G)/[2(3BþG)]. Fig. 4 shows the elastic modulus as a function of com-position x for Si1�xGex alloys. From Fig. 4, it can be find that Bulkmodulus B, shear modulus G and Young’s modulus E all decreasewith the increasing composition x. From silicon to germanium inP42/mnm structure, the Young’s modulus of Ge12 is decrease

-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

-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

xz plane

-0.3 -0.2 -0.1 0.0 0.1 0.2 0.3-0.3

-0.2

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-75 -50 -25 0 25 50 75-75

-50

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75

plane

-75 -50 -25 0 25 50 75-75

-50

-25

0

25

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75

-75 -50 -25 0 25 50 75-75

-50

-25

0

25

50

75

Poisson's ratio xy plane Poisson's ratio

Poisson's ratio yz plane shear modulus (GPa) xy

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 thexy 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.

Q. Fan et al. / Materials Science in Semiconductor Processing 43 (2016) 187–195 191

30.25% than that of Si12, and bulk modulus is 26.67%, shear mod-ulus is 29.17%, respectively. According to Pugh [41], a smaller B/Gvalue (B/Go1.75) for a solid represents a brittle manner but alarger B/G value (B/G41.75) usually represents a ductile manner.Moreover, Poisson’s ratio v is consistent with B/G, which refers toductile compounds usually with a larger Poisson’s ratio v

(v40.26) [42]. The value of B/G is smaller than 1.75 and the valueof v is smaller than 0.26 for Si–Ge alloys in Table 2. These showthat Si–Ge alloys are brittle, Ge12 has the most brittleness andSi8Ge4 has the least brittleness.

The anisotropy of the crystal lattice along different directions,the atomic arrangement of the periodicity and the degree of

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

Q. Fan et al. / Materials Science in Semiconductor Processing 43 (2016) 187–195192

density are not the same. This leads to the different physical andchemical properties of crystals in different directions, which arerepresented as the crystal anisotropy. The anisotropy in poly-crystalline materials can also be influenced to certain texturepatterns often produced during manufacturing of the material. Itcan be defined as a difference, when measured along differentaxes, in a material’s physical or mechanical properties. The de-pendence of Young’s modulus on the direction of load can be usedas an example. Most materials exhibit the anisotropy behavior. Theanisotropy of crystal is different in different directions, such aselastic modulus, hardness, fracture resistance, thermal expansioncoefficient, thermal conductivity, electrical resistivity, magneticsusceptibility and refractive index. It is well known that the ani-sotropy of elasticity is an important implication in engineeringscience and crystal physics. In this paper, we mainly discuss theanisotropy of elastic modulus of materials. The directional de-pendence of the anisotropy is calculated by the Elastic AnisotropyMeasures (ELAM) [43,44] code. The calculated Poisson’s ratio andshear modulus along different directions as well as the projectionsin different planes are shown in Fig. 5. Fig. 5 shows that thePoisson’s ratio and shear modulus in xy plane of Si1-xGex alloysexhibits to be more anisotropic than on other planes. The Si12 hasthe greatest anisotropy in Poisson’s ratio, Si8Ge4 has the smallestanisotropy in Poisson’s ratio. The anisotropy of shear modulus ofSi12, Si8Ge4, Si4Ge8 and Ge12 are as well as the anisotropy ofPoisson’s ratio. The Si12 has the greatest anisotropy in shearmodulus, while Si8Ge4 has the smallest anisotropy in shear

modulus. The 3D figures of the Young’s modulus for Si12, Si8Ge4,Si4Ge8 and Ge12 are shown in Fig. 6, and the 2D representation ofthe Young’s modulus for Si12, Si8Ge4, Si4Ge8 and Ge12 are shown inFig. 7. The surface in each graph denotes the magnitude of E alongdifferent directions. The 3D figure appears as a spherical shape foran isotropic structure, while the deviation from the sphericalshape exhibits the content of anisotropy [45]. From Fig. 6, it isobvious that the 3D figures evidently deviate in shape from thesphere along the xy plane than then xz plane, as the C33 larger thanthe C11 for Si12, which may result in the xy plane is more aniso-tropic than the xz plane in Si12. The maximal and minimum valuesof Young’s modulus both appear in the xy plane, but only theminimum values appear in the xz and yz planes. The ratio ofmaximal values and the minimum values are 1.337; 1.163; 1.181;1.173 for Si12, Si8Ge4, Si4Ge8 and Ge12, respectively, which meansthe Si12 exhibits the largest anisotropy. The elastic anisotropy of acrystal can be characterized by many different ways, for example,the universal anisotropic index AU. The universal elastic anisotropyindex AU for a crystal with any symmetry can be given by follows[46–48]: AU¼5 GV/GR þBV/BR–6. The calculated results of AU arelisted in Table 2. Table 2 shows that Si has the greatest anisotropyin AU, while Si8Ge4 has the smallest anisotropy. Consequently, Si12exhibits the greatest anisotropy in Poisson’s ratio, shear modulus,Young’s modulus and AU, and Si8Ge4 exhibits the smallestanisotropy.

The sound velocities are determined by the symmetry of thecrystal and propagation direction. Using the elastic constants, the

-100 -50 0 50 100

-100

-50

0

50

100

Emax=131 GPa

Si12

Emin=98 GPa

-100 -50 0 50 100

-100

-50

0

50

100

Emax=107 GPa

Si8Ge4

Emin=92 GPa

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-50

0

50

100

Emax=98 GPa

Si4Ge8

Emin=83 GPa

-100 -50 0 50 100-100

-50

0

50

100

Emax=88 GPa

Emin=75 GPa

Young's modulus (GPa) Young's modulus (GPa)

Young's modulus (GPa) Young's modulus (GPa)

Ge12

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.

Q. Fan et al. / Materials Science in Semiconductor Processing 43 (2016) 187–195 193

phase velocities of pure transverse and longitudinal modes of theSi12, Si8Ge4, Si4Ge8 and Ge12 can be calculated by following theprocedure of Brugger [49]. For example, the pure transverse andlongitudinal modes can only be found for [001], [110] and [111]directions in a cubic crystal, and the sound propagating modes inother directions are the quasi-transverse or quasi-longitudinalwaves. In the principal directions, the acoustic velocities in a Cubicsymmetry can be expressed by:

ρ[ ] = ( )v C100 / , 9l 11

ρ[ ] = [ ] = ( )v v C010 001 / , 10t t1 2 44

ρ[ ] = ( + + ) ( )v C C C110 2 /2 , 11l 11 12 44

ρ[ ] = ( − ) ( )v C C110 / , 12t1 11 12

ρ[ ] = ( )v C001 / , 13t2 12

ρ[ ] = ( + + ) ( )v C C C111 2 4 /3 , 14l 11 12 44

ρ[ ¯ ] = [ ¯ ] = ( − + ) ( )v v C C C112 112 /3 . 15t t1 2 11 12 44

In the principal directions, the acoustic velocities in a tetra-gonal symmetry are given by the following expressions:

ρ[ ] = ( )v C100 / , 16l 11

ρ[ ] = ( )v C010 / , 17t1 44

ρ[ ] = ( )v C001 / , 18t2 66

ρ[ ] = ( )v C001 / , 19l 33

ρ[ ] = [ ] = ( )v v C100 010 / , 20t t1 2 66

ρ[ ] = ( + + ) ( )v C C C110 2 /2 , 21l 11 12 66

ρ[ ] = ( − ) ( )v C C110 /2 , 22t1 11 12

-2

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0

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4

Ener

gy (e

V)

(a) Si12

Z A M G Z R X G

1.85 eV 1.90 eV

Z A M G Z R X G(b) Si8Ge4

1.65 eV

Ener

gy (e

V)

(c) Si4Ge8

Z A M G Z R X G

1.60 eV

(d) Ge12

Z A M G Z R X G

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

Table 4The calculated band gap (eV) of for Si12, Si8Ge4, Si4Ge8, Ge12 in P42/mnm structure.

PBE PBEsol CA-PZ HSE06 mBJ

Si12 1.24, 1.28a 1.15 1.17 1.85 2.03a

Si8Ge4 1.25 1.17 1.21 1.90Si4Ge8 1.03 0.98 1.02 1.65Ge12 0.71, 0.81a 0.68 1.12 1.61 1.46a

a Ref. [13].

Q. Fan et al. / Materials Science in Semiconductor Processing 43 (2016) 187–195194

ρ[ ] = ( )v C001 / , 23t2 44

where ρ is the density of Si–Ge alloys; vl is the longitudinal soundvelocity; vt1 and vt2 refer to the first transverse mode and thesecond transverse mode, respectively. The equations of Debyetemperature, longitudinal sound velocity and transverse andsound velocity are given by Ref [48,50]. The calculated sound vo-locity of Si–Ge alloys in P42/mnm structure together with Si, Ge indiamond structure are shown in Table 3. The density and elasticconstants of the Si12, Ge12 in P42/mnm structure are slightlysmaller than that of Si, Ge in diamond structure. Hence, the soundvelocities along different directions (such as [100], [010], [001],[110] and [1–10] directions) of Si12, Ge12 in P42/mnm structure aresmaller than that of Si, Ge in diamond structure. The Debye tem-perature of Si12, Ge12 in P42/mnm structure and Si, Ge in diamondstructure are also shown in Table 3. The Debye temperatures of Si,Ge in diamond structure have excellent agreements with othertheoretical results (Si: 636 K, Ge: 374 K) [51]. The Debye tem-peratures of Si12, Ge12 in the P42/mnm structure are slightlysmaller than that of Si, Ge in diamond structure.

It is well known that the electronic structure determines thefundamental physical and chemical properties of materials. Thecalculated electronic band structures for Si12, Si8Ge4, Si4Ge8 andGe12 in P42/mnm structure are presented in Fig. 8. The bottom ofthe conduction band occurs along the Z–A direction and the top ofthe valence band occurs at A point, which means that Si12 in theP42/mnm structure is a semiconductor with an indirect band gapof 1.85 eV within HSE06. However, both the valence band max-imum and the conduction band minimum locate at the A point forGe12 in P42/mnm structure, indicating that Ge12 in P42/mnmstructure is a direct band gap semiconductors with a band gap of1.61 eV within HSE06. The calculated results using differentfunctions of Si12, Si8Ge4, Si4Ge8 and Ge12 in P42/mnm structure arelisted in Table 4. The calculated results of Si12 and Ge12 in P42/mnm structure are 1.28 eV and 0.81 eV within GGA by Nguyenet al., respectively, and our calculated results are in excellentagreements with the results of Nguyen et al. [13]. Compared withthe results of Nguyen et al., the band gap of Si12 within HSE06 isslightly smaller within mBJ, but the band gap of Ge12 within HSE06is slightly larger within mBJ (see Table 4). For Si8Ge4, the con-duction band minimum is at (0.231 0.231 0.5) point along the Z�Adirection (see Fig. 8(b)), while the valence band maximum is lo-cated at (0.0 0.389 0.0) point along the X�G direction. For Si4Ge8,the conduction band minimum is at (0.250 0.250 0.5) point alongthe Z�A direction, while the valence band maximum is located atA point. In addition, Si8Ge4 and Si4Ge8 are indirect band gapsemiconductors with band gap of 1.90 and 1.65 eV, respectively.

4. Conclusions

In summary, systematic DFT calculations have been performedon Si12, Ge12 and Si–Ge alloys solid solutions, including the

Q. Fan et al. / Materials Science in Semiconductor Processing 43 (2016) 187–195 195

stability, mechanical properties, anisotropic properties and elec-tronic properties. The calculated lattice parameters and band gapof silicon and germaniumwith the reported results [13] are exactlythe same. The lattice parameters of Si–Ge alloys increase withgermanium concentration, showing a positive deviation from Ve-gard’s law, while the bulk modulus, Shear modulus and Young’smodulus decreases with composition x. For anisotropic properties,Si12 exhibits the greatest anisotropy in Poisson’s ratio, shearmodulus, Young’s modulus and AU, and Si8Ge4 exhibits the smal-lest anisotropy in Poisson’s ratio, shear modulus, Young’s modulusand AU. The electronic structure calculations reveal that Si12 in P42/mnm structure is a semiconductor with an indirect band gap of1.85 eV, Ge12 in P42/mnm structure is a direct band gap semi-conductors with the band gap of 1.61 eV, and Si8Ge4 and Si4Ge8 areindirect band gap semiconductors with the band gap of 1.90 and1.65 eV, respectively.

Acknowledgments

This work was supported by the Natural Science Foundation ofChina (No. 61474089), Open fund of key laboratory of complexelectromagnetic environment science and technology, ChinaAcademy of Engineering Physics (No. 2015-0214. XY.K).

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