Prediction of Novel Phase of Silicon and Si–Ge Alloys

Journal of Solid State Chemistry 233 (2016) 471–483

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Journal of Solid State Chemistry

journal homepage: www.elsevier.com/locate/jssc

Prediction of novel phase of silicon and Si–Ge alloys

Qingyang Fan a, Changchun Chai a, Qun Wei b,n, Yintang Yang a, Qi Yang a, Pengyuan Chen a, Mengjiang Xing c, Junqin Zhang a, Ronghui Yao b 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 Information Engineering & Automation, Kunming University of Science and Technology, Kunming 650051, PR China article info abstract

Article history: The structural, thermodynamic, elastic, anisotropic and electronic properties of P2221-Si have been Received 21 August 2015 studied using ﬁrst-principles calculations. The elastic constants are satisﬁed with mechanical stability Received in revised form criteria. The mechanical anisotropy is predicted by anisotropic constants Poisson's ratio, shear modulus, 5 November 2015 Young's modulus and three dimensional curved surface of Young's modulus. These results show that Accepted 13 November 2015 P222 -Si and Si–Ge alloys are anisotropic. The sound velocities in different directions and Debye tem- Available online 1 December 2015 1 perature for P2221-Si and Si–Ge alloys are also predicted. Electronic structure study shows that P2221-Si Keywords: is an indirect semiconductor with band gap of 0.90 eV. In addition, the band structures of Si–Ge alloys are Silicon investigated in this paper. Finally, we also calculate the thermodynamics properties and obtained the Si–Ge alloys relationships between thermal parameters and temperature. Mechanical properties & 2015 Elsevier Inc. All rights reserved. Electronic properties

1. Introduction calculations. They found that the band structures of bct and M4 phases of silicon show that they are semiconductors with an in- Silicon is an indispensable material for the modern industry direct band gap, which is twice smaller than that calculated for the because it has lots of unique physical and chemical characteristics. cubic silicon (Si-I, space group: Fd-3m). Utilizing first-principles For example, its excellent electronic and optical properties make it calculations, Hao et al. [14] studied the electronic and elastic an important optoelectronic material [1,2]; its desirable piezo- properties and mechanical properties of a new silicon allotrope resistance coefficients make it almost a perfect material for mi- (T12-Si) to enrich the relevant information. The results show that croelectro-mechanical transducers [3–5]. Many other crystal forms the T12-Si is mechanically anisotropic and has a lower bulk mod- of silicon under pressure have also been reported [6]. The crystal ulus and shear modulus than Si-I. Its hardness is 10.3 GPa, smaller structures of silicon and germanium were studied by energy dis- than that of Si-I (13.5 GPa). Analyses of the electronic properties persive X-ray diffraction at room temperature and pressures up to reveal that T12-Si is an indirect band gap crystal with a gap value of 50 GPa [7]. Silicon transforms to a primitive hexagonal (Si-V) 0.69 eV. Recently, six metastable allotropes of silicon with direct or – structure about 16 GPa, to an others phases Si-VI between 35 and quasidirect band gaps of 0.3 91.25 eV are predicted utilizing ab 40 GPa, and to hcp (Si-VII) about 40 GPa. A transition to the β-Sn initio calculations at ambient pressure by Wang et al. [15]. Five of them possess band gaps within the optimal range for high con- phase initiates at 11.270.2 GPa and two new phases coexist to verting efficiency from solar energy to electric power and also 12.570.2 GPa [8]. To extend the functionality of silicon in appli- have better optical properties than the Si-I phase. De and Pryor cations, a wide range of nanostructures, for instances, nanotubes [16] calculated the electronic band structure and dielectric func- [9], nanowires [10], nanorods [11], and nanoribbons [12],have tions for silicon in lonsdaleite phase and this phase has an indirect been prompted by the modern technology. Wu et al. [13] in- band gap of 0.95 eV. They also calculate the optical properties of vestigated the stabilities and electronic properties of two hy- silicon in the lonsdaleite phase using a transferable model em- pothetical allotropes of silicon, the body-centered tetragonal (bct) pirical pseudopotential method with spin–orbit interactions. and monoclinic (M4) phases utilizing density functional In this paper, a novel silicon phase (space group: P2221) P2221- Si with indirect band gap 0.90 eV is investigated. The original n Corresponding author. structure of P2221 Si is P2221-carbon in Refs. [17,18], with Si E-mail address: [email protected] (Q. Wei). substituting C. Furthermore, the detailed physical properties (such http://dx.doi.org/10.1016/j.jssc.2015.11.021 0022-4596/& 2015 Elsevier Inc. All rights reserved. 472 Q. Fan et al. / Journal of Solid State Chemistry 233 (2016) 471–483

Fig. 1. Unit cell crystal structures of P2221-silicon and Si–Ge alloys.

as structural properties, elastic properties, anisotropic and elec- Eruzerhof (PBE) [19],WuandCohen(WC)[20],PBEsol[21] and local tronic properties) of novel silicon allotropes are studied. In addi- density approximation (LDA) functional [22,23] form were adopted tion, the electronic and elastic properties of Si–Ge alloys in P2221 as the exchange and correlation interaction. The properties of the phase are also investigated in this paper. predicted P2221 phases were obtained via ultrasoft pseudopotentials [24] through the CASTEP code [25]. The spacing in the k-point Monkhorst-Pack grid was 0.03 Å1 (7 3 7) for Brillouin zone sampling [26]. The Broyden–Fletcher–Goldfarb–Shanno (BFGS) [27] 2. Calculated method minimization was used for the convergence criterion. Structural op- A plane-wave basis set with the energy cutoff of 340 eV was used. timization was performed until the enthalpy change per atom was 6 Generalized gradient approximation (GGA) within Perdew–Burke– less than 1 10 eV, the ionic forces on atoms were less than Q. Fan et al. / Journal of Solid State Chemistry 233 (2016) 471–483 473

Table 1

Calculated lattice parameters of P2221-Si and Si-I (in units of Å).

P2221 Lattice constants

WC PBE PBEsol CA-PZ

xa(Å) b (Å) c (Å) a (Å) b (Å) c (Å) a (Å) b (Å) c (Å) a (Å) b (Å) c (Å)

0 5.439 12.988 5.323 5.448 13.017 5.339 5.445 12.995 5.327 5.364 12.803 5.248 0.056 5.453 13.015 5.336 5.461 13.046 5.353 5.458 13.022 5.340 5.372 12.820 5.255 0.111 5.465 13.053 5.350 5.473 13.085 5.367 5.470 13.061 5.355 5.378 12.847 5.265 0.167 5.474 13.098 5.364 5.484 13.129 5.382 5.480 13.105 5.368 5.384 12.873 5.273 0.222 5.486 13.139 5.379 5.494 13.175 5.398 5.491 13.146 5.383 5.391 12.899 5.283 0.278 5.494 13.188 5.394 5.505 13.222 5.412 5.501 13.194 5.398 5.398 12.926 5.292

Fd-3m 5.460 5.465 5.466 5.374 5.402a 5.392a 5.429b 5.465c Exp. 5.431

a Ref. [14]. b Ref. [46]. c Ref. [13].

1.00 1.00

0.98 0.99

0.96

0.98 0.94

0.97 0.92 0 0 V X / 0.90 / X V 0.96 0.88

0.95 0.86

0 K 0.84 0.94 a/a 200 K 0 300 K b/b 0 400 K 0.82 c/c 0 600 K 0.93 0 3 6 9 1215180 2 4 6 8 1012141618 Pressure (GPa) Pressure (GPa)

Fig. 2. The lattice constants a/a0, b/b0, c/c0 compression as functions of pressure and temperature for P2221-silicon.

0.01 eV/Å, and the stress components were less than 0.02 GPa. The found in the silicon allotropes system. The crystal structure of thermodynamic properties of P2221-Si are calculated by the quasi- P2221-Si and Si–Ge alloys are shown in Fig. 1.TheSiatomsoccupy harmonic Debye model [28–32]. Utilizing the quasi-harmonic Debye the Wyckoff positions 4e (0.84569, 0.08485, 0.41830), 4e (0.72918, model, one can obtain the thermodynamic properties of P2221-Si at 0.71207, 0.00700), 2b (0.23378, 0.50000, 0.00000), 2 c (0.00000, pressures and temperatures. 0.81164, 0.25000), 2 c (0.00000, 0.38574, 0.25000), 2d (0.50000,

0.59759, 0.25000) and 2d (0.50000, 0.82627, 0.75000) in P2221-Si, respectively. For Si–Ge alloys, the Ge atoms occupy the positions 3. Results and discussion with the minimum energy. Using both GGA and LDA methods, the equilibrium lattice constants are determined by minimizing the total A thermodynamically stable silicon allotrope, orthorhombic-Si, is energy with respect to variation of the cell volume. The structural 474 Q. Fan et al. / Journal of Solid State Chemistry 233 (2016) 471–483

-106.7 c-Si Lonsdaleite Si -106.8 M-Si Cco-Si P222 -Si Z-Si 1 -106.9 P4 /ncm-Si tP16-Si 2

-107.0

-107.1 Enthalpy (eV/atom) -107.2

-107.3

-10 -5 0 5 10 15 20 Pressure (GPa)

Fig. 3. Calculated enthalpies of different silicon structures as a function of pressure.

6 Frequency (THz) 4

0 G Z T Y S X U R

Fig. 4. Phonon spectra for P2221-silicon.

Table 2

Calculated Elastic Constants Cij, bulk modulus B, shear modulus G and Young's modulus E of P2221-Si and Si-I (in units of GPa). Also shown is Poisson's ratio v.

C11 C12 C13 C22 C23 C33 C44 C55 C66 BGE v

P2221 0 144 52 54 150 48 141 65 64 52 83 54 133 0.23 0.056 131 49 51 149 48 145 54 66 64 80 54 132 0.22 0.111 140 49 50 146 48 130 65 63 55 79 54 132 0.22 0.167 137 48 49 143 46 128 63 62 53 77 53 129 0.22 0.222 135 48 48 141 44 126 62 61 52 76 52 127 0.22 0.278 132 47 47 140 45 125 60 60 49 75 50 123 0.23

Fd-3m GGA 154 56 79 88 64 155 0.21 Ref. [14] 163 58 80 93 68 164 0.21 Ref. [14] 162 63 77 96 65 159 0.23 Ref. [47] 167 65 81 Ref. [48] 166 64 80 Exp. [49] 166 64 80 Exp. [53] 99

parameters at zero pressure for P2221-Si are shown in Table 1.In increases with the increasing composition x.Inaddition,theresults Table 1, we can see that the calculated lattice parameter of Si-I obtained by the PBE method are closer to the experimental values (space group: Fd-3m) is in excellent agreement with the available than that obtained by the other methods. Therefore, we believe that experimental value. And the lattice parameters of Si–Ge alloys the PBE method is a more reasonable method and will focus on the Q. Fan et al. / Journal of Solid State Chemistry 233 (2016) 471–483 475

Fig. 5. 2D representation of Poisson's ratio in the xy plane (a), xz plane (b) and yz plane (c) for P2221-silicon. 2D representation of shear modulus in the xy plane (d), xz plane

(e) and yz plane (f) for P2221-silicon. The solid line represents the maximum and dashed line represents the minimum. 476 Q. Fan et al. / Journal of Solid State Chemistry 233 (2016) 471–483

0 GPa 5 GPa 10 GPa

125

100

-25

-50

-75

-100

-125 -125 -100 -75 -50 -25 0 25 50 75 100 125 xy plane

0 GPa 5 GPa 10 GPa 0 GPa 5 GPa 10 GPa 125 125

100 100

75 75

50 50

25 25

0 0

-25 -25

-50 -50

-75 -75

-100 -100

-125 -125 -125 -100 -75 -50 -25 0 25 50 75 100 125 -125 -100 -75 -50 -25 0 25 50 75 100 125 xz plane yz plane

Fig. 6. The directional dependence of the Young's modulus for P2221-silicon (a), 2D representation of Young's modulus in the xy plane (b), xz plane (c) and yz plane (d) for

P2221-silicon.

results from PBE calculations in the following discussions. and Si–Ge alloys are shown in Table 2. The ﬁrst and foremost, The relationship between lattice parameters and pressures these elastic constants in Table 2 satisfy traditional mechanical T¼0 K is shown in Fig. 2(a) and (b) presents the relations of the V/ stability conditions [32,33]. The calculated elastic constants of Si-I

V0 as a function of pressure P up to 18 GPa at T¼0, 200, 300, 400, are in excellent agreement with the available experimental values. and 600 K. We noted that, when the pressure increases, the The dynamic stability of these structures was veriﬁed by the compression along the c-axis is much larger than those along the phonon spectrum calculations, which show no imaginary fre- a-axis and b-axis in the basal plane. From Fig. 2(a), we can also quency along the whole Brillouin zone (see Fig. 4). So the elastic easily see that the compression of a-axis is the most difﬁcult. The constants and phonon calculations reveal P2221-Si is mechanically metastable P2221-Si is higher in energy than Si-I by 0.085 eV/atom and dynamically stable. The bulk modulus B and shear modulus G at zero pressure. Calculated enthalpies of different silicon struc- are calculated by the Voigt–Reuss–Hill approximation [34–36] and tures as a function of pressure are shown in Fig. 3. Si-I is the most the arithmetic average of Voigt and Reuss bounds is termed as the stable phase in the whole pressure range. tP16-Si is mechanically Voigt–Reuss–Hill approximations [36]. The Young's modulus E and and dynamically stable, whereas its energy is higher than Si-I by Poisson's ratio v are obtained from the following equations 0.277 eV/atom at zero pressure. [36–38]: E¼9BG/(3BþG), v¼(3B2G)/[2(3BþG)], respectively.

The calculated elastic constants and elastic modulus of P2221-Si From Table 2, we can found that bulk modulus, shear modulus and Q. Fan et al. / Journal of Solid State Chemistry 233 (2016) 471–483 477

Fig. 7. The directional dependence of the Young's modulus for Si–Ge alloys. 478 Q. Fan et al. / Journal of Solid State Chemistry 233 (2016) 471–483

Table 3 3 The density (in g/cm ), anisotropic sound velocities (in m/s), average sound velocity (in m/s) and the Debye temperature (in K) for the P2221-silicon and Si–Ge alloys.

Compound x 0 0.056 0.111 0.167 0.222 0.278

ρ 2.227 2.409 2.580 2.750 2.917 3.080

[100] [100]vl 8041 7374 7366 7058 6803 6547

[010]vt1 4832 5154 4617 4390 4222 3989

[001]vt2 5360 5234 4942 4748 4573 4414

[010] [010]vl 8206 7865 7523 7211 6953 6742

[100]vt1 4832 5154 4617 4390 4222 3989

[001]vt2 5402 4735 5019 4786 4610 4414

[001] [001]vl 7956 7758 7098 6822 6572 6371

[100]vt1 5360 5234 4942 4748 4573 4414

[010]vt2 5402 4735 5019 4786 4610 4414

vl 8343 7943 7650 7328 7059 6782

vt 4924 4735 4575 4390 4222 4029

vm 5456 5241 5063 4857 4672 4461

ΘD 590 563 545 519 500 478

Young's modulus decrease with increasing composition x. of the P2221-Si and Si–Ge alloys can be calculated from the single In the meantime, elastically anisotropic parameters have an crystal elastic constants following the procedure of Brugger [40]. important implication in engineering science. In the present work, The sound velocity and Debye temperature are two fundamental we investigated the elastic anisotropic of P2221-Si in Poisson's parameters for evaluating the chemical bonding characteristics ratio, shear modulus and Young's modulus. The directional de- and thermal properties of materials in materials science. The pendence of anisotropy was calculated by using the program sound velocities are determined by the symmetry of the crystal Elastics Anisotropy Measures (ELAM) [39]. The directional de- and propagation direction. In the principal directions the acoustic pendence of the Poisson's ratio at different pressure, the xy, xz, and velocities of orthorhombic symmetry can be expressed by [41]: yz planes for P2221-Si is plotted in Fig. 5(a)–(c), respectively. The []=100vClt11 /ρρ , []= 010 v 1 C 66 / , [] 001 anisotropic of Poisson's ratio in xy plane for P2221-Si increases with increasing pressure, but the curve of maximal and minimal vCtlt255=[]=[]=[]/ρρ . 010 vC 22166 / , 100 vC / ρ , 001 Poisson's ratio are turn to circle, no longer tetragonal. So the ani- vCtlt244=[]=[]=[]/.001ρρ vC 33155 /,100 vC /,010 ρ sotropic of Poisson's ratio in xy plane for P2221-Si at 10 GPa is slightly less than 5 GPa. Moreover, the anisotropic of Poisson's vCt244= /.ρ ()1 ratio in xz and yz planes for P222 -Si increases with increasing 1 ρ – pressure. The directional dependence of the shear modulus at where is the density of P2221-Si and Si Ge alloys; the calculated different pressure, the xy, xz, and yz planes for P222 -Si is plotted velocities of pure transverse and longitudinal modes of the P2221- 1 – in Fig. 5(d)–(f), respectively. The maximal and minimal values of Si and Si Ge alloys are listed in Table 3. From Table 3, we can easily to know that the density increase with increasing composition x, shear modulus for P2221-Si are 65, 67, 69 and 44, 42, 40 GPa at P¼0, 5, 10 GPa, respectively. From Fig. 5(d)–(f), we also can noted obviously, sound velocities decrease with increasing composition v Θ that the anisotropic of shear modulus in xy, xz, and yz planes plane x. The average sound velocity m and Debye temperature D can be approximately calculated by the following relations [38,42]: for P2221-Si increases with increasing pressure. The directional dependence of the Young's modulus and the projection in xy, xz, 1 ⎡ ⎛ N ρ ⎞⎤3 h ⎢ 3n ⎜ A ⎟⎥ and yz planes for P2221-Si are illustrated in Fig. 6. Emax/ ΘD = vm, k ⎣ 4π ⎝ M ⎠⎦ ()2 Emin(P¼0)¼145/114¼1.27, Emax/Emin(P¼5)¼152/107¼1.42, Emax/ B Emin(P¼10) ¼151/102¼1.48, the anisotropic of Young's modulus in xy, xz, and yz planes plane for P2221-Si increases with increasing 1 21− 1 pressure, too. v =[ ( + )] 3 , m 33 The directional dependence of the Young's modulus of Si–Ge 3 vvlt ()3 alloys are illustrated in Fig. 7. For an isotropic system, the 3D where h is Planck's constant, k is Boltzmann's constant, N is directional dependence would exhibit a spherical shape, while B A the deviation degree from the spherical shape reﬂects the Avogadro's number, n is the number of atoms in the molecule, M is ρ v v content of anisotropy. From Fig. 7(a)–(e)wecanseethatthe3D molecular weight, and is the density. l and t are the long- ﬁgures of the Young's modulus for Si–Ge alloys have kind of itudinal and transverse sound velocities, respectively, which can deviation in shape from the sphere, which indicates that the be obtained from Navier's equation [43]: – Young's modulus for these Si Ge alloys show anisotropy. In 4 1 G addition, the maximal values of Si Ge ,Si Ge , vBGvlt=(+ ),. = 0.944 0.056 0.889 0.111 3 ρρ ()4 Si0.833Ge0.167,Si0.778Ge0.222 and Si0.722Ge0.278 are 146, 145, 141, 139 and 135 GPa, respectively, and the minimal values of where B and G are isothermal bulk modulus and shear modulus,

Si0.944Ge0.056,Si0.889Ge0.111,Si0.833Ge0.167,Si0.778Ge0.222 and respectively. The longitudinal and transverse sound velocities of v ¼ v ¼ Si0.722Ge0.278 are 105, 105, 104, 103 and 102 GPa, respectively. So P2221-Si are smaller than Si-I ( t 6194 m/s, l 10159 m/s), be- the ratio of maximal and minimal are 1.39, 1.39, 1.37, 1.35 and cause of P2221-Si have the smaller elastic modulus. The Debye 1.32, that is to say, the anisotropic of Young's modulus decrease temperature is 590 K at P¼0 GPa and T¼0 K, and it is also smaller with the increasing the composition x. than Si-I (ΘD¼744 K). The effect of increasing pressure on long- The phase velocities of pure transverse and longitudinal modes itudinal and transverse sound velocities for Si–Ge alloys (x¼0.056, Q. Fan et al. / Journal of Solid State Chemistry 233 (2016) 471–483 479

Fig. 8. Electronic band structure of P2221-silicon and Si–Ge alloys.

0.111, 0.167, 0.222 and 0.278) are gradually weakened. is mainly due to the fact that it is based on simple model as- It is well known that the electronic structure determines the sumptions which are not sufﬁciently ﬂexible to accurately re- fundamental physical and chemical properties of materials. It is produce the exchange correlation energy and its charge derivative. known that the calculated band gap with DFT are usually under- However, this paper focuses on the band gap of the trend with the estimated by 30–50%, the true band gap must be larger than the composition x, so the underestimation is not effect on the con- calculated results. The underestimation of the band gap with GGA clusions. The calculated electronic band structure for P2221-Si is 480 Q. Fan et al. / Journal of Solid State Chemistry 233 (2016) 471–483

Table 4

Calculated the band gap of P2221-silicon and Si–Ge alloys.

Bandgap P2221 Fd-3m

x 0 0.056 0.111 0.167 0.222 0.278 0 WC 0.792 0.859 0.807 0.775 0.699 0.649 0.667 PBE 0.901 0.981 0.920 0.886 0.801 0.752 0.764 PBEsol 0.779 0.843 0.791 0.761 0.686 0.638 0.592 CA-PZ 0.768 0.887 0.847 0.826 0.761 0.729 0.568 Other calculations 0.47a, 0.52b, 0.55c Experimental 1.17d

a Ref. [14]. b Ref. [50]. c Ref. [51]. d Ref. [52].

10 10

9 9

8 8

7 7

6 6 ) -1 5 K 5 -5 0 K 0 GPa 200 K 6 GPa α (10 4 400 K 12 GPa 4 600 K 18 GPa 3 3

2 2

1 1

0 0 024681012141618 0 100 200 300 400 500 600 Pressure (GPa) Temperature (K)

Fig. 9. Temperature (a) and pressure (b) dependence of the thermal expansion coefﬁcient for P2221-silicon.

presented in Fig. 8(a). The top of the valence band and the bottom of Si-I than that obtained by the others method. From Table 4,we of the conduction band occur along the ΓY direction, indicating also noted that the band gap have the maximal value at compo- that P2221-Si is a semiconductor with indirect band gap of 0.90 eV, sition x¼0.056, and decrease with increasing composition x. which is larger than that of Si-I (0.67 eV). The calculated electronic The investigation on the thermodynamic properties of solid at band structure for Si–Ge alloys are presented in Fig. 8(b)–(f). The high pressure and high temperature is an interesting topic in the band gap show the maximal value at composition x¼0.056, condensed matter physics. The thermal properties of P2221-Si are however, the band gap of Si–Ge alloys decrease with increasing determined in the temperature range from 0 to 600 K, where the composition x. Table 4 list the band gap of P2221-Si, Si-I and Si–Ge quasi-harmonic model remains fully valid. The pressure effect is alloys thought others functional. We can note that the results investigated in the range of 0–18 GPa. In the quasi-harmonic De- obtained by the PBE method are closer to the experimental values bye model [37,38,44,45], the Debye temperature ΘD and the Q. Fan et al. / Journal of Solid State Chemistry 233 (2016) 471–483 481

Debye temaperture 18 720

539.5 16 700 561.5 14 680 583.5

12 605.5 660 627.5 10 640 649.5 8 671.5 620 693.5 Pressure (GPa) 6 600 715.5 0 K 580 4 200 K 300 K 560 2 400 K 500 K 540 0 0 100 200 300 400 500 600 024681012141618 Temperature (K) Pressure (GPa)

Fig. 10. Two-dimensional contour plots of Debye temperature versus pressure and temperature for P2221-silicon (a), pressure dependence of the Debye temperature for

P2221-silicon (b).

thermal expansion coefficient α are two key quantities. Fig. 9 then increase smoothly with increasing temperature. With the shows the variations of the thermal expansion with pressures and temperature increases, CV climbs to the limitation of Dulong–Petit temperatures. From Fig. 9(a), with the pressure increases, the law eventually. thermal expansion coefficient decreases almost exponentially at high temperature, and the higher the temperature is, the faster the thermal expansion coefficient decreases. This shows that the effect 4. Conclusions of temperature is much greater than that of pressure on the fi thermal expansion coef cient. When the pressure is 16 GPa, the In summary, potential P2221-Si metastabe phase was sys- thermal expansion at 600 K is just a little larger than that at 400 K, tematic and comprehensive investigated, such as crystal structure, which means that the temperature dependence of is very small at stability, elastic constants, anisotropic and electronic properties, high temperature and high pressure. by first-principle calculations combining with the quasi-harmonic The Debye temperature closely relates to many physical prop- Debye model. The elastic constants and phonon calculations reveal fi erties of solids, such as speci c heat, dynamic properties, and P2221-Si is mechanically and dynamically stable. Anisotropy ana- melting temperature. As shown in Fig. 10, the Debye temperature lyses demonstrate P2221-Si possesses a high degree of anisotropy of P2221-Si is plotted as the function of pressure and temperature. under high pressure. The anisotropic properties of sound velocities It is found that the Debye temperature increases with increasing also indicate the elastic anisotropy in these crystals. The Debye pressure and decreasing temperature. Furthermore, the char- temperature, heat capacity and the thermal expansion coefficient acteristic of Debye temperature versus pressure nearly show a of P2221-Si under high pressure and high temperature are re- linear relation at a given temperature. The Debye temperature of vealed for the first time. The present calculations provide funda-

P2221-Si is 570.32 K at 0 GPa and 300 K. Heat capacity belongs to mental information for a better understanding of the structural, one of the most important thermodynamic properties of solids. It elastic, anisotropic and thermodynamic characteristic of this in- is related to the temperature dependence of fundamental ther- teresting semiconductor material under high pressure and modynamic functions, and it is of key importance for linking temperature. thermodynamics with microscopic structure and dynamics. Here the two dimensional contour plots of the dependence of the heat capacity on pressure and temperature are displayed in Fig. 11,as Acknowledgments well as the heat capacity as a function of temperature at various pressures. It is obvious that the heat capacity at constant pressure This work was supported by the National Natural Science

(CP) and at constant volume (CV) increases sharply at To400 K, Foundation of China (No. 61474089), Open Fund of Key Laboratory 482 Q. Fan et al. / Journal of Solid State Chemistry 233 (2016) 471–483

C 18 V 30

16 0.000 Dulong-Petit limit 25 3.000 14 6.000 ) 12 20 -1 K

9.000 -1

10 12.00 15 (Jmol V C 8 15.00

Pressure (GPa) Pressure 18.00 10 6 21.00 4 Heat capacity 5 24.00 0 GPa 6 GPa 2 12 GPa 18 GPa 0 0 0 100 200 300 400 500 600 0 100 200 300 400 500 600 Temperature (K) Temperature (K) C P 18 30

0.000 16 3.488 25 14 6.975

12 10.46 20

13.95 10

17.44 15 p (J/mol/K) C 8 20.93

Pressure (GPa) 0 GPa 24.41 10 6 6 GPa 12 GPa

27.90 Heat capacity 4 18 GPa 5 2

0 0 0 100 200 300 400 500 600 0 100 200 300 400 500 600 Temperature (K) Temperature (K)

Fig. 11. Calculated speciﬁc volume CV and pressure heat capacity CP as a function of pressure for P2221-silicon at different temperature: CV contours (a), CV–T (b), CP contours

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