CHINESE JOURNAL OF PHYSICS

VOL. 44, NO. 6

DECEMBER 2006

Vibrational Properties of Si and Ge under High Pressures B. R. Wu∗ Center of General Education, Chang Gung University, Kwei-Shan, Tao-Yuan, Taiwan 333, R.O.C (Received July 24, 2006) Using a first-principles method, the vibrational properties of silicon and germanium under both ambient conditions and high pressure have been studied. The vibrational properties of silicon and germanium are very similar. The phonon frequencies of the first Brillouin zone center and zone boundary under ambient conditions agree very well with the previous experimental results. The frequencies of all modes exhibit blueshifts under high pressure except for the TA mode. The frequency of the TA mode exhibits a redshift under compression. The mode softening of the TA mode is related to the phase transformation of silicon and germanium from the cubic diamond phase to the β-Sn phase. The anharmonicity of the TA mode becomes more significant as the applied pressure increases, and it help to advance the phase transformation of silicon and germanium from the cubic diamond phase to the β-Sn phase. PACS numbers: 63.2.Dj, 64.70.Kb, 63.20.-e

I. INTRODUCTION

Silicon and germanium have been found to have many polymorphs. Most of the polymorphs of silicon and germanium are produced under high pressure [1]. That is just the reason why the high-pressure behavior of silicon and germanium has attracted attention and has been continuously studied over three decades. They exhibit remarkable similarities in their high pressure behavior. For the pressure-induced phase transitions, silicon and germanium show the following phase-transition sequence: cubic diamond (cd) → β-Sn → Imma → simple hexagonal (sh) → Cmca → hexagonal close packed (hcp). Silicon has a further hcp → face center cubic (fcc) transition. The transition pressures of Si are around 11 GPa for cd to β-Sn, 13 GPa for β-Sn to Imma, 15 GPa for Imma to sh, 38 GPa for sh to Cmca, 42 GPa for Cmca to hcp, and 80 GPa for hcp to fcc [2, 3]. The transition pressures of Ge are around 10 GPa for cd to β-Sn, 75 GPa for β-Sn to Imma, 85 GPa for Imma to sh, 100 GPa for sh to Cmca, and 170 GPa for Cmca to hcp [4, 5]. This reveals the much higher transition pressures of Ge than Si under high pressures, except for the transition pressure of the cd to β-Sn phases. The differences are caused by the core of Ge, which is larger and contains 3d electrons. It gives an additional interatomic repulsion and leads to higher transition pressures between the metallic phases. Among these pressure-induced phase transitions of Si and Ge, the most studied is the transition from the semiconducting phase to the metallic phase (cd → β-Sn). The in situ

http://PSROC.phys.ntu.edu.tw/cjp

454

c 2006 THE PHYSICAL SOCIETY

OF THE REPUBLIC OF CHINA

VOL. 44

B. R. WU

455

X-ray diffraction method is a widely used experimental method for studying the structural transformation and the equation of states of Si and Ge under high pressures [6, 7]. In addition to the X-ray diffraction method, Raman spectroscopy is also used to study the high pressure behavior of phonons at the center of the first Brillouin zone [6, 8, 9]. The phonon frequencies of Ge at the boundary of the first Brillouin zone are also investigated using the inelastic neutron scattering method under high pressure [10]. On the theoretical side, there are many ab initio studies focused on the calculations of the total energies and the enthalpy of several phases of Si and Ge, and the investigation of the transition pressures (for examples: Ref. [2], [4] and [11]). The Monte Carlo method and the molecular dynamic method are used to study the dynamics of the transition of the cd → β-Sn phases of Si [12]. The investigations of the high-pressure behavior of phonons of Si and Ge are less than the studies from the point of view of energies. Those studies are also limited to the phonon frequencies of the first Brillouin zone center mode of Si and Ge under compression [8, 13], while the study of phonons of the first Brillouin zone boundary under high pressures are not many [14]. To have a further understanding about the mechanism of the phase transition cd → β-Sn of Si and Ge, it is necessary to investigate the vibrational properties of the cubic diamond phase of Si and Ge under high pressure. In this paper, we will present a study of the vibrational properties at the first Brillouin zone center and zone boundary of the cubic diamond phase of Si and Ge under high pressure. As no data can be compared with the work on the high-pressure behavior of phonon frequencies of Si at the first Brillouin zone boundary, the phonon data at the first Brillouin zone boundary of Si at ambient pressure are also presented as a check of reliability.

II. CALCULATED METHODS

This study is based on the total energy calculations within the density functional theory [15] and the local density approximation [16]. The Ceperley-Alder form [17] is used for the exchange-correlation energy, and the norm-conserving pseudo-potential derived by Bachelet et al. [18] is taken for the silicon and germanium ion. The wavefunctions are expanded using plane waves and the calculations are performed in momentum space [19]. The k points chosen with the Monkhorst-Pack scheme [20] are applied to the k-point samplings of integration over the first Brillouin zone. The convergence for the k-point samplings and the energy cutoffs are tested: up to 343 and 1220 k points in the first Brillouin zone are made for Si and Ge, respectively, while up to 40 and 50 Rydbergs of the energy cutoffs corresponding to Si and Ge are carried out. The convergence tests of the energy cutoffs show that the error bars of the total energy (Etot ) is less than 7 and 5 meV/atom, and the error bars of the k-point samplings is less than 4 and 5 meV/atom for Si and Ge, respectively. In the calculation of the vibration frequencies, we use 64, 144, and 96 k points for both Si and Ge at the Γ, X, and L points in the first Brillouin zone, respectively, while the energy cutoff of 30 Ry is employed for both elements. The vibration frequencies are studied with the frozen phonon method [21]. We calculated the phonon frequencies using the displacement patterns corresponding to the snapshot of the

VIBRATIONAL PROPERTIES OF SI AND . . .

456

VOL. 44

TABLE I: The vibrational modes of silicon at ambient conditions. The mode Gr¨ uneisen parameter γ is defined as γi = −∂ ln ωi /∂ ln V . Vibration Frequency (cm−1 ) Mode Gr¨ uneisen parameter γ Modes This work Expt. Deviaton This work Expt. T2g (Γ) 517 520.2a, 523b , 519.5c 0.5∼1.1% 0.97 0.96d, 0.98c LOA (X) 411.0 415.0b, 417e 1% 0.98 TO (X) 462.2 456.3b 1.3% 1.51 1.5c TA (X) 141.2 152e 7.1% -1.66 -1.7f , -1.4c b LO (L) 412.4 411 0.3% 1.63 LA (L) 374.2 0.32 TO (L) 495.5 487b 1.7% 1.25 1.3c TA (L) 101.7 114.3e 11% -1.35 -1.3c a b c d e f Ref. [8] , Ref. [22] , Ref. [23] , Ref. [13] , Ref. [24] , Ref. [25] .

normal modes. We average the energy differences obtained from two opposite displacement directions in each pattern to eliminate the anharmonicity contributed by odd power terms (see Eq. (1)); then the lowest anharmonic term comes from the quartic term: ∆E =

1 1 1 K2 u2 + K3 u3 + K4 u4 + . . . . 2! 3! 4!

(1)

A very small displacement (u = 0.002d0 , where d0 is the equilibrium bond length) of the atom was chosen and the quartic term of Eq. (1) may be neglected. The phonon frequency is given by X 1 ∆E = Eu − E0 = mω 2 |ui |2 , 2 N

(2)

i

where Eu (E0 ) is the total energy of the distorted (undistorted) structure, and i runs over the atoms in the unit cell. The Gr¨ uneisen parameter of the ith mode is from the definition γi = −

V dωi , ωi dV

(3)

where ωi is the frequency of the ith vibration mode and V is the volume.

III. RESULTS AND DISCUSSION

We have studied the vibrational properties under ambient conditions and under high pressures. The cubic diamond structure is the most stable phase of silicon and germanium. For the cubic diamond structure, there is only one optical mode, T2g , in the Brillouin zone center; this mode is triply degenerate and is Raman-active. For the first Brillouin zone boundary modes, we present the phonon modes at the X and L points. At the X point, the

VOL. 44

B. R. WU

457

TABLE II: The vibrational modes of germanium at ambient conditions. Vibration Modes T2g (Γ) LOA (X) TO (X) TA (X) LO (L) LA (L) TO (L) TA (L) a Ref. [8] ,

Frequency (cm−1 ) Mode Gr¨ uneisen This work Expt. Deviation This work Theor. 298.4 300.7a, 304b 0.8%, 1.8% 1.02 240.3 240.3b 0.67 262.1 275.3b 4.8% 1.15 83.3f , 3.1%, -1.36f , 80.7 -1.48 b 80 0.9% -1.2g 237.4 244.7b 3% 1.49 b 223.6 222 0.7% 0.46 279.9 275.3b 1.7% 1.13 f b 63.4 65 , 63.3 2.5%, 0.1% -1.01 -1.18f b c d e f g Ref. [24] , Ref. [23] , Ref. [13] , Ref. [26] , Ref. [10] , Ref. [27] .

parameter γ Expt. 1.12c, 1.00d, 1.14e

-1.31f ,-0.35c,-0.23c , -0.38c, -1.53e

1.42c, 1.27c, 0.9c -1.52f ,-0.4c, -0.29c

longitudinal optical mode and the longitudinal acoustic mode are degenerate, this is called the LOA mode, and both the transverse optical (TO) mode and the transverse acoustic (TA) mode are doubly degenerate. At the L point in the first Brillouin zone, the transverse optical (TO) mode and the transverse acoustic (TA) mode are also doubly degenerate, and the longitudinal optical (LO) and the longitudinal acoustic (LA) modes are nondegenerate. Among these modes at the L point, the TA mode has the lowest phonon frequency. The phonon frequencies of Si and Ge under ambient conditions are given in Table I and II, respectively. Experimental values of the phonon frequencies of Si and Ge are also listed for comparison. Our calculated phonon frequencies for both Si and Ge agree with the experimental values very well. For Si and Ge, the phonon frequency of the T2g mode at the Brillouin zone center is the highest one; the phonon frequency of the TO mode is the highest and that of the TA mode is the lowest at both the X and L points. The TA mode at the L point has a lower frequency than at the X point for both Si and Ge. The frequency of the LO mode at the L point is slightly higher than that of the LOA mode at the X point for Si. Oppositely, the frequency of the LO mode at the L point is lower than that of the LOA mode at the X point, around 3 cm−1 for Ge. The high-pressure behavior of the vibrational properties at the first Brillouin zone center and zone boundary of Si and Ge are also presented. The mode Gr¨ uneisen parameters γi and the ω-P curves are studied, and the ω-P curves are also fitted to a quadratic polynomials, to have a quantitatively discussion and compare with the experimental results. The pressure P used in this paper is simply taken from the equation of state (EOS), which is the relation between pressure and volume under an isothermal process. The EOS of this calculation is obtained from the first derivative of the fitted total energy with respect to volume; the total energy is fitted to the Murnaghan equation of state [28]. The calculated mode Gr¨ uneisen parameters, γi of Si and Ge, are also shown in Table I and Table II, respectively. The calculated mode Gr¨ uneisen parameters of Si also agree with the experimental data very well. At the L point, the mode Gr¨ uneisen parameter of the LO mode is larger

458

VIBRATIONAL PROPERTIES OF SI AND . . .

VOL. 44

FIG. 1: Frequency of the T2g mode at the Brillouin zone center of silicon and germanium versus pressure. Solid and dashed lines are the results of this work. Open circles, triangles, and squares are the experimental results by Terrence and Liu [8], by Ulrich et al. [13], and by Olego and Cardonna [26], respectively.

than that of the TO mode, while the frequency of the LO mode is smaller than that of the TO mode. At the X point, the mode Gr¨ uneisen parameter and the frequency of the LOA mode are smaller than those of the TO mode, as the mixture of longitudinal and optical vibration characters lead to a lower frequency of the LOA mode. For germanium, it seems not easy to have good accuracy for measuring the phonon frequency by the inelastic neutron scattering method. For the TA modes at the X point, and the TO, TA modes at the L point, the mode Gr¨ uneisen parameters of germanium show large discrepancies from different experimental groups. Our result agrees with the experimental result of Klotz et al. [10], and also accords with their theoretical result for the TA mode at both the X and L points. However, the TA modes at the X and L points have negative mode Gr¨ uneisen parameters for both Si and Ge. This implies that there will be a phase transformation of Si and Ge under compression. The phonon frequencies under pressure are calculated up to 25 and 38 GPa for Si and Ge, respectively. The relationships of the mode frequencies versus the pressure at the Brillouin zone center of Si and Ge are drawn in Fig. 1, and the experimental results of previous studies [8, 13, 26] are also shown for comparison. The frequency shows a sublinear dependence on pressure, and exhibits blueshifts as the applied pressure increases for both Si and Ge. This accords with the results of previous studies by Terrence and Liu [8], by Ulrich et al. [13], and by Olego and Cardona [26]. How the mode frequencies at the X and

VOL. 44

B. R. WU

459

FIG. 2: Mode frequencies of silicon vary with pressure (a) at the X point, and (b) at the L point in the first Brillouin zone.

460

VIBRATIONAL PROPERTIES OF SI AND . . .

VOL. 44

TABLE III: The pressure coefficients (ω0 , ω1 , and ω2 ) of the phonon frequencies of silicon. The relation of the mode frequency and pressure exhibits sublinearly. The frequencies can be fitted to the quadratic polynomial of pressure (ω = ω2 P 2 + ω1 P + ω0 ). Units of pressure are in GPa, and the unit of frequency is cm−1 . Vibration Modes T2g (Γ)

(Theor.) (Expt.)

LOA (X) TO (X) TA (X) LO (L) LA (L) TO (L) TA (L) a Ref. [8] , b Ref. [23] , c Ref. [13] .

ω0 516.70 518.6a 519.5b 523.88c 411.3061 462.5954 139.7080 412.5352 374.4270 495.6129 99.8983

ω1 4.97 5.5a 5.2b 5.10c 3.9452 6.8412 -2.1302 6.3633 1.1400 5.8566 -1.1029

ω2 -0.0483 -0.0866a -0.07b -0.062c -0.0431 -0.0697 -0.1376 -0.0603 -0.0195 -0.0511 -0.1472

TABLE IV: The pressure coefficients of the phonon frequencies of germanium. The relation of the mode frequency and pressure exhibits sublinearly. The frequencies can be fitted to the quadratic polynomial of pressure (ω = ω2 P 2 + ω1 P + ω0 ). The unit of pressure is GPa, and the unit of frequency is cm−1 . Vibration Modes T2g (Γ)

(Theor.) (Expt.)

LOA (X) TO (X) TA (X) LO (L) LA (L) TO (L) TA (L) a Ref. [8] , b Ref. [25] ,

c

ω0 299.5820 302.6a 300.6b 304.64c 240.5578 266.0606 79.7157 239.0274 224.2422 281.4415 60.6569

ω1 3.4320 4.1a 3.85b 4.02c 2.0573 3.8974 -1.4266 4.4755 1.2987 4.0033 -0.4450

ω2 -0.0288 -0.07a -0.039b -0.059c -0.0218 -0.0458 -0.0321 -0.0433 -0.0152 -0.0387 -0.0326

Ref. [13] .

L points of Si vary with applied pressure are plotted in Fig. 2, these frequencies also all vary sublinearly with pressure. The frequencies of the LOA and TO modes at the X point (Fig. 2(a)) of Si show blueshifts as the pressure increases. The mode frequencies of the LOA and TO modes of Si shift from 411 and 462 cm−1 to 473 and 572 cm−1 , respectively. However, when the applied pressure increases, the frequency of the TA mode exhibits

VOL. 44

B. R. WU

461

redshifts. At the L point (Fig. 2(b)), all the mode frequencies of Si display blueshifts under pressure except for the TA mode. The mode frequencies of the LO and TO mode shift from 412 and 495 cm−1 to 515 and 592 cm−1 , respectively, as the pressure increases up to 20 GPa. The ω-P curve of the LA mode at the L point of Si is more flat than the curves of the other modes. Its frequency only shifts from 374 to 390 cm−1 as the pressure raises from ambient conditions to 20 GPa. The frequency of the TA mode of Si decreases under compression, and the frequency decreases to zero when the pressure reaches around 22 and 21 GPa at the X point and L point, respectively. Fig. 3 is the relationships of the mode frequency and pressure at the X and L points of Ge, and the experimental results [10] are also shown for comparison. Similar to the ω-P curves of Si, these curves also show sublinear characteristics, and all the mode frequencies exhibit blueshifts under compression, except for the TA mode at the first Brillouin zone boundary. These mode frequencies shift from 224-300 cm−1 to 251-346 cm−1 when the applied pressure is up to 34 GPa. The ω-P curve of the LA mode at the L points of Ge is also more flat than the other modes. The TA modes at the X and L points of Ge show redshifts under compression, that agrees very well with the previous experimental results study by Klotz et al. [10]. The frequency of the TA mode decrease to zero as the pressure increases up to around 28 and 34 GPa, corresponding to the TA modes at the X and L points. As all mode frequencies vary sublinearly with pressure, we fit the frequencies to a quadratic polynomial of pressure; the pressure coefficients are given in Table III and IV for Si and Ge, respectively. The pressure coefficients of the ω-P curve at the Γ point agree with the experimental results very well. The pressure coefficients of the ω-P curves are strongly mode dependent. At the L point of both Si and Ge, the linear pressure coefficient of the ω-P curve of the LO mode is greater than that of the TO mode, although the mode frequency of the LO mode is smaller than that of the TO mode. This indicates that the frequency of the LO mode is more sensitive to compressions than the TO mode. For the TO mode, the linear pressure coefficients of the ω-P curve at the X point is greater than at the L point, also the response of the frequency to pressure at the X point is larger than at the L point. From the relation of the linear pressure coefficients, we can see that the frequencies of the optical modes are more sensitive to pressure than those of the acoustic modes for both Si and Ge. The linear pressure coefficients of the ω-P curves of the phonon modes of Si are all larger than those of Ge, except for the LA mode at the L point. This implies that the mode frequencies of Si are more sensitive to pressure than those of Ge. Due to the mode softening of the TA modes, the linear pressure coefficients of the ω-P curves of the TA modes are negative. The TA mode at the X point also shows more sensitivity to pressure than at the L point. Fig. 4 and Fig. 5 correspond to the mode Gr¨ uneisen parameters γi of Si and Ge. The mode Gr¨ uneisen parameter of the T2g mode at the Γ point hardly changes under compressions. For the zone boundary modes, the mode Gr¨ uneisen parameters slightly decreased as the pressure increases, except for the TA modes. The TA modes have negative mode Gr¨ uneisen parameters, and the γT A drop rapidly under compression. For the TA modes of Si, the mode Gr¨ uneisen parameter at the L point drops more rapidly than at the X point. The two γT A -P curves at the L and X points cross around 5 GPa. For the

462

VIBRATIONAL PROPERTIES OF SI AND . . .

VOL. 44

FIG. 3: Mode frequencies of germanium vary with pressure (a) at the X point, and (b) at the L point in the first Brillouin zone. Open triangles are the experimental results by Klotz et al. [10].

VOL. 44

B. R. WU

463

FIG. 4: Mode Gr¨ uneisen parameters γ of silicon vary with pressure at the Γ, X, and L points in the first Brillouin zone.

TA modes of silicon and germanium, mode softening indicates that there will be a phase transformation under compression. Actually around 11 and 10 GPa, respectively, Si and Ge change their phase from a cubic diamond to a β-Sn structure. The vibration pattern of the TA mode at the L point shows the adjacent double layers moving along the < 111 > direction in opposite directions. The mode softening implies that the sliding of these planes become easier and easier as the applied pressure increases. Such a mode softening of the TA mode could induce a phase transformation for both Si and Ge from the cubic diamond phase to the β-Sn phase. The phonon frequency of the TA mode at the L point of Si will be reduced to zero around 21 GPa, this result accords with the previous study by Ga´al-Nagy et al. [14]. But they thought that the mode softening of the TA mode of Si has no relation with the phase transformation from the cubic diamond phase to the β-Sn phase; the reason being that the frequency of the TA mode does not reduce to zero around the transition pressure 11 GPa. However, we believe that the mode softening of the TA mode has an important relation with the phase transformation from the cubic diamond phase to the β-Sn phase, as it is not necessary for the frequency to be zero for a phase transformation to occurr. In the cases of III-V and II-VI compound semiconductors, one of the optical modes has mode softening under pressure and a phase transition occurs before the frequency of the optical mode reaches zero (see for example Ref. [29]). We have studied the total energy for various displacements of atoms of the TA mode, as shown in Fig. 6. As can be seen in Fig. 6, the energy increases as the ratio of the displacement (∆d) to bond length (d)

464

VIBRATIONAL PROPERTIES OF SI AND . . .

VOL. 44

FIG. 5: Mode Gr¨ uneisen parameters γ of germanium vary with pressure at the Γ, X, and L points in the first Brillouin zone.

increases, and the energy difference ∆E decreases at the same ∆d/d under compression. This reveals that as the pressure increases, the atoms move more easily. When the applied pressure is greater than 20.01 GPa, the total energy difference is negative, and the amount of the energy difference increases while the pressure increases. When the applied pressure is increased up to 20.01 GPa, the energy difference stays constant as ∆d/d is less than 2%. As the pressure is increased up to 22.11 GPa, the total energy decreases while ∆d/d increases, that is atoms of the two planes prefer to move to a new position. The slope of the curves decreases when the applied pressure increases. This indicates that the barrier becomes lower and lower as the pressure increases. At a temperature of 0◦ C, the thermal energy of each atom has 0.035 eV that can help atoms overcome the energy barrier, so the transition pressure shall be much lower than 22 GPa. The anharmonicity of the TA mode increases as the applied pressure increases. This implies that the anharmonicity of the TA mode promotes the phase transformation from the cubic diamond phase to the β-Sn phase for Si and Ge.

VOL. 44

B. R. WU

465

FIG. 6: Total energy differences (∆E = Eu − E0 ) of the TA mode at the L point in the first Brillouin zone of silicon versus the ratio of displacement to bond length. Here Eu (E0 ) is the total energy of the distorted (undistorted) structure, while ∆d is the displacement of the atom and d is the undistorted bond length. IV. CONCLUSIONS

Using a first-principles calculation, we have performed a study of the vibrational properties of Si and Ge under ambient conditions and under high pressure. The vibrational properties of silicon and germanium are quite similar. At ambient conditions, the phonon frequencies at the first Brillouin zone center and zone boundary agree with the experimental results very well. For the high-pressure behavior of the mode frequencies, the frequencies of all modes exhibit blueshifts as the applied pressure increases, except for the TA modes. Opposite to the other modes, the TA modes exhibit redshifts under compressions. The mode softening of the TA modes indicates that a phase transformation will occur. The anharmonicity of the TA modes under compression promotes the phase transformation of Si and Ge from the cubic diamond phase to the β-Sn phase.

Acknowledgments The author gratefully acknowledges the support from the National Science Council, R.O.C with Grant No. NSC-93-2112-M-182-003, and NSC-92-2112-M-182-005. We thank the National Center for High-Performance Computing, Hsin-Chu, Taiwan for providing part

466

VIBRATIONAL PROPERTIES OF SI AND . . .

VOL. 44

of the computing facilities.

References Electronic address: [email protected] [1] A. Mujica, A. Rubio, A. Mu˜ noz, and R. J. Needs, Rev. Mod. Phys. 75, 863 (2003), and references therein. [2] A. Mujica, S. Radescu, A. Mu˜ noz, and R. J. Needs, Phys. Stat. sol. (b) 223, 379 (2001). [3] M. I. McMahon and R. J. Nelmes, Phys. Rev. B 47, 8337 (1993). [4] A. Mujica, S. Radescu, A. Mu˜ noz, and R. J. Needs, J. Phys.: Condens. Matter 13, 35 (2001). [5] Y. K. Vohra, K. E. Brister, S. Desgreniers, and A. L. Ruoff, Phys. Rev. Lett. 56, 1944 (1986). [6] J. Z. Hu and I. L. Spain, Solid State Commun. 51, 263 (1984). [7] H. Olijnyk, S. K. Sikka, and W. B. Holzapfel, Phys. Lett. A 103, 137 (1984). [8] P. M. Terrence and L. Liu, J. Phys. Chem. Solids 52, 507 (1991). [9] Y. Gogotsi, G. Zhou, S. -S. Ku, and S. Cetinkunt, Semicond. Sci. Technol. 16, 345 (2001). [10] S. Klotz, J. M. Besson, M. Braden, K. Karch, P. Pavone, D. Strauch, and W. G. Marshall, Phys. Rev. Lett. 79, 1313 (1997). [11] H. Libotte and J. -P. Gaspard, Phys. Rev. B 62, 7110 (2000). [12] I. -H. Lee, J. -W. Jeong, and K. J. Chang, Phys. Rev. B 55, 5689 (1997). [13] C. Ulrich, E. Anastasssakis, K. Syassen, A. Debernardi, and Cardona, Phys. Rev. Lett. 78, 1283 (1997). [14] K. Ga´ al-Nagy, A. Bauer, M. Schmitt, K. Karch, P. Pavone, and D. Strauch, Phys. Stat. Sol. (b) 211, 275 (1999). [15] P. Hohenberg and W. Kohn, Phys. Rev. 136, B864 (1964). [16] W. Kohn and L. J. Sham, Phys. Rev. 140, A1133 (1965). [17] D. M. Ceperley and B. J. Alder, Phys. Rev. Lett. 45, 566 (1980). [18] G. B. Bachelet, D. R. Hamann, and M. Schl¨ uter, Phys. Rev. B 26, 4199 (1982). [19] J. Ihm, A. Zunger, and M.L. Cohen, J. Phys. C: Solid State Phys. 12, 4409 (1979). [20] H. J. Monkhorst and J. D. Pack, Phys. Rev. B 13, 5188 (1976). [21] K. Kunc and R. M. Martin, Phys. Rev. B 24, 2311 (1981). [22] J. Kukda, D. Strauch, P. Pavone, and Y. Ishii, Phys. Rev. B 50, 13347 (1994). [23] B. A. Weinstein and G. J. Piermarini, Phys. Rev. B 12, 1172 (1975), and references therein. [24] G. Nilsson and G. Nelin, Phys. Rev. B 6, 3777 (1972). [25] J. Prechtel, J. Kalus, E. Luscher, L. Pintschovius, and R. Ghoshi, Phys. Stat. Solidi (b) 93, 653 (1979). [26] D. Olego and M. Cardonna, Phys. Rev. B 25, 1151 (1982). [27] T. Soma, Y. Saitoh, and H. Matsuo, Solid State Commun. 39, 913 (1981). [28] F. D. Murnaghan, Proc. Natl. Acad. Sci. USA 30, 244 (1944). ¨ Ozg¨ ¨ ur, Y. I. Alivov, C. Liu, A. Teke, M. A. Reshchikov, S. Doan, V. Avrutin, S.-J. Cho, [29] U. and H. Morkoc, J. Appl. Phys. 98, 041301 (2005). ∗

VOL. 44, NO. 6

DECEMBER 2006

Vibrational Properties of Si and Ge under High Pressures B. R. Wu∗ Center of General Education, Chang Gung University, Kwei-Shan, Tao-Yuan, Taiwan 333, R.O.C (Received July 24, 2006) Using a first-principles method, the vibrational properties of silicon and germanium under both ambient conditions and high pressure have been studied. The vibrational properties of silicon and germanium are very similar. The phonon frequencies of the first Brillouin zone center and zone boundary under ambient conditions agree very well with the previous experimental results. The frequencies of all modes exhibit blueshifts under high pressure except for the TA mode. The frequency of the TA mode exhibits a redshift under compression. The mode softening of the TA mode is related to the phase transformation of silicon and germanium from the cubic diamond phase to the β-Sn phase. The anharmonicity of the TA mode becomes more significant as the applied pressure increases, and it help to advance the phase transformation of silicon and germanium from the cubic diamond phase to the β-Sn phase. PACS numbers: 63.2.Dj, 64.70.Kb, 63.20.-e

I. INTRODUCTION

Silicon and germanium have been found to have many polymorphs. Most of the polymorphs of silicon and germanium are produced under high pressure [1]. That is just the reason why the high-pressure behavior of silicon and germanium has attracted attention and has been continuously studied over three decades. They exhibit remarkable similarities in their high pressure behavior. For the pressure-induced phase transitions, silicon and germanium show the following phase-transition sequence: cubic diamond (cd) → β-Sn → Imma → simple hexagonal (sh) → Cmca → hexagonal close packed (hcp). Silicon has a further hcp → face center cubic (fcc) transition. The transition pressures of Si are around 11 GPa for cd to β-Sn, 13 GPa for β-Sn to Imma, 15 GPa for Imma to sh, 38 GPa for sh to Cmca, 42 GPa for Cmca to hcp, and 80 GPa for hcp to fcc [2, 3]. The transition pressures of Ge are around 10 GPa for cd to β-Sn, 75 GPa for β-Sn to Imma, 85 GPa for Imma to sh, 100 GPa for sh to Cmca, and 170 GPa for Cmca to hcp [4, 5]. This reveals the much higher transition pressures of Ge than Si under high pressures, except for the transition pressure of the cd to β-Sn phases. The differences are caused by the core of Ge, which is larger and contains 3d electrons. It gives an additional interatomic repulsion and leads to higher transition pressures between the metallic phases. Among these pressure-induced phase transitions of Si and Ge, the most studied is the transition from the semiconducting phase to the metallic phase (cd → β-Sn). The in situ

http://PSROC.phys.ntu.edu.tw/cjp

454

c 2006 THE PHYSICAL SOCIETY

OF THE REPUBLIC OF CHINA

VOL. 44

B. R. WU

455

X-ray diffraction method is a widely used experimental method for studying the structural transformation and the equation of states of Si and Ge under high pressures [6, 7]. In addition to the X-ray diffraction method, Raman spectroscopy is also used to study the high pressure behavior of phonons at the center of the first Brillouin zone [6, 8, 9]. The phonon frequencies of Ge at the boundary of the first Brillouin zone are also investigated using the inelastic neutron scattering method under high pressure [10]. On the theoretical side, there are many ab initio studies focused on the calculations of the total energies and the enthalpy of several phases of Si and Ge, and the investigation of the transition pressures (for examples: Ref. [2], [4] and [11]). The Monte Carlo method and the molecular dynamic method are used to study the dynamics of the transition of the cd → β-Sn phases of Si [12]. The investigations of the high-pressure behavior of phonons of Si and Ge are less than the studies from the point of view of energies. Those studies are also limited to the phonon frequencies of the first Brillouin zone center mode of Si and Ge under compression [8, 13], while the study of phonons of the first Brillouin zone boundary under high pressures are not many [14]. To have a further understanding about the mechanism of the phase transition cd → β-Sn of Si and Ge, it is necessary to investigate the vibrational properties of the cubic diamond phase of Si and Ge under high pressure. In this paper, we will present a study of the vibrational properties at the first Brillouin zone center and zone boundary of the cubic diamond phase of Si and Ge under high pressure. As no data can be compared with the work on the high-pressure behavior of phonon frequencies of Si at the first Brillouin zone boundary, the phonon data at the first Brillouin zone boundary of Si at ambient pressure are also presented as a check of reliability.

II. CALCULATED METHODS

This study is based on the total energy calculations within the density functional theory [15] and the local density approximation [16]. The Ceperley-Alder form [17] is used for the exchange-correlation energy, and the norm-conserving pseudo-potential derived by Bachelet et al. [18] is taken for the silicon and germanium ion. The wavefunctions are expanded using plane waves and the calculations are performed in momentum space [19]. The k points chosen with the Monkhorst-Pack scheme [20] are applied to the k-point samplings of integration over the first Brillouin zone. The convergence for the k-point samplings and the energy cutoffs are tested: up to 343 and 1220 k points in the first Brillouin zone are made for Si and Ge, respectively, while up to 40 and 50 Rydbergs of the energy cutoffs corresponding to Si and Ge are carried out. The convergence tests of the energy cutoffs show that the error bars of the total energy (Etot ) is less than 7 and 5 meV/atom, and the error bars of the k-point samplings is less than 4 and 5 meV/atom for Si and Ge, respectively. In the calculation of the vibration frequencies, we use 64, 144, and 96 k points for both Si and Ge at the Γ, X, and L points in the first Brillouin zone, respectively, while the energy cutoff of 30 Ry is employed for both elements. The vibration frequencies are studied with the frozen phonon method [21]. We calculated the phonon frequencies using the displacement patterns corresponding to the snapshot of the

VIBRATIONAL PROPERTIES OF SI AND . . .

456

VOL. 44

TABLE I: The vibrational modes of silicon at ambient conditions. The mode Gr¨ uneisen parameter γ is defined as γi = −∂ ln ωi /∂ ln V . Vibration Frequency (cm−1 ) Mode Gr¨ uneisen parameter γ Modes This work Expt. Deviaton This work Expt. T2g (Γ) 517 520.2a, 523b , 519.5c 0.5∼1.1% 0.97 0.96d, 0.98c LOA (X) 411.0 415.0b, 417e 1% 0.98 TO (X) 462.2 456.3b 1.3% 1.51 1.5c TA (X) 141.2 152e 7.1% -1.66 -1.7f , -1.4c b LO (L) 412.4 411 0.3% 1.63 LA (L) 374.2 0.32 TO (L) 495.5 487b 1.7% 1.25 1.3c TA (L) 101.7 114.3e 11% -1.35 -1.3c a b c d e f Ref. [8] , Ref. [22] , Ref. [23] , Ref. [13] , Ref. [24] , Ref. [25] .

normal modes. We average the energy differences obtained from two opposite displacement directions in each pattern to eliminate the anharmonicity contributed by odd power terms (see Eq. (1)); then the lowest anharmonic term comes from the quartic term: ∆E =

1 1 1 K2 u2 + K3 u3 + K4 u4 + . . . . 2! 3! 4!

(1)

A very small displacement (u = 0.002d0 , where d0 is the equilibrium bond length) of the atom was chosen and the quartic term of Eq. (1) may be neglected. The phonon frequency is given by X 1 ∆E = Eu − E0 = mω 2 |ui |2 , 2 N

(2)

i

where Eu (E0 ) is the total energy of the distorted (undistorted) structure, and i runs over the atoms in the unit cell. The Gr¨ uneisen parameter of the ith mode is from the definition γi = −

V dωi , ωi dV

(3)

where ωi is the frequency of the ith vibration mode and V is the volume.

III. RESULTS AND DISCUSSION

We have studied the vibrational properties under ambient conditions and under high pressures. The cubic diamond structure is the most stable phase of silicon and germanium. For the cubic diamond structure, there is only one optical mode, T2g , in the Brillouin zone center; this mode is triply degenerate and is Raman-active. For the first Brillouin zone boundary modes, we present the phonon modes at the X and L points. At the X point, the

VOL. 44

B. R. WU

457

TABLE II: The vibrational modes of germanium at ambient conditions. Vibration Modes T2g (Γ) LOA (X) TO (X) TA (X) LO (L) LA (L) TO (L) TA (L) a Ref. [8] ,

Frequency (cm−1 ) Mode Gr¨ uneisen This work Expt. Deviation This work Theor. 298.4 300.7a, 304b 0.8%, 1.8% 1.02 240.3 240.3b 0.67 262.1 275.3b 4.8% 1.15 83.3f , 3.1%, -1.36f , 80.7 -1.48 b 80 0.9% -1.2g 237.4 244.7b 3% 1.49 b 223.6 222 0.7% 0.46 279.9 275.3b 1.7% 1.13 f b 63.4 65 , 63.3 2.5%, 0.1% -1.01 -1.18f b c d e f g Ref. [24] , Ref. [23] , Ref. [13] , Ref. [26] , Ref. [10] , Ref. [27] .

parameter γ Expt. 1.12c, 1.00d, 1.14e

-1.31f ,-0.35c,-0.23c , -0.38c, -1.53e

1.42c, 1.27c, 0.9c -1.52f ,-0.4c, -0.29c

longitudinal optical mode and the longitudinal acoustic mode are degenerate, this is called the LOA mode, and both the transverse optical (TO) mode and the transverse acoustic (TA) mode are doubly degenerate. At the L point in the first Brillouin zone, the transverse optical (TO) mode and the transverse acoustic (TA) mode are also doubly degenerate, and the longitudinal optical (LO) and the longitudinal acoustic (LA) modes are nondegenerate. Among these modes at the L point, the TA mode has the lowest phonon frequency. The phonon frequencies of Si and Ge under ambient conditions are given in Table I and II, respectively. Experimental values of the phonon frequencies of Si and Ge are also listed for comparison. Our calculated phonon frequencies for both Si and Ge agree with the experimental values very well. For Si and Ge, the phonon frequency of the T2g mode at the Brillouin zone center is the highest one; the phonon frequency of the TO mode is the highest and that of the TA mode is the lowest at both the X and L points. The TA mode at the L point has a lower frequency than at the X point for both Si and Ge. The frequency of the LO mode at the L point is slightly higher than that of the LOA mode at the X point for Si. Oppositely, the frequency of the LO mode at the L point is lower than that of the LOA mode at the X point, around 3 cm−1 for Ge. The high-pressure behavior of the vibrational properties at the first Brillouin zone center and zone boundary of Si and Ge are also presented. The mode Gr¨ uneisen parameters γi and the ω-P curves are studied, and the ω-P curves are also fitted to a quadratic polynomials, to have a quantitatively discussion and compare with the experimental results. The pressure P used in this paper is simply taken from the equation of state (EOS), which is the relation between pressure and volume under an isothermal process. The EOS of this calculation is obtained from the first derivative of the fitted total energy with respect to volume; the total energy is fitted to the Murnaghan equation of state [28]. The calculated mode Gr¨ uneisen parameters, γi of Si and Ge, are also shown in Table I and Table II, respectively. The calculated mode Gr¨ uneisen parameters of Si also agree with the experimental data very well. At the L point, the mode Gr¨ uneisen parameter of the LO mode is larger

458

VIBRATIONAL PROPERTIES OF SI AND . . .

VOL. 44

FIG. 1: Frequency of the T2g mode at the Brillouin zone center of silicon and germanium versus pressure. Solid and dashed lines are the results of this work. Open circles, triangles, and squares are the experimental results by Terrence and Liu [8], by Ulrich et al. [13], and by Olego and Cardonna [26], respectively.

than that of the TO mode, while the frequency of the LO mode is smaller than that of the TO mode. At the X point, the mode Gr¨ uneisen parameter and the frequency of the LOA mode are smaller than those of the TO mode, as the mixture of longitudinal and optical vibration characters lead to a lower frequency of the LOA mode. For germanium, it seems not easy to have good accuracy for measuring the phonon frequency by the inelastic neutron scattering method. For the TA modes at the X point, and the TO, TA modes at the L point, the mode Gr¨ uneisen parameters of germanium show large discrepancies from different experimental groups. Our result agrees with the experimental result of Klotz et al. [10], and also accords with their theoretical result for the TA mode at both the X and L points. However, the TA modes at the X and L points have negative mode Gr¨ uneisen parameters for both Si and Ge. This implies that there will be a phase transformation of Si and Ge under compression. The phonon frequencies under pressure are calculated up to 25 and 38 GPa for Si and Ge, respectively. The relationships of the mode frequencies versus the pressure at the Brillouin zone center of Si and Ge are drawn in Fig. 1, and the experimental results of previous studies [8, 13, 26] are also shown for comparison. The frequency shows a sublinear dependence on pressure, and exhibits blueshifts as the applied pressure increases for both Si and Ge. This accords with the results of previous studies by Terrence and Liu [8], by Ulrich et al. [13], and by Olego and Cardona [26]. How the mode frequencies at the X and

VOL. 44

B. R. WU

459

FIG. 2: Mode frequencies of silicon vary with pressure (a) at the X point, and (b) at the L point in the first Brillouin zone.

460

VIBRATIONAL PROPERTIES OF SI AND . . .

VOL. 44

TABLE III: The pressure coefficients (ω0 , ω1 , and ω2 ) of the phonon frequencies of silicon. The relation of the mode frequency and pressure exhibits sublinearly. The frequencies can be fitted to the quadratic polynomial of pressure (ω = ω2 P 2 + ω1 P + ω0 ). Units of pressure are in GPa, and the unit of frequency is cm−1 . Vibration Modes T2g (Γ)

(Theor.) (Expt.)

LOA (X) TO (X) TA (X) LO (L) LA (L) TO (L) TA (L) a Ref. [8] , b Ref. [23] , c Ref. [13] .

ω0 516.70 518.6a 519.5b 523.88c 411.3061 462.5954 139.7080 412.5352 374.4270 495.6129 99.8983

ω1 4.97 5.5a 5.2b 5.10c 3.9452 6.8412 -2.1302 6.3633 1.1400 5.8566 -1.1029

ω2 -0.0483 -0.0866a -0.07b -0.062c -0.0431 -0.0697 -0.1376 -0.0603 -0.0195 -0.0511 -0.1472

TABLE IV: The pressure coefficients of the phonon frequencies of germanium. The relation of the mode frequency and pressure exhibits sublinearly. The frequencies can be fitted to the quadratic polynomial of pressure (ω = ω2 P 2 + ω1 P + ω0 ). The unit of pressure is GPa, and the unit of frequency is cm−1 . Vibration Modes T2g (Γ)

(Theor.) (Expt.)

LOA (X) TO (X) TA (X) LO (L) LA (L) TO (L) TA (L) a Ref. [8] , b Ref. [25] ,

c

ω0 299.5820 302.6a 300.6b 304.64c 240.5578 266.0606 79.7157 239.0274 224.2422 281.4415 60.6569

ω1 3.4320 4.1a 3.85b 4.02c 2.0573 3.8974 -1.4266 4.4755 1.2987 4.0033 -0.4450

ω2 -0.0288 -0.07a -0.039b -0.059c -0.0218 -0.0458 -0.0321 -0.0433 -0.0152 -0.0387 -0.0326

Ref. [13] .

L points of Si vary with applied pressure are plotted in Fig. 2, these frequencies also all vary sublinearly with pressure. The frequencies of the LOA and TO modes at the X point (Fig. 2(a)) of Si show blueshifts as the pressure increases. The mode frequencies of the LOA and TO modes of Si shift from 411 and 462 cm−1 to 473 and 572 cm−1 , respectively. However, when the applied pressure increases, the frequency of the TA mode exhibits

VOL. 44

B. R. WU

461

redshifts. At the L point (Fig. 2(b)), all the mode frequencies of Si display blueshifts under pressure except for the TA mode. The mode frequencies of the LO and TO mode shift from 412 and 495 cm−1 to 515 and 592 cm−1 , respectively, as the pressure increases up to 20 GPa. The ω-P curve of the LA mode at the L point of Si is more flat than the curves of the other modes. Its frequency only shifts from 374 to 390 cm−1 as the pressure raises from ambient conditions to 20 GPa. The frequency of the TA mode of Si decreases under compression, and the frequency decreases to zero when the pressure reaches around 22 and 21 GPa at the X point and L point, respectively. Fig. 3 is the relationships of the mode frequency and pressure at the X and L points of Ge, and the experimental results [10] are also shown for comparison. Similar to the ω-P curves of Si, these curves also show sublinear characteristics, and all the mode frequencies exhibit blueshifts under compression, except for the TA mode at the first Brillouin zone boundary. These mode frequencies shift from 224-300 cm−1 to 251-346 cm−1 when the applied pressure is up to 34 GPa. The ω-P curve of the LA mode at the L points of Ge is also more flat than the other modes. The TA modes at the X and L points of Ge show redshifts under compression, that agrees very well with the previous experimental results study by Klotz et al. [10]. The frequency of the TA mode decrease to zero as the pressure increases up to around 28 and 34 GPa, corresponding to the TA modes at the X and L points. As all mode frequencies vary sublinearly with pressure, we fit the frequencies to a quadratic polynomial of pressure; the pressure coefficients are given in Table III and IV for Si and Ge, respectively. The pressure coefficients of the ω-P curve at the Γ point agree with the experimental results very well. The pressure coefficients of the ω-P curves are strongly mode dependent. At the L point of both Si and Ge, the linear pressure coefficient of the ω-P curve of the LO mode is greater than that of the TO mode, although the mode frequency of the LO mode is smaller than that of the TO mode. This indicates that the frequency of the LO mode is more sensitive to compressions than the TO mode. For the TO mode, the linear pressure coefficients of the ω-P curve at the X point is greater than at the L point, also the response of the frequency to pressure at the X point is larger than at the L point. From the relation of the linear pressure coefficients, we can see that the frequencies of the optical modes are more sensitive to pressure than those of the acoustic modes for both Si and Ge. The linear pressure coefficients of the ω-P curves of the phonon modes of Si are all larger than those of Ge, except for the LA mode at the L point. This implies that the mode frequencies of Si are more sensitive to pressure than those of Ge. Due to the mode softening of the TA modes, the linear pressure coefficients of the ω-P curves of the TA modes are negative. The TA mode at the X point also shows more sensitivity to pressure than at the L point. Fig. 4 and Fig. 5 correspond to the mode Gr¨ uneisen parameters γi of Si and Ge. The mode Gr¨ uneisen parameter of the T2g mode at the Γ point hardly changes under compressions. For the zone boundary modes, the mode Gr¨ uneisen parameters slightly decreased as the pressure increases, except for the TA modes. The TA modes have negative mode Gr¨ uneisen parameters, and the γT A drop rapidly under compression. For the TA modes of Si, the mode Gr¨ uneisen parameter at the L point drops more rapidly than at the X point. The two γT A -P curves at the L and X points cross around 5 GPa. For the

462

VIBRATIONAL PROPERTIES OF SI AND . . .

VOL. 44

FIG. 3: Mode frequencies of germanium vary with pressure (a) at the X point, and (b) at the L point in the first Brillouin zone. Open triangles are the experimental results by Klotz et al. [10].

VOL. 44

B. R. WU

463

FIG. 4: Mode Gr¨ uneisen parameters γ of silicon vary with pressure at the Γ, X, and L points in the first Brillouin zone.

TA modes of silicon and germanium, mode softening indicates that there will be a phase transformation under compression. Actually around 11 and 10 GPa, respectively, Si and Ge change their phase from a cubic diamond to a β-Sn structure. The vibration pattern of the TA mode at the L point shows the adjacent double layers moving along the < 111 > direction in opposite directions. The mode softening implies that the sliding of these planes become easier and easier as the applied pressure increases. Such a mode softening of the TA mode could induce a phase transformation for both Si and Ge from the cubic diamond phase to the β-Sn phase. The phonon frequency of the TA mode at the L point of Si will be reduced to zero around 21 GPa, this result accords with the previous study by Ga´al-Nagy et al. [14]. But they thought that the mode softening of the TA mode of Si has no relation with the phase transformation from the cubic diamond phase to the β-Sn phase; the reason being that the frequency of the TA mode does not reduce to zero around the transition pressure 11 GPa. However, we believe that the mode softening of the TA mode has an important relation with the phase transformation from the cubic diamond phase to the β-Sn phase, as it is not necessary for the frequency to be zero for a phase transformation to occurr. In the cases of III-V and II-VI compound semiconductors, one of the optical modes has mode softening under pressure and a phase transition occurs before the frequency of the optical mode reaches zero (see for example Ref. [29]). We have studied the total energy for various displacements of atoms of the TA mode, as shown in Fig. 6. As can be seen in Fig. 6, the energy increases as the ratio of the displacement (∆d) to bond length (d)

464

VIBRATIONAL PROPERTIES OF SI AND . . .

VOL. 44

FIG. 5: Mode Gr¨ uneisen parameters γ of germanium vary with pressure at the Γ, X, and L points in the first Brillouin zone.

increases, and the energy difference ∆E decreases at the same ∆d/d under compression. This reveals that as the pressure increases, the atoms move more easily. When the applied pressure is greater than 20.01 GPa, the total energy difference is negative, and the amount of the energy difference increases while the pressure increases. When the applied pressure is increased up to 20.01 GPa, the energy difference stays constant as ∆d/d is less than 2%. As the pressure is increased up to 22.11 GPa, the total energy decreases while ∆d/d increases, that is atoms of the two planes prefer to move to a new position. The slope of the curves decreases when the applied pressure increases. This indicates that the barrier becomes lower and lower as the pressure increases. At a temperature of 0◦ C, the thermal energy of each atom has 0.035 eV that can help atoms overcome the energy barrier, so the transition pressure shall be much lower than 22 GPa. The anharmonicity of the TA mode increases as the applied pressure increases. This implies that the anharmonicity of the TA mode promotes the phase transformation from the cubic diamond phase to the β-Sn phase for Si and Ge.

VOL. 44

B. R. WU

465

FIG. 6: Total energy differences (∆E = Eu − E0 ) of the TA mode at the L point in the first Brillouin zone of silicon versus the ratio of displacement to bond length. Here Eu (E0 ) is the total energy of the distorted (undistorted) structure, while ∆d is the displacement of the atom and d is the undistorted bond length. IV. CONCLUSIONS

Using a first-principles calculation, we have performed a study of the vibrational properties of Si and Ge under ambient conditions and under high pressure. The vibrational properties of silicon and germanium are quite similar. At ambient conditions, the phonon frequencies at the first Brillouin zone center and zone boundary agree with the experimental results very well. For the high-pressure behavior of the mode frequencies, the frequencies of all modes exhibit blueshifts as the applied pressure increases, except for the TA modes. Opposite to the other modes, the TA modes exhibit redshifts under compressions. The mode softening of the TA modes indicates that a phase transformation will occur. The anharmonicity of the TA modes under compression promotes the phase transformation of Si and Ge from the cubic diamond phase to the β-Sn phase.

Acknowledgments The author gratefully acknowledges the support from the National Science Council, R.O.C with Grant No. NSC-93-2112-M-182-003, and NSC-92-2112-M-182-005. We thank the National Center for High-Performance Computing, Hsin-Chu, Taiwan for providing part

466

VIBRATIONAL PROPERTIES OF SI AND . . .

VOL. 44

of the computing facilities.

References Electronic address: [email protected] [1] A. Mujica, A. Rubio, A. Mu˜ noz, and R. J. Needs, Rev. Mod. Phys. 75, 863 (2003), and references therein. [2] A. Mujica, S. Radescu, A. Mu˜ noz, and R. J. Needs, Phys. Stat. sol. (b) 223, 379 (2001). [3] M. I. McMahon and R. J. Nelmes, Phys. Rev. B 47, 8337 (1993). [4] A. Mujica, S. Radescu, A. Mu˜ noz, and R. J. Needs, J. Phys.: Condens. Matter 13, 35 (2001). [5] Y. K. Vohra, K. E. Brister, S. Desgreniers, and A. L. Ruoff, Phys. Rev. Lett. 56, 1944 (1986). [6] J. Z. Hu and I. L. Spain, Solid State Commun. 51, 263 (1984). [7] H. Olijnyk, S. K. Sikka, and W. B. Holzapfel, Phys. Lett. A 103, 137 (1984). [8] P. M. Terrence and L. Liu, J. Phys. Chem. Solids 52, 507 (1991). [9] Y. Gogotsi, G. Zhou, S. -S. Ku, and S. Cetinkunt, Semicond. Sci. Technol. 16, 345 (2001). [10] S. Klotz, J. M. Besson, M. Braden, K. Karch, P. Pavone, D. Strauch, and W. G. Marshall, Phys. Rev. Lett. 79, 1313 (1997). [11] H. Libotte and J. -P. Gaspard, Phys. Rev. B 62, 7110 (2000). [12] I. -H. Lee, J. -W. Jeong, and K. J. Chang, Phys. Rev. B 55, 5689 (1997). [13] C. Ulrich, E. Anastasssakis, K. Syassen, A. Debernardi, and Cardona, Phys. Rev. Lett. 78, 1283 (1997). [14] K. Ga´ al-Nagy, A. Bauer, M. Schmitt, K. Karch, P. Pavone, and D. Strauch, Phys. Stat. Sol. (b) 211, 275 (1999). [15] P. Hohenberg and W. Kohn, Phys. Rev. 136, B864 (1964). [16] W. Kohn and L. J. Sham, Phys. Rev. 140, A1133 (1965). [17] D. M. Ceperley and B. J. Alder, Phys. Rev. Lett. 45, 566 (1980). [18] G. B. Bachelet, D. R. Hamann, and M. Schl¨ uter, Phys. Rev. B 26, 4199 (1982). [19] J. Ihm, A. Zunger, and M.L. Cohen, J. Phys. C: Solid State Phys. 12, 4409 (1979). [20] H. J. Monkhorst and J. D. Pack, Phys. Rev. B 13, 5188 (1976). [21] K. Kunc and R. M. Martin, Phys. Rev. B 24, 2311 (1981). [22] J. Kukda, D. Strauch, P. Pavone, and Y. Ishii, Phys. Rev. B 50, 13347 (1994). [23] B. A. Weinstein and G. J. Piermarini, Phys. Rev. B 12, 1172 (1975), and references therein. [24] G. Nilsson and G. Nelin, Phys. Rev. B 6, 3777 (1972). [25] J. Prechtel, J. Kalus, E. Luscher, L. Pintschovius, and R. Ghoshi, Phys. Stat. Solidi (b) 93, 653 (1979). [26] D. Olego and M. Cardonna, Phys. Rev. B 25, 1151 (1982). [27] T. Soma, Y. Saitoh, and H. Matsuo, Solid State Commun. 39, 913 (1981). [28] F. D. Murnaghan, Proc. Natl. Acad. Sci. USA 30, 244 (1944). ¨ Ozg¨ ¨ ur, Y. I. Alivov, C. Liu, A. Teke, M. A. Reshchikov, S. Doan, V. Avrutin, S.-J. Cho, [29] U. and H. Morkoc, J. Appl. Phys. 98, 041301 (2005). ∗