All-Electron GW Calculations of Silicon, Diamond, and Silicon Carbide

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1 Materials Transactions, Vol. 51, No. 12 (2010) pp to 2156 #2010 The Japan Institute of Metals All-Electron GW Calculations of Silicon, Diamond, and Silicon Carbide Soh Ishii, Shohei Iwata and Kaoru Ohno Department of Physics, Yokohama National University, Yokohama , Japan All-electron GW calculations of Si, diamond, and SiC were carried out to obtain the quasiparticle energy spectra, for the first time, including both valence and core levels, corresponding to the ultraviolet and X-ray photoemission spectra (UPS and XPS). We used an allelectron mixed basis approach, in which wave functions are expanded with both plane waves and numerical atomic orbitals. In particular for the core states, our results obtained using the perturbative (non-self-consistent) GW approach are in good agreement with available experimental data. The quasiparticle energies of core levels so obtained are also in good agreement with experiments when we do not take into account the renormalization factor z. Moreover, we refer to the bandwidth of diamond and compare with experiments. [doi: /matertrans.m ] (Received September 6, 2010; Accepted September 28, 2010; Published November 25, 2010) Keywords: first principles calculation, semiconductor, electronic structure, energy band, excitation energy, GW approximation 1. Introduction The first principles calculations on the basis of the density functional theory (DFT) 1) and the local density approximation () 2) were extensively used in the solid state physics. They are capable to correctly predict the ground state properties of variety of materials. In spite of this great success, the eigenvalues are not capable to predict the energy band gap of insulators correctly. The typically underestimates experimental band gaps by about 30%50%. In order to correctly determine quasiparticle energy spectra of materials, one may look for the poles of the oneparticle Green s function calculated in the framework of the many-body perturbation theory. The GW approximation () is the first order approximation of the electron selfenergy with respect to the dynamically screened Coulomb interaction, W. In this approximation, the self-energy operator can be symbolically expressed by igw. 3) This method has been extended to the ab initio calculations, 4,5) and applied to a variety of real systems including crystals, surfaces and clusters. 6) The results are successfully compared with experiments. These ab initio GW calculations were mainly based on the pseudopotential approach combined with plane-wave expansion method 4,5) or the linear combination of atomic orbitals (LCAO) using Gaussian-orbitals. 7) Recently all-electron GW calculations on the basis of the full-potential linear muffin-tin orbital (FLMTO) approach, the full-potential augmented plane wave (FLAPW) approach and the projector augmented wave (PAW) approach appeared in the investigation of semiconductors. 8 10) All the results have been, however, restricted to the valence electrons only, and no result has been reported on the quasiparticle energies of core electron levels. Only exception is the semicore calculation by means of the pseudopotential Gaussian-orbital approach. 11) Instead, from the viewpoint of the all-electron calculation, it was pointed out that there is a systematic error in the previous pseudopotential GW calculations due to the ignorance of the core-valence exchange diagram. 9,12,13) Moreover, it was claimed that self-consistent GW calculations are required to obtain good quasiparticle energies compared with experiments. 9,12,13) The aim of the present paper is to show that the non-selfconsistent treatment of the all-electron GW calculations can give satisfactory results for the quasiparticle energy spectra including not only the valence but also the core levels of typical semiconductors, silicon, diamond and silicon carbide. As will be shown below, our perturbative (non-self-consistent) treatment can provide good quasiparticle energies compared with experiments. Moreover, recently Yokoya et al. measured experimental bandwidth of heavily boron-doped superconducting diamond by using angle resolved photoelectron spectroscopy (ARPES), comparing experiments with band calculations. 14) According to their results, bandwidth is smaller than that of experiments by about 10%. We discuss the bandwidth of diamond in section IV. In the present calculations, we use the all-electron mixed basis approach, in which one particle wave function is expanded both plane waves (PWs) and numerical atomic orbitals (AOs). 15,16) We have successfully applied this approach to the all-electron GW calculations of quasiparticle energies of molecules and clusters ) Although Ref. 20) deals with GaAs crystal as well as GaAs clusters, this is the first time report on the all-electron mixed basis GW calculations of energy bands and core levels of typical semiconductor crystals. 2. Methodology In our all-electron mixed basis approach, the AOs are generated by a modified Herman Skillman s code 22) which uses logarithmic mesh in radial direction and by smoothly cutting the tail within the non-overlapping atomic sphere. This approach can well describe both the spatially extended states and the localized states with relatively small number of basis functions. 15,16) In the present work, we perform both the k-point sampling and the q-point sampling in our allelectron GW code 17 20) as described below. Our formulation of the ab-initio all-electron is essentially based on the paper by Hybertsen and Louie, 4) who used the pseudopotential approach. We start from a calculation at the level and determine the wave functions

2 All-Electron GW Calculations of Silicon, Diamond, and Silicon Carbide 2151 ji and the energy eigenvalues ". In the many-body perturbation theory, the quasiparticle energies E qp can be obtained by solving the Dyson equation Z ðt þ V nuc þ V H Þjiþ dr 0 ðr; r 0 ; E qp Þji ¼ E qp ji: ð1þ Here, T, V nuc, V H, and are the kinetic energy operator, the nucleus Coulomb potential, Hartree potential, and the selfenergy operator, respectively. This self-consistent equation is correct if ðr; r 0 ;!Þ varies slowly with! over the corresponding small range around! ¼ E qp.21) Here, we employ an approximation, where the self-energy operator is given by ðr; r 0 ;!Þ ¼ i Z 1 d! 0 Gðr; r 0 ;! þ! 0 ÞWðr; r 0 ;! 0 Þe i!0 ; ð2þ 2 1 by ignoring the 3-point vertex function. Here, G, W, and denote the one-particle Green s function, the dynamically screened Coulomb interaction, and a positive infinitesimal number, respectively. Usually, undressed Green s function, G 0, and undressed dynamically screened interaction, W 0, are used instead of dressed ones, G and W, respectively. In what follows we shall not distinguish G and G 0, and W and W 0. W 0 is evaluated within the random phase approximation (RPA) as is usual. 4) In the present study, we use the eigenvalue, " n 0 k, and wave function, n 0 kðrþ to evaluate G: Gðr; r 0 ;!Þ¼ X n 0 kðrþ n 0 k ðr0 Þ ; ð3þ! " n 0 k n 0 k i n 0 k where n 0 ¼ 0 þ for " n 0 < F and n 0 ¼ 0 for " n 0 > F, provided that F is the Fermi energy. The dynamically screened Coulomb interaction in Fourier space is related to the dielectric matrix by W GG 0ðq;!Þ¼½" 1 Š GG 0ðq;!Þvðq þ G 0 Þ; where vðq þ GÞ ¼4=jq þ Gj 2 is the Coulomb potential in Fourier space ( is the volume of the unit cell), and " GG 0ðq;!Þ is the dielectric matrix defined by " GG 0ðq;!Þ¼ GG 0 vðq þ GÞP GG 0ðq;!Þ; ð5þ with the polarizability function within the RPA: P GG 0ðq;!¼ 0Þ ¼ X nn 1 khje iðqþgþr jn 1 k þ qihn 1 k þ qje iðqþg0 Þr 0 ji ð4þ f ð" n 1 kþqþ f ð" Þ " n1 kþq " ; ð6þ where G and G 0 are reciprocal lattice vectors, and f ð"þ denotes the Fermi-Dirac distribution function. The matrix elements in the numerator involve the intermediate states jn 1 k þ qi and the summation over k should be performed in the whole Brillouin zone. We evaluate this polarizability function for many different q points; for details, see below. The Fourier transform of eq. (4) to real space is then, Wðr; r 0 ;!Þ ¼ X X e iðqþgþr W G;G 0ðq;!Þe iðqþg 0 Þr 0 : ð7þ q G;G 0 We divide the dynamically screened Coulomb interaction W into two parts as W ¼ v þ½w vš, where v is the bare Coulomb interaction. Then the self-energy operator is also divided into two parts: One is the Fock exchange term given by x ðr; r 0 Þ¼ i Z 2 vðr r0 Þ e i!0 Gðr; r 0 ;! 0 Þd! 0 : ð8þ and the other is the correlation term given by c ðr; r 0 ;!Þ ¼ i Z d! 0 e i!0 Gðr; r 0 ;! þ! 0 Þ½Wðr; r 0 ;! 0 Þ vðr r 0 ÞŠ: ð9þ 2 The expectation values of the Fock exchange contribution are given by x; ¼hj x ðr; r 0 Þji ¼ Xocc X hje iðqþgþr jn 1 k qihn 1 k qje iðqþgþr0 jivðq þ GÞ; ð10þ n 1 qg where the summation with respect to n 1 should include core levels, while for the correlation part of the self-energy, the generalized plasmon-pole (GPP) model 4) is used to bypass the calculation of the! dependence of the dielectric matrices and the! 0 -integration in eq. (2). Then the expectation values of the correlation part are given by c; ðeþ ¼hj c ðr; r 0 ; EÞji ¼ Xocc X X hje iðqþgþr jn 1 k qihn 1 k qje iðqþg 0 Þr 0 ji 1 2 GG 0ðqÞ n 1 q 2 ~! GG 0 GG 0ðqÞ½E " n1 k q þ ~! GG 0ðqÞŠ vðq þ G0 Þ þ Xemp X X hje iðqþgþr jn 1 k qihn 1 k qje iðqþg 0 Þr 0 ji 1 2 GG 0ðqÞ n 1 q 2 ~! GG 0 GG 0ðqÞ½E " n1 k q ~! GG 0ðqÞŠ vðq þ G0 Þ; ð11þ where ~! GG 0 and 2 GG 0ðqÞ are the same functions as those defined in the paper by Hybertsen and Louie. 4) Here we note that the summation with respect to n 1 in the first term includes only valence levels and that the valence electron density (not the all electron density) should be used to determine these quantities because the screening is caused by the valence electrons only. The matrix elements in the numerator of (6), (10), and (11) involve the intermediate states jn 1 k þ qi and jn 1 k qi; and a summation over whole Brillouin zone with respect to k is needed in (6), while a summation over whole Brillouin zone with respect to q is, in principle, needed in (10) and (11). The latter q-point

3 2152 S. Ishii, S. Iwata and K. Ohno silicon Table 1 The eigenvalues " and the quasiparticle energies E (or ) evaluated in the vicinity of the band gap of silicon (in units of ev). Contributions to the quasiparticle energies are also shown: xc; ¼hj xc ji, x; ¼hj x ji, and c; ð" Þ¼hj cð" Þji are the expectation values of the exchange-correlation potential, the exchange part [eq. (10)] and the correlation part [eq. (11)] of the self-energy operator, respectively. There is no meaning in the absolute values of " and E. " xc; x; c;n ð" Þ E 0 2c :04 9:55 3: c :69 7:17 3: v :58 12:60 0: v 4:53 12:32 17: :25 4:32 X 4c :46 6:08 5: X 1c :12 6:18 3: X 4v :51 12: X 1v 0:37 13:06 15: :16 0:24 L 3c :87 6:00 3: L 1c :59 8:13 3: L 0 3v :27 12:81 0: L 1v :00 14: L 0 2v 2:00 12:97 16: :80 1:85 Si2p 83:48 46:95 65: :91 91:36 Si2s 125:94 45:99 71: :64 141:37 sampling can, however, be restricted only inside the irreducible wedge with the 1/48 (or 1/24 for SiC) volume of the whole zone, if we average the degenerate complex of the final quasiparticle energies over the levels to be degenerate and over the stars of the symmetry k point under consideration in (10) and (11), as first pointed out in Ref. 4). Finally, the quasiparticle energies are obtained in terms of the first-order perturbation theory as or E " " þhjð" Þ xc ji; ð12þ þ z " hjð" Þ xc ji; ð13þ z " ¼½1 ð@ð!þ=@!þ " Š 1 : ð14þ In the present study, although we employ both eq. (12) and eq. (13) to evaluate the QP energies, the resulting quasiparticle energies do not change significantly for valence and conduction bands. In such cases, it only makes all quasiparticle energies shift constantly in the same direction. However, for core energy levels, eq. (12) and eq. (13) lead to different results (see Sec. 4). 3. Conditions of Calculations In the present calculation, we assume the smallest fcc unit cell of the edge length of 3.84 Å, 2.52 Å, and 3.08 Å, respectively, for Si, diamond, and SiC inside which two atoms are located. We use 10 (for silicon), 8 (for diamond), and 9 (for silicon carbide) numerical AOs and 1471 PWs corresponding to the cutoff energy of 48 Ry (for silicon), 111 Ry (for diamond), and 75 Ry (for silicon carbide). For the evaluation of P GG 0ðq;!¼ 0Þ and c;, 120 (for silicon) or 100 (for diamond and silicon carbide) states are used in the summation over n (and n 1 ) in eqs. (6) and (11), and 339 GðG 0 Þ corresponding to the cutoff energy of 19 Ry (for silicon), 43 Ry (for diamond) and 29 Ry (for silicon carbide) are used. On the other hand, in the calculation of x;, G corresponding to the cutoff energy of 431 Ry (for silicon), 1001 Ry (for diamond) and 673 Ry (for silicon carbide) are used to take into account correctly the core contribution. For level calculations, we use two special k points to achieve a convergence within 0.01 ev. On the other hand, for the calculation of " GG 0ðq; 0Þ, x; and c; ð" Þ, we use 23 (for silicon), 13 (for diamond), and 25 (for silicon carbide) q points including point in the irreducible wedge of the Brillouin zone (for the point q ¼ 0, we used the treatment described in Appendix B of Ref. 4)), and 32 k points including point in the whole Brillouin zone. 4. Results and Discussion Tables 1, 2, and 3 list the separate contributions to the GW quasiparticle energies and the final result E or calculated with eq. (12) or eq. (13) for several valence and conduction levels at the, X, and L points of silicon, diamond, and silicon carbide, respectively. The listed contributions, ", xc;, x;, and c; ð" Þ represent the expectation values of the (Kohn-Sham) Hamiltonian, the exchange-correlation potential, and the exchange ( x ) and correlation ( c ) parts of the selfenergy operator, respectively. In the present case, we do not have any meaning in the absolute values of E as well as " because an ambiguity exists in the value of energy zero. As for the level symbols, 0 25v is the top of the valence band, X 1c is located near the bottom of the conduction band. From these tables, we find that the correction to the, j x; þ c; ð" Þ xc;j is at most 1.5 ev for valence levels although it is large for core levels. This means that the for silicon and diamond crystals is not so bad compared to the case of clusters ) Also we find that, for the 2p and 2s core levels, the values at different k points are the same up to the two digits in ev, i.e., they have almost no dispersion.

4 diamond Table 2 The eigenvalues " and the quasiparticle energies E (or ) estimated at the levels in the vicinity of the band gap of diamond (in units of ev). Contributions to the quasiparticle energies are also shown: xc; ¼hj xc ji, x; ¼hj x ji, and c; ð" Þ¼hj cð" Þji are the expectation values of, respectively, the exchange-correlation potential, the exchange part [eq. (10)] and the correlation part [eq. (11)] of the self-energy operator. There is no meaning in the absolute values of " and E. " All-Electron GW Calculations of Silicon, Diamond, and Silicon Carbide 2153 xc; x; c;n ð" Þ E 0 2c :60 10:86 6: c :74 10:14 3: v :06 16:46 0: v 7:93 16:19 22: :57 8:47 X 1c :85 8:91 3: X 4v :77 17: X 1v :97 20: L 1c :06 11:00 4: L 3c :34 8:92 4: L 0 3v :57 17: L 1v :57 18: :17 0:15 L 0 2v 2:07 17:27 22: :50 2:38 C1s 249:82 57:96 92: :13 266:37 SiC Table 3 The eigenvalues " and the quasiparticle energies E (or ) estimated at the levels in the vicinity of the band gap of silicon carbide (in units of ev). Contributions to the quasiparticle energies are also shown: xc; ¼hj xc ji, x; ¼hj x ji, and c; ð" Þ¼hj cð" Þji are the expectation values of, respectively, the exchange-correlation potential, the exchange part [eq. (10)] and the correlation part [eq. (11)] of the self-energy operator. There is no meaning in the absolute values of " and E. " xc; x; c;n ð" Þ E 15c :35 8:01 4: c :42 9:33 4: v :15 15: v 4:44 14:48 19: :40 4:41 X 3c :84 7:36 3: X 1c :66 7:14 2: X 5v :24 15: X 3v :86 16: X 1v :24 20: L 1c :21 6:03 4: L 3c :92 6:73 4: L 1c :72 8:91 4: L 3v :05 15: L 1v :05 16: L 1v 0:73 15:77 20: :63 0:66 Si2p 80:01 46:99 64: :46 84:93 Si2s 128:72 47:09 72: :16 133:67 C1s 253:13 58:18 93: :09 268:13 We show the band structures of Si and diamond in Fig. 1 and Fig. 2. The band gaps of silicon and diamondwithin the () are 0.43 ev (1.19 ev) and 4.0 ev (5.0 ev) respectively. The corresponding experimental energy gaps are 1.17 ev and 5.48 ev, respectively. The present improves the eigenvalues. The present GW calculations agrees well with other GW calculations. As for valence bands, the systematically enlarges bandwidth of the. In Tables 4, 5, and 6, we compare our results of silicon, diamond, and silicon carbide with the preexisting results using the pseudopotential plane-wave approach 4) and the pseudopotential Gaussian-orbital approach 7,11) (and also the PAW approach 10) and the FLMTO approach 8) for silicon) as well as experiments ) The final results indicated by E and are fairly in good agreement with previous calculations and experiments as for valence and conduction bands. As for core energy levels, the difference between quasiparticle energies with and without the renormalization factor z is larger than that of valence and conduction bands. For example, in the case of diamond, the corresponding value of 1s level is about 5 ev. Equation (12), at least phenomenologically, leads to better results compared to eq. (13).

5 2154 S. Ishii, S. Iwata and K. Ohno Fig. 1 Band structure of silicon in ev. Triangle and circle show the eigenvalues and GW quasiparticle energies, respectively. In the figure, we take the top of valence bands as a reference value. See also Table 4. Fig. 2 Band structure of diamond in ev. White circle and red circle show the eigenvalues and GW quasiparticle energies, respectively. Although we show the absolute value of the eigenvalues and GW quasiparticle energies, in this case we do not have any physical meanings. See also Table 5. silicon Table 4 Quasiparticle energies at symmetry points of silicon in units of ev. For each symmetry point, the values of,, and experiments (Expt.) are listed. In the column of, the symbols, PAW, PPPW, and PPGO denote, respectively, the previous theoretical results using the PAW approach, 10Þ the pseudopotential plane-wave approach 4Þ and the pseudopotential Gaussian-orbital approach. 7Þ The present results are indicated as E and. The top of the valence bands is taken to be zero as a reference value. PAW 10Þ FLMTO 8Þ PPPW 4Þ PPGO 7Þ E Expt. 23Þ 0 2c , Þ 15c , Þ 0 25v v 11:96 11:85 12:1 12:04 12:15 11:95 12:13 12:5 0:6 X 4c X 1c , Þ X 4v 2:89 2:9 2:99 2:93 3:14 3:08 2:9, 25Þ 3:3 0:2 26Þ X 1v 7:79 7:74 7:95 8:06 8:04 L 3c :15 0:1 27Þ L 1c , 28Þ 2:4 0:15 27Þ L 0 3v 1:22 1:16 1:27 1:25 1:25 1:23 1:2 0:2, 1:5 29Þ L 1v 7:01 6:97 7:18 7:14 7:31 7:26 6:7 0:2 L 0 2v 9:43 9:57 9:79 9:70 9:70 9:65 9:3 0:4 Si2p 90:91 94:8 11Þ 100:81 99: Þ Si2s 133:27 152:54 149: Þ Let us discuss the reason in terms of the Ward identities, which is exact and related to renormalized Green s function and vertex function. In general, the generalized Ward identity 34) is known but, for simplicity, we focus on a Ward identity (several Ward identities are known) expressed as z 1 q ¼ ðq ¼ 0;!! 0Þ; ð15þ because other Ward identities are not directly related to the present study. This equation is exact in the limit q ¼ 0 and then!! 0. It is easy to find that z and cancel each other when evaluating the quasiparticle energies because the electron self-energy operator is exactly defined by ¼ igw. In other words, that the effect of z becomes large means that the effect of vertex function also becomes large. However, all the GW calculations including present calculations do not take into account the effect of because correct behaviour of is not yet known and such a kind of calculations is no longer GW calculation. Our calculated results with and without z show that the inclusion of only z is not enough and we may have to employ in the calculation. As a result, this kind of cancellation error may occur when evaluating the quasiparticle energies of core levels within the GW approximation. On the contrary, in valence levels, z does not contribute quasiparticle energies significantly.

6 Table 5 Quasiparticle energies at symmetry points of diamond in units of ev. For each symmetry point, the values of, and experiments (Expt.) are listed. In the column of, the symbols, PPPW and PPGO denote, respectively, the previous theoretical results using the pseudopotential plane-wave approach 4Þ and the pseudopotential Gaussian-orbital approach. 7Þ The present results are indicated as E and. diamond All-Electron GW Calculations of Silicon, Diamond, and Silicon Carbide 2155 PPPW 4Þ PPGO 7Þ E Expt. 23Þ 0 2c :3 0:5 31Þ 15c v v 21:32 23:0 22:80 23:03 22:75 24:2 1, 21 1 X 1c X 4v 6:31 6:88 6:69 6:81 X 1v 12:60 13:80 14:09 13:95 L 1c L 3c L 0 3v 2:80 3:02 3:06 3:06 L 1v 13:39 14:4 14:27 14:63 14:48 12:8 0:3 31Þ L 0 2v 15:46 17:3 16:95 16:96 16:71 15:2 0:3 31Þ C1s 263:21 285:59 280: Þ SiC Table 6 Quasiparticle energies at symmetry points of silicon carbide in units of ev. For each symmetry point, the values of,, and experiments (Expt.) are listed. In the column of, the symbols, PAW and PPGO, denote, respectively, the previous theoretical results using the PAW approach 10Þ and the pseudopotential Gaussian-orbital approach. 7Þ The present results are indicated as E and. The top of the valence bands is taken to be zero as a reference value. PAW 10Þ PPGO 7Þ E Expt. 23Þ 15c c v v 14:93 16:44 16:03 15:88 X 3c , 4.7 X 1c X 5v 3:19 3:62 3:54 3:49 3:6 X 3v 7:72 8:64 8:25 8:20 X 1v 9:76 11:24 10:68 10:57 L 1c L 3c L 1c L 3v 1:04 1:21 1:05 1:14 1:16 L 1v 8:56 9:42 9:09 9:04 L 1v 11:22 12:75 12:26 12:13 Si2p 90:73 98:09 96:40 100:45 33Þ Si2s 138:41 150:79 145:14 151:7 33Þ C1s 263:48 285:72 279:60 283:3 33Þ Next, let us discuss the validity of the RPA in core region. The is essentially equivalent to the RPA in the correlation term, c. The RPA is exact at long range limit and therefore in the vicinity of nuclei the RPA is not correct in principle even if the local-field corrections in Ref. 4) are taken into account. Note that the local-field corrections in Ref. 4) and local-field effect in the electron liquid theory are completely different. The former is within the RPA and the latter corresponds to vertex corrections, namely, going beyond the RPA. Nevertheless, the present calculations give fairly good results compared with experiments. In principle, we have to employ some approximations beyond the RPA, namely, vertex corrections for W given by eq. (4) in the calculations of core levels. Unfortunately, at the moment, we do not know whether the present good results comes from the invalidity of the RPA because the vertex corrections of the GW self-energy also play a role. Finally, let us refer to the band width of diamond. Recently, Yokoya et al. measured experimental bandwidth of heavily boron-doped superconducting diamond by using angle resolved photoelectron spectroscopy (ARPES), comparing experiments with band calculations. 14) They

7 2156 S. Ishii, S. Iwata and K. Ohno proposed that bandwidth is smaller than that of experiments by about 10%. Our results of the bandwidth is about 21.3 ev, comparable with other calculations within the. On the other hand, the present GW approach enlarges the bandwidth by about 10%. It seems that the present GW calculations improves the band structure. 5. Conclusions We have newly implemented the k-point and q-point samplings in the all-electron mixed basis GW code and applied it to silicon, diamond, and silicon carbide crystals. We have solved the Dyson equation first iteratively within the. In the calculations, we have also employed the generalized plasmon-pole model to skip the numerical integration with frequency. The obtained quasiparticle energies at both valence bands and conduction bands are in good agreement with other ab-initio GW calculations and experiments. We have also performed the GW calculations of core energy levels. The present GW calculations significantly improved the eigenvalues, although the RPA is theoretically invalid in this region. At the core energy levels, the quasiparticle energies strongly depend on the renormalization factor z: quasiparticle energies obtained via eq. (12) are in much better agreement with experiment, compared with the corresponding values via eq. (13). The present GW calculations also reproduced the bandwidth of valence bands of diamond. Acknowledgments This work has been partially supported by the Grant-in-Aid for Scientific Research on Priority Areas (No ) from the Ministry of Education, Culture, Sports, Science and Technology and for Scientific Research B (No ) from Japan Society for the Promotion of Science. The authors acknowledge the support of the HITACHI SR11000 supercomputing facilities at the Information Initiative Center of Hokkaido University. REFERENCES 1) P. Hohenberg and W. Kohn: Phys. Rev. 136 (1964) A864 A871. 2) W. Kohn and L. J. Sham: Phys. Rev. 140 (1965) A1133 A ) L. Hedin: Phys. Rev. 139 (1965) A796 A828. 4) M. S. Hybertsen and S. G. Louie: Phys. Rev. B 34 (1986) ) R. W. Godby, M. Schlüter and L. J. Sham: Phys. Rev. Lett. 56 (1986) ) See, for example, K. Ohno, K. Esfarjani and Y. Kawazoe: Computational Materials Science: From Ab Initio to Monte Carlo Methods, (Springer-Verlag, Berlin, Heidelberg, 1999). 7) M. Rohlfing, P. Krüger and J. Pollmann: Phys. Rev. B 48 (1993) ) T. Kotani and M. v. Schilfgaarde: Solid State Commun. 121 (2002) ) W. Ku and A. G. Eguiluz: Phys. Rev. Lett. 89 (2002) ) S. Lebegue, B. Arnaud, M. Alouani and P. E. Bloechl: Phys. Rev. B 67 (2003) ) M. Rohlfing, P. Krüger and J. Pollmann: Phys. Rev. B 57 (1997) ) S. V. Faleev, M. vanschilfgaarde and T. Kotani: Phys. Rev. Lett. 93 (2004) ) A. N. Chantis, M. vanschilfgaarde and T. Kotani: Phys. Rev. Lett. 96 (2006) ) T. Yokoya, T. Nakamura, T. Matsushita, T. Muro, Y. Takano, M. Nagao, T. Takenouchi, H. Kawarada and T. Oguchi: Nature 438 (2005) ) K. Ohno, F. Mauri and S. G. Louie: Phys. Rev. B 56 (1997) ) T. Ohtsuki, K. Ohno, K. Shiga, Y. Kawazoe, Y. Maruyama and K. Masumoto: Phys. Rev. Lett. 81 (1998) ) S. Ishii, K. Ohno, Y. Kawazoe and S. G. Louie: Phys. Rev. B 63 (2001) ) S. Ishii, K. Ohno, Y. Kawazoe and S. G. Louie: Phys. Rev. B 65 (2002) ) S. Ishii, K. Ohno, V. Kumar and Y. Kawazoe: Phys. Rev. B 68 (2003) ) E. Kikuchi, S. Iwata, S. Ishii and K. Ohno: Phys. Rev. B 76 (2007) ) L. Hedin and S. Lundqvist: Solid State Physics 23 (1969) ) F. Herman and S. Skillman: Atomic Structure Calculations, (Prentice- Hall, Englewood Cliffs, NJ, 1963). 23) Numerical Data and Functional Relationships in Science and Technology, ed. by K.-H. Hellwege and O. Madelung, Landolt-Börnstein, New Series, Group III, Vols. 17a and 22a, (Springer, Berlin, 1982). 24) J. E. Ortega and F. J. Himpsel: Phys. Rev. B 47 (1993) ) W. E. Spicer and R. C. Eden: Proc. Ninth Int. Conf. on the Physics of Semiconductors, Moscow, 1968, ed. by S. M. Ryvkin (Nauka, Leningrad, 1968) Vol. 1, p ) A. L. Wachs, T. Miller, T. C. Hsieh, A. P. Shapiro and T.-C. Chiang: Phys. Rev. B 32 (1985) ) D. Straub, L. Ley and F. J. Himpsel: Phys. Rev. Lett. 54 (1985) ) R. Hulthen and N. G. Nilsson: Solid State Commun. 18 (1976) ) F. J. Himpsel, P. Heimann and D. E. Eastman: Phys. Rev. B 24 (1981) ) Silicon 2p, 2s XPS experiment. 31) F. J. Himpsel, J. F. van der Veen and D. E. Eastman: Phys. Rev. B 22 (1980) ) Diamond 1s XPS experiment. 33) A. R. Chourasia: Surf. Sci. Spectra 8 (2001) ) S. Engelsberg and J. R. Schrieffer: Phys. Rev. 131 (1963)